Conductance (probability)

In graph theory the conductance of a graph measures how "well-knit" the graph is: it controls how fast a random walk on converges to its stationary 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.

The conductance of a cut $(S, \bar S)$ in a graph is defined as:

:$\varphi(S) = \frac$

where the are the entries of the adjacency matrix for , so that

:$a(S) = \sum_ \sum_ a_$

is the total number (or weight) of the edges incident with . is also called a volume of the set $S \subseteq V$.

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

: $\phi(G) = \min_\varphi(S).$

Equivalently, conductance of a graph is defined as follows:

: $\phi(G) := \min_\frac.\,$

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

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 $\Phi_S$ 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 $\Phi_S$ 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.

See also

* Resistance distance
* Percolation theory
* Krackhardt E/I Ratio

References

*
*
*{{cite book , author=Fan Chung , authorlink = Fan Chung , title=Spectral Graph Theory , series=CBMS Lecture Notes , volume=92 , publisher=AMS Publications , date=1997 , isbn=0-8218-0315-8 , page=212
* A. Sinclair. Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhauser, Boston-Basel-Berlin, 1993.
* D. Levin, Y. Peres, E. L. Wilmer: Markov Chains and Mixing Times

Markov processes
Algebraic graph theory
Matrices
Graph invariants