Cheeger Inequalities for General Edge-Weighted Directed Graphs
نویسندگان
چکیده
We consider Cheeger Inequalities for general edge-weighted directed graphs. Previously the directed case was considered by Chung for a probability transition matrix corresponding to a strongly connected graph with weights induced by a stationary distribution. An Eulerian property of these special weights reduces these instances to the undirected case, for which recent results on multi-way spectral partitioning and higher-order Cheeger Inequalities can be applied. We extend Chung’s approach to general directed graphs. In particular, we obtain higher-order Cheeger Inequalities for the following scenarios: (1) The underlying graph needs not be strongly connected. (2) The weights can deviate (slightly) from a stationary distribution.
منابع مشابه
Spectral graph theory: Cheeger constants and discrepancy∗
In this third talk we will discuss properties related to edge expansion. In particular, we will define the Cheeger constant (which measures how easy it is to cut off a large piece of the graph) and state the Cheeger inequalities. We also will define and discuss discrepancy for undirected and directed graphs. We also state the Perron-Frobenius Theorem which is a useful tool in spectral graph the...
متن کاملCheeger Inequalities for Submodular Transformations
The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations. In this pape...
متن کاملLaplacians of graphs and Cheeger inequalities
We define the Laplacian for a general graph and then examine several isoperimetric inequalities which relate the eigenvalues of the Laplacian to a number of graphs invariants such as vertex or edge expansions and the isoperimetric dimension of a graph.
متن کاملOn Cheeger-type inequalities for weighted graphs
We give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger-type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger-type bound available.
متن کاملA Local Clustering Algorithm for Connection Graphs
We give a clustering algorithm for connection graphs, that is, weighted graphs in which each edge is associated with a d-dimensional rotation. The problem of interest is to identify subsets of small Cheeger ratio and which have a high level of consistency, i.e. that have small edge boundary and the rotations along any distinct paths joining two vertices are the same or within some small error f...
متن کامل