复杂网络上的经典随机游动与网络复杂性
Classical random walks over complex networks and network complexity
摘要：
本文将具有任意度分布的复杂网络上经典随机游动的定态看作是处于热平衡的状态。通过将态的分布确定为正则系综，我们能够定义随机行走系统的温度和哈密顿量。然后，我们计算亥姆霍兹自由能，平均能量，和熵的热平衡状态。结果表明，平均能量的能量均分。发现熵由两部分组成。第一部分随着步行者数量的增加而减少。熵的第二部分仅取决于网络的拓扑结构，并且当更多的边或节点被添加到网络时将增加。我们比较了熵的拓扑部分和一些网络描述符，发现拓扑熵可以作为网络复杂性的度量。此外，我们还讨论了步行者在下一个时间步有先验概率停留在同一个节点上的情形，发现先验概率的影响等价于增加网络中每个节点的度。
英文摘要：
In this paper we view the steady states of classical random walks over complex networks with an arbitrary degree distribution as states in thermal equilibrium. By identifying the distribution of states as a canonical ensemble, we are able to define the temperature and the Hamiltonian for the random walk systems. We then calculate the Helmholtz free energy, the average energy, and the entropy for the thermal equilibrium states. The results shows equipartition of energy for the average energy. The entropy is found to consist of two parts. The first part decreases as the number of walkers increases. The second part of the entropy depends solely on the topology of the network, and will increase when more edges or nodes are added to the network. We compare the topological part of entropy with some of the network descriptors and find that the topological entropy could be used as a measure of network complexity. In addition, we discuss the scenario that a walker has a prior probability of resting on the same node at the next time step, and find that the effect of the prior resting probabilities is equivalent to increasing the degree for every node in the network.
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