Birth and death master equation for the evolution of complex networks

Master equations for the evolution of complex networks with positive (birth) and negative (death) transition probabilities per unit time are analyzed. Explicit equations for the time evolution of the total number of nodes and for the relative node frequencies are given. It is shown that, in the continuous limit, the master equation reduces to a Fokker-Planck equation (FPE). The basic dynamical function for its stationary solution is the ratio between its drift and diffusion coefficients. When this ratio is approximated by partial fractions (Padé's approximants), a hierarchy of stationary solutions of the FPE is obtained analytically, which are expressed as an exponential times the product of powers of monomials and binomials. It is also shown that if the difference between birth and death transition probabilities goes asymptotically to zero, the exponential factor in the solution is absent. Fits to real complex network probability distribution functions are shown. Comparison with rank-ordered data shows that, in general, the value of this exponential factor is close to unity, evidencing crossovers among power-law scale invariant regimes which might be associated to an underlying criticality and are related to a generalization of the beta distribution. The time dependent solution is also obtained analytically in terms of hyper-geometric functions. It is also shown that the FPE has similarity solutions. The limitations of the approach here presented are also discussed.