Computing cliques and cavities in networks

Abstract Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to simplices and cliques of different orders. Notably, cavities consisting higher-order play an important role in brain functions. Since searching for maximum is NP-complete problem, we use k-core decomposition determine the computability a given network. For computable network, design search method with implementable algorithm finding orders, obtaining also Euler characteristic number. Then, compute Betti numbers by using ranks boundary matrices adjacent cliques. Furthermore, optimized Finally, apply neuronal network C. elegans data from one typical dataset, find all its some providing basis further mathematical analysis computation structure function.