Vitanov Nikolay K, Vitanov Kaloyan N, Kantz Holger
Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria.
Max-Planck Institute for the Physics of Complex Systems, Noethnitzerstr. 38, 01187 Dresden, Germany.
Entropy (Basel). 2020 Oct 31;22(11):1240. doi: 10.3390/e22111240.
We discuss the motion of substance in a channel containing nodes of a network. Each node of the channel can exchange substance with: (i) neighboring nodes of the channel, (ii) network nodes which do not belong to the channel, and (iii) environment of the network. The new point in this study is that we assume possibility for exchange of substance among flows of substance between nodes of the channel and: (i) nodes that belong to the network but do not belong to the channel and (ii) environment of the network. This leads to an extension of the model of motion of substance and the extended model contains previous models as particular cases. We use a discrete-time model of motion of substance and consider a stationary regime of motion of substance in a channel containing a finite number of nodes. As results of the study, we obtain a class of probability distributions connected to the amount of substance in nodes of the channel. We prove that the obtained class of distributions contains all truncated discrete probability distributions of discrete random variable ω which can take values 0,1,⋯,N. Theory for the case of a channel containing infinite number of nodes is presented in Appendix A. The continuous version of the discussed discrete probability distributions is described in . The discussed extended model and obtained results can be used for the study of phenomena that can be modeled by flows in networks: motion of resources, traffic flows, motion of migrants, etc.
我们讨论了物质在包含网络节点的通道中的运动。通道的每个节点都可以与以下对象交换物质：(i) 通道的相邻节点，(ii) 不属于该通道的网络节点，以及 (iii) 网络环境。本研究的新观点在于，我们假设通道节点之间的物质流与以下对象之间存在物质交换的可能性：(i) 属于网络但不属于通道的节点，以及 (ii) 网络环境。这导致了物质运动模型的扩展，并且扩展模型包含之前的模型作为特殊情况。我们使用物质运动的离散时间模型，并考虑物质在包含有限数量节点的通道中的稳态运动。作为研究结果，我们得到了一类与通道节点中物质数量相关的概率分布。我们证明，所得到的分布类包含离散随机变量 ω 的所有截断离散概率分布，ω 可以取值 0,1,⋯,N。包含无限数量节点的通道情况的理论在附录 A 中给出。所讨论的离散概率分布的连续版本在 中描述。所讨论的扩展模型和得到的结果可用于研究可以用网络流建模的现象：资源运动、交通流、移民运动等。
