Conductance (graph)

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 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.

An undirected graph G and a few example cuts with the corresponding conductances

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

${\displaystyle \varphi (S)={\frac {\sum _{i\in S,j\in {\bar {S}}}a_{ij}}{\min(a(S),a({\bar {S}}))}}}$

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

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

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

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

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

Equivalently, conductance of a graph is defined as follows:

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

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.

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 ${\displaystyle S}$ of the capacity of ${\displaystyle S}$ divided by the ergodic flow out of ${\displaystyle S}$. 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 ${\displaystyle \Phi _{S}}$ 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 ${\displaystyle \Phi _{S}}$ over sets ${\displaystyle S}$ 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

• Béla Bollobás (1998). Modern graph theory. GTM. 184. Springer-Verlag. p. 321. ISBN 0-387-98488-7.
• Kannan, R.; Vempala, S.; Vetta, A. (May 2004). "On clusterings: Good, bad and spectral" (PDF). J. ACM. 51 (3): 497–515. doi:10.1145/990308.990313.
• 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