Hierarchical Cutting of Complex Networks Performed by Random Walks

TL;DR

This work addresses how a uniform random walk, by deleting each traversed edge, progressively dismantles a network and reveals a hierarchical binary structure captured as a dendrogram. It introduces sequential and parallel cutting dynamics and analyzes the resulting component sizes $(n,m)$ and permanence $P$, along with region-based balance metrics. Across ER, BA, and GEO networks, GEO graphs more frequently produce balanced, sizeable component pairs under parallel cuts, while permanence times are broadly similar across network types. The findings highlight network-type dependencies in dismantling behavior with implications for exploration, resilience, and design of geometric versus non-geometric networks.

Abstract

Several interesting approaches have been reported in the literature on complex networks, random walks, and hierarchy of graphs. While many of these works perform random walks on stable, fixed networks, in the present work we address the situation in which the connections traversed by each step of a uniformly random walks are progressively removed, yielding a successively less interconnected structure that may break into two components, therefore establishing a respective hierarchy. The sizes of each of these pairs of sliced networks, as well as the permanence of each connected component, are studied in the present work. Several interesting results are reported, including the tendency of geometrical networks sometimes to be broken into two components with comparable large sizes.

Figures (9)

• Figure 1: Illustration of the basic event in the considered gradual slicing of a network by a respective random walk. One of the obtained connected components (a) is broken into two connected components (b) and (c) by an agent moving from node $\alpha$ to node $\beta$.
• Figure 2: The two types of cutting dynamics considered in the present work when a current component is broken into two new connected components. In the sequential walk (a), a single agent remains in only the new connected component into which it moves but is assigned to a randomly chosen node. Parallel walk (b) involves assigning agents to randomly chosen nodes of each of the two new connected components. The green arrows show the agent movements within the connected components up to their respective separation.
• Figure 3: The possible pairs $(n,m)$ that can be obtained by the adopted network slicing procedure are restricted to the triangular region $ABC$. Henceforth, we divide this region into two respective sub-regions ADEB (blue) and DCE (green), with the latter region corresponding to pairs of broken components having more comparable sizes. Observe that $N$ is the number of nodes in the original network.
• Figure 4: The average $\pm$ standard deviation of the total duration of each sequential random walk (single agent) respectively to: (a) ER network, (b) BA network, and (c) GEO network. These results consider 10,000 walks starting at different nodes (randomly chosen) for 50 networks of each type.
• Figure 5: Degree distribution (average $\pm$ standard deviation, considering 500 networks of each type) for: (a) ER network, (b) BA network, and (c) GEO network.