Forest Expression of Networks and their Applications

Complex networks with hierarchical relationships and sparse structures among nodes are pervasive in both social and physical sciences. Examples include hierarchical power distribution networks and tree-like structures centered around hub nodes in disease transmission networks. A common challenge in such studies is to visualize or analyze the underlying hierarchy. Existing analytical tools are limited, often relying on ego networks or handcrafted features. In this article, we propose a novel framework that represents each connected component of a network as a tree, and the entire network as a forest. This forest-based representation enables intuitive 3D visualizations and introduces new structural features that facilitate meaningful network comparisons. For dynamic network systems, we further develop pixel-based visualizations of these features to provide scalable overviews. Numerical studies across a range of cases demonstrate that the proposed approach offers powerful graphical and analytical tools for network analysis.