The Martingale Problems In A Class Of Random Networks

Complex networks can describe various complex systems in the real world. They are alsopowerful tools to study topological structures and dynamical properties of these systems. In recentyears, researchers from home and abroad pay much attention to complex networks. They have done alot of studies and found that almost all of the real networks possess similar properties, such as thescale free property and the high clustering property. The studies of the formation mechanism of thenetworks have become a hot topic. Researchers attempt to model the networks that satisfy the aboveproperties and to give the mathematical description of the properties. Previous models assume that thetime is discrete, while time is continuous in fact. In addition, some papers don’t explain thedifferentiability of the degree of a node. From mathematical perspective, networks can be seen asgraphs consisting of nodes and edges, where nodes represent different individuals in the real networksand edges represent relationship between individuals. Many network evolution models can beregarded as Markov processes with the state space consisting of graphs. Different from the generalstochastic processes, graph-valued stochastic processes study graphs where nodes and edges changerandomly. In this paper, we study a simple network model where we add a new node with two edgesor rewire an edge at every operation. First, we give the description of the model. The number of theoperation is dominated by a homogeneous Poisson process, and the degree of a node and the sum ofthe degrees of the neighbors are continuous time stochastic processes. Then, we give a sufficientcondition for the degree of a node to possess submartingale property. It provides the evidence forthe differentiability of the degree of a node. Under this condition, we get the differential equation thatthe degree of a node satisfies and its solution. Moreover, the degree distribution of the network isobtained. Finally, we give a sufficient condition for the sum of the degrees of the neighbors to possesssubmartingale property.