Complex networks are critical in many scientific, technological, and societal contexts due to their ability to represent and analyze intricate systems with interdependent components. Often, after labeling the nodes of a network with a community detection algorithm, its modular organization emerges, allowing a better understanding of the underlying structure by uncovering hidden relationships. In this paper, we introduce a novel information-geometric framework for the filtering and decomposition of networks whose nodes have been labeled. Our approach considers the labeled network as the outcome of a Markov random field modeled by a q-state Potts model. According to information geometry, the first and second order Fisher information matrices are related to the metric and curvature tensor of the parametric space of a statistical model. By computing an approximation to the local shape operator, the proposed methodology is able to identify low and high information nodes, allowing the decomposition of the labeled network in two complementary subgraphs. Hence, we call this method as the LO-HI decomposition. Experimental results with several kinds of networks show that the high information subgraph is often related to edges and boundaries, while the low information subgraph is a smoother version of the network, in the sense that the modular structure is improved.

翻译：复杂网络因其能够表征和分析具有相互依赖组件的复杂系统，在众多科学、技术和社会场景中至关重要。通常，在使用社区检测算法对网络节点进行标记后，其模块化组织得以显现，通过揭示隐藏关系使人们能更好地理解底层结构。本文提出了一种新颖的信息几何框架，用于对已标记节点的网络进行过滤与分解。我们的方法将被标记网络视为由q态Potts模型建模的马尔可夫随机场的结果。根据信息几何理论，一阶和二阶费希尔信息矩阵与统计模型参数空间的度量张量和曲率张量相关联。通过计算局部形状算子的近似值，所提出的方法能够识别低信息节点与高信息节点，从而将标记网络分解为两个互补的子图。因此，我们将此方法称为LO-HI分解。在多种网络上的实验结果表明，高信息子图通常与边缘和边界相关，而低信息子图则是网络的光滑化版本，其模块化结构得到了增强。