We introduce and analyze NetOTC, a procedure for the comparison and soft alignment of weighted networks. Given two networks and a cost function relating their vertices, NetOTC finds an appropriate coupling of their associated random walks having minimum expected cost. The minimizing cost provides a numerical measure of the difference between the networks, while the optimal transport plan itself provides interpretable, probabilistic alignments of the vertices and edges of the two networks. The cost function employed can be based, for example, on vertex degrees, externally defined features, or Euclidean embeddings. Coupling of the full random walks, rather than their stationary distributions, ensures that NetOTC captures local and global information about the given networks. NetOTC applies to networks of different size and structure, and does not the require specification of free parameters. NetOTC respects edges, in the sense that vertex pairs in the given networks are aligned with positive probability only if they are adjacent in the given networks. We investigate a number of theoretical properties of NetOTC that support its use, including metric properties of the minimizing cost and its connection with short- and long-run average cost. In addition, we introduce a new notion of factor for weighted networks, and establish a close connection between factors and NetOTC. Complementing the theory, we present simulations and numerical experiments showing that NetOTC is competitive with, and sometimes superior to, other optimal transport-based network comparison methods in the literature. In particular, NetOTC shows promise in identifying isomorphic networks using a local (degree-based) cost function.
翻译:我们引入并分析计算网络加权网络的比较和软调整程序NetOTC。鉴于两个网络及其脊椎相关的成本函数,NetOTC发现与其相关的随机行走相配,其预期成本最低。成本最小化提供了网络之间差异的数字计量,而最佳运输计划本身提供了两个网络脊椎和边缘的可解释、概率一致。使用的成本功能可以基于例如脊椎度、外部定义特征或Euclidean嵌入等。将完全随机行走而不是固定分布相加,确保NetOTC获取关于特定网络的本地和全球信息。NetOTC适用于不同规模和结构的网络,并不要求自由参数的规格。NetOTC尊重边缘,即特定网络的脊椎配对只有在与特定网络相邻的情况下才与正概率一致。我们调查了支持其使用的NetOTC的理论属性,包括降低成本和固定分布分布的固定分布,确保NetOTC获取本地和全球信息信息信息。 NetOTC应用不同规模和结构的网络的网络网络,并且不要求自由参数。 NetOTC尊重边缘, 只有在与当前网络的最佳网络进行成本分析,我们当前和数字的模型中,我们之间可以确定一个成本和数字的模型。