Modularity is a key organizing principle in real-world large-scale complex networks. Many real-world networks exhibit modular structures such as transportation infrastructures, communication networks, and social media. Having the knowledge of the shortest paths length distribution between random pairs of nodes in such networks is important for understanding many processes, including diffusion or flow. Here, we provide analytical methods which are in good agreement with simulations on large scale networks with an extreme modular structure. By extreme modular, we mean that two modules or communities may be connected by maximum one link. As a result of the modular structure of the network, we obtain a distribution showing many peaks that represent the number of modules a typical shortest path is passing through. We present theory and results for the case where interlinks are weighted, as well as cases in which the interlinks are spread randomly across nodes in the community or limited to a specific set of nodes.

中文翻译：

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极端模块化网络中的距离分布

模块化是现实世界中大型复杂网络的关键组织原则。许多现实世界的网络都展示了模块化的结构，例如交通基础设施，通信网络和社交媒体。了解此类网络中节点随机对之间的最短路径长度分布对于了解许多过程（包括扩散或流动）非常重要。在这里，我们提供的分析方法与具有极端模块化结构的大型网络上的仿真非常吻合。极端模块化意味着两个模块或社区最多可以通过一个链接连接。由于网络的模块化结构，我们获得了一个分布，该分布显示了许多峰值，这些峰值代表典型的最短路径通过的模块数量。