Complex biological, neuroscience, geoscience and social networks exhibit heterogeneous self-similar higher order topological structures that are usually characterized as being multifractal in nature. However, describing their topological complexity through a compact mathematical description and deciphering their topological governing rules has remained elusive and prevented a comprehensive understanding of networks. To overcome this challenge, we propose a weighted multifractal graph model capable of capturing the underlying generating rules of complex systems and characterizing their node heterogeneity and pairwise interactions. To infer the generating measure with hidden information, we introduce a variational expectation maximization framework. We demonstrate the robustness of the network generator reconstruction as a function of model properties, especially in noisy and partially observed scenarios. The proposed network generator inference framework is able to reproduce network properties, differentiate varying structures in brain networks and chromosomal interactions, and detect topologically associating domain regions in conformation maps of the human genome.

复杂的生物、神经科学、地球科学和社会网络表现出异质自相似的高阶拓扑结构，这些结构通常具有多重分形的性质。然而，通过紧凑的数学描述来描述它们的拓扑复杂性并破译它们的拓扑控制规则仍然难以捉摸，并阻碍了对网络的全面理解。为了克服这一挑战，我们提出了一种加权多重分形图模型，该模型能够捕获复杂系统的潜在生成规则并表征它们的节点异质性和成对交互。为了用隐藏信息推断生成度量，我们引入了一个变分期望最大化框架。我们证明了作为模型属性函数的网络生成器重建的鲁棒性，尤其是在嘈杂和部分观察到的场景中。所提出的网络生成器推理框架能够重现网络特性，区分大脑网络和染色体相互作用中的不同结构，并检测人类基因组构象图中的拓扑关联域区域。