Self-concordant function

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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.Template:Fact 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 $(S,\overline{S})$ in a graph is defined as:

$\phi (S)=\frac{\sum _{i\in S,j\in \overline{S}}{a}_{ij}}{\min (a(S),a(\overline{S}))}$

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:

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

Equivalently, conductance of a graph is defined as follows:

$\varphi (G):={\min }_{S\subseteq V;0\le a(S)\le a(V)/2}\frac{\sum _{i\in S,j\in \overline{S}}{a}_{ij}}{a(S)}.$

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 ${\mathrm{\Phi }}_S$ 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 ${\mathrm{\Phi }}_S$ 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
• Krackhardt E/I Ratio

References

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• 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