1-planar graph

Gene

Combinatorics

Laplacian matrix

Discrete mathematics

Physics

Graph

Mathematical analysis

Philosophy

Psychology

Eigenvalues and eigenvectors

Chordal graph

Maximal independent set

Chemistry

Social connectedness

Sierpinski triangle

Reciprocal

Mathematics

Fractal

Linguistics

Psychotherapist

Biochemistry

Quantum mechanics

Robustness (evolution)

Indifference graph

Algebraic connectivity

IEEE Transactions on Cybernetics

The hierarchical graphs and Sierpiński 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 nonfractal and small-world, while the Sierpiński graphs are fractal and “large-world.” Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński 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.

Consensus in Self-Similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence