Claudio Castellano has published Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects in Physical Review X.
In many social and biological systems, the pattern of interactions is described by complex networks—mathematical constructions composed of points (vertices) representing individuals, joined by lines (edges), standing for pairwise interactions between them. These structures are important because they affect the behavior of the dynamical processes they mediate. In the case of epidemic spreading, the global structure of the interaction pattern sets the epidemic threshold, i.e., the minimum value of the probability that an individual transmits the disease to one of her contacts that is sufficient to induce a global disease outbreak. The possibility to make predictions about this threshold based on simple network properties is of paramount importance in epidemiological applications. We provide a physical interpretation of the largest eigenvalue of the adjacency matrix of networks, which allows an estimation of the epidemic threshold and other dynamical processes in many types of real networks.
The value of the epidemic threshold is, in general, related to the largest eigenvalue of the adjacency matrix of the network. Previous works provided an estimate of this eigenvalue for a specific class of networks. Here, we find a much more general expression, which is shown to be accurate for over 100 real-world networked systems. Our result allows us to understand the physical origin of the new expression, pointing out that its value actually depends on the competition between two different types of subnetworks in the larger structure.
This new understanding immediately gives new predictions about the behavior of dynamical processes on networks, as well as providing hints about the effect of intervention strategies, such as immunization of individuals to curb the spreading of a disease.