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The organisation of a network in a maximal set of nodes having at least $k$ neighbours within the set, known as $k$-core decomposition, has been used for studying various phenomena. It has been shown that nodes in the innermost $k$-shells play a crucial role in contagion processes, emergence of consensus, and resilience of the system. It is known that the $k$-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 $k$-core decomposition of some empirical networks as well as that of some randomised counterparts, and examine the extent to which the $k$-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 $k$-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 $k$-shells into a small number of communities.
Core-periphery structure and community structure are two typical meso-scale structures in complex networks. Though the community detection has been extensively investigated from different perspectives, the definition and the detection of core-periphe
Multiplex networks are convenient mathematical representations for many real-world -- biological, social, and technological -- systems of interacting elements, where pairwise interactions among elements have different flavors. Previous studies pointe
Community detection is expensive, and the cost generally depends at least linearly on the number of vertices in the graph. We propose working with a reduced graph that has many fewer nodes but nonetheless captures key community structure. The K-core
We present a new layout algorithm for complex networks that combines a multi-scale approach for community detection with a standard force-directed design. Since community detection is computationally cheap, we can exploit the multi-scale approach to
Community structure is one of the most relevant features encountered in numerous real-world applications of networked systems. Despite the tremendous effort of scientists working on this subject over the past few decades to characterize, model, and a