Accelerating networks: Effects of preferential connections

Networks are commonly observed structures in complex systems with interacting and interdependent parts that self-organize. For nonlinearly growing networks, when the total number of connections increases faster than the total number of nodes, the network is said to accelerate. We propose a systematic model for the dynamics of growing networks represented by distribution kinetics equations. We define the nodal-linkage distribution, construct a population dynamics equation based on the association-dissociation process, and perform the moment calculations to describe the dynamics of such networks. For nondirectional networks with finite numbers of nodes and connections, the moments are the total number of nodes, the total number of connections, and the degree (the average number of connections per node), represented by the average moment. Size independent rate coefficients yield an exponential network describing the network without preferential attachment, and size dependent rate coefficients produce a power law network with preferential attachment. The model quantitatively describes accelerating network growth data for a supercomputer (Earth Simulator), for regulatory gene networks, and for the Internet.