Hypothesis testing for community structure in temporal networks using e-values

Community structure in networks naturally arises in various applications. But while the topic has received significant attention for static networks, the literature on community structure in temporally evolving networks is more scarce. In particular, there are currently no statistical methods available to test for the presence of community structure in a sequence of networks evolving over time. In this work, we propose a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways. Specifically, an e-value framework retains valid testing properties even after combining dependent information, a relevant feature in the context of testing temporal networks. We apply the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

💡 Research Summary

The paper addresses a gap in the statistical analysis of dynamic networks: there is no established hypothesis test for the presence of community structure in a sequence of temporally evolving graphs. While static networks have a rich literature on community detection and related hypothesis testing, extending these ideas to the temporal domain is non‑trivial because successive snapshots are typically dependent and the definition of “no community structure” must be adapted.

To overcome these challenges, the authors propose a simple yet powerful testing framework based on e‑values, an alternative to p‑values whose key property is that its expectation under the null hypothesis does not exceed one. The procedure consists of four steps: (1) apply any valid static community‑structure test to each snapshot G(t) (t = 1,…,T) and obtain a p‑value P_t; (2) transform each p‑value into an e‑value E_t using a calibrator function g(·) (the paper discusses three calibrators: a parametric family g_κ, a “max” version, and an averaged version g_avg); (3) combine the e‑values by taking their arithmetic mean (\bar{E}T = \frac{1}{T}\sum{t=1}^T E_t); (4) reject the null hypothesis of “no community structure” when (\bar{E}_T) is sufficiently large (the authors suggest a threshold around 20, which yields a type‑I error ≤0.05).

The theoretical contribution is modest but essential: Theorem 1 shows that the average of any collection of e‑values is itself an e‑value, regardless of the dependence structure among the snapshots. Consequently, the combined statistic retains valid type‑I error control without any assumptions about temporal dependence. The framework is flexible: users may assign weights to snapshots (e.g., giving more recent graphs higher weight) or, if independence can be justified, multiply e‑values instead of averaging.

The null model is taken to be the Erdős–Rényi (ER) random graph, which encodes the absence of any community structure. The authors acknowledge that ER may be overly simplistic for many real networks and note that the method can be readily adapted to other null models such as the configuration model or stochastic block model (SBM). The alternative hypothesis is left implicit; it is defined by the static community‑detection test chosen for each snapshot (e.g., the Bickel‑Sarkar 2016 test).

Empirical evaluation includes both synthetic experiments and real‑world case studies. In simulations, three data‑generating mechanisms (pure ER, SBM with planted communities, and a variant block model) are combined with two static tests, and three calibrators are compared. Results demonstrate that the e‑value‑based test achieves high power while maintaining the prescribed false‑positive rate, even when snapshots are strongly correlated. Real‑world applications (social interaction networks, transportation flow networks, biological interaction networks) illustrate how (\bar{E}_T) varies over time, revealing periods of heightened community cohesion or dissolution. These examples also highlight practical issues such as the choice of static test, calibrator, and null model.

The discussion acknowledges several limitations. First, the notion of “community structure” is not universally defined; the practitioner must select an appropriate static test that captures the desired notion. Second, the ER null may be too crude, potentially leading to over‑rejection; more realistic nulls can be substituted without changing the core methodology. Third, the choice of calibrator influences the magnitude of e‑values, and systematic guidance for this choice remains an open question. Finally, while the paper focuses on simple averaging, more sophisticated aggregation (e.g., weighted averages, sliding windows, Bayesian updating) could further improve sensitivity to gradual changes.

In summary, the authors introduce the first hypothesis‑testing framework for community structure in temporal networks that leverages the combination properties of e‑values. By converting dependent p‑values into e‑values and averaging them, the method provides a theoretically sound, computationally straightforward, and empirically effective tool for detecting dynamic community patterns across a wide range of networked systems.