Here we propose a new method to compare the modular structure of a pair of node-aligned networks. The majority of current methods, such as normalized mutual information, compare two node partitions derived from a community detection algorithm yet ignore the respective underlying network topologies. Addressing this gap, our method deploys a community detection quality function to assess the fit of each node partition with respect to the other network’s connectivity structure. Specifically, for two networks A and B, we project the node partition of B onto the connectivity structure of A. By evaluating the fit of B’s partition relative to A’s own partition on network A (using a standard quality function), we quantify how well network A describes the modular structure of B. Repeating this in the other direction, we obtain a two-dimensional distance measure, the bi-directional (BiDir) distance. The advantages of our methodology are three-fold. First, it is adaptable to a wide class of community detection algorithms that seek to optimize an objective function. Second, it takes into account the network structure, specifically the strength of the connections within and between communities, and can thus capture differences between networks with similar partitions but where one of them might have a more defined or robust community structure. Third, it can also identify cases in which dissimilar optimal partitions hide the fact that the underlying community structure of both networks is relatively similar. We illustrate our method for a variety of community detection algorithms, including multi-resolution approaches, and a range of both simulated and real world networks.

在这里，我们提出了一种新的方法来比较一对节点对齐的网络的模块化结构。当前的大多数方法（例如归一化的互信息）会比较从社区检测算法得出的两个节点分区，而忽略相应的基础网络拓扑。为解决这一差距，我们的方法部署了一个社区检测质量功能，以评估每个节点分区相对于另一个网络的连接结构的适合性。具体来说，对于两个网络A和B，我们将B的节点分区投影到A的连通性结构上。通过评估B的分区相对于A在网络A上自己的分区的适合度（使用标准质量函数），我们可以量化网络A描述了B的模块化结构。我们获得了二维距离度量，即双向（BiDir）距离。我们方法论的优点是三方面的。首先，它适用于试图优化目标函数的各种社区检测算法。其次，它考虑了网络结构，特别是社区内部和社区之间的连接强度，因此可以捕获具有类似分区但其中一个可能具有更明确或更强大的社区结构的网络之间的差异。第三，它还可以识别出不同的最佳分区掩盖了两个网络的底层社区结构相对相似这一事实的情况。我们将说明用于多种社区检测算法的方法，包括多分辨率方法，