Computing cliques and cavities in networks

This paper presents methods for computing cliques and cavities in complex networks. The authors address the NP-completeness of finding maximum cliques by using k-core decomposition to assess network computability. For computable networks, they propose a search method and algorithm for identifying cliques of various orders and calculating the Euler characteristic number. Betti numbers are computed using boundary matrices of adjacent cliques. An optimized algorithm is designed for finding cavities, and the approach is applied to the *C. elegans* neuronal network, revealing its cliques and cavities.

Dinghua Shi, Zhifeng Chen, Xiang Sun, Qinghua Chen, Chuang Ma, Yang Lou, Guanrong Chen