Clique Community Persistence: A Topological Visual Analysis Approach for Complex Networks

Abstract

Complex networks require effective tools and visualizations for their analysis and comparison. Clique communities have been recognized as a powerful concept for describing cohesive structures in networks. We propose an approach that extends the computation of clique communities by considering persistent homology, a topological paradigm originally introduced to characterize and compare the global structure of shapes. Our persistence-based algorithm is able to detect clique communities and to keep track of their evolution according to different edge weight thresholds. We use this information to define comparison metrics and a new centrality measure, both reflecting the relevance of the clique communities inherent to the network. Moreover, we propose an interactive visualization tool based on nested graphs that is capable of compactly representing the evolving relationships between communities for different thresholds and clique degrees. We demonstrate the effectiveness of our approach on various network types.

Memory sequencing reveals heritable single cell gene expression programs associated with distinct cellular behaviors

“…Within this network, we sought tightly connected groups of nodes within this network known as network communities . We performed k -clique community detection Jonsson et al, 2006;Palla et al, 2005) with k=4 on binarized networks created by thresholding the original weighted network at decreasing values (Giusti et al, 2015;Rieck et al, 2018) . More specifically, a k -clique is a collection of k nodes that are all-to-all connected and a k-clique community is a collection of k -cliques that are connected through adjacent k -cliques (two k-cliques are adjacent if they share k-1 nodes).…”

Persistence homology of networks: methods and applications

“…This filtration is originally defined in [24] for just the 0-dimension, however it can be extended to higher dimensions as well. Let G = (V, E ) be a graph with a vertex weight function ω : V → R. For this filtration, we use vertex weights, instead of the edge weights, as threshold values.…”

“…This filtration is used to detect the evolution of k-clique communities for a fixed k in [24]. In this filtration, we assume the graph G = (V, E ) has a vertex weight function ω :…”

“…Cliques and cycles are important structural features of complex networks to describe their cohesive structures. Rieck et al [24] use persistent homology to detect clique communities and their evolution in weighted networks. Persistent diagram is created using the vertex-based clique filtration (VBCL) and the k-clique filtration (kCL).…”

“…Ignacio et al [44] analyze the patterns and shapes in remittance and migration networks as a directed weighted network via persistent homology to identify flow patterns [41] VR 0 dim PD Brain networks [39] WS 0-2 dim PD Brain Networks [42] VR 0-dim PD Brain Networks [44] VR 0-2 dim PD Migration and remittance networks [37] VR 1 dim PB Simulated idiotypic networks [43] TMP 1-2 dim PD Co-occurrence networks [45] POW 0 dim PB Co-occurrence networks [24] VBCL,kCL 0 dim PD Co-occurrence, brain and collaboration network between multiple countries. They detect both local and global patterns that highlight simultaneous interactions among multiple nodes.…”