Modeling Real-World Networks Using Intrinsic Vertex Fitness ERGMs

In this work, we present a Fitness Exponential Random Graph Model (FERGMs) that brings together concepts from statistical mechanics and network science to describe complex networks beyond the traditional scale-free framework. By assigning each node an intrinsic ''fitness'' and viewing network formation as a thermodynamic process, we develop two complementary variants: a Configuration Model (CM) with bounded fitness and a Generalized Configuration Model (GCM) with unbounded fitness. These model are regulated by a temperature-like parameter $\tau$ and a chemical potential $\mu$, respectively. Together, these parameters control the balance between randomness and structural organization, as well as the overall link density in the network. We obtain closed-form expressions for the degree distributions across distinct thermodynamic regimes, showing that they become Poisson-like at high temperatures and transition to structured, weakly scale-free forms at low temperatures. We then examine 12 heterogeneous real-world networks—including social (YouTube, Orkut), technological (Skitter, web-Google), collaborative (DBLP), and infrastructural (road networks) systems—and find that FERGM surpasses conventional power-law models, lowering the relative error in average-degree prediction from as high as 768\% to nearly zero, while also capturing clustering and higher-order statistics. This framework offers a principled, interpretable, and empirically supported alternative to scale-free models, opening up new opportunities for network design, inference, and analysis in domains ranging from epidemiology to computational social science.