Mean size of avalanches on directed random networks with arbitrary degree distributions

The mean size of unordered binary avalanches on infinite directed random networks may be determined using the damage propagation function introduced by [B. Samuelsson and J. E. S. Socolar, Phys. Rev. E 74, 036113 (2006)]. The derivation of Samuelsson and Socolar explicitly assumes a Poisson distribution of out-degrees. It is shown here that the damage propagation function method may be used whenever the in-degree and out-degree of network nodes are independently distributed; in particular, it is not necessary that the out-degree distribution be Poisson. The general case of correlated in- and out-degrees is discussed and numerical simulations (on large finite networks) are compared with the theoretical predictions (for infinite networks).