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\lecturehead{CS 7880}{Network Algorithms}{Spring 2016}{19
January 2016}{Rajmohan Rajaraman}

\begin{itemize}
\item
  Random walks and electrical networks
\item
  Random walk measures and graph parameters
\end{itemize}

\section{Random walks and electrical networks}
We present an intriguing connection between random walks and
electricity flow in networks.  This phenomena has been known for
several years, going back to the nineteenth century.  A comprehensive
manuscript on this connection is by Doyle and Snell, available online.
Our coverage of this connection will closely follow Lovasz's survey on
random walks.  We will introduce the notion of effective resistance
between two nodes in a graph, which has found several applications
recently, including in graph sparsification.  More recently, the
connection between electrical flows and random walks has been
instrumental in obtaining near-linear time algorithms for network
flows --- considered among the breakthrough results in the field.

Consider an undirected graph $G$, and let $s$ and $t$ denote distinct
vertices in $G$.  Now, view this graph $G$ as an electrical network:
assume that each edge corresponds to a resistor of resistance
1$\Omega$, and there is current $i$ entering $s$ and exiting $t$ such
that the difference in voltage between $t$ and $s$ is $1$.
Specifically, we let the voltage $\phi(s)$ at $s$ be $0$, and the
voltage at $\phi(t)$ at $t$ be $1$.  Let $\phi(v)$ denote the voltage
at any vertex $v$.  Let $i_{xy}$ denote the current flowing from $x$
to $y$; note that $i_{yx} = -i_{xy}$.  

By Ohm's Law, we have, for every edge $(u,v)$:
\[
\phi(v) - \phi(u) = i_{uv}.
\]
Adding the above equation, over all incident edges of a vertex $v \neq
s,t$, we obtain
\[
d(v) \phi(v) - \sum_{(u,v) \in E} \phi(u) = \sum_{(u,v) \in E} i_{uv} = 0,
\]
where the last equality follows from Kirchoff's Law.  We thus have
\begin{eqnarray}
\label{eqn:harmonic}
\phi(v)  = \frac{1}{d(v)} \sum_{(u,v) \in E} \phi(u)
\end{eqnarray}
A function $f$ satisfying Equation~\ref{eqn:harmonic} and the
conditions $\phi(s) = 0, \phi(t) = 1$ is referred to as a harmonic
function with poles $s$ and $t$.

We now establish that an appropriately defined probability function
for a random walk is also a harmonic function with poles $s$ and $t$.
Let $p(u)$ denote the probability that a random walk starting at $u$
visits $t$ before $s$.  Clearly, $p(s) = 0$ and $p(t) = 1$.  For any
$v \neq s, t$, we obtain
\[
p(v) = \frac{1}{d(v)} \sum_{(u,v) \in E} p(u).
\]
We leave the following observation as an exercise.
\begin{Observation}
There is a unique function satisfying Equation~\ref{eqn:harmonic}.
\end{Observation}
We thus obtain that $\phi$ and $p$ are the same functions.  This
equivalence leads to the following beautiful result connecting the
commute times of a random walk and effective resistances.  For any two
vertices $s$ and $t$, the {\em commute time}\/ $C(s,t)$ is the sum of
the hitting times $H(s, t) + H(t,s)$.  Note that unlike hitting time,
commute time is symmetric, and, in fact, forms a metric.  We define
the {\em effective resistance}\/ $R_{st}$ between $s$ and $t$ in $G$
as the potential difference between $s$ and $t$ if a unit current
enters the network at $s$ and leaves at $t$; alternatively, it is the
inverse of the current $i$ that enters $s$ and leaves $t$ if the
potential difference between $s$ and $t$ is set to $1$.  That is:
\begin{eqnarray}
\label{eqn:eff_res}
R_{st} = \frac{1}{\sum_{(s,v) \in G} i_{sv}} = \frac{1}{\sum_{(s,v) \in G} \phi(v)} = \frac{1}{\sum_{(s,v) \in G} p(v)}
\end{eqnarray}

\begin{Theorem}
For any graph $G$ and vertices $s$ and $t$, we have
\[
C(s,t) = 2m R_{st}.
\]
\end{Theorem}
\begin{Proof}
Let $p$ denote the probability that a random walk starting at $s$
visits $t$ before returning to $t$.  Let $\tau$ denote the return time
to $s$, and let $\sigma$ denote the time it takes to return to $u$
after visiting $t$.  Clearly $\tau \le \sigma$.  We know that
\[
E[\tau] = \frac{2m}{d(s)}
\]
and
\[
E[\sigma] = C(s,t).
\]
We also have the following relationship between $\sigma$ and $\tau$.
\[
E[\sigma - \tau] = (1 - p) E[\sigma].
\]
Simplifying yields 
\[
p = \frac{E[\tau]}{E[\sigma]} = \frac{2m}{d(s) C(s,t)}.
\]
Now $p$ is also given as follows.
\[
p = \frac{1}{d(s)} \sum_{(s,v) \in E} p_v = \frac{1}{R_{st} d(s)},
\]
where the last equality follows from Equation~\ref{eqn:eff_res}.
Putting the preceding two equations for $p$ together yields $C(s,t) =
2m R_{st}$.
\end{Proof}

\section{Random walks and graph measures}
We have seen that the time taken for a random walk to mix within
$\eps$ distance to the stationary distribution is upper bounded by
$O(\ln n/\eps)/(1 - \lambda_2)$, where $\lambda_2$ is the second
largest eigenvalue (in absolute value) of the random walk matrix.  The
term $1 - \lambda_2$ is referred to as the spectral gap.  The larger
the spectral gap, the smaller the mixing time and hence the faster the
random walk mixes.

A family of regular graphs with constant degree is referred to as a
family of {\em expanders}\/ if the spectral gap of every graph in the
family is bounded away from $0$.  There are other ``combinatorial''
notions of expansion as well.  A graph $G = (V, E)$ has a {\em vertex
  expansion}\/ of $\alpha$ if the maximum, over all subsets $S$ of
size at most $|V|/2$, of the ratio $|N(S)|/|S|$, where $N(S)$ is the
set of neighbors of vertices of $S$ in $V - S$.  We will consider
other notions too, including edge expansion and conductance.  For a
family of constant-degree regular graphs, a graph has a positive
constant spectral gap, if and only if it has a positive constant
vertex expansion (and vice versa), and if and only if it has a
positive constant edge expansion (and vice versa), and if and only if
it has a positive constant conductance (and vice versa).

For any sufficiently large $n$, a random 4-regular graph over $n$
vertices is an expander with probability at least $1/2$.  This result
is due to Pinkser.  We will prove a weaker result for random bipartite
graphs.
\end{document}
