Distribution dynamics of evolving networks

How networks come into existence and how they change with time are fundamental issues in numerous physical and social systems. Based on the concept of a nodal-linkage distribution, we propose a unified population dynamics approach for the evolution of networks to random (single-scale) or power law (scale-free) conformations. The functional form of the rate coefficients for addition or removal of links governs the asymptotic forms, which are independent of initial states. The population balance equation, cast either as an integrodifferential equation, a difference–differential equation, or a partial differential equation, can be solved by standard methods, including a moment method. Growth and dissolution rate coefficients that vary with a power of the linkage number yield an asymptotic power law distribution of links. Rate coefficients independent of linkage number yield exponential (random) networks.