Abstract
Entropy is an important index for describing the structure, function, and evolution of network. The existing research on entropy is primarily applied to undirected networks. Compared with an undirected network, a directed network involves a special asymmetric transfer. The research on the entropy of directed networks is very significant to effectively quantify the structural information of the whole network. Typical complex network models include nearest-neighbour coupling network, small-world network, scale-free network, and random network. These network models are abstracted as undirected graphs without considering the direction of node connection. For complex networks, modeling through the direction of network nodes is extremely challenging. In this paper, based on these typical models of complex network, a directed network model considering node connection in-direction is proposed, and the eigenvalue entropies of three matrices in the directed network is defined and studied, where the three matrices are adjacency matrix, in-degree Laplacian matrix and in-degree signless Laplacian matrix. The eigenvalue-based entropies of three matrices are calculated in directed nearest-neighbor coupling, directed small world, directed scale-free and directed random networks. Through the simulation experiment on the real directed network, the result shows that the eigenvalue entropy of the real directed network is between the eigenvalue entropy of directed scale-free network and directed small-world network.
Highlights
In recent years, research pertaining to complex network topologies has garnered significant attention
We propose a novel entropy eigenvalue-based of the adjacency matrix, in-degree Laplacian matrix and indegree signless Laplacian matrix
We investigate the directional characteristic of node connections in directed complex networks by modelling directed nearest-neighbour coupling, directed small-world, directed scale-free, and directed random networks
Summary
Research pertaining to complex network topologies has garnered significant attention. Many indicators reflect the features of a complex network topology, such as network node degree, clustering coefficient, density, and network diameter, etc [1,2,3]. Insufficient information for mapping the overall network topology is a concern to scholars. In this regard, network entropy [4] is a crucial method. In 1955, Rashevsky [10] first proposed the concept of graph entropy, which is based on the vertices symmetric structure of chemical molecules. The definition of entropy proposed above comes from different fields. Graph entropy based on the in-direction of node connections is challenging for directed complex networks. It is more difficult to research graph entropy in directed networks than in undirected networks
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