Complex networks have complete subgraphs such as nodes, edges, triangles,
etc., referred to as cliques of different orders. Notably, cavities consisting
of higher-order cliques have been found playing an important role in brain
functions. Since searching for the maximum clique in a large network is an
NP-complete problem, we propose using k-core decomposition to determine the
computability of a given network subject to limited computing resources. For a
computable network, we design a search algorithm for finding cliques of
different orders, which also provides the Euler characteristic number. Then, we
compute the Betti number by using the ranks of the 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 in one dataset, and find all of its cliques and some
cavities of different orders therein, providing a basis for further
mathematical analysis and computation of the structure and function of the C.
elegans neuronal network.