Fuzzy Statistical Analysis on Complex Network Characterisation: Bridging the Ihara Zeta Function and Thermodynamic Partition Functions

Complex networks usually demonstrate intrinsic uncertainty, imprecision, and ambiguity in their structural and dynamical characteristics, which positions them as suitable candidates for analysis through fuzzy systems. The statistical characterization of these networks has been associated with both the Ihara Zeta function, which evaluates prime cycles, and partition functions from thermodynamics. However, traditionally, these functions have been viewed as separate entities, thereby neglecting their potential interconnection. This paper establishes a link between the Ihara Zeta function in algebraic graph theory and the partition function in statistical mechanics to elucidate network structure within the fuzzy systems framework. This linkage offers a fresh perspective on the relationship between microscopic structure and macroscopic properties of networks, taking into account the inherent uncertainty and ambiguity frequently encountered in real-world applications. We derive thermodynamic quantities, such as entropy, that correlate with the configurations of prime cycles of varying lengths and extend these ideas to fuzzy entropy measures to accommodate imprecise or incomplete information. Employing the n-th derivative of the Ihara Zeta function facilitates the computation of prime cycle quantities within a network. This method is conceptually similar to using the partition function within the Bose-Einstein statistical model. The derived entropy measures enable us to investigate phase transitions in network structure, especially in fuzzy conditions, where transitions may not be distinctly defined. Numerical experiments and empirical data validate the efficacy of our approach in characterizing network structure, thus offering a holistic understanding of both microscopic and macroscopic network attributes in the context of fuzzy systems.