Research of ergodic heterogeneous "greedy-vertices" resource networks

A modification of the graph threshold dynamic model "resource network" – "resource network with greedy vertices" is considered. At each discrete-time moment, the vertices of the graph transfer the resource to each other through the edges with limited throughputs. First they transfer the available resource to their own loops, and then the rest of the resource is distributed to the outgoing edges according to the rules of the standard resource network (in proportion to the throughputs taking into account their limitations). The topology of an ergodic heterogeneous resource network with "greedy" vertices is considered – the network is described by a strongly connected directed graph. It is shown that when the values of the total resource are not larger than the sum of the throughputs of all loops, the network with "greedy" vertices stops, which is impossible for a standard ergodic resource network and is caused by the existence of modified loops; the nature of the network stop is investigated. For the values of the total resource that are larger than the sum of the throughputs of all loops, it is proved that from a certain time moment the network will function equivalently to the corresponding standard resource network.