Conductance (graph)

From Wikipedia, the free encyclopedia

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.[citation needed] 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 {\sum _{{i\in S,j\in {\bar S}}}a_{{ij}}}{\min(a(S),a({\bar S}))}}

where the a_{{ij}} are the entries of the adjacency matrix for G, so that

a(S)=\sum _{{i\in S}}\sum _{{j\in V}}a_{{ij}}

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:

\phi (G)=\min _{{S\subseteq V}}\varphi (S).

Equivalently, conductance of a graph is defined as follows:

\phi (G):=\min _{{S\subseteq V;0\leq a(S)\leq a(V)/2}}{\frac {\sum _{{i\in S,j\in {\bar S}}}a_{{ij}}}{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.

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

References

  • Béla Bollobás (1998). Modern graph theory. GTM 184. Springer-Verlag. p. 321. ISBN 0-387-98488-7. 
  • Kannan, R.; Vempala, S., and Vetta, A. (May 2004). "On clusterings: Good, bad and spectral". J. ACM. 51, 3: 497–515. 
  • Fan Chung (1997). Spectral Graph Theory. CBMS Lecture Notes 92. AMS Publications. p. 212. ISBN 0-8218-0315-8. 
  • 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
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.