Dynamic Analysis of transportation systems under small perturbations

In the present paper, we assess the dynamical response of a network under small perturbations that do not modify the network topology by means of a centrality index, called Icentr (see section 3.1 and Mussone et al. (2022) for its definition). In its design, this centrality index accepts weights on both nodes and edges, and so it can be iteratively applied, so to simulate a discrete linear dynamical system. It is associated to a suitable matrix M , that depends only on the topology of the network and on the edge weights. We prove that the dynamical system converges, independent of the network, and compute the limit point. We apply it to a dataset of 34 underground transportation networks from cities throughout the world and investigate the convergence type of a single network and of a single node. We show that the nodes of the network tend to cluster according to their convergence type and that the convergence rate of a single network is exponential in the number of iterations. Furthermore, we show that the ratio | λ 2 / λ 1 | where λ 1 ≥ | λ 2 | are the two largest module eigenvalues of M , is suitable to compare the dynamical behavior of different networks.