Partition of Networks into Basins of Attraction

TL;DR

This paper investigates how networks can be partitioned into basins of attraction based on steepest ascent to high-degree nodes, revealing different behaviors in scale-free networks and deriving distributions for basin sizes.

Contribution

It introduces a method to partition networks into basins of attraction based on degree ascent, providing analytical expressions and distributions for various network types.

Findings

Large gamma networks have many small basins, while small gamma networks have a giant basin.

Derived expressions for the expected number of basins and their size distribution.

Extended the analysis to regular networks and chains with weighted nodes.

Abstract

We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or "attracted" to its neighbor of maximal degree, as long as the degree is increasing. A node that has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent $\gamma$: for small $\gamma$ (broad distribution) networks are dominated by a giant basin, whereas for large $\gamma$ (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.