Spectral detection of simplicial communities via Hodge Laplacians

Abstract

While the study of graphs has been very popular, simplicial complexes are relatively new in the network science community. Despite being are a source of rich information, graphs are limited to pairwise interactions. However, several real world networks such as social networks, neuronal networks etc. involve simultaneous interactions between more than two nodes. Simplicial complexes provide a powerful mathematical way to model such interactions. Now, the spectrum of the graph Laplacian is known to be indicative of community structure, with nonzero eigenvectors encoding the identity of communities. Here, we propose that the spectrum of the Hodge Laplacian, a higherorder Laplacian applied to simplicial complexes, encodes simplicial communities. We formulate an algorithm to extract simplicial communities (of arbitrary dimension). We apply this algorithm on simplicial complex benchmarks and on real data including social networks and language-networks, where higher-order relationships are intrinsic. Additionally, datasets for simplicial complexes are scarce. Hence, we introduce a method of optimally generating a simplicial complex from its network backbone through estimating the true higher-order relationships when its community structure is known. We do so by using the adjusted mutual information to identify the configuration that best matches the expected data partition. Lastly, we demonstrate an example of persistent simplicial communities inspired by the field of persistence homology.

Hyper-diffusion on multiplex networks

“…(38) for the eigenvalues of L and then generalise the results for the eigenvalues of L norm (Eq. (43). Assume that the zero eigenvalue of L has degeneracy s − 1.…”

“…The research on multiplex and higher-order networks has rapidly expanded in recent years and it has emerged that going beyond simple pairwise interactions significantly enrich our ability to describe the interplay between network structure and dynamics. Indeed dynamics on multiplex network [1] and on higher-order networks [5,6,8] display significantly different behaviours than the corresponding dynamics defined on single pairwise networks and has effect on percolation [10,11], contagion models [12][13][14][15][16][17] game theory models [18], synchronisation [19][20][21][22][23][24][25][26][27] and diffusion models [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Dynamics on higher-order networks: a review

“…The efficiency of this mechanism is validated for edge-label community detection and clustering with time-stamped data. Simplicial communities are detected from real-world data of social networks while showing that the spectra of the Hodge Laplacian encodes the communities [ 44 ]. A stochastic generative model is introduced to hypernetwork clustering with heterogeneous node degree and hyperlink size distribution [ 45 ]; this is shown to be highly scalable and efficient with the utilization of synthetic and various real-world data.…”

Dynamics on higher-order networks: A review

“…The efficiency of this mechanism is validated for edge-label community detection and clustering with time-stamped data. Simplicial communities are detected from real-world data of social networks while showing that the spectra of the Hodge Laplacian encodes the communities [44]. A stochastic generative model is introduced to hypernetwork clustering with heterogeneous node degree and hyperlink size distribution [45], which is shown to be highly scalable and efficient with the utilization of synthetic and various real-world data.…”