[Paper Review] Adjusted chi-square test for degree-corrected block models

Linfan Zhang, Arash A. Amini|arXiv (Cornell University)|Dec 30, 2020

Complex Network Analysis Techniques64 references4 citations

TL;DR

This paper proposes an adjusted chi-square test for goodness-of-fit in degree-corrected stochastic block models (DCSBM), leveraging a compressed adjacency matrix conditional on degrees to enable scalability in large sparse networks. The test maintains asymptotic validity under null hypotheses when the harmonic mean of node degrees grows, demonstrates high power against diverse alternatives—including latent-variable models—and enables consistent community detection via sequential testing, with empirical validation on real networks like Facebook-100 showing widespread lack of fit for small-community DCSBM.

ABSTRACT

We propose a goodness-of-fit test for degree-corrected stochastic block models (DCSBM). The test is based on an adjusted chi-square statistic for measuring equality of means among groups of $n$ multinomial distributions with $d_1,\dots,d_n$ observations. In the context of network models, the number of multinomials, $n$, grows much faster than the number of observations, $d_i$, corresponding to the degree of node $i$, hence the setting deviates from classical asymptotics. We show that a simple adjustment allows the statistic to converge in distribution, under null, as long as the harmonic mean of $\{d_i\}$ grows to infinity. When applied sequentially, the test can also be used to determine the number of communities. The test operates on a compressed version of the adjacency matrix, conditional on the degrees, and as a result is highly scalable to large sparse networks. We incorporate a novel idea of compressing the rows based on a $(K+1)$-community assignment when testing for $K$ communities. This approach increases the power in sequential applications without sacrificing computational efficiency, and we prove its consistency in recovering the number of communities. Since the test statistic does not rely on a specific alternative, its utility goes beyond sequential testing and can be used to simultaneously test against a wide range of alternatives outside the DCSBM family. In particular, we prove that the test is consistent against a general family of latent-variable network models with community structure.

Motivation & Objective

• To develop a goodness-of-fit test for degree-corrected stochastic block models (DCSBM) that remains valid under non-i.i.d. settings where node degrees vary significantly.
• To address the challenge of testing DCSBM fit in large sparse networks where classical asymptotics fail due to growing number of nodes relative to individual degrees.
• To enable consistent community detection by sequentially applying the test to determine the optimal number of communities.
• To create a test that is robust and powerful against a wide range of alternatives beyond the DCSBM family, including latent-variable network models.
• To provide a scalable, computationally efficient tool for exploratory network analysis, especially for real-world networks where DCSBM may not be a good fit.

Proposed method

• The test uses an adjusted chi-square statistic to compare multinomial means across groups, where each group corresponds to a node’s degree-conditional edge distribution.
• It operates on a compressed version of the adjacency matrix, grouping rows based on a (K+1)-community assignment when testing for K communities, enhancing power without sacrificing efficiency.
• The adjustment to the chi-square statistic ensures convergence in distribution under the null hypothesis as long as the harmonic mean of node degrees tends to infinity.
• The method conditions on observed degrees, making it suitable for sparse networks and preserving the network’s degree heterogeneity.
• The test is applied sequentially to determine the number of communities by identifying the smallest K where the null hypothesis of a K-community DCSBM is not rejected.
• The approach is extended to Poisson count arrays, bipartite, and directed networks by adapting the multinomial framework to appropriate exponential family distributions.

Experimental results

Research questions

• RQ1 Can a goodness-of-fit test for DCSBM be constructed that remains valid when the number of nodes grows faster than individual node degrees?
• RQ2 Does the proposed adjusted chi-square test maintain asymptotic validity under the null hypothesis when node degrees are heterogeneous and sparse?
• RQ3 Can the test be used to consistently recover the true number of communities in a network, even when the model is misspecified?
• RQ4 How powerful is the test against alternatives outside the DCSBM family, such as latent-variable models with community structure?
• RQ5 Can the test be applied effectively to real-world networks where DCSBM is commonly assumed but may not be a good fit?

Key findings

• The adjusted chi-square test converges in distribution under the null hypothesis as long as the harmonic mean of node degrees tends to infinity, ensuring asymptotic validity in sparse settings.
• The test demonstrates high power against a broad class of alternatives, including latent-variable network models with community structure, even without specifying the direction of deviation.
• In the Facebook-100 dataset, a DCSBM with fewer than 25 communities is strongly rejected in almost all networks, indicating widespread lack of fit.
• The test's statistic itself serves as a powerful exploratory tool: community profile plots reveal structural patterns such as single or multiple elbows, aiding in visualizing community structure.
• Sequential application of the test consistently recovers the number of communities, with FNAC+ and AS tests achieving near-perfect accuracy in distinguishing DCSBM from DCLVM alternatives when sample size is large.
• The test remains effective even when the true model has the same number of communities as the null (e.g., K=4), showing it can detect model misspecification beyond just community count differences.