Abstract
Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders. Notably, cavities consisting of higher-order cliques play an important role in brain functions. Since searching for maximum cliques is an NP-complete problem, we use k-core decomposition to determine the computability of a given network. For a computable network, we design a search method with an implementable algorithm for finding cliques of different orders, obtaining also the Euler characteristic number. Then, we compute the Betti numbers by using the ranks of boundary matrices of adjacent cliques. Furthermore, we design an optimized algorithm for finding cavities of different orders. Finally, we apply the algorithm to the neuronal network of C. elegans with data from one typical dataset, and find all of its cliques and some cavities of different orders, providing a basis for further mathematical analysis and computation of its structure and function.
Highlights
Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders
Motivated by all the above observations, this paper studies the fundamental issue of computability of a complex network, based on which the investigation continues to find higher-order cliques and their Euler characteristic number, as well as all the Betti numbers and higher-order cavities
The proposed approach starts from k-core decomposition[14] and, through finding cliques of different orders, performs a sequence of computations on the ranks of the corresponding boundary matrices, to obtain all the Betti numbers
Summary
Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders. Higher-order cliques and smallest cavities are basic components of totally homogenous networks. Higher-order cycles of a connected undirected network include cliques and cavities.
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