César Terrero

The kinetic theory of graphs (KTG) studies the evolution of complex networks from a dynamical perspective, modeling the formation of edges as a statistical process of “collisions” or “aggregation,” analogous to molecular kinetics in a gas. Unlike static combinatorial approaches, KTG provides a framework for analyzing large-scale networks as they evolve over time ($t$).

In this talk, we present several applications of networks as representations of complex systems, focusing on the phase transitions that arise during their evolution, from both topological and dynamical viewpoints. We argue that, while traditional gas kinetics assumes indistinguishable molecules, understanding the emergent structural order in complex networks requires differentiating agents based on both local and global interaction rules.

Using the Erdös–Rényi process in the large- limit as a baseline—modeled through Smoluchowski-type equations—we implement a procedure to objectively identify the effective degrees of freedom. These are characterized by the concentration of specific substructures, such as floating and rooted trees, and their connection to special agents and the influence of global rules.