Optimal Graph Clustering without Edge Density Signals
Authors: Maximilien Dreveton
Elaine S. Liu
Matthias Grossglauser
Patrick Thiran
Venue: NeurIPS 2025 poster ()
Keywords: community detection
unsupervised learning
stochastic block model
spectral clustering
Primary area: general_machine_learning
TL;DR: We establish the theoretical limits of clustering under the Popularity-Adjusted Block Model and show that even without edge-density signals, cluster recovery is possible by leveraging differences in intra- and inter-cluster popularity parameters

Abstract: This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches.

Optimal Graph Clustering without Edge Density Signals
Maximilien Dreveton
Elaine Liu
EPFL
EPFL
maximilien wtt.:iiin.ndreeett@@gaaall. com
elainelludstannfrd.cco edu
Matthias Grossglauser EPFL
Patrick Thiran EPFL patrick thiiineetiirr ch
matthias, matthias. bra.hhaa..gggosgggguurrrreppp...
Abstract
This paper establishes the theoretical limits of graph clustering under the Popularity- Adjusted Block Model (PABM), addressing limitations existing models. con- trast the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Correche Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution the characterization of the optimal error rate for clustering under PABM. which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recov- ery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability indepen- dently of global edge densities. Finally, because PABM exhibits richer structure, its expected adjacency matrix has rank between & and k22, where is the number of clusters. As result, spectral embeddings based on the top eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating 2 eigenvectors outperform traditional spectral approaches.
Introduction
Graph clustering the task of partitioning the vertex set of a graph into non-overlapping groups such that vertices within the same group exhibit similar patterns properties. As fundamental task in the statistical analysis of networks, graph clustering plays key role revealing the underlying structure and functional organization of complex networks (Avrachenkov and Dreveton 2022).
Most graph clustering algorithms are based on the assumption that vertices within the same cluster more densely connected than vertices in different communities. In other words, intra-cluster edge-density higher than inter-cluster edge-density. Under this premise, metrics such as modularity, graph cuts, their variants commonly used to motivate and design graph clustering algorithms. However, these methods fundamentally rely on the edge density as their primary input signal. This leads to natural question: Is edge density essential for recovering clusters, signals be exploited instead? In this work, we demonstrate that the connection patterns individual vertices be exploited to recover clusters, even when intra-cluster and inter-cluster edge densities are equal.
other structural
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Random graphs with cluster structure: block models with and without degree heterogeneity. Random graphs with cluster structure are often modeled using block models. Let 2 E [k"" be a vector representing the cluster assignments of each vertex. For all the random graphs that we consider, the adjacency matrix
{0, (0.1)*xm assumed to symmetric with zero diagonal and As Agi ~ Ber(P.)) for all > J. where P.j E [0, 1] is the probability of an edge between vertices and j. The simplest block model supposes that
[P = Po 1 otherwise.
(1.1)
This model is often called the planted partition model, the stochastic block model (SBM) with homogeneous interactions A known drawback of this model is that all vertices share the same expected degree. To mitigate this issue, Karrer and Newman (2011) proposed the degree-corrected block model (DCBM), where
SH ====2
Pg =
(1.2)
0,0,9 otherwise.
The quantities .,,-.. On are the corrececorrection parameters. To ensure identifiability, these pa- rameters are normalized such that denotes the size of cluster
0g = (z) for all a E [At], where na = =Ki: 2; = a}l
However, the degree-correction parameter 0g uniformly inflates or deflates the connection probabilities of vertex i across all clusters. As a result, vertices with a large pnggre-coorection parameter have more edges both within their own cluster and with other clusters. This makes impossible to model vertices that exhibit higher connectivity exclusively within their own cluster. To mitigate this issue, Sengupta and Chen 2018) introduced the popularity adjusted block model (PABM), where
in AITD
2 = 2z+
PJ =
otherwise.
In this model, the quantity X!n (resp.. Agut) is the popularity of vertex with other vertices within its own cluster (resp.. with vertices in other clusters). These coefficients are normalized such that E: a:ma ATT (z) and Ei a, Aque = no (=) for all a E [A]. This model allows for a vertex to be highly popular among its cluster (high A,"),, but to be not necessarily popular (Aput = 1) or even be very unpopular (small Aont) with vertices in other clusters.
Optimal clustering error rate: from edge-density to popularity patterns An important question to assess the difficulty of the clustering task in a block model is the derivation of the optimal error rale. By optimal error rate, we refer the minimum possible error that the best algorithm achieves when attempting to recover the true cluster assignment of all vertices. This error rate is typically measured in terms of the misclassification rate rate-thar that is. the proportion of vertices incorrectly assigned to their true clusters, up to a permutation of the labels. The optimal error rate reflects the information-theowe theoretic limits of the clustering task, because characterizes how well one could possibly do even with unlimited computational power, given the amount of signal and noise in the data. It also provides benchmark evaluate existing algorithms and guides the development of new methods that approach (either theoretically or empirically) these theoretical limits.
Studying the effect of the different model parameters (such sparsity degree heterogeneity) on the error rate offers deep insight into the fundamental difficulty of the graph clustering problem across different network settings. Consider SBM with k clusters of same size n/ and homogeneous interactions as in [I.I. When 1/n < p. < 1, the optimal error rate is asymptotically (Zhang and Zhou 2016)
exp (- (VD (VD-v7)))
v
'A block model is said to have homogeneous interactions the entries P. depends only on whether & 2, or 2 * Zy: otherwise, the model is said to have heterogeneous interactions. Our work focuses on models with heterogeneous interactions, with homogeneous interactions treated as special case. However, for simplicity, in the Introduction present results only for the homogeneous setting.

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As and represent the intra-cluster and inter-cluster edge densities, respectively, the key quantity (vP vq)² in the expression above captures the influence of edge density: the larger the gap between p and g. the easier it is to recover the clusters.
Next, consider DCBM with & clusters of same size n/k and homogeneous interactions in (.2) Under some technical conditions on the degree-correction parameters, Gao et al. (2018) establishes that, when p. o(1) with p/q- O1) and p w(1/n), the optimal error rate is asymptotically
1 Σ exp (-0
0: 12 (Vp Va)"
Compared to the standard SBM, the difficulty of clustering now varies across vertices and is quantified by the term exp 10 /D /g)² /k), which depends on each vertex E [n] and is monotonically decreasing in 0g. The optimal eтror rate corresponds to the average of these quantities over all vertices. This highlights the effect of degree heterogeneity: vertices with larger expected degree are easier to cluster, as their neighborhoods contain more information.
However, the same key quantity (Vr- vT)² representing the edge-density signal shows up in the DCBM error rate. Indeed, as mentioned earlier, the degree-correction parameters uniformly inflate or deflate the connection probabilities. As result, the value of 0; impacts the clustering difficulty of vertex in a predictable and monotonic way. This no longer holds in the PABM, which introduces a richer and more nuanced structure. The first major contribution of this work is characterize the optimal error rate for clustering under the PABM. As the general expression is somewhat involved, we begin with the simplest case of clusters of equal size. In this setting, we establish that the optimal error rate is given by
2
Σ jE(n)
in NinNjo
Aput yout
exp
As in the DCBM, the error rate PABM is expressed as an average over the difficulty of clustering each individual vertex. However, in PABM, these per-vertex difficulties have more intricate form, and we provide further insight in Sections 2.2 and 2.4 A particularly important observation the following: suppose p- g, that the expected numbers of intra-cluster and of inter-cluster edges are equal. In this case, the SBM and DCBM reduce to the Rénōs-Rééyi and Chung- Lu models, respectively, and cluster recovery is fundamentally impossible. Remarkably, this is not true for PABM: cluster recovery may still be possible provided the popularity coefficients AIT and Agnt are different. This reveals novel aspect of degree heterogeneity captured by PABM but missed by DCBM: local differences in intra- and inter-cluster popularity enhance the separability of clusters, even when traditional global edge-density signals vanish. Another phenomenon, more subtle, occurs in PABM: the optimal error rate is not monotonically increasing when the number of inter-cluster edges increases. We rigorously establish these phenomena in Examples D and 2 and illustrate them in our numerical simulations.
Higher-order eigenvectors for clustering with popularity patterns Finally, we perform numerical experiments to evaluate the effectiveness of spectral clustering methods. When the adjacency matrix A is sampled from a block model, it can be decomposed as A = P + +X, where P is a low-rank matrix encoding the underlying structure, and X is random noise matrix with zero-mean sub- Gaussian entries. This decomposition forms the basis of spectral methods for graph clustering, where the general approach is to apply clustering algorithm (such k-means) low-dimensional emhedding derived from a low-rank approximation of A.
In classical models like SBM and DCBM, when p * q. the rank of P is equal to the number of clusters However, in PABM, the situation is more complex: the rank of P can be greater than k. but cannot be greater than 62 This implies that embeddings based solely on the top-k eigenvectors may miss important structural information. To address this, recent works propose spectral algorithms that incorporate k² eigenvectors to better capture the richer structure of PABM (Noroozi et al. 2021 Koo et al. 2023). Our numerical experiments demonstrate that these methods outperform traditional spectral approaches that rely only on k eigenvectors, both on synthetic and real datasets.
In the numerical section, we illustrate two surprising results discussed in the theoretical section: the non-monotonic behavior of the error with respect to edge density, and the ability to recover clusters
3

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even when p = q. While it would have been possible to use greedy algorithm to approximate the MLE, we opted for spectral methods because of their widespread use and of their well-established effectiveness for clustering in block models. The experiments demonstrate that the phenomena highlighted in the theoretical section also arise when using spectral algorithms. They show that these behaviors are not merely mathematical artifacts stemming from the increased complexity of PABM relative to DCBM, but that they do occur in practice and are observable in real-world settings.
The paper is structured as follows. We derive the optimal error rate in PABM and provide some examples in Section 2 We present our numerical experiments in Section 3 We discuss some related works in Section Finally, we conclude in Section
Notations Ber(p), Exp(A) and Uni(a, b) denote the Bernoulli distribution with parameter p, the exponential distribution with parameter X, and the uniform distribution over the interval [a. bj. We use the Landau notations o and O, and write f ((g) when g = o(f) and f (Z(g) when g = Of)).
2 Optimal Error Rate in Popularity-Adjusted Block Models
2.1 Model Definition and Parameter Space
We consider n vertices partitioned into 2 disjoint blocks. The partition is encoded by a vertex- labeling vector 2" = (=i,... ... ,x,)) E [K]"* so that 2 indicates the cluster of vertex i. These n vertices interact pairwise, giving rise undirected edges, and these pairwise interactions are grouped by symmetric matrix A E (0, 1)axn called the adjacency matrix. The Popularity Adjusted Block Model supposes that, conditionally on the block structure, the upper-diagonal elements (Asz)is are independent Bernoulli random variables such that, conditionally on a and zj.
Au
I 27,25 ~ Ber (pndiez Ase; Ajt;Bet3] Bet=;)
(2.1)
where (M)))E are the popularity parameters and B E B is the connectivity matrix across clusters. The parameter Pn. controls the graph sparsity, as the average degree is of order npn when the following assumption is made.
Assumption 1. The quantities Bub and dia are constant (so they do not scale with for all i E [n] and a, b E [k:]
Given a realization of PABM, we aim to infer the latent block structure 2", Let & = ((A) be an estimate of 2², and define the clustering error as
loss (2", (2",2)
min Ham (2", TT 2), rESym(k)
(2.2)
where Sym(k) is the set of permutations of [A] and Ham(..) is the Hamming distance. We interested in the expected loss of an estimator, namely E [loss(2 2=, {(A)),, where the expectation is taken with respect to the random variable A sampled from (2.1
2.2 A Key information-Theored Divergence
For any E [k]", denote Piş(=) Prddiz Ajz, Brazy. To understand the difficulty of correctly clustering given vertex i, we introduce an alternative cluster labeling ge E [K]] such that zje 2] for all j A i, while 2/-6
[k] \ { In other words, the cluster labeling z' agrees with for all vertices except for i, which is placed in cluster instead of being in cluster 2²., To shorten the notations, let p = P(=*) and Pise = P(zaa) The difficulty of correctly recovering the cluster of vertex i depends on how hard it to statistically distinguish whether the observed graph was generated from the true model Pa or from the alternative model pio This a classical hypothesis testing problem: the more similar the distributions induced by p and pia, the less distinguishable two graphs drawn from these two models are, and thus the harder it is to infer the correct cluster assignment for vertex i. The statistical difficulty of this test is quantified by the Chernoff divergence A(i, a), which measures the exponential rate at which the error probability decays when testing

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between these two competing models. More precisely,
A(i, = max (1 - Ren, (0(0)))
X jzL
Ber
Pii
X Ber (Pi))
(2.3)
f
where Ren, the Rényi divergence order t. Moreover, by using the linearity of Rényi divergence with respect to multiplication and the sparsity of the model (that is, P.j o(1) for all i,j), we have
(i, (1 o(1))
max (E(0.1)
Σ J#1
( (1- (P (Pt))))
Among all alternative models pia the most challenging to distinguish from the true model p the one with the smallest Chernoff divergence A(i, a). We thus define
Chernoff(i, = min A(i,a) a#
which captures the hardest hypothesis testing problem associated with recovering the cluster of vertex i. Intuitively speaking, the larger the value of Chernoff(i, (i), the easier it is correctly recover 27, as all alternative models defined above are sufficiently different from P The following assumption asserts that for every E (n]. the quantity Chernof[ =*) is unbounded. This assumption is necessary to ensure that the recovery of 2 is asymptotically possible.
Assumption 2. Suppose that min4e]u] Chernoff =*) (1).
2.3 Main Result: Optimal Error Rate in PABM
For any [k]" denote by na(z) Eie(n) 1f2s a} the size of the cluster such that E. Fa and define
[k]. Let [0, 1]
6n = 2
{ [K]": (2(2) E [[1 e) (1 t)ma]
[ Va E [m]]
Let A (Naa)iE()).e(() be a matrix with non-negative coefficients such that ||A 1 = noa and kxc be matrix of full rank.
B E B
Theorem 1 (Lower-bound the clustering error). Let x* Zn( E) and A being sampled from 2.1 Suppose Assumption 2 holds. Then, there exists some o(1) such that
1+9
inf E [loss(=", 2)] 2
(1 e) mina Ta
Cheroo((),2"))
7
iE(m)
where the inf laken over all estimators & 2(A).
Theorem (Achievability). Let x" E (x and being sampled from tions 1 and 2 hold. Then, there exists an estimator such that
2.1) Suppose Assump-
14n
[loss(=",)) S
Chernoff(i,z")
rE(m)
for some 7 = o(1)
The gap between the lower bound (Theorem I and the achievability (Thcorem stems only from second-order terms. Indeed, the sequences appearing in Theorems and are not identical. Moreover, the multiplicative factor mine) of constant order can be absorbed into the sequence n, as the term EiE(n) e Chernof((i."")) vanishes as n > oc. We chose to display this factor explicitly in our bounds so that the sequence does not depend on the parameter e.
We show in Section 2.4 how Theorems 1 and recover known results in SBM and DCBM, and in Section 2.5 how they reveal novel properties that did not exist previous models, when they are specialized to PABM. Table summarizes all three classes of block models considered in this paper.
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SBM
DCBM
PABM
Popa
0,03popm
x AJ PoPn Xq 23 9upa. otherwise.
if 21 2j.
Homogeneous
P.j
Poj
90pa
0,0,9upm
Heterogeneous
Pa= Po
P. O,,,
Ps Aiz, Aja, Br
Table 1: Different expressions of the elements of the matrix P for the block model variants considered in this paper. The quantity Pr is related to the graph sparsity (as the expected degree is of order np,)). All other quantities are strictly positive and independent of n.
2.4 Recovering Known Optimal Error Rates in SBM and DCBM
Inhomogeneous SBM
Let A/a 1 for all E [n] and a E [k), that recover the SBM with inhomogeneous interactions in which P. = Pn se By linearity of the Rényi divergence, we have
R
a(i,a) = max (1 - t) tE(0.1)
nob Ren, (Ber (pn Bub) Ber (pnB)z)))
b=1
(1 + o(() npn max 1)
Σ b=1
Tb (Bab + (1-1) (1-1)B223 Bao Bzzbb
(E(0.1)
CHAs(a,=))
The quantity CHAs(a, b) is called the Chernoff-Hellinger divergence (Abbe and Sandon 2015). When npn w(1), we observe that
Σ "p. (wAte)) CHas(a,2)) = (()oo(())m min CHas(o.)) kE|n]
and we recover the instance-optimal enror rate SBM with inhomogeneous interactions (Yun and Proutière 2016). Finally, the Chernoff- Hellinger divergence has simple expression in the case of homogeneous interactions. Indeed, when B&s = pol(a = b) + 4a + 6) and the clusters are of equal-size (T. = 1/k), the divergence simplifies to min CH asEbE(()
"e- (vPo 40
Pegree-Correctee Block Model Suppose that Aia; 0; so that the PABM boils down to DCBM with homogeneous interactions in which P.j = 0,0, 22; Pr. For the simplicity of the discussion. we consider cluster of equal-size (ie. Ta = 1/k for all a E [k)), and homogeneous interactions (i.e., Bab palfa b} golfa # b)). Consider vertex i in cluster 2f and let [k])
We have A (ii a) = 0.n (Vpo /4o)² where we used Σ: =;= 0 = 1. Thus, we recover the asymptotic optimal error-rate 1 Σ, "ga (vpo-vm)² established in Gao et 2018).
2.5 Optimal Error Rate in Homogeneous PABM
We now show how Theorems I and 2 when applied to PABM, reveal novel phenomena. Suppose Bab fa = b) + 40 {a + b) and Aia 1= a + {2 F a). We have
TPPm
Σ (v Am min 5a where 86. n/h $: a#=;
Aout Xout 20
2k
Proposition 3. Consider a PABM with homogeneous interactions with Polg O, and equal-size communities. Suppose that xont ... Xout = 1 and that the coefficients X Nin are sampled
from Uni(1 - 1.1 1 c) with e E (0, Denote "e- = +ेc ((1 c)3/22 - (1 - c33/2). We have
Σ
Cheruoff(i..") = =11 + o())e " go (01--)))
[E(n]

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where
k
nen po - -E- nen pou2 (e
Ve
ko
(erf
(npn pou+) erf (nPn pou-)) ( k
Jn
2cporpn
2c
T2P240
with u+ - vI Vo 90/Po and erf(() - 2/ e- - di is the Gauss error function.
Although the expression of Jn is quite involved, we can give two interesting particular cases. Firstly, to highlight the effect of degree heterogeneity, suppose that po go. In this extreme case where we expect the same number of interactions within and across clusters, the only information comes from the degree heterogeneity. Therefore, many existing graph clustering algorithms are expected to fail (as indeed shown in Section 3 However, the quantity Chernoff (i, =*) does not vanish. Therefore, even in this extreme setting, consistent recovery is possible, as highlighted in the following example. This stands out in stark contrast to the standard and degree-corrected block models, where setting Po = go causes the model to collapse into an Erdös- Erdös-Rényi graph and Chung-Lu graph, respectively- both of which contain no information about the underlying cluster structure.
Example 1. Consider the setting of Proposition where Po 90- eore= 0 the model reduces to an Erdōs-Rényi Rényi graph with edge adge-probability p, and thus no recovery is possible. However, 0,
Cheroof((sa") -1+ (1)) "E"
((11)))-="rll po(1-2))
7I
EE|M
Observe that 2 is strictly positive and increasing c E (0, 1]. Thus, if > the optimal error rate satisfies 1 3 Lie Cheruoff(i.z") =(1), yielding that cluster recovery is possible. Moreover, this rate is monotonically decreasing in c.
In the following example, we fix c E (0, 1) and po > 0, and we let qo vary between 0 and po-
Example Consider the setting of Proposition with exp(- - k Po5(1 - -2)) x Jn is not monotonically increasing in E, but instead first increases with &, reaches some maximum value, and then decreases. We illustrate this Figure Appendix D33
qo/Po E (0.1), The quantity
The intuition behind Example 2 is as follows, As = the graph is disconnected and the k largest components are aligned with the clusters. Hence, the difficulty of clustering is at its lowest and can only increases with ç. as additional edges are inter-cluster edges and act as noise. However, when becomes large enough, the difference between the intra- and inter-connectivity patterns, governed by the A and Aout, becomes more pronounced. As result, this provides additional information that can be exploited for clustering (as in Example [T. This leads to trade-off between the benefit brought by the absence of any inter-cluster edges (for learning from well-separated clusters) and the benefit brought by their presence large numbers (for learning from popularity patterns). This non- monotonic behavior specific to PABM and does not occur in DCBM. Finally, this non-monotyniccty not an artifact of setting xout = 1 and sampling the A from a uniform distribution. This choice was made because of the difficulty to derive a closed-form expression for the optimal error rate. In Appendix D.3 we show numerically that these phenomena persist under alternative distributions for the coefficients X and out
3 Numerical Experiments
In this section, we numerically evaluate the performance of several existing variants of spectral clustering on both synthetic and real-world datasets Specificallyy, compare the following variants:
sbm: Lloyd's algorithm applied to the embedding formed by the & largest (in magnitude) eigenvectors of the adjacency matrix (see Algorithm 2):;
dcbm: Lloyd's algorithm applied an estimate P of the connectivity matrix P, constructed using the l largest (in magnitude) eigenvectors of the adjacency matrix (see Algorithm 3):
pabm: subspace clustering applied to the embedding formed by the largest (in magnitude) eigenvectors of the adjacency matrix (see Algorithm 5);
OOr code is available at pattps://ggiiee http:///......eeeeee/eeeeeennn/a pabm
7

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osc: the spectral clustering variant described in Algorithm
sklearm: Lloyd's iss algorithm applied to the embedding formed by the & smallest eigenvectors of the graph's ss normalized Laplacian, corresponding to the implementation available in the scikit-learn library (see Algorithm
The shm and debm variants are tailored for graphs generated from SBM and DCBM, respectively, and are known to recover clusters accurately under these models (Zhang) 2024 Gao et al. 2018). In contrast, PABM exhibits more complex structure, as the rank the matrix can exceed k, but cannot be greater than k-2 We refer to (Koo et al. 2023 Noroozi et al. 2021) and to Section E.4 the Appendix for examples. To accommodate this higher-rank structure, the pabm and ose variants rely embedding based on k-2 eigenvectors rather than the traditional k, allowing them to capture the higher-rank structure of PABM more effectively.
In the finalization phase of the manuscript, we became aware of two more algorithms designed for community recovery in PABM, namely Thresholded Cosine Spectral Clustering (tesc) and Greedy Subspace Projection Clustering (gspc), introduced in Yuan et al. (0255) and in Bhadra et al. (2025). respectively. To avoid overburdening this section, we refer the interested reader to the Appendix [E.3 for description of these algorithms. We also provide in the Appendix the pseudo-code of all the algorithms.
In all experiments, we report the accuracy, defined as one minus the loss in 2.2 It is equal to the proportion of correctly clustered vertices.
3.1 Synthetic Data Sets
We first consider homogeneous PABM whose interaction probabilities are given by
{ AM AA p { Aagu Xount Ep
if 2f 2j. otherwise.
Pg
(3.1)
The parameter p E (0, 1) controls the overall sparsity of the network, while the parameter & E [0. controls the fraction of edges across clusters (in particular, & O implies no inter-cluster edges while & implies the same expected number of edges between any pair of clusters). As in Examples and we let Aqut 1 and sample the coefficients A from the uniform distribution in (1 1 c).
In Figure Ta we g and vary c. This precisely the setting of Example [ We observe that pabm and osc variants, which are specifically designed for PABM, recover the clusters when is large enough, whereas the variants tailored for SBM and DCBM fail to do so. This illustrates that pabm and osc successfully learn the clusters without edge-density signal by using the difference in the individual vertex degree connectivity patterns. In Figure [bb we set с = 0.8 and let & vary. We observe that the acuracy of pabm and osc is not monotonically decreasing with &. In fact, it goes to a minimum value before increasing again. This illustrates the phenomenon described in Example 2 In contrast, the accuracy obtained by sbm and dcbm variants monotonically decreases, because increasing & from 0 to monotonically decreases the edge-density signal
To further highlight the impact of the embedding dimension on the clustering accuracy, we plot in Figure 2 the accuracy of the different spectral clustering variant as a function of embedding dimension d. We observe that the performance of pabm and osc improves significantly as the dimension increases from d = to = 6, after which reaches a plateau. In contrast, the performance of the sbm and debm variants remains unchanged with increasing d.
3.2 Real Data Sets
In this section, we show on real datasets that spectral algorithms that use more eigenvectors such as pabm and outperform the traditional variants that use only & eigenvectors. Table in Appendix F.3 summarizes some statistics of the dataset used. Table 2 shows the accuracy obtained by the different variants of spectral clustering the real data sets.
In both cases, the accuracy achieved by sklearn matches that of dcbm and omitted from the figures.
8

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1.0
1.0
sbm
sbm
0.9
dcbm
dcbm
0.9
Accuracy
0.8
pabm
Accuracy
pabm
oSc
0.7-
0.8-
0.7
0.5
0.6
0.0
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
c
(a) =
(b) 0.8
Figure I: Performance of graph clustering on homogeneous PABM, where the matrix P is given in Equation (3.]). We sampled graphs with 900 vertices in k 3 clusters of same size, average edge density p = 0.05. In both figures, the X are iid sampled from Uni(1 - c, 1 + and XQ = for all i. Accuracy is averaged over 15 realizations, and error bars show the standard errors.
1.0
Accuracy
Accuracy
0.8
0.6
sbm
0.6
sbm
dcbm
dcbm
pabm
0.4
0.4
10 embedding dimension
4
8
12
4
10
12
embedding dimension
(a) Xout
(b) X? Uni(0, 2)
Figure Effect the embedding dimension on the performance of graph clustering on homogeneous PABM, where the matrix P is given in Equation (3.I). We sampled graphs with 900 vertices in = 3 clusters of same size, average edge density م = 0.05. In both figures, the A!" are iid sampled from Uni(0, 2). Accuracy is averaged over 15 realizations, and error bars show the standard errors.
sbm
debm
pabm
osc
tesc
gspe
sklearn
politicalBlogs
0.63
0.95
0.91
0.95
0.65
0.95
0.52
liveJournal-top
0.56
0.61
0.99
0.59
0.98
0.60
0.99
citeseer
0.27
0.38
0.45
0.56
0.33
0.51
0.58
cora
0.34
0.37
0.47
0.47
0.30
0.42
0.27
mnist
0.44
0.54
0.88
0.74
0.11
0.79
0.78
fashionmnist
0.22
0.41
0.63
0.61
0.60
0.50
0.60
c0far10
0.17
0.43
0.74
0.58
0.49
0.62
0.71
Table 2: Accuracy of several spectral clustering variants on real data sets.
Related Work
Optimal clustering error rate. A rich line of research focused on characterizing the optimal error rates for clustering in stochastic block models and their variants. Early results established the minimax error rate in the SBM (Zhang and Zhou| 2016). while later work extended these insights more general models such as the degree-corrected block model (Gao et al. 2018). Further developments have addressed more complex network structures, such as categorical edge types (Yun and Proutière! 2016), weighted interactions (Xu et al. 2020), and more general interaction patterns (Avrachenkov al. 2022). These studies leverage information-thoree uncover the fundamental limits of clustering error. Parallel developments have taken place in the
tools to derive minimax bounds and to

---

mixture model literature, where optimal error rates have been studied extensively, particularly in
Gaussian mixture models (Lu and Zhou 2016 Cai al. 2019 Chen and Zhang 2024) and in more general mixture models (Dreveton et al. 2024.. In both settings, central objective is to understand how the separation between components governs the intrinsic difficulty of the clustering task.
Clustering with higher-order eigenvectors. Several studies have identified the benefits of incorpo- rating higher-order eigenvectors beyond the first & in spectral graph clustering. In networks whose connections depend on both cluster membership and spatial position, Avrachenkov et al. (2021) demonstrated that the second eigenvector of the graph Laplacian typically aligns with the geometric structure rather than with the cluster structure. As a result, traditional spectral methods that rely solely on the leading eigenvectors often produce geometric partitioning that fails to accurately capture the underlying cluster structure. Their analysis reveals that incorporating additional eigenvectors beyond the conventional first i can provide crucial information for distinguishing between geometric proximity and actual cluster membership.
In sparse networks with strong degree heterogeneity- some vertices have significantly higher degree than otherss-spectra clustering based on the top & eigenvectors of the adjacency matrix often fails. In such cases, the leading eigenvectors tend to localize around high-degree vertices. rather than capturing the underlying cluster structure. Trimming- based approaches have been proposed to mitigate this issue by down-weighting or removing influential high-degree vertices (Le al.) 2017). Alternatively, using the normalized Laplacian shifts the problem: its leading eigenvectors may become concentrated on peripheral substructures, such as dangling trees, while the cluster signal may still lie in higher-order eigenvectors. To address this, regularization techniques have been introduced to stabilize the spectral embedding and improve clustering performance (Qin and Rohe 2013).
Although the previous paragraphs illustrate two different settings where higher-order eigenvectors crucial for uncovering cluster structure, they also share key limitation: the leading eigenvectors are largely uninformative, and only the higher-order ones carry meaningful clustering information. PABM is fundamentally different, as potentially all 1.2 eigenvectors can be informative for clustering. This richer spectral structure opens new avenues for designing more effective spectral algorithms.
5 Conclusion
We established the optimal error rate for clustering under the PABM, providing precise information- theoretic characterization of the fundamental limits of clustering in this rich and flexible model. Our results highlight how heterogeneity in vertex popularity fundamentally alters the clustering landscape. and how this is reflected in the spectral structure of the network. While our analysis provides solid theoretical foundation, several important questions remain open. A deeper theoretical understanding of practical algorithms such as OSC and subspace clustering remains a key challenge. Another important direction for future work is model selection: developing principled methods distinguish between models such as DCBM and PABM, and to infer key parameters like the number i of clusters or the rank of the connection probability matrix P. Addressing these challenges is essential to translate theoretical insights into robust, data-driven tools for network analysis.
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12

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A Additional Discussion of Theoretical Results
A.1 Instance-Optimal versus Minimax Setting
In our study of optimal clustering rates in PABM, we did not made any assumption on the matrix B (beyond being symmetric). As a result, our analysis derives the optimal error rate for a specific instance of PABM (similar to the instance-wise analysis in Yun and Proutière (2016) for the edge- labeled SBM) rather than a minimax error rate (as Zhang and Zhou (2016) for SBM and Gao et al. (018 for DCBM). Both approaches are valuable: the minimax framework requires defining a parameter space to which E belongs (typically the space of matrices having diagonal values larger than or equal to p and off-diagonal values smaller than or equal to g. but offers no guarantees when the matrix lies outside this space, while the instance optimal-rate restricts to a specific but arbitrary matrix B.
Moreover, we wish to emphasize important point: a rate-optimal algorithm in the minimax setting may not be rate-optimal for specific instances, even when those instances fall within the defined parameter space. For example. Lloyd's algorithm is minimax-optimal over the class of sub- Gaussinn mixture models Lu and Zhou (2016). but it fails to be instance-optimal for Gaussian mixture models with anisotropic covariance structures Chen and Zhang (2024).
For the parameter space described two paragraphs above (matrix B with diagonal values larger than or equal to p and off- off-diagonal values smaller than or equal to q), the worst-case rate for SBM and DCBM arises when B. = p for all E [k] and Bad = for all a &f b, leading to a minimax rate involving the term (vP- - / In contrast, for PABM, the situation more complex because of the additional dependence on the individual parameters Xin- As result, we do not believe that a simple closed-form expression for the minimax rate in PABM is attainable. Indeed, as shown in Example 2 reducing the gap between p and q does not necessarily increases the optimal error-rate.
A.2 Overview of the Proofs
The overall structure of the proofs for Theorems I and 2, which establish the optimal error rate, is similar to that used for SBM and DCBM (in Zhang and Zhou (2016) and Gao et al. (2018), respectively). However, the PABM setting introduces additional technical complexity that requires a more refined analysis.
(i) For the lower-bound, a first challenge is to address the minimum over all permutations in the definition of the error loss. Hence, rather than directly examining inf ke Fln==loss(2², ,2]]. we follow previous works such as ZZhang and Zhou (2016); Gao et al. (018) and focus on a sub- problem infzez EE [[[osss(=",)) 2)], where 2 C [ is chosen such that loss -. _) = Ham 2)/n for all ==, 8 E Z. This sub-problem is simple enough to analyze, while still capturing the hardness of the original clustering problem. Next, we use result from Dreveton et al. (2024) to show that the Bayes risk inf PP, + =;} for the misclustering of a single vertex / is asymptotically e
More precisely, (Dreveton et al. 2024 Lemma 2) establishes the worst-case error rate for binary hypothesis testing problem where the observed random variable is drawn from either distribution fi or fa (corresponding to hypothesis H1 and H2, respectively). Both fi and f2 are arbitrary and known probability density functions. By the Neyman-Pearson Pearson lemma, the likelihood ratio test (equivalent to the MLE in this context) minimizes the probability of error, thereby ruling out all other estimators for this problem. The error of the MLE is then upper-bounded using Chernoff's 's method and lower-bounded using large deviation argument. In our setting, the hypothesis is formulated in Equation B.I.. where fi and f2 are product distributions of Bernoulli random variables.
(ii) The proof of the achievability is however more involved, and required a new approach. It begins. similarly to prior work on block models, by upper-bounding E [oss(2*, (oasss""))))))) 2)] (where é the MLE) by Σn P(Hamm("* ( m). Thus, the core difficulty relies in upper-bound the quantities ((L(E) > [(²")) for any such that Ham 2) m (where [(2) denotes the likelihood of a given an observation of A). This is more challenging in PABM than in SBM and DCBM. Indeed, unlike in SBM (and to some extent DCBM), the likelihood ratio L(z)/L(z*) cannot be easily simplified. As for SBM and DCBM, we rely on Chernoff bounds to obtain (T) > I(=*)) S Efe [o{(L())/L()))
for any t > O. But, in SBM and DCBM, one can use f = 1/2 and obtain clean exponential bounds
13

---

whose terms are exp(- axp(-80.0lyP.. ). For PABM, the optimal to use depends intricately on the misclassified set {u: Zu F 2 and thus on itself. To address this, we adopt a more refined approach: we decompose the upper bound into three components Ti (1), T2()), and (2(+), and select a tailored value of each labeling 2. This additional complexity distinguishes our analysis from earlier work and reflects the greater structural richness of PABM compared to SBM and DCBM.
A.3 Extension when Assumption 2 Fails
The situation when Assumption 2 fails is slightly delicate. Suppose firstly that Assumption fails such that max; Chernoll =") O(1). In that case, the optimal clustering error cannot vanish. Indeed, using arguments similar to those in Zhang and Zhou (2016), we can establish that the optimal error rate is lower-bounded by non-zero constant
O. More generally, we introduce the set S {i E [m]: Chernoff(i, =²) = O(1)} of vertices having a non-vanishing error of being misclustered. Assumption 2 fails whenever S * 0. By refining the proof of Theorem we can obtain a lower-bound for the clustering error of any algorithm of the form:
1+0
- e) mina na
|S] n
inf E [loss(2", [2),
Chernoff((aa"))
|SC|
YES
This decomposition reflects that constant fraction of nodes (those in S) are intrinsically hard to classify, while the rest exhibit standard exponential error decay. When Assumption 2holds, S 0 and the lower bound matches the result in Theorem [ (And observe that the case max; Chernoff =*) O(1) discussed earlier is equivalent to |S| n, and we recover inf E [loss(2", 2)] 2 c for some non-vanishing constant e 0.)
Showing that the MI.E attains this bound when S * 0 appears plausible but requires additional technical work.
B Proof of Theorem 1
B.1 Clustering one Vertex at Time: the Genie-aided Problem
Let i E [n]] and suppose genie gives you zΣ,- i.e., the community labels of all nodes but i. Denote H&" a = a the hypothesis that node i belongs to the cluster a E [K:]. Letting X = Ag being the i-th row of the adjacency matrix, the hypothesis testing resumes to
H
X
Ber Shadis
(B.1)
The worst-case error of a testing procedure Φ: f0, 1]n-1 - f1, ... ,k] is
r())
= max (Ø(X) = a Hb).
azb
By the Neyman Neyman-Pearson Pearson lemma, we have MLE arg ming r()) where
II (1- 1indini Pro BoaB 1-1 (pndiez (pndicsAyalaaz NyaBejo) Ass
(A..) arg max aEik] jti
Recall that the quantity (in A) defined in 2.3 by
A(i, a) = sup (1 - R)n,
X j+H
Ber
(rddis
X
Ber (Prdiez >sst)
rE(0.1)
j#1
Dreveton et al. 2024 Lemma 2) shows that for all a + 2 we have
P (Φ (L a 22, = = (1+)
14

---

provided that A(i,a) = (1). Furthermore, if A a) = w (log k), union bounds imply that
P ( (At) = (1+o (1
(B.2)
where
Chernoff 22, = min A a)
of
is the Chernoff information associated with this hypothesis testing problem.
B.2 Lower-bounding the Optimal Error
Proof of Theorem 7 For simplicity, we shorten (T, by 2. Let 2" E be the true cluster membership vector. We denote the set of vertices in cluster by [=(=*) {i E [n] 2 a) Following the same proof strategy as previous works on clustering block models (Gao et al. 2018 Dreveton al.) 2024), we define clustering problem over a subset of [k]" to avoid the issues of label permutations in the definition of the loss function (22)). For every cluster a E [k], we define the set Ta of the [[a(=")] n(1-e))mmn vertices belonging to cluster and having the largest Chernoff (i, 2"). We motivate this as follows. A vertex i with a large Chernoff 2*) implies that genie provides ="; (the community labels of all vertices but i), the inference of 2 is easy. Hence, the set T, contains the vertices belonging to the cluster a that are the easiest to cluster, and therefore good estimator & should correctly infer the vertices belonging T., In contrast, vertices with small Chernofl (i, =") may be impossible to cluster, even with the best estimator, and these are the vertices that matter in deriving the lower-bound. Let T = Uae[k] Ia and define new parameter space Z C Z
2 {2: 2; = 2 for all i T and [=(=)]
[(1 c) (1 e)Hal}-
This new space 2 is composed of all vectors E Z that only differ from x" on the indices i's that do not belong to T. By definition of T, these vertices are the hardest to cluster. By construction of ž, we have for any 2, z' E 2
n(1 em
Ham(z, 2/) Σ 1f2
# S [T
k
4k
Because 2 2, have by definition of Z that min the previous inequality ensures that Ham(z,z') < 2-1 min thus apply Lemma establish that
[a(=)] 2 (1 c)m Therefore,
[ [=)] for all 2, z' E 2. We can
_ Hamm(zzz)) V2, d' E loss(2,
1 Σ1
1( 2f}.
(B.3)
n
n 1E
For any estimator é, we can build an estimator
2' 2 such that
2 if T, otherwise, &
and this estimator satisfies loss(>", 2') < loss(=", , 2). Therefore,
1
inf loss(=", (,",
> inf loss(2" (20, 8))
inf Ham(=", 2),
SE2
n #ez
where the last equality follows from B.3 Hence, we obtain
inf E Loss(z", (z", 2))
inf
P (2, +
1
Σ
inf (2, # 2f).
#1
iET
iET
From Equation
B.2 we have
inf (2, + x)) 2
15

---

for some 7/ o(1). Let 17 = max; We obtain
inf E loss( (2" .) RE
T
(L+n))heroof((.222
T
iETe
(1+t)Cherooft(.."")
n
iEin]
Tii
- (1--n)Chernoffiix.
where the second inequality uses the fact that collects the indices the vertices with the smallest
Chernoff =") ), and the last line uses T" a(l-e] (by definition of T)).
Finally, note that we can always chose 7 to be nonnegative and thus the function x + + convex. Hence, by Jensen's ' inequality, we have
1+1
Σ AE|n]
Chermoft(b.e")) 1+n
Σ
(her")ff(i;e"))
2
72
71
iE|ml
B.3 Additional Lemma
Lemma 4 (Lemma C.5 Avrachekkoo et al. 2022)). Let 21, E [K]" such that Ham T2 oz)) < minae[[] (2)() for some T" E Sym(k). Then I" the unique minimizer of T Symnk) H Ham 22).
C Proof of Theorem 2
Warm-up: notations and MILE Let = [kr] be any vertex labeling. We denote L(z) (A the likelihood of given the observation A. We study the performance of the maximum likelihood estimator 2 = ((A)) defined by
2 arg max L(z).
=E(
where ties are broken arbitrarily. Hence, by definition, the MLE is any estimator 2 such that
I(2) 2 I2()) for all 2 E [k]].
Moreover, have
Ham(z*, o _00 oz)) 2))
n EE [loss(e* 2]] = E
min [k)))(()))
- E [Hamm=", =)]
We also recall (see Dreveton et al.
2023
Lemma 7)) that, for any 2 E [k]] we have
(oss(z, x') < - 1/k).
Therefore,
m211-//k
nE [loss(2.. [x)] <
Σ
mB (Ham(z", 2) = m.)
16

---

For technical reasons that will become clear in the end of the proof, we first need to split the sum into two parts. Let my 1, whose value will be determined later. We have
n 11-11 1/R)
E [aam(z",
=
Σ m=l
mlP (Ham(=" m) +
Σ
miP (Ham(=", m)
m
mL
x mo +
Σ m0+1
miP (Ham(=", m)
Let denote mm the set of vertex labeling [k]] such that Ham(2", (2),,) m. By definition of the maximum likelihood and by union bounds, we have
E 2m) I P (32 E Zm: L(z) 2 L(z")) & Σ 2 L(="))
2E2m
Hence, by combining the previous inequalities, we obtain
m(1-1/k)
E [loss(=", 200;(===,000))) 2)] s mo
Σ
Σ P(T(E)2 1(31)
(C.1)
=E
A large part of the rest of the proof is devoted to upper-bound ΣxEzm. (T](2) 2 I(=*)) for an arbitrary m. We first observe that
I(=)
II P (Ais 24, 2z) i<j
II (PndizzAjaz Brazz) Au (1- Pmiiis Brizz) 1-Asi
In all the following, to avoid overburdening the notations, we denote P.j. = pnddie Aiz Bres. We also introduce
T(2,2") {(i.)) < and (2 24) * (2],2
We have
Ai
1-Ais
L(z) = [(2r)
pz
332
p
1 - P
Kjj
(3)))
Therefore, by Chernoff bounds, we have for any 0,
log L(s) b(a*)
P (L(x)) L(z))) P
P2 Ais log pt
+(1-A,)) 1-
E
c
K
(1,,))
(1 -1- Ren PG).
(C.2)
(.,)))
For case of the exposition, we start by deriving an upper bound on Σ P (L(()) 2 L(x*)) in the simplest case m We the general case m > later.
Case m
1. Observe that
Zi {a [k]": Ham(z, = {2ta u [n], [k] {2)1),
17

---

where 2nm = 2 for all * and 2 = a. Hence,
P (Ham(=", = = ( E [m], 3a E [K] \ {2,]: I( > I(x"))
E PL((E)) > L(z"))
(C.3)
[=
Moreover, for any [n] and E [K] \ { have
(L((E)) > L(z))) )) s e Eo Ren, (PE)" .Pis).
This last inequality is valid for any 0. Applying it with t arg g00 Ejfu Ren, (Pa) Pūj), we obtain
P (L( I(=*)) Σ 6 Aue S e (CCheroof(a.=")
because Chernol[ x*) = milla;;;; Ann. Hence, using we have
2)) =) S k
Cheruoff(u,z")
(ii) Case 2 2 Consider now such that Ham(z, =*) m. Introduce u1,-.. Um the m vertices satisfying Zup * Zup for all p E [m]. By definition, for any v of {ui,-.. Uph we have
2p z
Observe that
T(z,2") = {(i,5): + and (24,2z) + (2), 25)] S1 U..
where
((i,g):2; and j #
S2 = {(i,]): 24 2 and 2j + }.
Thus, we have
Σ
(1- Ren, (P P.j) = 1
Σ
(1 - Ren, (P.
2
K (K,))T((,22)
(iJ)eT(z,*))
1
Σ (1 (i-t)Ren, (P. P.) +
Σ
(1 t Re (Pij. Pij))
=
2
(i,))S1
(1.))
Ti(1)
T2(t)
Notice further that
(1(t)
Σ
Ea
R)R, (Pi, Pij)
(C.4)
iE (E(",." (bm)) 1ØL
and
Tz(t)=
Σ
Σ
(1- (1-t) t)Ren, (Pij, P.j)
igfui." am jEfun. umn
Σ
Σ
(1
t) Ren, (P. P.j))
iEfui." (amm jdd toe Hmmt
(1)
Σ
Σ
(1- - Ren; (Pij, Pij)
iE ( Mm jElug..... am
8z1
18

---

Combined to the Chernoff bounds CC2), this leads
P (L 2 L(z')) 6 ( (T(1()))
(C.5)
where (t) is given by Ta(1)
E E
(1- - Ren PPj.PP)
(C.6)
iE(MIr" (m (E
jz2
Note that Ep bounded away from one, when (1) equals O. We also recall that, for any B E (0.1) with a -. B, and any probability distributions f and g, we have (ann Erven and Harremoës 2014 Theorem 16)
Let us lower-bound Ti. For E [m]] denote lp - arg max max -))
Ren, (P P upd)
1, the objective function inside the argmax
a B
B Rens(S.9) s Ren. S Rena(f.g)-
Denote t" = min ... tm Without loss of generality, suppose that t" = t1. Using the previous inequality with tn and [ t1, we have,
Σ
Rens, (Ppjs p(nss)
Σ
Ren, (Pig)PPa))
2
(E
ME(m|Mupt
for any p E [m]. Thus,
t1)
Σ
Rent (P (Pigs: Paps)
2
1 - t (1 (1-tp) tp)
Σ jEIn]\(up)
Reni, pp Pupgi:Ppp
1 tp
jE(H]Mup)
1 t1 Chernoff (u.).
tp
by definition of lp- Because all the lp are bounded away from and i = min(41, ... 4m J. we have
1-h 2 C for some constant C 2 1. Recalling the definition of Tj in (C.4), we obtain
1-tp
m Chernoff (up)
(t))
p=1 tp
m
Chernof[(ur)
Chernoff
lp
p=2
m
Chernoff(u)) +
Σ
Chernoff(u,)
(C.7)
p=2
We now upper-bound (t1),, defined in (C.6). By Assumption
[ all the Rényi divergences are of the
same order. Thus, there exists quantity C such that C² lnd
Ta(l)) <
Σ
C² m
(1 Ren,, (Pžz P.))
sE(uI Menc
< Cf m
Chernoff (up,
(C.8)
2n
(E(1, m))
In the rest the proof, we denote 8m
C% m 2n
By combining (C77 and
C.8 with the Chernoff bound C.5) we have
P (L l(z)) Chernoff(u)) (C- 6mm En Cheruoff(ap).
19

---

We recall that & 2m if and only if there exists set {ui,-.. Um of m distinct vertices such that
Za = zu = u E {ui, "mm)
Moreover, for any such set {" Jm there exists (A: - 1)th ways to construct a 2m. Hence, Cheruoff((u)
Σ ( l(=)) s (k 1)m
Σ
Chernoff(up)
(k: 1) 1)m
e (heruoff(u))
Σ
e (C-m8m Dp Chernof((v))
261
(ug."" _0m)
In the previous inequality, the second summation is over all set {u2, :. Um of m elements belonging to {u.) There
n
1
c(n - 1)
m-1
ways of choosing such set. We finally obtain
e(k 1)(n e- (C 1 minee))) Chernoff(
m-1
(L(2)) L(2))
Chernof((u))
m
2m2m
ekn
-(C-8m) - min Chernoff(i) m
Chernof((u)
m
Ending the proof.
Going back to
C.1
have
n(1-1/k)
E [loss 2² 2)]
s mo
Chernoff(us)
Σ
mQm-1
(C.9)
m=
where Qm
ebn m-1 e (C-6m) minzg(s] (i)rnoff()) Denote also = Em
Chernoff(ws) and
2enke min, Chernoff(i,a") and recall C 2 1. We also introduce
[2enke - (C-5 1/a)) min, Chernof((i,=")
By assumption, we have (n1-11/k) < 1 - and thus mi = o(n). Observe that
1
Qm $ 2
Vm E {mi + 1,,.. ,n(..,nn111111 - 1/k)}
and thus
m(1-1/k)
Σ
mQm
m-1
m
8
m7
m-1
mi +
2m
<
(C.10)
m= +1
m=m1++ +1
by using properties on geometric sums (see Lemma 5
We still need to upper-bound Em=ma+1 mQ Let my C6m min; (hernof(((."")
Observe
that, for any mg S m S mi, we have Qm I 1/2. Then, we are left with two cases.
(a) If mo I 1, then chose mo = Then, we simply have
m1
00
1
m-
m
m-
by using Lemma 5 as above. By combining (C.9 and C.10)
we have
R
E [osss (2" -22
8
20

---

(b) Otherwise, chose mo [ml] Then, we upper-bound Σ m mQm-1 by 4 as above, and we obtain from (C.9) that
E [loss(=", - (mo + 8R)
Moreover, mi [moo] S 2mño. This gives
R x (2mo + 8R) I 22kee Côm, min, Cheruoff(i.z") +8)
E [[oss(>", , 2)]
Observe that this last upper-bound is also an upper-bound for E [loss(=', 2]] in the case (a). To finish proof, we recall that my o(n) and thus om, - o1) by definition of 6m.
Additional Lemma This lemma and its proof are taken from Avrachenkov et al. 2022 Lemma A.8), and reproduced here for the sake of completeness. Lemma 5, For any integer M 2 1 and any number 0 S s < 1,
M
ms
(1 - s) -M
M
Proof. Denote S Em msm By differentiating 2m 8 (1 (1-8)-1-) - with respect s. we find that
8 S =
msm1 = (1- - 22 + (1 -
m=M
from which we see that
S = s= s) + (MM s---)
(1-s)2 s)2 (1 (1-s(1 s(1 - 1/M))
Ms
The upper bound now follows from 1- - s(1 1/M) < 1. The lower bound is immediate, corre- sponding to the first term of the nonnegative series.
D Proof of Proposition 3 and Examples and
D.1 Chernoff divergence for Homogeneous PABM
We start with the following lemma.
Lemma Consider a PABM with homogeneous interactions, and k equal-size communities. Suppose the coefficients X .... Xin (resp., Aq ... Xout are sampled iid from distribution Din (resp., Douth, where Dín and Dout are two distributions supported on B. and with mean 1. Let E [n]. We have
27
Chernoff(i, 2*) (1 o(1)) nPn
where Y and Y' are two independent random variables sampled from Din and Douts respectively.
Proof. We apply the law of large number to the quantity defined in the equation above Proposition 3
Lemma 7. Consider the same setting and notations in Lemma 6 We also suppose that the distributions Dm and Dout have pdf fD.. and fD... with respect to the Lebesgue measure. Denote Yin = E /Y] and Yout = ElVY), where Y Diu and Y ~ Dout- Finally, suppose that >
and let { qo Po- We have
nPa k
1 Xin - 2T 2/t) = (1+
exp
21

---

where
Jn //
exp (- k Po (x+ Ey Your
nPn Your VEVIU) )) fr.m (D (v)dzdy.
Proof. From Lemma 6 we have
(V VExputY")
Chernoff(i, (L + o(1)) nPn PoE
nPn Po (1 (1))
X Exput 2% Yout VE
Xin Aout
As A3" ~ Din and Aont
Dout, computing 1 E; -Chernoft(i,=") resumes to compute
lim 100 1=1
Σ
exp
(-"
[nPp
nPn k
(x Ey 2n Your VEVzy))
In particular, exp (- "=pppp po (ar + Ey Yout v8 Viyyy) is bounded by [0. 1], hence its variance is also upper bounded by Let Jn be the expectation of this quantity over Dins Dout Centering the variable, we bound the total variance
n 2 Var (cxp (- nPn po (x+ Ey 2% Yout Yout VEVry) Jn)
Σ
oc.
Kolmogorov's variance criterion for averages Kallenberg 2021
Lemma 5.22) implies
nPn Po x + Ey - Yout
Your v VEvzy) - Jn) a.S. O.
Therefore the limit converges to its expectation almost surely,
n2n po
x Ey 27 Yout V VEVay)) Jn.
lim <00 n
exp
where
Jn [fen exp
(- nPp Po
1 Ey 2nn Yout v VEVzy)) frn (D) (v)d
D.2 Proof of Proposition 3
To prove Proposition 3 we apply Lemma in the particular case where Din is the uniform distribution Uni(1 c, 1 + c) and Dout is the Dirac distribution 1. Hence, the integral Jn given in Lemma becomes
k f exp (-MPu Po (a + - VEV=)) (x)dd,
13Pn (oxp(-"0
Jn =
where fe.. (z) 1.(x (1 c, c)), and the lower and upper limits of the integral are 1 - and c, respectively. For simplicity we write y = / Mr where M nPnPo/ k.. We perform the
following change of variable: Vā - dy / M x: 1/2 and da. 2 dy = 24 dy. The
lower and upper integration limits become y- / M /I and y+ VMIIc Changing
22

---

variables and completing the square gets us
++
2y M
Jn = 2c
cxp
(-1² MM MYVEE
dy
1
() exp
(-(y -VVEVM + ME( - -1) 2y dy
2c
M
exp(ME(77) - 11)
(ut
exp (-(y- - 7in
Vin->in/VVipp VEVMT)) ydy.
cM
Again, substitute a = Nav / VVVM M to get
Jn = 2xp(M -
1))
exp (-u²) (u+ Nn / EVVM du
cM
exp(MM(>70 -1)) 1)) cM
(f()) exp (-u²) udu + 7i VEVM // exp (-u²) du
where u
Nn En) and 4+ = VM(V++C- - V). The first integral can by solved by-parts, and the later we recognize as the Gauss error function. Hence,
Jn exp(ME(1)) - 1))
(2 (exp(( exp(- -2) + 2 VEM ((er(((()))) - erl(u-)))
c.M
where erf(t) 2/V7 e- 12 dt. Morcover, the quantity Yin Ey [VY] can be computed explicitly. We obtain Yin 2c Ji Vxdr se (1 c) (1 (1-c)²) ) We denote this last quantity by Yes to emphasize that it depends only
D.3 Discussion Relative to Examples 1 [and]
This involved expression of Jn computed in Proposition 3 is well-behaved and practically interesting for particular values of & and c. As such, a few remarks are order.
First (resuming Example 1. when & = 1, we have u -I- -----r e- for all e E (0, 1]. Moreover, M + oc as n + oo, and u.ş + Foc. So simplifies to the much simpler expression
and u4 v 0
Jn Te
kn npnPo
(- nPn Po -
-22)) -2))
exp
h
which only depends on e (recall %/ 30
(1 c)) (1 - -1) and is monotonically decreasing E (0, 1]. This agrees with the following intuitive fact: as increases, the higher variance in the popularity heterogeneity aids recovery.
Another interesting case is when fix in Example As increases from 0 to 1, J first monotonically increase, then monotonically decrease and approaches as & + 1. In particular, & = corresponds to disconnected communities, hence clustering is trivial. As increases, the additional inter-cluster edges act as noise to our classification task. On the other hand, a very large & allows us to better learn from the popularity patterns as qo gets closer and closer to po, and leverage from the variance introduced by c. Especially, uz - Foo when E > Eo for some constant E (0, 1). Hence in this regime, classification is easy, as J,, Jev/5 exp(
exp(-ME(1 MM((1 (2))
as - oo and , - 1 0. This phenomena illustrates interesting duality of the role of inter-cluster edges-thes they act as noise below a threshold &o. yet serves to emphasize the popularity variance introduced by c above the same threshold.
cVM
To better illustrate this two phenomenon, we plot in Figure 3 the error rates obtained for homogeneous PABM where oot = 1 and the n are sampled from Uni( - c, 1+c). This illustrate the phenomenon
23

---

highlighted by the Examples 1 [nd 2
(i) the error rate do not vanish when the edge-density signal
disappear and (ii) the error rate is not monotonously decreasing with the edge-density signal.
Optimal error rate
1.0
Optimal error rate
0.12
0.8
0.10
0.6
0.08
0.06
0.04
0.2
0.02
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(a) g
(b) 0.8
Figure 3: Optimal error rate on PABM with homogeneous interactions. The matrix P is given in Equation (3.1,, and we let 900 vertices, k 3 clusters of same size, average edge density 0.05, and interaction probabilities p = P and q = Ep. In both figures, the quantities A are iid sampled from Di, Uni(1 c, + c) and the X are all equal
In Figure 3a let ç an vary c, while Figure 3b we let c 0.8 and we vary g. The optimal error rates are computed using the formula obtained in Proposition
To show that these phenomena not artifact of setting the Xour all equal to I and sampling the A from a particular distribution, we also provide in Figure plot of the optimal error rate (as given by the formula derived in Proposition when the coefficients X and oour are sampled from different distributions.
Optimal error rate
0.08
0.10
0.09
0.07
0.08
0.06
0.07
0.06-
0.05
otima
0.05-
0.04
0.04-
0.03
0.03
0.0
0.4
0.6
1.0
0.0
0.2
0.6
0.8
E
(a) exponential
(b) lognormal
Figure Numerical values obtained for the optimal error rate 1 Σ, exp( Chernoff (i, =)) on
PABM with homogeneous interactions. The matrix / is given in Equation (3.1. and we let n = 900 vertices, & 3 clusters of same size, average edge density p 0.05, and interaction probabilities p= and Ep. In both figures, the quantities Atn and oout are iid sampled from distribution DD Figure 4a D is the exponential distribution with mean Figure 4b) D the log- fog-normal distribution with parameters (μ4, a) = (-1/2, 1 1) (chosen so that the mean of the distribution is 1).
E Description of the Algorithms
E.1 Variants of Spectral Clustering with h Eigenvectors
Algorithms
and provide the sklearn, sbm, and dcbm variants of spectral clustering, respectively.
E.2 Variants of Spectral Clustering with A² Eigenvectors
In this section, describe two algorithms proposed in the litterature for clustering PABM.

---

Algorithm 1: Spectral Clustering: scikit-learn
Input:
Adjacency matrix A E R4"", number of clusters h Predicted community memberships E [k]"
Output:
Let D = diag( (1.)) be the degree matrix
Compute the normalized Laplacian - Ln AD-1/2AD112
Compute the & eigenvectors of L associated to its k smallest eigenvalues. Construct V R
using these eigenvectors as its columns.
Let E [k]" be the output of Lloyd's algorithm (to solve the k-means problem) on the cloud of k-dimensional points V2 semm
Algorithm 2: Spectral Clustering: standard block model variant
Input: Adjacency matrix A E R, number of clusters ht Output: Predicted community memberships
[k]"
Compute the k eigenvectors " , U/ of A associated to its k largest eigenvalues (in absolute value) |o| 2 ... 2 lo Let V = (v1, U&} Bnx* and Σ diag(0 .Tk) R
2 Let E [K]" be the output of Lloyd's algorithm (to solve the k-means problem) on the cloud k-dimensional points ((VE))))))))
Algorithm 3: Spectral Clustering: degree-corrected block model variant
Input: Adjacency matrix A R number of clusters k
Output: Predicted community memberships 2 E [k]"
Compute the & eigenvectors u1, U% of A associated to its k largest eigenvalues (in absolute value) |oi| 2 ... 2 |ox| Let = (vI, ......... ,x)) B and Σ = diag(o), ,0k) E R
2 Let P = VEVT
Let So {i E [n]: IP 0}. Define P. P3.11 for E Si and P. = P. for i E So- Let E [k] 12 be the output of Lloyd's algorithm (to solve the k-means problem) on the cloud of n-dimensional points (P, (P,)) (note that we assign the vertices of Sp arbitrarily).
Orthogonal Spectral Clustering Koo et al. 2023) observed that PABM is a special case of the Generalized Random Dot Product Graph (GRDPG) for which the latent position vectors lie in distinct orthogonal subspaces, each subspace corresponding to a community. This leads to Algorithm 4
Algorithm 4: Orthogonal Spectral Clustering
Adjacency matrix A E R number of clusters k Output: Predicted clusters é [k]" Compute the eigenvectors of A associated to its k(k + 1)/2 most positive eigenvalues and k(k 1) most negative eigenvalues. Construct V E R ==**² using these eigenvectors as its columns.
Input:
2 Compute = | E Rxn applying entry-wise.
3 Let E [k]" be the output of spectral clustering (see Algorithm adjacency matrix is B.
1 applied on the graph whose
Subspace Spectral Clustering Noroozi al. 2222 proposes another approach to cluster PABM. In particular, they notice that the expected adjacency matrix of PABM has rank between k and 2?2 and is composed of subspaces. In particular, two vertices in the same community belong to the same subspace. This motivates the usage of subspace clustering, opposed to k-means, for clustering the cloud of point obtained via the spectral embedding. For subspace clustering, we use the implementation provided in You et al. (2016) and available at httph:://// gi thub. www/ /hnn/goonyYo suussaae- clustering] and we refer to Elhamifar and Vidal 2013) for an introduction on (sparse) subspace clustering. We summarized this in Algorithm
25

---

Algorithm 5: Subspace Clustering on Spectral Embedding
Input: Adjacency matrix A E R number of clusters k, embedding dimension d (default: d = k?2)
Output: Predicted clusters 2 [k]"
Compute the d eigenvectors 01.""" of A associated to its d largest eigenvalues (in absolute
value) |0n| 2 ... 2 loal. Construct V = (Vi,... (a) Ba* and Σ = diag( 11, "......0 .dd).
2 Let & E [k]" be the output of subspace clustering on the cloud of d-dimensional points
((VE), ())()))
E.3 Additional Clustering Algorithms
The algorithm from Bhadra et al. (2025) is an iterative community detection method designed for the Popularity-Adjuster Block Model (PABM). begins by computing an adjacency spectral embedding of the network into low ciw-dimensional space of dimension d (where typically d d= kP)) For each tentative community, a subspace is estimated via singular value decomposition of the node embeddings in that cluster. The algorithm then greedily reassigns nodes to the community whose subspace yields the smallest projection error, thereby minimizing the objective function. This process iterates until node assignments stabilize, yielding a community structure tailored to the PABM. Although the original paper does not assign name to the algorithm, we refer to it as Greedy Subspace Projection Clustering (gspc), Algorithm provides the pseudo-code.
Algorithm 6: Greedy Subspace Projection Clustering (gspc)
Input: Adjacency matrix A R number of communities K, embedding dimension d (default: d k-2), initial cluster labels 20)) E [k]r
Output: Final cluster labels E [A]]
Compute adjacency spectral embedding X R Bnd from A;
Initialize cluster labels & + 2(0); repeat
for k - to K do Extract X < {ax l k);
Compute leading d left singular vectors Uk of X
7
for i <- 1 to n do
for k - 1 to K do
Compute projection loss Lik <- ||2; U2U x;1l::
10
Update 2 + arg ming Liki
until no label changes maximum iterations reached;
12
return
Thresholded Cosine Spectral Clustering (fcsc), proposed in Yuan et al. (2025). begins by computing the top 622 eigenvectors of the adjacency matrix to capture structural information. Cosine similarities between eigenvector rows then calculated and thresholded to suppress noise. Finally, Lloyd's algorithm applied to the thresholded similarity representation to output the predicted cluster labels.
Finally, Yuan et al. (2025) also proposes refine the cluster labels obtained by tcsc. This leads to Refined Thresholded Cosine Spectral Clustering (r-tesc), which improve upon the initial labels from Icsc by re- re-estimating block connection probabilities and then reassigning vertices to clusters according to a profile likelihood criterion. This refinement step reduces misclassification and yields more accurate community recovery. Pseudo-code for fese is provided in Algorithm 7 and the reader refered to Yuan et al. 2025 Theorem 2) for the refinement step.
E.4 Rank Analysis in PABM
For simplicity, let us consider PABM with & = 3 blocks, and suppose that vertices are ordered such that the first ni vertices are the first cluster, the next n2 vertices are the second cluster, and the last nig
n2 72 vertices are in the third cluster. For any vertex E [n]. we denote by r, its
26

---

Algorithm 7: Thresholded Cosine Spectral Clustering (tcsc)
Input: Adjacency matrix A E R number communities h Output: Predicted clusters 2 E [K]*
Compute the top-K eigenvectors of A and form U E ==*K""
2 For each pair rows U,, U, compute the cosine similarity Sij I 1
(U.0)))
Apply thresholding: set St 0 if Sig where is data-driven threshold. Apply Lloyd's algorithm (k-means) to the rows of 8 to obtain the cluster labels
rank-indexing of its cluster (that is, r; = if i is in cluster 1, r; = --- nı if i in cluster 2, and Ti ni n2 is in cluster 3). Denote A (a.b) the matrix of size na ny-by-1 such that A/a.b) Aip.
We also assume that Bab plfa b} + alfa + bf with p & Then, the matrix P is given by
(A a (A a1 (A0.3))
(1
(A
==
(2 (2
q (2.3.3) (A (1,2))
(A() (A(2.)))
PA (3.3) (A (3.
Thus, the matrix P is composed of k blocks of rank one. Excluding trivial cases, the rank of P can take any value between k: == 3 and k = 9. For example, if all the vectors A ( are all-1 vectors, then P has rank 1. But, if A (1,1) contains entries that are not all equal to 1, the rank of P increases to 4. Similarly, if both (2,1) and A (2.2) contain non-constant entries, the rank of P becomes 5, and so on.
Additional Numerical Experiments
F1 Performance of tese and gspe
In this section. compare the accuracy obtained by tesc and gspe with the accuracy of pabm and osc (and of sklearn as baseline). We sample PABM with homogeneous interactions, and take the same parameters as in Section 3.1
1.0-
1.0
tcsc
tcsc
gspc
0.9
Accuracy
Accuracy
sklearn
sklearn
0.8
0.6
0.7
0.6
0.4-
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(a) E =
(b) c 0.8
Figure 5: Performance of graph clustering on homogeneous PABM, where the matrix P is given in Equation (3.I). We sampled graphs with 900 vertices in A 3 clusters of same size, average edge density p = 0.05. In both figures, the AT are iid sampled from Uni(1 - c, 1 and Aone = for all i. Accuracy is averaged over 15 realizations, and error bars show the standard errors.
27

---

1.0
1.0-
Accuracy
0.8
sklearn
Accuracy
0.8
sklearn
tcsc
ritscc
pabm
0.6
osc
0.6
osc
0.4
0.4
10
12
4
10 embedding dimension
12
embedding dimension
(a) X
(b) X Uni(0,2)
Figure 6: Effect of the embedding dimension on the performance of graph clustering on homogeneous
PABM, where the matrix P is given Equation (3.1). We sampled graphs with n 900 vertices in == 3 clusters of same size, average edge density p = 0.05. In both figures, the A are iid sampled from (ni 2). Accuracy is averaged over 15 realizations, and error bars show the standard errors.
F22 Numerical Experiments on Heterogeneous PABM
We generate the coefficients (ea)EEio) independently from each other and from a distribution with mean and bounded support so that supi,a Aia < 1///p. and let
Arx; P Dixy Ep
if = otherwise.
Psj
(F.I)
To generate the Aia we consider the following three distributions: Pareto with exponent 1.5, log- normal with location 0 and shape and exponential with parameter 1. The support of these distribu- tions is unbounded. To avoid having values too low and too large for the coefficients Xia, we sample a random variable via following one of these three distributions, and let
Via
Uia E [Tmin: Tmax)
Aia
Tmin
if via
Tmin:
Tmax
if Via
Tmax-
In all experiments, Tmin
0.05 and Tmax 5. Finally, we normalize the Aio to that
Σ, Ala = 1 for all a E [k:]. Figure 7 show that pabm and osc almost always outperform the shm and dcbm variants.
1.0
0.9
skleam
0.8
0.8
dohm
Accuracy
skleam
0.7
pabm
Accurac
0.8
skleam
0.6
drbm
0.6
tesc
0.7
debm
0.5
0.4
05c
0.4
0.6
osc
tesc
0.3
g5pc
gspc
0.2
20
1.5
(a) Pareto
(b) log-normal
(c) exponential
Figure 7: Performance of clustering algorithms on heterogeneous PABM. where the matrix P is given in (F. with p 0.05, and the >ia coefficients sampled as described in the text. The curve show the average accuracy on 10 realization of PABM with n = 2000 vertices in 5 clusters of same size. Error bars show the standard emors (over 15 realizations).
28

---

F.3 Real Data Sets Description
Table 3 provides some statisties about the graph used. For all graphs, we only considered the largest connected components, Moreover, for LiveJournal data set, we extract the two largest clusters. Finally, for MNIST. FashionMNNIST and Cifar10, first embed the images into low-dimensional space and we consider the k-nearest neighbor graph (with k: = 10) obtained from 10, 000 images. We use the embedding provided in the graphlearning package
data set
k
d
d2
(d)2
Reference
political blog
1,222
16,714
2
27.3
38.4
Adamic and Glance (2005)
LiveJournal-tpp2
2,766
24,138
2
17.5
31.8
Backstrom et al, 2006
citeseer
2,110
3,668
6
3.5
4.0
Getoor
2005
cora
2,485
5,069
7
4.1
5.4
Getoor
2005
MNIST
10,000
85,938
10
17.2
5.0
LeCun (1998)
FashionMNIST
10,000
83,486
10
16.7
4.0
Xıao et al. (2017)
CIFAR-10
10,000
97,044
10
19.4
8.8
Krizhevsky et 2009)
Table 3: Summary of some statistics of the real data sets considered. The quantities n, El. and
refer the number of vertices n, of edges, and of clusters. The quantities d and / d² (d)² refer to the average and standard deviation of the degrees, respectively.
29

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The abstract and the introduction clearly state all the results of the paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We discuss some limitations in the related work section as well as in the conclusion. We also mention in the numerical section than pabm and ose tend to be more computationally intensive than the other spectral clustering variants.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sellsspeciiiationn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
30

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Answer: [Yes]
All theorems are carefully stated and the assumptions are also explained and discussed.
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proof' in the paper should be numbered and cross referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: All information to reproduce the experimental results is available the paper (some details are in the Appendix).
Guidelines:
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If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily new algorithm, the paper should make it clear how to reproduce that algorithm.
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(c) If the contribution is a new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reprodueibility may be tricky in some cases, in which authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
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Reviews:
[{"review_id": "1BVfo7YZNS", "reviewer": "Reviewer_fUf2", "summary": "This work studies the problem of graph clustering—the purpose is to let intra-cluster edge-density to be higher than inter-cluster edge-density. Unlike models such as SBM and DCBM, this work focuses on a more general model PABM. PABM allows vertices to be more popular inside their clusters and unpopular with vertices in other clusters.\n\nThe primary contribution of this work is to characterize the clustering complexity under PABM via the optimal error rate. The analysis shows that clustering difficulty not only depends on the gap between $p$ and $q$ (the expected intra-cluster and inter-cluster connection probabilities), but also exhibits nuanced variation across vertices—manifesting a more intricate form compared to DCBM.\n\nIn addition, this work experimentally examines on the effectiveness of different spectral clustering methods. Traditional spectral algorithms often decompose the adjacency matrix $A$ into a low-dimensional embedding matrix $P$ and a random noise matrix $X$: $A=P+X$, and then use $k$ eigenvectors of $P$ to do clustering. However, to better leverage richer structure present in PABM, recent works incorporate $k^2$ eigenvectors of $P$ instead. The experiments in this work provide evidence that these recent methods surpass the classical approaches in this context.", "questions": "* Section 2 is overly dense with mathematical notation and formulas; adding more intuition or high-level discussion (e.g., for Theorem 1 and 2) would significantly improve readability and accessibility. \n\n* Could you clarify your technical contributions more precisely? The literature on optimal clustering error rates in the SBM and related models is extensive. It would be helpful to clearly state which existing techniques your work builds upon and which aspects represent novel contributions.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n* The paper provides a theoretical analysis of the optimal clustering error rate of a more general random model PABM. The analysis appears to be non-trivial.\n\nWeaknesses:\n\n* The paper emphasizes that their result implies \"cluster recovery remains possible in PABM even when traditional edge-density signals (the gap between $p$ and $q$) vanish, provided intra- and inter-cluster popularity coefficients differ.\" However, I find this somewhat misleading. The PABM model introduces many additional parameters, and even when the gap between $p$ and $q$ disappears, node popularity parameters (such as $\\lambda_i^{\\text{in}}$, $\\lambda_i^{\\text{out}}$, etc.) still influence the expected number of intra- and inter-cluster edges. As a result, these parameters can implicitly encode edge-density signals, even in the absence of a direct $p$-$q$ gap.\n\n2. There is no theoretical analysis of spectral clustering methods under the PABM model—only heuristic arguments are provided. In particular, the experimental section on spectral methods appears disconnected from the theoretical results on the optimal error rate in PABM.\n\n3. The main technical contribution of the paper is unclear to me.", "was_revised": true, "final_justification": "I think the paper is borderline.", "rebuttal": "Dear Reviewer fUf2,\n\nWe thank you for your time in evaluating our submission and we are grateful for your comments. Please find below responses to the weaknesses and questions raised in your review.\n\nWeaknesses:\n\n1.  - Consider a PABM with homogeneous interactions and $k$ clusters of equal-size $n/k$. Because the quantities $\\lambda_{i}^{\\rm in}$ and $\\lambda_i^{\\rm out}$ are normalized, the expected number of edges within a cluster is $\\frac{n}{k} (\\frac{n}{k}-1)p \\approx \\left( \\frac{n}{k} \\right)^2 p$ whereas the number of edges between any two clusters is $\\left( \\frac{n}{k} \\right)^2 q$. Therefore, when $p=q$, the expected edge density is the same within and across clusters. This is what we mean by the absence of an edge-density signal: there is no signal at the **global level** as the difference in expected edge densities between clusters equals zero. However, even when $p=q$, PABM retains a cluster-identifying signal at the **local level**, encoded in the individual edge probabilities $P_{ij}$. This local structure enables cluster recovery in PABM, in contrast to SBM and DCBM, which lack such a local signal under equal edge densities. \n    - To distinguish the cluster assignments of two vertices $i$ and $j$, global and local signals play different roles. When $p \\ne q$, the global signal can be leveraged across all models by simply comparing the number of neighbors that $i$ and $j$ have in each of the $k$ communities. In contrast, to use the much weaker local signal (only present in PABM), one needs to compare the full interaction vectors $P\\_{i \\\\cdot} = (P_{iu})\\_{u \\\\in [n] }$ and $P\\_{j \\\\cdot}  = (P_{ju})\\_{u \\\\in [n] }$, which encode the individual edge probabilities between $i, j$ and all other vertices.\n\n2. In the numerical section, we illustrate two surprising phenomena discussed in the theoretical section: the non-monotonic behavior of the error with respect to edge density, and the ability to recover clusters even when $p=q$. While it would have been possible to use a greedy algorithm to approximate the MLE, we opted for spectral methods because of their widespread use and established effectiveness for clustering in block models. In particular, our experiments demonstrate that the phenomena highlighted in the theoretical section also arise when using spectral algorithms. This shows that these behaviors are not merely mathematical artifacts stemming from the increased complexity of PABM relative to DCBM, but do occur in practice and are observable in real-world settings.\n\n3. Please refer to our response to Question 2 for a detailed explanation of the technical difficulties involved.\n\nQuestion.\n\n1. We acknowledge that Section 2 is dense with mathematical notation. However, this level of technical detail is unavoidable given the greater complexity of PABM compared to SBM and DCBM. To help for readability, we present our main results in the introduction for the homogeneous versions of the model, and we include several illustrative examples in Section 2.4. We hope that these choices make the material more accessible to readers. \n\n2. We acknowledge that we did not include an overview of the proof techniques (partly due to space constraints) and we will add it in the revised version. While the overall structure of the proofs for Theorems 1 and 2, which establish the optimal error rate, is similar to that used for SBM and DCBM, the PABM setting introduces additional technical complexity that requires a more refined analysis.\n   - For the lower-bound, a first challenge is to address the minimum over all permutations in the definition of the error loss. Hence, rather than directly examining $\\inf_{\\hat{z}} \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, we follow previous work such as [Zhang \\& Zhou 2016] and [Gao, Ma, Zhang, Zhou 2018] and focus on a sub-problem $\\inf_{ \\hat{z} \\in \\mathcal{Z} } \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, where $\\mathcal{Z} \\subset [k]^n$ is chosen such that $\\mathrm{loss}(z^{\\*}, \\hat{z}) = \\mathrm{Ham}( z^{\\*}, \\hat{z} )$ for all  $z^{\\*}, \\hat{z} \\in \\mathcal{Z}$. The idea is that this sub-problem is simple enough to analyze, while still capturing the hardness of the original clustering problem. Next, we use a result from [Dreveton, Gözeten, Grossglauser, Thiran 2024] to show that the Bayes risk $\\inf_{\\hat{z}_i} \\mathbb{P} ( \\hat{z}_i \\ne z_i^{\\*})$ for the misclustering of a single vertex $i$ is asymptotically $e^{-(1+o(1)) \\mathrm{ Chernoff}(i,z^{\\*})}$. (We refer to answer of Question 1 of reviewer 2qS9 for an overview of this result.) We acknowledge that this part follows to some extent from existing and established techniques in the literature on optimal error rates in block models and mixture models. \n   - The proof of the upper bound is however more involved, and required a new approach. It begins, similarly to prior work on block models, by upper-bounding $\\mathbb{E} [ \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$ (where $\\hat{z}$ is the MLE) by $\\sum_m \\mathbb{P}( \\mathrm{Ham}(z^{\\*}, \\hat{z}) = m)$. Thus, the core difficulty relies in upper-bound the quantities $\\mathbb{P}(L(z) > L(z^{\\*}))$ for any $z$ such that $\\mathrm{Ham}(z^{\\*}, z) = m$ (where $L(z)$ denotes the likelihood of $z$ given an observation of $A$). This is more challenging in PABM than in SBM and DCBM. Indeed, unlike in SBM (and to some extent DCBM), the likelihood ratio $\\frac{L(z)}{L(z^{\\*})}$ cannot be easily simplified. As for SBM and DCBM, we rely on Chernoff bounds to obtain $\\mathbb{P}(L(z) > L(z^{\\*})) \\le \\mathbb{E} [ e^{t \\log (L(z) /  L(z^{\\*}) } ]$ for any $t>0$. But, in SBM and DCBM, one can use $t = 1/2$ and obtain clean exponential bounds whose terms are $\\exp(- \\theta_u \\theta_v (\\sqrt{p}-\\sqrt{q})^2)$. For PABM, the optimal $t$ to use depends intricately on the misclassified set $\\{ u \\colon z_u \\ne z_u^*\\}$, and thus on $z$ itself. To address this, we adopt a more refined approach: we decompose the upper bound into three components $T_1(t)$, $T_2(t)$, and $T_3(t)$, and select a tailored value of $t$ for each labeling $z$. This additional complexity distinguishes our analysis from earlier work and reflects the greater structural richness of PABM compared to SBM and DCBM."}, {"review_id": "bUJvFybUKN", "reviewer": "Reviewer_V4fn", "summary": "The primary goal of this work is to demonstrate that one can exploit the individual vertex popularity parameters to separate the clusters even when the traditional edge density signal may not be present. The authors establish the optimal clustering rate for the PABM, an extension of the stochastic block model that incorporates the vertex popularity parameters. They highlight the tension between the connectivity patterns signal and the individual vertex popularity.", "questions": "See the concerns listed under weaknesses. In addition:\n- Isn't stating and emphasizing \"clustering without edge density signals\" misleading? All the information in the model are still in the edge densities. In the example in Section 1, even when $p=q$, the edge densities $P_{ij}$ carry information about the clusters since they are still different for a pair $(i,j)$ when $z_i = z_j$ vs. the opposite case. It is just that the difference is not homogeneous across node-pairs (in contrast to the block model).", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\nThe paper is clearly written with well-organized explanations and includes helpful examples that illustrate key ideas.\n\nThe authors show a clear and thoughtful connection between their work and already existing results in the field of optimal clustering. The alignment with known theoretical results for SBM/DCSBM helps reinforce its relevance and correctness.\n\nThe paper presents technically solid results, and the theorems are stated clearly with supporting intuition.\n\n## Weaknesses\n\nWhile the results are correct and technically sound, they lack novelty in the sense that the main conclusions are somewhat expected given prior work, and the proof techniques follow somewhat standard arguments without introducing significant new ideas. \n\nThe experimental results are interesting, however, they appear somewhat detached from the theoretical setting (which uses the practically intractable MLE), making it difficult to assess how well the empirical findings validate the paper’s formal guarantees. In addition, a comparison of the spectral clustering methods would be more informative and fair if it more clearly isolated the effect of the embedding dimensionality $k$, allowing for a better understanding of its impact on performance.", "was_revised": true, "final_justification": "The authors pointed to some novel aspects of their work which I initially missed (e.g. the non-monotonic behavior of the error with respect to edge density).", "rebuttal": "Dear Reviewer V4fn,\n\nWe thank you for your time in evaluating our submission and we are grateful for your comments. Please find below responses to the weaknesses and questions raised in your review. \n\n### Weaknesses\n\n1. - The optimal error rate in PABM could, to some extent, have been anticipated by experts in the field. However, while the proof for the lower-bound builds on prior work, establishing the upper-bound involves nontrivial and novel technical components, as detailed in our responses to other questions. \n   -  More importantly, our work goes beyond extending the optimal error rates to a more general model. Indeed, our work reveals two counter-intuitive phenomena that have not been previously observed: (i) the non-monotonic behavior of the clustering error with respect to edge density, and (ii) the possibility to recover the clusters even when $p = q$. These insights, discussed following Proposition 3 and illustrated in Examples 1 and 2, deepen our understanding of the complexity of graph clustering under PABM and were overlooked in earlier work. Finally, the derivations in Proposition 3, even if conducted in a simplified setting, are technically involved and further highlight the subtlety of these phenomena. \n\n2. - In the numerical section, we illustrate two surprising phenomena discussed in the theoretical section: the non-monotonic behavior of the error with respect to edge density, and the ability to recover clusters even when $p=q$. While it would have been possible to use a greedy algorithm to approximate the MLE, we opted for spectral methods because of their widespread use and established effectiveness for clustering in block models. In particular, our experiments demonstrate that the phenomena highlighted in the theoretical section also arise when using spectral algorithms. This shows that these behaviors are not mere mathematical artifacts stemming from the increased complexity of PABM relative to DCBM, but that they do occur in practice and are observable in real-world settings.\n   -   Moreover, we conducted additional experiments, where we vary the embedding dimension $d$ and report the results in the table below (we intend to include a figure with these results in the revised version). Under the same parameter settings as in Figure 1, we observed that the performance of *pabm* and *osc*  improves significantly as the dimension increases from $d=3$ to $d=6$, after which it reaches a plateau. In contrast, the performance of the *sbm* and *dcbm* variants remains unchanged with increasing $d$. \n\n| Embedding dimension $d$ | 3    | 5    | 7    | 9    | 11   |\n|-------------------------|------|------|------|------|------|\n| **sbm**                 | 0.63 | 0.65 | 0.65 | 0.64 | 0.65 |\n| **dcbm**                | 0.69 | 0.69 | 0.67 | 0.68 | 0.66 |\n| **pabm**                | 0.39 | 0.86 | 0.89 | 0.94 | 0.97 |\n| **osc**                 | 0.66 | 0.98 | 0.99 | 0.99 | 0.97 |\n\n**Table**: Accuracy of different spectral clustering variants as a function of the embedding dimension $d$.  \nExperiments are done with the same model parameters as Figure 1a of the paper ($n=900$ vertices in $k=3$ communities, and we used $\\xi = 1$ and $c = 1$). Standard errors are typically less than 0.01 and thus not shown. \n\n\n### Question\n1.  Consider a PABM with homogeneous interactions and $k$ clusters of equal-size $n/k$. Because the quantity $\\lambda_{i}^{\\rm in}$ and $\\lambda_i^{\\rm out}$ are normalized, the expected number of edges within a cluster is $\\frac{n}{k} (\\frac{n}{k}-1)p \\approx \\left( \\frac{n}{k} \\right)^2 p$ while the number of edges between any two clusters is $\\left( \\frac{n}{k} \\right)^2 q$. Therefore, when $p=q$, the expected edge density is the same within and across clusters. This is what we mean by the absence of an edge-density signal: there is no signal at the \\textit{global level} as the difference in expected edge densities between clusters equals zero. However, even when $p=q$, PABM retains a cluster-identifying signal at the \\textit{local level}, encoded in the individual edge probabilities $P_{ij}$. This local structure enables cluster recovery in PABM, in contrast to SBM and DCBM, which lack such a local signal under equal edge densities. \nTo distinguish the cluster assignments of two vertices $i$ and $j$, global and local signals play different roles. When $p \\ne q$, the global signal can be leveraged across all models by simply comparing the number of neighbors that $i$ and $j$ have in each of the $k$ communities. In contrast, to use the much weaker local signal (only present in PABM), one needs to compare the full interaction vectors $P\\_{i \\\\cdot} = (P_{iu})\\_{u \\\\in [n] }$ and $P\\_{j \\\\cdot}  = (P_{ju})\\_{u \\\\in [n] }$, which encode the individual edge probabilities between $i, j$ and all other vertices."}, {"review_id": "IaUeCp8yEe", "reviewer": "Reviewer_2qS9", "summary": "This paper studies the Popularity-Adjusted Block Model (PABM) that is a variant of the classic Stochastic Block Model (SBM) by introducing popularity for each vertex with other vertices within its own clusters and with vertices in other clusters. The authors establish a lower-bound on the expected clustering error that coincides with the known optimal error-rates for inhomogeneous SBM and Degree-Corrected Block Model (DCBM) when the clusters are of equal size. They also establish an upper bound on the clustering error that is nearly tight with the lower-bound (up to a mild multiplicative factor). Motivated by the high-order interaction of vertices in PABM, they perform numerical experiments to evaluate the effectiveness of spectral clustering methods with various embedding dimensions. The results demonstrate the tradeoff between cluster information and noise in dimension selection.", "questions": "-\tQ1. It seems that Lemma 2 from the literature (Dreveton et al., 2024) is crucial to the proof of Theorem 1 since it relates the MLE to the Chernoff divergence. What is the intuition behind this lemma? Can you give an intuitive explanation on how to deny all estimators in this proof?\n-\tQ2. Can you provide high-level descriptions for the proofs of Theorems 1 and 2, and give more technical comparisons to the previous work?\n-\tQ3. What is the formal definition of the accuracy in the experiments?", "limitations": "-\tThe authors have not addressed the limitations, but discussed some open problems in the Conclusion section.\n-\tThe definitions of Renyi divergence and $\\otimes$ are missing. The derivation process of the explicit expression of $\\Delta(i,a)$ in Line 154 is not given.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- S1. It is a solid theoretical paper.\n- S2. The optimal lower and upper bounds are strong.\n\nWeaknesses:\n- W1. Since some key steps in the proof of the lower bound derive from previous literature, I am a bit confused to the technical contributions of this part. See the questions for more information.\n- W2. Because lacking high-level descriptions, the main ideas in the proofs (especially in the lower bound proof) are not clear enough for me.", "was_revised": true, "final_justification": "I think this paper should be accepted if the authors could clarify the essential contributions of their work.", "rebuttal": "Dear Reviewer 2qS9,\n\nWe thank you for your time in evaluating our submission and we are grateful for your comments. Please find below responses to the questions raised in your reviews. \n\n1. Lemma 2 in [Dreveton, Gözeten, Grossglauser, Thiran 2024] establishes the worst-case error rate for a binary hypothesis testing problem where the observed random variable is drawn from either distribution $f_1$ or $f_2$ (corresponding to hypothesis $H_1$ and $H_2$, respectively). Both $f_1$ and $f_2$ are arbitrary and known probability density functions. By the Neyman–Pearson lemma, the likelihood ratio test (equivalent to the MLE in this context) minimizes the probability of error, thereby ruling out all other estimators for this problem. The error of the MLE is then upper-bounded using Chernoff’s method and lower-bounded using a large deviation argument. In our setting, this hypothesis testing formulation corresponds to Equation (A.1), line 386, where $f_1$ and $f_2$ are product distributions of Bernoulli random variables.\n We acknowledge that the result at line 390 currently appears without sufficient context, and we will revise the paper to ensure this argument is fully self-contained and clearly motivated.\n2. We acknowledge that we did not include an overview of the proof techniques (partly due to space constraints) and we will add this in the revised version. While the overall structure of the proofs for Theorems 1 and 2, which establish the optimal error rate, is similar to that used for SBM and DCBM, the PABM setting introduces additional technical complexity that requires a more refined analysis.\n   - For the lower-bound, a first challenge is to address the minimum over all permutations in the definition of the error loss. Hence, rather than directly examining $\\inf_{\\hat{z}} \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, we follow previous work such as [Zhang \\& Zhou 2016] and [Gao, Ma, Zhang, Zhou 2018] and focus on a sub-problem $\\inf_{ \\hat{z} \\in \\mathcal{Z} } \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, where $\\mathcal{Z} \\subset [k]^n$ is chosen such that $\\mathrm{loss}(z^{\\*}, \\hat{z}) = \\mathrm{Ham}( z^{\\*}, \\hat{z} )$ for all  $z^{\\*}, \\hat{z} \\in \\mathcal{Z}$. The idea is that this sub-problem is simple enough to analyze, while still capturing the hardness of the original clustering problem. Next, we use a result from [Dreveton, Gözeten, Grossglauser, Thiran 2024] to show that the Bayes risk $\\inf_{\\hat{z}_i} \\mathbb{P} ( \\hat{z}_i \\ne z_i^{\\*})$ for the misclustering of a single vertex $i$ is asymptotically $e^{-(1+o(1)) \\mathrm{ Chernoff}(i,z^{\\*})}$. (We refer to our answer of Question 1 above for an overview of this result.) We acknowledge that this part follows to some extent from existing and established techniques in the literature on optimal error rates in block models and mixture models. \n   - The proof of the upper bound is however more involved, and required a new approach. It begins, similarly to prior work on block models, by upper-bounding $\\mathbb{E} [ \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$ (where $\\hat{z}$ is the MLE) by $\\sum_m \\mathbb{P}( \\mathrm{Ham}(z^{\\*}, \\hat{z}) = m)$. Thus, the core difficulty relies in upper-bound the quantities $\\mathbb{P}(L(z) > L(z^{\\*}))$ for any $z$ such that $\\mathrm{Ham}(z^{\\*}, z) = m$ (where $L(z)$ denotes the likelihood of $z$ given an observation of $A$). This is more challenging in PABM than in SBM and DCBM. Indeed, unlike in SBM (and to some extent DCBM), the likelihood ratio $\\frac{L(z)}{L(z^{\\*})}$ cannot be easily simplified. As for SBM and DCBM, we rely on Chernoff bounds to obtain $\\mathbb{P}(L(z) > L(z^{\\*})) \\le \\mathbb{E} [ e^{t \\log (L(z) /  L(z^{\\*}) } ]$ for any $t>0$. But, in SBM and DCBM, one can use $t = 1/2$ and obtain clean exponential bounds whose terms are $\\exp(- \\theta_u \\theta_v (\\sqrt{p}-\\sqrt{q})^2)$. For PABM, the optimal $t$ to use depends intricately on the misclassified set $\\{ u \\colon z_u \\ne z_u^*\\}$, and thus on $z$ itself. To address this, we adopt a more refined approach: we decompose the upper bound into three components $T_1(t)$, $T_2(t)$, and $T_3(t)$, and select a tailored value of $t$ for each labeling $z$. This additional complexity distinguishes our analysis from earlier work and reflects the greater structural richness of PABM compared to SBM and DCBM.\n3. In the theoretical section, we focus on the *loss*, defined in Equation (2.2) as the proportion of misclustered vertices. In contrast, the numerical experiments report the *accuracy*, defined as the proportion of correctly clustered vertices, which is simply one minus the loss. This choice aligns with common clustering evaluation metrics used by practitioners (such as ARI and NMI), which also take the value 1 under perfect recovery. We acknowledge that the definition of accuracy and of the Rényi divergence were missing in the original submission and will include them in the revised version."}, {"review_id": "yVg26rVXzM", "reviewer": "Reviewer_fKLt", "summary": "The authors investigate the information-theoretic limits of the almost exact recovery problem under the ​​Popularity-Adjusted Block Model (PABM)​​, a variant of the ​​Degree-Corrected Stochastic Block Model (DCSBM)​​ that accounts for differences in popularity between in-cluster and out-cluster edges. Specifically, they derive an information-theoretic lower bound on the misclassification rate and establish its achievability. Additionally, the authors provide empirical validation of the spectral clustering method applied to this model.", "questions": "1) Rank Analysis in Spectral Methods\n\nIn the empirical studies, the authors show that the rank $r$ of the proposed model ranges in $[K, K^2]$, whereas for traditional models (e.g., SBM, DCSBM, Gaussian mixtures), this value is exactly K. This necessitates the use of higher-order eigenvectors in spectral methods. The reviewer is curious about how the rank $r$ relates to the underlying parameters of the model—could the authors provide further theoretical or empirical insights into this dependency?  \n\n2) Extension of Spectral Method Error Rates\n\n[Zhang, 2024] demonstrates that simple spectral clustering (without MLE) can achieve exponentially small error rates for the naive SBM. The reviewer wonders whether this result can be extended to the proposed PABM.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nThe work is solid, and the problem is of significant interest to the research community. The authors establish both the information theoretical limitation and the achievability by leveraging maximum likelihood estimators (MLE). In Theorem $1$, they derive a lower bound on the theoretical limitation using Chernoff divergence, and in Theorem $2$, they demonstrate its achievability up to a constant factor of $(1-\\eps)/4$. Additionally, in Section 2.4, the authors elucidate the connections and distinctions between their proposed model and previously established influential models, highlighting the generality of their approach by showing that their results can recover those of earlier work.\n\nWeaknesses:\n1) It would be better to highlight the technical challenges  in the proof compare with previous work, especially with [Dreveton et al 2024, Gao et al 2018]. \n\n2)  It seems that there is still a constant gap between the converse (lower bound) and achievability (upper bound). Could the authors elaborate more on this gap and compare it with related literatures. i.e , [Dreveton et al 2024, Gao et al 2018, Yun and Proutière 2016].\n\n3) While the achievability is established through the optimal MLE estimator, its practical implementation remains challenging due to severe non-convexity. Previous works typically address this limitation by employing spectral method results as initial estimates, followed by iterative refinement through MLE updates across all nodes.", "was_revised": true, "final_justification": "The authors answered my questions.", "rebuttal": "Dear Reviewer fKLt,\n\nWe thank you for your time in evaluating our submission and we are grateful for your comments. Please find below responses to the weaknesses and questions raised in your review. \n\nWeaknesses\n1. We acknowledge that we did not include an overview of the proof techniques (partly due to space constraints) and we will add this in the revised version. While the overall structure of the proofs for Theorems 1 and 2, which establish the optimal error rate, is similar to that used for SBM and DCBM, the PABM setting introduces additional technical complexity that requires a more refined analysis.\n   - For the lower-bound, a first challenge is to address the minimum over all permutations in the definition of the error loss. Hence, rather than directly examining $\\inf_{\\hat{z}} \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, we follow previous work such as [Zhang \\& Zhou 2016] and [Gao, Ma, Zhang, Zhou 2018] and focus on a sub-problem $\\inf_{ \\hat{z} \\in \\mathcal{Z} } \\mathbb{E}[ n^{-1} \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$, where $\\mathcal{Z} \\subset [k]^n$ is chosen such that $\\mathrm{loss}(z^{\\*}, \\hat{z}) = \\mathrm{Ham}( z^{\\*}, \\hat{z} )$ for all  $z^{\\*}, \\hat{z} \\in \\mathcal{Z}$. The idea is that this sub-problem is simple enough to analyze, while still capturing the hardness of the original clustering problem. Next, we use a result from [Dreveton, Gözeten, Grossglauser, Thiran 2024] to show that the Bayes risk $\\inf_{\\hat{z}_i} \\mathbb{P} ( \\hat{z}_i \\ne z_i^{\\*})$ for the misclustering of a single vertex $i$ is asymptotically $e^{-(1+o(1)) \\mathrm{ Chernoff}(i,z^{\\*})}$. (We refer to our answer of Question 1 of reviewer 2qS9 for an overview of this result.) We acknowledge that this part follows to some extent from existing and established techniques in the literature on optimal error rates in block models and mixture models. \n   - The proof of the upper bound is however more involved, and required a new approach. It begins, similarly to prior work on block models, by upper-bounding $\\mathbb{E} [ \\mathrm{loss}(z^{\\*}, \\hat{z}) ]$ (where $\\hat{z}$ is the MLE) by $\\sum_m \\mathbb{P}( \\mathrm{Ham}(z^{\\*}, \\hat{z}) = m)$. Thus, the core difficulty relies in upper-bound the quantities $\\mathbb{P}(L(z) > L(z^{\\*}))$ for any $z$ such that $\\mathrm{Ham}(z^{\\*}, z) = m$ (where $L(z)$ denotes the likelihood of $z$ given an observation of $A$). This is more challenging in PABM than in SBM and DCBM. Indeed, unlike in SBM (and to some extent DCBM), the likelihood ratio $\\frac{L(z)}{L(z^{\\*})}$ cannot be easily simplified. As for SBM and DCBM, we rely on Chernoff bounds to obtain $\\mathbb{P}(L(z) > L(z^{\\*})) \\le \\mathbb{E} [ e^{t \\log (L(z) /  L(z^{\\*}) } ]$ for any $t>0$. But, in SBM and DCBM, one can use $t = 1/2$ and obtain clean exponential bounds whose terms are $\\exp(- \\theta_u \\theta_v (\\sqrt{p}-\\sqrt{q})^2)$. For PABM, the optimal $t$ to use depends intricately on the misclassified set $\\{ u \\colon z_u \\ne z_u^*\\}$, and thus on $z$ itself. To address this, we adopt a more refined approach: we decompose the upper bound into three components $T_1(t)$, $T_2(t)$, and $T_3(t)$, and select a tailored value of $t$ for each labeling $z$. This additional complexity distinguishes our analysis from earlier work and reflects the greater structural richness of PABM compared to SBM and DCBM.\n2.  A gap between the second-order terms in the lower and upper bounds is unavoidable, as the MLE is not Bayes optimal for the loss function $\\mathrm{loss}(z^{\\*},\\hat{z})$, and is only asymptotically optimal. This phenomenon is also observed in prior works. For instance, in [Zhang \\& Zhou, 2016], the error rate for SBM takes the form $\\exp( - (1+\\eta) nI/(\\beta k) )$ where $\\eta = o(1)$. However, the sequences $\\eta$ appearing in the lower and upper bounds are not necessarily identical. Notably, for any constant $C > 0$, as $I \\rightarrow \\infty$, we have $C \\exp( - (1+\\eta) nI/(\\beta k) ) = \\exp( - (1+\\eta') nI/(\\beta k) )$ where $\\eta' = \\eta - \\frac{\\beta k}{nI} \\log C = o(1)$. Thus, a constant multiplicative factor can be absorbed into the $o(1)$ term inside the exponent.  The same reasoning applies [Yun and Proutière, 2016] (see, for example, the remark following Theorem 2: “the number of misclassified items scales at least as $n \\exp(-(1 + o(1)) n D(\\alpha, p))$”, where the $o(1)$ term is left unspecified), to DCBM [Gao et al., 2018], and to our own results.  In particular, the sequences $\\eta$ appearing in Theorems 1 and 2 are not the same, and the multiplicative factor $(1-\\epsilon) \\min_a \\pi_a /4$ can be absorbed into the sequence $\\eta$, as this factor is of constant order and the term $\\frac{1}{n} \\sum_i e^{-\\mathrm{Chernoff}(i)}$ goes to zero. We chose to make this term explicit in our bounds to avoid having the sequence $\\eta$ depend on the parameter $\\epsilon$. In the revised version, we will include a remark following Theorem 1 to clarify that the gap between the lower and upper bounds is purely due to second-order terms. \n3. We acknowledge that the spectral clustering algorithm used in the numerical section does not come with theoretical guarantees. While a two-step procedure is an appealing direction, it presents a significant challenge in the context of PABM. \n Specifically, the second step via an iterative refinement requires a consistent estimator of the model parameters (which is derived from the initial clustering). In SBM, this is feasible, because the model is parameterized by $\\binom{k}{2}$ edge probabilities (see [Zhang \\& Zhou 2016] and [Yun \\& Proutière 2016]). However, this becomes impossible in DCBM which involves an additional $N$ individual degree parameters $\\theta_1, \\cdots, \\theta_N$. To address this, [Gao, Ma, Zhang, Zhou 2018] proposed a different strategy: for each vertex $i$, they count the number of neighbors vertex $i$ has in each community, normalize these counts by the corresponding community sizes, and assign $i$ to the community with the highest normalized count. While this method bypasses direct estimation of the model parameters, it critically relies on the assumption that intra-cluster edge density is higher than inter-cluster density. Notably, this refinement breaks down in DCBMs with non-homogeneous interactions, where the connection probabilities $p_{ab}$ across cluster pairs are arbitrary (and we are not aware of any existing work that establishes rate-optimality for a polynomial-time algorithm in such a setting). For the same reason, this refinement also breaks down in PABM with homogeneous interactions where $p=q$. Deriving a rate-optimal polynomial-time algorithm for PABM is a crucial avenue for future research. \n\nQuestions:\n1. For simplicity, let us consider a PABM with $k=3$ blocks of equal-size, and suppose that vertices are ordered such that the first $n_1$ vertices are in the first cluster, the next $n_2$ vertices are in the second cluster, and the last $n_3 = n - n_1 - n_2$ vertices are in the third cluster. For any vertex $i \\in [n]$, we denote by $r_i$ its rank-indexing of its cluster (that is, $r_i = i$ if $i$ is in cluster 1, $r_i = i-n_1$ if $i$ is in cluster 2, and $r_i = i-n_1-n_2$ if $i$ is in cluster 3). Denote $\\Lambda^{(a,b)}$ the matrix of size $n_a \\times 1$ such that $\\Lambda_{r_{i}}^{(a,b)}  = \\lambda_{i b}$. We also assume that $B_{ab} = p 1(a=b) + q 1(a \\ne b)$ with $p \\ne q$. Then, the matrix $P$ is given by \n \\begin{align*}\n  P = \n  \\begin{pmatrix}\n    p \\\\Lambda^{(1,1)} (\\Lambda^{(1,1)} )^T \\& \n    q \\Lambda^{(1,2)} (\\Lambda^{(2,1)} )^T \\& \n    q \\Lambda^{(1,3)} (\\Lambda^{(3,1)} )^T \\\\\\\\\n    q \\Lambda^{(2,1)} (\\Lambda^{(1,2)} )^T \\& \n    p \\Lambda^{(2,2)} (\\Lambda^{(2,2)} )^T \\& \n    q \\Lambda^{(2,3)} (\\Lambda^{(3,2)} )^T \\\\\\\\\n    q \\Lambda^{(3,1)} (\\Lambda^{(1,3)} )^T \\& \n    q \\Lambda^{(3,2)} (\\Lambda^{(2,3)} )^T \\& \n    p \\Lambda^{(3,3)} (\\Lambda^{(3,3)} )^T\n  \\end{pmatrix}.\n \\end{align*}\nThus, the matrix $P$ is composed of $k^2 = 9$ blocks of rank one. Excluding trivial cases, the rank of $P$ can take any value between $k = 3$ and $k^2 = 9$.  For example, if all the vectors $\\Lambda^{(a,b)}$ are all-1 vectors, then $P$ has rank $1$. But, if $\\Lambda^{(1,1)}$ contains entries that are not all equal to 1, the rank of $P$ increases to $4$. Similarly, if both $\\Lambda^{(1,1)}$ and $\\Lambda^{(1,2)}$ contain non-constant entries, the rank of $P$ becomes 5, and so on. We acknowledge that we did not explain in detail how the result $rank(P) \\in [k,k^2]$ arises, as we only relied instead on prior works. We plan to provide a more explicit explanation in the revised version.  \n2. This is indeed a natural follow-up question to our work. The version of spectral clustering studied in [Zhang, 2024] involves applying $k$-means to the embedding obtained from the top $k$ eigenvectors, and its optimality has been established only for SBM with homogeneous interactions. However, it is not immediately clear that this version of spectral clustering remains rate-optimal for SBM with heterogeneous interactions. Furthermore, our numerical experiments show that this version of spectral clustering fails to achieve the optimal rate in PABM (and possibly in DCBM as well, where normalization of the embedding (as done in [Gao, Ma, Zhang, Zhou 2018]) is typically required to recover optimal performance)."}, {"review_id": "llZERnogCN", "reviewer": "Reviewer_r5oC", "summary": "This paper concerns with the optimal error rate of Popularity Adjusted Block Model (PABM), which is a generalization of Degree-Corrected Block Model (DCBM). In a DCBM, each vertex has its popularity parameter which affects its connection probability to everyone else. In a PABM of k communities, there will be k such parameters for a specific vertex, which affect its connection probability to vertices in difference communities correspondingly. An optimal error rate is identified with matching upper and lower bound. Empirical studies include a few existing algorithms, which verify a point that looking at the top k^2 eigenvectors might be the right algorithm for spectral clustering on PABM with k communities.", "questions": "**Q1:** What is the conclusion when Assumption 2 does not hold? \n\nI would naturally expect either the optimal error rate identified by Theorem 1&2 holds even when the divergence is constant order, which would reflect a constant error rate as the prior work on DCBM [Gao et al. ‘18] has their minimax rate holds even to the constant average degree regime (and thus only a constant fraction of recovery is possible), or there is a negative result saying that when Assumption 2 does not hold, the optimal error rate is at least some constant, like in [Zhang and Zhou ‘16]. The Chernoff-Hellinger divergence for SBM also characterizes the information-theoretic error rate when exact recovery is not possible but almost exact recovery is possible (an estimator has o(n) errors with high probability).\n\n**Q2:** What is the assumption on the dependency between matrix $B$ and $p,q$?\n\nI understand that $B$ takes only $p$ and $q$ when the model is homogeneous (in this case, $\\lambda_{iz_j}$ takes two possible values as well). Is the matrix $B$ having diagonal values greater than or equal to $p$ and the off-diagonal values less than or equal to $q$ ($\\*$), or is it the case that the result holds even without this constraint? My understanding is the rate identified in Theorem 1 & 2 does not depend on any assumptions on $B$, putting assumptions on the dependency between $B$ and $p,q$ would give it a specific formula, like the perfect homogeneity gives the formula between line 88 and line 89. Does the same specific rate hold for the ($\\*$) case I described above?\n\n**Q3:** Can the main result be formalized into a minimax rate or an information-theoretic result? How should I understand their relations to the result in this paper?\n\nI was trying to understand this optimal error rate result as something quite independent to those minimax rates and information-theoretical analysis, like the result in [1]. But obviously some minimax rates as well as the Chernoff-Hellinger divergence [Abbe and Sandon '15], which determines the information-theoretic threshold for exact recovery on SBM, can be seemingly read from the main result.\n\n--\n\n[1] Zhang, Anderson Ye. \"Fundamental limits of spectral clustering in stochastic block models.\" IEEE Transactions on Information Theory (2024).", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths:**\n\nThe authors provide $z^*$ depending matched error bounds, and identify two interesting phenomena in Example 1 and Example 2, which are: *a)* Non-trivial clustering is possible even when $p=q$ and \n    *b)* The minimax rate is not monotonically increasing when the number of inter-cluster edges increases.\n\n\n**Weaknesses:**\n\n1) The result does not exactly match the form of minimax rates in the prior works such as [Zhang and Zhou ‘16] and [Gao et al. ‘18].\n\n2) This work does not provide an efficient algorithm that provably achieves the optimal error rate identified in this work.\n\n3) The opposite side of Assumption 2 is not formally discussed.\n\n4) Some parts of the writing are a bit unclear to me.\n\n    a) In the Introduction, PABM is introduced under the homogeneous setting, which has parameters $p$ and $q$. In Section 2, PABM is defined under the general heterogeneous setting, with matrix B. However, the dependency between these two sets of parameterizations is not formally discussed.\n\n    b) There is no formal definition of those various terms of recovery (e.g. consistent recovery, cluster recovery, and recovery). From my understanding, they all refer to $\\lim_{n\\rightarrow \\infty} P(\\hat{z}=z^*)=1$ (Just for clarification, I would refer to it as exact recovery in this review).", "was_revised": true, "final_justification": "The rebuttal addressed my main concerns so I raised my score.", "rebuttal": "Dear reviewer r5oC,\n\nWe thank you for your time in evaluating our submission and we are grateful for your comments. Please find below responses to the questions raised in your review. \n\n1. The situation when Assumption 2 fails is indeed delicate and deserves a more thorough discussion. \n    - First, we would like to clarify that the results in [Gao et al., 2018] do not fully cover the constant average degree regime. Specifically, their lower-bound (Theorem 2 of Gao et al 2018) assumes that $p \\\\| \\\\theta \\\\|^2_{\\\\infty} = o(1)$. When $p=\\\\Theta(1)$, this condition requires $\\\\| \\\\theta \\\\|^2_{\\\\infty} = o(1)$ which is incompatible with the normalization condition $\\\\frac{1}{n_a} \\\\sum_{i : z_i^{\\*} = a} \\\\theta_i \\\\in (1-\\\\delta, 1+\\\\delta)$ imposed on the degree-correction parameters. Their upper bound (Theorem 1 of Gao et al 2018) is more flexible as it only assumes $\\\\| \\theta \\\\|_{\\infty} = o(n/k)$. But, this result still requires $I \\rightarrow \\infty$ which, in the case of constant $p,q$, implies that the set $S = \\{ i \\in [n] \\colon \\theta_i = O(1) \\}$ satisfies $|S| = o(n)$. Recalling that the probability of misclustering a vertex $i$ is at least $\\exp(-\\theta_i( \\sqrt{p}-\\sqrt{q})^2)$, the set $S$ is composed of the vertices having a non-vanishing probability of being misclassified by the best estimator.  \n\n    - Turning to our results, when Assumption 2 fails such that $\\max_i \\mathrm{Chernoff}(i,z^{\\*}) = O(1)$, then the optimal clustering error cannot vanish. Indeed, using arguments similar to those in [Zhang \\& Zhou, 2016], we can establish that the optimal error rate is lower-bounded by a non-zero constant $c > 0$. More generally, if we define, analgously to DCBM, the set $S = \\\\{ i \\in [n] \\colon \\mathrm{Chernoff}(i,z^{\\*}) = O(1) \\\\}$ of vertices having a non-vanishing error of being misclustered. Then, by refining our lower-bound argument, we obtain a lower bound of the form: $\\frac{1}{|S^c|} \\sum_{i \\in S^c} e^{ - \\mathrm{Chernoff}(i,z^{\\*}) } + c \\frac{|S|}{n}$. This decomposition reflects that a constant fraction of nodes (those in $S$) are intrinsically hard to classify, while the rest exhibit standard exponential error decay. When Assumption 2 holds, $|S| = 0$ and the lower bound matches the result in our Theorem 1. We will include this refined lower bound and discussion in the revised version. Extending the upper bound to this more general setting is plausible, but involves technical subtleties that we are currently investigating.\n\n2. - We do not make any assumption on the matrix $B$ (beyond being symmetric). As a result, our analysis derives the optimal error rate for a specific instance of PABM (similar to the instance-wise analysis in [Yun and Proutière, 2016] for the edge-labeled SBM) rather than a minimax error rate (as in [Zhang \\& Zhou, 2016] for SBM and [Gao, Ma, Zhang, Zhou, 2018] for DCBM). Both approaches are valuable: the minimax framework requires defining a parameter space to which $B$ belongs (typically the space (*) mentioned in the review), but offers no guarantees when $B$ lies outside this space, while the instance optimal-rate restricts to a specific but arbitrary matrix $B$. \n    -  Moreover, we wish to emphasize an important point: a rate-optimal algorithm in the minimax setting may not be rate-optimal for specific instances—even when those instances fall within the defined parameter space. For example, Lloyd’s algorithm is minimax-optimal over the class of sub-Gaussian mixture models [Lu \\& Zhou, 2016. Statistical and computational guarantees of Lloyd’s algorithm and its variants. arXiv:1612.02099], but it fails to be instance-optimal for Gaussian mixture models with anisotropic covariance structures [Chen \\& Zhang, 2024. Achieving optimal clustering in Gaussian mixture models with anisotropic covariance structures. NeurIPS 2024]. \n    -   For the parameter space (*), the worst-case rate for SBM and DCBM arises when $B_{aa} = p$ for all $a \\in [k]$ and $B_{ab} = q$ for all $a \\ne b$, leading to a minimax rate involving the term $(\\sqrt{p} - \\sqrt{q})^2$. In contrast, for PABM, the situation is more complex because of the additional dependence on the individual parameters $\\lambda_{ia}$. As a result, we do not believe a simple closed-form expression for the minimax rate in PABM is attainable. Indeed, as shown in Example 2, reducing the gap between $p$ and $q$ does not necessarily increases the optimal error-rate. \n\n3. -  To express our result in a minimax setting, one would need to define a parameter space $\\Theta$, and analyze the maximum error-rate over all instances of PABM belonging to this space. Because the MLE is (asymptotically) rate-optimal for each instance of PABM, the minimax rate is the maximum error rate over $\\Theta$. However, identifying this worst-case rate is, in general, intractable for most meaningful choices of $\\Theta$. For this reason, we focused on analyzing specific instances of PABM and did not pursue a full minimax characterization.\n     -   Our results recover known expressions in special cases (for example, the Chernoff-Hellinger divergence in SBM arises from a particular instance of a SBM, and the minimax rate for DCBM is also recovered when all intra-cluster connection probabilities equal $p$ and all inter-cluster probabilities equal $q$). In principle, one could derive information-theoretic results such as thresholds for exact recovery by analyzing when $n \\\\, \\mathrm{loss}(z^*, \\hat{z}) \\rightarrow 0 $. However, for PABM, this does not lead to a clean, interpretable formula, except in the case of SBM where the threshold is precisely given by the Chernoff-Hellinger divergence."}]

Author rebuttal:

Meta-review: This paper presents a strong theoretical contribution to the field of graph
clustering by establishing the information-theoretic limits of community
detection under the Popularity-Adjusted Block Model (PABM). The key insight
is that PABM, a generalization of the well-studied SBM and DCBM, allows for
cluster recovery even when traditional global edge-density signals are
absent, by leveraging more nuanced local differences in node connectivity
patterns. The authors provide tight, matching upper and lower bounds on the
optimal clustering error rate, a technically sound and significant result.

Initially, the reviewers, while recognizing the technical solidity, raised
several important concerns. These included the clarity of the novel
technical contributions compared to prior work, the potentially misleading
framing of "clustering without edge density signals," and a perceived
disconnect between the theoretical analysis based on the maximum likelihood
estimator and the experiments using spectral methods.

However, the authors provided rebuttals that thoroughly addressed these
points. They clarified that their main claim refers to the absence of a
global density signal, effectively shifting the focus to richer, local
structural information. Crucially, they detailed the novel technical
aspects of their proofs, particularly for the upper bound, which requires a
more refined analysis than for simpler models. Their arguments successfully
demonstrated that the empirical results, while using a practical algorithm,
serve to validate the non-intuitive theoretical phenomena uncovered by
their analysis. This comprehensive response led multiple reviewers to raise
their scores, resulting in a strong consensus for acceptance.

And the current submission is filled with mathematics, while missing
important explanations that tell readers the motivations, importance and
understandings of these theoretical analyses.

Overall, it should be noted that the current state of this submission has
not yet met the quality demand of NeurIPS. The final version must soundly
integrate these explanations and additional clarifications to meet the
quality of NeurIPS publication.
Decision: Accept (poster)