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Conductance (graph)

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In graph theory the conductance of a graph G=(V,E) measures how "well-knit" the graph is: it controls how fast a random walk on G converges to a uniform distribution. The conductance of a graph is often called the Cheeger constant of a graph as the analog of its counterpart in spectral geometry. Since electrical networks are intimately related to random walks with a long history in the usage of the term "conductance", this alternative name helps avoid possible confusion.

Contents

The conductance of a cut ( S , S ¯ ) in a graph is defined as:

φ ( S ) = ∑ i ∈ S , j ∈ S ¯ a i j min ( a ( S ) , a ( S ¯ ) )

where the a i j are the entries of the adjacency matrix for G, so that

a ( S ) = ∑ i ∈ S ∑ j ∈ V a i j

is the total number (or weight) of the edges incident with S.

The conductance of the whole graph is the minimum conductance over all the possible cuts:

ϕ ( G ) = min S ⊆ V φ ( S ) .

Equivalently, conductance of a graph is defined as follows:

ϕ ( G ) := min S ⊆ V ; 0 ≤ a ( S ) ≤ a ( V ) / 2 ∑ i ∈ S , j ∈ S ¯ a i j a ( S ) .

For a d-regular graph, the conductance is equal to the isoperimetric number divided by d.

Generalizations and applications

In practical applications, one often considers the conductance only over a cut. A common generalization of conductance is to handle the case of weights assigned to the edges: then the weights are added; if the weight is in the form of a resistance, then the reciprocal weights are added.

The notion of conductance underpins the study of percolation in physics and other applied areas; thus, for example, the permeability of petroleum through porous rock can be modeled in terms of the conductance of a graph, with weights given by pore sizes.

Conductance also helps measure the quality of a Spectral clustering. The maximum among the conductance of clusters provides a bound which can be used, along with inter-cluster edge weight, to define a measure on the quality of clustering. Intuitively, the conductance of a cluster(which can be seen as a set of vertices in a graph) should be low. Apart from this, the conductance of the subgraph induced by a cluster(called "internal conductance") can be used as well.

Markov chains

For an ergodic reversible Markov chain with an underlying graph G, the conductance is a way to measure how hard it is to leave a small set of nodes. Formally, the conductance of a graph is defined as the minimum over all sets S of the capacity of S divided by the ergodic flow out of S . Alistair Sinclair showed that conductance is closely tied to mixing time in ergodic reversible Markov chains. We can also view conductance in a more probabilistic way, as the minimal probability of leaving a small set of nodes given that we started in that set to begin with. Writing Φ S for the conditional probability of leaving a set of nodes S given that we were in that set to begin with, the conductance is the minimal Φ S over sets S that have a total stationary probability of at most 1/2.

Conductance is related to Markov chain mixing time in the reversible setting.

References

Conductance (graph) Wikipedia
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