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.

Current Flow Group Closeness Centrality for Complex Networks

Abstract: The problem of selecting a group of vertices under certain constraints that maximize their joint centrality arises in many practical scenarios. In this paper, we extend the notion of current flow closeness centrality (CFCC) to a set of vertices in a graph, and investigate the problem of selecting a subset S to maximizes its CFCC C(S), with the cardinality constraint |S| = k. We show the NP-hardness of the problem, but propose two greedy algorithms to minimize the reciprocal of C(S). We prove the approximation ratios by showing the monotonicity and supermodularity. A proposed deterministic greedy algorithm has an approximation factor and cubic running time. To compare with, a proposed randomized algorithm gives -approximation in nearly-linear time, for any ? > 0. Extensive experiments on model and real networks demonstrate the effectiveness and efficiency of the proposed algorithms, with the randomized algorithm being applied to massive networks with more than a million vertices.

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

Abstract: Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12]. We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite graph with n vertices and all k >= 2, the k-th largest eigenvalue is at most 1-Omega(k^3/n^3), which extends the only other such result known, which is for k=2 only and is due to [LO81]. This upper bound improves to 1-Omega(k^2/n^2) if the graph is regular. We generalize these results, and we provide sharp bounds on the spectral measure of various classes of graphs, including vertex-transitive graphs and infinite graphs, in terms of specific graph parameters like the volume growth. As a consequence, using the entire spectrum, we provide (improved) upper bounds on the return probabilities and mixing time of random walks with considerably shorter and more direct proofs. Our work introduces spectral embedding as a new tool in analyzing reversible Markov chains. Furthermore, building on [Lyo05], we design a local algorithm to approximate the number of spanning trees of massive graphs.

Perfect state transfer in products and covers of graphs

Abstract: A continuous-time quantum walk on a graph is represented by the complex matrix , where is the adjacency matrix of and is a non-negative time. If the graph models a network of interacting qubits, transfer of state among such qubits throughout time can be formalized as the action of the continuous-time quantum walk operator in the characteristic vectors of the vertices. Here, we are concerned with the problem of determining which graphs admit a perfect transfer of state. More specifically, we will study graphs whose adjacency matrix is a sum of tensor products of -matrices, focusing on the case where a graph is the tensor product of two other graphs. As a result, we will construct many new examples of perfect state transfer.

Spectral simplex method

Abstract: We develop an iterative optimization method for finding the maximal and minimal spectral radius of a matrix over a compact set of nonnegative matrices. We consider matrix sets with product structure, i.e., all rows are chosen independently from given compact sets (row uncertainty sets). If all the uncertainty sets are finite or polyhedral, the algorithm finds the matrix with maximal/minimal spectral radius within a few iterations. It is proved that the algorithm avoids cycling and terminates within finite time. The proofs are based on spectral properties of rank-one corrections of nonnegative matrices. The practical efficiency is demonstrated in numerical examples and statistics in dimensions up to 500. Some generalizations to non-polyhedral uncertainty sets, including Euclidean balls, are derived. Finally, we consider applications to spectral graph theory, mathematical economics, dynamical systems, and difference equations.