Multi-layer Stochastic Co-block Model

• The paper introduces ML-ScBM, a framework that models asymmetric latent structures by assigning separate sender and receiver communities for each network layer.
• It employs spectral co-clustering with debiased aggregation of Gram matrices to overcome community cancellation in multilayer settings.
• The work provides statistical guarantees and sequential tests for estimating distinct numbers of sender and receiver communities, highlighting the benefits of multiple layers.

Searching arXiv for recent and directly relevant papers on multi-layer stochastic co-block models and neighboring multilayer block-model formulations. Multi-layer stochastic co-block model (ML-ScBM) denotes a class of latent block models for multi-layer directed or bipartite networks in which each layer has its own block connectivity matrix, but the latent structure is asymmetric: nodes are assigned separate sender/row and receiver/column communities, and the two community counts may differ. In the directed aligned-node setting, a layer-$\ell$ edge satisfies $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$; in the bipartite degree-corrected setting, $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$. The model therefore differs from a standard multilayer SBM, which uses one common partition, by explicitly representing outgoing and incoming patterns or row and column communities separately (Qing, 25 Feb 2026, Su et al., 2023, Qing, 2024).

1. Core concept and asymmetric latent structure

The defining feature of ML-ScBM is the separation between sending and receiving roles. In the formulation for multi-layer directed networks, one observes $L$ directed layers on the same $n$ nodes, with adjacency matrices

$$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$

and

$$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$

Here $g^s$ gives sender community labels, $g^r$ gives receiver community labels, and $B^{(\ell)}\in[0,1]^{K_s\times K_r}$ is the layer-specific block probability matrix. The key structural point is that a node may send edges like one community but receive edges like another, so $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$0 is natural rather than exceptional (Qing, 25 Feb 2026).

In the directed co-clustering formulation of spectral community detection, the same asymmetry is written in row/column notation. There are $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$1 row clusters and $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$2 column clusters, membership matrices $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$3 and $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$4, and layer-specific block matrices $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$5. The model is

$$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$6

with population matrix

$$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$7

In this representation, row clusters encode who sends similarly, while column clusters encode who receives similarly (Su et al., 2023).

For multi-layer bipartite networks, the asymmetry is attached to two distinct node sets rather than to two roles played by the same node set. The degree-corrected model uses $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$8 row nodes, $$\mathbb P(A^{(\ell)}_{ij}=1)=B^{(\ell)}(g^s(i),g^r(j))$$9 column nodes, row membership matrix $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$0, column membership matrix $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$1, degree parameters $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$2, and layer-specific block matrices $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$3. The model is

$\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$4

This formulation reduces to the degree-corrected stochastic co-block model when $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$5, to the stochastic co-block model when the degree parameters are constant, and to multilayer SBM when the network is undirected and $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$6 (Qing, 2024).

2. Canonical multilayer formulations

The directed ML-ScBM and the multilayer degree-corrected stochastic co-block model share the assumption that layers are conditionally independent given latent communities, but they organize asymmetry differently. In the aligned-node directed setting, the same node set appears on both sides of each adjacency matrix, and asymmetry comes from distinct sending and receiving clusterings. In the bipartite setting, each layer is an $\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$7 matrix, and asymmetry is anchored in separate row and column node types (Su et al., 2023, Qing, 2024).

Population geometry is central to both formulations. For directed co-clustering, the row-side and column-side aggregate population matrices are

$\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$8

and

$\Omega_l=\Theta_r Z_r B_l Z_c^\top \Theta_c$9

The leading eigenvector matrices of these population operators are

$L$0

If the aggregated block matrix is full rank, distinct clusters correspond to distinct rows of $L$1 or $L$2; if it is rank deficient, cluster recovery still remains possible under explicit separation conditions. This relaxes the common assumption that the embedding rank must equal the number of clusters (Su et al., 2023).

For the multilayer degree-corrected stochastic co-block model, the key population matrices are

$L$3

A key lemma states that, under suitable rank conditions, the row-normalized eigenvectors of these matrices have exactly one distinct row per community. In particular, if $L$4, then

$L$5

and when $L$6, different row communities are separated by $L$7 in the normalized eigenspace (Qing, 2024).

These formulations show that ML-ScBM is not defined merely by multiple adjacency matrices. Its distinctive object is a multilayer collection of block operators with separate latent structures on the two sides of interaction.

3. Spectral co-clustering and debiased aggregation

A major algorithmic theme in ML-ScBM is that naive layer aggregation can be misleading. In the directed setting, direct summation $L$8 can suffer from community cancellation: different layers may offset each other and destroy cluster signal. To avoid this, the spectral method developed in (Su et al., 2023) aggregates Gram matrices rather than adjacency matrices: $L$9 Because diagonal terms create bias, the method subtracts degree diagonals and forms the debiased sum-of-Gram matrices

$n$0

The algorithm, called DSoG, computes leading eigenvectors of $n$1 and $n$2, then applies $n$3-means to their rows. It also allows the embedding ranks $n$4 and $n$5, so rank need not equal the number of sender or receiver communities (Su et al., 2023).

The bipartite analogue is NcDSoS, normalized spectral co-clustering based on debiased sum of squared matrices. Its observed operators are

$n$6

followed by spectral decomposition, row normalization, and $n$7-means on the normalized eigenvector rows. The paper explicitly contrasts NcDSoS with non-debiased variants, arguing that the diagonal degree terms inflate the spectral norm error and that subtracting $n$8 and $n$9 gives a tighter approximation to the population matrices (Qing, 2024).

These constructions indicate that second-order aggregation is structurally important in multilayer asymmetric networks. This suggests that, in ML-ScBM, the informative signal often lives more naturally in layer-aggregated covariance-type operators than in raw adjacency sums.

4. Estimating the numbers of sender and receiver communities

A distinctive inferential problem in ML-ScBM is that the number of sender communities and the number of receiver communities must be estimated separately. In (Qing, 25 Feb 2026), this is formulated as a sequence of goodness-of-fit tests for candidate pairs $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$0: $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$1 The alternative is underfitting: the candidate model merges true sender or receiver communities.

The test is built from a normalized residual matrix. With estimated labels $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$2 and estimated block probabilities $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$3, the empirical residual matrix is

$$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$4

with

$$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$5

The largest singular value is used because it measures the strongest leftover low-rank structure after fitting the candidate model. Under the correct model, $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$6 is near $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$7; under underfitting, $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$8 diverges to infinity (Qing, 25 Feb 2026).

Two procedures are then built from this dichotomy. MLDiGoF searches candidate pairs in lexicographic order, first by increasing $$A^{(\ell)} \in \{0,1\}^{n\times n}, \qquad \ell=1,\dots,L,\quad A^{(\ell)}_{ii}=0,$$9 and then by smaller $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$0, and stops at the first pair for which $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$1, with default threshold $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$2. MLRDiGoF instead examines ratios

$$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$3

and returns the first candidate where $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$4, with default $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$5. Both procedures are proved to consistently recover $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$6 under the multi-layer stochastic co-block model (Qing, 25 Feb 2026).

This testing framework addresses a limitation that remains explicit in several spectral co-clustering papers, namely that $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$7 and $$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$8 are treated as known inputs (Qing, 2024).

5. Statistical guarantees and the benefit of multiple layers

A recurring theoretical conclusion is that additional layers improve recovery when the latent row/column structure is shared across layers and the block matrices vary by layer. In the directed spectral co-clustering analysis, the misclassification bounds for row and column clustering are

$$\mathbb{P}\!\left(A^{(\ell)}_{ij}=1\right)=B^{(\ell)}\!\bigl(g^s(i),g^r(j)\bigr), \qquad i\neq j.$$9

and

$g^s$0

under the stated assumptions. The bounds improve as $g^s$1 increases, as $g^s$2 increases, and as the eigenspace separation parameters $g^s$3 increase. The paper emphasizes that multi-layers would bring benefits to the clustering performance, and that weak signal in one layer can be compensated by stronger signal in other layers (Su et al., 2023).

For the multilayer degree-corrected stochastic co-block model, the central theorem yields consistency, and under balanced communities and homogeneous degree scaling the cleaner corollary gives

$g^s$4

$g^s$5

If $g^s$6, then

$g^s$7

These rates make the role of $g^s$8 explicit: error decreases as the number of layers increases (Qing, 2024).

For model selection, (Qing, 25 Feb 2026) proves a sharp dichotomy for $g^s$9: under the null,

$g^r$0

while under underfitting,

$g^r$1

The sequential procedures based on this dichotomy are consistent for the true $g^r$2.

Together, these results indicate that multilayer information improves inference in at least two distinct senses: it reduces clustering error for fixed community counts, and it supports separate estimation of sender and receiver community numbers.

6. Relation to neighboring multilayer block models

ML-ScBM is often discussed alongside a broader multilayer block-model literature, but several neighboring models are not co-block models in the strict sense. The multilayer SBM of (Valles-Catala et al., 2014) treats an observed network as the aggregate of hidden interaction layers and allows different group assignments in different layers for the same node set, yet the asymmetry is across layers rather than across node types. The strata multilayer SBM of (Stanley et al., 2015) clusters layers into strata that share a common SBM, coupling node-to-community and layer-to-stratum assignments, but it still uses SBM structure rather than separate row and column communities. The goodness-of-fit test in (Qing, 7 Aug 2025) and the connectivity-matrix inference theory in (Su et al., 2024) concern standard multilayer SBM with common node memberships across layers, not a co-block model.

The same distinction appears in mixed-membership and weighted multiplex settings. The multi-layer mixed membership stochastic block model of (Qing, 2024) uses a common mixed-membership matrix $g^r$3 across layers and does not introduce distinct row/column memberships. The multiplex Dirichlet stochastic block model of (Promskaia et al., 2024) models directed compositional edge weights and has a clear co-block flavor through expected block-to-block shares $g^r$4, but it uses a single shared partition of nodes across layers rather than separate sender and receiver clusterings. The partially exchangeable SBM of (Durante et al., 2024) is designed for node-colored multilayer networks and introduces a hierarchical partition prior coherent with within- and across-layer block-connectivity structures, yet its novelty lies in partial exchangeability rather than in stochastic co-blocking.

This suggests a useful terminological boundary. “Multi-layer stochastic co-block model” is most precise when it refers to multilayer asymmetric block models with distinct latent structure on the two sides of interaction, as in directed sender/receiver models and bipartite row/column models (Qing, 25 Feb 2026, Su et al., 2023, Qing, 2024). By contrast, many multilayer SBMs are related in spirit because they share latent structure across layers, but they do not represent two-sided asymmetry and therefore occupy a different part of the multilayer block-model landscape.