Polaris: Sampling from the Multigraph Configuration Model with Prescribed Color Assortativity
Giulia Preti, Matteo Riondato, Aristides Gionis, Gianmarco De Francisci Morales
TL;DR
Polaris addresses polarization analysis by defining a null ensemble of colored multigraphs that strictly preserves the degree sequence and Joint Color Matrix (JCM) of an observed network. It provides two Metropolis-Hastings–based samplers, Polaris-B and Polaris-C, to draw from this constrained space using DES and JDES moves, with theoretical guarantees of ergodicity and correctness. Empirical results show that traditional configuration-model nulls poorly preserve color assortativity, while Polaris-C delivers higher acceptance rates and faster mixing across datasets with many colors. The work enables statistically sound significance testing of polarization patterns in social networks and lays groundwork for broader analyses of color-based interaction structures.
Abstract
We introduce Polaris, a network null model for colored multi-graphs that preserves the Joint Color Matrix. Polaris is specifically designed for studying network polarization, where vertices belong to a side in a debate or a partisan group, represented by a vertex color, and relations have different strengths, represented by an integer-valued edge multiplicity. The key feature of Polaris is preserving the Joint Color Matrix (JCM) of the multigraph, which specifies the number of edges connecting vertices of any two given colors. The JCM is the basic property that determines color assortativity, a fundamental aspect in studying homophily and segregation in polarized networks. By using Polaris, network scientists can test whether a phenomenon is entirely explained by the JCM of the observed network or whether other phenomena might be at play. Technically, our null model is an extension of the configuration model: an ensemble of colored multigraphs characterized by the same degree sequence and the same JCM. To sample from this ensemble, we develop a suite of Markov Chain Monte Carlo algorithms, collectively named Polaris-*. It includes Polaris-B, an adaptation of a generic Metropolis-Hastings algorithm, and Polaris-C, a faster, specialized algorithm with higher acceptance probabilities. This new null model and the associated algorithms provide a more nuanced toolset for examining polarization in social networks, thus enabling statistically sound conclusions.