Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins

In this paper, we study the structure and dynamical properties of protein contact networks with respect to other biological networks, together with simulated archetypal models acting as probes. We consider both classical topological descriptors, such as modularity and statistics of the shortest paths, and different interpretations in terms of diffusion provided by the discrete heat kernel, which is elaborated from the normalized graph Laplacians. A principal component analysis shows high discrimination among the network types, by considering both the topological and heat kernel based vector characterizations. Furthermore, a canonical correlation analysis demonstrates the strong agreement among those two characterizations, providing thus an important justification in terms of interpretability for the heat kernel. Finally, and most importantly, the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey. Notably, the heat trace decay of an ensemble of varying-size proteins denotes subdiffusion, a peculiar property of proteins.