Cheeger constant (graph theory)


In mathematics, the Cheeger constant of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling. The graph theoretical notion originated after the Cheeger isoperimetric constant of a compact Riemannian manifold.
The Cheeger constant is named after the mathematician Jeff Cheeger.

Definition

Let be an undirected finite graph with vertex set and edge set. For a collection of vertices, let denote the collection of all edges going from a vertex in to a vertex outside of :
Then the Cheeger constant of, denoted, is defined by
The Cheeger constant is strictly positive if and only if is a connected graph. Intuitively, if the Cheeger constant is small but positive, then there exists a "bottleneck", in the sense that there are two "large" sets of vertices with "few" links between them. The Cheeger constant is "large" if any possible division of the vertex set into two subsets has "many" links between those two subsets.

Example: computer networking

In applications to theoretical computer science, one wishes to devise network configurations for which the Cheeger constant is high even when is large.
For example, consider a ring network of computers, thought of as a graph. Number the computers clockwise around the ring. Mathematically, the vertex set and the edge set are given by:
Take to be a collection of of these computers in a connected chain:
So,
and
This example provides an upper bound for the Cheeger constant, which also tends to zero as. Consequently, we would regard a ring network as highly "bottlenecked" for large, and this is highly undesirable in practical terms. We would only need one of the computers on the ring to fail, and network performance would be greatly reduced. If two non-adjacent computers were to fail, the network would split into two disconnected components.

Cheeger Inequalities

The Cheeger constant is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph. The so-called Cheeger inequalities relate the Eigenvalue gap of a graph with its Cheeger constant. More explicitly
in which is the maximum degree for the nodes in and is the spectral gap of the Laplacian matrix of the graph.