Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs

Abstract: In this article we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for PDEs. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, the Mean Curvature equation, and the Eikonal operator on the graph.

Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an adaptive multi-layer strategy

Abstract: Summary Functions of eigenvalues of the graph Laplacian matrix L, especially the extremal non-trivial eigenvalues, the algebraic connectivity λ2 and the spectral radius λn, have been shown to be important in determining the performance in a host of consensus and synchronisation applications. In this paper, we focus on formulating an entirely distributed control law for the control of edge weights in an undirected graph to solve a constrained optimisation problem involving these extremal eigenvalues. As an objective for the distributed control law, edge weights must be found that minimise the spectral radius of the graph Laplacian, thereby maximising the robustness of the network to time delays under a simple linear consensus protocol. To constrain the problem, we use both local weight constraints that weights must be non-negative, and a global connectivity constraint, maintaining a designated minimum algebraic connectivity. This ensures that the network remains sufficiently well connected. The distributed control law is formulated as a multilayer strategy, using three layers of successive distributed estimation. Adequate timescale separation between the layers is of paramount importance for the proper functioning of the system, and we derive conditions under which the distributed system converges as we would expect for the centralised control or optimisation system to converge. © 2017 The Authors International Journal of Robust and Nonlinear Control published by John Wiley & Sons Ltd.

A linear generative model for graph structure

Abstract: This paper shows how to construct a linear deformable model for graph structure by performing principal components analysis (PCA) on the vectorised adjacency matrix. We commence by using correspondence information to place the nodes of each of a set of graphs in a standard reference order. Using the correspondences order, we convert the adjacency matrices to long-vectors and compute the long-vector covariance matrix. By projecting the vectorised adjacency matrices onto the leading eigenvectors of the covariance matrix, we embed the graphs in a pattern-space. We illustrate the utility of the resulting method for shape-analysis.