空间网络集成的熵
Entropy of Spatial Network Ensembles
摘要：
本文从图熵的角度分析了空间网络集成的复杂性。在数学上，我们将空间网络建模为软随机几何图，即，具有两个随机性来源的图，即在空间中随机定位的节点和在节点对之间独立形成的链接，其中概率由它们的相互距离的指定函数（“对连接函数”）给出。我们考虑节点位置以及成对连接中出现随机性的一般情况（即，对于给定的对距离，对应的边缘状态是随机变量）。经典的随机几何图和指数图模型在一定的范围内可以恢复。我们推导出一个简单的空间网络集成的熵的范围，并计算条件熵的集成给定的硬和软（概率）对连接函数的节点位置分布。在这种形式主义下，我们推导出连接函数，在一般的约束下产生最大熵。最后，我们应用我们的分析框架来研究两个实际的例子：ad hoc无线网络和美国航班网络。通过对这些例子的研究，我们说明了这两个表现出的属性是指示近最大熵合奏。
英文摘要：
We analyze complexity in spatial network ensembles through the lens of graph entropy. Mathematically, we model a spatial network as a soft random geometric graph, i.e., a graph with two sources of randomness, namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function (the "pair connection function") of their mutual distance. We consider the general case where randomness arises in node positions as well as pairwise connections (i.e., for a given pair distance, the corresponding edge state is a random variable). Classical random geometric graph and exponential graph models can be recovered in certain limits. We derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft (probabilistic) pair connection functions. Under this formalism, we derive the connection function that yields maximum entropy under general constraints. Finally, we apply our analytical framework to study two practical examples: ad hoc wireless networks and the US flight network. Through the study of these examples, we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles.
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