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

The hierarchical graphs and Sierpinski 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 Sierpinski graphs are fractal and largeworld. Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpinski 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.