Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matrix Concentration for Random Signed Graphs and Community Recovery in the Signed Stochastic Block Model

Sawyer Jack Robertson

TL;DR

This work develops sharp matrix concentration results for random signed graphs, establishing that both the adjacency and Laplacian matrices concentrate near their means under tractable sparsity and sign-assignment conditions. Applying these tools to the two-community signed stochastic block model, the authors show that the normalized signed Laplacian's spectral gap concentrates near $2s$ and that the sign of the leading eigenvector provides a weakly consistent estimator for balanced community detection (i.e., $\pm 1$ synchronization). They analyze both dense ($p,q=\Theta(k^{-1/2})$) and sparse ($p,q=\Theta(\log k/k)$) regimes, deriving explicit eigenvalue and eigenvector concentration bounds and validating them with experiments. The results bridge matrix concentration theory with structured random graphs, offering principled spectral procedures for community detection in signed networks and insights into synchronization under random sign perturbations.

Abstract

We consider graphs where edges and their signs are added independently at random from among all pairs of nodes. We establish strong concentration inequalities for adjacency and Laplacian matrices obtained from this family of random graph models. Then, we apply our results to study graphs sampled from the signed stochastic block model. Namely, we take a two-community setting where edges within the communities have positive signs and edges between the communities have negative signs and apply a random sign perturbation with probability $0< s <1/2$. In this setting, our findings include: first, the spectral gap of the corresponding signed Laplacian matrix concentrates near $2s$ with high probability; and second, the sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for the balanced community detection problem, or equivalently, the $\pm 1$ synchronization problem. We supplement our theoretical contributions with experimental data obtained from the models under consideration.

Matrix Concentration for Random Signed Graphs and Community Recovery in the Signed Stochastic Block Model

TL;DR

This work develops sharp matrix concentration results for random signed graphs, establishing that both the adjacency and Laplacian matrices concentrate near their means under tractable sparsity and sign-assignment conditions. Applying these tools to the two-community signed stochastic block model, the authors show that the normalized signed Laplacian's spectral gap concentrates near and that the sign of the leading eigenvector provides a weakly consistent estimator for balanced community detection (i.e., synchronization). They analyze both dense () and sparse () regimes, deriving explicit eigenvalue and eigenvector concentration bounds and validating them with experiments. The results bridge matrix concentration theory with structured random graphs, offering principled spectral procedures for community detection in signed networks and insights into synchronization under random sign perturbations.

Abstract

We consider graphs where edges and their signs are added independently at random from among all pairs of nodes. We establish strong concentration inequalities for adjacency and Laplacian matrices obtained from this family of random graph models. Then, we apply our results to study graphs sampled from the signed stochastic block model. Namely, we take a two-community setting where edges within the communities have positive signs and edges between the communities have negative signs and apply a random sign perturbation with probability . In this setting, our findings include: first, the spectral gap of the corresponding signed Laplacian matrix concentrates near with high probability; and second, the sign of the first eigenvector of the Laplacian matrix defines a weakly consistent estimator for the balanced community detection problem, or equivalently, the synchronization problem. We supplement our theoretical contributions with experimental data obtained from the models under consideration.
Paper Structure (17 sections, 22 theorems, 77 equations, 11 figures)

This paper contains 17 sections, 22 theorems, 77 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Outline of this article
  4. Background
  5. Signed graphs, community detection, and synchronization
  6. Random graphs and the SSBM
  7. Concentration for signed adjacency and Laplacian matrices
  8. Signed adjacency matrices
  9. Signed Laplacian matrices
  10. Application: Spectral properties of the SSBM
  11. The dense setting
  12. The sparse setting
  13. Experimental data
  14. Missing proofs
  15. Proofs from \ref{['sec:background']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $A$ be a random signed adjacency matrix on $n$ nodes. Assume $\max_{i, j}|s_{ij} - 1/2| > c_0$ for some $c_0>0$. Assume that for some $\alpha\geq c_1\log{n}$ and $c_1 >0$. Then for any $r>0$ there exists $C = C(r, c_0, c_1) >0$ such that for $n$ sufficiently large and with probability at least $1 - n^{-r}$, it holds

Figures (11)

  • Figure 1: A diagram describing the SSBM in the two-community setting (see \ref{['defn:signed-bisection']}).
  • Figure 2: Three signed graphs sampled from $\mathsf{SignedBlock}_{k=50}(10\log{k}/k, \log{k}/k, s)$ for various choices of $s$. Red edges (resp. blue edges) are negatively signed (resp. positively signed).
  • Figure 3: Eigenvalue histograms corresponding to the normalized signed Laplacian matrices of the three graphs shown in (a). We note the correspondence between the least eigenvalue of each graph and $2s$.
  • Figure 5: A histogram of the eigenvalues of an observed random signed adjacency matrix $A$ and Laplacian matrix $\mathcal{L}$ obtained from the dense setting. Here, we take $k=10^3$, $\gamma_1=10$, $\gamma_2=1$, $s=0.1$. In this case, $\lambda_{2k}(A) \approx 278.59$ and $\lambda_1(\mathcal{L})\approx 0.2$, illustrating the concentration near $(1-2s)(\gamma_1+\gamma_2)\sqrt{k}\approx 278.28$ and $2s=0.2$, respectively (see \ref{['cor:eigvals-adjacency', 'cor:eigvals-concentration-lap-dense']}).
  • Figure 6: A histogram of the eigenvalues of an observed random signed adjacency matrix $A$ and Laplacian matrix $\mathcal{L}$ obtained from the sparse setting. Here, we take $k=10^3$, $\gamma_1=10$, $\gamma_2=1$, $s=0.1$. In this case, $\lambda_{2k}(A) \approx 61.61$ and $\lambda_1(\mathcal{L})\approx 0.19$, illustrating the concentration near $(1-2s)(\gamma_1+\gamma_2)\log{k}\approx 60.78$ and $2s=0.2$, respectively (see \ref{['cor:sharp-eigvals-adjacency', 'cor:eigvals-concentration-lap']}).
  • ...and 6 more figures

Theorems & Definitions (34)

  • Theorem 1.1: Informal statement of \ref{['th:adjacency-matrix']}
  • Theorem 1.2: Informal statement of \ref{['cor:eigvals-concentration-lap-dense']} and \ref{['cor:eigvals-concentration-lap']}
  • Theorem 1.3: Informal statement of \ref{['cor:eigvals-concentration-lap-dense']} and \ref{['cor:eigvals-concentration-lap']}
  • Theorem 2.1: Lei and Rinaldo, lei2015consistencysupplei2015consistency
  • Definition 2.2: Signed stochastic block model
  • Definition 2.3: Signed stochastic bisection model
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 24 more