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 its all 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.

中文翻译：

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计算网络中的群体和空腔

复杂的网络具有完整的子图，例如节点，边线，三角形等，称为不同顺序的集团。值得注意的是，已发现由高阶团组成的空腔在脑功能中起着重要作用。由于在大型网络中搜索最大团簇是一个NP完全问题，因此我们建议使用k核分解来确定给定网络在有限计算资源的情况下的可计算性。对于可计算网络，我们设计了一种搜索算法来查找不同顺序的集团，该算法还提供了欧拉特征数。然后，我们通过使用相邻团的边界矩阵的秩来计算贝蒂数。此外，我们设计了一种用于查找不同阶腔的优化算法。最后，我们将该算法应用于C的神经元网络。