Random networks are heterogeneous exhibiting a multi-scaling law

Unlike the scale-free (SF) architecture, random networks of the Erdos–Renyi (ER) are also called homogeneous networks consisting of the same kind of nodes because these nodes approximately have same degrees, which follow a Poisson distribution. This paper tries to demonstrate that this random network is actually heterogeneous, i.e., it is composed of different kinds of nodes by introducing a new metric, called B-index to quantify a node, defined as the extent that the node’s removal can break down its neighborhood. One interesting phenomenon is observed on random networks, i.e., the distribution of B-index can be roughly divided into many sub-distributions with different scalings exhibiting a multi-scaling law. This phenomenon appears, fluctuates and disappears as the changes of random networks. In addition, the analysis of spreading dynamics on the susceptible infected recovered (SIR) model suggests that the nodes with the highest B-index can alleviate the overlap problem of infected area of multiple origins, and are more influential for the spreading of epidemics, especially for the small number of origins.