异质弱相依网络鲁棒性研究

Robustness of interdependent networks with heterogeneous weak inter-layer links

传统研究认为网络间相依边的引入使网络鲁棒性大幅降低,但现实相依网络的鲁棒性往往优于理论结果.通过观察现实相依网络的级联失效过程,发现节点不会因相依节点失效而损失所有连接边,且由于网络节点的异质性,每个节点的连接边失效概率也不尽相同.针对此现象,提出一种异质弱相依网络模型,与传统网络逾渗模型不同,本文认为两个弱相依节点的其中一个失效后,另一个节点的连接边以概率 γ 失效而不是全部失效,并且不同节点连接边失效概率 γ 会因节点的异质性而不同.通过理论分析给出模型基于生成函数的逾渗方程,求解出任意随机分布异质对称弱相依网络的连续相变点.仿真结果表明方程的理论解与随机网络逾渗模拟值相符合,网络鲁棒性随着弱相依关系异质程度的增大而提高.

The robustness of complex networks plays an important role in human society. By further observing the networks on our planet, researchers find that many real systems are interdependent. For example, power networks rely on the Internet to transfer operation information, predators have to hunt for herbivores to refuel themselves, etc. Previous theoretical studies indicate that removing a small fraction of nodes in interdependent networks leads to a thorough disruption of the interdependent networks. However, due to the heterogeneous weak inter-layer links, interdependent networks in real world are not so fragile as the theoretical predictions. For example, an electronic components factory needs raw materials which are produced by a chemical factory. When the chemical factory collapses, the electronic components factory will suffer substantial drop in the production, however, it can still survive because it can produce some other raw materials by itself to sustain its production of some products. What is more, because of the heterogeneity on real industry chains, different electronic components factories produce different kinds of products, which still guarantees the diversity of electronic goods on the whole. In this paper, we develop a framework to help understand the robustness of interdependent networks with heterogeneous weak inter-layer links. More specifically, in the beginning, a fraction of 1 – p nodes are removed from network A and their dependency nodes in network B are removed simultaneously, then the percolation process begins. Each connectivity link of a node with weak inter-layer dependency is removed with a probability γ after the failure of its counterpart node. The γ values for different nodes are various because of heterogeneity. At the end, the nodes can survive as long as one of the remaining connectivity links reaches the giant component. We present an analytical solution for solving the giant component size and analyzing the crossing point of the phase transition of arbitrary interdependent random networks. For homogeneous symmetric Erdos-Rényi networks, we solve the continuous transition point and the critical point of γ. The simulation results are in good agreement with our exact solutions. Furthermore, we introduce two kinds of γ distributions to analyze the influence of heterogeneous weak inter-layer links on the robustness of interdependent networks. The results of both distributions show that with the increase of heterogeneity, the transition point pc decreases and the networks become more robust. For the first simple γ distribution, we also find the percolation transition changes from discontinuous one to continuous one by improving the heterogeneity. For the second Gaussian γ distribution, a higher variance makes the interdependent networks more difficult to collapse. Our work explains the robustness of real world interdependent networks from a new perspective, and offers a useful strategy to enhance the robustness by increasing the heterogeneity of weak inter-layer links of interdependent networks.