Interplay between $$k$$-core and community structure in complex networks

Abstract

The organisation of a network in a maximal set of nodes having at least k neighbours within the set, known as $$k$$ k -core decomposition, has been used for studying various phenomena. It has been shown that nodes in the innermost $$k$$ k -shells play a crucial role in contagion processes, emergence of consensus, and resilience of the system. It is known that the $$k$$ k -core decomposition of many empirical networks cannot be explained by the degree of each node alone, or equivalently, random graph models that preserve the degree of each node (i.e., configuration model). Here we study the $$k$$ k -core decomposition of some empirical networks as well as that of some randomised counterparts, and examine the extent to which the $$k$$ k -shell structure of the networks can be accounted for by the community structure. We find that preserving the community structure in the randomisation process is crucial for generating networks whose $$k$$ k -core decomposition is close to the empirical one. We also highlight the existence, in some networks, of a concentration of the nodes in the innermost $$k$$ k -shells into a small number of communities.

A Transcriptome Community-and-Module Approach of the Human Mesoconnectome

“…We found a logarithmic correlation between k-core and eigenvector, with an apparent convergence to a k-core value (72 and 86 for D1 and D2, respectively), something previously suggested as typical in multilayer networks caused by the occurrence of efficient information distributor nodes within one layer that is prevalent between layers [98] in the mesoconnectome. Furthermore, the pronounced presence of k-cores leads towards a modular topology of the mesoconnectome, a feature that has been reported in [99]. Observing such behavior on a bottom-up model based on brain transcriptomics provides support for a multi-scale brain model [36].…”

Analysing Twitter semantic networks: the case of 2018 Italian elections

“…5, 6, 7. Such a structure is known as core-periphery and is present in many social, economic and financial systems [48]. Remarkably, the nodes belonging to the innermost shell overlap with the core ones computed with the multinomial version of the surprise, as proved by computing the Jaccard index 8 over the two sets of nodes.…”

Networked partisanship and framing: a socio-semantic network analysis of the Italian debate on migration

“…An interesting phenomenon that is observed when analysing monthly semantic networks is the co-existence of different types of mesoscopic structures. As pointed out in other works [49], communities are found to co-exist with k-shells in several systems: in particular, the innermost k-shells is frequently sub-divided into (a small number) of communities further partitioning it. These sub-communities play an important role: within our partisan communities-induced semantic networks, this role has been interpreted as a natural division of the most debated topics animating the same 'global' discussion.…”