Research of complete homogeneous "greedy-vertices" resource networks

The modification of the graph dynamic model "resource network" – "resource network with greedy vertices" is described. In this model, the graph vertices at each discrete-time moment exchange resources through the edges with limited throughput, first passing the available resource to themselves via the loop and then distributing the remaining resource to adjacent vertices according to the "standard" resource network rules. These are two rules with threshold switching: if the vertex resource exceeds the total throughput of all its outgoing edges, it gives away the full throughput to each edge; otherwise, it gives the entire available resource, distributing it in proportion to the throughputs of the outgoing edges. The process of functioning of a complete homogeneous resource network with "greedy" vertices at different values of total resource and different initial states is investigated. Possible network states are described; a non-standard state – the shutdown of the network has been identified. Two total resource thresholds, separating zones of different network behavior, have been found: the first threshold divides the zones of "insufficient" and "sufficient" resources, the second divides the zones of "sufficient small" and "sufficient large" resources. For each zone the functioning of the network is described and the asymptotic states and flows are investigated. The examples demonstrating numerical experiments are given for all typical situations.