Topological energy of networks

Energy is an important network indicator defined by the eigenvalues of an adjacency matrix that includes the neighbor information for each node. This article expands the definition of network energy to include higher-order information between nodes. We use resistance distances to characterize the distances between nodes and order complexes to extract higher-order information. Topological energy ( T E), defined by the resistance distance and order complex, reveals the characteristics of the network structure from multiple scales. In particular, calculations show that the topological energy can be used to distinguish graphs with the same spectrum well. In addition, topological energy is robust, and small random perturbations of edges do not significantly affect the T E values. Finally, we find that the energy curve of the real network is significantly different from that of the random graph, thus showing that T E can be used to distinguish the network structure well. This study shows that T E is an indicator that distinguishes the structure of a network and has some potential applications for real-world problems.

Automated summary

Energy, defined by the eigenvalues of an adjacency matrix, is an important network indicator that incorporates the neighbor information for each node. This article extends the definition of network energy to include higher-order information between nodes, using resistance distances and order complexes. The topological energy (T E), defined by these measures, reveals the characteristics of network structure at multiple scales and can distinguish graphs with the same spectrum.