Maximum entropy network ensembles have been very successful in modelling sparse network topologies and in solving challenging inference problems. However the sparse maximum entropy network models proposed so far have fixed number of nodes and are typically not exchangeable. Here we consider hierarchical models for exchangeable networks in the sparse limit, i.e. with the total number of links scaling linearly with the total number of nodes. The approach is grand canonical, i.e. the number of nodes of the network is not fixed a priori: it is finite but can be arbitrarily large. In this way the grand canonical network ensembles circumvent the difficulties in treating infinite sparse exchangeable networks which according to the Aldous-Hoover theorem must vanish. The approach can treat networks with given degree distribution or networks with given distribution of latent variables. When only a subgraph induced by a subset of nodes is known, this model allows a Bayesian estimation of the network size and the degree sequence (or the sequence of latent variables) of the entire network which can be used for network reconstruction.
翻译:在模拟稀有的网络地形和解决具有挑战性的推论问题方面,最大英特罗比网络组合非常成功。然而,迄今提出的稀少的英特罗比网络模型固定了节点数量,而且通常不能互换。在这里,我们考虑的是在稀少限度内交换网络的等级模式,即与节点总数成线缩放的链接总数。这个方法非常粗糙,即网络节点的数目没有先验地固定:它有限,但可能任意很大。这样,大型金特罗比网络组合可以绕过处理无穷无尽的可交换网络的困难,而根据 Aldous-HOoversorem 的理论,这些网络必须消失。这个方法可以处理具有一定程度分布或网络的隐藏变量分布的网络。当只知道一个节点子引出的子图时,这个模式允许对网络的网络大小和程度序列(或潜在变量序列)进行巴伊斯估计,而整个网络可用于网络重建。