Community-Size Biases in Statistical Inference of Communities in Temporal Networks

TL;DR

The paper investigates how priors over community assignments shape Bayesian inference of communities in temporal networks, identifying biases that favor moderate-sized communities under uniform or Markov-based priors. It introduces a layerwise-exchangeable count-splitting (LECS) prior that uses cross-layer exchangeability and geometric retention to reduce localization of community sizes over time. The authors provide theoretical results showing LECS mitigates localization and demonstrate through simulations that LECS-based temporal SBMs outperform existing Markov-process priors in recovering small or large communities. The work highlights the importance of realistic generative models for mesoscale inference in evolving networks and offers practical, open-source tools for benchmarking. The LECS framework also opens avenues for extending similar exchangeability-based priors to other temporal mesoscale structures.

Abstract

In the study of time-dependent (i.e., temporal) networks, researchers often examine the evolution of communities, which are sets of densely connected sets of nodes that are connected sparsely to other nodes. An increasingly prominent approach to studying community structure in temporal networks is statistical inference. In the present paper, we study the performance of a class of statistical-inference methods for community detection in temporal networks. We represent temporal networks as multilayer networks, with each layer encoding a time step, and we illustrate that statistical-inference models that generate community assignments via either a uniform distribution on community assignments or discrete-time Markov processes are biased against generating communities with large or small numbers of nodes. In particular, we demonstrate that statistical-inference methods that use such generative models tend to poorly identify community structure in networks with large or small communities. To rectify this issue, we introduce a novel statistical model that generates the community assignments of the nodes in given layer (i.e., at a given time) using all of the community assignments in the previous layer. We prove results that guarantee that our approach greatly mitigates the bias against large and small communities, so using our generative model is beneficial for studying community structure in networks with large or small communities. Our code is available at https://github.com/tfaust0196/TemporalCommunityComparison.

Figures (11)

• Figure 1: An example of the community-size frequency histogram for a 50-node monolayer network. The horizontal axis indicates the number of nodes in the network with community assignment $1$, and the vertical axis indicates the observed frequency of community assignments with that number of nodes in community $1$.
• Figure 1: Heat maps of (a) an example of actual community structure and (b) an illustration of the permuted community structure that we observe commonly at local maxima of $\mathbb{P}(g|A)$ in a 100-node network with 5 layers. Each rectangle in a heat map corresponds to one node-layer $(i,\ell)$. Dark blue rectangles signify the community assignment $g_{(i,\ell)} = 1$, and light blue rectangles signify the community assignment $g_{(i,\ell)} = 2$.
• Figure 1: Heat maps of the seeded community structure that we use to generate the adjacency structure $A$ for (a) community-1 size $q = 50$ and (b) community-1 size $q = 90$. Each rectangle in a heat map corresponds to one node-layer $(i,\ell)$. Dark blue rectangles signify the community assignment $g_{(i,\ell)} = 1$, and light blue rectangles signify the community assignment $g_{(i,\ell)} = 2$.
• Figure 2: An example of the community-size histograms in a 50-node temporal network with $5$ layers. In both panels (a) and (b), the horizontal axis indicates the number of node-layers with community assignment $1$, and the vertical axis indicates the observed frequency of community assignments with that number of node-layers in community $1$. In the histograms in (a), we consider only the node-layers in the indicated layer when counting the number of nodes in community $1$. In the histogram in (b), we consider all of the node-layers in the network when counting the number of nodes in community $1$.
• Figure 2: The mean NMI for our LECS-prior-based approach and the Bazzi et al. approach with and without multilayer swaps for several values of the community-1 size $q$.

Theorems & Definitions (17)

• Proposition 3.1
• Theorem 4.1
• Proof 1
• Corollary 4.2
• Proof 2: Proof of Corollary \ref{['ArashCorollary']}
• Lemma A.1
• Proof 3
• Proposition B.1
• Proof 4
• Proposition C.1: Uniform distribution on community assignments