Will electrical cyber–physical interdependent networks undergo first-order transition under random attacks?

• The “partial one-to-one correspondence” interdependent networks model is proposed.
• Two novel inter-similarity indexes are proposed.
• Connectivity vulnerabilities of three Chinese electrical cyber–physical interdependent networks are investigated.
• Chinese electrical cyber–physical interdependent networks will not undergo first-order transition under random attacks.
• The connectivity vulnerability of electrical cyber–physical interdependent networks can be reflected by its single-layer network.

Whether the realistic electrical cyber–physical interdependent networks will undergo first-order transition under random failures still remains a question. To reflect the reality of Chinese electrical cyber–physical system, the “partial one-to-one correspondence” interdependent networks model is proposed and the connectivity vulnerabilities of three realistic electrical cyber–physical interdependent networks are analyzed. The simulation results show that due to the service demands of power system the topologies of power grid and its cyber network are highly inter-similar which can effectively avoid the first-order transition. By comparing the vulnerability curves between electrical cyber–physical interdependent networks and its single-layer network, we find that complex network theory is still useful in the vulnerability analysis of electrical cyber–physical interdependent networks.