Extreme vulnerability of high-order organization in complex networks

The robustness and vulnerability of complex networks have always been an important research field, and there are many corresponding phenomena in the real world. For example, natural disasters may destroy power facilities and lead to the fragmentation of the electricity network, resulting in paralysis of network functions. The traditional research on the robustness of networks is to study the connected sub-clusters of low-order (original) networks. Recently, the conception of high-order organization was introduced to complex networks and it was pointed out that the functionality of networks is implemented by their high-order organization. Therefore, when the networks are attacked, one should observe the fragmentation of both low-order networks and high-order networks. In other words, when attacked, even if the low-order network is robust, but the network structure of its corresponding high-order network may have been severely damaged. In this case, it will still be difficult for the networks to maintain its functions. In this paper, based on percolation theory, we conducted random and malicious node and edge attacks on low-order networks. We gradually remove nodes or links from the low-order networks and we then analyze how the network fragmentation of the low-order network and the high-order network change after the attack. The results show that high-order networks are significantly more fragile than low-order networks. © 2021 Elsevier B.V.