Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
Authors: Manuel Lecha
Andrea Cavallo
Francesca Dominici
Ran Levi
Alessio Del Bue
Elvin Isufi
Pietro Morerio
Claudio Battiloro
Venue: ICLR 2026 Poster (2026)
Keywords: Topological Deep Learning
Graph Neural Networks
Primary area: learning on graphs and other geometries & topologies

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, such as simplicial or cell complexes. 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 theoretically characterize SSNs by proving they are strictly more expressive than standard graph and TDL models, and they are able to recover several topological descriptors. Building on previous evidence that such descriptors are critical for characterizing brain activity, we then introduce a new principled framework for brain dynamics representation learning centered on SSNs. Empirically, we test SSNs on 4 distinct tasks across 13 datasets, spanning from brain dynamics to node classification, showing competitive performance. Notably, SSNs consistently 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 topological models for learning from structured brain data, establishing a unique real-world case study for TDL. Code and data are uploaded as supplementary material.

Published as a conference paper at ICLR 2026
DIRECTED SEMI-SIMPLICIAL LEARNING WITH APPLICATIONS TO BRAIN ACTIVITY DECODING
Manuel Lecha'
Andrea Cavallo? Francesca Dominici
Ran Levi'
Alessio Del Bue"
Elvin Isufi Pietro Morerio'
Claudio Battiloro
Istituto Italiano di Tecnologia Delff University of Technology Harvard University
'University of Aberdeen
ABSTRACT
Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learn- ing (TDL) addresses this limitation by leveraging combinatorial topological spaces, such as simplicial or cell complexes. However, existing TDL models are restricted to undirected settings and fail capture the higher-order directed patterns preva- lent 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), principled class of TDL models that operate on semi- mimiiiiillicall sets, combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, SSNS, which dynamically select the most informative relations in learn- able manner. We theoretically characterize SSNs by proving they are strictly more expressive than standard graph and TDL models, and they are able to recover several topological descriptors. Building on previous evidence that such descriptors are critical for characterizing brain activity, we then introduce a new rigorous frame- work for brain dynamics representation learning centered on SSNs. Empirically, we test SSNs on 4 distinct tasks across 13 datasets, spanning from brain dynamics to node classification, showing competitive performance. Notably, SSNs consistently achieve saate-of-the-ad performance on brain dynamics classification tasks, outper- forming the second-bess model by up to 27%. and message passing GNNs by up 50% in accuracy. Our results highlight the potential of topological models for learn- ing from structured brain data, establishing unique real-world case study for TDL.
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 (Barabási, 2002). Graph Neural Networks (GNNs) (Scarselli et al., 2008), deep learning models operating on graph-structured data, have shown remarkable performance on several tasks from different domains, such as computational chemistry (Gilmer et al., 2017; Jumper et al., 2021), social network analysis (Xia et al., 2021; Kipf & Welling, 2017), and neuroscience (Bessadok et al., 2022). 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 (Bruna al., 2014; Gori et al., 2005). Most GNNs are Message Passing Neural Networks (MPNNs) (Gilmer et al., 2017), which learn meaningful representations of node or edge data via local aggregation governed by the underlying graph connectivity.
However, many complex real-world systems exhibit higher-order interactions that go beyond simple pairwise relationships (Battiston et al., 2020; Millán et al., 2025). Such dependencies are naturally modeled by Combinatorial Topologicul Spaces (CTSs), mathematical objects such as simplicial or cell complexes that generalize graphs by encoding multi-way way interactions as sets of nodes, i.e., the simplices/cells. CTSs induce set-containment relations among higher-order entities via hicrarchical inclusion, extending conventional adjacency in graphs (Grady & Polimeni, 2010) and enabling pow- erful algebraic topological tools (Barbarossa & Sardellitti, 2020). Topological Deep Learning (TDL) (Battiloro, 2024; Bodnar, 2023; Hajij et al., 2022b; Papamarkou et al., 2024) builds on this structure,
"Corresponding authors. Emails: ma manuel. 1echa@1 it.iit, cbatti loro@hsph www.waarvae.....

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X 6(x², X Wa wn(X'))
K, x', R
Semi-Simplicial Set and Relations
Semi-Simplicial Neural Network
Figure 1: Overview of the Semi-Simpllcial Neural Networks framework for brain dynamics classification. Given a connectome sample, represented directed graph (digraph), and its corre- sponding binary activity response to external stimulation, we jointly model them as an attributed semi- simplicial set IC that captures higher-order co-activation patterns X We then select set of higher- order directed relations R. induced by the topology of KC, and process X' with Semi- Mimi-iiiliiial Neural Networks (SSNs), where the relations in define how information is propagated and updated. Our experiments show that this modeling choice is crucial for accurately predicting the eliciting stimulus.
extending GNNs into architectures that operate directly on CTSs. Compared to standard GNNs, TDL models have demonstrated stronger expressivity within the Weisfeiler-Leman (WL) hierarchy (Horn et al., 2022; Bodnar et al., 2021b), improved ability to capture long-range dependencies (Battiloro et al., 2025a; Giusti et al., 2023b;a), and robustness heterophilic regimes (Bodnar et al., 2022).
Yet, simplicial complexes and other commonly used CTSs (Battiloro et al., 2024b; Bodnar al., 2021a; Hajij et al., 2022b) remain limited when directionality governs system dynamics. Here, directionality refers to structural asymmetry; for example, directed edge has source and target, and its reverse a different edge, with one not implying the other-unlike orientation, which merely assigns sign to the same undirected edge (e.g., positive several GNN variants incorporate directionality at the level of directed graphs (digraphs) (Tong et al., 2020a; Zhang al., 2021; Rossi et al., 2024), the challenge of modeling higher-order directional structure in TDL remains largely unexplored. Building on the foundations of Riihimäki (2023), the short paper of Lecha al. (2025) introduced Directed Simplicial Neural Networks (Dir-SNNs), based manner.
negative flux; App. C.3 C.3.C.4). While
on directed cliques, totally ordered node sequences, connected in a direction-present While this work formalized message passing on directed simplicial complexes, is confined to narrow class of spaces, lacks theoretical guarantees, and provides only limited synthetic validation.
Brain networks are among the most notable real-world domains requiring methods capable of processing higher-order, directional information (Giusti et al., 2016). Digraphs naturally capture the asymmetric flow information in neuronal communication, from presynaptic to postsynaptic neurons, but fail to encode the higher-order co-activation patterns critical for understanding brain function. Higher-order directed motifs are abundant across scales (Sizemore et al., 2018; 2019), carry functional meaning (Nolte al., 2019), and form spiking assemblies that enhance the representational fidelity of neural activity (Reimann et al., 2017; Ecker et al., 2024). To capture such structures, the emerging field of Neurotopology (Conceição semi-simplicial sets (Hatcher, 2005), general CTSs exemplified by directed flag complexes and tournaplexes (Gove et al., 2021). Conceição al. (2022) proposed topological featurization pipeline that aggregates invariants across meaningful neuron subpopulations, and together with Reimann al. (2022), showed that such descriptors can recover stimulus identity in a biologically realistic neocortical model (Markram et al., 2015), even when traditional approaches fail. However, these handcrafted pipelines are fundamentally limited: they fix representational power in advance through predefined invariants, rely carefully tuned sampling heuristics to select informative neuron subsets, and lack robustness under perturbations or shuffled activity (Conceição et al., 2022).
al., 2022; Reimann et al., 2017) employed
Therefore, the absence of general, formal, and comprehensive TDL framework that leverages higher- order directionality motivates the methodological contributions of this work. Moreover, connecting these developments to neurotopology creates unique opportunity learn meaningful representations of dynamical brain activity, and motivates our applied contributions. For a detailed discussion of related work, Appendix B.
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Contribution. The methodological and applied advances of this work are listed below, and they address several open problems in TDL as identified by Papamarkou et al. (2024).
CL. We introduce Semi-simplicial Neural Networks (SSNs), the first TDL models explicitly designed for semi-simplicial sets. The rich variety of ways SSNs propagate information across simplices is formalized via dice-map-inducee relations collected in relational algebra and generalizing common (directed) graph and topological adjacencies. To address scalability and efficiency (Open Problem 6 in (Papamarkou al., 2024)), we further propose Routing-SSNs (R-SSNs), which employ learnable routing mechanism (Shazeer et al., 2017; Wang et al., 2023) to dynamically select the top-k most relevant relations, reducing parameter count and inference time. Theoretically, we prove that SSNs are strictly more expressive than peessag-ppsssing GNNs (Gilmer et al., 2017), Directed GNNs (Dir- GNNs) (Rossi et al., 2024), and Pessaae-Passiigg Simplicial Neural Networks (MPSNNs) (Bodnar et al., 2021b) in the WL. hierarchy (Section 3).
C2. We propose a novel brain dynamies. Its core elements are Dynamical Activity Complexes co-activation: a neuron complexes endowed with binary, time-evolving, features encoding neuronal group is active at time / if all constituent neurons fire simultaneously at L. Critically, we formally prove that SSNs operating on DACs are uniquely capable, among existing graph and TDL models, of recovering broader class of topologiaal invariants known to characterize brain network activ- ity (Reimann et al., 2022), thanks to their ability to jointly encode directionality and higher-order structure (Open Problem 9 in (Papamarkou et al., 2024)). See Section 4 for details.
framework for higher-order representation learning of simplicial
C3. We test SSNs on 4 distinct tasks across 13 datasets, spanning from brain dynamics to node classification, showing competitive performance. The brain dynamics classification tasks represent unique, meaningful, real-world case study for TDL, where SSNs consistently achieve state-of-the-art results (see Figure 1, Open Problem 1 in (Papamarkou et al., 2024)). As such, the data and tasks also serve as a competitive public benchmark for graph- based and TDL models (Open Problem 2 of (Papamarkou et al.. 2024)). Specifically, following Conceição et al. (2022); Reimann al. (2022), we leverage stimulus-responss response activity from biologically realistic neocortical microcir- cuit (Markram et al., 2015), evaluating: (i) classification from fixed topological brain samples-i.e., a feature classification task, and (ii) classification from randomly sampled neuron neighborhoods-iic.. graph/complex classification task. SSNs achieve accuracy gains over 50% compared to baseline MPNNs (Gilmer et al., 2017). Full details are in Section 5.
2 PRELIMINARIES
In this section, revisit the foundational building blocks of our framework. We begin with semi- simplicial sets, which extend simplicial complexes by allowing multiple simplices over the same vertex set and support directionality. We then discuss directed simplicial complexes as key subclass, with directed flag complexes arising canonically from digraphs. Next, we introduce a rich class of ceee-mpp-innuuse relations, which generalize standard topological adjacencies and extend them to higher-order directed structures. Finally, we formalize data integration via attributed semi-simplicial sets, focusing on time-varying binary features relevant brain dynamics.
(a)
(b)
(c)
(d)
Figure 2: (a) A simplicial complex of dimension 2; (b) a cemi-simplicial set of dimension 2; (c) a
digraph; (d) its associated directed flag complex.
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Simplicial Complexes. A simplicial complex is a pair IC = (V, E), where V is a finite set of vertices and E is a collection of non-emply finite subsets of V satisfying two properties: (P1) for every U e V. the singleton {o} is in Σ; and (P2) if a E Σ and 7 is non-empty subset of o, then E Σ. The dimen- sion of simplex a S V is dim(a) = |o| -o|-1; the dimension of the complex is N = m dim(o).
We denote the set of n-simplices by Σ,, = {o E Σ dim(a) = nn. thus Σ = -U n=0 E, An example of 2-dimensional simplicial complex is shown in Figure 2(a), with vertices V {0, 1, 2, 3). edges Σ1 {{0, 1). (0, [1.2), (2, 3). (1.3)} }. and 2-simplices Σg { 1.2), (1)
Semi-Simplicial Sets. Semi-simplicial sets generalize simplicial complexes by allowing multiple distinct simplices to share the same vertex set. Figure 2 illustrates this: in (b), the 2-simplices (0, 1, and (0. 2, 1) represent two different triangles over the same vertex set {0, 1, whereas in (a) simplicial complex admits only one simplex on {0, 1, 2]. This added flexibility is crucial for modeling directionality. For instance, in a digraph the edges (0, 1) and (1.0) 0) are distinct, even though both correspond to {0, 1). Formally, a semi-simplicial set S consists of (i) sets ) where S,, is the set of n-simplices, and (ii) face maps di S, + Sp for n > 0 and 0 I S n, specifying how simplices are glued. These maps satisfy the simplicial identity d,d, d for i < j. ensuring consistency of the face structure. We denote the total set of simplices by S = U Sn.
Directed Simplicial Complexes. A directed simplicial complex K. is a semi-simplicial set in which each simplex encodes a fully transitive directed structure. Formally, each set of n-simplices K., consists of ordered (n + 1)-tuples o (vo..... vn)) such that (v,, 12) is directed edge for every i << j. Directed simplicial complexes of dimensions I and 2 are illustrated in Figures 2(c-d). The face maps de KC,, + Kn act by deleting the i-th vertex, i.c., dff (o) = (Va..... ................................ 0, U2). ). For example, the 1-simplex (0. 1) in Figure 2(c) has faces dj((0,
(1) and d ((0. 1)) (0). We require IC to contain all faces of its simplices, i.e., satisfy properties (P1) and (P2) described above.
Directed Flag Complexes. Given a digraph G = E), the induced directed flag complex Kgg (Lüt- gehetmann et al., 2020) the directed simplicial complex whose n-simplices are precisely the transitive (n++1)-clquue (v;, 03) E E' for all i < j. In other words, each directed clique in G is promoted to a simplex. Figures 2(c-d) illustrate digraph and its directed flag complex. Importantly, this construction is injective on isomorphism classes: non-isomorphic digraphs map to non-isomorphic complexes, ensuring that structural information from the original digraph is lost (see Appendix C.I).
of G-that is, ordered tuples (vo....,",,) of distinct vertices such that
Face Map Relations. We make simplices interact in a snacctin-sensitive way using relations induced by face maps, generalizing binary adjacency relations in graphs, digraphs, and simplicial complexes. Each face map d. defines binary relation Rar = [(TT, o) I a E Sn, di' (a) = r). linking an n-simplex to its i-th face. Collectively, these relations gen- erate a face-map relation algebra Rd. closed under standard relational operations union U, intersection n, composition (chaining relations), and converse (reversing them; see App. C.5). For example, in directed simplicial complexes, for n 2 1, Rui,, = R o Ra; = ((,,T) df
O
T
Figure 3: (Top) Directed 2-simplices and T related by dg(o)) = d3 = e, where e denotes the shared edge high- lighted in red. Hence (a.T) E R22.0.2- (Bottom) Composing Raja yields directed path across 2-simplices.
(r) relates two n-simplices whose i-th and j-th facets coincide (Fig. (Top)). Compositions of such relations natu- rally define directed paths across simplices (Fig. 3 (Bottom)). Further illustrative examples are given in App. C.6.
Attributed Semi Semi-Simplicial Sets. An attributed semi-simplicial set is a pair Sp = FS, where S is a semi-simplicial set and F is a map F S - F assigning each simplex E S feature vector 2g = F()) E R" For a chosen indexing' of the n-simplices, we define the n-feature matrix X, [oo, i Tois dimensions to obtain the global feature matrix X == [Xp: XN] E Rl(s In this work, we often focus on the special case of dynamic binary data, where F = B {0. 10. C RT. A binary dynamics map B assigns to each simplex binary activation vector vf E {0, 1]7 indicating whether it is active (1) or inactive (0) at each of T discrete time steps.
|l..Ixmm. Extending the indexing globally, we concatenate across all
GGiven finite set 8, a bijection (1,..., |||} + S.
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3 SEMI-SIMPLICIAL NEURAL NETWORKS
TDL architectures typically represent higher-ordet relations as unordered vertex sets. In simplicial complexes, each vertex set admits at most simplex, which prevents assigning distinct features to different orderings of the same vertices. While this is appropriate for symmetric interactions (Benson al., 2016; Battiston et al., 2020), it breaks down when order is essential, as brain dynam- ics (Markram et al., 2015), where the same clique of neurons may encode different information depending on firing order. Ignoring directionality creates two expressiveness gaps:
(a) Information loss. Symmetrizing directed simplicial complex into an undirected one is non- injective: distinct motifs (e.g., transitive cliques versus directed cycles) can collapse the same unordered simplex (see Appendix C.3).
(b) Limited adjacencies. TDL models typically define interactions via subset containment (simplices communicate only via shared vertices or hierarchical inclusion), which underutilizes directional higher-order propagation.
To address these limitations, we propose SSNs, a principled class of TDL architectures that operate directly on attributed semi-simplicial sets. This section supports Contribution CI.
Semi-Simplicial Neural Networks (SSNs). SSNs propagate information using face-mapp seeeuua--nduued induced relations R RRg. leveraging the algebraic structure of semi-simplicial sets to capture directional motifs and their higher-order interactions. The I-th SSN layer updates features X as
X o X wn(x)))
RER
where wR is relation-dependent learnable function, X aggregates multiple messages per simplex across relations, and é a learnable update function. This formulation subsumes a wide family of models: if wR MPNN-D (Rossi et al., 2024), the SSN is message passing architecture; if wR is masked self-attention, the SSN is a transformer-like architecture.
As concrete example, for digraph the relations
Rm = Rag o Rdy
Rout = Ray o R
recover the standard in- and out-adjacency matrices An and Aouts respectively. Setting R {Ri, Rout WR MPNN-D, and X =. the resulting SSN coincides with Dir-GNN (Rossi ct al., 2024), i.c.,
X o X + X'
Further details are given in Appendix E.I.
Routing SSNs. While SSNs fully exploit the combinatorial structure of semi-simplicial sets, two key challenges arise: (1) Scalability: explicitly modeling many relations can lead to parameter growth, as each relation requires its weights; and (2) Relevance of relations: not all relations contribute equally to learning, and some may be redundant or uninformative. To address these limitations, we propose Routing SSNs (R-SSNs), which incorporate learnable gating mechanism (Fedus et al., 2022; Shazeer et al., 2017; Wang et al., 2023) to dynamically select the top-k relations from predefined relation classes.
Given face-map-induced relations R, let PR = { R, be partition of R. that groups relations into semantic classes, enabling different interaction types be modeled separately. For instance, one class may represent interdimensional communication (e.g.. messages from 2-simplices their 1-faces), while another captures intradimennional communication (e.g., direction- relations between 2-simplices). The 1-th R-SSN layer updates features X as
x of X
GR(X') wxX)))
ŘEP RER
where GR(X') E [0, is gating function that outputs normalized scores for relations within each class. We instantiate routing via a top-k mechanism, setting Gx(X') = for all but the highest-scoring relations in each class. Aggregation across relation classes is performed by O A detailed formulation of the routing mechanism, including gating strategies and the auxiliary losses used prevent routing collapse, is given in Appendix F.2.
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3.1 THEORETICAL PROPERTIES OF SSNs
We next provide a theoretical characterization of SSNs, focusing on their generality, WL- expressivity (Xu et al., 2019), and permutation equivariance under simplex reindexing.
Generality. SSNs unify and extend prior models across graphs and (directed) simplicial complexes.
Proposition 1. Semi-simplicial neural networks (SSNs) subsume directed message-passingg GNNs (Rossi et al., 2024), messagge-pssingg GNNs (Gilmer et al., 2017) on undirected graphs, message-passinng simplicial neural networks (Bodnar el al., 202 lb) on undirected simplicial com- plexes and Directed Simplicial Neural Networks (Lecha et al., 2025) directed simplicial complexes.
WL Expressivity. The Weisfeiler-Lenn (WL) test (Weisfeiler & Leman, 1968) (see App.D.2) bounds the discriminative power of GNNs (Xu et al., 2019). Graph liftings (e.g.. directed flag com- plexes) enrich representations with higher-order relations, mapping isomorphic graphs to isomorphic CTSs and separating non-isomorphic ones. Hence, TDL models can surpass standard graph- based models in expressivity (Horn et al., 2022). We now show that SSNs go further, strictly exceeding both Dir- GNNs and MPSNNs.
Theorem 1.
There exist SSNs strictly more expressive than directed graph neural networks (Dir- GNNs) (Rossi et al., 2024) at distinguishing non isomoooo divected graphs.
Theorem 2. There exist SSNs strictly more expressive than message-passiee simplicial neural networks (MPSNNs) (Bodnar et al., b021b) at distinguishing non-isomorphic directed simplicial complexes.
Permutation Equivariance. Matrix representations of semi-simplicial sets require arbitrary simplex indexing (see Section 2), which has no semantic meaning. SSNs must therefore be equivariant (or invariant) to simplex permutations, ensuring representations depend only on the underlying domain (Bronstein et al., 2021).
Theorem 3 (Informal). Consider an attributed semi-simplicial set with collection of face- map-induced relations R C Rd. An SSN layer, defined in equation I, is equivariant to simplex reindexing if. for each relation R E RR the message function wn and the aggregation operator permutation equivariant.
The proofs of the statements in this section, and a detailed complexity analysis of SSNs are reported in App.D) and App.F, respectively.
4 TOPOLOGICAL DEEP REPRESENTATION LEARNING FOR BRAIN DYNAMICS
We introduce principled framework for representation learning dynamical brain activity, com- patible with graph- and TDL architectures. At its core are Dynamical Activity Complexes (DACs): directed simplicial complexes that encode neuronal co-activation patterns over time. We prove that SSNs operating on DACS can recover a broad class of invariants, quantities preserved under equivalence (isomorphism; see Ap).C.1), that characterize brain activity (Reimann al., 2017) and are defined in A.2. Crucially. this class strictly exceeds the invariants recoverable by graph-based or traditional TDL models. This section supports Contribution C2.
Brain Activity Modeling, Brain network activity modeling comprises two components: structural connectivity and dynamics. Structural connectivity is modeled by directed graph G (V, E), with nodes V as neurons and edges (u, v) E E denoting synapses from presynaptic u to postsynaptic e. Dynamics are represented by binary functions B {B - B"), encoding firing (1) and quiescent (0) states across T discrete time bins under simulus-driven inputs. Further details in App. G.I.
Neurotopology at Glance. Given binary dynamics B on digraph 9 = (V, F), each time bin t defines an activation state B,, partitioning neurons into active V = {v B. = }} and inactive vo,t The active set induces a subgraph gz. = G[V at time t, together with its directed flag complex Kgg... capturing co-activation. Applying topological invariant T across time yields temporal signature of activity, T(Fkg)= = [T(gg..)...... T(Kgur)) E R Reimann et al. (2017) showed that distinct stimuli induce distinct temporal signatures, measurable via invariants such as Euler characteristic or clique counts. Building on this, Conceição et al. (2022) introduced a topological featurization pipeline for stimuli classification within G, aggregating signatures from neuron samples

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with uncommon activity patterns. While effective in realistic neocortical models (Markram et al., 2015), this pipeline faces key limitations: (i) it relies on predefined invariants, fixing representational power in advance; (īi) it requires carefully designed sampling strategies to select informative neuron subsets, making results sensitive to the chosen subgraphs and to structural variability across samples; and (iii) it has been reported to lack robustness under shuffled or perturbed activity (Conceição et al., 2022). By contrast, our framework allows learning directly from arbitrary topologies, removing the dependence on predefined invariants and sampling heuristics.
Dynamical Activity Complex. We introduce Dynamical Activity Com- plexes (DACs), representation that integrates higher-order structural connectivity with evolving brain activity in a form directly amenable to end-to-end learning. A DAC is a binary directed simplicial complex built from a sampled brain digraph and its associated binary dynam- ics, encoding activity over T discrete time steps. Formally, let GB be a dynamic binary digraph, where each vertex v E V is assigned T T-dimensional binary feature vector B(v) BT. The DAC is the attributed directed flag complex Kg,B (Kg, B), with attribute map
0800
01
0TOTic]
B(r) min Bi (v)..... min (w) E EB" VEO
(3)
NEE
so that a simplex a is active at time t iff all its vertices are active at t. This construction captures higher-order co-activation motifs: for instance, in Fig. 4, the 2-simplex activates only at t = =3, the unique time all its vertices fire simultaneously. Crucially, DACs (i) preserve isomorphisms, and (ii) encode the full temporal sequence of functional complexes, retaining the ability recover established invariants in neurotopology pipelines (Propositions 4-5 in Appendix G.2).
Figure 4: (Top) Dynamic binary graph; (Bottom) its DAC. Binary vectors indicat- ing simplex activation over 4 time steps.
Relevant Invariants. Neurotopological studies (Reimann et al., 2017: Conceição et al., 2022: Reimann et al., 2022) have empirically validated the set of topological invariants T (size, ec, td, dir, hodir, rc] as particularly effective for charac- terizing brain activity, as measured by predictive performance in ciimllus-nduueed activity classification. Briefly: the size (size) invariant counts active neighbors per node at each time step, i.e., synaptically connected active neurons; the directionality (dir) invariant measures the imbalance between active outgoing and incoming edges per node, i.c., post- versus pre-synaptic neigh- bors; the higher-order directionality (hodir) extends this to n-simplices, comparing outgoing and incoming active cliques; the reciprocity (rc) invariant counts active nodes with active neighbors in both directions, i.e., neurons that are simultaneously pre- and post-synaptic; the transitive degree (td) invariant counts active directed 2-simplices incident to a node, i.e transitive synaptic triads; and the Euler characteristic (ec) is the alternating sum of active simplices across dimensions. By construction of DACs, this Euler characteristic coincides with the one in Proposition 6. See App. G.3 for formal definitions and details.
Model
size dir hodir
td ec
GNN
x
x
x
x
Dir-GNN
x
x
MPSNN
V
SSN (Ours)
Table 1: Topological invariant computation across architectures.
Theorem 4. Let Gn be a dynamic binary digraph with corresponding DAC K.g. For every invariant of ceoee--ap-inuueed relations RT C "Rd and a Semi-Simplicial Neural
T E TT, there exists Network SSN as in equation I such that:
SSN =
R T
Moreover, the class of invariants recoverable by SSNs strictly exceeds that of mnessage-passngg neural networks (Gilmer et al., 2017), directed GNNs (Rossi et al., 2024), and message paassigg simplicial networks (Bodnar et al., 2021b).
A formal proof is given in Appendix G.4. Table summarizes invariant computation across models, showing that SSNs uniquely recover the complete set of critical invariants necessary for compre- hensive brain activity characterization. This establishes SSNs as a strict generalization of prior approaches, enabling fully data-driven, localized, and meaningful representations of brain dynamics.
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5 NUMERICAL RESULTS
We empirically validate SSNs on four tasks across 13 datasets: two neural stimulus classification tasks (six datasets), edge regression (four datasets), and node classification (three datasets). Results for edge regression and node classification are reported in Appendix H.2 and Appendix H.3, respectively. In the main body, we show that the theoretical guarantees of SSNs (Sections 3 3-4) translate into measurable gains on srain-stimulus classification, a task for which we provide a principled justification for using SSNs and introduce dedicated framework (Section 4).
Our brain network experiments test two hypotheses: (i) modeling directed higher-order structure and interactions increases expressivity, enabling SSNs to distinguish stimuli that collapse under graph- based or undirected TDL. models; and (ii) SSNs extend neurotopology pipelines by reliably inferring stimulus identity from arbitrary topological samples, robust to structural variability and permutations. We evaluate these hypotheses in two settings of increasing difficulty; Task 5.1, feature classification with fixed topology and varying dynamics; and Task 5.2. graph/complex classification where both topology and dynamies vary. This section supports Contribution C3.
We build on the well-established NMC- model (Markram et al., 2015; Markram, 2006), biologically detailed microcircuit of the somatosensory cortex in a two-week-old rat with simulated responses to external stimuli. The dataset consists of a large structural digraph g = (V, E) with [V] = 31,346 neurons and |E] 7,803,528 synapses, together with 4495 binary dynamics func- tions. Following Conceição et al. (2022), neuronal activity is discretized into T- 2 time bins, yielding {B: V + B ) where each B; assigns to every neuron E V a two-dimensional binary vector encoding its activation response to one stimulus. Each stimulus corresponds to a uniformly random sample from eight distinct thalamocortical input pat- terns. Table 2 highlights the abundance of higher-order directed motifs, and further dataset details are provided in App. G.1.
Data Overview.
Type
Count
Neurons
31,346
Edges
7,803,528
Triangles
76,936.601
Tetrahedra
65,939,554
Pentachorons
7,637,507
Table 2: Counts of di-
rected simplices in the Mi- crocircuit complex Kg.
Experimental Setup. We evaluate SSNs and their routing- based variant, R-SSNs (top-k = 2 relations, except (4, 125pm where I 4), against comprchensive set of set-, graph-, and topology- based baselines, as recommended in the recent position paper Dechler-Speichee et al., 2025). Specifically, we compare with: (i) Deep Sets (DS) (Zaheer et al., 2017); (ii) pessage-passingg GNNs (Gilmer et al., 2017: Hamilton et al., 2017); (iii) Directed GNNs (Dir-GNNs) (Rossi et al., 2024); and (iv) message- passing Simplicial Neural Networks (MPSNNs) (Bodnar et al., 2021b). This selection isolates the contributions of handling relational, directional, and higher-order structure jointly. We also benehmark against the topological featurization pipeline of Conceição et al. (2022), which computes invariants followed by an SVM classifier (see App.G.3). To ensure faimess, we match parameter budgets across baselines by scaling hidden dimensions: DS-256, GNN-256, Dir-GNN-256, and the undirected higher-order baseline MPSNN-64. For SSNs, R-SSNs, and MPSNNs (both standard and 64-dim variants), we include simplices up to dimension two (triangles). SSNs use standard boundary/coboundary relations intra-dimensionally, combined with directed up/down adjacencies inter-dimensionally (see App. C.6). Hyperparameters for all models are tuned via grid search. SSNs and R-SSNs are instantiated as pesssag-passingg SSNs, are all graph- and higher-order baselines, using SAGE-style (Hamilton et al., 2017) message functions. Additional results with attention-basse baselines (Veličković et al., 2018) are reported in App. 1.1.2.
5.1 TASK 1: CLASSIPYING DYNAMICAL BRAIN ACTIVITY IN FIXED VOLUMETRIC SAMPLES
Given a fixed brain structure and time-evolving neuronal activation patterns, the goal is to classify the stimulus that triggered the observed dynamics. Following Reimann et al. (2022), we extract volumetric samples U C V defined by neuronal population centroid c (4.8) 8) and sampling radius r E (125pm, 175mm, 325mm)). Each sample induces a subgraph Gu = G(U]. and we restrict the binary dynamics B, lu, lifting them as in equation 3 to obtain 4,495 lifted dynamics B u per sample. The task is then to classify each lifted dynamic by its associated stimulus identity.
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Model
(4.125m)
(4, (4.52mm
(8, (7)mm))
M=1
M=3
=
# Par.
Par. (%)
SVP
42.14 1.19
35.91 = 3.36
45.32 + 1.68
27.94 + 0.94
27.87 = 0.89
28,86
0.42
312
0.3%
DS
26.63 = 0.10
19.47 = 1.16
27.31 +: 0.34
23.28 0.48 24.29
0.38
25.09 1: 0.14
1,680
2%
DS-256
25.21 1.11
19.52 +: 1.44
25.12 +: 2.12
25.24 + 0.32
24.76
0.33
25.87 4: 0.19
70,672
68%
GNN
26.00 1,10
21.56 + 1,14 34.02 4:335
25.43 0.43
26.03
0.69
26.49 + 1.02
5,392
5%
GNN-256
24.70 1.31
23.02 + 2.1
33.47 4.15
24.40 0.51
27.60
0.91
28.27 + 0.17
70,672
68%
DirGNN
36.71 2.00
48.11
1.87
53.72 2.89
25.21 0.18
31,08
1.04
33.00 3.66
9,744
9%
DisGNN-2566
50.89 +: 13,00
60.02
0.75
63.52 0.70
0.54
35.21
0.83
137.232
133%
MPSNN
43.33 6.95
42.65
2.65
0.50
32.10
0.77
33.68
3.04
23,888
23%
MPSNN-64
46.85 6.71
54.24
7.06
64,02 4.91
1,31
34.91
0.73
42.23 +
90,768
88%
R-SSN (Ours)
57.32 5.25
79.64
1.84
70.66
1.56
28.68 0.79
40.29 1.12
48.20
1: 0.70
18,084
18%
SSN (Ours)
75.13
1.28
87.16 + 1,36 78.32 + 7.03
46.73 + 1.16
61.35
1.07
64.72 + 0.54
103,184
100%
Gain 2-nd Best
1.24.24
27.144
14.30
17.225 %
1:26.14 %
1 22.44 %
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, 125jmm), (4, 325mmm), and (8. 175mmm) correspond to fixed volumetric brain samples (i.e., fixed topology, see Section 5.1). Columns M = 1, .M = 3. and M 5 correspond to the number of neuron neighborhoods M sampled within the (4, 325mmm volumetric sample (i.e., varying topology, see Section 5.2). The top 1s, 2nd and 3rd best results are highlighted. Active parameter counts (#Par.) and relative active parameter percentages (Par. %) at inference are reported with respect to SSNs.
We focus three volumetric samples. (c.r) E {(4, 125pm), (8, 175pm), (4, 325mm)) These correspond to the most and least discriminative volumes under traditional dimensionality-redutiii pipelines (Reimann et al., 2022), along with additional case test robustness to volumetric variability. All models employ permutation-invarian mean readout for compatibility with Task 2, where structural variability is explicit. In Table (columns (4, 125mmm) (4, 325mm), and (8, 175jm)), we report mean accuracy and standard deviation over five splits (60% train, 20% validation, 20% test). Additional robustness analyses and attention-bru based experiments are provided Appendix H.1.2.
5.2 TASK 2: CLASSIPYING NEURON NEIGHBOURHOOF DYNAMICAL ACTIVITY COMPLEXES
We also consider more challenging setting in which heterogeneous neuron samples and their synaptic connections induce topological variability, moving from feature classification (Task 5.1) to graph/complex classification. Each neuron n in G defines a neighborhood subgraph 9,, ([V()))- For each binary dynamic, we sample M neuron-neighborhood without replacement from the 25 largest within a given volumetric sample U, following Reimann et al. (2022). Restricting each dynamic B, to its M sampled subgraphs and lifting via equation 3 yields M 4,495 DACs.
subgraphs uniformly at random
We use the (4, 325mm) component with M E (1,3.5) to test two hypotheses: (i) whether SSNs can robustly infer stimulus identity from arbitrary neighborhoods, and (ii) whether they exhibit strong sample efficiency, since smaller M yields fewer training examples. Table 3 (columns M 1, 5) reports mean accuracy and standard deviation across five splits (60% train. 20% validation, 20% test). Robustness to volumetric and temporal resolution is discussed in Appendix H.
5.3 DISCUSSION
Results in Table 3 show that SSNs accurately classify stimulus identity in fixed brain volumes (Task 5.1) and remain effective under arbitrary neighborhood topologies (Task 5.2), where structural variability is introduced. Their gains stem from jointly modeling higher-order directed motifs, prevalent at multiple scales in brain networks (Sizemore et al., 2018; Tadié et al., 2019; Sizemore et al., 2019; Andjelković et al., 2020), and direction-aware interactions, both of which are critical for capturing neural activity and function (Nolte et al., 2019; Reimann et al., 2017: Wang et al., 2010; Ecker et al., 2024: Reimann et al., 2022). These empirical findings mirror our theoretical results (Sections 3.1 and 4), which establish that SSNs surpass Dir-GNN and MPSNNs in WL-expressivity (Theorems 1-2) and recover broader class of (neuro)topological invariants (Theorem 4).

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Published as a conference paper at ICLR 2026
The baselines further highlight the limitations of existing approaches. Standard GNNs and DSs perform worst, reflecting their inability to capture higher-order and directional dependencies. SVMs on topological features perform better, confirming the importance of higher-order directed connec- tivity; however, they remain constrained by hand-crafted preprocessing and predefined invariants, suggesting that richer representations which model directionality and hierarchy in isolation, also underperform relative to SSNs, indicating that either aspect alone is insufficient.
be learned directly from data. Dir-GNNs and MPSNNs,
In the most challenging regime (M = 1), with structural variability and extreme data scarcity (one neuron neighborhood per dynamic). SSNs outperform all baselines by at least 17%, underscoring the value of strong inductive biases for sample efficiency (Bronstein et al., 2021). Importantly SSNs achieve these gains with parameter counts comparable to or smaller than the baselines (Table 3, columns #Par. and Par.%). R-SSNs consistently rank second or third across all settings while using substantially fewer parameters, enabling faster training and inference. Detailed complexity and runtime analyses are provided in Appendix F and Appendix H.
6 CONCLUSION
We introduced Semi-Simplicial Neural Networks (SSNs), a new class of topological deep learning (TDL) architectures for data structured as semi-simplicial sets. SSNs generalize graph-based and existing TDL models by propagating information over simplices via face-map induced relations that can be hierarchical, directed, and asymmetric. They subsume prior architectures and provide strictly stronger Weiseeiler-Leman (WI.) expressivity.
iice-maa--nd
By capturing a broader class of topological invariants of neural activity, SSNs establish a principled link between neurotopology and deep learning. Experiments biologically realistic neocortical model support these theoretical advantages in practice: SSNs consistently outperform state-of-the-a baselines on stimulus classification. Limitations and directions for future work are discussed in App. A. Overall, our theoretical and methodological contributions, together with an open-source codebase, lay the groundwork for scalable directed higher-order representation learning and suggest SSNs as promising foundation for TDL models in complex scientific and real-world domains, including neuroscience.
REPRODUCIBILITY STATEMENT
We provide information needed to reproduce our results. For the primary neural binary dynamics classification tasks, we describe the experimental setup in Section 5 and report full hyperparameter configurations Appendix H.1. We also provide detailed protocols and hyperparameters for the addi- tional baselines and ablations, including the lopological-feature baseline and iitention-based variants.
For the Edge Traffic Regression and Node Classification tasks, we report the experimental pro- tocols and hyperparameter settings in Appendix H.2 and Appendix H.3, respectively. Hardware specifications and computational resource requirements are documented in Appendix E.3.
Our code is available httgs://tthhb..oom/Mmmmeeccccc
Data available at hutp:///eenodoorr//ccoosts/////
ACKNOWLEDGEMENTE
The work of is supported by the TU Delft AI Labs programme, the NWO OTP GraSPA proposal #19497, and the NWO VENI proposal 222.032. The work of AC is supported by the NWO OTP GraSPA proposal #19497. The work of RL is partially supported by an EPSRC grant EP/Y028872/1. The work of FD and CB is supported by the National Institutes of Health Grant IRO1ES037156-01.
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APPENDIX CONTENTS
Limitations & Future Directions
21
B Related Works
20
C Preliminaries
23
C.I Algebraic Background
23
C.2 Relational Preliminaries
23
C.3 Directionality and Symmetry in Relational and Combinatorial Structures
24
C.4 Direction vs. Orientation in Combinatorial Topological Structures
25
C.5 Face-Map Relation Algebras
26
C.6 Examples of Common Graph and Topological Face-Map Relations
26
C.7 Invariants, Colourings and the Weisaeiler-Leman Test
30
Theoretical properties SSNs
29
D.1
Generality
29
D.2
WL Expressivity
30
D.3
Permutation Equivariance
32
E
Implementation Details
33
E.1
SSNs
33
E.2
R-SSNs
33
E.3
Computational Resources
34
Computational Complexity
35
Topological Deep Representation Learning for Brain Dynamics
35
G.1
Data
35
G.2 Dynamical Activity Complexes
36
G.3 Dynamical Activity Complexes Invariants
37
G.4
Topological Neural Networks and Dynamical Activity Complexes Invariants
38
H
Additional Numerical Results
40
H.1
Dynamical Brain Activity Classification
40
H.1.1
Experimental Details
40
H.1.2
Additional Results
42
H.1.3
Mon-Innariant Readouts
43
H.1.4
Increased Temporal Resolution and Variation in Volumetric Sampling
43
H.2
Edge Regression Traffic Task
44
H.3
Node Classification
45
A LIMITATIONS & FUTURE DIRECTIONS
In settings where data does not faithfully reflect inherent directionality- because the inductive biases from topology and connectivity misalign with actual information flows, or because the system itself lacks directional Structure-SSNg may introduce unnecessary memory overhead due 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: 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 neurotopological perspective, these techniques cannot yet be applied in vivo, as they require access to detailed connectomic data-thal is, structural information about the brain's '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 (Markram al., 2024:
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2025), where 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-suce as tournaplexes and other directed combinatorial structures- are largely unexplored and may provide complementary inductive biases. Incorporating temporal dynamics and signed interactions (c.g., excitatory vs. inhibitory) (Millán al., 2025) is another key avenue. We emphasize that our contribution 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.
B RELATED WORKS
Topological Deep Learning. TDL builds directly on pioneering efforts in Topological Signal Processing (TSP) (Barbarossa & Sardellitti, 2020; Schaub et al., 2021; Roddenberry et al., 2022; Sardellitti et al., 2021; Isufi et al., 2025; Battiloro, 2024). which underscore the importance of modeling multi-way relationships. Extensions of the WL test simplicial and regular cell complexes (Bodnar et al., b;a) have demonstrated that message passing on these higher-order structures outperforms its graph- counterpart. Both convolutional (Ebli et al., 2020; Yang et al., 2022; Hajij et al., 2020; Yang & Isufi, 2023; Roddenberry et al., 2021; Hajij et al., 2022a; Yan & Kuruoglu, 2025) and attentional (Battiloro et al., 2024b; Giusti et al., 2022: Goh al., 2022: Giusti al., 2023a) architectures over simplicial and cell complexes have been introduced. Furthermore, message passing and diffusion on cellular sheaves over graphs (Hansen & Ghrist, 2019; Hansen & Gebhart, 2020; Bodnar et al., 2022; Battiloro et al., 2024c; Barbero et al., 2022) have proven particularly effective in heterophilic settings. Alternatives that eschew message passing altogether for simplicial complexes have been detailed in (Madhu et al.; Gurugubelli & Chepuri, 2024; Maggs et al., 2024), while an approach to infer a latent regular cell complex for downstream improvement has been introduced in (Battiloro et al., 2024a). Gaussian Processes simplicial and cell complexes have been presented in (Yang et al., 2024; Alain et al., 2024). Comprehensive reviews of TDL can be found in (Papillon et al., 2023; 2025). A combinatorial framework for TDL combinatorial complexes (CCs) was proposed in (Hajij et al., 2022b). CCs relax the structural constraints of classical topological domains by treating cells as ranked, unordered subsets of vertices. This generality makes CCs broadly applicable but prevents them from distinguishing vertex orderings or encoding coherent face maps. As consequence, CCs cannot represent directional or order-sensitive higher-order interactions, and structures such as directed simplicial complexes. ordered motifs, or tournaplexes fall outside their scope. semi-simpliciall sets (SSSs) provide exactly this missing structure: they support multiple ordered simplices on the same vertex set together with coherent face maps satisfying the simplicial identities, enabling principled modeling of directionality and order-aware higher-order interactions. These structural differences imply that CCs and SSSs capture different domains of objects, and neither framework subsumes the other. An extension of the CC framework was recently proposed in (Hajij al., 2025). A fairly general topological expressivity analysis of TDL models is given in (Eitan et al., 2025). Quantum simplicial neural networks have been introduced in (Piperno et al., 2025), continuous variants in (Finizade al., 2025), and geometrically equivariant architectures over simplicial and broader combinatorial spaces in (Fijkelboom et al., 2023; Battiloro et al., 2025b). Finally, software libraries and benchmarks for TDL have been released in (Hajij et al., 2024; Telyatnikov et al., 2025; Papillon et al., 2024; Ballester et al., 2024).
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 (Li et al., 2016; Abbass, 2018). Directed GNNs for node classifica- tion (Rossi et al., 2024; Tong et al., 2020b), for instance, have demonstrated superior performance on heterophilic benchmarks (Rossi et al., 2024; Koke & Cremers, 2023). Spectral approaches ex- learning to directed graphs: DiGCN uses a Personalized PageRank-basee
tend Laplacian-bact operator (Tong et al., 2020a), while MAGNet introduces the Magnetic Laplacian, encoding edge directionality in the complex phase (Zhang et al., 2021). This formulation was later extended to the Maggntic Laplacian, addressing key limitations and enabling principled learning signed directed graphs (Fiorini et al., 2023). Extending this works, Fuchsgruber et al. (2025) propose the Magnetic Edge Laplacian for edge-level regression tasks in traffic forecasting. Notably, they
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introduce both an orientation-equiis equivariant variant -eeiant--eeiieddd for prrectioon-ensitivvy signals such as traffic flow- and an orientation invvrriat generalizing previous adge-based methods. Beyond pairwise structures, recent works extend to non-uniform, non-hierarchical settings using neural networks on directed hypergraphs (Fior- ini et al., 2024; Ma et al., 2024), with applications in traffic forecasting (Luo al., 2022) and human pose estimation (Cui et al., 2024). Bernárdez et al. (2025) further show that hierarchical, order-aware relations-built on the combinatorial framework of Hajij et al. (2022b)-better capture communi- cation patterns in computer network systems. Finally, the preliminary work of Lecha et al. (2025) introduces Directed Simplicial Neural Networks, formalizing higher-order topological directionality for learning on directed simplicial complexes based on the theoretical foundations of Riihimäki (2023).
for ddrectioonagnostee signals such as speed
The manifold hypothesis posits that neuronal activity is constrained to lower- dimensional subspace, shaped by the structural limitations imposed by synaptic connectivity (Cun- ningham & Yu, 2014; Gallego et al., 2017: Curto & Morrison, 2019; Chambers & MacLean, 2016; Rubinov & Sporns, 2010; Bargmann & Marder, 2013; Bassett & Sporns, 2017). This concept builds on the foundational principle that structure shapes function, suceinetly captured by Hebb's rule: "neurons that fire together, wire together." " Yet, precisely how structural connectivity gives rise to neural manifolds remains major open question. Biologically grounded computational models, such as the Blue Brain's 's (Markram, 2006) rat neocortical microcircuit (NMC) (Markram et al., 2015), offer valuable platform for investigating this relationship, providing detailed representation of brain structure modeled as a directed graph. Notably, recent work of Reimann et al. (2022) con- firmed that the NMC model adheres the manifold hypothesis. However, while directed graphs effectively capture the inherent directionality of synaptic transmission from presynaptic to postsy- naptic neurons (Reimann et al., 2017) their dyadic nature limits their capacity to represent polyadic interactions and higher-order co-activation patterns that are critical to neural computation. Experi- mental 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 digeer-dimensional cliques, suggesting that such patterns reflect greater functional com- plexity and information abstraction. Simplices--hgher-diiee higher-dimensionll analogs of cliques- have been identified as overexpressed motifs across multiple scales of brain networks (Sizemore et al., 2018; 88; Tadić et al., 2019; Sizemore al., 2019; Andjelković al., 2020), and have been linked to functional relevance (Nolte et al., 2019). Alternatively, co-fluctuation frameworks such as (Santoro et al., 2024) derive meaningful weighted, undirected simplicial complexes by algebraically combining low-order fMRI signals to construct higher-order time series. 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 (Tadić et al., 2019). Crucially, neurons embedded in higher- order directed simplices with correlated activity exhibit reduced sensitivity to noise and stronger alignment with the underlying neural manifold (Cunningham & Yu, 2014; Gallego et al., 2017), ulti- mately improving representational fidelity (Nolte et al., 2019; Reimann et al., 2017; Wang et al., 2010). 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 cogh-coople ones support resilience- bbtth coexisting within the same network (Ecker et al., 2024). These insights have fueled the rise of Neurotopology, a field modeling brain activity through topological spaces derived from spike co-activation patterns. Seminal works (Conceição et al., 2022: Reimann et al., 2017) propose modeling neural dynamics as üime-indexee sequences of directed flag complexes (Lütgehetmann et al., 2020) and touraaplxxss (Gove et al., 2021). derived from subdigraphs induced by active neurons. Brain dynamics are then characterized by computing topological invariants over these evolving structures. Building on this framework, Reimann al. (2022) demonstrated that stimulus identity can be accurately decoded from the NMC model using this topological leatures-even when conventional methods fail. These results highlight topologiaal featurization as a robust and complementary alternative to traditional manifold-based approaches.
Neurotopology.
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C PRELIMINARIES
C.I ALGEBRAIC BACKGROUND
Groups. A group is pair (G.-), consisting of set G and binary operation satisfying the following axioms: (A1) Associativity: for all a, b, E G we have a- (b-c)= = (a c; (A2) Identity: there exists element E G' such that there exists an element a & G such that a a-ta = aa
ac = for all G; (A3) Inverse: for each a G
Given a set S, the set of all bijections p: S together with the composition of functions forms group known as the symmetric group of S, denoted by SSm(
Isomorphisms. Let X and Y be two structured objects of the same type. An isomorphism & X - Y a bijective map with inverse و Y + X that preserves all relevant structure. If such a map exists, say that X and Y are isomorphic, denoted X = Y. Two sets S and S' are isomorphic if there exists a bijection between them. Two directed graphs G = (V, E) and G' = (V', E") are isomorphic if there exists bijection vertex sets V and V are attributed-en e.g., equipped with dynamic binary functions B : - {0. 1) we additionally require that the attributes are preserved under the inverse map: for all a E V. B(u)= B' (1 (v--(()) In general for any two sets equipped with collection of relations if set isomorphism preserves relation. Specifically, for directed simplicial complexes K and K', an isomorphism is bijection vb on the vertex that preserves simplices: a tuple (vo...., 2a) E Σn is simplex in I. if and only if (vb(vo)..... ..... (v,))) Σ% is a simplex in K',
VV such that (u, v) E if and only if (v(u), v(v)) E E'. If the
and B'; V + {0, 1 we
Automorphisms. An automorphism of a structured object S is isomorphism from S onto itself. The set of all automorphisms of S, denoted by (S), forms a group under composition. Indeed, composition is associative, the identity map serves the identity element, and each automorphism is invertible. If S is a plain set, then any bijection (permutation) S + S is an automorphism, so Aut(S) Sym(S) the full symmetric group on S.
C.2 RELATIONAL PRELIMINARIES
Binary Relations. Let S be set. A binary relation R on S is any subset R C S x S. A binary relation is symmetric if, for all a, T E S, (a,T) E R implies (T, E R; transitive if, for all o, T, K S. (0, E R and (k, r) E R together imply (a, E R; and irreflexive if (a,o) & R for every a S. We further say that is n-uniform every element of S is related to exactly n other elements. Since relations subsets of S x S, canonical examples include the universal relation 1 1= S x S, the empty relation 0 = 0, and the identity relation id {(o σ) a E S}.
Basic Operations. Standard set operations extend naturally to relational operations:
Rn R' {(a, r) e S x $ (a, r) E R and (a, r) E R'), R R' ((a,r) E S S (o,T) E or (a, E R³).
Furthermore, relational composition and converse operations defined as:
Ro R' = ((a,7) e S S 3x E S such that (a, e R and (k, E R}},
R T = ((0,T) E Sx (T,O) E R}.
Characteristic function. Analogous to how subsets can be identified with their indicator functions, relation R C S x S can be uniquely represented by its characteristic function Xa S x S - {0, 1}. This function assigns a binary value each ordered pair (a, r) E S S: returns 1 if (o, and O otherwise:
11 if (a, R, 0. (a, $ R.
XR((O, r))
Matricial Representation. Equipping the set with an ordering S [o...... ... )n), the matricial representation of R is the n. matrix (Ax); xa((0,, a))) We call it relation matrix or adjacency matrix. Such matrix representations underpin computational implementations, relational learning frameworks, and message-passing neural network architectures (Veličković, 2022).
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n-ary Relations. An n-ary relation over set S is subset R S", that is, collection of n-tuples drawn from For instance, ternary relations R - S x S x S arise by extending binary relations via the natural join operation:
Re R' = (O, E x (a, E R and (k, E R-)
This operation yields ternary relation by joining the tuples from R and It' their shared middle element y, thus composing relational information across three elements of the set S.
C.3 DIRECTIONALITY AND SYMMETRY IN RELATIONAL AND COMBINATORIAL TOPOLOGICAL STRUCTURES
Directionality. A relation of arity n + 1 over vertex set V is intrinsically directional when it is stored as an ordered tuple
Un) and membership depends on the order of its entries. Such relation is symmetric if membership is invariant under permutations: for every (oo....,0m) E R and every permutation E Sym(n + 1). the tuple (o))))..... pp(n) also belongs to R. Otherwise, the relation is asymmetric. Binary relations provide the most familiar case. An asymmetric binary relation R C V x V-where (a, r) E R does not necessarily entail (7, a) E R-is naturally represented by directed graph = (V, E) with R E. In contrast, undirected graphs correspond to symmetric binary relations: the presence of (a, T) guarantees (T, so one may equivalently replace ordered pairs with unordered pairs fo, F} without loss of topological information. This viewpoint extends seamlessly to higher-orde ooder structures. A directed simplicial complex I. can be regarded as an asymmetric relational structure over V, equipped with families of asymmetric relations I, S V*+1 of varying arities, which collect directed simplices of different dimensions. Thus, the paymmetric-symmet distinction provides a unifying language for characterizing directed versus undirected combinatorial objects.
Symmetrization. Given an irreflexive relation on set (equivalently, simple directed graph G - (V, E)), its symmetrization is defined as Raym R U R or in graph-theoretic terms, Eaym EU ET. This construction associates an undirected graph Gaym with the original directed graph via canonical projection (or quotient map)
[(u, u)) {u, (u,v) E Esym.
There is a one-to-one correspondence between directed symmetric graphs and undirected graphs under this map. The same principle extends naturally to directed simplicial complexes. Given directed simplicial complex K--wtth directed graphs as 1-skeleton map æ K Ksym by sending each n-simplex (vo...., Un) to the unordered set (un,..., --. vn). While this map preserves the simplicial structure (i.e., maps directed simplicial complexes to valid abstract simplicial complexes), discards ordering information by identifying every ordered simplex with all of its permutations. This loss of orientation leads to a reduction in expressive power, which we make precise in the following proposition.
define its symmetrization
2
(a)
(b)
(c)
Figure 5: (a) A pair of non-isomorphic digraphs Ç and g'. (b) their corresponding non-isomorphic directed flag complexes Kgg and Kgg, respectively, and (c) the resulting symmetrized simplicial complex Ksym. which is isomorphic in both cases, i.e., T T(Cg) = ((Krr).
Proposition 2. The symmetrization map x K. - Kwyyn a simplicial structure forgetful map, that is, preserves isomorphisms but is not injective on isomorphism classes.
Proof. Let z be an isomorphism between IC and K' and let (oy = (v(v) u E o}. Note that, sym is well defined since different orderings of the same vertex set in mapped to the same set
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Vsym O, implying preservation of simplicial structure (simplices are sent to simplices). Since & is bijective and π does not collapse vertices (only their ordering), it follows that t'yy is bijective simplicial map preserving the face structure. Hence, Vsym is an isomorphism between Kaym and yyyym and π is exist non-isomorphic directed simplicial complexes IC and K'' such that <T()) = π <T(C", implying the non-injectivity. Finally, note that the number of simplices in Kaym equals that in K. if and only if no two distinct simplices in K share the same underlying vertex set-that is, each simplex has unique ordering. Otherwise, collapses the redundant orderings, reducing the total face count (see Figure 5).
(u(v): E o). Moreover, by construction
pwwtoopppiim--eeerviin Clearly, there
Symmetrization naturally extends to attributed directed simplicial complexes via a fixed permutation- invariant aggregation of features.
Transitivization. Conversely, given a simplicial complex XC = EV endowed with an arbitrary total order < on its vertex set V. one naturally obtains unique directed simplicial complex Kair by assigning to each simplex E Σ the directed simplex de = (No, f1,.... U,y,, where the vertices of are arranged in increasing order (ig < U1 <... < vn)- Consequently, for every pair of vertices vi, Vi with < the directed edge (14, vÿ) is present in Kair. By construction, every simplex thus forms a transitive clique at the graph level.
C.4 DIRECTION VS. ORIENTATION IN COMBINATORIAL TOPOLOGICAL STRUCTURES
We clarify the conceptual distinction between directionality and orientation, two notions that are often conflated.
Directionality. As defined in Section C.3, a relation of arity n + 1 over vertex set V is directional when it stored as an ordered tuple (Vo,-.. Un) and membership depends on the order. For instance, in a directed graph the edge (0. 1) does not imply (1, Reversing an edge (0, 1) vs. (1,0) or permuting a triangle (0. 1,2) vs. (2, 0) yields different simplex that may encode different information. This property enables models to capture higher-order asymmetric motifs-such as brain co-activations (Sec. 4), traffic flows (App. H.2), or citation graphs (App. H.3). By contrast, symmetrization collapses distinct motifs (c.g., transitive cliques vs. cycles) into single entity (Sec. C.3, Prop. 2, Fig. 5), erasing essential directional dependencies. Thus, directionality encodes irreducible asymmetry in the data.
Orientation. Given an unordered k-simplex o = {uo...., )x an orientation is a choice of equivalence class of orderings of its vertices under permutations. Formally, if E Alt(k + C Sym(k + 1) is an even permutation, then
[vo VU........ Un] (Up(o)..... .... (p(k))-
The alternating group Alt(k Lttkk+1) 1) consists of permutations that can be expressed as even number of transpositions. Hence each k-simplex has exactly two orientations, corresponding to +1. Orientation is thus valuee structure, while directionality involves the full symmetric group.
Oriented vs. directed edge. An oriented 1-simplex [o. 1] assigns sign: a positive orientation may encode flux from 0 to 1, and negative orientation flux from to 0, thus usually serving reference direction current flow. Both are the same edge, differing only by sign. directed edge instead distinguishes (0, 1) and (1, two distinct simplices, each possibly carrying different information.
Oriented vs. directed triangle. The unordered simplex (0, 1,2) admits two orientations: clockwise (0, 1, (1, 0) (,, 1) and counter-clockwise (0, 2, 1) ~ 1, ~ (1, Orientation collapses the 3! orderings into two parity classes. A directed 2-simplex, however, distinguishes all six ordered triples, allowing models treat distinct transitive cliques fundamentally different.
Orientation imposes a sign convention on an undirected simplex, while directionality generates genuinely distinct simplices on the same vertex set. As consequence, models that ignore asymmetry cannot distinguish higher-order motifs, and their adjacency is restricted to set inclusion, preventing direction-awant propagation of information.
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C.5 FACE-MAP RELATION ALGEBRAS
Let S be semi-simplicial set with simplices S and face maps de : Sa - Sn-1. Each face map naturally induces irreflexive binary relation:
Ryy ((a,+) r) a Sn, ()) =] E P(Sx
which relates each simplex a E S, with its i-th facet in Sn-1-
Positive Face Map Induced Relation Algebra. The set of all binary relations on S, denoted as P(S x S), forms positive relation algebra closed under the operations union (U), intersection (N), relational composition (o), and converse (T). Specifically, the structure:
(P(S S), U, n, 9, T,id, 0,
satisfies standard axioms in (Tarski, 1941). Formally, we define the face map relation algebra Ra the smallest positive relation subalgebra of P S) generated by the collection {Ray 2 0,,0 S i S n), equipped with standard operations U, n, 9, T together with relations id, 0, 1.
For theoretical purposes, we consider ternary relations from natural joins R 14 R' from R, R' E Ra.- These higher-order relations play role in generalization of Crisfeiler-Lemane -tppe procedures (see Appendix D.2).
C.6 EXAMPLES OF COMMON GRAPH AND TOPOLOGICAL RACE-MAP RELATIONS
Overall, mace-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 (Veličković, 2022; Rossi et al., 2024; Bodnar et al., 2021b) (see Appendix D.1). In this section, we provide some of this common examples. In particular, we introduce the relation set used in our SSN experiments, combining coundarycco-bouddar maps with directed up/down adjacencies for intradimensional communication.
Let be a semi-simplicial set with set of simplices S. define the following face map-induced binary relations:
Ruiii == Ryy o Ra = ((a,7) T) d) d E Rd:
which relates two n-simplices and if their i-th and j-th facets coincide, respectively (see Figure 6). natural extension, often employed in message-passing architectures, is the following ternary relation:
RALAJ RuL.a Ra: (,) | (
d](()) = xx E Rds
(5)
which explicitly includes the shared lower-dimensional face x. between simplices and 7. Similarly.
T
Figure 6: Two 2-simplices (triangles) a and r are related such that db(a) = -B(()) = where e denotes the shared edge highlighted in red. Therefore, (a, r) R2
we define:
Ratid = Ragix RT ((a,T) |3x 3x E
S, (k) and dx T} E Ra (6)
which relates a and T if they are, respectively, the i-th and j-th facets of the same (n + 1)-simplex k (see Figure 7). Its corresponding ternary extension is defined as:
Ropsa == Ratii ba R RT--1 ((a,TRK) 3R E S, (x)) and (k) = T E Ra, (7)
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which explicitly includes the common higher-dimensien simplex From this point forward, will use the same notation for both binary and ternary relations when the context is clear, as all constructions valid in both cases. In Section D.2, unless otherwise specified, relations will be assumed to be in their ternary form. Notably, when i # j, these relations are asymmetric, thereby
Two 1-simplices (edges) a (red) and (brown) are related as facets common (n + 1)- simplex w (light purple triangle). In (left), (a, T) R11,2.0. whereas in (right), (α,T) R11,211-
Figure 7:
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 O-simplices (nodes), the standard directed up adjacency relations can be derived as follows:
Rin Rot,u,1 { (u, u) Ee E, (e(e) and di (e) v Rd
(8)
indicating that u the target of (i.e., the source of and sink). Moreover,
Rout == R, = (u, v)) Be E (e) = u and (e) v Rdd
(9)
meaning the source of (i.e., v is the target of e, and w the source). These relations converses of each other: (Rm)) = Rout, and thus:
Raym
Rin Rout E Rd,
(10)
induces a symmetric directed graph Gsym = (V, Esym) in one-to-one correspondence with the undirected graph Gsym EV.
Directed Paths and Higher Order Directionality. A directed path of length in a digraph (V, F)) is a sequence of vertices (vi, .,...., 2/a)
such that (1;, ti+1) E E for every 1 S < k. Similarly, given a set of simplices S a semi-simplicial set equipped with a face-map induced relation R E Ra. R-path of lenght & in the semi-simplicial set is sequence of simplices (01, 02. okk such that (a;, 0++1) E for every 1 i k. Notably, (01,0k) E Ro where R denotes the k-fold composition of R with itself. Composing different instances of R,4.i.j yields different higher-order simplicial paths, connecting directed simplices in distinct direction-preserving manner (Riihimäki, 2023; Lecha al., 2025) (see Figure 8).
(b)
Figure 8: (a) An R2 directed path (red arrow) in consistently directed two-dimensional structure with unique source and sink, and (b) An R21,12 path (red arrow) depicting a circular flow around source vertex.
Boundary & Converse Boundary. Let = (V,E) be a simplicial complex with maximal dimension N, and let Kdir denote the associated directed simplicial complex induced by a fixed total order on V. For each dimension a with 1 S n < N, the nth boundary relation defined as:
B, =
Ray
(11)
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which relates each n-simplex all its (n dimensionnl faces. Conversely, for N - 2 the n-th converse boundary relation is defined as
n+1 C
U ( 1=
(12)
relating each n-simplex to all (n + dimeinensional simplices containing it as a facet. Notably, for any 0 S m < < N the composition of maps Bn.m Bm+1 Bm++2 o.... Bn relating an n-simplex with its m-dimensional faces. Similarly, for the converse boundary relations, define Cm,n Cn-1
Cm+1 Cm, relating n-simplex with all m-simplices containing it.
Aggregated Lower and Upper Adjacencies. For each dimension 2 n > O, we define the collection of all possible n-lower adjacency relations as
R4n {Rinis (1,5) E [m]?)
(13)
Accordingly, define
Ryn = URI
(14)
which relates simplex a to all simplices 7 sharing a common facet with a. Then, for dimensions N 1 2 n 2 we define the collection all possible n-th upper adjacency relations by
Rtn {Rinss (i,5) in] and i # j)
(15)
Finally, we define the aggregated upper adjacency relation
Rin = U
(16)
which relates simplex to all simplices that share common higher-dimensional simplex containing both as facets. Notably, B,, Cn. Ris, Ryn. coincide with the four types of adjacent simplices defined in (Bodnar al., 2021b) under which MPSNN operate. Under this setting, denote Un = (Bn, On, Ru } and D, = (Bn, C. U4a URn with the understanding that only those relations which are defined are included. Accordingly,
u
U Un m
and
D=
U D..
(17)
VL 0
defined. Throughout all experiments. SSNs use the relation set D equation 17 2-dimensional semi-simplicial sets, combining coreddaryyo-bouddaa maps with all directed up/down adjacencies for intradimensional communication.
C.7 INVARIANTS, COLORINGS AND THE WEISFEILER LEMAN TEST
Invariants. An invariant for a given class of objects is function assigning to each object a value that remains unchanged under isomorphisms. Formally, X = Y, then f(X) = f(Y). Objects with identical values under 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 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 & = (V, E) or directed simplicial complex K. (V, potentially with dynamic binary functions, vertex invariant
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is a map e: V - Corc:V {0.1)" + C invariant under isomorphisms, meaning c(u) c()(t)) for all vertices v under isomorphism a/. Vertex invariants extend naturally to graph or complex invariants through a permutation-inve invariant aggregation, i.e, functions that are invariant under set isomorphism. Precisely, given a vertex invariant c, an induced binary graph invariant can be defined as: <(9B) ((((((v) v V)]), where is a permutation-invarian function. Consequently, if GB _ G/. then c(GB) = c(gn)) Real-valued vector vertex invariants arranged ordered sequences, c(Gg) = (c(U)),... )(N)) according to a fixed vertex ordering V = (UI..... uN } constitute graph features, which are generally not invariant, they explicitly depend on the chosen vertex ordering.
Graph coloring. Let G = (V, F) be a graph. A graph coloring is a vertex invariant c e V - N
which assigns to each vertex a positive integer (its color). For any n E N, the preimage c- Vn partitions the vertex set into subsets of vertices sharing the same color property. The pair Ge = (g,c) called the colored graph and is attributed graph.
Color Refinement. A color assigment c' is refiment of if for every pair e E V, c' c implies c(u) c(v). In other words, vertices that share the same color under c' must also share the same color under c. Thus, e partitions the vertex set into a potentially richer variety of color classes. Specifically, they are equivalent ex _ c, if for every pair u. E V. c'u) = c'(u) if and only if c(u) = c(v).
The Weisfeiler-Leman Test. Given a graph G, define an initial coloring cO e.g., setting (x) = for all v E V. This yields the colored graph Gn = (g, c,); clearly g = g' if and only if Go _ 9-- For a vertex и E Go. define the invariant dg V x N- + M(N) (here, M(N) denotes the set of finite multisets of N) by (u)
(fc'(v) (u,v) v) E E}}. i.e., assigning the multiset of colors of u's neighbours the relation E. Then, we obtain a refined coloring c by setting c') has (u), (u) E N, where hash: N x M(N) + N is an injective function. The refinement process continues until the coloring stabilizes; that is, until c'+ c'(x) for all E V. Once stability is reached, denote the stable coloring by e and the resulting colored graph by G. (G,c). Assume that the image of c consists of l distinct colors (which, without loss of generality, can be labeled {1..... k)). A colored graph invariant h is then obtained by counting the vertices of each color: h(G.) = (1 c (1) l.....I c (k) D. Thus, for a pair of graphs G and G' we have h(G.) h(C)), then G 전 G' This algorithmic procedure, yielding incomplete graph invariants discriminating some non-isomorphic pairs, is the Weisfeiler- test (Weisfeiler & Leman, 1968).
D 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 Equation 1, in which each relation relation-specifie transforma- tion WR follows the directional passsage-passingg paradigm introduced in the MPNN-D module (Rossi et al., 2024). this setting, messages are inherently asymmetric and propagated along the relation R: that is, if (a, r) E R, then wn updates the representation of 7 using information from but not the other way around. For the remainder of this section, we adopt MPSSNs as our reference architecture for theoretical analysis.
D. GENERALITY
Proposition 1. Semi-simplicial neural networks (SSNs) subsume directed message-passing
GNNs (Rossi et al., 2024) on directed graphs, paassage GNNs (Gilmer et al., 2017) on undirected graphs, message-passing simplicial neural networks (Bodnar et al., 2021b) simplicial complexes and Directed Simplicial Neural Networks (Lecha et al., 2025) on directed simplicial complexes.
Proof. Let WR denote the message propagation operator associated with face-map-induced relation R. as defined in Equation 1. We instantiate each w following the directional message-passiie scheme of the MPNN-D framework (Rossi et al., 2024), which propagates information along R: specifically, if (a, T) E R, then Wn updates the embedding of T using the features of a. First, consider an attributed directed graph Gr. the SSN is instantiated with the relation set R [Ri:; Rout) and the directional
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propagation rules of MPNN-D, the resulting model corresponds exactly to Dir-GNN (Rossi et al., 2024). Next, if GF undirected- -viewed -vewwe as a symmetric directed graph- tten choosing R == { yields Raym- -MPSSN that recovers the standard message-passiingg GNN (Veličković, 2022; Gilmer et al., 2017) operating over Gp. Let K be an attributed simplicial complex, and let F be its associated directed simplicial complex induced by a fixed total order on the vertices (see Appendix C.5). Then, an MPSSN operating on Kdir, with the undirected relation set (as defined in Equation 17) corresponds exactly to message-passing simplicial neural network (MPSNN) (Bodnar al., 2021b) acting on Kp. Finally, let Kr be an attributed directed simplicial complex, MPSSN operating on the set of relations as in (as defined in Equation 17) corresponds exactly to a passsag Directed Simplicial Neural Network (Dir-SNN) (Lecha et al., 2025). Therefore, by appropriately selecting the relation set R C Rd and instantiating the message propagation operators {WR) ) accordingly, the SSN framework unifies and generalizes MPNNs, Dir-GNNs, MPSNNs and Dir-SNNs.
D.2 WEISFEILER-LEMAN EXPRESSIVENESS
Relational Leimmn Test (R- Let S be set equipped with a collection of binary relations R. We extend the standard Weisfeiler-Leman (WL) test to handle multiple relation types. following the formulation of Barcelo et al. (2022). Initialize coloring ca, e.g., c² (a) 0 for all S. For each R E R. iteration 1 2 0, and S, define
(0) ((c((T) (a, r) R));
the multiset of colors of elements related to a under R. The refinement step
c+1 (a) = hash (c()), (c/((o) RER)
(18)
where hash: N x M(N)R + N is injective. This reduces to the classical 1-WL test (Weisfeiler Leman, 1968) when [R| =
Refinement by Unions. For finite family A of relations on S, its finite union-closure is
(A)
{
{UR] FSA, F finite
REF
That is, (A) consists of all finite unions of relations from A. Given families A, B of relations, we say that A refines B by finite unions if B S (A).
Lemma 1. Let be a set, and let A and B be finite collections of relations on S. Suppose A is a refinement by finite unions of B. Then, A-WL at least as expressive as B-WL.
Proof. Let a and b denote the colorings of A-WL and B-WL, respectively. We show by induction that a refines b at each iteration I. Initially, we set (a) = (g) = 0 for all a E S, trivially satisfying the base case. Assume the claim holds iteration 1.If a a (o) a (T), injectivity of hash implies
dn(a) = (T) for all R E A.
By Lemma in (Bevilacqua et al., 2022), equal multisets under finer coloring remain equal under any coarser coloring; using the inductive hypothesis (that a' refines b²), we get
(l) = (T) for all R E A.
Now fix R' E B. Since A retines by unions, there exists AR S A such that R' = -UREA R. By the WL definition, the neighbor multiset for a union relation is the multiset union with multiplicities added (bag sum) of the components:
(x) = b'a(o).
BEAR
Therefore,
Un (a)
Ule(T) (T). (T).
REAR
Hence the arguments to hash at iteration coincide and (') = =b++1(7) induction, A-WL is least expressive as B-WL.
This completes the n
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Semi-Simplicial Weisfeiler-Lemall Test (SSWL). Let S be attributed semi-simplicial set with set of simplices S, and let _ Ry be a collection of dace-map-ine induceed relations. The Semi-Simplicial Weisfeiler-Leman Test (SSWL) is the R-WL test equation 18 applied to S.
Lemma 2. Let S be the set of attributed simplices of semi- simpliamll set S endowed with collection of fuce-map-nduued relations 'R C RRa- Consider an R-MPSSN defined by equation 1. Suuppsse the message module wn, the aggregator ② and the update function all injective. Then, R-MPSSN is as expressive as the R-SSWL test.
Proof. Let c and denote the colorings produced by the R-SSWL and the R-MPSSN, respectively. We show by induction that e refines x. At O, without loss of generality, we initialize (o) (o) 0 for all a E S, establishing the base case trivially. Assume the induction hypothesis holds iteration Consider two n--smmllices 0, E such that (a+ = ( By the definition of R--SSL, this implies identical multisets of colors aggregated from neighboring simplices at iteration l. By the induction hypothesis, these multisets coincide under x', ensuring equality in the arguments of the R-SSN, thereby yielding (o) (T). By induction, refines Conversely, xf x (T). the injectivity of the composition of injective operators wp. X and o, implics identical multisets iteration 1. By the induction hypothesis, the hash function receives identical inputs, resulting in c (a) = d++ (r). Thus, c = x, proving that R-MPSSN and R-SSWL are equally expressive.
(a) g
g'
(c) Kag
(d) Kgg
Figure 9: A pair of non-isomorphic directed graphs, shown in (a) and (b), along with their correspond- ing directed flag complexes in (c) and (d), respectively. While these digraphs can be distinguished by SSNs operating Kgg and Kg', they cannot be distinguished by Dir-GNN: (Rossi et al., 2024).
SSN vs. Dir-GNN. We now prove Theorem 1.
Recall from App. C.2 that relation R n-uniform if each element related to exactly n others.
Theorem 1. There exist SSNs that are strictly more expressive than directed graph neural networks (Dir-GNNs) (Rossi et al., 2024) at distinguishing non-isomorphic directed graphs.
Proof. Let G be a directed graph and Kgg its corresponding directed flag complex, with maximal di- mension 2. Consider the MPSSN defined on Kg with face-map-induced relations Do (Rins Rom)) and 'D as in 17. Then, the proof follows from Theorem 1. of (Lecha et al., 2025). To build intuition, Fig. 12 depicts two non-isomorphic directed graphs in which both Rin and Ront are 2-uniform relations-that is, each node has exactly two incoming and two outgoing neighbors. Assuming constant activation features across all vertices, these graphs cannot be distinguished by Dir-GNNs. In contrast, SSNs separate them by exploiting structural differences in their associated directed flag complexes.
MPSSNs vs. MPSNNs. We prove that there exist instances of MPSSNs that are strictly more powerful than MPSNNs (Bodnar et al., 2021b) distinguishing directed simplicial complexes.
Lemma 3. Let K. be directed simplicial complex with set of simplices E, and let U and D be collections of face-map-indues induced relations defined as in Equation 17. Then, D-SSWL is at least as expressive as U-SSWL.
Proof. Since D is refinement by unions of 24, the result immediately follows from Lemma 1. -7
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T
T
T
(a) Kaa
(b) Kab
(c) Kaym
Figure 10: Two directed simplicial complexes: (a) C.. (b) Ks, and (c) their shared symmetrized
simplicial complex Kaym. Despite K. * Ks. both are mapped the same undirected complex under symmetrization.
Lemma 4. There exists pair of directed simplicial complexes that are distinguishable by D-SSWL but indistinguishabi by U-SSWI.
Proof. Let K.,, Ks. and Kaym be in Figure 10, with constant activation values assigned to all sim- plices. By construction, the U-SSWL updates on K. and K,, are equivalent to running SWL (Bodnar et al., 2021b) on the shared symmetrized complex Kaym. and thus produce identical outputs. In contrast, D-SSWL distinguishes K. and Kcs due to their differing directional structure.
Corollary 1. D-SSWL is strictly more expressive than U-SSWL.
Proof. Direct consequence Lemmas 3 and 4.
Lemma 5. There exist D-SSNs that are strictly more expressive than U-SSNs in distinguishing non-isomorphic directed simplicial complexes (Bodnar al., 202 Ib).
Proof. Follows directly from Corollary and Lemma 2.
Theorem 2. There exist SSNs that are strictly more expressive in distinguishing non-isomorphic directed simplicial complexes than MPSNNs (Bodnar et al., 2021b) under symmetrization.
Proof. Follows directly from Lemma 5 and Proposition 2, which ensures the symmetrization map π be forgetful.
D.3 PERMUTATION EQUIVARIANCE AND INVARIANCE
Let SF be attributed set equipped with relations R. Every permutation p Aut(S) induces an isomorphic relabelled structure (S F) (p- S, p- R, p F) where p R {p(o), (a,r)
R} for all R E R and F = F a pー We say that map & SR (SR)' is:
Permutation equivariant if, for every E Aut(S), holds that & o po s.
Permutation invariant if, for every P E ASt(S), it satisfies e Φ.
Consider an attributed semi-simplicial set S with attributed simplices Sr and face-mane face-map-induced relations R. Representing S as matrix requires assigning an arbitrary global indexing (ordering) to simplices S = (oI..... N } (see Section 2). Neural network models processing these data must therefore exhibit invariance equivariance under relabeled (permutations of indices) isomorphic structures. Specifically, iimiliimpliiial Neural Networks (SSNs) operate on an attributed set of simplices with man-inuen-d relations SR. necessitating consistent behavior under reindexing of the entire structure, including both the relations and feature assignments. We prove that SSN layers indeed satisfy this property:
Theorem 3.
Let SR be an attributed semi-simplicial set with lice-map-inducedd relations R. Consider an SSN layer defined in Equation equation I. for each relation R E R, the mapping WR and the aggregator X are permutation equivariant, then the SSN layer is permutation equivariant with respect to simplex reindexing. That is, for all p E Aut(S), we have:
SSN op po SSN.
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Proof. Let X denote the matrix of features at layer I, and define the aggregated features X & ( By assumption, each wa is permutation equivariant and & is permutation equivari- ant (or invariant, depending on definition), so for any E Aut(S), ② RER wR (p(x)()) p( Let Φ be the pointwise update applied by the SSN layer: [ = ologg 2dh)). Since @ acts independently on each simplex, for any p.
ppplzo,), p(="))) = o ( a Fop()) )
4+1 2000
p(z'xt))
Hence ((p(X'), p(x'))) pl p(o(X' .x) Combining the equivariance of aggregation and the update yields SSN op po SSN. This completes the proof.
E IMPLEMENTATION DETAILS
E. SEMI-SIMPLICIAL NEURAL NETWORKS (SSNs)
We implement Semi-Simplicial Neural Networks (SSNs) leveraging PyTorch Geometric's HeteroConv wrapper (Paszke et al., 2019) to efficiently compute convolutions over heteroge- neous graph structures. Consider a semi-simplicial
S equipped with an attributed of simplices ST. and face-map mase-mmap-nnee relations R. Nodes correspond to simplices in S, where each node type is defined as NodeT
subset of relations linking simplices of dimension to those of dimension I, with each relation R2 E R2 being binary relation Rt C S, X S. The EdgeTTypes 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 f, is defined by modifying the HeteroConv layer as follows:
= (o) Formally, Ri R represents
dim(S)
X
X
W (X))
(19)
RER
This formulation involves two-step aggregation procedure: initially, messages from each source dimension s are independently aggregated via relation-speciff mappings wn- Subsequently, these dimension-specifie contributions merged to update the features of simplices at dimension t. Importantly, the HeteroConv wrapper enables various convolutional operations, such as SAGE, GAT, GCN, for implementing the relational mappings wi.
E.2 ROUTING SEMI-SIMPLICHE NEURAL NETWORKS (R-SSNs)
Let SR denote attributed semi-simplicial set equipped with collection of face-map map-induced relations R C Ra. Consider a partition Pr = (R...... Rr } of R into n distinct relation classes. The 1-th layer of Routing Semi-Simplicial Neural Network (R-SSN) updates the features X as follows:
x1+1 (x!
(o(X))-wx(X'))
(20)
REPa RE
where GR(X') E 1] is gating function that outputs normalized weighting scores for each expert. This gating mechanism employs top-k selection regime, dynamically identifying the & most relevant experts for each subset of relations E PR during message aggregation. The operator + represents the final aggregation of expert representations. Following the details outlined in Appendix E. we partition R into relations {R:), where each binary relation Rt C S, x S. For computational efficiency and practical implementation, adapt the HeteroConv framework to express the R-SSN update at layer for the n-dimensional feature representation:
dim(S)
x!+1 o6 (xxx.
G(rl) wn(X)))
(21)
==0
RER:
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Here, the representation z = P(X.) is obtained through pooling operator P R., - Rd where Kn = |Sp| and D' is the dimensionality of features at layer For each source dimension we denote M [R the number of available relation experts. The soft gating function G: RD + [0, Ml computes normalized scores for expert selection via a top-k mechanism, defined explicitly as:
G(2)) Softmax (eakmmax(TopKggaaee(((()
(22)
where the gating scores are calculated as:
gates(x) = x Wg + Softplus(x Wn),
(23)
with noise (O001) 1) and learnable parameters Wg, Wn E D' that modulate clean and noisy gating scores, respectively. It is well-established that expert selection mechanisms can lead imbalance issues, with certain experts disproportionately favored during training (Shazeer et al., 2017: Bengio et al., 2015). To address this, we adopt soft constraint from (Shazeer et al., 2017), incorporating a regularization term in the training loss to encourage equitable distribution of samples among experts. Specifically, for each training sample we compute the probability P(z, i) that the gating function G(z); 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 gates (z) ). where gates(z) (z) = x W + Softplus(z: W.), to be larger than the k-th greatest gating score, excluding itself, i.e.:
P(x, Pr(gates(x): > kth excluddnngggaaeexx)..kk
(24)
where N 1) and kth kth_ excludxcg computes the k-th largest element of gates(z) excluding the i-th element. Following (Shazeer et al., 2017), we can simplify this to:
P(æ, i) = & (x (x kth, excluding(gates(>> k,
(25)
Softplus Wn);
where 4 is the cumulative density function of the standard normal distribution. We now define the load vector (Xa,(X), i.e., an estimator of the number of samples assigned to each expert given batch X, whose components are
Load(X); = Σ P(r,i).
(26)
Finally, the additional loss term Lioud is:
Liond CVo (Load(X))*
(27)
where (.) computes the coefficient of variation and Apead is hyperparameter that weights the contribution of this term 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 practice (Shazeer et al., 2017).
E.3 COMPUTATIONAL RESOURCES
Experiments conducted on 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 managed using Weights & Biases.
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F COMPUTATIONAL COMPLEXITY
We analyze the effect of refining relations relational message passing architectures, as we are interested in moving from undirected directed (i.e., direction-aware) relations. In these models, separate message is computed per relation instance of given relation type. Formally, let S be finite attributed set (e-g., the set of attributed simplices of semi-simplicial set). The asymptotic complexity is governed by three quantities that may grow with the input: N, the number of elements in S; D, the hidden feature dimension; and E, the total number of relation instances (edges), each corresponding to message. In contrast, P. the number of relation types, is treated as an iopuu--nddpendent constant determined by the model design.
Example (Dir-GNN). Consider directed, attributed graph G = EV, Here, the attributed set is S the set of nodes. The relation set is B Runs Rout corresponding incoming and outgoing edges. Thus, = [B] = 2. While N [V] and E El may grow with the input size, P remains fixed by construction.
Let S be an attributed set of N elements with matrix form X E NND'' and let B be a collection of P relations on S, such that for each relation E. = Σn Fr. Let H be an SSN layer of X = H(X', B) where Xi+1 E N x Di+1 with wn an MPNN-D module (Rossi et al., 2024). Then, its forward pass time complexity
Proposition 3.
a set of En elements and
To = O(ND2 + ED).
Proof. Each relation R E B involves two operations: (i) dense projection of the N node fea- tures, which requires X'WR E [Nx is D (ii) A message-passsiggg step over the edges ER of the relation, each edge transmitting a D -dimensionn message, with cost OlE D(+1), Summing over all P relations yields: Tss = E nes (ONDD'D())) OEE, HUPNDDDD+-EDDEE Assuming D se DI+1 D we obtain:
TB = O(PND2 + ED).
Moreover, if treating P small constant, argue:
Tg ONND2 ED).
Corollary 2. Let S be set, and let and B be two collections of relations on S, such that A is finite refinement by unions of B. Then, T Ts.
Proof. By Proposition 3, Ts = O(ND2 ED). Since A refines B by splitting each coarse relation into at most constant number Q of finer ones, the same summation relations yields T O(QND² + E D). Then, TA = O(N + E D) TB. Thus the refinement does not change the asymptotic forward-pass cost.
Example (Dir-GNN): The forward-pan pass cost of one SSN layer applied to this setup is, by our derived bound:
T O(PND2 + ED) 0(2ND2 + ED).
As is standard in Big-O notation, constants can be absorbed, yielding:
2 + ED).
stated corollary in the proof of our theorem. We highlight that this bound exactly matches the complexity reported for Dir-GNN (Sec. 3, p. 6, (Rossi et al., 2024)), showing consistency between our general analysis and this specific case.
G TOPOLOGICAL DEEP REPRESENTATION LEARNING FOR BRAIN DYNAMICS
G.1 DATA
We build upon a simulation that was run on a Blue Brain Project (Markram, 2006), a biologically validated digital reconstruction of microcireuit in the somatosensory cortex of owo-week-old rat
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(the NMC-model) (Markram et al., 2015) used in subsequent neurotopological studies (Reimann et al., 2017; Conceição et al., 2022; Reimann et al., 2022). The model involves two fundamental components: structural connectivity of the circuit and neuronal binary dynamics. First, structural connectivity is modeled by a directed graph 9 = (V, F.), with neurons represented by nodes V and directed edges (u, v) E F. denoting synaptic connections from presynaptic neuron
postsynaptic neuron v. Second, a collection of binary dynamics B = {B V + {o, } 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 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 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 = 4 (mean + 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 set of 4495 binary dynamics B = {V - {0, () In particular, following Conceição et al. (2022), we focus on the time subinterval At = [10 ms, 60 ms). where spiking activity is mostly concentrated. This interval is subdivided into two 25-ms segments, yielding a set of 4495 binary dynamics B = {V - {0, 1) Here, (v) = indicates that a neuron v is active during the t-th 25-ms segment (with t {0, 1}) in the experiment.
G.2 DYNAMICAL ACTIVITY COMPLEXES
In Section 4, 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-basc based and TDL models. To establish the soundness of this lifting, we verify two funda- mental properties. First, isomorphism preservation: if two dynamic binary digraphs are isomorphic, their lifted representations must also isomorphic. This ensures that equivalent neural dynamics yield identical higher-order structures. Second, consistency with the neurotopological pipeline: 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 GB, or more generally dynamic binary directed simplicial complex Sg, let V denote the set of active vertices at time f, and I the corresponding set of active simplices.
First, the following proposition guarantees that isomorphic digraph dynamics yield identical higher- order co-activation structures.
Proposition 4. Let Gn _ S' then K.g.B = K.g..B
Proof. Recall that two binary graphs are isomorphic if there exists 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 σ, assign the activation pattern
B(o) min (v), .... min (v) E BT. LuEa
(28)
where (v) E {0, 1) denotes the activation of vertex at time t. Since the minimum function min is permutation-invariant,, invariant, any reindexing 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-invarian aggregation function would suffice.
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Second, we show that the lifted structure encodes the full time series of functional complexes as subcomplexes.
Proposition 5. Let Gn be dynamic binary graph and g. its associated DAC. If G[V is the functional digraph with funcfional complex Kight,, then:
Kig.t KGg.
Proof. A simplex belongs to E exactly when vertices are active at time t. Therefore E is closed under taking faces implying that gg. [g.B(>**.] forms subcomplex of gg In particular, simplices of gg [E correspond precisely the cliques of the functional digraph G!.,, thereby Kgu.+ = Kg.8(Et4
G.3 TOPOLOGICAL INVARIANTS FOR DYNAMICAL ACTIVITY COMPLEXES
Let I be dynamic binary digraph and Kg & its corresponding Dynamical Activity Complex (DAC). We describe the dynamic binary topological invariants frequently employed in neurotopological stud- ies of (Reimann et al., 2017; Conceição et al., 2022; Reimann et al., 2022), leveraging their structure and associated face-map-induced relations. These invariants play central role in characterizing the evolving topological structure of brain activity. Given face-map-induced relation I GB or KB. we define its hohop composition restricted to the active simplices at time t as
Rl.ck ((a,T) r) E R | | E "".")
where I. 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 Raym be defined as in Equation 10. Define the functional size of vertex as:
size(u, u, k)) = [ (w) t
counting active k- k-hop neighbors per node each time step, i.e., k-hop synaptically connected active neuron. This invariant, despite its simplicity, effectively distinguishes stimuli and outperforms other invariants in practice (Conceição al., 2022).
Euler Characteristic. For a dynamic binary directed simplicial complex KB the functional Euler Characteristic is defined as the alternating sum:
ec(KCn) = [1 M
n=0
where N is the maximal dimension of K. For a dynamic binary digraph Gn, the functional Fuler char- acteristic of its associated DAC ec coincides with the time series of the Fuler characteristic of the functional flag complexes (X(Kg.+))i= computing the alternating sum of active simplices across dimensions at each time step (see Proposition 6). The variation in the amplitude of the Euler charac- teristic time series of functional complexes effectively characterises stimuli (Reimann et al., 2017). Moreover, it has been reported as a top- lop-performing feature for stimulus classification (Conceição al., 2022).
Proposition 6. Let GB be dynamic binary graph and Kg. its associated DAC. Then,
ec(Kgg,) = (x((gr.))
Proof. By definition co(Kg,,)) co(Kgg,b B)) = (x(Ceu. oF
From Proposition 5 it follows that
Transitive Degree. Focusing on 2-dimensional directed flag complexes (consistent with our experi- mental setup in Section 5), let Co be defined in Section C.5. The transitive degree of vertex is the number of directed 3-cliques (equivalently, 2-simplices) containing e:
ud(v) [(C))()))]]
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the active directed 2-simplices (transitive synaptic triads) containing vertex each time
Graph Level Directionality, Let Rm and Rout be as defined in Equations 8 and 9. he active k-hop in-degree and out-degree of vertex defined as:
indeg(u, u, k) = [| (.a, (w)ll and outdeg(u, k) = [ 1 (w)l
respectively counting the number of active k-hop incoming and outgoing neighbours. The resulting k-hop functional signed degree is:
dir(u, k) indeg(u, k) outdeg(u, k)),
quantifying local asymmetry in neuronal activity (Gove et al., 2021).
High-Order Directionality. Let Riniis be defined as Equation 4. For n-simplex (n 0), define the (i,j)-th k-hop functional signed degree as:
degn.ig(o,k) = [1Rt)))
which relates n--smplices to their lower (i, k-hop adjacent active neighbors. For * j. define:
hodir,,i.jor, k) d (a) (o),...(()..
This invariant was shown to be critical for characterizing structural directed flag complexes in brain connectivity (Riihimäki, 2023).
Reciprocity, Let Rye R n Rout, The k--hpp functional reciprocal degree of vertex u is defined
as:
rc(u, k)
Rl.rk (w)l
counting active vertices simultaneously serving as in- and out-ncighbors of u. This invariant effec-
tively differentiates stimulus classes (Conceição et al., 2022).
Following the procedure described App. D.2, for any simplex- or vertex-level invariant t E {size, dir, hodir, rc, td], we define the corresponding global graph complex-level invariant as
t(Kg, B) = s ((((t()))))v))
where e is permutation-invariian aggregation function.
G.4 TOPOLOGICAL NEURAL NETWORKS AND DYNAMICAL ACTIVITY COMPLEX INVARIANTS
From now on, let denote T {sizo, cc, td, dir, hodir, rc} denote the collection invariants.
Lemma 6. Let KB be DAC with labeled set of simplices S and corresponding binary feature matrix E BT where each row encodes the activation pattern of simplex over T discrete time steps. For every invariant T T there exists subcollection RT RR, of face- induced relations and corresponding operators } RERR such that
T(KB) d( Σ AR(ARX)),
RERT
where o is permutatioo-innaarat invariant function.
Proof. For a fixed relation It consider the matrix product (ARX). The (i,t)-th entry is
[s]
(ARX); = (AR)'
Σ
1(xy==
1=
This sum counts the number active simplices Jj at time t that belong to R(o:). In other words, (AnX); =I faz E R(o,) X = 1} | R!.'(a,) where we define R (o;) the set of simplices in R(a,) with an active feature at time step t. Hence, the i-th row is (A = (1 R& R² l..... R!,T(o)) DT= counting active neighbours of under the relation R across the T time steps. For each relation R. we choose RR to further process the count matrix ARX. We set 2 = Id to be the identity for relations R Ran: Roym, Brey Co,2, Ri ) the negation AR... - /d for the relation
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Rout and Aid, := (-1)* sum (a row-sum with sign adjustment) for id,, 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 RT R be the subcollection of relations relevant for the invariant T. Define the aggregated representation as Z _ LR An(An (.) Since the order of simplices in S is arbitrary, apply a permutation invariant function & (such as the row-sum, row-mean, or row-max) to Z yielding an invariant quantity: T(KB) = (Z)) This completes the proof.
Corollary 3. Let Kg be DAC. There exists a Semi-Simplicial Neural Network (SSN) that computes each invariant in T.
Proof. For each relation R, define wR(X): == ARXW, with W ITXT (the T' T identity matrix). Next, let an operator x act on the output of (x(X) corresponding to the operator Ag. Then, define an aggregation operator + as the summation over the subcollection Rr R necessary for the computation of given invariant: + RER 2 (x(()) Σ RER, AR (A(AxX). Finally, applying
permutation invariant function & to this aggregated output yields the invariant:
T(KB) d
wR(X)).
RERT R
Thus, an SSN structured in this way compute each invariant in T.
(a) g
(b) g'
(c) Gsym
Figure 11: Two directed graphs: (a) 9, (b) g', and (c) their shared symmetrized graph Gsym- Despite g $ g', both are mapped to the same undirected complex under symmetrization.
Our strategy is straightforward: construct non-isomorphic objects that (i) are distinguishable by invariants but (ii) remain indistinguishable to the corresponding architecture due to bounded expres- sivity-or, or, when required, become isomorphic under symmetrization æ (App. C.3). The following lemmas establish this through explicit examples. Throughout, symmetrization implicitly extends to constantly attributed simplices, where permutations of vertices collapse into single canonical simplex with identical constant features.
Lemma 7, MPNNs and MPSNNs cannol compute invariants dir, hodir and rc.
Proof. Directionality. Let V {0, 1, 2} and define two binary digraphs with constant activa- tion B(i = for all E V. Graph G (Fig. 11())) with edges E (0, 1). (0, 2), (1,2) forms a transitive 3-clique with dir(Gn) = (1-2,0, 2, corresponding respectively to source (ver- tex 0), an intermediate vertex (vertex 1), and a sink (vertex 2). Graph G' (Fig. 11(b)) with the edge set E' = {00, 1), (1,2), (2, (2,0)} 0)} forms 3-cycle. Every vertex has equal in- and out-degree, giving dir(G))) (9) = ((g)) (Fig. 11(c)).
((0,0,0)} Thus dir separates G and G', but their symmetrizations coincide:
Higher-Order Directionality: On V = f0, 1,2,3) consider Kgg and Kgr from Fig. 5(b). both with constant activations. Their higher order directional signatures differ, e.g.
hodirz,,.2 - # {{0**)} hodirg...
as Kg contains two directionally consistent 2-simplicial paths, while Kg yields disconnected simplices; yet their symmetrizations coincide, (KGg)) = T(Co;))
Reciprocity. Let GB with edges E {0, 1), (0. 2), (1, (2) and GB with E' {(0, 1), (0,2), (1,2)}- Only 9B contains reciprocal edges, hence re(gn) # re(g))) Still, 7 (Gn) _ (9%)), and the same holds for their directed flag complexes.
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In all cases, symmetrization erases the distinctions. Since MPNNS and MPSNNS operate only on the symmetrized iT-) objects, they cannot compute these invariants.
Recall from App. C.2 that relation R is n-uniform if each element related to exactly a others. Lemma 8. MPNNs and Dir-GNNs cannot compute invariants cc, td, and hodir.
Proof. Consider the binary directed complexes Kg and Kgg Fig. (a))(b), both with constant activation. Each has vertices and 14 edges, but Kgg contains exactly 7 transitive 3-cliques, whereas Kg;, includes none. Hence their invariants differ:
(o(KGe)= ((3*]) ({0*T)) tc(Kgg))
= - # -7 7 14 = co(Kotg)))
=Km * {{0)} = 2..2ir,...2Kgga)
Dir-GNNs propagate only through Rin and Rout. each a 2-uniform relation (every node connected two neighbors). MPNNs, by contrast, operate via Reym. a 4-uniform relation obtained by merging them. With constant activations, such uniformity makes neighborhood multisets indistinguishable. preventing these models from recovering the invariants above.
Theorem 4. Let GB be dynamic binary digraph with corresponding DAC Kgg B. For every invariant T E T, there exists set of face-maa-ineeee man-induccd relations RT C. RRs and a Semi Semi-Simplicial Neural Network SSN as in Equation equation such that:
SSN RT) = TKKg,t))
Moreover, the class of invariants recoverable by SSNs strictly exceeds that of message-ppsssnggg neural networks (Gilmer et al., 2017), directed GNNs (Rossi et al., 2024), and message-passinn simplicial networks (Bodnar et al., 2021b).
Proof. Follows immediately from Corollary 3 and Lemmas and 8.
H ADDITIONAL NUMERICAL RESULTS
SSN Relations. Throughout, we operate with the collection of relations equation 17 on semi- simplicial sets of dimension 2, comprising standard boundary/co-boundary maps together with all directed up/down adjacencies enabling intradimensional communication.
H.1 DYNAMICAL BRAIN ACTIVITY CLASSIFICATION
We provide comprehensive details on the dataset, hyperparameter configurations, ToPoFeat+SVM baseline and runtimes for the main experiments in Sections 5.1 and 5.2. Additionally, we report results for an attention-based variant of SSN and we present numerical evidence demonstrating the robustness of our model under an alternative, non-invariant readout aeating-applicablle exclusively to tasks without induced structural variability, such as brain dynamics representation on fixed neuronal samples.
H.1.1 EXPERIMENTAL DETAILS
Fixed Volumetric Samples. Directed simplices have been shown to be overexpressed motifs in brain networks at all scales (Sizemore et al., 2018; Tadić et al., 2019; Sizemore al., 2019; Andjelković al., 2020). Table 4 presents the structural statistics of the simplicial complexes derived from three representative volumetric samples: (4, 125mm), (4, 325mm) and (8, 175μm) used our experimen- tal evaluation. These statistics highlight the intrinsic structural complexity and high-dimensional
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Cell Type
(4, 11555m)
(1.325mm)
(1754m)
Nodes
600
600
600
Edges
19,209
7,450
10,852
Triangles
62,481
5,322
14,141
Terrahedra
26,450
349
2,240
Pentachorons
1,757
61
Hexaterous
Table 4: Simplex count per dimension for each volumetric sample used the experiments.
organization present within localized regions of the neocortical microcireuit, underscoring the rele- vance of higher-order topological representations in modeling neural computation.
Hyperparameter Configuration. For all tested models (SSNs and competitors), hyperparameters were optimized as follows: number of layers E {2, 4, 6}: hidden dimension E {16. 32, 64) for non-topological models (GNN, Dir-GNN), and {16, 32) for topological models (MPSNN, SSN). Additional settings were fixed across all models: dropout rate of 0.3; inner aggregation set to sum; outer aggregation set to mean; batch size of 16; Adam (Kingma Ba, 2014) optimizer with learning rate of 0.001; early stopping with patience of 25 validation steps; and validation performed at every training step. The best- pest-performing configurations across five splits are reported in Table 5, 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).
Model
Hid Dim
Layers
Params
Par. Ratio (%)
DS
2
1,680
2%
DS-256
256
2
70,672
68%
GNN
2
5,392
5%
GNN-256
70,672
Dir-GNN
2
9,744
9%
Di--GNN-256
256
2
137,232
133%
MPSNN
23,888
MPSNN-61
4
90,768
88%
R-SSN (Ours)
32
6
18,084
18%
SSN (Ours)
103.184
100%
Table 5: Model architecture details. Parameter counts (# Params) and relative percentages (Par. Ratio %) are reported compared to SSNs.
Runtime. Table 6 reports the average runtime per epoch for both validation and training. On larger
complexes, the routing mechanism R-SSN proves advantageous: for the (4, 125, jum) case (=s 60k: simplices). R-SSN requires only about 70% of the time of standard SSN, while for (8, 175, jm) (N 27k simplices), the time is reduced to 88%. For the smaller (4, um) complex ( 13k simplices), by contrast, the validation speedup is marginal, and during training SSN is slightly faster than R-SSN. This runtime benefit stems from the fact that the number of active experts (and, consequently, active relations) is decided advance and can be significantly less than the number of relations present in SSN even at training time. As result, overall computation is reduced compared to the full model. Moreover, in backpropagation only the gradients of the weights tied active experts need to be updated, i.e., those that actually contribute to the output of the network. At the same time, R-SSNs introduce a small extra cost due to the gating mechanism parameters. The net advantage arises only when the savings from discarding relation experts outweigh the overhead of gating. In practice, this effect depends on complex size: for small complexes, eliminating relations may not fully offset the added ve 5.8k parameters of the gating mechanism. For larger datasets, however, the reduction in computation is substantial, showing that R-SSN scales more efficiently and provide meaningful training speed improvements as the number of simplices increases.
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Valid. Avg. Runtime (seepoch)
Train Avg. Runtime (veepoch)
Model
(4. (4,12)
(1, 325pm)
(8, 1755mm
[4. 125pm)
(4. (1,52m
(8,17))
DS
23.37 0.09
8.42 0.04
9.60 0.17
94.18 + 0.46
39.56 1.50
33,96 + 0.32
DS-256
23.68 0.07
8:46 0.03
94.85
39.40
33.92
GNN
21.35 0.05
7.87 +.1.5
0.26
32.08 0.56
35.70
GNN-256
22.04 1: 0.41
7.66 0.19
8.19
104.94 0.77
33.16 = 0.15
42.46 1: 2.01
Dir-GNN
21.49 + 0.01
0.13
8.90 0.13
86.18 + 0.10
32.17 = 0.53
3641 + 0.32
Dir DR--GNN--56
22.50 1 0.49
7.79 0:032
8.39 0.20
103.6 1: 2.3
33.75 1: 0.07
40.23 1: 0.84
MPSNN
37.18 0,25
9,98 0.05
12.38 + 0.06
190.5 0.8
44.71 0.13
61.08 + 0.15
MPSNN-64
52.36 0.17
11.12 0.16
15.60 0.19
298.8 1: 0.48
53.35 0.48
81.60 1: 0.19
(Ours)
52.18 0.15
9.98
300.3
$2.68 0.51
91.22 0.39
R-SSN (Ours)
37.79
9.30 0:099
54.18 0.63
80.93 0.51
Table 6: Avg. runtime per validation epoch (left) and training epoch (right) for different model configurations.
H.1.2 ADDITIONAL RESULTS.
Topological features baseline. To assess the impact of end-to-end feature extraction, we com- pare with linear SVM on feature vector of precomputed topological invariants defined in Appendix G.3. For each DAC, we compute the following invariants: Euler characteristic (ec), Graph- level directionality (dir), Neighborhood size (size) and higher-order directionality (hodir), computed on the (0,1) relation for edges and on the (0,1), (1,2), (0.2) relations for triangles. Node-level invari- ants (size, dir, hodir) are summed to obtain complex-level values. The dir, size and hodir invariants can be computed for different neighborhood orders K. All features are computed across two time bins and concatenated, resulting time bins, has size 1, size and hodir have size K and hodir has size 4K. The parameter count for the SVM classifier is based on the adopted one-vs-al as-ll classification strategy, which trains one SVM for each class. Each linear SVM has 2(6K + 1) parameters (i.e., the input feature size plus one), which leads to a total of 16(6K' + 1) + parameters. In Table 3 we report the number for K which leads to the best performance. The results in Table 7 show that increasing K leads to better classification performance, corroborating the importance of considering higher-order relations in the complex.
feature vector of size 2(6K' + 1) per sample, where 2 are the
1255mm;
325mm)
(8,17ppm)
29,11
a
27.94
0.94
2697
0,97
27.60
0.38
39.60
0.82
35.18
1.68
43.22 1 1.28
27.63
0.89
27.23 1 0.97
28:44
0.27
27.76
28.86 0.42
Table 7: Accuracy for the SVP baselines for varying K (%, higher is better t). The top 1 2nd and 3rd results are highlighted.
Attention-based SSN. We further evaluate SSNs and baselines by using GAT message-passing scheme (Veličković et al., 2018) as wn in equation instead of GraphSAGE (used in Table 3). Table 8 shows that SSN largely outperforms all baselines also in this configuration on the (4, 325mm) and (M = 3) datasets, corroborating its improved capability to leverage higher-order directed connectivity information under different messagg-passing schemes.
Model
(4, (4:320mm)
GAT
24.42
0.62
DAT
42.03 0.31
28.59
0.62
MPSNN-GAT
SSN-GAT (Ours)
37.45 3.58
$1.27
1.85
135423%
11.7.2%
Table 8: Accuracy for GAT peessage-passing scheme (%, higher is better 1). The top 1s, 2nd and 3rd results are highlighted. Gain reports the absolute accuracy improvement (1%% of our model relative to the best performing baseline.
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H.1.3 NON-INVARIANT READOUTS.
In this work, we develop model capable of robustly processing arbitrary localized structural regions within neural microcireuits, 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 microcireuit 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 indexing across samples, we employ permutation-invarian readouts. Remarkably, our model demonstrates robustness to shufiled spike trains 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.
Model
(4,125m
(4, 325mm)
(8, 175mmm
DS
86.65 + 1.06
92,26 + 1.07
88.72 + 0.82
GNN
77.67 + 1.96
87.88 + 0.45
84.98 + 1.43
Dir-GNN
85.07 +: 0.91
90.94 +: 0.81
87.42 + 1.43
MPSNN
86,95 + 0.52
92,09 + 1.29
88,72 + 1.20
SSN (Ours)
87.63 + 0.43
92.02 +: 0.82
88.98 + 1.10
Gain
0.68%
J. 1.0.4%
1 0.26%
Table 9: Binary dynamics classification results (%, higher is better 1) across volumetric samples. The top 1st 2nd and 3rd results are highlighted. Gain reports the absolute accuracy improvement (t) or drop (J %) of our model relative to the best performing baseline.
Hyperparameter Configuration. For all evaluated models (SSNs and baselines), hyperparam- eters were optimized over the following grid: number of layers E {2, 4, 6). hidden dimension {16,32, 64), and dropout rate {0.3, 0.8). The following settings were fixed across all mod- els: inner aggregation set to sum, outer aggregation to mean, batch size of 16, and the Adam optimizer (Kingma& & Ba, 2014) with learning rate of 0.001. Early stopping was applied with 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 9.
Results. In this setting, all baselines perform significantly better than in our main experiments (see Table 3), 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 4). Moreover, methods that rely on a fixed neuron ordering are intrinsically limited those specific samples, making the models in Table strong baselines only in these constrained scenarios.
1.1.4 INCREASED TEMPORAL RESOLUTION AND VARIATION IN VOLUMETRIC SAMPLING
We provide additional numerical results for Section 5.2, evaluating two alternative scenarios: (i) increased temporal resolution (T = 4 time bins), and (ii) variation in volumetric sampling.
Experimental Settings. We extend our evaluations in the more challenging data-scarce setting (N-
1), exploring two additional regimes: component (4, 325mm) with increased temporal resolution (T'=4 4 time bins), and component (4, 125ym) 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 10 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
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Model
125
T
DS
22.06 0.26
23.13 0.39
DS-256
21.62 0.61
23.65 0.36
GNN
24.45 0.63
22.87 =: 0.49
GNN-256
24.03 0.60
22.36 0.18
Di-GNN
Dir GNN-266
25.03 1.35
22.67 0.35
MPSNN
25.88
0.66
24.79 =: 1.21
MPSNN-64
26.44 1.26
24.43 0.34
SSN (Ours)
46.30 2.11
39.14 8.09
Guän
19.86%
14.35%
Table Binary Dynamical Complex classification results (%, higher better t). The top 1s 2nd and 3rd results are highlighted. Absolute accuracy Gain over the second-best model are also reported.
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.
H.2 EDGE REGRESSION TRAFFIC TASK
(a) Riy.0.0
(b) Rag
(c) RAg.O.1
(d) R
Figure 12: Examples of the four lower directed edge adjacencies used for SSN in the experiments on edge flow prediction. The first one (a) connects each edge to those sharing the same destination node; the second one (b) connects each edge those sharing the same source node; the third one (c) connects each edge to those leaving from its destination node; the fourth one (d) connects each edge those entering in source node.
Model
Anaheim
Barcelona
Chicago
Winnipeg
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
Hidge+Dir
0.081
0.146
0.110
0.159
EIGN
0.130
0.061
0.090
SSN (Ours)
0.077
0.130
0.045
0.116
Gain
0.008
0.000
0.001
1 0.026
Traffic dynamics regression results (RMSE, lower better J). The top 1s, 2nd and 3rd results are highlighted. Absolute Gain over the best-performing baseline model is also reported (negative J better).
Table 11:
We evaluate our model on traffic assignment problems using traffic datasets from (for Research Core Team, Accessed 06.08.2000 20.244 (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 (Patriksson, 2015). 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 edge data, making the problem topological. We frame the task traffic simulation
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problem following Fuchsgruber et al. (2025): given the traffic flows 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 (Fuchsgruber et al., 2025). In detail, we define the following edge features (if available); capacity, length, free flow time (i.c., travel time with 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 (Fuchsgruber al., 2025), if street is traversed in both directions, we do 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 To compare our results with those in (Fuchsgruber et al., 2025), where undirected edges have 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 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:
Lrain aLdir + (1 11-0 adffff
(29)
where Lair a regression loss (e.g., MSE) on the directed flows, Laift is the loss between the predicted and ground truth flow differences for undirected edges and a E [0, 1] is a hyperparameter. We test our SSN using different combinations of the 4 directed edge adjacencies illustrated in Figure 12, and we report results for the best one consisting of adjacencies (b) and (c) in Figure 12 plus the undirected edge adjacency consisting of the union of (b) and (c). We compare with six baselines:
MLP; LineGraph, spectral GNN applied node-level; HodgeGNN (Roddenberry & Segarra, 2019), based on the edge Laplacian; Hodge+Inv, a variant of HodgeGNN modeling orientation-invariant features as orientation equivariant; Hodge+Dir, variant of HodgeGNN that treats all edges as directed; EIGN (Fuchsgruber et al., 2025), 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 (Fuchsgruber et al., 2025) as well as the experimental setting.
Hyperparameter Configuration. We selected the hyperparameters through grid search among the following values: convolution strategy E. {GCN, SAGE), batch normalization E (TTue, False]. hidden size E (32, 64, 128], a E {0, 0.5,,,,, number of layers E {1, 5, 9). We fixed the following other parameters for all models: inner aggregation set to 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 12.
Results. Table 11 shows the MSE the edge flow prediction for SSN and baselines. SSN outper- forms all baselines on Chicago and matches the best persfoerforinn 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 improvement the importance of considering directed adjacencies on this task.
undirected approaches which, again, validates
Dataset
Hid dim
# Layers
Conv.
BN
Anaheim
32
9
GCN
False
1
Barcelona
64
5
GCN
True
Chicago
64
5
SAGE
False
0
Winnipeg
64
9
GCN
False
0.5
Table 12: Model architecture details for edge-level datasets.
H.3 NODE CLASSIFICATION
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-mald particularly challenging for GNNs. Rossi et al. (2024) showed that modeling graphs directed
these settings
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Dataset
Type
Path Length
# Rt-Paths
(%1,0,0%%)
(%)
(o.)
(1)
355,808
57.50
12.05
12.05
18.41
Cora MT.
Homophilic
31,672.710
93.99
0.70
0.70
4.60
Citeseer
Homophilic
76,344
77.22
6.46
6.46
9.87
2.665.146
98.06
0.23
0.23
1.48
222.096
14,90
30.23
30.23
24.65
Roman-Empire
Heterophillic
1,384,820
7.93
27.29
27.29
37.50
Table 13:
Higher-order edge directionality statistics for the Cora-ML, Citeseer (Bojchevski & Günnemann, 2018) and Roman Caman--mpire (Platonov et al., 2023) 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-MI. and Citeseer exhibit predominantly homophilic patterns, with directionality concentrated in edge pairs sharing common source target, and markedly lower prevalence of fully directed paths.
Model
Roman Empiire
Model
Cora-MI.
GuguSAGE
91. 0.27
GNN
87.06 1: 1.47
94.34 0.56
Di-CINN (2024)
91. 0.32
Dir-GNN
86.60 + 1.43
94.09 + 0.48
Polynomer
92.55 01.77
MPSNN
86.58 :: 1.41
94.14 :: 0.83
MPSNN
0.63
SSN (Ours)
86.64 + 1,30
94.52 + 0.53
SSN (Ours)
93.52 0.28
Gain
0.4.2%
0.188 %
Gain
1 0.96 16
Table 14: Node classification results (accuracy %, higher is better 1) on the Jooann-Emiing dataset (Left column) (Platonov et al., 2023) and on Cora-ML and Citeseer (Right column) datasets (Bo- jehevski Günnemann, 2018). The top two performing methods are highlighted as follows: 1st and 2nd Gain reports the accuracy improvement (1 %) drop (J 9%) of our model relative to the best performing baseline. In the Roman-Ein Eamare table, the 2nd entry corresponds to the previous state-of-the-an method.
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 Ronan-Empire dataset, whose distinctive chain-like topology makes especially well- suited for higher-order directed modeling. We evaluate our method on this heterophilic benchmark, alongside two widely used homophilic citation graphs: Cora-ML and Citescer (Bojchevski & Günnemann, 2018). Our empirical findings align with the conclusions of Rossi et al. (2024): directionality whether higher-order or not-offerr minimal gains in traditional strongly homophilic benchmarks but yields substantial improvements in highly directed and heterophilic Roman Roman-Empire dataset (see Table 13). In particular, we observe that higher-order directionality plays a key role in boosting performance in such settings, setting state of the art on the Roman-Famire dataset (cf. Table 14).
Roman Empire (A Chain-Like Graph). The Goman-Empire graph of Platonov al. (2023) 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 R 0.05), enriched with shortcut edges that capture long-range grammatical dependencies and characterized by a chain-dominate topology featuurin the smallest average per-node degree (2.91) and largest diameter (6824) among commonly used benchmark datasets. Table 13 quantifies the higher-order edge directionality: at path length 1. fully directed edge pairs (RL1...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 Semi-Sine mimpiiiimliaa Neural Network (SSN). While traditional undirected topological models inherently neglect directionality, and recent approaches like Dir-Gi Dir-GNN (Rossi et al., 2024) leverage only first- order directional cues, SSN explicitly captures higher-order directed simplicial motifs, effectively modeling long-range syntactic chains and dependencies.
an ideal stress test for our
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Baselines. For the Roman-Empire Empire dataset, benchmark against several strong baselines: Dir- GNN (Rossi et al., 2024), which previously established the efficacy of direction-aware aware models; Graph- SAGE (Hamilton al., 2017), with extensive hyperparameter tuning by Luo et al. (2024)- the most competitive configurations for undirected graph-based methods; and Polynormer (Deng al., 2024), Graph Transformer (GT) that currently holds state-of-the-art performance. The tuning methodology in (Luo et al., 2024) aligns with that of Polynormer (Deng et al., 2024). Previous results are taken directly from their respective works, Additionally, we evaluate MPSNN (Bodnar al., 2021b), enabling 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 (Hamilton et al., 2017), Dir-GNN (Rossi et al., 2024), and MPSNN (Bodnar et al., 2021b), as detailed in Table 14.
Experimental Setup. For the Roman-Empire dataset, we adopt the data splits from (Platonov et al., 2023). For Cora-ML and Citeseer, we use 10 random splits with a 50/25/25 rrin-validation-tess ratio, reporting mean accuracy and standard deviation across splits (see Table 14). 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 Co- We propagate embeddings across four directional edge relationships (R R11,0.1. R11,100 and R11,1,1), updating node embeddings using boundary operator B1.
Hyperparameters. Consistent with (Rossi et al., 2024) and (Luo et al., 2024), we employ concatenationn-bssee Jumping Knowledge. Differing from (Luo al., 2024), who utilized hid- den dimensions of 256 (GraphSAGE) and 512 (GAT and GCN), we uniformly set a smaller hidden dimension of 128 and utilize BAGEEllike aggregations (WR) across all five relations. Hyperparameter search covers the number of layers (5. 7,9 9), dropout rates (0.3, 0.5, 0.7). J. inner aggregation as max, outer aggregation as sum, and the Adam optimizer (learning rate 0.01).
Results. Table 14 demonstrates that our SSN sets a new state of the art on the Roman-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 (Luo et al., 2024) 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- persfrrerormnn method on Cora-ML. These results align with the findings of (Rossi et al., 2024), 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 13, which reveal that message passing in both Cora-ML and Citeseer predominantly occurs along edges with shared sources or targets a structural hallmark of citation networks. This is accompanied by a marked collapse 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.
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Reviews:
[{"review_id": "nGYAHhY4ZR", "reviewer": "Reviewer_DBfX", "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.\nThe 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).", "questions": "- Which face-map–induced relation classes (e.g., interdimensional, intradimensional, directed adjacency) contribute most to performance?\n\n- The framework mainly uses binary activations for brain data. How would SSNs handle continuous-valued features or multimodal signals?\n\n- How will the routing behave under noise or shuffled connectivity?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well written and easy to follow.\n\n- The paper provides strong theoretical results on WL-expressivity, permutation equivariance, and invariant recovery, establishing clear theoretical advantages over prior models. \n\n- 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. \n\n- 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. \n\n- The Routing-SSN mechanism provides a practical path to scaling directed topological models while retaining expressivity and efficiency.", "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.\n\n- 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.\n\n- 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.\n\n- 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.\n\n[1] Joshi, Chaitanya K. \"Transformers are graph neural networks.\" arXiv preprint arXiv:2506.22084 (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "eQuCTpWxBz", "reviewer": "Reviewer_jgH5", "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.", "questions": "Can the proposed SSNs be generalized to combinatorial complexes with cells of dimension greater than two?\n\nCan DACs be extended from binary activation to real-valued activity?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 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.\n\n- 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. \n\n- 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.", "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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\n[1] Higher-order connectomics of human brain function reveals local topological signatures of task decoding, individual identification, and behavior (Nature Communications, 2024)\n\n[2] Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs\n\n[3]Topological Deep Learning: Going Beyond Graph Data\n\n[4] TopoTune: A Framework for Generalized Combinatorial Complex Neural Networks", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6DWipEt6GW", "reviewer": "Reviewer_7KeL", "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.", "questions": "1. How should practitioners pick the top-k in Routing-SSN beyond grid search?\n2. Can routing scores or other signals identify which directed motifs drove a prediction?", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper has a well-motivated focus on directionality and multi-way interactions, important for brain data.\n- General framework that subsumes GNNs, directed GNNs, and simplicial networks.\n- Theoretical results on WL-expressivity and recovery of neuro-topological invariants.\n- Strong empirical gains on brain-dynamics classification.\n\nOverall, 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.\n\nThat 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.\n\n**References:** \n\n[1] Cin++: Enhancing topological message passing. Giusti et.al. 2023\n\n[2] Cycle invariant positional encoding for graph representation learning. Yan et al. 2024\n\n[3] Attending to topological spaces: The cellular transformer. Ballester et. al 2024\n\n[4] Position: Graph learning will lose relevance due to poor benchmarks Bechler-Speicher et al., 2025", "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.\n- 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.\n\n**References:**\n\n[1] Cin++: Enhancing topological message passing. Giusti et.al. 2023\n\n[2] Cycle invariant positional encoding for graph representation learning. Yan et al. 2024\n\n[3] Attending to topological spaces: The cellular transformer. Ballester et. al 2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)