Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding

## Abstract

Abstract Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning (TDL) addresses this limitation by leveraging combinatorial topological spaces. However, existing TDL models are restricted to undirected settings and fail to capture the higher-order directed patterns prevalent in many complex systems, e.g., brain networks, where such interactions are both abundant and functionally significant. To fill this gap, we introduce Semi-Simplicial Neural Networks (SSNs), a principled class of TDL models that operate on semi-simplicial sets—combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, which dynamically select the most informative relations in a learnable manner. We prove that SSNs are strictly more expressive than standard graph and TDL models. We then introduce a new principled framework for brain dynamics representation learning, grounded in the ability of SSNs to provably recover topological descriptors shown to successfully characterize brain activity. Empirically, SSNs achieve state-of-the-art performance on brain dynamics classification tasks, outperforming the second-best model by up to 27%, and message passing GNNs by up to 50% in accuracy. Our results highlight the potential of principled topological models for learning from structured brain data, establishing a unique real-world case study for TDL. We also test SSNs on standard node classification and edge regression tasks, showing competitive performance. We will make the code and data publicly available.

## 1 Introduction

Networks are commonly represented as graphs, i.e., a set of nodes and a set of unordered pairs of nodes–the edges–modeling pairwise interactions [1]. Graph Neural Networks (GNNs) [2, 3, 4], deep learning models operating on graph-structured data, have shown remarkable performance on several tasks from different domains, such as computational chemistry [5, 6], social network analysis [7, 8], and neuroscience [9], to name a few. The success of GNNs is mainly due to their ability to synergize the flexibility of deep learning models with the inductive bias encoded in the graph structure [10, 11]. Most GNNs are Message Passing Networks (MPNNs) [12], i.e., they learn meaningful representations of node or edge data via local aggregation processes governed by the underlying graph connectivity.

Figure: Figure 1: Overview of the Semi-Simplicial Neural Networks framework for Brain Dynamics Classification. Given a structured brain sample and its corresponding response to external stimulation, we jointly model them as an attributed semi-simplicial set $\mathcal{K}$ that captures higher-order co-activation patterns $X^{l}$. A set of relations $\mathcal{R}$ induced by the complex’s topology is then selected, and the structure, data, and relations are processed by *Semi-Simplicial Neural Networks* (SSNs)—a novel class of architectures capable of leveraging higher-order directional information. Our experiments demonstrate that this approach is essential for accurately predicting the originating stimulus.
Refer to caption: extracted/6484153/fig/gabstract.png

However, many complex real-world systems exhibit higher-order interactions beyond simple pairwise relationships [13, 14]. These intricate dependencies are better represented by Combinatorial Topological Spaces (CTSs)— mathematical objects, such as simplicial or cell complexes, that generalize graphs by encoding multi-way dependencies as sets of nodes, referred to as simplices/cells. This combinatorial structure induces sophisticated set-containment relationships among higher-order entities via hierarchical inclusion, extending the conventional notion of node adjacency in graphs [15] and providing a powerful set of algebraic topological tools [16]. Topological Deep Learning (TDL) [17, 18, 19, 20] leverages this information to extend GNNs into principled architectures directly operating on CTSs. TDL models demonstrate higher expressivity in the Weisfeiler–Leman (WL) hierarchy [21, 22, 23, 24, 25, 26], enhanced capacity to capture long-range interactions [21, 27, 28], and improved performance in heterophilic scenarios compared to traditional GNNs [29, 30].

Simplicial complexes and other commonly used CTSs [23, 24, 19] fall short when prescribed directionality fundamentally shapes system dynamics. Although recent works introduced GNNs that incorporate directionality through directed graphs (digraphs) [31, 32, 33], very little is known about how to integrate and leverage higher-order directional information of the data in TDL models. Building on the theoretical foundation laid by [34], the short paper [35] introduced a formal notion of higher-order topological directionality—directed cliques, defined as totally ordered node sequences connected in a direction-preserving manner—for learning on directed simplicial complexes constructed from these directed motifs [36, 37, 38]. The authors proposed Directed Simplicial Neural Networks (Dir-SNNs) exploiting these asymmetric relations among simplices. However, the framework in [35] solely proposes message passing architectures, considers a narrow class of spaces–directed simplicial complexes–and relationships, lacks in-depth theoretical and practical insights, and is supported by limited synthetic empirical validation.

Brain networks are among the most notable real-world systems requiring methods capable of processing higher-order, directional information [39]. While digraphs capture the inherent directionality of neuronal communication—from presynaptic to postsynaptic neurons—they fail to fully represent critical neural co-activation patterns. Higher-order directed motifs formed by neurons are abundant across scales [40, 41, 42, 43], hold functional significance [44] and constitute spiking correlation assemblies that enhance the representational fidelity of brain activity [44, 45, 46, 47]. To model these structures, seminal works in the emerging field of Neurotopology [48, 45] employed semi-simplicial sets—general combinatorial objects exemplified by directed flag complexes [45, 49] and tournaplexes [50]. In [48], the authors introduced a non-learnable featurization pipeline that characterizes time-varying brain activity through topological invariants computed on brain networks modeled as semi-simplicial sets and aggregated across neuronal subpopulations. Extending this approach, [51] successfully recovered stimulus identities in a biologically realistic neocortical model [52, 53], even when traditional methods failed. However, these handcrafted pipelines overlook the potential of topology-aware, data-driven learning frameworks also robust to structural variability.

The lack of a general, formal, and exhaustive TDL framework leveraging higher-order directionality motivates the methodological advancements in this work. Linking them with Neurotopology—along with its limitations and open challenges—offers a unique opportunity to learn meaningful representations of dynamical brain activity. For a detailed discussion of related work, see Appendix [C](https://arxiv.org/html/2505.17939v2#A3).

Contribution. We establish a series of methodological and applied contributions. Our work also helps to partially or significantly address some of the most pressing open problems in TDL presented in the recent position paper [20].

C1. We introduce *Semi-simplicial Neural Networks* (SSNs), a novel class of TDL models explicitly designed to process and learn from data on semi-simplicial sets. SSNs propagate and integrate information across simplices via relations induced by face maps–peculiar mappings that specify how simplices are hierarchically glued together–, formally organized in a tractable mathematical tool, a relational algebra. To address scalability and efficiency (Open Problem 6 in [20]), we propose Routing-SSNs (R-SSNs), a variant employing a learnable routing mechanism [54, 55] that dynamically selects the top-$k$ most relevant relations among simplices, significantly reducing active parameters and inference time. We prove that SSNs are more expressive than message passing GNNs [12], Directed GNNs (Dir-GNNs) [33] and Message Passing Simplicial Neural Networks (MPSNNs) [22], as measured by the Weisfeiler–Lehman hierarchy. See Section [3](https://arxiv.org/html/2505.17939v2#S3) for a detailed description.

C2. We propose a novel *topology-grounded framework* for higher-order representation learning of brain dynamics. Its core elements are *Dynamical Activity Complexes (DACs)*—binary, time-evolving, directed simplicial complexes encoding neuronal co-activation: a neuron group is active at time $t$ if all constituent neurons fire simultaneously at $t$. Critically, we formally prove that SSNs operating on DACs are uniquely capable—among existing graph and TDL models—of recovering a broader class of topological invariants known to characterize brain network activity [45, 48, 51], thanks to their ability to jointly encode directionality and higher-order structure (Open Problem 9 in [20]). See Section [4](https://arxiv.org/html/2505.17939v2#S4) for a detailed explanation of the framework.

C3. We provide a *unique real-world case study for TDL*, by showing that SSNs achieve state-of-the-art results in neural stimuli classification tasks (see Figure [1](https://arxiv.org/html/2505.17939v2#S1.F1)) (Open Problem 1 in [20]). As such, the data and tasks we employ also serve as a competitive public benchmark for evaluating graph-based and TDL models (Open Problem 2 of [20]). In particular, following [48, 51], we apply structured stimuli to a biologically realistic neocortical microcircuit model [53], addressing two progressively challenging neuroscientific tasks: (1) stimulus classification from dynamics observed within fixed volumetric brain samples, i.e., a feature classification task, and (2) stimulus classification based on randomly sampled neuron neighborhoods, i.e., a graph/complex classification task. SSNs demonstrate significant effectiveness, achieving peak accuracy improvements exceeding 50% over baseline MPNNs [12]. Finally, the versatility of SSNs is also proved by competitive performance on node classification and edge regression tasks [56]. See Section [5](https://arxiv.org/html/2505.17939v2#S5) for a comprehensive description of the experimental setup and results.

## 2 Background

In this section, we first revisit the foundational domains of interest—namely, semi-simplicial sets, which generalize simplicial complexes to support directionality. We then formalize data integration via attributed semi-simplicial sets, with a particular focus on time-varying binary features. Next, we define a rich class of relations induced by face maps, forming a relational algebra that underpins deep learning on semi-simplicial structures. Finally, we revisit directed simplicial complexes—a specific subclass of semi-simplicial set—and directed flag complexes, which are directed simplicial complexes constructed from digraphs.

Simplicial Complexes. A simplicial complex $\widetilde{\mathcal{K}}$ is a pair $\widetilde{\mathcal{K}}=(V,\Sigma)$, where $V$ is a finite set of vertices and $\Sigma$ is a collection of non-empty finite subsets of $V$ satisfying two properties: (P1) for every $v\in V$, the singleton $\{v\}$ is in $\Sigma$; and (P2) if $\sigma\in\Sigma$ and $\tau$ is a non-empty subset of $\sigma$, then $\tau\in\Sigma$. The dimension of a simplex $\sigma\subseteq V$ is $\dim(\sigma)=|\sigma|-1$; the dimension of the complex is $N=\max_{\sigma\in\Sigma}\dim(\sigma)$. We denote the set of $n$-simplices by $\Sigma_{n}=\{\sigma\in\Sigma\mid\dim(\sigma)=n\}$, thus $\Sigma=\bigcup_{n=0}^{N}\Sigma_{n}$. Notably, permuting the vertices of a simplex does not alter its identity, making simplices suitable only for modeling permutation-invariant multi-way interactions [13].

Semi-Simplicial Sets. Semi-simplicial sets generalize simplicial complexes by allowing multiple distinct simplices to share the same set of vertices. For example, in a directed graph (digraph), the edges $(1,0)$ and $(0,1)$ are distinct, even though both share the same vertex set $\{0,1\}$. Thus, this flexibility is essential for modeling directional structures. A semi-simplicial set $\mathcal{S}$ consists of (i) a collection of sets $\{S_{n}\}_{n=0}^{N}$, where each $S_{n}$ is called the set of $n$-simplices and (ii) face maps $d_{i}:S_{n}\to S_{n-1}$ for $n>0$ and $0\leq i\leq n$, specifying how simplices are hierarchically glued. These face maps satisfy the simplicial identity $d_{i}d_{j}=d_{j-1}d_{i}$ for $i<j$, ensuring a well-defined and consistent face structure [36]. We denote the set of simplices with $S=\bigcup_{n=0}^{N}S_{n}$.

Attributed Semi-Simplicial Sets. Let $\mathbb{F}$ be a feature space embedded in $\mathbb{R}^{m}$, so that each element $x\in\mathbb{F}$ can be uniquely represented as an $m$-dimensional real vector. Let $\mathcal{S}$ be a semi-simplicial set and $S^{\mathbb{F}}=\{F:S\to\mathbb{F}\}$ be the set of mappings assigning to each simplex $\sigma\in S$ a feature vector $F(\sigma)=x_{\sigma}\in\mathbb{F}\subseteq\mathbb{R}^{m}$. An *attributed semi-simplicial set* is thus a pair $S_{F}=(S,F)$, with $F\in S^{\mathbb{F}}$. Assuming a fixed labelling on the set of $n$-simplices $S_{n}=(\sigma_{1},\sigma_{2},\dots,\sigma_{|S_{n}|})$, we define the $n$-feature matrix as $X_{n}\coloneqq[x_{\sigma_{1}};\dots;x_{\sigma_{|S_{n}|}}]\in\mathbb{R}^{|S_{n}
|\times m}.$ Letting $l(n)=\sum_{i=0}^{n-1}|S_{i}|$ be the cumulative count of simplices up to dimension $n-1$, we extend this labelling globally by re-indexing each simplex $\sigma_{i}\in S_{n}$ as $\sigma_{i+l(n)}\in S$. The global feature matrix for $\mathcal{S}_{F}$ is then defined as $X\coloneqq[X_{0};,\dots;X_{N}]\in\mathbb{R}^{|S|\times m}$. In this work, we are particularly interested in *dynamic binary data*, i.e., $\mathbb{F}=\mathbb{B}^{T}\coloneqq\{0,1\}^{T}\subset\mathbb{R}^{T}$. A dynamic binary function $B\in S^{\mathbb{B}^{T}}$ assigns each simplex a time-varying binary activation state $x_{\sigma}\in\{0,1\}^{T}$, indicating whether it is active (1) or inactive (0) at each of the $T$ discrete time steps. The resulting $n$-dimensional feature data take then the form of binary matrices $X_{n}\in\mathbb{R}^{|S_{n}|\times T}$ for $0\leq n\leq N$.

Face Map Relations. To define deep learning models operating on attributed semi-simplicial sets, we need to specify a collection of relations $\mathcal{R}$ that governs how features are fused and propagated across simplices. To this aim, we endow the set of simplices $S$ with relations induced directly by the face maps of the underlying semi-simplicial structure. Each face map $d_{i}^{n}$ gives rise to a binary relation $R_{d_{i}^{n}}=\{(\tau,\sigma)\mid\sigma\in S_{n},\;d_{i}^{n}(\sigma)=\tau\}\in
\mathcal{P}(S\times S)$ that associates each simplex $\sigma\in S_{n}$ with its $i$-th face $\tau\in S_{n-1}$. We define the collection of *face map-induced relation algebra* $\mathcal{R}_{d}$ as the smallest positive relation-subalgebra of $\mathcal{P}(S\times S)$ containing $\{R_{d_{i}^{n}}\mid n\geq 0,0\leq i\leq n\}$ and closed under union $(\cup)$, intersection $(\cap)$, relational composition $(\circ)$, and relational converse $(\top)$. This algebraic framework enables the systematic construction of incidence patterns and higher-order adjacency structures—crucial for modeling directional dependencies. See Appendix [D.4](https://arxiv.org/html/2505.17939v2#A4.SS4) for the formal algebraic details and Appendix [D.5](https://arxiv.org/html/2505.17939v2#A4.SS5) for illustrative examples.

Figure: Figure 2: (Top) Digraph; (Bottom) Directed flag complex.
Refer to caption: extracted/6484153/fig/before_dfg.png

Directed Simplicial Complexes. A directed simplicial complex $\mathcal{K}$ is a semi-simplicial set in which each simplex encodes a fully transitive directed structure. Formally, each set of $n$-simplices $\mathcal{K}_{n}$ consists of ordered $(n+1)$-tuples $\sigma=(v_{0},\dots,v_{n})$ such that $(v_{i},v_{j})$ is a directed edge for every $i<j$ (see $2$-simplex $(0,1,2)$ in Figure [2](https://arxiv.org/html/2505.17939v2#S2.F2), bottom). The face maps $d_{i}^{n}:\mathcal{K}_{n}\to\mathcal{K}_{n-1}$ act by removing the $i$-th vertex, yielding $d^{n}_{i}(\sigma)=(v_{0},\ldots,\hat{v}_{i},\ldots,v_{n})$ where $\hat{v}_{i}$ denotes omission of the vertex in position $i$. We require $\mathcal{K}$ to be closed under face maps, i.e., it must satisfy properties (P1) and (P2) described above. Given a directed simplicial complex $\mathcal{K}$ and a dynamic binary function $B\in S^{\mathbb{B}^{T}}$, we define the pair $\mathcal{K}_{B}=(\mathcal{K},B)$ as a *dynamic binary directed simplicial complex*.

Directed Flag Complexes. A directed flag complex $\mathcal{K}_{\mathcal{G}}$ [45, 49] is a directed simplicial complex built from a digraph $\mathcal{G}$. In particular, given a digraph $\mathcal{G}=(V,E)$, its directed flag complex $\mathcal{K}_{\mathcal{G}}$ is the directed simplicial complex whose $n$-simplices consist precisely of the transitive $(n+1)$-cliques in $\mathcal{G}$—that is, ordered $(n+1)$-tuples of distinct vertices $(v_{0},\dots,v_{n})$ such that $(v_{i},v_{j})\in E$ for all $i<j$. See Figure [2](https://arxiv.org/html/2505.17939v2#S2.F2) for an example. Notably, this construction preserves the full structure of the underlying graph while mapping non-isomorphic graphs to non-isomorphic complexes, i.e., it is injective on isomorphism classes (see Appendix [D.1](https://arxiv.org/html/2505.17939v2#A4.SS1)).

## 3 Semi-Simplicial Neural Networks

Topological Deep Learning (TDL) architectures [17, 18] represent higher-order relations as unordered vertex sets—e.g., simplices or cells in simplicial or cell complexes. Consequently, feature propagation and aggregation are based solely on subset containment: simplices interact via shared vertices or hierarchical inclusion, with no notion of directionality. Although this unordered representation suits symmetric interactions [57, 13], it becomes problematic when interaction order is fundamental,
as in brain dynamics [53]. Specifically, disregarding vertex order leads to two significant limitations: (a) *Information Loss.* Transforming directed simplicial complexes into undirected ones through symmetrization is a non-injective transformation that collapses distinct motifs (e.g., transitive cliques and directed cycles) into indistinguishable structures (see Appendix [D.3](https://arxiv.org/html/2505.17939v2#A4.SS3))
Specifically, symmetrizing distinct, non-isomorphic digraphs can yield isomorphic flag complexes; likewise, directed flag complexes may become indistinguishable under symmetrization;
(b) *Underexploitation of Face-Map Structure*. Traditional TDL architectures inadequately leverage the rich structure provided by face maps, thus failing to fully exploit directional dependencies and complex interactions among directed motifs [34, 35]. To address these limitations, we present a series of principled developments that support Contribution C1.

We introduce Semi-Simplicial Neural Networks (SSNs), a novel, principled class of TDL architectures operating on attributed semi-simplicial sets. SSNs propagate information through a collection of face-map-induced relations $\mathcal{R}\subseteq\mathcal{R}_{d}$. By leveraging the inherent structure of semi-simplicial sets, SSNs can capture local directional motifs and their intricate interactions.

SSNs. Let $\mathcal{S}^{\mathcal{R}}_{F}$ be an attributed semi-simplicial set equipped with a collection of face-map-induced relations $\mathcal{R}\subseteq\mathcal{R}_{d}$. The $l$-th layer of an SSN updates the features $X^{l}$ according to

$$ $X^{l+1}=\phi\,\!\Bigl{(}X^{l},\;\bigotimes_{R\in\mathcal{R}}\;\omega_{R}\! \bigl{(}X^{l}\bigr{)}\Bigr{)},$ (1) $$

where $\omega_{R}$ is a relation-dependent learnable function, $\bigotimes$ is an aggregation operator that synchronizes the possibly multiple messages arriving on a single simplex across multiple relations into a single message and $\phi$ is a learnable update function. Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)) induces a wide class of architectures, since $\omega_{R}$ can be any learnable function. For example, an SSN can be implemented as a message-passing network if $\omega_{R}$ is an instance of a directed MPNN module (MPNN-D) [33] running on the digraph induced by $S$ and $R$, where the nodes are the simplices in $S$ and the edges are the elements of the relation $R$. Similarly, a transformer-like model can be obtained by choosing $\omega_{R}$ to be a masked multi-head self-attention module with the mask specified by $R$. See Appendix [F.1](https://arxiv.org/html/2505.17939v2#A6.SS1) for more details.

Routing SSNs. While SSNs fully exploit the combinatorial structure of semi-simplicial sets, two key challenges arise: (1) *Scalability*—naively expanding and refining relational modeling can lead to parameter explosion, as each relation requires its own set of learnable weights; and (2) *Relevance of Relations*—not all relations equally contribute to learning; some may be redundant or less informative. To address these limitations, we propose Routing SSNs (R-SSNs), which incorporate a learnable gating [58, 54, 55] to dynamically select the top-$k$ relations from predefined relation classes. Formally, let $\mathcal{S}_{F}^{\mathcal{R}}$ denote an attributed semi-simplicial set equipped with a collection of face-map-induced relations $\mathcal{R}\subset\mathcal{R}_{d}$.
Let $\mathcal{P}_{\mathcal{R}}=\{\mathcal{R}_{1},\dots,\mathcal{R}_{n}\}$ be a partition of $\mathcal{R}$ into $n$ distinct relation classes. The $l$-th layer of an R-SSN updates the features $X^{l}$ as follows:

$$ $X^{l+1}=\phi\,\!\Bigl{(}X^{l},\;\bigoplus_{\hat{\mathcal{R}}\in\mathcal{P}_{ \mathcal{R}}}\bigotimes_{R\in\hat{\mathcal{R}}}G_{R}(X^{l})\cdot\omega_{R}(X^{ l})\Bigr{)},$ (2) $$

where $G_{R}$ is a gating function producing normalized weighting scores for each direction, i.e., $G_{R}(X^{l})\in[0,1]$. The gating is instantiated as a top-$k$ routing mechanism, i.e., the scores are different from zero only for the top-$k$ most relevant relations within each subset $\hat{\mathcal{R}}\in\mathcal{P}_{\mathcal{R}}$. The final integration across relational classes is performed via $\bigoplus$. A detailed formulation of the routing mechanism, along with the gating strategies and tailored loss functions employed to prevent routing collapse, is presented in Appendix [F.2](https://arxiv.org/html/2505.17939v2#A6.SS2).

### 3.1 Theoretical properties of SSNs

In this section, we characterize SSNs theoretically, focusing on their generality, Weisfeiler–Leman (WL) expressivity [59], and permutation equivariance under simplex relabeling.

Generality. SSNs subsume and generalize a substantial class of existing architectures.

###### Proposition 1 .

Semi-simplicial neural networks (SSNs) subsume directed message-passing GNNs [33] on directed graphs, message-passing GNNs [12] on undirected graphs, message-passing simplicial neural networks [22] on simplicial complexes and Directed Simplicial Neural Networks [35] on directed simplicial complexes.

WL Expressivity. The expressive power of TDL message-passing architectures is commonly assessed by their ability to distinguish non-isomorphic combinatorial topological spaces (CTSs) [22]. The classical Weisfeiler–Leman test [60] (see Appendix [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2)) provides an upper bound on the discriminative power of GNNs [59]. Graph liftings, such as the directed flag complex, enrich graph representations with higher-order relations, ensuring that isomorphic graphs map to isomorphic CTSs and non-isomorphic graphs to distinct CTSs. Consequently, TDL models may be more expressive than graph-based models [22, 24, 25, 26].

###### Theorem 1 .

There exist SSNs strictly more expressive than directed graph neural networks (Dir-GNNs) [33] at distinguishing non-isomorphic directed graphs.

###### Theorem 2 .

There exist SSNs strictly more expressive than message-passing simplicial neural networks (MPSNNs) [22] at distinguishing non-isomorphic directed simplicial complexes.

Permutation Equivariance. Representing an attributed semi-simplicial set in matrix form requires assigning arbitrary labels to its simplices (see Section [2](https://arxiv.org/html/2505.17939v2#S2)), which defines the representation of their associated relations. As these labels carry no semantic meaning, neural networks must be equivariant (or invariant) to permutations of simplex indices. That is, the output must transform consistently under relabeling, ensuring that learned representations depend solely on the underlying domain [61].

###### Theorem 3 (Informal) .

Consider an attributed semi-simplicial set with a collection of face-map–induced relations $\mathcal{R}\subseteq\mathcal{R}_{d}$. An SSN layer, as defined in Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)), is equivariant to simplex relabeling if, for each relation $R\in\mathcal{R}$, the message function $\omega_{R}$ and the aggregation operator $\bigotimes$ are permutation equivariant.

Further details, including the formal statements and proofs, are provided in Appendix [E](https://arxiv.org/html/2505.17939v2#A5). Furthermore, a computational complexity analysis of both SSNs and R-SSNs is provided in Appendix [G](https://arxiv.org/html/2505.17939v2#A7), showing that complexity grows linearly with respect to the number of relations in SSNs, while remaining fixed in R-SSNs under a top-$k$ selection.

###### Proposition 1 .

###### Theorem 1 .

###### Theorem 2 .

###### Theorem 3 (Informal) .

## 4 Topological Deep Representation Learning for Brain Dynamics

We introduce a principled and novel representation learning framework for modeling dynamical brain activity, compatible with both graph-based and TDL architectures. To this end, we capture the evolving structure–activity organization through novel objects, Dynamical Activity Complexes (DACs), dynamic binary directed simplicial complexes that explicitly encode neuronal co-activation patterns over time. We prove that SSNs operating on DACs can recover a broad class of topological invariants known to characterize brain activity [45], and that this set strictly exceeds the invariants recoverable by graph and TDL models. This section supports Contribution C2.

Brain Activity Modeling. The modeling of brain network activity involves two fundamental components: structural connectivity and brain dynamics. First, *structural connectivity* is typically modeled by a directed graph $\mathcal{G}=(V,E)$, with neurons represented by nodes $V$ and directed edges $(u,v)\in E$ denoting synaptic connections from presynaptic neuron $u$ to postsynaptic neuron $v$. Second, a collection of *binary dynamics* $\mathcal{B}=\{B:V\rightarrow\mathbb{B}^{T}\}$ encodes neuronal activity over time, capturing neuronal firing (1) and quiescent (0) states across $T$ discrete time bins, under various stimulus-driven input patterns originating from thalamocortical afferents. The *structural graph* $\mathcal{G}$ induces a directed flag complex referred to as the *structural complex* $\mathcal{K}_{\mathcal{G}}$ [45]. Additional details in Appendix [H.1](https://arxiv.org/html/2505.17939v2#A8.SS1).

Figure: Table 1: Topological invariant computation across architectures.

Neurotopology at a Glance. Given a binary dynamics $B$ on a structural digraph $\mathcal{G}=(V,E)$, each time bin (component) $t$ defines a neuron activation state $B_{t}$, partitioning the vertices into active $V^{1,t}$ and inactive $V^{0,t}$ subsets, with $V^{1,t}=\{v\in V\mid B_{t}(v)=1\}$. The subgraph induced by active neurons $\mathcal{G}^{1,t}=\mathcal{G}[V^{1,t}]$ defines the *functional graph* at time $t$. Neural responses to external stimuli are thus characterized by the temporal sequence of functional graphs $\mathcal{F}_{\mathcal{G}}=[\mathcal{G}^{1,1},\dots,\mathcal{G}^{1,T}]$ and their corresponding *functional complexes*, $\mathcal{F}_{\mathcal{K}_{\mathcal{G}}}=[\mathcal{K}_{\mathcal{G}^{1,1}},\dots
,\mathcal{K}_{\mathcal{G}^{1,T}}]$. Given a topological invariant $\operatorname{T}$—formally defined in Appendix [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2)—one computes $\operatorname{T}(\mathcal{K}_{\mathcal{G}^{1,t}})$ at each time step $t$. The resulting time-dependent *topological signature* is given by $\operatorname{T}(\mathcal{F}_{\mathcal{K}_{\mathcal{G}}})=[\operatorname{T}[
\mathcal{K}_{\mathcal{G}^{1,1}}],\dots,\operatorname{T}[\mathcal{K}_{\mathcal{
G}^{1,T}}]]\in\mathbb{R}^{T}$. Seminal work [45] demonstrated that distinct stimuli produce unique temporal topological signatures, distinguishable via variations in Euler characteristics, clique counts, and homology groups. Given subsets of neurons that are geometrically or topologically localized within the structural graph $\mathcal{G}$, denoted by $\{U_{i}\}_{i=0}^{N}$, their induced subgraphs $\{\mathcal{G}_{i}=\mathcal{G}[U_{i}]\}_{i=0}^{N}$ yield a set of local time-dependent topological signatures $\{\operatorname{T}(\mathcal{F}_{\mathcal{K}_{\mathcal{G}_{i}}})\}_{i=0}^{N}$. Concatenating these local signatures provides a comprehensive *local-to-global* topological representation of the entire network’s activity. Building on this approach, recent works [48, 51] successfully recovered stimulus identity in a biologically accurate neocortical model [53], even when traditional methods failed—highlighting the effectiveness of topological representations. However, these methodologies rely on fixed topological signatures aggregated across multiple neuronal samples, which limits their applicability in settings that require localized decoding and makes them sensitive to structural variability across samples.

Figure: Figure 3: (Top) Dynamic binary graph; (Bottom) associated DAC. Binary vectors indicate simplex activation over 4 time steps.
Refer to caption: extracted/6484153/fig/before_dac.png

Dynamical Activity Complex. We bridge the expressive power of topological representations with the data-driven flexibility of deep learning by systematically integrating higher-order structural connectivity with evolving brain activity. To formalize this, we introduce *Dynamical Activity Complexes* (DACs), binary directed simplicial complexes built from a sampled structural brain digraph and its associated binary dynamics, encoding brain activity over $T$ discrete time steps. Formally, let $\mathcal{G}_{B}$ be a dynamic binary digraph, where each vertex $v\in V$ is associated with a $T$-dimensional binary feature vector $B(v)\in\mathbb{B}^{T}$. A DAC is the attributed directed flag complex $\mathcal{K}_{\mathcal{G},\tilde{B}}=(\mathcal{K}_{\mathcal{G}},\tilde{B}),$ where the attribute map $\tilde{B}$ is given by

$$ $\tilde{B}(\sigma)=\Bigl{[}\min_{v\in\sigma}B_{1}(v),\,\dots,\,\min_{v\in\sigma }B_{T}(v)\Bigr{]}\in\mathbb{B}^{T},$ (3) $$

i.e., each simplex $\sigma$ is assigned a binary feature vector $x_{\sigma}\in\mathbb{B}^{T}$, where the $t$-th component is 1 if every vertex $v\in\sigma$ is active at time $t$. Therefore, aggregating neuron states via a component-wise logical AND captures the higher-order co-activation neural patterns. Figure [3](https://arxiv.org/html/2505.17939v2#S4.F3), (bottom) shows that the $2$-simplex is active only at $t$ = 3 as it is the unique time all its neurons are active. Crucially, this construction (i) *preserves isomorphisms*, and (ii) *encodes the temporal sequence of functional complexes*, ensuring compatibility with existing neurotopological pipelines (see Propositions [5](https://arxiv.org/html/2505.17939v2#Thmproposition5) and [6](https://arxiv.org/html/2505.17939v2#Thmproposition6) in Appendix [H.2](https://arxiv.org/html/2505.17939v2#A8.SS2)).

Relevant Invariants. Neurotopological studies [45, 48, 51] empirically validated the set of topological invariants $\mathcal{T}=\{\operatorname{size},\operatorname{ec},\operatorname{td},
\operatorname{dir},\operatorname{hodir},\operatorname{rc}\}$ as particularly effective for characterizing brain activity, as measured by predictive performance in stimulus-induced activity classification. These correspond, respectively, to: vertex active neighbor count, Euler characteristic evolution, transitive motif participation, net directional flow, higher-order directional flow, and reciprocal connection count. Formal definitions are provided in Appendix [H.3](https://arxiv.org/html/2505.17939v2#A8.SS3). We next show that SSNs applied to DACs are the only TDL architectures capable of computing the full set $\mathcal{T}$ of invariants [48, 51].

###### Theorem 4 .

Let $\mathcal{G}_{B}$ be a dynamic binary digraph, with corresponding Dynamical Activity Complex $\mathcal{K}_{\mathcal{G},\tilde{B}}$. Then, for every invariant $\operatorname{T}\in\mathcal{T}$, there exists a set of face-map-induced relations $\mathcal{R}_{\operatorname{T}}\subset\mathcal{R}_{d}$ and a Semi-Simplicial Neural Network $\operatorname{SSN}$ as in Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)) such that:

$$ $\operatorname{SSN}\bigl{(}X,\mathcal{R}_{\operatorname{T}}\bigr{)}= \operatorname{T}\bigl{(}\mathcal{K}_{\mathcal{G},\tilde{B}}\bigr{)}.$ $$

Moreover, at least one invariant $\operatorname{T}\in\mathcal{T}$ cannot be computed by message-passing neural networks [12], directed graph neural networks [33], or message-passing simplicial neural networks [22].

A formal proof of Theorem [4](https://arxiv.org/html/2505.17939v2#Thmtheorem4) is provided in Appendix [H.4](https://arxiv.org/html/2505.17939v2#A8.SS4). Table [1](https://arxiv.org/html/2505.17939v2#S4.T1) summarizes the capability of different models to compute these topological invariants, highlighting that our SSN framework uniquely recovers the complete set of critical invariants necessary for comprehensive brain activity characterization. This result suggests SSNs as a legitimate generalization of prior methods, enabling the fully data-driven extraction of meaningful, localized, and robust representations of brain dynamics.

###### Theorem 4 .

## 5 Numerical Results

In this section, we demonstrate the effectiveness of SSNs in classifying brain stimuli from neural activation patterns. Building on the theoretical framework in Sections [3](https://arxiv.org/html/2505.17939v2#S3) and [4](https://arxiv.org/html/2505.17939v2#S4), our experimental evaluation aims to: (i) establish the *ability* to accurately infer stimulus identity from *localized, sample-specific brain dynamics*—each inducing structurally distinct higher-order topologies—and (ii) show that SSNs—through their ability to model *directed higher-order structure*—
uniquely offer the relational expressivity essential for neurotopological applications. To this end, we consider two progressively challenging scenarios: first, a feature classification task with fixed topology and varying dynamics (Task [5.1](https://arxiv.org/html/2505.17939v2#S5.SS1)); second, a graph/complex classification task where both the dynamics and topology vary (Task [5.2](https://arxiv.org/html/2505.17939v2#S5.SS2)). To further demonstrate the broad applicability of SSNs, we show that they achieve competitive results on edge regression [56] and node classification tasks, reported in Appendix [I.3](https://arxiv.org/html/2505.17939v2#A9.SS3) and Appendix [I.4](https://arxiv.org/html/2505.17939v2#A9.SS4), respectively. This section supports Contribution C3.

Figure: Table 2: Counts of directed simplices in the Microcircuit complex $\mathcal{K}_{\mathcal{G}}$.

Data Overview. We build on the dataset introduced in [45, 48, 51], derived from the Blue Brain Project’s [52] biologically validated digital reconstruction of a microcircuit in the somatosensory cortex of a two-week-old rat (the NMC-model) [53]. Partitioning neuronal activation into discrete time bins ($T=2$), the dataset comprises a large structural digraph $\mathcal{G}=(V,E)$ with $|V|=31,346$ and $|E|=7,803,528$, and a collection of $4495$ binary dynamics functions $\{B_{i}:V\to\mathbb{B}^{2}\}_{i=1}^{4495}$. Each function assigns to every neuron vertex $v\in V$ a two-dimensional binary vector that encodes its activation response to an input stimulus. Specifically, each stimulus corresponds to a uniformly random sample from eight distinct synaptic input patterns delivered via thalamocortical afferents. See Appendix [H.1](https://arxiv.org/html/2505.17939v2#A8.SS1) for additional details. Table [2](https://arxiv.org/html/2505.17939v2#S5.T2) confirms the abundance of higher-order directed motifs.

Baselines and Experimental Setting. We evaluate SSNs and their routing-based variant, R-SSNs (using only the top $k$=2 relations except for $(4,125\mu\text{m})$ on which $k$=4), against set-based, graph-based, and topological baselines, as prescribed by the recent position paper [62]: Deep Sets [63] (DS), message-passing GNNs [3, 64], Directed GNNs (DirGNNs) [33], and message-passing Simplicial Neural Networks (MPSNNs) [22]. Through these choices, we can fairly show the advantages and need of handling jointly relational, directed, and higher-order information. We also ensure fair comparisons in terms of parameter budget by scaling the hidden dimensions of the best-performing configurations of the set- and graph-based baselines (DS-256, GNN-256, and Dir-GNN-256) and the undirected higher-order baseline (MPSNN-64). In all the experiments, we consider simplices up to dimension two (triangles) for SSNs, R-SSNs, MPSNN, and MPSNN-64. The hyperparameters of all the models are selected via a grid search. Both SSNs and R-SSNs are instantiated as MPSSNs (see Section [3](https://arxiv.org/html/2505.17939v2#S3)). All the graph and higher-order baselines use SAGE-like [64] message functions.

**Table 3: 8-class stimulus classification accuracy (%, higher is better) based on dynamic binary brain activation responses corresponding to one of eight distinct thalamic input patterns. Columns $(4,125\mu\text{m})$, $(4,325\mu\text{m})$, and $(8,175\mu\text{m})$ correspond to fixed volumetric brain samples (i.e., fixed topology, see Section [5.1](https://arxiv.org/html/2505.17939v2#S5.SS1)). Columns $M=1$, $M=3$, and $M=5$ correspond to the number of neuron neighborhoods $M$ sampled within the $(4,325\mu\text{m})$ volumetric sample (i.e., varying topology, see Section [5.2](https://arxiv.org/html/2505.17939v2#S5.SS2)). The top $\mathbf{1^{\text{st}}}$, $\mathbf{2^{\text{nd}}}$, and $\mathbf{3^{\text{rd}}}$ best results are highlighted. Active parameter counts (#Par.) and relative active parameter percentages (Par. %) at inference are reported with respect to SSNs.**
| Model | $\mathbf{(4,125\boldsymbol{\mu m})}$ | $\mathbf{(4,325\boldsymbol{\mu m})}$ | $\mathbf{(8,175\boldsymbol{\mu m})}$ | M = 1 | M = 3 | M = 5 | # Par. | Par. (%) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| DS [63] | 26.63 $\pm$ 0.10 | 19.47 $\pm$ 1.16 | 27.31 $\pm$ 0.34 | 23.28 $\pm$ 0.48 | 24.29 $\pm$ 0.38 | 25.09 $\pm$ 0.14 | 1,680 | 2% |
| DS-256 [63] | 25.21 $\pm$ 1.11 | 19.52 $\pm$ 1.44 | 25.12 $\pm$ 2.12 | 25.24 $\pm$ 0.32 | 24.76 $\pm$ 0.33 | 25.87 $\pm$ 0.19 | 70,672 | 68% |
| GNN [5] | 26.00 $\pm$ 1.10 | 21.56 $\pm$ 1.14 | 34.02 $\pm$ 4.35 | 25.43 $\pm$ 0.43 | 26.03 $\pm$ 0.69 | 26.49 $\pm$ 1.02 | 5,392 | 5% |
| GNN-256 [5] | 24.70 $\pm$ 1.31 | 23.02 $\pm$ 2.1 | 33.47 $\pm$ 4.15 | 24.40 $\pm$ 0.51 | 27.60 $\pm$ 0.91 | 28.27 $\pm$ 0.17 | 70,672 | 68% |
| DirGNN [33] | 36.71 $\pm$ 2.00 | 48.11 $\pm$ 1.87 | 53.72 $\pm$ 2.89 | 25.21 $\pm$ 0.18 | 31.08 $\pm$ 1.04 | 33.00 $\pm$ 3.69 | 9,744 | 9% |
| DirGNN-256 [33] | 50.89 $\pm$ 13.00 | 60.02 $\pm$ 0.75 | 63.52 $\pm$ 0.70 | 25.43 $\pm$ 0.54 | 35.21 $\pm$ 0.83 | 39.41 $\pm$ 0.27 | 137,232 | 133% |
| MPSNN [22] | 43.33 $\pm$ 6.95 | 42.65 $\pm$ 3.71 | 53.12 $\pm$ 2.65 | 27.45 $\pm$ 0.50 | 32.10 $\pm$ 0.77 | 33.68 $\pm$ 3.04 | 23,888 | 23% |
| MPSNN-64 [22] | 46.85 $\pm$ 6.71 | 54.24 $\pm$ 7.06 | 64.02 $\pm$ 4.91 | 29.48 $\pm$ 1.31 | 34.91 $\pm$ 0.73 | 42.23 $\pm$ 1.82 | 90,768 | 88% |
| R-SSN (Ours) | 57.32 $\pm$ 5.25 | 79.64 $\pm$ 1.84 | 70.66 $\pm$ 1.56 | 28.68 $\pm$ 0.79 | 40.29 $\pm$ 1.12 | 48.20 $\pm$ 0.70 | 18,084 | 18% |
| SSN (Ours) | 75.13 $\pm$ 1.28 | 87.16 $\pm$ 1.36 | 78.32 $\pm$ 7.03 | 46.73 $\pm$ 1.16 | 61.35 $\pm$ 1.07 | 64.72 $\pm$ 0.54 | 103,184 | 100% |
| Gain over 2-nd Best | $\uparrow$ 24.24 % | $\uparrow$ 27.14 % | $\uparrow$ 14.30<br>% | $\uparrow$ 17.25 % | $\uparrow$ 26.14 % | $\uparrow$ 22.49 % | - | - |

### 5.1 Task 1: Classifying Dynamical Brain Activity in Fixed Volumetric Samples

Given a fixed structure in the brain and a time-evolving neuronal activation pattern on it, our goal is to classify the stimulus that triggered the observed dynamics. We extract volumetric samples and activations following [51]. Each volumetric sample $U\subset V$, parameterized by a spatial neuronal population centroid $c\in\{4,8\}$ and a sampling radius $r\in\{125\mu\text{m},175\mu\text{m},325\mu\text{m}\}$, defines a structural subgraph $\mathcal{G}_{U}=\mathcal{G}[U]$. We then lift each restricted binary dynamic $B_{i}|_{U}$ as described in ([3](https://arxiv.org/html/2505.17939v2#S4.E3)), yielding $4{,}495$ lifted dynamics $\tilde{B}_{i}|_{U}$ for each volumetric sample. Given a fixed $U$, the task is to classify each lifted dynamic according to its associated stimulus identity. We select the three volumetric samples parameterized by $(c,r)\in\{(4,125\mu\text{m})$, $(8,175\mu\text{m})$, $(4,325\mu\text{m})\}$, which correspond, respectively, to the most and least discriminative volumes under traditional dimensionality reduction methods [51], plus a third one to assess robustness to volumetric variability. All models use a permutation-invariant readout (mean) to ensure compatibility with the subsequent task, where it is required to accommodate structural variability. In Table [3](https://arxiv.org/html/2505.17939v2#S5.T3) (columns $(4,125\mu\text{m})$, $(4,325\mu\text{m})$, and $(8,175\mu\text{m})$), we report mean accuracy and standard deviation across 5 data splits, each of them grouped in 60% train, 20% validation, 20% test sets . Further details in Appendix [I.2](https://arxiv.org/html/2505.17939v2#A9.SS2).

### 5.2 Task 2: Classifying Neuron Neighbourhood Dynamical Activity Complexes

We consider a more challenging setting in which localized, heterogeneous samples of neurons and their synaptic connections within a fixed volumetric region induce topological variability—unlike the fixed-topology setting of Task [5.1](https://arxiv.org/html/2505.17939v2#S5.SS1)—thereby transitioning from feature classification to a graph/complex classification task. Specifically, each neuron $n$ in $\mathcal{G}$ defines a neighborhood subgraph $\mathcal{G}_{n}=\mathcal{G}[\mathcal{N}(n)]$. For each binary dynamic, we sample uniformly at random without replacement $M$ neuron-neighborhood subgraphs from the $25$ largest within a given volumetric sample $U$, following the protocol in [51]. This yields a total of $M\times 4{,}495$ DACs, obtained by lifting each binary dynamics function $B_{i}$—restricted to its $M$ sampled subgraphs—according to the procedure in ([3](https://arxiv.org/html/2505.17939v2#S4.E3)). We use the $(4,\ 325\mu\text{m})$ component with $M\in\{1,3,5\}$. This setup allows us to assess the model’s ability to infer stimulus identity from sample-specific neural dynamics under different data-scarcity conditions; smaller values of $M$ correspond to fewer training samples. In Table [3](https://arxiv.org/html/2505.17939v2#S5.T3) (columns $M=1$, $M=2$ and $M=3$), for all datasets, we report mean accuracy and standard deviation across 5 splits each of them again grouped in 60% train, 20% validation, 20% test sets. Results at higher temporal resolution and smaller volumes are provided in Appendix [I.2](https://arxiv.org/html/2505.17939v2#A9.SS2).

### 5.3 Discussion

Results in Table [3](https://arxiv.org/html/2505.17939v2#S5.T3) validate the capability of SSNs to classify stimulus identity in fixed brain volumetric samples (Task [5.1](https://arxiv.org/html/2505.17939v2#S5.SS1)), and demonstrate their effectiveness in highly localized scenarios—such as neuron neighborhoods (Task [5.2](https://arxiv.org/html/2505.17939v2#S5.SS2))—where structural variability is introduced across samples. We attribute the performance gains of SSNs over baseline methods to their ability to jointly model higher-order directed motifs—prevalent at multiple scales in brain networks [40, 41, 42, 43]—alongside direction-aware interactions, both of which are critical to accurately capturing neural activity and function [44, 45, 46, 47, 51]. These results align with our theoretical findings in Sections [3.1](https://arxiv.org/html/2505.17939v2#S3.SS1) and [4](https://arxiv.org/html/2505.17939v2#S4), which highlight the superior expressive power of SSNs in classifying directed graphs (Theorem [1](https://arxiv.org/html/2505.17939v2#Thmtheorem1)) and directed simplicial complexes (Theorem [2](https://arxiv.org/html/2505.17939v2#Thmtheorem2)), as well as their capacity to provably recover a broader class of neurotopological invariants (Theorem [4](https://arxiv.org/html/2505.17939v2#Thmtheorem4)). Notably, standard GNNs and DSs exhibit the lowest performance, reflecting their inability to capture higher-order and directional dependencies. Even Dir-GNNs and MPSNNs that independently model directionality and hierarchical structure, respectively, fall short compared to SSNs, indicating that modeling either aspect alone is insufficient. In the most challenging setting ($M=1$), characterized by structural variability and extreme data scarcity (only one neuron neighborhood sample per binary dynamic), SSNs outperform
all other models by at least 17%. This result underscores the importance of strong, expressive inductive biases in low-data regimes (cf. [61]).
Moreover, SSNs demonstrate remarkable parameter efficiency by achieving superior performance with a number of parameters comparable or even lower than the competitive baselines, as shown in columns #Par. and Par.(%).
Furthermore, R-SSNs
consistently rank as second- or third-best across all settings, utilizing significantly fewer parameters and achieving lower inference times. For a detailed analysis of computational complexity and runtime comparisons between SSNs and R-SSNs, see Appendix [G](https://arxiv.org/html/2505.17939v2#A7).

## 6 Conclusion

We introduced Semi-Simplicial Neural Networks (SSNs), a novel class of TDL architectures designed to learn from data structured as semi-simplicial sets. SSNs extend existing graph and TDL models by processing information over simplices through hierarchical, direction-aware, and potentially asymmetric relations defined by face maps. SSNs subsume existing TDL architectures and have enhanced WL-expressivity. By provably capturing a broader class of neurotopological invariants fundamental to characterizing neural activity, SSNs provide a powerful link between neurotopology and deep learning. Extensive experiments conducted on a biologically accurate stimulated brain network model confirm the effectiveness of SSNs, demonstrating superior stimulus classification performance compared to state-of-the-art methods. We discuss the limitations and future directions in Appendix [B](https://arxiv.org/html/2505.17939v2#A2). Our theoretical and methodological contributions, supported by an openly available codebase, lay a solid foundation for higher-order representation learning of brain dynamics, opening new promising avenues for TDL.

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Funding: This study was supported by funding to the Blue Brain Project, a research center of the Ecole polytechnique federale de Lausanne (EPFL), from the Swiss government’s ETH Board of the Swiss Federal Institutes of Technology. RL is supported by an EPSRC grant EP/P025072/ and a collaboration agreement 832 with Ecole Polytechnique Federale de Lausanne. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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## Appendix Contents

| A  Reproducibility Statement | [A](https://arxiv.org/html/2505.17939v2#A1) |
| --- | --- |
| B  Related Works | [C](https://arxiv.org/html/2505.17939v2#A3) |
| C  Limitations & Future Directions | [B](https://arxiv.org/html/2505.17939v2#A2) |
| D  Preliminaries | [D](https://arxiv.org/html/2505.17939v2#A4) |
| D.1  Algebraic Background | [D.1](https://arxiv.org/html/2505.17939v2#A4.SS1) |
| D.2  Relational Preliminaries | [D.2](https://arxiv.org/html/2505.17939v2#A4.SS2) |
| D.3  Directed vs. Undirected Combinatorial Structures | [D.3](https://arxiv.org/html/2505.17939v2#A4.SS3) |
| D.4  Face-Map Relation Algebras | [D.4](https://arxiv.org/html/2505.17939v2#A4.SS4) |
| D.5  Examples of Common Graph and Topological Face-Map Relations | [D.5](https://arxiv.org/html/2505.17939v2#A4.SS5) |
| D.6  Invariants, Colourings and the Weisfeiler–Leman Test | [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2) |
| E  Theoretical properties of SSNs | [E](https://arxiv.org/html/2505.17939v2#A5) |
| E.1  Generality | [E.1](https://arxiv.org/html/2505.17939v2#A5.SS1) |
| E.2  WL Expressivity | [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2) |
| E.3  Permutation Equivariance | [E.3](https://arxiv.org/html/2505.17939v2#A5.SS3) |
| F  Implementation Details | [F](https://arxiv.org/html/2505.17939v2#A6) |
| F.1  SSNs | [F.1](https://arxiv.org/html/2505.17939v2#A6.SS1) |
| F.2  R-SSNs | [F.2](https://arxiv.org/html/2505.17939v2#A6.SS2) |
| F.3  Computational Resources | [F.3](https://arxiv.org/html/2505.17939v2#A6.SS3) |
| G  Computational Complexity | [G](https://arxiv.org/html/2505.17939v2#A7) |
| H  Topological Deep Representation Learning for Brain Dynamics | [H](https://arxiv.org/html/2505.17939v2#A8) |
| H.1  Data | [H.1](https://arxiv.org/html/2505.17939v2#A8.SS1) |
| H.2  Dynamical Activity Complexes | [H.2](https://arxiv.org/html/2505.17939v2#A8.SS2) |
| H.2  Dynamical Activity Complexes Invariants | [H.3](https://arxiv.org/html/2505.17939v2#A8.SS3) |
| H.3  Topological Neural Networks and Dynamical Activity Complexes Invariants | [H.4](https://arxiv.org/html/2505.17939v2#A8.SS4) |
| I  Additional Numerical Results | [I](https://arxiv.org/html/2505.17939v2#A9) |
| I.1  Task 1: Classifying Dynamical Brain Activity in Fixed Volumetric Samples | [I.1](https://arxiv.org/html/2505.17939v2#A9.SS1) |
| I.2  Task 2: Classifying Neuron Neighbourhood Dynamical Activity Complexes | [I.2](https://arxiv.org/html/2505.17939v2#A9.SS2) |
| I.3  Edge Regression Traffic Task | [I.3](https://arxiv.org/html/2505.17939v2#A9.SS3) |
| I.4  Node Classification | [I.4](https://arxiv.org/html/2505.17939v2#A9.SS4) |

## Appendix A Reproducibility Statement

We provide all necessary details to ensure full reproducibility of our results. The experimental settings for the primary neural binary dynamics classification tasks are described in detail in Section [5](https://arxiv.org/html/2505.17939v2#S5), with complete hyperparameter configurations provided in Appendix [I.1](https://arxiv.org/html/2505.17939v2#A9.SS1) and  [I.2](https://arxiv.org/html/2505.17939v2#A9.SS2). For the Edge Traffic Regression and Node Classification tasks, experimental protocols and hyperparameter settings are reported in Appendix [I.3](https://arxiv.org/html/2505.17939v2#A9.SS3) and [I.4](https://arxiv.org/html/2505.17939v2#A9.SS4), respectively. Hardware specifications and computational resource requirements are documented in Appendix [F.3](https://arxiv.org/html/2505.17939v2#A6.SS3). Code and data will be soon available.

## Appendix B Limitations & Future Directions

In settings where data does not faithfully reflect inherent directionality—either because the inductive biases from topology and connectivity misalign with actual information flows, or because the system itself lacks directional structure—SSNs may introduce unnecessary memory overhead due to reduced parameter sharing across semantically distinct relations, without yielding significant accuracy gains. Nevertheless, as shown in our experiments, performance remains competitive and is not adversely affected. This limitation can be mitigated in two ways: (i) via our proposed routing mechanism, which adaptively selects the top-$k$ most informative relations during learning; and (ii) by investigating directed topological latent inference to recover meaningful directional structure. From a neurotopological perspective, these techniques cannot yet be applied in vivo, as they require access to detailed connectomic data—that is, structural information about the brain’s wiring. However, this limitation is expected to diminish with the increasing availability of densely sampled electron microscopy connectomes co-registered with neural activity data. More significantly, the emergence of generative pipelines for constructing biologically realistic simulated digital brains offers a transformative alternative. These developments open the door to novel applications, including the design of intelligent systems inspired by biologically grounded architectures [65, 66], where our model enables the learning of representations that capture high-level information abstractions within such systems. Beyond our current scope, several promising directions remain open, including the unsupervised discovery of neural activity motifs and the extension of our framework to broader neuroscience tasks. Alternative lifting strategies—such as tournaplexes and other directed combinatorial structures—are largely unexplored and may provide complementary inductive biases. Incorporating temporal dynamics and signed interactions (e.g., excitatory vs. inhibitory) [14] is another key avenue. We emphasize that our contribution is primarily theoretical and methodological: it establishes a connection between TDL and higher-order directed complex systems such as the brain. We do not foresee any potential for harm or misuse of this technology if applied responsibly.

## Appendix C Related Works

Topological Deep Learning. TDL builds directly on pioneering efforts in Topological Signal Processing (TSP) [16, 67, 68, 69, 70, 17], which underscore the importance of modeling multi-way relationships. Extensions of the WL test to simplicial and regular cell complexes [22, 24] have demonstrated that message passing on these higher-order structures outperforms its graph-based counterpart. Both convolutional [71, 72, 73, 74, 75, 76, 77] and attentional [23, 78, 79, 28] architectures over simplicial and cell complexes have been introduced. Furthermore, message passing and diffusion on cellular sheaves over graphs [80, 30, 29, 81, 82] have proven particularly effective in heterophilic settings. Alternatives that eschew message passing altogether for simplicial complexes have been detailed in [83, 84, 85], while an approach to infer a latent regular cell complex for downstream improvement has been introduced in [86]. Gaussian Processes on simplicial and cell complexes have been presented in [87, 88]. Comprehensive reviews of TDL can be found in [89, 90]. A combinatorial framework for TDL (combinatorial complexes), based on ranked, not necessarily uniform, set-type relations, was proposed in [19]. A fairly general topological expressivity analysis of TDL models is given in [91]. Quantum simplicial neural networks have been introduced in [92], continuous variants in [93], and geometrically equivariant architectures over simplicial and broader combinatorial spaces in [94, 95]. Finally, software libraries and benchmarks for TDL have been released in [96, 97, 98, 99].

Directed Graph and Topological Models. A growing body of work has incorporated directionality into neural architectures to address the inherent asymmetry of relational data in complex systems. Early spatial methods distinguish incoming and outgoing messages via separate learnable weights, explicitly modeling node roles [100, 101]. Directed GNNs for node classification [33, 102], for instance, have demonstrated superior performance on heterophilic benchmarks [33, 103]. Spectral approaches extend Laplacian-based learning to directed graphs: DiGCN uses a Personalized PageRank–based operator [31], while MAGNet introduces the Magnetic Laplacian, encoding edge directionality in the complex phase [32]. This formulation was later extended to the Sign-Magnetic Laplacian, addressing key limitations and enabling principled learning on signed directed graphs [104]. Extending this works, Fuchsgruber et al. [56] propose the Magnetic Edge Laplacian for edge-level regression tasks in traffic forecasting. Notably, they introduce both an orientation-equivariant variant—designed for direction-sensitive signals such as traffic flow—and an orientation-invariant variant—suited for direction-agnostic signals such as speed limits—thereby generalizing previous edge-based methods. Beyond pairwise structures, recent works extend to non-uniform, non-hierarchical settings using neural networks on directed hypergraphs [105, 106], with applications in traffic forecasting [107] and human pose estimation [108]. Bernardez et al. [109] further show that hierarchical, order-aware relations—built on the combinatorial framework of Hajij et al. [19]—better capture communication patterns in computer network systems. Finally, the preliminary work in [35] introduces Directed Simplicial Neural Networks, formalizing higher-order topological directionality for learning on directed simplicial complexes based on the theoretical foundations of Riihimäki et al. [34].

Neurotopology. The manifold hypothesis posits that neuronal activity is constrained to a lower-dimensional subspace, shaped by the structural limitations imposed by synaptic connectivity [110, 111, 112, 113, 114, 115, 116]. This concept builds on the foundational principle that structure shapes function, succinctly captured by Hebb’s rule: “neurons that fire together, wire together.” Yet, precisely how structural connectivity gives rise to neural manifolds remains a major open question. Biologically grounded computational models, such as the Blue Brain’s [52] rat neocortical microcircuit (NMC) [53], offer a valuable platform for investigating this relationship, providing a detailed representation of brain structure modeled as a directed graph. Notably, recent work [51] confirmed that the NMC model adheres to the manifold hypothesis. However, while directed graphs effectively capture the inherent directionality of synaptic transmission—from presynaptic to postsynaptic neurons [45]—their dyadic nature limits their capacity to represent polyadic interactions and higher-order co-activation patterns that are critical to neural computation. Experimental studies have revealed that neuronal connectivity exhibits significant non-random higher-order structure. For instance, it becomes increasingly unlikely for random activity to produce coherent patterns on higher-dimensional cliques, suggesting that such patterns reflect greater functional complexity and information abstraction. Simplices—higher-dimensional analogs of cliques—have been identified as overexpressed motifs across multiple scales of brain networks [40, 41, 42, 43], and have been linked to functional relevance [44]. Remarkably, human connectomes have been shown to contain undirected simplices of up to $16$ and even $20$ dimensions, despite comprising only $1,115$ nodes representing brain regions [41]. Crucially, neurons embedded in higher-order directed simplices with correlated activity exhibit reduced sensitivity to noise and stronger alignment with the underlying neural manifold [110, 111], ultimately improving representational fidelity [44, 45, 46]. Recent findings also suggest that neural circuits balance robustness and efficiency by organizing into subpopulations with distinct topologies: low-complexity simplices promote efficient communication, while high-complexity ones support resilience—both coexisting within the same network [47]. These insights have fueled the rise of Neurotopology, a field modeling brain activity through topological spaces derived from spike co-activation patterns. Seminal works [48, 45] propose modeling neural dynamics as time-indexed sequences of directed flag complexes [49] and tournaplexes [50], derived from subdigraphs induced by active neurons. Brain dynamics are then characterized by computing topological invariants over these evolving structures. Building on this framework, [51] demonstrated that stimulus identity can be accurately decoded from the NMC model using this topological features—even when conventional methods fail. These results highlight topological featurization as a robust and complementary alternative to traditional manifold-based approaches.

## Appendix D Preliminaries

### D.1 Algebraic Background

Groups. A *group* is a pair $(G,\cdot)$, consisting of a set $G$ and a binary operation $\cdot$ satisfying the following axioms: (A1) Associativity: for all $a,b,c\in G$ we have $a\cdot(b\cdot c)=(a\cdot b)\cdot c$; (A2) Identity: there exists an element $e\in G$ such that $ea=ae=a$ for all $a\in G$; (A3) Inverse: for each $a\in G$ there exists an element $a^{-1}\in G$ such that $a^{-1}a=aa^{-1}=e$. Given a set $S$, the set of all bijections $\rho:S\rightarrow S$ together with the composition of functions forms a group known as the *symmetric group of $S$*, denoted by $\operatorname{Sym}(S)$.

Isomorphisms. Let $X$ and $Y$ be two structured objects of the same type. An *isomorphism* $\psi:X\rightarrow Y$ is a bijective map with inverse $\psi^{-1}:Y\rightarrow X$ that preserves all relevant structure. If such a map exists, we say that $X$ and $Y$ are *isomorphic*, denoted $X\cong Y$. Two sets $S$ and $S^{\prime}$ are isomorphic if there exists a bijection between them. Two directed graphs $\mathcal{G}=(V,E)$ and $\mathcal{G}^{\prime}=(V^{\prime},E^{\prime})$ are isomorphic if there exists a bijection $\psi:V\rightarrow V^{\prime}$ such that $(u,v)\in E$ if and only if $(\psi(u),\psi(v))\in E^{\prime}$. If the vertex sets $V$ and $V^{\prime}$ are attributed—e.g., equipped with dynamic binary functions $B:V\to\{0,1\}^{T}$ and $B^{\prime}:V^{\prime}\to\{0,1\}^{T}$—we additionally require that the attributes are preserved under the inverse map: for all $v\in V$, $B(v)=B^{\prime}\bigl{(}\psi^{-1}(v)\bigr{)}$. In general for any two sets equipped with a collection of relations if set isomorphism preserves relation. Specifically, for directed simplicial complexes $\mathcal{K}$ and $\mathcal{K}^{\prime}$, an isomorphism is a bijection $\psi$ on the vertex set that preserves simplices: a tuple $(v_{0},\dots,v_{n})\in\Sigma_{n}$ is a simplex in $\mathcal{K}$ if and only if $(\psi(v_{0}),\dots,\psi(v_{n}))\in\Sigma^{\prime}_{n}$ is a simplex in $\mathcal{K}^{\prime}$.

Automorphisms. An *automorphism* of a structured object $\mathcal{S}$ is an isomorphism from $\mathcal{S}$ onto itself. The set of all automorphisms of $\mathcal{S}$, denoted by $\operatorname{Aut}(\mathcal{S})$, forms a group under composition. Indeed, composition is associative, the identity map serves as the identity element, and each automorphism is invertible. If $S$ is a plain set, then any bijection (permutation) $\rho:S\to S$ is an automorphism, so $\operatorname{Aut}(S)=\operatorname{Sym}(S)$ is the full symmetric group on $S$.

### D.2 Relational Preliminaries

Binary Relations. Let $S$ be a set. A binary relation $R$ over $S$ is defined as $R\subseteq S\times S$. Specifically, a relation is called *symmetric* if, for all $\sigma,\tau\in S$, $(\sigma,\tau)\in R$ implies $(\tau,\sigma)\in R$; it is *transitive* if, for all $\sigma,\tau,\kappa\in S$, $(\sigma,\kappa)\in R$ and $(\kappa,\tau)\in R$ imply $(\sigma,\tau)\in R$; and *irreflexive* if for all $\sigma\in S$, $(\sigma,\sigma)\not\in R$. As relations are essentially subsets of $S\times S$, natural examples include the universal relation $1=S\times S$, the empty relation $0=\emptyset$, and the identity relation $\mathrm{id}=\{(\sigma,\sigma)\mid\sigma\in S\}$.

Basic Operations. Standard set operations extend naturally to relational operations:

$$ $\displaystyle R\cap R^{\prime}$ $\displaystyle=\{(\sigma,\tau)\in S\times S\mid(\sigma,\tau)\in R\text{ and }( \sigma,\tau)\in R^{\prime}\},$ $\displaystyle R\cup R^{\prime}$ $\displaystyle=\{(\sigma,\tau)\in S\times S\mid(\sigma,\tau)\in R\text{ or }( \sigma,\tau)\in R^{\prime}\}.$ $$

Furthermore, relational composition and converse operations are defined as:

$$ $\displaystyle R\circ R^{\prime}$ $\displaystyle:=\{(\sigma,\tau)\in S\times S\mid\exists\kappa\in S\text{ such that }(\sigma,\kappa)\in R\text{ and }(\kappa,\tau)\in R^{\prime}\},$ $\displaystyle R^{\top}$ $\displaystyle:=\{(\sigma,\tau)\in S\times S\mid(\tau,\sigma)\in R\}.$ $$

Characteristic function. Analogous to how subsets can be identified with their indicator functions, a relation $R\subseteq S\times S$ can be uniquely represented by its characteristic function $\chi_{R}:S\times S\to\{0,1\}$. This function assigns a binary value to each ordered pair $(\sigma,\tau)\in S\times S$: it returns 1 if $(\sigma,\tau)\in R$ and 0 otherwise:

$$ $\chi_{R}((\sigma,\tau))=\begin{cases}1,\text{ if }(\sigma,\tau)\in R,\\ 0,\text{ if }(\sigma,\tau)\notin R.\end{cases}$ $$

Matricial Representation. Equipping the set $S$ with an ordering $S=\{\sigma_{1},\dots,\sigma_{n}\}$, the matricial representation of $R$ is the $n\times n$ matrix $(A_{R})^{i}_{j}=\chi_{R}((\sigma_{i},\sigma_{j}))$. We call it relation matrix or adjacency matrix. Such matrix representations underpin computational implementations, relational learning frameworks, and message-passing neural network architectures [117].

$n$-ary Relations. An *$n$-ary relation* over a set $S$ is a subset $R\subseteq S^{n}$, that is, a collection of $n$-tuples drawn from $S$. For instance, ternary relations $R\subseteq S\times S\times S$ arise by extending binary relations via the natural join operation:

$$ $R\bowtie R^{\prime}=\{\,(\sigma,\kappa,\tau)\in S\times S\times S\mid(\sigma, \kappa)\in R\text{ and }(\kappa,\tau)\in R^{\prime}\,\}.$ $$

This operation yields a ternary relation by joining the tuples from $R$ and $R^{\prime}$ on their shared middle element $y$, thus composing relational information across three elements of the set $S$.

### D.3 Directed vs. Undirected Combinatorial Structures

In general, an *$n$-ary relation* $R\subseteq S^{n}$ is said to be *symmetric* if membership is preserved under any permutation of its elements. That is, for every $n$-tuple $(\sigma_{1},\dots,\sigma_{n})\in R$ and every permutation $\rho\in\operatorname{Sym}(n)$, the permuted tuple $(\sigma_{\rho(1)},\dots,\sigma_{\rho(n)})$ also belongs to $R$. Combinatorial objects such as directed graphs and directed simplicial complexes can be viewed as *asymmetric relational structures* over a shared vertex set $V$, where the relations (of varying arities) encode intrinsic directionality. In contrast, their undirected counterparts can be interpreted as *symmetric relational structures*, in which permutations of tuple entries do not affect membership.

Asymmetric vs. Symmetric Relational Structures. An asymmetric binary relation on a set $S$, that is, a relation for which the presence of an ordered pair $(\sigma,\tau)\in R$ does not necessarily imply the presence of the reverse pair $(\tau,\sigma)$ is naturally represented by a directed graph $\mathcal{G}=(V,E)$; in this context, we identify $S$ with the vertex set $V$ and the relation $R$ with the edge set $E$. In contrast, undirected graphs are well-suited for representing symmetric relations, where the presence of $(\sigma,\tau)$ guarantees the presence of $(\tau,\sigma)$. In such cases, “forgetting” the order—by collapsing $(\sigma,\tau)$ and $(\tau,\sigma)$ into the unordered pair $\{\sigma,\tau\}$—incurs no loss of information. More generally, a directed simplicial complex $\mathcal{K}$ can be viewed as an asymmetric relational structure over a universe $V$, equipped with a family of asymmetric relations $\Sigma_{n}\subseteq V^{n+1}$ of increasing arity (unary, binary, ternary, etc.), which collect the simplices of different dimensions. The asymmetry is inherent: for instance, directed cycles cannot be represented within the framework of directed simplices.

Symmetrization. Given an irreflexive relation $R$ on a set $S$ (or equivalently, a simple directed graph $\mathcal{G}=(V,E)$), we define its symmetrization as $R_{\mathrm{sym}}=R\cup R^{T}$, or, in graph-theoretic terms, $E_{\mathrm{sym}}=E\cup E^{T}$. This construction associates an undirected graph $\mathcal{G}_{\mathrm{sym}}$ with the original directed graph. The correspondence is established via the canonical projection (or quotient map) defined by $\pi((u,v))=\{u,v\}$ for every $(u,v)\in E_{\mathrm{sym}}$. Notably, there is one-to-one correspondence between directed symmetric graphs and undirected graphs. This concept extends naturally to directed simplicial complexes. Given a directed simplicial complex $\mathcal{K}$—which may include directed graphs as a special case—we define its *symmetrization map* $\pi:\mathcal{K}\to\mathcal{K}_{\mathrm{sym}}$ by mapping each $n$-simplex $(v_{0},\dots,v_{n})$ to the unordered set $\{v_{0},\dots,v_{n}\}$ for all dimensions $n$. While this map preserves the simplicial structure (i.e., sends directed simplicial complexes to valid abstract simplicial complexes), it discards ordering information by identifying every ordered simplex with all of its permutations. This results in a potential loss of expressiveness, which we formalize in the following proposition.

Figure: (a)
Refer to caption: extracted/6484153/fig/digraphs.png

###### Proposition 2 .

The symmetrization map $\operatorname{\pi}:\mathcal{K}\to\mathcal{K}_{\mathrm{sym}}$ is a simplicial structure *forgetful map*, that is, preserves isomorphisms but is not injective on isomorphism classes.

###### Proof.

Let $\psi$ be an isomorphism between $\mathcal{K}$ and $\mathcal{K}^{\prime}$ and let $\psi_{\mathrm{sym}}(\sigma)=\{\psi(v):v\in\sigma\}$. Note that, $\psi_{\mathrm{sym}}$ is well defined since different orderings of the same vertex set in $\sigma$ are mapped to the same set $\{\psi(v):v\in\sigma\}$. Moreover, by construction $\pi\circ\psi=\psi_{\mathrm{sym}}\circ\pi$, implying preservation of simplicial structure (simplices are sent to simplices). Since $\psi$ is bijective and $\pi$ does not collapse vertices (only their ordering), it follows that $\psi_{\mathrm{sym}}$ is a bijective simplicial map preserving the face structure. Hence, $\psi_{\mathrm{sym}}$ is an isomorphism between $\mathcal{K}_{\mathrm{sym}}$ and $\mathcal{K}^{\prime}_{\mathrm{sym}}$, and $\operatorname{\pi}$ is isomorphims-preserving. Clearly, there exist non-isomorphic directed simplicial complexes $\mathcal{K}$ and $\mathcal{K}^{\prime}$ such that $\pi(\mathcal{K})\cong\pi(\mathcal{K}^{\prime}),$ implying the non-injectivity. Finally, note that the number of simplices in $\mathcal{K}_{\mathrm{sym}}$ equals that in $\mathcal{K}$ if and only if no two distinct simplices in $\mathcal{K}$ share the same underlying vertex set—that is, each simplex has a unique ordering. Otherwise, $\pi$ collapses the redundant orderings, reducing the total face count (see Figure [4](https://arxiv.org/html/2505.17939v2#A4.F4)).
∎

Symmetrization naturally extends to attributed directed simplicial complexes via a fixed permutation-invariant aggregation of features.

Transitivization. Conversely, given a simplicial complex $\mathcal{K}=(V,\Sigma)$ endowed with an arbitrary total order $<$ on its vertex set $V$, one naturally obtains a unique directed simplicial complex $\mathcal{K}_{\mathrm{dir}}$ by assigning to each simplex $\sigma\in\Sigma$ the directed simplex $\sigma_{<}=(v_{0},v_{1},\dots,v_{n})$, where the vertices of $\sigma$ are arranged in increasing order $(v_{0}<v_{1}<\cdots<v_{n})$. Consequently, for every pair of vertices $v_{i},v_{j}\in\sigma$ with $i<j$, the directed edge $(v_{i},v_{j})$ is present in $\mathcal{K}_{\mathrm{dir}}$. By construction, every simplex thus forms a transitive clique at the graph level.

###### Proposition 2 .

###### Proof.

### D.4 Face-Map Relation Algebras

Let $\mathcal{S}$ be a semi-simplicial set with simplices $S$ and face maps $d_{i}^{n}:S_{n}\to S_{n-1}$. Each face map naturally induces an irreflexive binary relation:

$$ $R_{d_{i}^{n}}=\{(\sigma,\tau)\mid\sigma\in S_{n},\;d_{i}^{n}(\sigma)=\tau\}\in \mathcal{P}(S\times S),$ $$

which relates each simplex $\sigma\in S_{n}$ with its $i$-th facet in $S_{n-1}$.

Positive Face Map Induced Relation Algebra. The set of all binary relations on $S$, denoted as $\mathcal{P}(S\times S)$, forms a *positive relation algebra* closed under the operations of union $(\cup)$, intersection $(\cap)$, relational composition $(\circ)$, and converse $(\top)$. Specifically, the structure:

$$ $\mathcal{R}=(\mathcal{P}(S\times S),\,\cup,\,\cap,\,\circ,\,\top,\mathrm{id}, \,0,\,1)$ $$

satisfies standard axioms in [118]. We define the *face map relation algebra* $\mathcal{R}_{d}$ as the smallest positive relation subalgebra of $\mathcal{P}(S\times S)$ generated by the collection $\{R_{d_{i}^{n}}\mid n\geq 0,,0\leq i\leq n\}$, equipped with standard operations $\cup,\,\cap,\,\circ,\,\top$ together with relations $\mathrm{id}$, 0 0, $1$.

For theoretical purposes, we consider ternary relations from natural joins $R\bowtie R^{\prime}$ from $R,R^{\prime}\in\mathcal{R}_{d}$. These higher-order relations play a role in our generalization of Weisfeiler–Leman-type procedures (see Appendix [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2)).

### D.5 Examples of Common Graph and Topological Face-Map Relations

Overall, face-map–induced relations generalize classical binary relations found in graphs, digraphs, and simplicial complexes, thereby providing a principled algebraic foundation for relational message-passing architectures [117, 33, 22] (see Appendix [E.1](https://arxiv.org/html/2505.17939v2#A5.SS1)). In this section, we provide some of this common examples.

Let $S$ be a semi-simplicial set with set of simplices $S$. We define the following face map–induced binary relations:

$$ $R_{n\downarrow,i,j}\coloneqq R_{d_{j}^{n}}^{\top}\circ R_{d_{i}^{n}}=\{(\sigma ,\tau)\mid d_{i}^{n}(\sigma)=d_{j}^{n}(\tau)\}\in\mathcal{R}_{d},\\$ (4) $$

which relates two $n$-simplices $\sigma$ and $\tau$ if their $i$-th and $j$-th facets coincide, respectively (see Figure [5](https://arxiv.org/html/2505.17939v2#A4.F5)). A natural extension, often employed in message-passing architectures, is the following ternary relation:

$$ $\tilde{R}_{n\downarrow,i,j}\coloneqq R_{n\downarrow,i,j}\bowtie R_{d_{i}^{n}}= \{(\sigma,\tau,\kappa)\mid d_{i}^{n}(\sigma)=d_{j}^{n}(\tau)=\kappa\}\in \mathcal{R}_{d},$ (5) $$

which explicitly includes the shared lower-dimensional face $\kappa$ between simplices $\sigma$ and $\tau$.

Figure: Figure 5: Two $2$-simplices (triangles) $\sigma$ and $\tau$ are related such that $d_{0}^{2}(\sigma)=d_{2}^{2}(\tau)=e$, where $e$ denotes the shared edge highlighted in red. Therefore, $(\sigma,\tau)\in R_{2\downarrow,0,2}$.
Refer to caption: extracted/6484153/fig/downij.png

Similarly, we define:

$$ $R_{n\uparrow,i,j}\coloneqq R_{d_{j}^{n+1}}\circ R_{d_{i}^{n+1}}^{\top}=\{( \sigma,\tau)\mid\exists\kappa\in S,\;d_{i}^{n}(\kappa)=\sigma\text{ and }d_{j} ^{n}(\kappa)=\tau\}\in\mathcal{R}_{d}$ (6) $$

which relates $\sigma$ and $\tau$ if they are, respectively, the $i$-th and $j$-th facets of the same $(n+1)$-simplex $\kappa$ (see Figure [6](https://arxiv.org/html/2505.17939v2#A4.F6)). Its corresponding ternary extension is defined as:

$$ $\tilde{R}_{n\uparrow,i,j}\coloneqq R_{n\uparrow,i,j}\bowtie R_{d_{i}^{n+1}}^{ \top}=\{(\sigma,\tau,\kappa)\mid\exists\kappa\in S,\;d_{i}^{n}(\kappa)=\sigma \text{ and }d_{j}^{n}(\kappa)=\tau\}\in\mathcal{R}_{d},$ (7) $$

which explicitly includes the common higher-dimensional simplex $\kappa$. From this point forward, we will use the same notation for both binary and ternary relations when the context is clear, as all constructions are valid in both cases. In Section [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2), unless otherwise specified, relations will be assumed to be in their ternary form.

Figure: Figure 6: Two $1$-simplices (edges) $\sigma$ (red) and $\tau$ (brown) are related as facets of a common $(n+1)$-simplex $\kappa$ (light purple triangle). In (left), $(\sigma,\tau)\in R_{1\uparrow,2,0}$, whereas in (right), $(\sigma,\tau)\in R_{1\uparrow,2,1}$.
Refer to caption: extracted/6484153/fig/upedges.png

Notably, when $i\neq j$, these relations are asymmetric, thereby inducing directed graphs over the set of simplices $S$. Well-known examples of directional face-map relations include those arising in graph and simplicial structures. For example, at the level of 0 0-simplices (nodes), the standard directed up adjacency relations can be derived as follows:

$$ $R_{\mathrm{in}}\coloneqq R_{0\uparrow,0,1}=\{\,(u,v)\mid\exists e\in E,\;d_{0} ^{1}(e)=u\text{ and }d_{1}^{1}(e)=v\,\},\in\mathcal{R}_{d}$ (8) $$

indicating that $u$ is the target of $v$ (i.e., $v$ is the source of $e$ and $u$ its sink). Moreover,

$$ $R_{\mathrm{out}}\coloneqq R_{0\uparrow,1,0}=\{\,(u,v)\mid\exists e\in E,d_{1}^ {1}(e)=u\text{ and }d_{0}^{1}(e)=v\,\}\in\mathcal{R}_{d},$ (9) $$

meaning $u$ is the source of $v$ (i.e., $v$ is the target of $e$, and $u$ the source). These two relations are converses of each other: $(R_{\mathrm{in}})^{\top}=R_{\mathrm{out}}$, and thus:

$$ $R_{\mathrm{sym}}=R_{\mathrm{in}}\cup R_{\mathrm{out}}\in\mathcal{R}_{d},$ (10) $$

induces a symmetric directed graph $\mathcal{G}_{\mathrm{sym}}=(V,E_{\mathrm{sym}})$ in one-to-one correspondence with the undirected graph $\mathcal{G}_{\mathrm{sym}}=(V,E)$.

Directed Paths and Higher Order Directionality. A directed path of length $k$ in a digraph $\mathcal{G}=(V,E)$ is a sequence of vertices $(v_{1},v_{2},\dots,v_{k})$ such that $(v_{i},v_{i+1})\in E$ for every $1\leq i<k$. Similarly, given a set of simplices $\mathcal{S}$ of a semi-simplicial set equipped with a face-map induced relation $R\in\mathcal{R}_{d}$, an $R$-path of lenght $k$ in the semi-simplicial set is a sequence of simplices $(\sigma_{1},\sigma_{2},\dots,\sigma_{k})$ such that $(\sigma_{i},\sigma_{i+1})\in R$ for every $1\leq i<k$. Notably, $(\sigma_{1},\sigma_{k})\in R^{\circ k}$, where $R^{\circ k}$ denotes the $k$-fold composition of $R$ with itself. Composing different instances of $R_{n\downarrow,i,j}$ yields different higher-order simplicial paths, connecting directed simplices in distinct direction-preserving manner [34, 35] (see Figure [7](https://arxiv.org/html/2505.17939v2#A4.F7)).

Figure: (a)
Refer to caption: extracted/6484153/fig/ho02.png

Boundary & Converse Boundary. Let $\mathcal{K}=(V,\Sigma)$ be a simplicial complex with maximal dimension $N$, and let $\mathcal{K}_{\mathrm{dir}}$ denote the associated directed simplicial complex induced by a fixed total order on $V$. For each dimension $n$ with $1\leq n\leq N$, the nth *boundary relation* is defined as:

$$ $B_{n}\coloneqq\bigcup_{i=0}^{n}R_{d_{i}^{n}},$ (11) $$

which relates each $n$-simplex to all its $(n-1)$-dimensional faces. Conversely, for $N-1\geq n\geq 0$, the $n$-th *converse boundary relation* is defined as

$$ $C_{n}\coloneqq\bigcup_{i=0}^{n+1}(R_{d_{i}^{n+1}})^{\top},$ (12) $$

relating each $n$-simplex to all $(n+1)$-dimensional simplices containing it as a facet. Notably, for any $0\leq m<n\leq N$ the composition of maps $B_{n,m}\coloneqq B_{m+1}\circ B_{m+2}\circ\cdots\circ B_{n},$ relating an $n$-simplex with its $m$-dimensional faces. Similarly, for the converse boundary relations, define $C_{m,n}\coloneqq C_{n-1}\circ\cdots C_{m+1}\circ C_{m},$ relating $n$-simplex with all $m$-simplices containing it.

Aggregated Lower and Upper Adjacencies. For each dimension $N\geq n>0$, we define the collection of all possible $n$-*lower adjacency* relations as

$$ $\mathcal{R}_{\downarrow n}=\{R_{\downarrow n,i,j}\mid(i,j)\in[n]^{2}\}.$ $$

Accordingly, we define

$$ $\displaystyle R_{\downarrow n}\coloneqq\bigcup\mathcal{R}_{\downarrow n},$ (13) $$

which relates a simplex $\sigma$ to all simplices $\tau$ sharing a common facet with $\sigma$. Then, for dimensions $N-1\geq n\geq 0$, we define the collection of all possible $n$-th *upper adjacency* relations by

$$ $\mathcal{R}_{\uparrow n}=\{R_{\uparrow n,i,j}\mid(i,j)\in[n]^{2}\text{ and }i \neq j\}.$ $$

Finally, we define the aggregated upper adjacency relation as

$$ $\displaystyle R_{\uparrow n}\coloneqq\bigcup\mathcal{R}_{\uparrow n},$ (14) $$

which relates a simplex $\sigma$ to all simplices $\tau$ that share a common higher-dimensional simplex containing both as facets. Notably, $B_{n}$, $C_{n}$, $R_{\downarrow n}$, $R_{\uparrow n}$, coincide with the four types of adjacent simplices defined in [22] under which MPSNN operate. Under this setting, we denote $\mathcal{U}_{n}=\{B_{n},C_{n},R_{\downarrow n},R_{\uparrow n}\}$ and $\mathcal{D}_{n}=\{B_{n},C_{n}\}\cup\mathcal{R}_{\downarrow n}\cup\mathcal{R}_{
\uparrow n}$ with the understanding that only those relations which are defined are included. Accordingly,

$$ $\mathcal{U}=\bigcup_{n=0}^{N}\mathcal{U}_{n}\qquad\text{ and }\qquad\mathcal{D }=\bigcup_{n=0}^{N}\mathcal{D}_{n},$ (15) $$

are defined.

### D.6 Invariants, Colorings and the Weisfeiler–Leman Test

Invariants. An *invariant* for a given class of objects is a function $\operatorname{f}$ assigning to each object a value that remains unchanged under isomorphisms. Formally, if $X\cong Y$, then $\operatorname{f}(X)=\operatorname{f}(Y)$. Objects with identical values under $\operatorname{f}$ share a common structural property. A classical example is set cardinality, a *complete invariant*: sets with the same cardinality are isomorphic. In contrast, vertex count is an *incomplete invariant* for graphs, as non-isomorphic graphs can share vertex counts, e.g., a complete and an edgeless graph.

From Local (vertex) to Global (graph/complex) Invariants. Given a digraph $\mathcal{G}=(V,E)$ or directed simplicial complex $\mathcal{K}=(V,\Sigma)$, potentially with dynamic binary functions, a *vertex invariant* is a map $\operatorname{c}:V\rightarrow\mathcal{C}$ or $\operatorname{c}:V\times\{0,1\}^{T}\rightarrow\mathcal{C}$ invariant under isomorphisms, meaning $c(v)=c(\psi(v))$ for all vertices $v$ under isomorphism $\psi$. Vertex invariants extend naturally to graph or complex invariants through a permutation-invariant aggregation, i.e, functions that are invariant under set isomorphism. Precisely, given a vertex invariant $\operatorname{c}$, an induced *binary graph invariant* can be defined as: $\operatorname{c}(\mathcal{G}_{B})=\phi\bigl{(}\{\{\operatorname{c}(v)\mid v\in
V
\}\}\bigr{)}$, where $\phi$ is a permutation-invariant function. Consequently, if $\mathcal{G}_{B}\cong\mathcal{G}^{\prime}_{B}$, then $\operatorname{c}(\mathcal{G}_{B})=\operatorname{c}(\mathcal{G}^{\prime}_{B})$. Real-valued vector vertex invariants arranged as ordered sequences, $\operatorname{c}(\mathcal{G}_{B})=(\operatorname{c}(v_{0}),\dots,\operatorname
{c}(v_{N}))$ according to a fixed vertex ordering $V=\{v_{1},\dots,v_{N}\}$ constitute graph features, which are generally not invariant, as they explicitly depend on the chosen vertex ordering.

Graph coloring. Let $\mathcal{G}=(V,E)$ be a graph. A graph coloring is a vertex invariant $\operatorname{c}:V\rightarrow\mathbb{N}$ which assigns to each vertex a positive integer (its color). For any $n\in\mathbb{N}$, the preimage $c^{-1}(n)=V_{n}$ partitions the vertex set into subsets of vertices sharing the same color property. The pair $\mathcal{G}_{c}=(\mathcal{G},\operatorname{c})$ is called the *colored graph* and is an attributed graph.

Color Refinement. A color assigment $\operatorname{c}^{\prime}$ is a *refiment* of $\operatorname{c}$ if for every pair $u,v\in V$, $\operatorname{c}^{\prime}(u)=\operatorname{c}^{\prime}(v)$ implies $\operatorname{c}(u)=\operatorname{c}(v)$. In other words, vertices that share the same color under $\operatorname{c}^{\prime}$ must also share the same color under $\operatorname{c}$. Thus, $\operatorname{c}$ partitions the vertex set into a potentially richer variety of color classes. Specifically, they are equivalent $\operatorname{c}^{\prime}\cong\operatorname{c}$ , if for every pair $u,v\in V$, $\operatorname{c}^{\prime}(u)=\operatorname{c}^{\prime}(v)$ if and only if $\operatorname{c}(u)=\operatorname{c}(v)$.

The Weisfeiler–Leman Test. Given a graph $\mathcal{G}$, define an initial coloring $\operatorname{c}^{0}$, e.g., setting $\operatorname{c}^{0}(v)=0$ for all $v\in V$. This yields the colored graph $\mathcal{G}_{0}=(\mathcal{G},\operatorname{c}^{0})$; clearly $\mathcal{G}\cong\mathcal{G}^{\prime}$ if and only if $\mathcal{G}_{0}\cong\mathcal{G}^{\prime}_{0}$. For a vertex $u\in\mathcal{G}_{0}$, define the invariant $\operatorname{c}^{l}_{E}:V\times\mathbb{N}\to\mathcal{M}(\mathbb{N})$ (here, $\mathcal{M}(\mathbb{N})$ denotes the set of finite multisets of $\mathbb{N}$) by $\operatorname{c}^{l}_{E}(u)=\{\{\operatorname{c}^{l}(v)\mid(u,v)\in E\}\}$, i.e., assigning the multiset of colors of $u$’s neighbours the relation $E$. Then, we obtain a refined coloring $\operatorname{c}^{l+1}$ by setting $\operatorname{c}^{l+1}(u)=\operatorname{hash}\bigl{(}\operatorname{c}^{l}(u),
\,\operatorname{c}_{E}^{l}(u)\bigr{)}\in\mathbb{N},$ where $\operatorname{hash}\colon\mathbb{N}\times\mathcal{M}(\mathbb{N})\to\mathbb{N}$ is an injective function. The refinement process continues until the coloring stabilizes; that is, until $\operatorname{c}^{l+1}(u)=\operatorname{c}^{l}(u)$ for all $u\in V$. Once stability is reached, denote the stable coloring by $\operatorname{c}$ and the resulting colored graph by $\mathcal{G}_{c}=(\mathcal{G},\operatorname{c})$. Assume that the image of $\operatorname{c}$ consists of $k$ distinct colors (which, without loss of generality, can be labeled $\{1,\dots,k\}$). A colored graph invariant $\operatorname{h}$ is then obtained by counting the vertices of each color: $\operatorname{h}(\mathcal{G}_{c})=(\mid\operatorname{c}^{-1}(1)\mid,\dots,\mid
\operatorname{c}^{-1}(k)\mid)$. Thus, for a pair of graphs $\mathcal{G}$ and $\mathcal{G}^{\prime}$ we have $\operatorname{h}(\mathcal{G}_{c})\neq\operatorname{h}(\mathcal{G}^{\prime}_{c})$, then $\mathcal{G}\not\cong\mathcal{G}^{\prime}$. This algorithmic procedure, yielding incomplete graph invariants discriminating some non-isomorphic pairs, is the Weisfeiler–Leman test [60].

## Appendix E Theoretical properties of SSNs

A *Message Passing Semi-Simplicial Neural Network* (MPSSN) is a specific instantiation of a semi-simplicial neural network (SSN), as defined in Equation [1](https://arxiv.org/html/2505.17939v2#S3.E1), in which each relation-specific transformation $\omega_{R}$ follows the directional message-passing paradigm introduced in the MPNN-D module [33]. In this setting, messages are inherently asymmetric and propagated along the relation $R$: that is, if $(\sigma,\tau)\in R$, then $\omega_{R}$ updates the representation of $\tau$ using information from $\sigma$, but not the other way around. For the remainder of this section, we adopt MPSSNs as our reference architecture for theoretical analysis.

### E.1 Generality

###### Proposition 3 .

Semi-simplicial neural networks (SSNs) subsume directed message-passing GNNs (Dir-GNNs) [33] on directed graphs, message-passing GNNs (MPNNs) [12] on undirected graphs, message-passing simplicial neural networks (MPSNNs) [22] on simplicial complexes and Directed Simplicial Neural Networks (Dir-SNN) on directed simplicial complexes.

###### Proof.

Let $\omega_{R}$ denote the message propagation operator associated with a face-map–induced relation $R$, as defined in Equation [1](https://arxiv.org/html/2505.17939v2#S3.E1). We instantiate each $\omega_{R}$ following the directional message-passing scheme of the MPNN-D framework [33], which propagates information along $R$: specifically, if $(\sigma,\tau)\in R$, then $\omega_{R}$ updates the embedding of $\tau$ using the features of $\sigma$. First, consider an attributed directed graph $\mathcal{G}_{F}$. If the SSN is instantiated with the relation set $\mathcal{R}=\{R_{\mathrm{in}},R_{\mathrm{out}}\}$ and the directional propagation rules of MPNN-D, the resulting model corresponds exactly to a Dir-GNN [33]. Next, if $\mathcal{G}_{F}$ is undirected—viewed as a symmetric directed graph—then choosing $\mathcal{R}=\{R_{\mathrm{sym}}\}$ yields an $R_{\mathrm{sym}}$-MPSSN that recovers the standard message-passing GNN [117, 12] operating over $\mathcal{G}_{F}$. Let $\mathcal{\tilde{K}}_{F}$ be an attributed simplicial complex, and let $\mathcal{\tilde{K}}_{\mathrm{dir},F}$ be its associated directed simplicial complex induced by a fixed total order on the vertices (see Appendix [D.4](https://arxiv.org/html/2505.17939v2#A4.SS4)). Then, an MPSSN operating on $\mathcal{\tilde{K}}_{\mathrm{dir},F}$ with the undirected relation set $\mathcal{U}$ (as defined in Equation [15](https://arxiv.org/html/2505.17939v2#A4.E15)) corresponds exactly to a message-passing simplicial neural network (MPSNN) [22] acting on $\mathcal{K}_{F}$. Finally, let $\mathcal{\tilde{K}}_{F}$ be an attributed directed simplicial complex, an MPSSN operating on the set of relations $\mathcal{D}$ as in (as defined in Equation [15](https://arxiv.org/html/2505.17939v2#A4.E15)) corresponds exactly to a message-passing Directed Simplicial Neural Network (Dir-SNN) [35]. Therefore, by appropriately selecting the relation set $\mathcal{R}\subseteq\mathcal{R}_{d}$ and instantiating the message propagation operators $\{\omega_{R}\}$ accordingly, the SSN framework unifies and generalizes MPNNs, Dir-GNNs, MPSNNs and Dir-SNNs.
∎

###### Proposition 3 .

###### Proof.

### E.2 Weisfeiler–Leman Expressiveness

Relational Weisfeiler–Leman Test. Let $S$ be a structured set equipped with a collection $\mathcal{R}$ of relations on $S$. We extend the standard Weisfeiler–Leman (WL) test by incorporating multiple relations in the procedure. Define an initial coloring $\operatorname{c}^{0}$, for instance, $\operatorname{c}^{0}(\sigma)=0$ for all $\sigma\in S$. For each $R\in\mathcal{R}$, each iteration $l\geq 0$, and each $\sigma\in S$, define the invariant $\operatorname{c}^{l}_{R}:S\to\mathcal{M}(\mathbb{N})$ by

$$ $\operatorname{c}^{l}_{R}(\sigma)=\{\{\operatorname{c}^{l}(\tau)\mid(\sigma, \tau)\in R\}\},$ $$

which is the multiset of colors of elements related to $\sigma$ under $R$. The refined coloring is then obtained by:

$$ $\operatorname{c}^{l+1}(\sigma)=\operatorname{hash}\Bigl{(}\operatorname{c}^{l} (\sigma),\,\bigl{(}\operatorname{c}^{l}_{R}(\sigma)\bigr{)}_{R\in\mathcal{R}} \Bigr{)},$ (16) $$

where $\operatorname{hash}\colon\mathbb{N}\times\mathcal{M}(\mathbb{N})^{|\mathcal{R}
|}\to\mathbb{N}$ is an injective function.

Union-Closure and Refinement. Given a collection of sets $\mathcal{A}$, the *union-closure* of $\mathcal{A}$ is defined as

$$ $\langle\mathcal{A}\rangle=\{\bigcup S\mid S\subseteq\mathcal{A}\}.$ $$

For two collections of relations $\mathcal{A}$ and $\mathcal{B}$ on a set $S$, we say that $\mathcal{A}$ is a *refinement by unions* of $\mathcal{B}$ if $\mathcal{B}\subseteq\langle\mathcal{A}\rangle$.

###### Lemma 1 .

Let $S$ be a set, and let $\mathcal{A}$ and $\mathcal{B}$ be finite collections of relations on $S$, such that $\mathcal{A}$ is a refinement by unions of $\mathcal{B}$. Then, $\mathcal{A}$-WL is at least as expressive as $\mathcal{B}$-WL.

###### Proof.

Let $\operatorname{a}$ and $\operatorname{b}$ denote the colorings of $\mathcal{A}$-WL and $\mathcal{B}$-WL, respectively. We show by induction that $\operatorname{a}$ refines $\operatorname{b}$ at each iteration $l$. Initially, we set $\operatorname{a}^{0}(\sigma)=\operatorname{b}^{0}(\sigma)=0$ for all $\sigma\in S$, trivially satisfying the base case. Suppose the hypothesis holds for iteration $l$. Assume $\operatorname{a}^{l+1}(\sigma)=\operatorname{a}^{l+1}(\tau)$ for $\sigma,\tau\in S$. By injectivity of $\operatorname{hash}$, we have $\operatorname{a}_{R}^{l}(\sigma)=\operatorname{a}_{R}^{l}(\tau)$ for each $R\in\mathcal{A}$. By Lemma 2 in [119], equal multisets under a finer coloring remain equal under a coarser coloring. Since $\mathcal{A}$ is a refinement by unions of $\mathcal{B}$, for each $R^{\prime}\in\mathcal{B}$ there exists a subcollection $\mathcal{A}_{R^{\prime}}\subseteq\mathcal{A}$ such that $R^{\prime}=\bigcup_{R\in\mathcal{A}_{R^{\prime}}}R$. Therefore,

$$ $\operatorname{b}^{l}_{R^{\prime}}(\sigma)=\bigoplus_{R\in\mathcal{A}_{R^{ \prime}}}\operatorname{b}^{l}_{R}(\sigma)=\bigoplus_{R\in\mathcal{A}_{R^{ \prime}}}\operatorname{b}^{l}_{R}(\tau)=\operatorname{b}^{l}_{R^{\prime}}(\tau),$ $$

where $\bigoplus$ denotes the union of multisets, summing up multiplicities. Consequently, arguments to the hash function at iteration $l$ coincide, yielding $\operatorname{b}^{l+1}(\sigma)=\operatorname{b}^{l+1}(\tau)$, as required.
∎

Semi-Simplicial Weisfeiler–Leman (SSWL) Test. Let $\mathcal{S}$ be a semi-simplicial set with set of simplices $S$. Let $\mathcal{R}\subseteq\mathcal{R}_{d}$ be a collection of face-map–induced relations. The *Semi-Simplicial Set Weisfeiler–Leman Test* is defined as the relational WL test (Equation ([16](https://arxiv.org/html/2505.17939v2#A5.E16))) applied to the attributed set $S$ equipped with relations $\mathcal{R}$.

###### Lemma 2 .

Let $S^{\mathcal{R}}$ be the set of simplices from a semi-simplicial set $\mathcal{S}$ equipped with face-map–induced relations $\mathcal{R}\subseteq\mathcal{R}_{d}$. Consider an $\mathcal{R}$-MPSSN defined by Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)). Suppose the message module $\omega_{R}$, the aggregator $\bigotimes$, and the update function $\phi$ are all injective. Then, an $\mathcal{R}$-MPSSN is as expressive as the $\mathcal{R}$-SSWL test.

###### Proof.

Let $\operatorname{c}$ and $\operatorname{x}$ denote the colorings produced by the $\mathcal{R}$-SSWL and the $\mathcal{R}$-MPSSN, respectively. We show by induction that $\operatorname{c}$ refines $\operatorname{x}$. At $l=0$, without loss of generality, we initialize $\operatorname{c}^{0}(\sigma)=\operatorname{x}^{0}(\sigma)=0$ for all $\sigma\in S$, establishing the base case trivially. Assume the induction hypothesis holds at iteration $l$. Consider two $n$-simplices $\sigma,\tau\in S$ such that $\operatorname{c}^{l+1}(\sigma)=\operatorname{c}^{l+1}(\tau)$. By the definition of $\mathcal{R}$-SSWL, this implies identical multisets of colors aggregated from neighboring simplices at iteration $l$. By the induction hypothesis, these multisets coincide under $\operatorname{x}^{l}$, ensuring equality in the arguments of the $\mathcal{R}$-SSN, thereby yielding $\operatorname{x}^{l+1}(\sigma)=\operatorname{x}^{l+1}(\tau)$. By induction, $\operatorname{c}$ refines $\operatorname{x}$. Conversely, if $\operatorname{x}^{l+1}(\sigma)=\operatorname{x}^{l+1}(\tau)$, the injectivity of the composition of injective operators $\omega_{R}$, $\bigotimes$ and $\phi$, implies identical multisets at iteration $l$. By the induction hypothesis, the hash function receives identical inputs, resulting in $\operatorname{c}^{l+1}(\sigma)=\operatorname{c}^{l+1}(\tau)$. Thus, $\operatorname{c}\cong\operatorname{x}$, proving that $\mathcal{R}$-MPSSN and $\mathcal{R}$-SSWL are equally expressive.
∎

Figure: (a) $\mathcal{G}$
Refer to caption: extracted/6484153/fig/2_u_t_graph.png

SSN vs. Dir-GNN. We now prove Theorem [1](https://arxiv.org/html/2505.17939v2#Thmtheorem1).

###### Theorem 5 .

There exist SSNs that are strictly more expressive than directed graph neural networks (Dir-GNNs) [33] at distinguishing non-isomorphic directed graphs.

###### Proof.

Let $\mathcal{G}$ be a directed graph and $\mathcal{K}_{\mathcal{G}}$ its corresponding directed flag complex, with maximal dimension $2$. Consider the MPSSN defined on $\mathcal{K}_{\mathcal{G}}$ with face-map–induced relations $\mathcal{D}_{0}=\{R_{\mathrm{in}},R_{\mathrm{out}}\}$ and $\mathcal{D}$ as in [15](https://arxiv.org/html/2505.17939v2#A4.E15). Then, the proof follows from Theorem 1. of [35]. To build intuition, Figure [11](https://arxiv.org/html/2505.17939v2#A9.F11) illustrates a pair of non-isomorphic directed graphs in which both $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$ correspond to $2$-uniform relations. Thereby, assuming constant activation features across all vertices, these graphs remain indistinguishable by Dir-GNNs, yet are successfully distinguished by SSNs due to structural differences in their corresponding flag complexes.
∎

MPSSNs vs. MPSNNs. We prove that there exist instances of MPSSNs that are strictly more powerful than MPSNNs [22] at distinguishing directed simplicial complexes.

Figure: (a) $\mathcal{K}_{a}$
Refer to caption: extracted/6484153/fig/dfg_a.png

###### Lemma 3 .

Let $\mathcal{K}$ be a directed simplicial complex with set of simplices $\Sigma$, and let $\mathcal{U}$ and $\mathcal{D}$ be collections of face-map–induced relations defined as in Equation [15](https://arxiv.org/html/2505.17939v2#A4.E15). Then, $\mathcal{D}$-SSWL is at least as expressive as $\mathcal{U}$-SSWL.

###### Proof.

Since $\mathcal{D}$ is a refinement by unions of $\mathcal{U}$, the result immediately follows from Lemma [1](https://arxiv.org/html/2505.17939v2#Thmlemma1).
∎

###### Lemma 4 .

There exists a pair of directed simplicial complexes that are distinguishable by $\mathcal{D}$-SSWL but indistinguishable by $\mathcal{U}$-SSWL.

###### Proof.

Let $\mathcal{K}_{a}$, $\mathcal{K}_{b}$, and $\mathcal{K}_{\mathrm{sym}}$ be as in Figure [9](https://arxiv.org/html/2505.17939v2#A5.F9), with constant activation values assigned to all simplices. By construction, the $\mathcal{U}$-SSWL updates on $\mathcal{K}_{a}$ and $\mathcal{K}_{b}$ are equivalent to running SWL [22] on the shared symmetrized complex $\mathcal{K}_{\mathrm{sym}}$, and thus produce identical outputs. In contrast, $\mathcal{D}$-SSWL distinguishes $\mathcal{K}_{a}$ and $\mathcal{K}_{b}$ due to their differing directional structure.
∎

###### Corollary 1 .

$\mathcal{D}$-SSWL is strictly more expressive than $\mathcal{U}$-SSWL.

###### Proof.

Direct consequence of Lemmas [3](https://arxiv.org/html/2505.17939v2#Thmlemma3) and [4](https://arxiv.org/html/2505.17939v2#Thmlemma4).
∎

###### Lemma 5 .

There exist $\mathcal{D}$-SSNs that are strictly more expressive than $\mathcal{U}$-SSNs in distinguishing non-isomorphic directed simplicial complexes [22].

###### Proof.

Follows directly from Corollary [1](https://arxiv.org/html/2505.17939v2#Thmcorollary1) and Lemma [2](https://arxiv.org/html/2505.17939v2#Thmlemma2).
∎

###### Theorem 6 .

There exist SSNs that are strictly more expressive in distinguishing non-isomorphic directed simplicial complexes than MPSNNs[22] under symmetrization.

###### Proof.

Follows directly from Lemma [5](https://arxiv.org/html/2505.17939v2#Thmlemma5) and Proposition [2](https://arxiv.org/html/2505.17939v2#Thmproposition2), which ensures the symmetrization map $\pi$ to be forgetful.
∎

###### Lemma 1 .

###### Proof.

###### Lemma 2 .

###### Proof.

###### Theorem 5 .

###### Proof.

###### Lemma 3 .

###### Proof.

###### Lemma 4 .

###### Proof.

###### Corollary 1 .

###### Proof.

###### Lemma 5 .

###### Proof.

###### Theorem 6 .

###### Proof.

### E.3 Permutation equivariance and invariance

Let $S_{F}^{\mathcal{R}}$ be an attributed set equipped with relations $\mathcal{R}$. Every permutation $\rho\in\operatorname{Aut}(S)$ induces an isomorphic relabelled structure $(S_{F}^{\mathcal{R}})_{\rho}=(\rho\cdot S,\rho\cdot R,\rho\cdot F)$ where $\rho\cdot R\;=\;\{\,(\rho(\sigma),\rho(\tau)):(\sigma,\tau)\in R\}$ for all $R\in\mathcal{R}$ and $\rho\cdot F\;=\;F\circ\rho^{-1}$. We say that a map $\phi:S_{F}^{\mathcal{R}}\rightarrow(S_{F}^{\mathcal{R}})^{\prime}$ is:

- •
*Permutation equivariant* if, for every $\rho\in\operatorname{Aut}(S)$, it holds that $\phi\circ\rho=\rho\circ\phi$.
- •
*Permutation invariant* if, for every $\rho\in\operatorname{Aut}(S)$, it satisfies $\phi\circ\rho=\phi$.

Consider an attributed semi-simplicial set $\mathcal{S}$ with attributed simplices $S_{F}$ and face-map-induced relations $\mathcal{R}$. Representing $S_{F}$ as a matrix requires assigning an arbitrary global labeling (ordering) to simplices $S=\{\sigma_{1},\dots,\sigma_{N}\}$ (see Section [2](https://arxiv.org/html/2505.17939v2#S2)). Neural network models processing these data must therefore exhibit invariance or equivariance under relabeled (permutations of labels) isomorphic structures. Specifically, Semi-Simplicial Neural Networks (SSNs) operate on an attributed set of simplices with face-map-induced relations $S_{F}^{\mathcal{R}}$, necessitating consistent behavior under relabeling of the entire structure, including both the relations and feature assignments. We prove that SSN layers indeed satisfy this property:

###### Theorem 7 .

Let $\mathcal{S}_{F}^{\mathcal{R}}$ be an attributed semi-simplicial set with face-map-induced relations $\mathcal{R}$. Consider an SSN layer defined as in Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)). If for each relation $R\in\mathcal{R}$, the mapping $\omega_{R}$ and the aggregator $\bigotimes$ are permutation equivariant, then the SSN layer is permutation equivariant with respect to simplex relabeling. That is, for all $\rho\in\operatorname{Aut}(S)$, we have:

$$ $\operatorname{SSN}\circ\rho=\rho\circ\operatorname{SSN}.$ $$

###### Proof.

Let $X^{l}$ denote the matrix of features at layer $l$, and define the aggregated features
$X^{m}=\bigotimes_{R\in\mathcal{R}}\omega_{R}\bigl{(}X^{l}\bigr{)}.$
By assumption, each $\omega_{R}$ is permutation equivariant and $\bigotimes$ is permutation equivariant (or invariant, depending on definition), so for any $\rho\in\operatorname{Aut}(S)$,
$\bigotimes_{R\in\mathcal{R}}\omega_{R}\bigl{(}\rho(X^{l})\bigr{)}=\rho\bigl{(}
X^{m}\bigr{)}.$
Let $\phi$ be the pointwise update applied by the SSN layer:
$\bigl{[}x^{l+1}_{\sigma_{i}}\bigr{]}=\phi\bigl{(}x^{l}_{\sigma_{i}},\,x^{m}_{
\sigma_{i}}\bigr{)}.$
Since $\phi$ acts independently on each simplex, for any $\rho$,

$$ $\phi\bigl{(}\rho(x^{l}_{\sigma_{i}}),\,\rho(x^{m}_{\sigma_{i}})\bigr{)}=\phi \bigl{(}x^{l}_{\sigma_{\rho(i)}},\,x^{m}_{\sigma_{\rho(i)}}\bigr{)}=x^{l+1}_{ \sigma_{\rho(i)}}=\rho\bigl{(}x^{l+1}_{\sigma_{i}}\bigr{)}.$ $$

Hence
$\phi\bigl{(}\rho(X^{l}),\,\rho(X^{m})\bigr{)}=\rho\bigl{(}\phi(X^{l},\,X^{m})
\bigr{)}.$
Combining the equivariance of aggregation and the update yields $\operatorname{SSN}\circ\rho\;=\;\rho\circ\operatorname{SSN}$. This completes the proof.
∎

###### Theorem 7 .

###### Proof.

## Appendix F Implementation Details

### F.1 Semi-Simplicial Neural Networks (SSNs)

We implement Semi-Simplicial Neural Networks (SSNs) leveraging PyTorch Geometric’s [HeteroConv](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.HeteroConv.html) wrapper [120] to efficiently compute convolutions over heterogeneous graph structures. Consider a semi-simplicial set $\mathcal{S}$ equipped with an attributed set of simplices $S_{F}^{\mathcal{R}}$ and face-map-induced relations $\mathcal{R}$. Nodes correspond to simplices in $S$, where each node type is defined as $\operatorname{NodeType}(\sigma)=\operatorname{dim}(\sigma)$. Formally, $\mathcal{R}_{t}^{s}\subseteq\mathcal{R}$ represents the subset of relations linking simplices of dimension $s$ to those of dimension $t$, with each relation $R_{t}^{s}\in\mathcal{R}_{t}^{s}$ being a binary relation $R_{t}^{s}\subset S_{s}\times S_{t}$. The $\operatorname{EdgeTypes}$ are thus structured as tuples $(s,R,t)$, encapsulating the relational interactions between simplices of varying dimensions.
The update rule for SSNs, targeting simplices of dimension $t$, is defined by modifying the HeteroConv layer as follows:

$$ $X_{t}^{l+1}=\phi\left(X_{t}^{l},\bigoplus_{s=0}^{\operatorname{dim}(\mathcal{S })}\bigotimes_{R\in\mathcal{R}_{t}^{s}}\omega_{R}(X_{s}^{l})\right).$ (17) $$

This formulation involves a two-step aggregation procedure: initially, messages from each source dimension $s$ are independently aggregated via relation-specific mappings $\omega_{R}$. Subsequently, these dimension-specific contributions are merged to update the features of simplices at dimension $t$. Importantly, the HeteroConv wrapper enables various convolutional operations, such as $\operatorname{SAGE}$, $\operatorname{GAT}$, or $\operatorname{GCN}$, for implementing the relational mappings $\omega_{R}$.

### F.2 Routing Semi-Simplicial Neural Networks (R-SSNs)

Let $\mathcal{S}_{F}^{\mathcal{R}}$ denote an attributed semi-simplicial set equipped with a collection of face-map-induced relations $\mathcal{R}\subset\mathcal{R}_{d}$. Consider a partition $\mathcal{P}_{\mathcal{R}}=\{\mathcal{R}_{1},\dots,\mathcal{R}_{n}\}$ of $\mathcal{R}$ into $n$ distinct relation classes. The $l$-th layer of a Routing Semi-Simplicial Neural Network (R-SSN) updates the features $X^{l}$ as follows:

$$ $X^{l+1}=\phi\,\!\Bigl{(}X^{l},\;\bigoplus_{\hat{\mathcal{R}}\in\mathcal{P}_{ \mathcal{R}}}\bigotimes_{R\in\hat{\mathcal{R}}}G_{R}(X^{l})\cdot\omega_{R}(X^{ l})\Bigr{)},$ (18) $$

where $G_{R}(X^{l})\in[0,1]$ is a gating function that outputs normalized weighting scores for each expert. This gating mechanism employs a top-$k$ selection regime, dynamically identifying the $k$ most relevant experts for each subset of relations $\hat{\mathcal{R}}\in\mathcal{P}_{\mathcal{R}}$ during message aggregation. The operator $\bigoplus$ represents the final aggregation of expert representations.
Following the details outlined in Appendix [F.1](https://arxiv.org/html/2505.17939v2#A6.SS1), we partition $\mathcal{R}$ into relations $\{R_{t}^{s}\}$, where each binary relation $R_{t}^{s}\subset S_{s}\times S_{t}$. For computational efficiency and practical implementation, we adapt the [HeteroConv](https://pytorch-geometric.readthedocs.io/en/latest/generated/torch_geometric.nn.conv.HeteroConv.html) framework to express the R-SSN update at layer $l$ for the $n$-dimensional feature representation:

$$ $X_{t}^{l+1}=\phi\,\!\Bigl{(}X_{t}^{l},\;\bigoplus_{s=0}^{\dim(\mathcal{S})} \bigotimes_{R\in\mathcal{R}_{t}^{s}}G(x_{s}^{l})\cdot\omega_{R}(X_{s}^{l}) \Bigr{)}.$ (19) $$

Here, the representation $x_{s}^{l}=\operatorname{P}(X_{s}^{l})$ is obtained through a pooling operator $\operatorname{P}:\mathbb{R}^{K_{n}\times D^{l}}\to\mathbb{R}^{D^{l}}$, where $K_{n}=|S_{n}|$ and $D^{l}$ is the dimensionality of features at layer $l$. For each source dimension $s$, we denote $M=|\mathcal{R}_{t}^{s}|$ as the number of available relation experts. The soft gating function $G:\mathbb{R}^{D^{l}}\to[0,1]^{M}$ computes normalized scores for expert selection via a top-$k$ mechanism, defined explicitly as:

$$ $G(x_{t}^{l})=\operatorname{Softmax}\bigl{(}\operatorname{TopK}(\operatorname{ gates}(x_{t}^{l}),k)\bigr{)},$ (20) $$

where the gating scores are calculated as:

$$ $\operatorname{gates}(x)=x\cdot W_{g}+\epsilon\cdot\operatorname{Softplus}(x \cdot W_{n}),$ (21) $$

with noise $\epsilon\sim\mathcal{N}(0,1)$ and learnable parameters $W_{g},W_{n}\in\mathbb{R}^{D^{l}\times M}$ that modulate clean and noisy gating scores, respectively. It is well-established that expert selection mechanisms can lead to imbalance issues, with certain experts disproportionately favored during training [54, 121]. To address this, we adopt a soft constraint from [54], incorporating a regularization term in the training loss to encourage equitable distribution of samples among experts. Specifically, for each training sample $x$, we compute the probability $P(x,i)$ that the gating function $G(x)_{i}$ remains active upon independently re-sampling noise for the $i$-th expert, holding other noises constant. This corresponds to the probability of the $i$-th gating score $\operatorname{gates}(x)_{i}$, where $\operatorname{gates}(x)=x\cdot W_{g}+\epsilon\cdot\operatorname{Softplus}\bigl
{(}x\cdot W_{n}\bigr{)}$, to be larger than the $k$-th greatest gating score, excluding itself, i.e.:

$$ $P(x,i)=Pr(\operatorname{gates}(x)_{i}>\operatorname{kth_excluding( \operatorname{gates}(x),k,i}))$ (22) $$

where $\epsilon\sim\mathcal{N}(0,1)$ and $\operatorname{kth_excluding}$ computes the $k$-th largest element of $\operatorname{gates}(x)$ excluding the $i$-th element.
Following [54], we can simplify this to:

$$ $P(x,i)=\Phi\left(\frac{(x\cdot W_{g})_{i}-\operatorname{kth_excluding( \operatorname{gates}(x),k,i})}{\operatorname{Softplus}\bigl{(}x\cdot W_{n} \bigr{)}_{i}}\right)$ (23) $$

where $\Phi$ is the cumulative density function of the standard normal distribution. We now define the load vector $\operatorname{Load}(X)$, i.e., an estimator of the number of samples assigned to each expert given a batch $X$, whose components are

$$ $\operatorname{Load}(X)_{i}=\sum_{x\in X}P(x,i).$ (24) $$

Finally, the additional loss term $\mathcal{L}_{\textnormal{load}}$ is:

$$ $\mathcal{L}_{\textnormal{load}}=\lambda_{\textnormal{load}}CV\left( \operatorname{Load}(X)\right)^{2}$ (25) $$

where $CV(\cdot)$ computes the coefficient of variation and $\lambda_{\textnormal{load}}$ is a hyperparameter that weights the contribution of this term in the total loss.
Minimizing this term corresponds to minimizing the variation of the number of samples assigned to each expert, i.e., balancing the load across experts, and has been shown beneficial in practice [54].

### F.3 Computational Resources

Experiments were conducted on a single NVIDIA A100, NVIDIA L40 GPU, NVIDIA A40 or NVIDIA V100 GPU. The total training time for all experiments was approximately two weeks. Hyperparameter tuning was managed using Weights & Biases.

## Appendix G Computational Complexity

###### Proposition 4 .

Let $\mathcal{S}$ be an attributed set of $N$ elements with matrix form $X^{l}\in\mathbb{R}^{N\times D^{l}}$, and let $\mathcal{B}$ be a collection of $P$ relations on $\mathcal{S}$, such that for each relation $R\in\mathcal{B}$ is a set of $E_{R}$ elements and $E=\sum_{R}E_{R}$. Let $\operatorname{H}$ be an $\operatorname{SSN}$ layer of $X^{l+1}=\operatorname{H}(X^{l},\mathcal{B})$ where $X^{l+1}\in N\times D^{l+1}$ with $\omega_{R}$ an MPNN-D module [33]. Then, its forward pass time complexity is

$$ $T_{\mathcal{B}}=\mathcal{O}(ND^{2}+ED).$ $$

###### Proof.

Each relation $R\in\mathcal{B}$ involves two operations: (i) A dense projection of the $N$ node features, which requires $X^{l}W_{R}\in\mathbb{R}^{N\times D^{(l+1)}}$ is $\mathcal{O}(ND^{l}D^{l+1})$. (ii) A message-passing step over the edges $E_{R}$ of the relation, each edge transmitting a $D^{l+1}$-dimensional message, with cost $\mathcal{O}(E_{R}\,D^{l+1})$. Summing over all $P$ relations yields: $T_{\mathcal{B}}=\sum_{R\in\mathcal{B}}[\mathcal{O}(ND^{l}D^{l+1})+\mathcal{O}(
E_{R}D^{l+1})]=\mathcal{O}(PND^{l}D^{l+1}+ED^{l+1})$. Assuming $D^{l}\approx D^{l+1}=D$ we obtain:

$$ $T_{\mathcal{B}}=\mathcal{O}(PND^{2}+ED).$ $$

Moreover, if treating $P$ as a small constant, one can argue:

$$ $T_{\mathcal{B}}=\mathcal{O}(ND^{2}+ED).$ $$

∎

###### Corollary 2 .

Let $\mathcal{S}$ be a set, and let $\mathcal{A}$ and $\mathcal{B}$ be two collections of relations on $\mathcal{S}$, such that $\mathcal{A}$ is a refinement by unions of $\mathcal{B}$. Then, $T_{\mathcal{A}}=T_{\mathcal{B}}$.

###### Proof.

By Proposition [4](https://arxiv.org/html/2505.17939v2#Thmproposition4), $T_{\mathcal{B}}=\mathcal{O}(ND^{2}+ED).$ Since $\mathcal{A}$ refines $\mathcal{B}$ by splitting each coarse relation into at most a constant number $Q$ of finer ones, the same summation over relations yields $T_{\mathcal{A}}=\mathcal{O}\bigl{(}QND^{2}+E\,D\bigr{)}$. Then, $T_{\mathcal{A}}=\mathcal{O}\bigl{(}ND^{2}+E\,D\bigr{)}=T_{\mathcal{B}}.$ Thus the refinement does not change the asymptotic forward-pass cost.
∎

Figure: Table 4: Avg. runtime per validation epoch for different SSN configurations

###### Proposition 4 .

###### Proof.

###### Corollary 2 .

###### Proof.

## Appendix H Topological Deep Representation Learning for Brain Dynamics

### H.1 Data

We build upon a simulation that was run on a Blue Brain Project [52], a biologically validated digital reconstruction of a microcircuit in the somatosensory cortex of a two-week-old rat (the NMC-model) [53] used in subsequent neurotopological studies [45, 48, 51]. The model involves two fundamental components: structural connectivity of the circuit and neuronal binary dynamics. First, *structural connectivity* is modeled by a directed graph $\mathcal{G}=(V,E)$, with neurons represented by nodes $V$ and directed edges $(u,v)\in E$ denoting synaptic connections from presynaptic neuron $u$ to postsynaptic neuron $v$. Second, a collection of *binary dynamics* $\mathcal{B}=\{B:V\rightarrow\{0,1\}^{T}\}$ is defined, where each dynamic $B$ encodes neuronal activity over time, capturing neuronal firing (1) and quiescent (0) states across $T$ discrete time bins, under $8$ stimulus-driven input patterns. The stimuli were delivered by activating thalamocortical afferent fibers—axons that carry sensory signals from the thalamus to the cortex, modeling how the brain receives external input. These synaptic input fibers were organized into bundles; specifically, the $2170$ input fibers were partitioned into $100$ spatially adjacent bundles using $k$-means clustering, reflecting the biological organization in which thalamic afferents target specific cortical zones. Each stimulus activated $10$ randomly selected bundles (approximately 10% of the afferents), ensuring that the same groups of fibers were targeted for each stimulus pattern. The activation followed an adapting, stochastic spiking process, which introduces variability and biological realism while preventing the memorization of fixed patterns. Stimuli were presented as a continuous stream: every $200$ ms, a decaying and adapting stochastic spiking process activated the corresponding fiber bundles during a predominant $10$ ms interval. The $200$ ms inter-stimulus interval was chosen based on the observation that the population response to each stimulus decayed to baseline within $100$ ms. Each stimulus pattern was repeated approximately $562\pm 4$ (mean $\pm$ std) times, yielding a total of $4495$ stimulus presentations. Each stimuli simulation time is segmented into a fixed number of $T$ time bins, for each bin the set of neurons that became active (i.e., exhibited a binary state of $1$) is recorded, thereby defining a set of $4495$ binary dynamics $\mathcal{B}=\{V\rightarrow\{0,1\}^{T}\}$. In particular, following [48], we focus on the time subinterval $\Delta t=[10\,\text{ms},60\,\text{ms}]$, where spiking activity is mostly concentrated. This interval is subdivided into two $25$-ms segments, yielding a set of $4495$ binary dynamics $\mathcal{B}=\{V\rightarrow\{0,1\}^{2}\}$. Here, $B_{t}(v)=1$ indicates that a neuron $v$ is active during the $t$-th $25$-ms segment (with $t\in\{0,1\}$) in the experiment.

### H.2 Dynamical Activity Complexes

In Section [4](https://arxiv.org/html/2505.17939v2#S4), we introduced a lifting procedure that maps dynamic binary digraphs to dynamic binary directed simplicial complexes. This transformation enables the representation of directed higher-order neural co-activation motifs in a structured format that is both expressive and compatible with graph-based and TDL models. To establish the soundness of this lifting, we verify two fundamental properties. First, isomorphism preservation: if two dynamic binary digraphs are isomorphic, their lifted representations must also be isomorphic. This ensures that equivalent neural dynamics yield identical higher-order structures. Second, consistency with the neurotopological pipeline: a Dynamical Activity Complex (DAC) must encode the time series of *functional complexes*—that is, the directed flag complexes derived from the subgraphs induced by active neurons at each time step. In this section, we formally prove both properties. Given a dynamic binary graph $\mathcal{G}_{B}$, or more generally a dynamic binary directed simplicial complex $\mathcal{S}_{B}$, let $V^{1,t}$ denote the set of active vertices at time $t$, and $\Sigma^{1,t}$ the corresponding set of active simplices.

First, the following proposition guarantees that isomorphic digraph dynamics yield identical higher-order co-activation structures.

###### Proposition 5 .

Let $\mathcal{G}_{B}\cong\mathcal{G}^{\prime}_{B}$ then $\mathcal{K}_{\mathcal{G},\tilde{B}}\cong\mathcal{K}_{\mathcal{G}^{\prime},
\tilde{B}}$.

###### Proof.

Recall that two binary graphs are isomorphic if there exists a bijection between their vertex sets that simultaneously preserves edge relations and node attributes. The directed flag complex lifting is known to be invariant under digraph isomorphisms. Therefore, it suffices to show that our assignment of binary activation sequences to simplices remains invariant under isomorphism.
Concretely, for each simplex $\sigma$, we assign the activation pattern

$$ $\tilde{B}(\sigma)=\Bigl{[}\min_{v\in\sigma}B_{1}(v),\,\dots,\,\min_{v\in\sigma }B_{T}(v)\Bigr{]}\in\mathbb{B}^{T}.$ (26) $$

where $B_{t}(v)\in\{0,1\}$ denotes the activation of vertex $v$ at time $t$. Since the minimum function $\min$ is permutation-invariant, any relabeling of vertices induced by a graph isomorphism preserves these activation sequences. Hence, the entire dynamic binary lifting commutes with graph isomorphisms, concluding the proof. More generally, any permutation-invariant aggregation function would suffice.
∎

Second, we show that the lifted structure encodes the full time series of *functional complexes* as subcomplexes.

###### Proposition 6 .

Let $\mathcal{G}_{B}$ be dynamic binary graph and $\mathcal{K}_{\mathcal{G},\tilde{B}}$ its associated DAC. If $\mathcal{G}^{1,t}=\mathcal{G}[V^{1,t}]$ is the functional digraph with functional complex $\mathcal{K}_{\mathcal{G}^{1,t}}$, then:

$$ $\mathcal{K}_{\mathcal{G}^{1,t}}=\mathcal{K}_{\mathcal{G},\tilde{B}}[\Sigma^{1, t}]$ $$

###### Proof.

A simplex belongs to $\Sigma^{1,t}$ exactly when all vertices are active at time $t$. Therefore $\Sigma^{1,t}$ is closed under taking faces implying that $\mathcal{K}_{\mathcal{G},\tilde{B}}[\Sigma^{1,t}]$ forms a subcomplex of $\mathcal{K}_{\mathcal{G},\tilde{B}}$. In particular, simplices of $\mathcal{K}_{\mathcal{G},\tilde{B}}[\Sigma^{1,t}]$ correspond precisely to the cliques of the functional digraph $\mathcal{G}^{1,t}$, thereby $\mathcal{K}_{\mathcal{G}^{1,t}}=\mathcal{K}_{\mathcal{G},\tilde{B}}[\Sigma^{1,
t}].$
∎

###### Proposition 5 .

###### Proof.

###### Proposition 6 .

###### Proof.

### H.3 Topological Invariants for Dynamical Activity Complexes

Let $\mathcal{G}_{B}$ be a dynamic binary digraph and $\mathcal{K}_{\mathcal{G},\tilde{B}}$ its corresponding Dynamical Activity Complex (DAC). We describe the dynamic binary topological invariants frequently employed in neurotopological studies [45, 48, 51], leveraging their structure and associated face-map-induced relations. These invariants play a central role in characterizing the evolving topological structure of brain activity. Given a face-map-induced relation $R$ on $\mathcal{G}_{B}$ or $\mathcal{K}_{B}$, we define its $k$-hop composition restricted to the active simplices at time $t$ as

$$ $R^{1,t,k}=\{(\sigma,\tau)\in R^{\circ k}\mid\tau\in\Sigma^{1,t}\},$ $$

where $\Sigma^{1,t}$ denotes the set of simplices active at time $t$. These restricted relational structures provide the foundation for extracting topological descriptors of time-evolving brain activity from DACs.

Size. Let $R_{\mathrm{sym}}$ be defined as in Equation [10](https://arxiv.org/html/2505.17939v2#A4.E10). Define the *$k$-hop functional size* of vertex $u$ as:

$$ $\operatorname{size}(u,k)=\big{[}|R^{1,t,k}_{\mathrm{sym}}(u)|\big{]}_{t=0}^{T}.$ $$

This invariant, despite its simplicity, effectively distinguishes stimuli and outperforms other invariants in practice [48].

Euler Characteristic. For a dynamic binary directed simplicial complex $\mathcal{K}_{B}$ the *functional Euler Characteristic* is defined as the alternating sum:

$$ $\operatorname{ec}(\mathcal{K}_{B})\;=\;\sum_{n=0}^{N}(-1)^{n}\,\big{[}|\Sigma^ {1,t}_{n}|\big{]}_{t=0}^{T},$ $$

where $N$ is the maximal dimension of $\mathcal{K}$. For a dynamic binary digraph $\mathcal{G}_{B}$, the functional Euler characteristic of its associated DAC $\operatorname{ec}(\mathcal{K}_{\mathcal{G}_{B}})$ coincides with the time series of the Euler characteristic of the functional flag complexes $[\operatorname{\chi}(\mathcal{K}_{\mathcal{G}^{1,t}})]_{t=1}^{T}$ (see Proposition [7](https://arxiv.org/html/2505.17939v2#Thmproposition7)). The variation in the amplitude of the Euler characteristic time series of functional complexes effectively characterises stimuli [45]. Moreover, it has been reported as a top-performing feature for stimulus classification [48].

###### Proposition 7 .

Let $\mathcal{G}_{B}$ be dynamic binary graph and $\mathcal{K}_{\mathcal{G},\tilde{B}}$ its associated DAC. Then,

$$ $\operatorname{ec}(\mathcal{K}_{\mathcal{G},\tilde{B}})=[\operatorname{\chi}( \mathcal{K}_{\mathcal{G}^{1,t}})]_{t=0}^{T}.$ $$

###### Proof.

By definition $\operatorname{ec}(\mathcal{K}_{\mathcal{G},\tilde{B}})=[\operatorname{\chi}(
\mathcal{K}_{\mathcal{G},\tilde{B}}[\Sigma^{1,t}])]_{t=0}^{T}$. From Proposition [6](https://arxiv.org/html/2505.17939v2#Thmproposition6) it follows that $\operatorname{ec}(\mathcal{K}_{\mathcal{G},\tilde{B}})=[\operatorname{\chi}(
\mathcal{K}_{\mathcal{G}^{1,t}})]_{t=0}^{T}.$
∎

Transitive Degree. Focusing on $2$-dimensional directed flag complexes (consistent with our experimental setup in Section [5](https://arxiv.org/html/2505.17939v2#S5)), let $C_{0,2}$ be defined as in Section [D.4](https://arxiv.org/html/2505.17939v2#A4.SS4). The *transitive degree* of vertex $v$ is the number of directed $3$-cliques (equivalently, $2$-simplices) containing $v$:

$$ $\operatorname{td}(v)=\big{[}|C_{0,2}^{1,t}(v)|\big{]}_{t=0}^{T}.$ $$

Graph Level Directionality. Let $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$ be as defined in Equations [8](https://arxiv.org/html/2505.17939v2#A4.E8) and [9](https://arxiv.org/html/2505.17939v2#A4.E9). he *active $k$-hop in-degree* and *out-degree* of a vertex $u$ are defined as:

$$ $\operatorname{indeg}(u,k)=\big{[}|R^{1,t,k}_{\mathrm{in}}(u)|\big{]}_{t=0}^{T} \text{ and }\operatorname{outdeg}(u,k)=\big{[}|R^{1,t,k}_{\mathrm{out}}(u)| \big{]}_{t=0}^{T},$ $$

respectively counting the number of active $k$-hop incoming and outgoing neighbours. The resulting *$k$-hop functional signed degree* is:

$$ $\operatorname{dir}(u,k)=\operatorname{indeg}(u,k)-\operatorname{outdeg}(u,k),$ $$

quantifying local asymmetry in neuronal activity [50].

High-Order Directionality. Let $R_{\downarrow n,i,j}$ be defined as in Equation [4](https://arxiv.org/html/2505.17939v2#A4.E4). For an $n$-simplex ($n>0$), define the $(i,j)$-th *$k$-hop functional signed degree* as:

$$ $\operatorname{deg}_{n,i,j}(\sigma,k)=\big{[}|R^{1,t,k}_{\downarrow n,i,j}( \sigma)|\big{]}_{t=0}^{T},$ $$

which relates $n$-simplices to their lower $(i,j)$ $k$-hop adjacent active neighbors. For $i\neq j$, define:

$$ $\operatorname{hodir}_{n,i,j}(\sigma,k)=\operatorname{deg}_{n,i,j}(\sigma)- \operatorname{deg}_{n,i,j}(\sigma).$ $$

This invariant was shown to be critical for characterizing structural directed flag complexes in brain connectivity [34].

Reciprocity. Let $R_{rc}=R^{\mathrm{in}}\cap R^{\mathrm{out}}$. The *$k$-hop functional reciprocal degree* of vertex $u$ is defined as:

$$ $\operatorname{rc}(u,k)=\big{[}|R^{1,t,k}_{\mathrm{rc}}(u)|\big{]}_{t=0}^{T},$ $$

counting active vertices simultaneously serving as in- and out-neighbors of $u$. This invariant effectively differentiates stimulus classes [48].

Following the procedure described in Appendix [E.2](https://arxiv.org/html/2505.17939v2#A5.SS2), for any simplex- or vertex-level invariant $\operatorname{t}\in\{\operatorname{size},\operatorname{dir},\operatorname{
hodir},\operatorname{rc},\operatorname{td}\}$, we define the corresponding global graph or complex-level invariant as

$$ $\operatorname{t}(\mathcal{K}_{\mathcal{G},\tilde{B}})=\phi(\{\{\operatorname{t }(v)\}\}_{v\in V}),$ $$

where $\phi$ is a permutation-invariant aggregation function.

###### Proposition 7 .

###### Proof.

### H.4 Topological Neural Networks and Dynamical Activity Complex Invariants

From now on, let denote $\mathcal{T}=\{\operatorname{size},\operatorname{ec},\operatorname{td},
\operatorname{dir},\operatorname{hodir},\operatorname{rc}\}$ denote the collection invariants.

###### Lemma 6 .

Let $\mathcal{K}_{B}$ be a DAC with a labeled set of simplices $S$ and corresponding binary feature matrix $X\in\mathbb{B}^{|S|\times T}$, where each row encodes the activation pattern of a simplex over $T$ discrete time steps. For every invariant $\operatorname{T}\in\mathcal{T}$ there exists a subcollection $\mathcal{R}_{\operatorname{T}}\subset\mathcal{R}_{d}$ of face-map–induced relations and corresponding operators $\{\lambda_{R}\}_{R\in\mathcal{R}_{T}}$ such that

$$ $\operatorname{T}(\mathcal{K}_{B})=\phi(\sum_{R\in\mathcal{R}_{T}}\lambda_{R}(A _{R}X)),$ $$

where $\phi$ is a permutation-invariant function.

###### Proof.

For a fixed relation $R$, consider the matrix product $(A_{R}X)$. The $(i,t)$-th entry is

$$ $(A_{R}X)^{i}_{t}=(A_{R})^{i}\cdot(X)_{t}=\sum_{j=0}^{\mid S\mid}\mathds{1}_{ \sigma_{j}\in R(\sigma_{i})}\cdot\mathds{1}_{(X)_{t}^{j}=1}.$ $$

This sum counts the number active simplices $\sigma_{j}$ at time $t$ that belong to $R(\sigma_{i})$. In other words, $(A_{R}X)^{i}_{t}=\mid\{\sigma_{j}\in R(\sigma_{i}):X^{j}_{t}=1\}\mid=\mid R^{1
,t}(\sigma_{i})\mid$, where we define $R^{1,t}(\sigma_{i})$ as the set of simplices in $R(\sigma_{i})$ with an active feature at time step $t$. Hence, the $i$-th row is $\bigl{(}A_{R}X\bigr{)}^{i}_{\cdot}=(\mid R^{1,0}(\sigma_{i})\mid,\dots,\mid R^
{1,T}(\sigma_{i})\mid)_{t=0}^{T}$, counting active neighbours of $\sigma_{i}$ under the relation $R$ across the $T$ time steps. For each relation $R$, we choose $\lambda_{R}$ to further process the count matrix $A_{R}X$. We set $\lambda_{R}=Id$ to be the identity for relations $R\in\{R_{sym},R_{in},R_{sym},R_{rc},C_{0,2},R_{\downarrow,i,j}\}$, the negation $\lambda_{R_{out}}=-Id$ for the relation $R_{out}$ and $\lambda_{\mathrm{id}_{s}}:=(-1)^{s}\operatorname{sum}$ (a row-sum with sign adjustment) for $\mathrm{id}_{s}$, denoting the identity relation restricted to elements of dimension $s$. These choices ensure that the processed outputs capture the necessary counts (with any required sign modifications) for computing the invariant. Let $\mathcal{R}_{\operatorname{T}}\subseteq\mathcal{R}$ be the subcollection of relations relevant for the invariant $\operatorname{T}$. Define the aggregated representation as $Z=\sum_{R\in\mathcal{R}_{\operatorname{T}}}\lambda_{R}(A_{R}X)$. Since the order of simplices in $S$ is arbitrary, we apply a permutation invariant function $\phi$ (such as the row-sum, row-mean, or row-max) to $Z$ yielding an invariant quantity: $\operatorname{T}(\mathcal{K}_{B})=\phi(Z)$. This completes the proof.
∎

###### Corollary 3 .

Let $\mathcal{K}_{B}$ be a DAC. There exists a Semi-Simplicial Neural Network (SSN) that computes each invariant in $\mathcal{T}$.

###### Proof.

For each relation $R\in\mathcal{R}$, define $\omega_{R}(X)\coloneqq A_{R}XW$, with $W=I_{T\times T}$ (the $T\times T$ identity matrix). Next, let an operator $\bigotimes$ act on the output of $\omega_{R}(X)$ corresponding to the operator $\lambda_{R}$. Then, define an aggregation operator $\bigoplus$ as the summation over the subcollection $\mathcal{R}_{\operatorname{T}}\subseteq\mathcal{R}$ necessary for the computation of a given invariant: $\bigoplus_{R\in\mathcal{R}_{\operatorname{T}}}\bigotimes_{R}\,\omega_{R}(X)=
\sum_{R\in\mathcal{R}_{\operatorname{T}}}\lambda_{R}\bigl{(}A_{R}X\bigr{)}$. Finally, applying a permutation invariant function $\phi$ to this aggregated output yields the invariant:

$$ $\operatorname{T}(\mathcal{K}_{B})=\phi\,(\bigoplus_{R\in\mathcal{R}_{ \operatorname{T}}}\bigotimes_{R}\,\omega_{R}(X)).$ $$

Thus, an SSN structured in this way can compute each invariant in $\mathcal{T}$.
∎

Figure: (a) $\mathcal{G}$
Refer to caption: extracted/6484153/fig/transitive.png

In the following, we consider counterexamples consisting of directed graphs, their associated DACs, and, more generally, directed simplicial complexes with constant activation. These structures cannot be distinguished by graph-based or topological architectures, yet topological invariants are able to discern differences among them. Any reference to symmetrization implicitly assumes its natural extension to constantly attributed simplices, wherein permutations of a simplex’s vertices collapse into a single canonical simplex with the same constant feature.

###### Lemma 7 .

MPNNs and MPSNNs cannot compute invariants $\operatorname{dir}$, $\operatorname{hodir}$ and $\operatorname{rc}$.

###### Proof.

Consider two binary digraphs on the vertex set $V=\{0,1,2\}$ with a constant activation, i.e., $B(i)=1$ for all $i\in V$ as in Figure [10](https://arxiv.org/html/2505.17939v2#A8.F10). Define $\mathcal{G}$ by the edge set $E=\{(0,1),(0,2),(1,2)\},$ forming a transitive $3$-clique. Its vertex functional directionality is $\operatorname{dir}(\mathcal{G}_{B})=\{\{-2,\,0,\,2\}\},$ corresponding respectively to a source (vertex 0), an intermediate vertex (vertex 1), and a sink (vertex 2). Next, define the second digraph $\mathcal{G}^{\prime}$ by the edge set $E^{\prime}=\{(0,1),(1,2),(2,0)\}$, which forms a $3$-cycle. Here, every vertex has equal in-degree and out-degree, yielding $\operatorname{dir}({\mathcal{G}^{\prime}_{B}})=\{\{0,0,0\}\}$. Although the two digraphs exhibit different directionality, $\pi(\mathcal{G})=\pi(\mathcal{G}^{\prime})=\mathcal{G}_{\mathrm{sym}}$. Further, let $\mathcal{G}$ and $\mathcal{G}^{\prime}$ the digraphs illustrated in in Figure [4](https://arxiv.org/html/2505.17939v2#A4.F4) (a) on vertex set $V=\{0,1,2,3\}$ with constant activation $B$. Define $\mathcal{G}_{B}$ with edges $E=\{(0,1),\,(0,2),\,(1,2),\,(2,1),\,(1,3),\,(2,3)\}$ and $\mathcal{G}^{\prime}_{B}$ with edges $E^{\prime}=\{(0,1),\,(0,2),\,(1,2),\,(2,1),\,(3,1),\,(3,2)\}$. Their functional directionalities are $\operatorname{dir}(\mathcal{G}_{B})=\{\{-2,\,0,\,0,\,2\}\}$ and $\operatorname{dir}(\mathcal{G}^{\prime}_{B})=\{\{-2,\,2,\,2,\,-2\}\}$, respectively. Thereby, their higher-order directionality also differ: $\operatorname{hodir_{2,0,2}}(\mathcal{K}_{\mathcal{G}_{B}})=\{\{1^{\times 2},0
^{\times 2}\}\}\neq\{\{0^{\times 4}\}\}=\operatorname{hodir_{2,0,2}}(\mathcal{
K}_{\mathcal{G}^{\prime}_{B}})$. Specifically, $\mathcal{K}_{\mathcal{G}_{B}}$ does not contain a $2$-dimensional $(0,2)$-cycle in the lower hemisphere (see Figure [4](https://arxiv.org/html/2505.17939v2#A4.F4)(b), left column). Despite these differences, their symmetrized directed flag complexes are isomorphic $\pi(\mathcal{K}_{\mathcal{G}^{\prime}})\cong\pi(\mathcal{K}_{\mathcal{G}^{
\prime}})=\mathcal{K}_{\mathrm{sym}}$. Hence, an MPSNN on $\mathcal{K}_{\mathrm{sym},B}$ using boundary relation $B$ (cf. Eq.[11](https://arxiv.org/html/2505.17939v2#A4.E11)), coboundary relation $C$ (cf. Eq.[12](https://arxiv.org/html/2505.17939v2#A4.E12)), and the lower and upper relational maps $R_{\downarrow}$ (cf. Eq.[13](https://arxiv.org/html/2505.17939v2#A4.E13)) and $R_{\uparrow}$ (cf. Eq.[14](https://arxiv.org/html/2505.17939v2#A4.E14)) cannot distinguish between the two directed simplicial complexes $\mathcal{K}_{\mathcal{G}_{B}}$ and $\mathcal{K}_{\mathcal{G}^{\prime}_{B}}$. Finally, define $\mathcal{G}_{B}$ with edge set $E={(0,1),\ (0,2),\ (1,2),\ (2,1)}$ and $\mathcal{G}_{B}^{\prime}$ with edge set $E^{\prime}={(0,1),\ (0,2),\ (1,2)}$. The only pair of reciprocal edges in $E$ is $(1,2)$ and $(2,1)$, meaning that $\mathcal{G}_{B}$ and $\mathcal{G}_{B}^{\prime}$ exhibit distinct reciprocal functional degree signatures. However, their symmetrized versions yield isomorphic graphs, as do their associated symmetrized directed flag complexes. Consequently, neither a standard MPNN nor an MPSNN can distinguish between them.
∎

###### Lemma 8 .

MPNNs and Dir-GNNs cannot compute invariants $\operatorname{ec}$, $\operatorname{td}$, and $\operatorname{hodir}$.

###### Proof.

Consider two binary directed simplicial complexes $\mathcal{K}_{\mathcal{G}_{B}}$ and $\mathcal{K}_{\mathcal{G}^{\prime}_{B}}$, corresponding to the dynamic binary complexes depicted in Figure [8](https://arxiv.org/html/2505.17939v2#A5.F8) (a) and (b), respectively, both with constant simplex activation. By construction, both complexes contain the same number of activated vertices (7) and edges (14). However, they differ fundamentally in their number of activated $2$-simplices: complex $\mathcal{K}_{\mathcal{G}_{B}}$ includes exactly $7$ transitive $3$-cliques, whereas $\mathcal{K}_{\mathcal{G}^{\prime}_{B}}$ includes none. This structural difference leads directly to distinct topological invariants: $\operatorname{tc}(\mathcal{K}_{\mathcal{G}_{B}})=\{\{3^{\times 7}\}\}\neq\{\{0
^{\times 7}\}\}=\operatorname{tc}(\mathcal{K}_{\mathcal{G}^{\prime}_{B}})$, $\operatorname{ec}(\mathcal{K}_{\mathcal{G}_{B}})=7-14+7=0\neq-7=7-14=
\operatorname{ec}(\mathcal{K}_{\mathcal{G}^{\prime}_{B}})$ and $\operatorname{hodir_{2,0,2}}(\mathcal{K}_{\mathcal{G}_{B}})=\{\{1^{\times 7}\}
\}\neq\{\{\emptyset\}\}=\operatorname{hodir_{2,0,2}}(\mathcal{K}_{\mathcal{G}_
{B}})$. A Dir-GNN can be viewed as an instance of an MPSSN operating over the relations $R_{\mathrm{in}}$ and $R_{\mathrm{out}}$, both of which correspond to 2-uniform relations in the underlying complexes. Likewise, an $R_{\mathrm{sym}}$-SSN (i.e., an MPNN) operates over a 4-uniform relation. Since both complexes are initialized with constant activations, neither MPNNs nor Dir-GNNs can capture or distinguish these topological invariants.
∎

###### Theorem 8 .

Let $\mathcal{G}_{B}$ be a dynamic binary digraph, with corresponding Dynamical Activity Complex $\mathcal{K}_{\mathcal{G},\tilde{B}}$. Then, for every invariant $\operatorname{T}\in\mathcal{T}$, there exists a set of face-map-induced relations $\mathcal{R}_{\operatorname{T}}\subset\mathcal{R}_{d}$ and a Semi-Simplicial Neural Network $\operatorname{SSN}$ as in Equation ([1](https://arxiv.org/html/2505.17939v2#S3.E1)) such that:

$$ $\operatorname{SSN}\bigl{(}X,\mathcal{R}_{\operatorname{T}}\bigr{)}= \operatorname{T}\bigl{(}\mathcal{K}_{\mathcal{G},\tilde{B}}\bigr{)}.$ $$

Moreover, at least one invariant $\operatorname{T}\in\mathcal{T}$ cannot be computed by message-passing neural networks [12], directed graph neural networks [33], or message-passing simplicial neural networks [22].

###### Proof.

Follows immediately from Corollary [3](https://arxiv.org/html/2505.17939v2#Thmcorollary3) and Lemmas [7](https://arxiv.org/html/2505.17939v2#Thmlemma7) and  [8](https://arxiv.org/html/2505.17939v2#Thmlemma8).
∎

###### Lemma 6 .

###### Proof.

###### Corollary 3 .

###### Proof.

###### Lemma 7 .

###### Proof.

###### Lemma 8 .

###### Proof.

###### Theorem 8 .

###### Proof.

## Appendix I Additional Numerical Results

### I.1 Task 1: Classifying Dynamical Brain Activity in Fixed Volumetric Samples

We provide comprehensive details on the dataset and hyperparameter configurations used in the main experiments in Section [5.1](https://arxiv.org/html/2505.17939v2#S5.SS1). Additionally, we present numerical evidence demonstrating the robustness of our model under an alternative, non-invariant readout setting—applicable exclusively to tasks without induced structural variability, such as brain dynamics representation on fixed neuronal samples.

Fixed Volumetric Samples. Directed simplices have been shown to be overexpressed motifs in brain networks at all scales [40, 41, 42, 43]. Table [5](https://arxiv.org/html/2505.17939v2#A9.T5) presents the structural statistics of the simplicial complexes derived from three representative volumetric samples: $(4,125\mu m)$, $(4,325\mu m)$, and $(8,175\mu m)$ used in our experimental evaluation. These statistics highlight the intrinsic structural complexity and high-dimensional organization present within localized regions of the neocortical microcircuit, underscoring the relevance of higher-order topological representations in modeling neural computation.

**Table 5: Simplex count per dimension for each volumetric sample used in the experiments.**
| Cell Type | $\mathbf{(4,125\boldsymbol{\mu m})}$ | $\mathbf{(4,325\boldsymbol{\mu m})}$ | $\mathbf{(8,175\boldsymbol{\mu m})}$ |
| --- | --- | --- | --- |
| Nodes | 600 | 600 | 600 |
| Edges | 19,209 | 7,450 | 10,852 |
| Triangles | 62,481 | 5,322 | 14,141 |
| Tetrahedra | 26,450 | 349 | 2,240 |
| Pentachorons | 1,757 | 4 | 61 |
| Hexaterons | 30 | 0 | 0 |

Hyperparameter Configuration. For all tested models (SSNs and competitors), hyperparameters were optimized as follows: number of layers $\in\{2,4,6\}$; hidden dimension $\in\{16,32,64\}$ for non-topological models (GNN, Dir-GNN), and $\in\{16,32\}$ for topological models (MPSNN, SSN). Additional settings were fixed across all models: dropout rate of $0.3$; inner aggregation set to $\operatorname{sum}$; outer aggregation set to $\operatorname{mean}$; batch size of $16$; Adam [122] optimizer with learning rate of $0.001$; early stopping with a patience of $25$ validation steps; and validation performed at every training step. The best-performing configurations across five splits are reported in Table [6](https://arxiv.org/html/2505.17939v2#A9.T6), along with their corresponding parameter counts. To ensure further fair comparisons, we also scaled the non-topological baselines (GNN-256, Dir-GNN-256 with hidden dimension 256) and the undirected topological baseline (MPSNN-64 with hidden dimension 64).

**Table 6: Model architecture details. Parameter counts (# Params) and relative percentages (Par. Ratio %) are reported compared to SSNs.**
| Model | Hid Dim | # Layers | # Params | Par. Ratio (%) |
| --- | --- | --- | --- | --- |
| DS [63] | 64 | 2 | 1,680 | 2% |
| DS-256 [63] | 256 | 2 | 70,672 | 68% |
| GNN [12] | 64 | 2 | 5,392 | 5% |
| GNN-256 [12] | 256 | 2 | 70,672 | 68% |
| Dir-GNN [33] | 64 | 2 | 9,744 | 9% |
| Dir-GNN-256 [33] | 256 | 2 | 137,232 | 133% |
| MPSNN [22] | 32 | 4 | 23,888 | 23% |
| MPSNN-64 [22] | 64 | 4 | 90,768 | 88% |
| R-SSN (Ours) | 32 | 6 | 18,084 | 18% |
| SSN (Ours) | 32 | 6 | 103,184 | 100% |

Robustness. In this work, we develop a model capable of robustly processing arbitrary localized structural regions within neural microcircuits, represented as Dynamical Activity Complexes (DACs). Unlike prior methods that rely on pooled neuronal samples, our approach addresses the inherent structural variability present in localized microcircuit data. As an initial evaluation, we assess our model on a task involving fixed topology but varying brain dynamics. To prevent artificially inflated accuracy due to consistent neuron labeling across samples, we employ permutation-invariant readouts. Remarkably, our model demonstrates robustness to shuffled spike trains as a natural consequence of its design, highlighting its ability to learn solely from topological activation patterns. To further challenge and validate its generalization capabilities, we additionally evaluate the model under non-invariant readout settings, where a consistent ordering of the fixed feature space eases the task of recovering stimulus identity through localized consistent activation patterns.

**Table 7: Binary dynamics classification results ($\%$, higher is better ↑) across volumetric samples. The top $\mathbf{1^{\text{st}}}$, $\mathbf{2^{\text{nd}}}$, and $\mathbf{3^{\text{rd}}}$ results are highlighted. Gain reports the absolute accuracy improvement ($\uparrow$%) or drop ($\downarrow$%) of our model relative to the best performing baseline.**
| Model | $\mathbf{(4,125\boldsymbol{\mu m})}$ | $\mathbf{(4,325\boldsymbol{\mu m})}$ | $\mathbf{(8,175\boldsymbol{\mu m})}$ |
| --- | --- | --- | --- |
| MLP | 86.65 $\pm$ 1.06 | 92.26 $\pm$ 1.07 | 88.72 $\pm$ 0.82 |
| GNN [64] | 77.67 $\pm$ 1.96 | 87.88 $\pm$ 0.45 | 84.98 $\pm$ 1.43 |
| Dir-GNN [33] | 85.07 $\pm$ 0.91 | 90.94 $\pm$ 0.81 | 87.42 $\pm$ 1.43 |
| MPSNN [22] | 86.95 $\pm$ 0.52 | 92.09 $\pm$ 1.29 | 88.72 $\pm$ 1.20 |
| SSN (Ours) | 87.63 $\pm$ 0.43 | 92.02 $\pm$ 0.82 | 88.98 $\pm$ 1.10 |
| Gain | $\uparrow$ 0.68% | $\downarrow$ 0.24% | $\uparrow$ 0.26% |

Hyperparameter Configuration.
For all evaluated models (SSNs and baselines), hyperparameters were optimized over the following grid: number of layers $\in\{2,4,6\}$, hidden dimension $\in\{16,32,64\}$, and dropout rate $\in\{0.3,0.8\}$. The following settings were fixed across all models: inner aggregation set to $\operatorname{sum}$, outer aggregation to $\operatorname{mean}$, batch size of $16$, and the Adam optimizer [122] with a learning rate of $0.001$. Early stopping was applied with a patience of 25 validation steps, and validation was conducted after every training iteration. The best-performing configurations averaged over five data splits are reported in Table [7](https://arxiv.org/html/2505.17939v2#A9.T7).

Results.
In this setting, all baselines perform significantly better than in our main experiments (see Table [3](https://arxiv.org/html/2505.17939v2#S5.T3)), reflecting the advantage of fixed structure settings for stimulus identification. Nevertheless, SSN achieves the highest accuracy in two of the three configurations and performs comparably in the remaining one, underscoring its ability to extract meaningful topological features even without structural variability. Notably, SSN’s performance correlates with the topological complexity of each volumetric sample (see Table [5](https://arxiv.org/html/2505.17939v2#A9.T5)). Moreover, methods that rely on a fixed neuron ordering are intrinsically limited to those specific samples, making the models in Table [7](https://arxiv.org/html/2505.17939v2#A9.T7) strong baselines only in these constrained scenarios.

### I.2 Task 2: Classifying Neuron Neighbourhood Dynamical Activity Complexes

We provide additional numerical results for Section [5.2](https://arxiv.org/html/2505.17939v2#S5.SS2), evaluating two alternative scenarios: (i) increased temporal resolution ($T=4$ time bins), and (ii) variation in volumetric sampling.

**Table 8: Binary Dynamical Complex classification results (%, higher is better ↑). The top $\mathbf{1^{\text{st}}}$, $\mathbf{2^{\text{nd}}}$, and $\mathbf{3^{\text{rd}}}$ results are highlighted. Absolute accuracy Gain over the second-best model are also reported.**
| Model | $\mathbf{125\mu\text{m}}$ | $\mathbf{T=4}$ |
| --- | --- | --- |
| DS [63] | 22.06 $\pm$ 0.26 | 23.13 $\pm$ 0.39 |
| DS-256 [63] | 21.62 $\pm$ 0.61 | 23.65 $\pm$ 0.36 |
| GNN [5] | 24.45 $\pm$ 0.65 | 22.87 $\pm$ 0.49 |
| GNN-256 [5] | 24.03 $\pm$ 0.60 | 22.36 $\pm$ 0.18 |
| Dir-GNN [33] | 23.46 $\pm$ 0.39 | 23.18 $\pm$ 0.23 |
| Dir-GNN-256 [33] | 25.03 $\pm$ 1.35 | 22.67 $\pm$ 0.35 |
| MPSNN [22] | 25.88 $\pm$ 0.66 | 24.79 $\pm$ 1.21 |
| MPSNN-64 [22] | 26.44 $\pm$ 1.26 | 24.43 $\pm$ 0.34 |
| SSN (Ours) | 46.30 $\pm$ 2.11 | 39.14 $\pm$ 8.09 |
| Gain | $\uparrow$ 19.86% | $\uparrow$ 14.35% |

Experimental Settings. We extend our evaluations in the more challenging data-scarce setting ($N=1$), exploring two additional regimes: component $(4,325\mu\text{m})$ with increased temporal resolution ($T=4$ time bins), and component $(4,125\mu\text{m})$ to assess volumetric consistency. This comprehensive experimental framework is designed to rigorously evaluate the robustness of our model under both temporal and volumetric sampling variability.

Results. Table [8](https://arxiv.org/html/2505.17939v2#A9.T8) shows that SSNs consistently achieve the highest classification accuracy across all evaluated settings, substantially outperforming baseline models in both the increased sub-neighborhood volume scenario and the finer temporal resolution setting ($T=4$). Notably, despite these variations, SSNs exhibit comparable performance to previously tested configurations, indicating that the model’s accuracy is largely invariant to changes in temporal resolution and sampling volume.

### I.3 Edge Regression Traffic Task

Figure: (a) $R_{1\downarrow,0,0}$
Refer to caption: extracted/6484153/fig/edges_a.png

**Table 9: Traffic dynamics regression results (RMSE, lower is better ↓). The top $\mathbf{1^{\text{st}}}$, $\mathbf{2^{\text{nd}}}$, and $\mathbf{3^{\text{rd}}}$ results are highlighted. Absolute Gain over the best-performing baseline model is also reported (negative $\downarrow$ is better).**
| Model | Anaheim | Barcelona | Chicago | Winnipeg |
| --- | --- | --- | --- | --- |
| MLP | 0.096 | 0.150 | 0.111 | 0.165 |
| LineGraph | 0.087 | 0.147 | 0.112 | 0.160 |
| HodgeGNN | 0.270 | 0.170 | 0.109 | 0.174 |
| Hodge+Inv | 0.096 | 0.146 | 0.108 | 0.162 |
| Hodge+Dir | 0.081 | 0.146 | 0.110 | 0.159 |
| EIGN | 0.069 | 0.130 | 0.061 | 0.090 |
| SSN (Ours) | 0.077 | 0.130 | 0.045 | 0.116 |
| Gain | $\uparrow$ 0.008 | 0.000 | $\downarrow$ 0.016 | $\uparrow$ 0.026 |

We evaluate our model on a traffic assignment problems using 4 traffic datasets from [123] (Anaheim, Barcelona, Chicago and Winnipeg).
These datasets contain the street networks from the corresponding cities and the relative oriented traffic flows from the Traffic Assignment Problem [124].
Some streets can be traversed in both ways, others only in one way, providing the problem setting with an inherent notion of directionality. Furthermore, traffic flows are edge data, making the problem topological.
We frame the task as a traffic simulation problem following [56]: given the traffic flows on a portion of the streets and the network topology, the goal is to predict the flows on the remaining streets.

Experimental Settings. We follow the same data-preprocessing as [56].
In detail, we define the following edge features (if available): capacity, length, free flow time (i.e., travel time with no congestion), B factor and power, (calibration parameters for the Traffic Assignment Problem), if there is a toll on the link, and the link type, e.g., highway.
Differently from [56], if a street is traversed in both directions, we do not transform it into a unique undirected edge, but we maintain the two directed edges in opposite directions, keeping their original directed flows.
We whiten features and normalize the target flows between 0 and 1.
To compare our results with those in [56], where undirected edges have a unique flow, we consider the difference between the flow predictions of the two directed edges in opposite directions at evaluation time (i.e., to compute the RMSE).
For training, instead, we design a loss to account for both the intrinsic directionality of the problem and the existence of directed edges in opposite directions. Specifically, our training loss is composed of two terms:

$$ $\mathcal{L}_{\textnormal{train}}=\alpha\mathcal{L}_{\textnormal{dir}}+(1- \alpha)\mathcal{L}_{\textnormal{diff}}$ (27) $$

where $\mathcal{L}_{\textnormal{dir}}$ is a regression loss (e.g., MSE) on the directed flows, $\mathcal{L}_{\textnormal{diff}}$ is the loss between the predicted and ground truth flow differences for undirected edges and $\alpha\in[0,1]$ is a hyperparameter.
We test our SSN using different combinations of the 4 directed edge adjacencies illustrated in Figure [11](https://arxiv.org/html/2505.17939v2#A9.F11), and we report results for the best one consisting of adjacencies (b) and (c) in Figure [11](https://arxiv.org/html/2505.17939v2#A9.F11) plus the undirected edge adjacency consisting of the union of (b) and (c).
We compare with six baselines: MLP; LineGraph, a spectral GNN applied at node-level; HodgeGNN [125], based on the edge Laplacian; Hodge+Inv, a variant of HodgeGNN modeling orientation-invariant features as orientation equivariant; Hodge+Dir, a variant of HodgeGNN that treats all edges as directed; EIGN [56], a GNN for edge signals that explicitly differentiates between orientation and direction invariance and equivariance for edge features. All the results for the baselines are taken from [56] as well as the experimental setting.

Hyperparameter Configuration. We selected the hyperparameters through a grid search among the following values: convolution strategy $\in\{\operatorname{GCN},\operatorname{SAGE}\}$, batch normalization $\in\{\operatorname{True},\operatorname{False}\}$, hidden size $\in\{32,64,128\}$, $\alpha\in\{0,0.5,1\}$, number of layers $\in\{1,3,5,9\}$. We fixed the following other parameters for all models: inner aggregation set to $\operatorname{mean}$, single batch, dropout rate of 0.1, learning rate of 0.01, 1500 epochs with early stopping with a patience of 80 validation steps and validation performed every training step. We report the best parameter configurations for each dataset in Table [10](https://arxiv.org/html/2505.17939v2#A9.T10).

Results. Table [9](https://arxiv.org/html/2505.17939v2#A9.T9) shows the MSE on the edge flow prediction for SSN and baselines. SSN outperforms all baselines on Chicago and matches the best-performing model on Barcelona, demonstrating the capability of its directed relations scheme to model edge flow invariance and equivariance to directionality.
On Anaheim and Winnipeg, SSN is outperformed by EIGN, but is close or better than the second baseline, showing an improvement w.r.t. undirected approaches which, again, validates the importance of considering directed adjacencies on this task.

**Table 10: Model architecture details for edge-level datasets.**
| Dataset | Hid dim | # Layers | Conv. | BN | $\mathbf{\alpha}$ |
| --- | --- | --- | --- | --- | --- |
| Anaheim | 32 | 9 | $\operatorname{GCN}$ | $\operatorname{False}$ | 1 |
| Barcelona | 64 | 5 | $\operatorname{GCN}$ | $\operatorname{True}$ | 1 |
| Chicago | 64 | 5 | $\operatorname{SAGE}$ | $\operatorname{False}$ | 0 |
| Winnipeg | 64 | 9 | $\operatorname{GCN}$ | $\operatorname{False}$ | 0.5 |

### I.4 Node Classification

**Table 11: Higher-order edge directionality statistics for the Cora-ML, Citeseer [126] and Roman-Empire [127] datasets. The analysis reveals a pronounced chain-like structure in the Roman-Empire graph, characterized by a higher proportion of fully directed edge paths. In contrast, Cora-ML and Citeseer exhibit predominantly homophilic patterns, with directionality concentrated in edge pairs sharing a common source or target, and a markedly lower prevalence of fully directed paths.**
| Dataset | Type | Path Length | # $R$-Paths | $R_{\downarrow 1,0,0}(\%)$ | $R_{\downarrow 1,0,1}(\%)$ | $R_{\downarrow 1,1,0}(\%)$ | $R_{\downarrow 1,1,1}(\%)$ |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Cora ML | Homophilic | 1 | 355,808 | 57.50 | 12.05 | 12.05 | 18.41 |
| 2 | 31,672,710 | 93.99 | 0.70 | 0.70 | 4.60 |  |  |
| Citeseer | Homophilic | 1 | 76,344 | 77.22 | 6.46 | 6.46 | 9.87 |
| 2 | 2,665,146 | 98.06 | 0.23 | 0.23 | 1.48 |  |  |
| Roman-Empire | Heterophilic | 1 | 222,096 | 14.90 | 30.23 | 30.23 | 24.65 |
| 2 | 1,384,820 | 7.93 | 27.29 | 27.29 | 37.50 |  |  |

**Table 12: Node classification results (accuracy in %, higher is better ↑) on the $\operatorname{Roman-Empire}$ dataset (Left column) [127] and on $\operatorname{Cora-ML}$ and $\operatorname{Citeseer}$ (Right column) datasets [126]. The top two performing methods are highlighted as follows: $\mathbf{1^{\text{st}}}$ and $\mathbf{2^{\text{nd}}}$. Gain reports the accuracy improvement ($\uparrow$%) or drop ($\downarrow$%) of our model relative to the best performing baseline. In the $\operatorname{Roman-Empire}$ table, the $\mathbf{2^{\text{nd}}}$ entry corresponds to the previous state-of-the-art method.**
| Model | Roman-Empire |
| --- | --- |
| GraphSAGE [128] | $91.06\pm 0.27$ |
| Dir-GNN [33] | $91.23\pm 0.32$ |
| Polynormer [129] | 92.55 $\pm$ 0.37 |
| MPSNN [22] | $88.76\pm 0.63$ |
| SSN (Ours) | 93.52 $\pm$ 0.28 |
| Gain | $\uparrow$ 0.96 % |

We further assess SSN’s performance on a node classification task involving both homophilic and heterophilic graphs. In homophilic graphs, nodes with the same label tend to be connected, while in heterophilic graphs, connected nodes typically belong to different classes—making these settings particularly challenging for GNNs. Rossi et al. [33] showed that modeling graphs as directed in heterophilic contexts can induce meaningful relationships that effectively increase the graph’s homophily, highlighting the potential benefits of leveraging directional information. In this work, we focus on the $\operatorname{Roman\text{-}Empire}$ dataset, whose distinctive chain-like topology makes it especially well-suited for higher-order directed modeling. We evaluate our method on this heterophilic benchmark, alongside two widely used homophilic citation graphs: $\operatorname{Cora\text{-}ML}$ and $\operatorname{Citeseer}$ [126]. Our empirical findings align with the conclusions of [33]: directionality—whether higher-order or not—offers minimal gains in traditional strongly homophilic benchmarks but yields substantial improvements in highly directed and heterophilic $\operatorname{Roman\text{-}Empire}$ dataset (see Table [11](https://arxiv.org/html/2505.17939v2#A9.T11)). In particular, we observe that higher-order directionality plays a key role in boosting performance in such settings, setting a new state of the art on the $\operatorname{Roman\text{-}Empire}$ dataset (cf. Table [12](https://arxiv.org/html/2505.17939v2#A9.T12)).

Roman Empire (A Chain-Like Graph). The $\operatorname{Roman\text{-}Empire}$ graph [127] is constructed from the Roman Empire Wikipedia article, where each node represents a word token and directed edges encode either immediate word succession or syntactic dependency. This process yields a highly directed, heterophilic graph (heterophily score $\approx$ 0.05), enriched with shortcut edges that capture long-range grammatical dependencies and characterized by a chain-dominated topology—featuring the smallest average per-node degree ($2.91$) and largest diameter ($6824$) among commonly used benchmark datasets. Table [11](https://arxiv.org/html/2505.17939v2#A9.T11) quantifies the higher-order edge directionality: at path length $1$, fully directed edge pairs ($R_{\downarrow 1,0,1}$) account for $30.23\%$ of one-hop interactions—more than double the proportion observed in Cora-ML ($12.05\%$) and Citeseer ($6.46\%$), where edges typically share a common source or target. This structure makes Roman-Empire an ideal stress test for our Semi-Simplicial Neural Network (SSN). While traditional undirected topological models inherently neglect directionality, and recent approaches like Dir-GNN [33] leverage only first-order directional cues, SSN explicitly captures higher-order directed simplicial motifs, effectively modeling long-range syntactic chains and dependencies.

Baselines. For the $\operatorname{Roman-Empire}$ dataset, we benchmark against several strong baselines: Dir-GNN [33], which previously established the efficacy of direction-aware models; GraphSAGE [64], with extensive hyperparameter tuning by Luo et al. [128]—yielding the most competitive configurations for undirected graph-based methods; and Polynormer [129], a Graph Transformer (GT) that currently holds state-of-the-art performance. The tuning methodology in [128] aligns with that of Polynormer [129]. Previous results are taken directly from their respective works. Additionally, we evaluate MPSNN [22], enabling a comprehensive comparison across models that incorporate, or omit, higher-order and directional interactions. For the homophilic datasets Cora-ML and Citeseer, we benchmark SSN against standard baselines: GNN [64], Dir-GNN [33], and MPSNN [22], as detailed in Table [12](https://arxiv.org/html/2505.17939v2#A9.T12).

Experimental Setup. For the $\operatorname{Roman\text{-}Empire}$ dataset, we adopt the data splits from [127]. For Cora-ML and Citeseer, we use 10 random splits with a 50/25/25 train-validation-test ratio, reporting mean accuracy and standard deviation across splits (see Table [12](https://arxiv.org/html/2505.17939v2#A9.T12)). In our SSN model, we restrict simplicial dimension to 1, lifting node-pair embeddings into common directed simplices (syntactic or citation-based connections) via the boundary converse operator $C_{0}$. We propagate embeddings across four directional edge relationships ($R_{\downarrow 1,0,0}$, $R_{\downarrow 1,0,1}$, $R_{\downarrow 1,1,0}$, and $R_{\downarrow 1,1,1}$), updating node embeddings using boundary operator $B_{1}$.

Hyperparameters. Consistent with [33, 128], we employ concatenation-based Jumping Knowledge. Differing from Luo et al. [128], who utilized hidden dimensions of 256 (GraphSAGE) and 512 (GAT and GCN), we uniformly set a smaller hidden dimension of $128$ and utilize SAGE-like aggregations ($\omega_{R}$) across all five relations. Hyperparameter search covers the number of layers $\{5,7,9\}$, dropout rates $\{0.3,0.5,0.7\}$, inner aggregation as $\operatorname{max}$, outer aggregation as $\operatorname{sum}$, and the Adam optimizer (learning rate = $0.01$).

Results. Table [12](https://arxiv.org/html/2505.17939v2#A9.T12) demonstrates that our SSN sets a new state of the art on the $\operatorname{Roman\text{-}Empire}$ dataset, surpassing the Polynormer Graph Transformer baseline. SSN notably improves accuracy by $2.46\%$ over classical graph methods on best tuning known performed in [128] and outperforms Dir-GNN by $2.29\%$. This highlights SSN’s superior ability to leverage higher-order directionality in effectively capturing complex relational structures in heterophilic, chain-like graphs. Additionally, SSN consistently outperforms MPSNN, reinforcing the critical importance of explicitly modeling directed higher-order structures. For the homophilic datasets, SSN achieves competitive performance—surpassing all baselines by $0.18\%$ on Citeseer and remaining within $0.42\%$ of the best-performing method on Cora-ML. These results align with the findings of Rossi et al. [33], confirming that incorporating directionality yields minimal benefit in strongly homophilic settings. An additional possible explanation for the limited gains from higher-order directionality is provided by the edge directionality statistics reported in Table [11](https://arxiv.org/html/2505.17939v2#A9.T11), which reveal that message passing in both $\operatorname{Cora\text{-}ML}$ and $\operatorname{Citeseer}$ predominantly occurs along edges with shared sources or targets—a structural hallmark of citation networks. This is accompanied by a marked collapse in the proportion of fully directed paths. As a result, the diversity of directional relationships becomes largely redundant, diminishing the marginal utility of higher-order directed modeling in these scenarios.

## Peer Reviews

confidence: 4
contribution: 3
initial_rating: 8
presentation: 3
questions: - Which face-map–induced relation classes (e.g., interdimensional, intradimensional, directed adjacency) contribute most to performance?

- The framework mainly uses binary activations for brain data. How would SSNs handle continuous-valued features or multimodal signals?

- How will the routing behave under noise or shuffled connectivity?
soundness: 3
strengths: - The paper is well written and easy to follow.

- The paper provides strong theoretical results on WL-expressivity, permutation equivariance, and invariant recovery, establishing clear theoretical advantages over prior models. 

- The integration with brain dynamics modeling via Dynamical Activity Complexes (DACs) connects deep learning with neurotopology in a rigorous, data-driven way, replacing handcrafted invariants. 

- SSN is able to achieve large accuracy gains (up to 50% over GNNs) on challenging stimulus classification tasks, validating the theory with well-controlled baselines and consistent robustness. 

- The Routing-SSN mechanism provides a practical path to scaling directed topological models while retaining expressivity and efficiency.
summary: This paper introduces Semi-Simplicial Neural Networks (SSNs), a new class of topological deep learning models for directed higher-order structures. SSNs operate on semi-simplicial sets using face-map–induced relations to propagate messages between simplices, extending both GNNs and simplicial neural networks. A scalable variant, Routing-SSN, dynamically selects relevant relations.
The authors prove that SSNs are strictly more expressive than directed GNNs and simplicial networks under the WL hierarchy and can recover a richer family of topological invariants relevant to brain activity. Experiments on 13 datasets show large gains over prior models (up to 50% over GNNs and 27% over other TDL baselines).
weaknesses: - Recent literature has shown that transformers can be considered message passing neural networks [1]. However, results using multi-head attention over node features are missing. Since SSNs generalize message passing to relation-aware updates, such attention-based comparisons would clarify whether the proposed relational framework yields benefits beyond architectural scaling, or the transformer model is able to learn relations.

- The ablation on relation classes (e.g., face-map–induced vs. interdimensional vs. intradimensional) is not reported. Understanding which classes contribute most to performance would help assess the inductive bias of the relational algebra and the necessity of routing.

- Several of the paper’s most important theoretical results, including the corollaries, are deferred to long appendices. Condensing or summarizing them in the main text would improve accessibility and emphasize the technical depth.

- On standard node classification benchmarks (e.g., cora), SSNs do not outperform existing baselines, and these graphs are relatively small. This weakens the general claim that SSNs are broadly superior rather than domain-specific to brain networks. Results on datasets like OGB would emphasize the scalability and generalizability of the model.

[1] Joshi, Chaitanya K. "Transformers are graph neural networks." arXiv preprint arXiv:2506.22084 (2025).

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: Can the proposed SSNs be generalized to combinatorial complexes with cells of dimension greater than two?

Can DACs be extended from binary activation to real-valued activity?
soundness: 3
strengths: - The paper gives clean algebraic definitions, distinguishes orientation vs directionality, and proves that SSNs strictly contain the expressive power of GNNs/Dir-GNNs/MPSNNs while being able to recover key invariants.

- Using semi-simplicial sets allows the model to treat different vertex orderings as distinct simplices, preserving directionality information that is lost in traditional TDL architectures. 

- The experiments span three application domains: brain dynamics, traffic flow regression, and node classification. The results are consistent with the theoretical claims, showing that the advantage of SSNs increases with the strength of directionality in the data.
summary: The paper proposes Semi-Simplicial Neural Networks (SSNs), a TDL architecture that operates on semi-simplicial sets rather than classical simplicial complexes, enabling joint modeling of directionality and higher-order structure. It further introduces Dynamical Activity Complexes (DACs) to represent time-varying neuronal co-activation and provides formal results showing SSNs can recover a broad class of topological invariants. Experiments span brain-dynamics classification, traffic edge-regression, and node classification.
weaknesses: 1. Although the paper presents DACs as a novel construct, similar formulations of dynamic higher-order connectivity already exist in prior work such as [1]. The paper should clarify what is genuinely new.

2. The semi-simplicial formalism is conceptually close to combinatorial complexes [2] and several Combinatorial-Complex Neural Networks (CCNNs) already exist [3], [4]. The paper lacks a conceptual and empirical comparison, leaving ambiguity as to whether SSNs represent a subset, superset, or merely a re-parameterization of these frameworks.

3. The latest higher-order model compared is MPSNN (2021). More recent approaches such as combinatorial-complex networks, cell-complex GNNs, or hypergraph GNNs are not included.

4. In the brain-related experiments, 2-simplices (triangles) are included, but in edge regression and node classification, only 1-simplices are used. The absence of experiments with higher-order (>2) simplices weakens the empirical evidence supporting the claimed expressivity advantages.

[1] Higher-order connectomics of human brain function reveals local topological signatures of task decoding, individual identification, and behavior (Nature Communications, 2024)

[2] Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs

[3]Topological Deep Learning: Going Beyond Graph Data

[4] TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks

---

confidence: 1
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. How should practitioners pick the top-k in Routing-SSN beyond grid search?
2. Can routing scores or other signals identify which directed motifs drove a prediction?
soundness: 3
strengths: - The paper has a well-motivated focus on directionality and multi-way interactions, important for brain data.
- General framework that subsumes GNNs, directed GNNs, and simplicial networks.
- Theoretical results on WL-expressivity and recovery of neuro-topological invariants.
- Strong empirical gains on brain-dynamics classification.

Overall, I find the paper to be strong. It introduces a novel framework that is well-grounded in theory, effectively subsuming several pioneering baselines. The experimental evaluation also appears comprehensive, covering a wide range of datasets, tasks, and key baselines, which collectively serve as an implicit ablation study for the proposed architecture.

That said, I am not deeply familiar with the field of brain activity modeling, and from my view it seems that the baselines considered are primarily taken from [4] (e.g., GNNs, DeepSets, MPSNN, etc.), potentially missing other relevant "high order message passing approaches" approaches such as [1, 2, 3]. Given my limited expertise in this domain of brain activity—particularly regarding datasets and evaluation protocols—I would prefer to calibrate my final assessment after reviewing feedback from other reviewers who may have more specialized knowledge in this area.

**References:** 

[1] Cin++: Enhancing topological message passing. Giusti et.al. 2023

[2] Cycle invariant positional encoding for graph representation learning. Yan et al. 2024

[3] Attending to topological spaces: The cellular transformer. Ballester et. al 2024

[4] Position: Graph learning will lose relevance due to poor benchmarks Bechler-Speicher et al., 2025
summary: The paper introduces Semi-Simplicial Neural Networks (SSNs), which learn on directed, higher-order structures (semi-simplicial sets) rather than just pairwise edges. A scalable variant (Routing-SSN) learns which relations to use. The authors prove SSNs are more expressive than standard GNNs and existing topological models and can recover key topological descriptors of brain activity. On simulated neocortical stimulus-decoding tasks, SSNs achieve state-of-the-art results and sizable gains over baselines.
weaknesses: - There is no comparison with other topological or higher-order message-passing baselines (e.g., [1, 2, 3]), if they are applicable on these tasks. If these methods are indeed not relevant or comparable in this context, it would be helpful for the authors to clarify why that is the case.
- Minor weakness: the paper lacks an ablation study showing which types of relations contribute most to performance or how sensitive results are to the choice of relations.

**References:**

[1] Cin++: Enhancing topological message passing. Giusti et.al. 2023

[2] Cycle invariant positional encoding for graph representation learning. Yan et al. 2024

[3] Attending to topological spaces: The cellular transformer. Ballester et. al 2024


Decision: Accept (Poster)