The Ihara zeta function as a partition function for network structure characterisation

作     者：Wang, Jianjia Hancock, Edwin R. 

作者机构：Xian Jiaotong Liverpool Univ Sch AI & Adv Comp Suzhou 215412 Peoples R China Univ York Dept Comp Sci York YO10 5GH England 

出 版 物：《SCIENTIFIC REPORTS》 (Sci. Rep.)

年 卷 期：2024年第14卷第1期

页      面：1-16页

核心收录：

基　　金：Natural Science Foundation of the Jiangsu Higher Educational Institution of China [24KJB510049] Research Development Fund [RDF-23-01-044] 

主　　题：Complex networks Ihara zeta function Partition function 

摘      要：Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose-Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations.