Anand Kartik, Krioukov Dmitri, Bianconi Ginestra
Bank of Canada, 234 Laurier Ave West, Ottawa, Ontario K1A 0G9, Canada.
Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jun;89(6):062807. doi: 10.1103/PhysRevE.89.062807. Epub 2014 Jun 11.
The entropy of network ensembles characterizes the amount of information encoded in the network structure and can be used to quantify network complexity and the relevance of given structural properties observed in real network datasets with respect to a random hypothesis. In many real networks the degrees of individual nodes are not fixed but change in time, while their statistical properties, such as the degree distribution, are preserved. Here we characterize the distribution of entropy of random networks with given degree sequences, where each degree sequence is drawn randomly from a given degree distribution. We show that the leading term of the entropy of scale-free network ensembles depends only on the network size and average degree and that entropy is self-averaging, meaning that its relative variance vanishes in the thermodynamic limit. We also characterize large fluctuations of entropy that are fully determined by the average degree in the network. Finally, above a certain threshold, large fluctuations of the average degree in the ensemble can lead to condensation, meaning that a single node in a network of size N can attract O(N) links.
网络集合的熵表征了编码在网络结构中的信息量，可用于量化网络复杂性以及在真实网络数据集中观察到的给定结构属性相对于随机假设的相关性。在许多真实网络中，单个节点的度并非固定不变，而是随时间变化，但其统计属性（如度分布）得以保留。在此，我们刻画了具有给定度序列的随机网络的熵分布，其中每个度序列是从给定的度分布中随机抽取的。我们表明，无标度网络集合熵的主导项仅取决于网络规模和平均度，且熵是自平均的，这意味着其相对方差在热力学极限下消失。我们还刻画了完全由网络中的平均度决定的熵的大幅波动。最后，在某个阈值之上，集合中平均度的大幅波动会导致凝聚，即大小为N的网络中的单个节点可以吸引O(N)条边。
