Consensus in Self-similar Hierarchical Graphs and Sierpi\'nski Graphs: Convergence Speed, Delay Robustness, and Coherence

TL;DR

This paper analyzes consensus dynamics on hierarchical and Sierpiński graphs, revealing how their structural differences influence convergence speed, robustness to delays, and coherence through spectral properties of their Laplacian matrices.

Contribution

It provides explicit recursive formulas and closed-form solutions for eigenvalues of Laplacian matrices in both graph types, highlighting their impact on consensus performance.

Findings

Hierarchical graphs have higher algebraic connectivity than Sierpiński graphs.

Convergence speed and robustness differ significantly between the two graph types.

Spectral properties directly relate to consensus efficiency and stability.

Abstract

The hierarchical graphs and Sierpi\'nski graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are non-fractal and small-world, while the Sierpi\'nski graphs are fractal and "large-world". Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpi\'nski graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.