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.

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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.