Zero forcing number of graphs with a power law degree distribution

TL;DR

This paper studies how the zero forcing number, a measure related to controllability in graphs, varies in graphs with power law degree distributions, revealing its dependence on graph diameter and generation model.

Contribution

It analyzes the zero forcing number in power law graphs generated by different models, linking it to graph diameter and controllability implications.

Findings

Zero forcing number approaches graph size as gamma approaches 2 in preferential attachment graphs.

In deactivation model graphs, the zero forcing number remains smaller than the graph size regardless of gamma.

Graph diameter scaling influences the controllability of dynamical systems on these graphs.

Abstract

The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbour colour forcing rule. The zero forcing number provides topological information about linear algebra on graphs, with applications to the controllability of linear dynamical systems and quantum walks on graphs among other problems. Here, I investigate the zero forcing number of undirected simple graphs with a power law degree distribution $p_k\sim k^{-\gamma}$. For graphs generated by the preferential attachment model, with a diameter scaling logarithmically with the graph size, the zero forcing number approaches the graph size when $\gamma\rightarrow2$. In contrast, for graphs generated by the deactivation model, with a diameter scaling linearly with the graph size, the zero forcing number is smaller than the graph size independently of $\gamma$. Therefore the scaling of the graph diameter with the graph size is another factor determining the controllability of dynamical systems.