Nonsymmetric Resource Networks. The Study of Limit States
A network model called a resource network and represented by an oriented weighted graph with loops is considered. In the bidirectional resource network: i) any two vertices are either not adjacent or connected by a pair of oppositely directed edges; ii) resources are assigned to vertices, which have unlimited volumes; the weights of arcs indicate their capacities. The total resource is constant, while resources at the vertices are reallocated according to certain rules in discrete time. The limit states of networks with arbitrary initial distribution of resource are analyzed. The threshold value T is proved to exist: when the total resource value is less than T the network with loops corresponds to a regular Markov chain; when the total resource value exceeds T the Markov property does not hold. The classification of vertices, depending on their ability to accumulate resources is given. The vertices capable to accumulate the amount of resource, surpassing their total output capacity was is were called the potential attractors. The criterion of attractiveness of vertices is formulated. The formulae of resource value at every vertex in limit state expressing its dependence on limit probabilities of corresponding Markov chain, the total resource value and total output capacity are derived. (original abstract)
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