text:
\chapter{The resistance metric}
\label{sec:effective-resistance-metric}

\headerquote{The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science.}{---~D.~Hilbert}

We now introduce the natural notion of distance on $(\Graph, \cond)$: the resistance metric $R$. While not as intuitive as the more common shortest-path metric, it reflects the topology of the graph more accurately and may be more useful for modeling and practical applications. The effective resistance is intimately related to the random walk on $(\Graph,\cond)$, the Laplacian, and the Dirichlet energy form \cite{Kig03,Lyons:ProbOnTrees,LevPerWil08,Soardi94,Kig01,Str06,DoSn84}.

In \S\ref{sec:Resistance-metric-on-finite-networks}, we give several formulations of this metric (Theorem~\ref{thm:effective-resistance-metric}), each with its own advantages. Many of these are familiar from the literature: \eqref{eqn:def:R(x,y)-Lap} from \cite{Pow76b} and \cite[\S8]{Peres99}, \eqref{eqn:def:R(x,y)-energy} from \cite{DoSn84}, \eqref{eqn:def:R(x,y)-diss} from \cite{DoSn84,Pow76b}, \eqref{eqn:def:R(x,y)-R}--\eqref{eqn:def:R(x,y)-S} from \cite{Kig03,Kig01,Str06}.

In \S\ref{sec:Resistance-metric-on-infinite-networks}, we extend these formulations to infinite networks. Due to the possible presence of nontrivial harmonic functions, some care must be taken when adjusting these formulations. It turns out that there are two canonical extensions of the resistance metric to infinite networks which are distinct precisely when $\Harm \neq 0$ (cf. \cite{Lyons:ProbOnTrees} and the references therein): the ``free'' resistance and the ``wired'' resistance. We are able to clarify and explain the difference in terms of the reproducing kernels for \HE and for \Fin, and also in terms of Dirichlet vs. Neumann boundary conditions; see Remark~\ref{rem:wired-vs-free-as-boundary-conditions}. We also explain the discrepancy in terms of projections in \HE and attempt to relate this to conditioning of the random walk on the network; see \S\ref{sec:probabilistic-interp-of-v_x-and-f_x} and Remark~\ref{rem:Schur-does-network-reduction}. Additionally, we introduce trace resistance and harmonic resistance and relate these to the free and wired resistances. (Note: unlike the others, harmonic resistance is not a metric.) In the limit, the trace resistance coincides with the free resistance.

\section{Resistance metric on finite networks}
\label{sec:Resistance-metric-on-finite-networks}

We make the standing assumption that the network is finite in \S\ref{sec:Resistance-metric-on-finite-networks}. However, the results actually remain true on any network for which $\Harm = 0$.

\begin{defn}\label{def:effective-resistance}
  If one amp of current is inserted into the \ERN at $x$ and withdrawn at $y$, then the \emph{(effective) resistance} $R(x,y)$ is the voltage drop between the vertices $x$ and $y$.
  \glossary{name={$R$},description={resistance metric},sort=r,format=textbf}
\end{defn}

\begin{theorem}\label{thm:effective-resistance-metric}
  The resistance $R(x,y)$ has the following equivalent formulations:
  \linenopax
  \begin{align}
    R(x,y)
    &= \dist_\Lap(x,y) := \{v(x)-v(y) \suth \Lap v = \gd_x-\gd_y\}
    \label{eqn:def:R(x,y)-Lap} \\
    &= \dist_\energy(x,y) := \{\energy(v) \suth \Lap v = \gd_x-\gd_y\} \label{eqn:def:R(x,y)-energy} \\
    &= \dist_\diss(x,y) := \min\{\diss(\curr) \suth \curr \in \Flo(x,y)\} \label{eqn:def:R(x,y)-diss} \\
    &= \dist_R(x,y) := 1/\min
    \nolimits_{v \in \dom\energy}\{\energy(v) \suth v(x)=1, v(y)=0\} \label{eqn:def:R(x,y)-R} \\
    &= \dist_\gk(x,y) := \min\nolimits_{v \in \dom\energy}\{\gk \geq 0 \suth |v(x)-v(y)|^2 \leq \gk \energy(v)\} \label{eqn:def:R(x,y)-S} \\
    &= \dist_s(x,y) := \sup_{v \in \dom \energy}\{|v(x)-v(y)|^2 \suth \energy(v) \leq 1\} \label{eqn:def:R(x,y)-sup}.
  \end{align}
  \begin{proof}
    \eqref{eqn:def:R(x,y)-Lap}$\iff$\eqref{eqn:def:R(x,y)-energy}. We may choose $v$ satisfying $\Lap v = \gd_x-\gd_y$ by Theorem~\ref{thm:Pot-is-nonempty-by-current-flows}.
    Then
    \linenopax
    \begin{align}
      \energy(v)
      = \sum_{z \in \verts} v(z) \Lap v(z)
      = \sum_{z \in \verts} v(z) (\gd_x(z) - \gd_y(z))
      = v(x) - v(y),
    \end{align}
    where first equality is justified by Theorem~\ref{thm:E(u,v)=<u,Lapv>}.

\begin{comment}
    \eqref{eqn:def:R(x,y)-diss}$\iff$\eqref{eqn:def:R(x,y)-energy}.
    Since \diss obtains its minimum on $\Flo(x,y)$, we have $\dist_\diss(x,y) = \diss(\curr_0)$ for some minimizing $\curr_0 \in \Flo(x,y)$. Then there is a corresponding potential $v_0 \in \Pot(x,y)$ by Theorem~\ref{thm:minimal-flows-are-induced-by-minimizers} Theorem~\ref{thm:minimal-flows-are-induced-by-minimizers}, and $\diss(\curr_0) = \energy(v_0)$ by Lemma~\ref{thm:energy=dissipation}. So
    \linenopax
    \begin{equation}\label{eqn:diss-dist-is-larger-than-energy-dist}
      \dist_\diss(x,y) = \energy(v_0) \geq \dist_\energy (x,y).
    \end{equation}
    By similar reasoning, $\dist_\energy(x,y) = \energy(v_0)$ for some minimizing $v_0 \in \Pot(x,y)$. Theorem~\ref{thm:minimal-flows-are-induced-by-minimizers} again indicates that $\energy(v_0) = \diss(\curr_0)$ for some minimizing $\curr_0 \in \Flo(x,y)$, and
    the other inequality follows.
\end{comment}

    \eqref{eqn:def:R(x,y)-energy}$\iff$\eqref{eqn:def:R(x,y)-diss}. Note that every $v \in \Pot(x,y)$ corresponds to an element $\curr \in \Flo(x,y)$ via Ohm's Law by Lemma~\ref{thm:potential-induces-current}, and $\energy(v)=\diss(\curr)$ by the same lemma. Also, this current flow is minimal by Theorem~\ref{thm:minimal-flows-are-induced-by-minimizers}.

    \eqref{eqn:def:R(x,y)-energy}$\iff$\eqref{eqn:def:R(x,y)-R}.
    Suppose that $\Lap v = \gd_x - \gd_y$. Since $\energy(v+k)=\energy(v)$ and $\Lap(v+k)=\Lap v$ for any constant $k$, we may adjust $v$ by a constant so that $v(y)=0$. Define
    \linenopax
    \begin{align*}
      u := \frac{v - v(x)}{v(x)-v(y)}
    \end{align*}
    so that $u(x)=0$ and $u(y)=1$. Observe that \eqref{eqn:def:R(x,y)-Lap} gives $\energy(v)=v(x)-v(y)$, whence
    \linenopax
    \begin{align*}
      \energy(u)
      = \energy(v)/(v(x)-v(y))^2
      = (v(x)-v(y))^{-1} \geq \min \energy(u).
    \end{align*}
    This shows $\energy(v) \leq [\min \energy(u)]^{-1}$ and hence $\dist_\energy \leq \dist_R$.

    For the other inequality, suppose $u$ minimizes $\energy(u)$, subject to $u(x)-u(y)=0$. Then by Theorem~\ref{thm:E(u,v)=<u,Lapv>} and the same variational argument as described in Remark~\ref{rem:variational-approach}, we have
    \linenopax
    \begin{align*}
      \energy(\gr,u) = \sum_{z \in \verts} \gr(z) \Lap u(z) = 0,
    \end{align*}
    for every function \gr for which $\gr(x) = \gr(y) = 0$. It follows that $\Lap u(z) = 0$ for $z \neq x,y$, and hence $\Lap u = \gx \gd_x + \gh \gd_y$. Observe that $\energy(u) = \energy(-u) = \energy(1-u)$, and so the same result follows from minimizing \energy with respect to the conditions $u(y)=1$ and $u(x)=0$. This symmetry forces $\gh = -\gx$ and we have $\Lap u = \gx \gd_x - \gx \gd_y$. Now for $v = \frac1\gx u$ one has $\Lap v = \gd_x - \gd_y$, and so
    \linenopax
    \begin{align*}
      \energy(u) = \gx^2 \energy(v) = \gx^2 (v(x)-v(y)) = \gx(u(x)-u(y)) = \gx,
    \end{align*}
    where the second equality follows by \eqref{eqn:def:R(x,y)-Lap}. Then $\gx = \frac1{\energy(v)} = \energy(u)$, whence $\dist_\energy \geq \dist_R$.

    \eqref{eqn:def:R(x,y)-R}$\iff$\eqref{eqn:def:R(x,y)-S}. Starting with \eqref{eqn:def:R(x,y)-S}, it is clear that
    \linenopax
    \begin{align*}
      \dist_\gk(x,y)
      &= \inf\{\gk \geq 0 \suth \tfrac{|v(x)-v(y)|^2}{\energy(v)} \leq \gk , v \in \dom \energy\} \\
      &= \sup\{\tfrac{|v(x)-v(y)|^2}{\energy(v)}, v \in \dom \energy, v \text{ nonconstant}\}.
    \end{align*}
    Given a nonconstant $v \in \dom \energy$, one can substitute $u:=\frac{v}{|v(x)-v(y)|}$ into the previous line to obtain
    \linenopax
    \begin{align*}
      \dist_\gk(x,y)
      &= \sup\{\tfrac{|u(x)-u(y)|^2\cancel{|v(x)-v(y)|^2}} {\energy(u)\cancel{|v(x)-v(y)|^2}}, v \in \dom \energy, v \text{ nonconstant}\} \\
      &= \sup\{\tfrac1{\energy(u)}, u \in \dom \energy, |u(x)-u(y)|=1\}\\
      &= 1/\inf\{\energy(u), u \in \dom \energy, |u(x)-u(y)|=1\}.
    \end{align*}
    Since we can always add a constant to $u$ and multiply by $\pm1$ without changing the energy, this is equivalent to letting $u$ range over the subset of $\dom\energy$ for which $u(x)=1$ and $u(y)=0$ and we have \eqref{eqn:def:R(x,y)-R}.

    \eqref{eqn:def:R(x,y)-S}$\iff$\eqref{eqn:def:R(x,y)-sup}.
    It is immediate that \eqref{eqn:def:R(x,y)-S} is equivalent to
    \linenopax
    \begin{align*}
      \sup\left\{\frac{|v(x)-v(y)|^2}{\energy(v)} \suth \energy(v) < \iy\right\}.
    \end{align*}
    For any $v \in \dom \energy$, define $w:= v/\sqrt{\energy(v)}$ so that $|w(x)-w(y)|^2 = |v(x)-v(y)|^2 \energy(v)^{-1/2}$ with $\energy(w) = 1$. Clearly then $|w(x)-w(y)|^2 \leq \dist_s(x,y)$. The other inequality is similar.
  \end{proof}
\end{theorem}

The equivalence of \eqref{eqn:def:R(x,y)-diss} and \eqref{eqn:def:R(x,y)-Lap} is shown elsewhere (e.g., see \cite[\S{II}]{Pow76b}) but the reader will find some gaps, so we have included a complete version of this proof for completeness. The terminology ``effective resistance metric'' is common in the literature (see, e.g., \cite{Kig01} and \cite{Str06}), where it is usually given in the form \eqref{eqn:def:R(x,y)-R}. The formulation \eqref{eqn:def:R(x,y)-S} will be helpful for obtaining certain inequalities in the sequel. It is also clear that $\dist_s$ of \eqref{eqn:def:R(x,y)-sup} is the norm of the operator $L_{xy}$ defined by $L_{xy} u := u(x) - u(y)$, see Lemma~\ref{thm:L_x-is-bounded} and Theorem~\ref{thm:free-resistance}.

\begin{remark}\label{rem:minimal-energy-is-obtained}
  Taking the minimum (rather than the infimum) in \eqref{eqn:def:R(x,y)-diss}, etc, is justified by Theorem~\ref{thm:energy-obtains-min}. The same argument implies that the energy kernel on \Graph is uniquely determined.
\end{remark}

\begin{remark}[Resistance distance via network reduction]
  \label{rem:Resistance-distance-via-network-reduction}
  Let $H$ be a (connected) planar subnetwork of a finite network $G$ and pick any $x,y \in H$. Then $H$ may be reduced to a trivial network consisting only of these two vertices and a single edge between them via the use of three basic transformations: (i) series reduction, (ii) parallel reduction, and (iii) the $\nabla$-\textsf{Y} transform. Each of these transformations preserves the resistance properties of the subnetwork, that is, for $x,y \in G \less H$, $R(x,y)$ remains unchanged when these transformations are applied to $H$. The effective resistance between $x$ and $y$ may be interpreted as the resistance of the resulting single edge. An elementary example is shown in Figure~\ref{fig:network-reduction}. A more sophisticated technique of network reduction is given by the Schur complement construction defined in Remark~\ref{rem:Resistance-distance-via-Schur-complement}.
\end{remark}

  \begin{figure}
    \centering
    \includegraphics{network-reduction.pdf}
    \caption{\captionsize Effective resistance as network reduction to a trivial network. This basic example uses parallel reduction followed by series reduction; see Remark~\ref{rem:Resistance-distance-via-network-reduction}.}
    \label{fig:network-reduction}
  \end{figure}

The following result is not new (see, e.g. \cite[\S2.3]{Kig01}), but the proof given here is substantially simpler than most others found in the literature.

\begin{lemma}\label{thm:R-is-a-metric}
  $R$ is a metric.
  \begin{proof}
    Symmetry and positive definiteness are immediate from \eqref{eqn:def:R(x,y)-energy}, we use \eqref{eqn:def:R(x,y)-Lap} to check the triangle inequality. Let $v_1 \in \Pot(x,y)$ and $v_2 \in \Pot(y,z)$. By superposition, $v_3 := v_1 + v_2$ is in $\Pot(x,z)$. For $s \nbr t$, it is clear that $v_3(s)-v_3(t) = v_1(s)-v_1(t) + v_2(s)-v_2(t)$. By summing along any path from $x$ to $z$, one sees that this remains true for $s \notnbr t$, whence
    \linenopax
    \begin{align*}
      R(x,z)
      = v_3(x)-v_3(z)
      &= v_1(x)-v_1(z) + v_2(x)-v_2(z) \\
      &\leq v_1(x)-v_1(y) + v_2(y)-v_2(z)

      = R(x,y) + R(y,z),
    \end{align*}
    where the inequality follows from Proposition~\ref{thm:v-min-at-a,max-at-w}.
  \end{proof}
\end{lemma}

\section{Resistance metric on infinite networks}
\label{sec:Resistance-metric-on-infinite-networks}

There are difficulties with extending the results of the previous section to infinite networks. The existence of nonconstant harmonic functions $h \in \dom \energy$ implies the nonuniqueness of solutions to $\Lap u = f$, and hence \eqref{eqn:def:R(x,y)-Lap}--\eqref{eqn:def:R(x,y)-diss} are no longer well-defined. Two natural choices for extension lead to the free resistance $R^F$ and the wired resistance $R^W$. In this section, we attempt to explain the somewhat surprising phenomenon that one may have $R^W(x,y) < R^F(x,y)$.
\begin{enumerate}
  \item In Theorem~\ref{thm:free-resistance}, we show how $R^F$ corresponds to choosing solutions to $\Lap u = \gd_x - \gd_y$ from the energy kernel, and how it corresponds to currents which are decomposable in terms of paths. The latter leads to a probabilistic interpretation which provides for a relation to the trace of the resistance discussed in \S\ref{sec:trace-resistance}.
  \item In Theorem~\ref{thm:wired-resistance}, we show how $R^W$ corresponds to projection to \Fin. Since this corresponds to minimization of energy, it is naturally related to capacity.
\end{enumerate}
See also Remark~\ref{sec:probabilistic-interp-of-v_x-and-f_x}. Both of these notions are methods of specifying a \emph{unique} solutions to $\Lap u = f$ in some way. The disparity between $R^F$ and $R^W$ is thus explained in terms of boundary conditions on \Lap as an unbounded self-adjoint operator on \HE in Remark~\ref{rem:wired-vs-free-as-boundary-conditions}. For an alternative approach, see \cite{Kig03}, where the author uses a potential-theoretic formulation (axiomatic harmonic analysis) to explain the discrepancy between $R^F$ and $R^W$ in terms of domains. (This will also be apparent from our approach, see Remark~\ref{rem:comparison-to-resistance-forms}.)

To compute effective resistance in an infinite network, we will need three notions of subnetwork: free, wired, and trace. Strictly speaking, these may not actually be subnetworks of the original graph; they are networks associated to a full subnetwork.
  Throughout this section, we use $H$ to denote a finite full subnetwork of \Graph, $\verts[H]$ to denote its vertex set, and $H^F$, $H^W$, and $H^\trc$ to denote the free, wired, and trace networks associated to $H$ (these terms are defined in other sections below).

\begin{defn}\label{def:relative-resistance}
  If $H$ is a subnetwork of \Graph which contains $x$ and $y$, define $R_H(x,y)$ to be the resistance distance from $x$ to $y$ as computed within $H$. In other words, compute $R_H(x,y)$ by any of the equivalent formulas of Theorem~\ref{thm:effective-resistance-metric}, but extremizing over only those functions whose support is contained in $H$.
\end{defn}
  \glossary{name={$R_H(x,y)$},description={relative resistance as computed with respect to the subnetwork $H$},sort=RH,format=textbf}

We will always use the notation $\{G_k\}_{k=1}^\iy$ to denote an \emph{exhaustion} of the infinite network \Graph. Recall from Definition~\ref{def:exhaustion-of-G} that this means each $G_k$ is a finite connected subnetwork of \Graph, $G_k \ci G_{k+1}$, and $\verts = \bigcup \verts[G_k]$. Since $x$ and $y$ are contained in all but finitely many $G_k$, we may always assume that $x,y \in G_k$. Also, we assume in this section that the subnetworks are full --- this is not necessary, but simplifies the discussion and causes no loss of generality.

\begin{defn}\label{def:full-subnetwork}
  Let $\verts[H] \ci \verts$. Then the \emph{full subnetwork} on \verts[H] has all the edges of \Graph for which both endpoints lie in \verts[H], with the same conductances.
\end{defn}

\section{Free resistance}

\begin{defn}\label{def:free-resistance}
  For any subset $\verts[H] \ci \verts$, the \emph{free subnetwork} $H^F$ is just the full subnetwork $H$. That is, all edges of \Graph with endpoints in \verts[H] are edges of $H$, with the same conductances.
  Let $R_{H^F}(x,y)$ denote the effective resistance between $x$ and $y$ as computed in $H=H^F$, as in Definition~\ref{def:relative-resistance}.

  The \emph{free resistance} between $x$ and $y$ is then defined to be
  \linenopax
  \begin{align}\label{eqn:def:free-resistance}
    R^F(x,y) := \lim_{k \to \iy} R_{G_k^F}(x,y),
  \end{align}
  where $\{G_k\}$ is any exhaustion of \Graph.
\end{defn}
  \glossary{name={$R^F(x,y)$},description={free resistance metric},sort=RF,format=textbf}

\begin{remark}\label{rem:free-resistance}
  The name ``free'' comes from the fact that this formulation is free of any boundary conditions or considerations of the complement of $H$, in contrast to the wired and trace formulations of the next two sections. See \cite[\S9]{Lyons:ProbOnTrees} for further justification of this nomenclature.

  One can see that $R_{H^F}(x,y)$ has the drawback of ignoring the conductivity provided by all paths from $x$ to $y$ that pass through the complement of $H$. This provides some motivation for the wired and trace approaches below.
\end{remark}

\begin{comment}\label{rem:free-as-projection}
  Let $\HE = \HE(\Graph$ be the energy space associated to \Graph, and $\HE(H^F)$ be the energy space associated $H^F$. Then
  \linenopax
  \begin{align*}
    \HE(H^F)
    = P_{H^F}(\HE)
    = \spn\{P_{\verts[H]} v_x\}_{x \in \verts[H]}.
  \end{align*}
\end{comment}

\begin{defn}\label{def:L_xy}
  Fix $x, y \in \Graph$ and define the linear operator $L_{xy}$ on \HE by $L_{xy}v := v(x)-v(y)$.
\end{defn}

\begin{remark}
  \label{rem:R(x,y) = |L{xy}|}
  Theorem~\ref{thm:free-resistance} is the free extension of Theorem~\ref{thm:effective-resistance-metric} to infinite networks; it shows that $R(x,y) = \|L_{xy}\|$ and that $R(x,o)$ is the best possible constant $k=k_x$ in Lemma~\ref{thm:L_x-is-bounded}. In the proof, we use the notation $\charfn{\cpath}$ for a current which is the characteristic function of a path, that is, a current which takes value 1 on every edge of $\cpath \in \Paths(x,y)$ and 0 on all other edges. Then $\curr = \sum \gx_\cpath \charfn{\cpath}$ indicates that \curr decomposes as a sum of currents supported on paths in \Graph.
\end{remark}
  \glossary{name={$\curr = \sum \gx_\cpath \charfn{\cpath}$},description={current decomposed as a sum over paths},sort=I,format=textbf}
  \glossary{name={\cpath},description={path},sort=G}

\begin{theorem}\label{thm:free-resistance}
  For an infinite network \Graph, the free resistance $R^F(x,y)$ has the following equivalent formulations:
  \linenopax
  \begin{align}
    R^F(x,y)
    &= (v_x(x) - v_x(y)) - (v_y(x) - v_y(y)) \label{eqn:def:R^F(x,y)-Lap} \\
    &= \energy(v_x-v_y) \label{eqn:def:R^F(x,y)-energy} \\
    &= \min \{\diss(\curr) \suth \curr \in \Flo(x,y) \text{ and } \curr = \textstyle \sum_{\cpath \in \Paths(x,y)} \gx_\cpath \charfn{\cpath}\} \label{eqn:def:R^F(x,y)-diss} \\
    &= 1 / \min\{\energy(u) \suth\vstr[2.5] u(x)=1, u(y)=0, u \in \dom \energy\} \label{eqn:def:R^F(x,y)-R} \\
    &= \inf\{\gk \geq 0 \suth |v(x)-v(y)|^2 \leq \gk \energy(v), v \in \dom \energy\} \label{eqn:def:R^F(x,y)-S} \\
    &= \sup\{|v(x)-v(y)|^2 \suth \energy(v) \leq 1, v \in \dom \energy\} \label{eqn:def:R^F(x,y)-sup}
  \end{align}
  \begin{proof}
    To see that \eqref{eqn:def:R^F(x,y)-energy} is equivalent to \eqref{eqn:def:free-resistance}, fix any exhaustion of \Graph and note that
    \linenopax
    \begin{align*}
      \energy(v_x-v_y) = \lim_{k \to \iy} \frac12 \sum_{s,t \in G_k} \cond_{st}((v_x-v_y)(s) - (v_x-v_y)(t))^2 = \lim_{k \to \iy} R_{G_k^F}(x,y),
    \end{align*}
    where the latter equality is from Theorem~\ref{thm:effective-resistance-metric}. Then for the equivalence of formulas \eqref{eqn:def:R^F(x,y)-Lap} and \eqref{eqn:def:R^F(x,y)-energy}, simply compute
    \linenopax
    \begin{align*}
      \energy(v_x-v_y)
      &= \la v_x-v_y, v_x-v_y \ra_\energy
       = \la v_x, v_x \ra_\energy - 2\la v_x, v_y \ra_\energy + \la v_y, v_y \ra_\energy
    \end{align*}
    and use the fact that $v_x$ is \bR-valued; cf.~\cite[Lemma~2.22]{DGG}.

    To see \eqref{eqn:def:R^F(x,y)-diss} is equivalent to \eqref{eqn:def:free-resistance}, fix any exhaustion of \Graph and define
    \linenopax
    \begin{align*}
      \Flo(x,y)\evald{H} := \{\curr \in \Flo(x,y) \suth \curr = \textstyle\sum\nolimits_{\cpath \ci H} \gx_\cpath \charfn{\cpath}\}.
    \end{align*}
    From \eqref{eqn:def:R(x,y)-diss}, it is clearly true for each $G_k$ that
    \linenopax
    \begin{align*}
      R_{G_k^F}(x,y) = \min \{\diss(\curr) \suth \curr \in \Flo(x,y) \text{ and } \curr = \textstyle \sum_{\cpath \ci G_k} \gx_\cpath \charfn{\cpath} \}.
    \end{align*}
    Since $\Flo(x,y)\evald{\Graph} = \bigcup_k \Flo(x,y)\evald{G_k}$, formula \eqref{eqn:def:R^F(x,y)-diss} follows. Note that \diss is a quadratic form on the closed convex set $\Flo(x,y)\evald{\Graph}$ and hence it attains its minimum.

    The equivalence of \eqref{eqn:def:R^F(x,y)-R} and \eqref{eqn:def:R^F(x,y)-sup} is \cite[Thm.~2.3.4]{Kig01}.

    As for \eqref{eqn:def:R^F(x,y)-S} and \eqref{eqn:def:R^F(x,y)-sup}, they are both clearly equal to $\|L_{xy}\|$ (as described in Remark~\ref{rem:R(x,y) = |L{xy}|}) by the definition of operator norm; see \cite[\S5.3]{Rud87}, for example. To show that these are equivalent to $R^F$ as defined in \eqref{eqn:def:free-resistance}, define a subspace of \HE consisting of those voltages whose induced currents are supported in a finite subnetwork $H$ by
    \linenopax
    \begin{equation}\label{eqn:def:HE-F-restr}
      \HE\evald[F]{H} = \{u \in \dom \energy \suth u(x)-u(y)=0 \text{ unless } x,y \in H\}.
    \end{equation}
    This is a closed subspace, as it is the intersection of the kernels of a collection of continuous linear functionals $\|L_{st}\|$, and so we can let $Q_k$ be the projection to this subspace. Then it is clear that $Q_k \leq Q_{k+1}$ and that $\lim_{k \to \iy} \|u - Q_k u\|_\energy = 0$ for all $u \in \HE$, so
    \linenopax
    \begin{align}\label{eqn:defn-Q_k}
      R_{G_k^F}(x,y)
      &= \| L_{xy} \|_{\HE|_{G_k} \to \bC }
       = \| L_{xy} Q_k\|,
    \end{align}
where the first equality follows from \eqref{eqn:def:R(x,y)-S} (recall that $G_k$ is finite) and therefore
    \linenopax
    \begin{align*}
      R^F(x,y) = \lim_{k \to \iy} R_{G_k^F}(x,y)
      &= \lim_{k \to \iy} \|L_{xy} Q_k\|
      = \left\| \lim_{k \to \iy} L_{xy} Q_k \right\|
      = \|L_{xy}\|.
      \qedhere
    \end{align*}
  \end{proof}
\end{theorem}

  In view of the previous result, the free case corresponds to consideration of only those voltage functions whose induced current can be decomposed as a sum of currents supported on paths in \Graph. The wired case considered in the next section corresponds to considering all voltages functions whose induced current flow satisfies Kirchhoff's law in the form \eqref{eqn:Kirchhoff-nonhomog}; this is clear from comparison of \eqref{eqn:def:R^F(x,y)-diss} to \eqref{eqn:def:R^W(x,y)-diss}. See also Remark~\ref{rem:wired-vs-free-as-Kirchhoff-conditions+}.

  Formula \eqref{eqn:def:R^F(x,y)-Lap} turns out to be useful for explicit computations. Explicit formulas for the effective resistance metric on \bZd are obtained from \eqref{eqn:def:R^F(x,y)-Lap} in \cite[\S14.2]{OTERN}; compare to \cite[\S{V.2}]{Soardi94}.

\begin{remark}\label{rem:martingales-in-Hilbert-space}
  In Theorem~\ref{thm:free-resistance}, the proofs that $R^F$ is given by \eqref{eqn:def:R^F(x,y)-diss} or \eqref{eqn:def:R^F(x,y)-S} stem from essentially the same underlying martingale argument. In a Hilbert space, a martingale is an increasing sequence of projections $\{Q_k\}$ with the martingale property $Q_k = Q_k Q_{k+1}$. Recall that conditional expectation is a projection. In this context, Doob's theorem \cite{Doob53} then states that if $\{f_k\} \ci \sH$ is such that $f_k = Q_k f_j$ for any $j \geq k$, then the following are equivalent:
\begin{enumerate}[(i)]
  \item there is a $f \in \sH$ such that $f_k = Q_k f$ for all $k$
  \item $\sup_k \|f_k\| < \iy$.
\end{enumerate}
  For \eqref{eqn:def:R^F(x,y)-diss}, we are actually projecting to subspaces of \HD, the Hilbert space of currents introduced as the \emph{dissipation space} in \S\ref{sec:H_energy-and-H_diss}.
  In \cite[\S9.1]{Lyons:ProbOnTrees}, the free resistance $R^F(x,y)$ is defined directly  via this approach (and similarly for $R^W(x,y)$).

  In view of the previous result, the free case corresponds to consideration of only those voltage functions whose induced current can be decomposed as a sum of currents supported on paths in \Graph. The wired case considered in the next section corresponds to considering all voltages functions whose induced current flow satisfies Kirchhoff's law \eqref{def:Kirchhoff's-law}; this is clear from comparison of \eqref{eqn:def:R^F(x,y)-diss} to \eqref{eqn:def:R^W(x,y)-diss}. See also Remark~\ref{rem:wired-vs-free-as-Kirchhoff-conditions+}.

  Formula \eqref{eqn:def:R^F(x,y)-Lap} turns out to be useful for explicit computations; we use it to obtain explicit formulas for the effective resistance metric on \bZd in Theorem~\ref{thm:R(x,y)-on-Zd}.
\end{remark}

The following result is also a special case of \cite[Thm.~2.3.4]{Kig01}.

\begin{prop}\label{thm:free-resistance-is-a-metric}
  $R^F(x,y)$ is a metric.
  \begin{proof}
    One has $R_{G_k^F}(x,z) \leq R_{G_k^F}(x,y) + R_{G_k^F}(y,z)$ for any $k$, so take the limit.
  \end{proof}
\end{prop}

From Theorem~\ref{thm:free-resistance}, it is clear that the triangle inequality also has the formulation
\linenopax
\begin{equation*}
  \energy(v_x-v_z) \leq \energy(v_x-v_y) + \energy(v_y-v_z), \q\forall x,y,z \in \verts,
\end{equation*}
which is easily shown to be equivalent to
\linenopax
\begin{equation*}
  v_x(z) + v_y(z) \leq v_z(z) + v_x(y), \q\forall x,y,z \in \verts,
\end{equation*}
using the convention $v_x(o)=0$.

The next result is immediate from \eqref{eqn:def:R^F(x,y)-S} and appears also in \cite{Kig03}.

\begin{cor}\label{thm:vx-is-Lipschitz}
  Every function in \HE is H\"{o}lder continuous with exponent $\frac12$.
\end{cor}

It is known from \cite{Nelson64} that the Gaussian measure of Brownian motion is supported on the space of such functions and this will be useful later; cf.~Remark~\ref{rem:boundary-of-G-from-Minlos} and the beginning of \S\ref{sec:Gel'fand-triples-and-duality}. It is somewhat subtle to determine if $R(x,\cdot)$ is in \HE.

\begin{comment}\label{conj:w(x)=R(o,x)}
  On a general \ERN $(\Graph, \cond)$, the function
  \linenopax
  \begin{equation}\label{eqn:w(x)=R(o,x)}
    w(x) = c R(o,x)
  \end{equation}
  satisfies the equation $\Lap w = -\gd_o$ for the appropriate choice of constant $c$. Therefore, the existence of a monopole is equivalent to $\energy(R(o,x)) < \iy$. Since Corollary~\ref{thm:vx-is-Lipschitz} shows that every element of \HE is H\"{o}lder with exponent $\frac12$, we also expect that $R(o,x)$ is H\"{o}lder-$\frac12$ if and only if $\energy(R(o,x)) < \iy$.
\end{comment}

\section{Wired resistance}

\begin{defn}\label{def:wired-resistance}
  Given a finite full subnetwork $H$ of \Graph, define the wired subnetwork $H^W$ by identifying all vertices in $\verts \less \verts[H]$ to a single, new vertex labeled \iy. Thus, the vertex set of $H^W$ is $\verts[H] \cup \{\iy\}$, and the edge set of $H^W$ includes all the edges of $H$, with the same conductances. However, if $x \in \verts[H]$ has a neighbour $y \in \verts \less \verts[H]$, then $H^W$ also includes an edge from $x$ to \iy with conductance
  \linenopax
  \begin{align}\label{eqn:cond-to-iy}
    \cond_{x\iy} := \sum_{y \nbr x, \, y \in H^\complm} \negsp[13]\cond_{xy}.
  \end{align}

  Let $R_{H^W}(x,y)$ denote the effective resistance between $x$ and $y$ as computed in $H^W$, as in Definition~\ref{def:relative-resistance}.
  The \emph{wired resistance} is then defined to be
  \linenopax
  \begin{align}\label{eqn:def:wired-resistance}
    R^W(x,y) := \lim_{k \to \iy} R_{G_k^W}(x,y),
  \end{align}
  where $\{G_k\}$ is any exhaustion of \Graph.
\end{defn}
  \glossary{name={$R^W(x,y)$},description={wired resistance metric},sort=RW,format=textbf}
  \glossary{name={$R_H(x,y)$},description={relative resistance as computed with respect to the subnetwork $H$},sort=RH}

\begin{remark}\label{rem:wired-resistance}
  The wired subnetwork is equivalently obtained by ``shorting together'' all vertices of $H^\complm$, and hence it follows from Rayleigh's monotonicity principle that $R^W(x,y) \leq R^F(x,y)$; cf. \cite[\S1.4]{DoSn84} or \cite[\S2.4]{Lyons:ProbOnTrees}. The reader will see by comparison to Theorem~\ref{thm:wired-resistance} that the wired resistance $R^W$ is also the effective resistance associated to the resistance form $(\energy,\Fin)$ of \cite{Kig03}; see Remark~\ref{rem:comparison-to-resistance-forms}. However, the wired resistance is \emph{not} related to the ``shorted resistance form'' of \cite[\S3]{Kig03} (see Prop.~3.6 in particular).

  The justification for \eqref{eqn:cond-to-iy} is that the identification of vertices in $G_k^\complm$ may result in parallel edges. Then \eqref{eqn:cond-to-iy} corresponds to replacing these parallel edges by a single edge according to the usual formula for resistors in parallel.
\end{remark}

\begin{theorem}\label{thm:wired-resistance}
  The wired resistance may be computed by any of the following equivalent formulations:
  \linenopax
  \begin{align}
    R^W(x,y)
    &= \min_v \{v(x)-v(y) \suth \Lap v = \gd_x-\gd_y, v \in \dom\energy\}
    \label{eqn:def:R^W(x,y)-Lap} \\
    &= \min_v \{\energy(v) \suth \Lap v = \gd_x-\gd_y, v \in \dom\energy\} \label{eqn:def:R^W(x,y)-energy} \\
    &= \min_\curr \{\diss(\curr) \suth \curr \in \Flo(x,y), \diss(\curr) < \iy\} \label{eqn:def:R^W(x,y)-diss} \\
    &= 1/\min\{\energy(u) \suth u(x)=1, u(y)=0, u \in \Fin\} \label{eqn:def:R^W(x,y)-R} \\
    &= \inf\{\gk \geq 0 \suth |v(x)-v(y)|^2 \leq \gk \energy(v), v \in \Fin\} \label{eqn:def:R^W(x,y)-S} \\
    &= \sup\{|v(x)-v(y)|^2 \suth \energy(v) \leq 1, v \in \Fin\} \label{eqn:def:R^W(x,y)-sup}
  \end{align}
  \begin{proof}
    Since \eqref{eqn:def:R^W(x,y)-S} and \eqref{eqn:def:R^W(x,y)-sup} are both clearly equivalent to the norm of $L_{xy}:\Fin \to \bC$ (where again $L_{xy}u - u(x)-u(y)$ as in Remark~\ref{rem:R(x,y) = |L{xy}|}), we begin by equating them to \eqref{eqn:def:wired-resistance}. From Definition~\ref{def:Fin}, we see that
    \linenopax
    \begin{align}\label{eqn:def:HE-W-restr}
      \HE\evald[W]{H}
      := \{u \in \HE \suth \spt u \ci H\}
    \end{align}
    is a closed subspace of \HE. Let $Q_k$ be the projection to this subspace. Then it is clear that $Q_k \leq Q_{k+1}$ and that $\lim_{k \to \iy} \| P_{\Fin}u - Q_k u\|_\energy = 0$ for all $u \in \HE$. Each function $u$ on $H^W$ corresponds to a function $\tilde u$ on \Graph whose support is contained in $H$; simply define
    \linenopax
    \begin{align*}
    \tilde u(x) =
    \begin{cases}
       u(x), & x \in H, \\
       u(\iy_H), & x \notin H.
    \end{cases}
    \end{align*}
    It is clear that this correspondence is bijective, and that
    \linenopax
    \begin{align*}
      R_{G_k^W}(x,y)
      &= \| L_{xy} \|_{\HE|_{G_k}^W \to \bC }
       = \| L_{xy} Q_k\|,
    \end{align*}
    where the first equality follows from \eqref{eqn:def:R(x,y)-S} (recall that $G_k$ is finite) and therefore
    \linenopax
    \begin{align*}
      R^W(x,y) = \lim_{k \to \iy} R_{G_k^W}(x,y)
      &= \lim_{k \to \iy} \|L_{xy} Q_k\|
      = \|L_{xy} P_{\Fin}\|,
    \end{align*}
    which is equivalent to \eqref{eqn:def:R^W(x,y)-S}.

    To see \eqref{eqn:def:R^W(x,y)-Lap} is equivalent to \eqref{eqn:def:R^W(x,y)-energy}, note that the minimal energy solution to $\Lap u = \gd_x - \gd_y$ lies in \Fin, since any two solutions must differ by a harmonic function. Let $u$ be a solution to $\Lap u = \gd_x - \gd_y$ and define $f = \Pfin u$. Then $f \in \Fin$ and $\Lap f = \gd_x - \gd_y$ implies
    \linenopax
    \begin{align}
      \|f\|_\energy^2
      = \sum_{z \in \verts} f(z) \Lap f(z)
      = \sum_{z \in \verts} f(z) (\gd_x-\gd_y)(z)
      = f(x) - f(y).
      \label{eqn:thm:wired-resistance:eqn1}
    \end{align}

    To see \eqref{eqn:def:R^W(x,y)-Lap}$\leq$ \eqref{eqn:def:R^W(x,y)-S}, let \gk be the optimal constant from \eqref{eqn:def:R^W(x,y)-S}. If $u \in \Fin$ is the unique solution to $\Lap u = \gd_x - \gd_y$, then
    \linenopax
    \begin{align*}
      \gk = \sup_{u \in \Fin} \left\{\frac{|u(x)-u(y)|^2}{\energy(u)}\right\}
      \geq \frac{|u(x)-u(y)|^2}{\energy(u)}
      = u(x)-u(y),
    \end{align*}
    where the last equality follows from $\energy(u) = u(x) - u(y)$, by the same computation as in \eqref{eqn:thm:wired-resistance:eqn1}.
    For the reverse inequality, note that with $L_{xy}$ as just above,
    \linenopax
    \begin{align*}
      \frac{|u(x)-u(y)|^2}{\energy(u)}
      = \left| L_{xy} \left(\tfrac{u}{\energy(u)^{1/2}}\right)\right|^2
      = \left| \left\la v_x-v_y, \tfrac{u}{\energy(u)^{1/2}}\right\ra_\energy\right|^2,
    \end{align*}
    for any $u \in \Fin$. Note that Lemma~\ref{thm:wired-resistance-is-a-metric} allows one to replace $v_x$ by $f_x = \Pfin v_x$, whence
    \linenopax
    \begin{align*}
      \frac{|u(x)-u(y)|^2}{\energy(u)}
      &\leq \energy(f_x-f_y) \energy\left(\tfrac{u}{\energy(u)^{1/2}}\right)
      = \energy(f_x-f_y)
    \end{align*}
    by Cauchy-Schwarz. The infimum of the left-hand side over nonconstant functions $u \in \Fin$ gives the optimal \gk in \eqref{eqn:def:R^W(x,y)-S}, and thus shows that \eqref{eqn:def:R^W(x,y)-S} $\leq$ \eqref{eqn:def:R^W(x,y)-energy}.

    To see \eqref{eqn:def:R^W(x,y)-energy} is equivalent to \eqref{eqn:def:R^W(x,y)-diss}, recall that \curr minimizes \diss over $\Flo(x,y)$ if and only if $\curr = \drp u$ for $u$ which minimizes \energy over $\{v \in \dom \energy \suth \Lap v = \gd_x - \gd_y\}$; see \cite[Thm.~3.26]{OTERN}, for example. Apply this to $\curr = \drp f$, where $f = \Pfin u$ is the minimal energy solution to $\Lap u = \gd_x - \gd_y$.

    The equivalence of \eqref{eqn:def:R^W(x,y)-R} and \eqref{eqn:def:R^W(x,y)-sup} is directly parallel to the finite case and may also be obtained from \cite[Thm.~2.3.4]{Kig01}.
  \end{proof}
\end{theorem}

\begin{remark}[$R^F$ vs. $R^W$ explained in terms of boundary conditions on \Lap]
  \label{rem:wired-vs-free-as-boundary-conditions}
  Observe that both spaces
  \linenopax
  \begin{align*}
    \HE\evald[F]{H} = \{u \in \HE \suth u(x)-u(y)=0 \text{ unless } x,y \in H\}
  \end{align*}
  and
  \linenopax
  \begin{align*}
    \HE\evald[W]{H} = \{u \in \HE \suth \spt u \ci H\}
  \end{align*}
  consist of functions which have no energy outside of $H$. The difference is that if the complement of $H$ consists of several connected components, then $u \in \HE|_H^F$ may take a different constant value on each one; this is not allowed for elements of $\HE|^W_H$. Therefore, $\HE|_H^F$ corresponds to Neumann boundary conditions and $\HE|_H^W$ corresponds to Dirichlet boundary conditions. That is, from the proofs of Theorem~\ref{thm:free-resistance} and Theorem~\ref{thm:wired-resistance}, we see
  \begin{enumerate}
    \item $R_{H^F}(x,y) = u(x)-u(y)$ where $u$ is the solution to $\Lap u = \gd_x - \gd_y$ with Neumann boundary conditions on $H^\complm$, and
    \item $R_{H^W}(x,y) = u(x)-u(y)$ where $u$ is the solution to $\Lap u = \gd_x - \gd_y$ under Dirichlet boundary conditions on $H^\complm$.
  \end{enumerate}

\end{remark}

\begin{remark}\label{rem:wired-vs-free-as-Kirchhoff-conditions+}
  While the wired subnetwork takes into account the conductivity due to all paths from $x$ to $y$ (see Remark~\ref{rem:free-resistance}), it is overzealous in that it may also include paths from $x$ to $y$ that do not correspond to any path in \Graph (see Remark~\ref{rem:martingales-in-Hilbert-space}). On an infinite network, this leads to current flows in which some of the current travels from $x$ to \iy, and then from \iy to $y$. Consider the example of \cite{Morris2003}: let \Graph be \bZ with $\cond_{n,n+1} =1$ for each $n$. Then define $J$ by
  \linenopax
  \begin{align*}
    J(n,n-1) =
    \begin{cases}
      1, & n \neq 1 \\
      0, & n=1.
    \end{cases}
  \end{align*}
  If a unit current flow from 0 to 1 is defined to be a current satisfying $\sum_{y \nbr x} \curr(x,y) = \gd_x - \gd_y$, then $J$ is such a flow which ``passes through \iy'' (of course, $J$ certainly not of finite energy).
\end{remark}

The proof of the next result follows from the finite case, exactly as in Theorem~\ref{thm:free-resistance-is-a-metric}.
\begin{theorem}\label{thm:wired-resistance-is-a-metric}
  $R^W(x,y)$ is a metric.
\end{theorem}

\section{Harmonic resistance}
\label{sec:Harmonic-resistance}

\begin{defn}\label{def:harmonic-resistance}
  For an infinite network $(G,\cond)$ define the \emph{harmonic resistance} between $x$ and $y$ by
  \linenopax
  \begin{align}\label{eqn:def:harmonic-resistance}
    R^\hrm(x,y) := R^F(x,y) - R^W(x,y).
  \end{align}
\end{defn}

The next result is immediate upon comparing Theorem~\ref{thm:free-resistance} and Theorem~\ref{thm:wired-resistance}.

\begin{theorem}\label{thm:harmonic-resistance}
  With $h_x = \Phar v_x$ as in Remark~\ref{rem:3-repkernels}, the harmonic resistance is equal to
  \linenopax
  \begin{align}
    R^\hrm(x,y)
    &= (h_x(x) - h_x(y)) - (h_y(x) - h_y(y)) \label{eqn:def:R^ha(x,y)-Lap} \\
    &= \energy(h_x-h_y) \label{eqn:def:R^ha(x,y)-energy} \\

    &= \tfrac1{\min\{\energy(v) \suth v(x)=1, v(y)=0\}}  - \tfrac1{\min\{\energy(f) \suth f(x)=1, f(y)=0, f \in \Fin\}} \label{eqn:def:R^ha(x,y)-R} \\
    &= \inf\{\gk \geq 0 \suth |h(x)-h(y)|^2 \leq \gk \energy(h), h \in \Harm\} \label{eqn:def:R^ha(x,y)-S} \\
    &= \sup\{|h(x)-h(y)|^2 \suth \energy(h) \leq 1, h \in \Harm\} \label{eqn:def:R^ha(x,y)-sup}
  \end{align}
  \begin{comment}
    This result mostly follows upon comparing Theorem~\ref{thm:free-resistance} and Theorem~\ref{thm:wired-resistance}.
    Note from the proofs of Theorem~\ref{thm:free-resistance} and Theorem~\ref{thm:wired-resistance} that
    \begin{align*}
      \energy(h_x - h_y)
      &= \energy(v_x - v_y) - \energy(f_x - f_y)
       = \|L_{xy}\| - \|L_{xy} \Pfin\|
       = \|L_{xy} \Phar \|
    \end{align*}
    by orthogonality, and then
    \begin{align*}
      \|L_{xy} \Phar \|
      &= \sup\{ |u(x) - u(y)|^2 \suth \energy(u) \leq 1, u \in \Harm\} \\
      &= \sup\left\{ \frac{|u(x) - u(y)|^2}{\energy(u)} \suth u \in \Harm \less \{0\}\right\}.
    \end{align*}
    Given a nonconstant $u \in \Harm$, one can substitute $h := \frac{u}{|u(x)-u(y)|}$ into the previous line to obtain
    \begin{align*}
      \|L_{xy} \Phar \|
      &= \sup\left\{ \frac{|h(x) - h(y)|^2}{\energy(h)} \suth |h(x)-h(y)|=1, h \in \Harm \right\} \\
      &= \sup\left\{ \frac{1}{\energy(h)} \suth |h(x)-h(y)|=1, h \in \Harm \right\},
    \end{align*}
    which is equal to the right-hand side of \eqref{eqn:def:R^ha(x,y)-R}. Now $\energy(v_x - v_y) = \energy(f_x - f_y) + \energy(h_x - h_y)$ by orthogonality, whence
    \begin{align*}
      \tfrac1{\min\{\energy(v) \suth v(x)=1, v(y)=0\}}
      &= \tfrac1{\min\{\energy(v) \suth v(x)=1, v(y)=0\}}  + \tfrac1{\min\{\energy(v) \suth v(x)=1, v(y)=0\}}
    \end{align*}
    follows from \eqref{eqn:def:R^W(x,y)-R} and \eqref{eqn:def:R^ha(x,y)-R}.
  \end{comment}
\end{theorem}

  \version{}{\marginpar{Idea for precise definition: $g_o$ is constant on this portion.}}
\begin{remark}\label{rem:harmonic-resistance-not-a-metric}\label{rem:Rharm-has-no-form}
  Note that $R^\hrm$ is not the effective resistance associated to a resistance form, as in Remark~\ref{rem:comparison-to-resistance-forms}, since (RF5) may fail. If $R^\hrm$ \emph{were} the effective resistance associated to a resistance form, then \cite[Prop.~2.10]{Kig03} would imply that $R^\hrm(x,y)$ is a metric, but this can be seen to be false by considering basic examples. See Example~\ref{exm:geometric-integers}, e.g. The same remarks also apply to the  boundary resistance $R^{\bdy}(x,y)$, discussed just below.

\end{remark}

\begin{defn}\label{def:boundary-resistance}
  For an infinite network $(G,\cond)$ define the \emph{boundary resistance} between $x$ and $y$ by
  \linenopax
  \begin{align}\label{eqn:def:boundary-resistance}
    R^{\bdy}(x,y) := \frac1{R^W(x,y)^{-1} - R^F(x,y)^{-1}}.
  \end{align}
\end{defn}

  Intuitively, some portion of the wired/minimal current from $x$ to $y$ passes through infinity; the quantity $R^{\bdy}(x,y)$ gives the voltage drop ``across infinity''; see Remark~\ref{rem:shortcut-through-infinity}. From this perspective, infinity is ``connected in parallel''. The boundary $\bd G$ in \cite{bdG} is a more rigorous definition of the set at infinity.

\begin{theorem}\label{thm:boundary-resistance}
  The boundary resistance is equal to
  \linenopax
  \begin{align}
    R^\bdy(x,y) =
    \frac{R^W(x,y)R^F(x,y)}{R^{\hrm}(x,y)}.
    \label{eqn:def:R^hrm(x,y)-R}
  \end{align}
  In particular, the resistance across the boundary is infinite if $\Harm = 0$.
  \begin{proof}
    From \eqref{eqn:def:harmonic-resistance} one has $R^F(x,y) = 1/(R^W(x,y)^{-1} - R^\bdy(x,y)^{-1})$, which gives
    \begin{align*}
      \frac1{\energy(v_x-v_y)}
        &=  \frac1{\energy(f_x-f_y)} - \frac1{R^\bdy(x,y)}
    \end{align*}
    by Theorem~\ref{thm:free-resistance} and Theorem~\ref{thm:wired-resistance}, and hence
    \begin{align*}
      \frac1{R^\bdy(x,y)}
        &= \frac1{\energy(f_x-f_y)} - \frac1{\energy(v_x-v_y)}.
    \end{align*}
    Now solving for $R^\bdy$ gives
    \begin{align}
      R^\bdy(x,y)
      = \frac{\energy(f_x-f_y)\energy(v_x-v_y)}{\energy(h_x-h_y)},

    \end{align}
    and the conclusion follows from \eqref{eqn:def:R^F(x,y)-energy}, \eqref{eqn:def:R^W(x,y)-energy}, and \eqref{eqn:def:R^ha(x,y)-energy}.
  \end{proof}
\end{theorem}

\section{Trace resistance}
\label{sec:trace-resistance}

The third type of subnetwork takes into account the connectivity of the complement of the subnetwork, but does not add anything extra.  The name ``trace'' is due to the fact that this approach comes by considering the trace of the Dirichlet form \energy to a subnetwork; see \cite{FOT94}. Several of the ideas in this section were explored previously in \cite{Kig01,Kig03,Metz}; see also \cite{Kig09}.

The discussion of the trace resistance and trace subnetworks requires some definitions relating the transition operator (i.e. Markov chain) \Prob to the probability measure $\prob^{(\cond)}$ on the space of (infinite) paths in \Graph which start at a fixed vertex $a$. Such a path is a sequence of vertices $\{x_n\}_{n=0}^\iy$, where $x_0=a$ and $x_n \nbr x_{n+1}$ for all $n$.

\begin{defn}\label{def:Gamma(a,b)}
  Let $\Paths(a)$ be the space of all paths \cpath beginning at the vertex $a \in \verts$, and let $\Paths(a,b) \ci \Paths(a)$ be the subset of paths that reach $b$, and that do so \emph{before} returning to $a$:
  \linenopax
  \begin{align}\label{eqn:def:Paths(a,b)}
    \Paths(a,b) := \{\cpath \in \Paths(a) \suth b=x_n \text{ for some $n$, with } x_k \neq a \text{ for } 1 \leq k \leq n\}.
  \end{align}
\end{defn}

\begin{defn}\label{def:prob^cond}
   The space $\Paths(a)$ carries a natural probability measure $\prob^{(\cond)}$ defined by
  \linenopax
  \begin{align}\label{eqn:prob^cond}
    \prob^{(\cond)}(\cpath) := \prod_{x_i \in \cpath} p(x_{i-1},x_i),
  \end{align}
  where $p(x,y) = \cond_{xy}/\cond(x)$.
  The construction of $\prob^{(\cond)}$ comes by applying Kolmogorov consistency to the natural cylinder-set Borel topology that makes $\Paths(a)$ into a compact Hausdorff space.
\end{defn}

\begin{defn}\label{def:hitting-time}
  Let $X_m$ be a random variable which denotes the (vertex) location of the random walker at time $m$. Then let $\gt_x$ be the \emph{hitting time} of $x$, that is, the random variable which is the expected time at which the walker first reaches $x$:
  \linenopax
  \begin{align}\label{eqn:def:hitting-time}
    \gt_x := \min\{m \geq 0 \suth X_m = x\}.
  \end{align}
  More generally, $\gt_H$ is the time at which the walker first reaches the subnetwork $H$. For a walk started in $H$, this gives $\gt_H = 0$.
\end{defn}

\subsection{The trace subnetwork}

It is well-known that networks $\{(\Graph,\cond)\}$ are in bijective correspondence with reversible Markov processes $\{\Prob\}$; this is immediate from the \emph{detailed balance equations} which follow from the symmetry of the conductance:
\linenopax
\begin{align*}
  \cond(x) p(x,y) = \cond_{xy} = \cond_{yx} = \cond(y) p(y,x).
\end{align*}
It follows from $\Lap = \cond(\one-\Prob)$ that networks are thus in bijective correspondence with \emph{Laplacians}, if one defines a Laplacian as in \eqref{eqn:def:laplacian}. That is, a Laplacian is a symmetric linear operator which is nonnegative definite, has kernel consisting of the constant functions, and satisfies $(\Lap\gd_x)(y) \leq 0$ for $x \neq y$. In other words, every row (and column) of $\trc(\Lap,H)$ sums to 0. (This is the negative of the definition of a \emph{Laplacian} as in \cite{Kig01} and \cite{CdV98}.)

In this section, we exploit the bijection between Laplacians and networks to define the trace subnetwork. For $\verts[H] \ci \Graph$, the idea is as follows:
\linenopax
\begin{align*}
  G \;\longleftrightarrow\; \Lap
  \;\limmode[\text{take the trace to \verts[H]}]\; \trc(\Lap, \verts[H])
  \;\longleftrightarrow\; H^\trc.
\end{align*}

\begin{defn}\label{def:trace-resistance}
  The \emph{trace} of \Graph to $\verts[H]$ is the network whose edge data is defined by the trace of \Lap to \verts[H], which is computed as the Schur complement of the Laplacian of $H$ with respect to \Graph. More precisely, write the Laplacian of \Graph as a matrix in block form, with the rows and columns indexed by vertices, and order the vertices so that those of $H$ appear first:
  \linenopax
  \begin{align}\label{eqn:Lap-block-decomp}
    \Lap =
      \begin{array}{l} \scalebox{0.70}{$H$} \\ \scalebox{0.70}{$H^\complm$} \end{array}
      \negsp[12]
      \left[\begin{array}{ll} A & B^T \\ B & D \end{array}\right],
  \end{align}
  where $B^T$ is the transpose of $B$. If $\ell(G) :=\{f:\verts \to \bR\}$, the corresponding mappings are
  \linenopax
  \begin{align}
    A:&\ell(H) \to \ell(H) & B^T&:\ell(H^\complm) \to \ell(H) \notag \\
    B:&\ell(H) \to \ell(H^\complm) & D&:\ell(H^\complm) \to \ell(H^\complm).
    \label{eqn:action-of-Lap-quadrants}
  \end{align}
\glossary{name={$\trc(\Lap, \verts[H])$},description={trace of the Laplacian to a subnetwork $H$},sort=tr,format=textbf}
  It turns out that the Schur complement
  \linenopax
  \begin{equation}\label{eqn:def:Schur-complement}
    \trc(\Lap, \verts[H]) := A - B^T D^{-1} B
  \end{equation}
  is the Laplacian of a subnetwork with vertex set $\verts[H]$; cf.~\cite[\S2.1]{Kig01} and Remark~\ref{rem:trace-valid-for-subsets}.\footnote{It will be clear from \eqref{eqn:Schur-complement-as-sum} that $D^{-1}$ always exists in this context, and hence \eqref{eqn:def:Schur-complement} is always well-defined. Furthermore, the existence of the trace is given in \cite[Prop.~2.10]{Kig03}; it is known from \cite[Lem.~2.1.5]{Kig01} that $D$ is invertible and negative semidefinite.} A formula for the conductances (and hence the adjacencies) of the trace is given in Theorem~\ref{thm:edges-of-H^S}. Denote this new subnetwork by $H^\trc$.

  If $\verts[H] \ci \verts$ is finite, then for $x,y \in H$, the trace of the resistance on $H$ is denoted $R_{H^\trc}(x,y)$, and defined as in Definition~\ref{def:effective-resistance}. The \emph{trace resistance} is then defined to be
  \linenopax
  \begin{align}\label{eqn:trace-resistance}
    R^\trc(x,y) := \lim_{k \to \iy} R_{G_k^\trc}(x,y),
  \end{align}
  where $\{G_k\}$ is any exhaustion of \Graph.
\end{defn}
  \glossary{name={$R^\trc(x,y)$},description={trace resistance metric},sort=Rt,format=textbf}
  \glossary{name={$R_H(x,y)$},description={relative resistance as computed with respect to the subnetwork $H$},sort=RH}

\begin{remark}\label{rem:trace-resistance-name}
  The name ``trace'' is due to the fact that this approach comes by considering the trace of the Dirichlet form \energy on a subnetwork; see \cite{FOT94}.
\end{remark}

Recall that $\Lap = \cond - \Trans = \cond(\id - \Prob)$, where \Trans is the transfer operator and $\Prob = \cond^{-1}\Trans$ is the probabilistic transition operator defined   \glossary{name={\Prob},description={probabilistic transition operator},sort=P} \glossary{name={\Trans},description={transfer operator},sort=T}
\glossary{name={\cond},description={conductance (multiplication) operator},sort=c}
pointwise by
  \linenopax
  \begin{align}
    \Prob u(x) = \sum_{y \nbr x} p(x,y) u(y),
    \q\text{for }
    p(x,y) = \frac{\cond_{xy}}{\cond(x)}.
    \label{eqn:def:p(x,y)}
  \end{align}
  \glossary{name={$p(x,y)$},description={transition probability of the random walk on the network: $\cond_{xy}/\cond(x)$},sort=prob}
  The function $p(x,y)$ gives transition probabilities, i.e., the probability that a random walker currently at $x$ will move to $y$ with the next step. Since
  \linenopax
  \begin{align}\label{eqn:def:reversible}
    \cond(x) p(x,y) = \cond(y) p(y,x),
  \end{align}
  the transition operator \Prob determines a \emph{reversible} Markov process with state space \verts; see \cite{LevPerWil08,Lyons:ProbOnTrees}. Note that the harmonic functions (i.e., $\Lap h=0$) are precisely the fixed points of \Prob (i.e., $\Prob h = h$). The proof of the next theorem requires a couple more definitions which relate \Prob to the probability measure $\prob^{(\cond)}$ on the space of paths in \Graph. Recall from Definition~\ref{def:paths} that a path is a sequence of vertices $\{x_n\}_{n=0}^\iy$, where $x_0=a$ and $x_n \nbr x_{n+1}$ for all $n$.

\begin{defn}\label{def:Gamma(a,b)}
  Let $\Paths(a)$ denote the space of all paths \cpath beginning at the vertex $a \in \verts$.
    \glossary{name={$\Paths(a)$},description={the space of all paths in $G$ beginning at the vertex $a$},sort=G,format=textbf}
Then $\Paths(a,b) \ci \Paths(a)$ consists of those paths that reach $b$, and \emph{before} returning to $a$:
  \linenopax
  \begin{align}\label{eqn:def:Paths(a,b)}
    \Paths(a,b) := \{\cpath \in \Paths(a) \suth b=x_n \text{ for some $n$, and } x_k \neq a, 1 \leq k \leq n\}.
  \end{align}
  If $a,b \in \bd H$, then we write
  \linenopax
  \begin{align}\label{eqn:def:Paths(a,b)-outside-H}
    \Paths(a,b)\evald{H^\complm}
    := \{\cpath \in \Paths(a,b) \suth x_i \in H^\complm, 0 < i < \gt_b\},
  \end{align}
  for the set of paths from $a$ to $b$ that do not pass through any vertex in $\verts[H]$.
\end{defn}

\begin{remark}\label{rem:trivial-paths}
  Note that if $x,y \in \bd H$ are adjacent, then any path of the form $\cpath = (x,y,\dots)$ is trivially in $\Paths(a,b)\evald{H^\complm}$.
\end{remark}

\begin{defn}\label{def:prob^cond}
   The space $\Paths(a)$ carries a natural probability measure $\prob^{(\cond)}$ defined by
  \linenopax
  \begin{align}\label{eqn:prob^cond}
    \prob^{(\cond)}(\cpath) := \prod_{x_i \in \cpath} p(x_{i-1},x_i),
  \end{align}
  where $p(x,y)$ is as in \eqref{eqn:def:p(x,y)}.
  \glossary{name={\cpath},description={path},sort=G}
  \glossary{name={$\prob(\cpath)$},description={probability of a path},sort=P,format=textbf}
  \glossary{name={$\prob^{(\cond)}$},description={a measure on the space of all paths in $G$},sort=Pc,format=textbf}
  The construction of $\prob^{(\cond)}$ comes by applying Kolmogorov consistency to the natural cylinder-set Borel topology that makes $\Paths(a)$ into a compact Hausdorff space; cf. \S\ref{sec:Probabilistic-interpretation} for further discussion.
\end{defn}

\begin{defn}\label{def:prob[a->b]}
  Let $\prob[a \to b]$ denote the probability that a random walk started at $a$ will reach $b$ before returning to $a$. That is,
  \glossary{name={$\prob[a \to b]$},description={probability of the random walk started from $a$ reaching $b$},sort=P,format=textbf}
  \linenopax
  \begin{align}\label{eqn:def:prob-a-to-b}
    \prob[a \to b] := \prob^{(\cond)}(\Paths(a,b)).
  \end{align}
  Note that this is equivalent to
  \linenopax
  \begin{align}\label{eqn:def:prob-a-to-b-as-hitting}
    \prob[a \to b] = \prob_a[\gt_b < \gt_a] := \prob[\gt_b < \gt_a \,|\, x_0=a],
  \end{align}
  where $\gt_a$ is the \emph{hitting time} of $a$, i.e., the expected time of the first visit to $a$, \emph{after} leaving the starting point. If $a,b \in \bd H$, then we write
  \linenopax
  \begin{align}\label{eqn:def:prob-a-to-b-outside-H}
    \prob[a \to b]\evald{H^\complm}
    := \prob^{(\cond)}\left(\Paths(a,b)\evald{H^\complm}\right),
  \end{align}
  \glossary{name={$\complement$},description={$H^\complm$ is the complement of $H$ in $G$},sort=C}
  that is, the probability that a random walk started at $a$ will reach $b$ via a path lying outside $H$ (except for the start and end points, of course).
\end{defn}

\begin{remark}[More probabilistic notation]\label{rem:prob-notation}\label{rem:notation-restriction-to-Gk}
  The formulation in \eqref{eqn:def:prob-a-to-b-outside-H} is conditioning $\prob^{(\cond)}(\Paths(a,b))$ on avoiding $H$; the notation is intended to evoke something like ``$\prob[a \to b \,|\, \cpath \ci H^\complm]$''. However, this would not be correct because $a,b \in H$ and \cpath may pass through $H$ after $\gt_b$.

  In Theorem~\ref{thm:edges-of-H^S}, we use the following common notation as in \cite{Spitzer} or \cite{Woess00}, for example. All notations are for the random walk started at $x$.
  \begin{align*}
    \Prob^n(x,y) = p^{(n)}(x,y) =\prob_x[X_n = y] &\q\text{prob. the walk is at $y$ after $n$ steps} \\
    G(x,y) = {\textstyle\sum}_{n=0}^\iy p^{(n)}(x,y) &\q\text{exp. number of visits to $y$} \\
    f^{(n)}(x,y) = \prob_x[\gt_y = n] &\q\text{prob. the walk first reaches $y$ on the \nth step} \\
    F(x,y) = {\textstyle\sum}_{n=0}^\iy f^{(n)}(x,y) &\q\text{prob. the walk ever reaches $y$}
  \end{align*}
  Note that if the walk is killed when it reaches $y$, then $p^{(n)}(x,y) = f^{(n)}(x,y)$ because the first time it reaches $y$ is the only time it reaches $y$. Therefore, when the walk is conditioned to end upon reaching a set $S$, one has $G(x,y) = F(x,y)$ for all $y \in S$.
\end{remark}

\begin{theorem}\label{thm:edges-of-H^S}
  For $\verts[H] \ci \verts$, the conductances in the trace subnetwork $H^\trc$ are given by
  \linenopax
  \begin{align}\label{eqn:conductances-in-HS}
    \cond_{xy}^\trc = \cond_{xy} + \cond(x) \prob[x \to y]\evald{H^\complm}.
  \end{align}
  \glossary{name={$\cond_{xy}^\trc$},description={conductances in the trace subnetwork},sort=c}
  Consequently, the transition probabilities in the trace subnetwork are given by
  \linenopax
  \begin{align}\label{eqn:transition-probabilities-in-HS}
    p^\trc(x,y) = p(x,y) + \prob[x \to y]\evald{H^\complm}.
  \end{align}
\begin{proof}
  Using subscripts to indicate the block decomposition corresponding to $H$ and $H^\complm$ as in \eqref{eqn:Lap-block-decomp}, the Laplacian may be written as
  \linenopax
  \begin{align*}
    \Lap =
    \left[\begin{array}{cc} \cond_A(\one - \Prob_A) & -\cond_A\Prob_{B^T} \\ -\cond_D\Prob_{B} & \cond_D(\one - \Prob_{D}) \end{array}\right],
    \qq \text{for} \qq
    \cond =
      \begin{array}{l} \scalebox{0.70}{$H$} \\ \scalebox{0.70}{$H^\complm$} \end{array}
      \negsp[12]
      \left[\begin{array}{ll} \cond_A &  \\  & \cond_D \end{array}\right].
  \end{align*}
  Then the Schur complement is
  \linenopax
  \begin{align}
    \trc(\Lap,H)

    &= \cond_A - \cond_A \Prob_A - \cond_A \Prob_{B^T} (\id - \Prob_D)^{-1} \cond_{D}^{-1} \cond_{D} \Prob_B \notag \\
    &= \cond_A - \cond_A\left(\Prob_A + \Prob_{B^T} \left(\sum_{n=0}^\iy \Prob_D^n\right) \Prob_B \right) \notag \\
    &= \cond_A(\id - \Prob_\xX).
    \label{eqn:Schur-complement-as-sum}
  \end{align}
  Note that $\Prob_D$ is substochastic, and hence the RW has positive probability of hitting $\bd G_k$, whose vertices act as absorbing states. This means that the expected number of visits to any vertex in $H^\complm$ is finite and hence the matrix $\Prob_\xX$ has finite entries.

  Meanwhile, using $\Prob_A(x,y)$ to denote the \nth[(x,y)] entry of the matrix $\Prob_A$, and $\gt_H^+$ as in Definition~\ref{def:prob[a->b]}, we have
  \linenopax
  \begin{align}
    \prob[x \to y]\evald{H^\complm}
    &= \prob^{(\cond)}\left(\Paths(x,y)\evald{H^\complm}\right) \notag \\
    &= \prob^{(\cond)}\left(\bigcup_{k=1}^\iy \{\cpath \in \Paths(x,y)\evald{H^\complm} \suth \gt_H^+ = k \}\right) \notag \\
    &= \prob^{(\cond)}\left(\{\cpath \in \Paths(x,y)\evald{H^\complm} \suth \gt_H^+ = 1 \}\right) \\
      &\hstr[10] + \sum_{k=2}^\iy \prob^{(\cond)}\left(\{\cpath \in \Paths(x,y)\evald{H^\complm} \suth \gt_H^+ = k \}\right) \notag \\
    &= \Prob_A(x,y) +\sum_{n=0}^\iy \sum_{s,t} \Prob_{B^T}(x,s) \Prob_D^n(s,t) \Prob_B(t,y) \label{eqn:prob-a-to-b-avoiding-H-computation}\\
    &= \Prob_\xX(x,y). \notag
  \end{align}
  To justify \eqref{eqn:prob-a-to-b-avoiding-H-computation}, note that by \eqref{eqn:action-of-Lap-quadrants}, $\Prob_D^n$ corresponds to steps taken in $H^\complm$. Therefore,
  \linenopax
  \begin{align*}
    \left(\Prob_{B^T} \left(\sum_{n=0}^\iy \Prob_D^n\right) \Prob_B\right)(x,y)
    = \Prob_{B^T} \Prob_D^0 \Prob_B(x,y) + \Prob_{B^T}\Prob_D^1 \Prob_B(x,y) + \dots

  \end{align*}
  is the probability of the random walk taking a path that steps from $x \in H$ to $H^\complm$, meanders through $H^\complm$ for any finite number of steps, and finally steps to $y \in H$. Since $y \notin H^\complm$,
  \linenopax
  \begin{align*}
    \Prob_{B^T} \Prob_D^k \Prob_B(x,y)
    = \prob_x[X_{k+2} = y]
    = \prob_x[\gt_y = k+2] ,
  \end{align*}
because the walk can only reach $y$ on the last step, as in Remark~\ref{rem:prob-notation}. It follows by classical theory (see \cite{Spitzer}, for example) that the sum in \eqref{eqn:prob-a-to-b-avoiding-H-computation} is a probability (as opposed to an expectation, etc.) and justifies the probabilistic notation $\Prob_\xX$ in \eqref{eqn:Schur-complement-as-sum}.
  Note that $\Prob_A(x,y)$ corresponds to the one-step path from $x$ to $y$, which is trivially in $\Paths(x,y)\evald{H^\complm}$ by \eqref{eqn:def:Paths(a,b)-outside-H}.
  Since $\Prob_A(x,y) = p(x,y) = \cond_{xy}/\cond(x)$, the desired conclusion \eqref{eqn:conductances-in-HS} follows from combining \eqref{eqn:Schur-complement-as-sum}, \eqref{eqn:prob-a-to-b-avoiding-H-computation}, and \eqref{eqn:def:prob-a-to-b-outside-H}. Of course, \eqref{eqn:transition-probabilities-in-HS} follows immediately by dividing through by $\cond(x)$.
\end{proof}
\end{theorem}

  The authors are grateful to Jun Kigami for helpful conversations and guidance regarding the proof of Theorem~\ref{thm:edges-of-H^S}.

\begin{remark}\label{rem:edges-of-HS}
  It is clear from \eqref{eqn:conductances-in-HS} that the edge sets of $\inn H$ and $\inn H^\trc$ are identical, but the conductance between two vertices $x,y \in \bd H^\trc$ is greater iff there is a path from $x$ to $y$ that does not pass through $H$. Indeed, if there is a path from $x$ to $y$ which lies entirely in $H^\complm$ except for the endpoints, then $x$ and $y$ will be adjacent in $H^\trc$, even if they were not adjacent in $H$.
\end{remark}

\begin{remark}[The trace construction is valid for general subsets of vertices]
  \label{rem:trace-valid-for-subsets}\label{rem:edges-of-HS}
  While Definition~\ref{def:trace-resistance} applies to a (connected) subnetwork of \Graph, it is essential to note that Theorem~\ref{thm:edges-of-H^S} applies to arbitrary subsets \verts[H] of \verts.

  It is clear from \eqref{eqn:conductances-in-HS} that the edge sets of $\inn H$ and $\inn H^\trc$ are identical, but the conductance between two vertices $x,y \in \bd H^\trc$ is greater iff there is a path from $x$ to $y$ that does not pass through $H$. Indeed, if there is a path from $x$ to $y$ which lies entirely in $H^\complm$ except for the endpoints, then $x$ and $y$ will be adjacent in $H^\trc$, even if they were not adjacent in $H$.
\end{remark}

\begin{remark}[Resistance distance via Schur complement]
  \label{rem:Schur-does-network-reduction}
  \label{rem:Resistance-distance-via-Schur-complement}
  A theorem of Epifanov states that every finite planar network with vertices $x,y$ can be reduced to a single equivalent conductor via the use of three simple transformations: parallel, series, and $\nabla$-\textsf{Y}; cf. \cite{Epifanov66,Truemper89} as well as \cite[\S2.3]{Lyons:ProbOnTrees} and \cite[\S7.4]{CdV98}. More precisely,
  \begin{enumerate}[(i)]
    \item Parallel. Two conductors $\cond_{xy}^{(1)}$ and $\cond_{xy}^{(2)}$ connected in parallel can be replaced by a single conductor $\cond_{xy} = \cond_{xy}^{(1)} + \cond_{xy}^{(2)}$.
    \item Series. If $z$ has only the neighbours $x$ and $y$, then $z$ may be removed from the network and the edges $\cond_{xz}$ and $\cond_{yz}$ should be replaced by a single edge $\cond_{xy} = (\cond_{xz}^{-1}+\cond_{yz}^{-1})^{-1}$.
    \item $\nabla$-\textsf{Y}. Let $t$ be a vertex whose only neighbours are $x,y,z$. Then this ``\textsf{Y}'' may be replaced by a triangle (``$\nabla$'') which does not include $t$, with conductances
      \linenopax
      \begin{align*}
        \cond_{xy} = \frac{\cond_{xt}\cond_{ty}}{\cond(t)}, \;
        \cond_{yz} = \frac{\cond_{yt}\cond_{tz}}{\cond(t)}, \;
        \cond_{xz} = \frac{\cond_{xt}\cond_{tz}}{\cond(t)}.
      \end{align*}
      This transformation may also be inverted, to replace a $\nabla$ with a \textsf{Y} and introduce a new vertex.
  \end{enumerate}
  It is a fun exercise to obtain the series and $\nabla$-\textsf{Y} formulas by applying the Schur complement technique to remove a single vertex of degree 2 or 3 from a network. Indeed, these are both special cases of the following: let $t$ be a vertex of degree $n$, and let $H$ be the (star-shaped) subnetwork consisting only of $t$ and its neighbours. If we write the Laplacian for just this subnetwork with the \nth[t] row \& column last, then
  \linenopax
  \begin{align*}
    \Lap|_H =
    \left[\begin{array}{rrrc}
      \cond_{x_1t} & \dots & 0 & -\cond_{x_1t} \\
      \vdots & \ddots & \vdots & \vdots \\
      0 & \dots & \cond_{x_nt} & -\cond_{x_nt} \\
      -\cond_{x_1t} & \dots & -\cond_{x_nt} & \cond(t)
    \end{array}\right]
  \end{align*}
  and the Schur complement is
  \linenopax
  \begin{align*}
    \trc(\Lap|_H,H\less\{t\})
    =
    \left[\begin{array}{rrrc}
      \cond_{x_1t} & \dots & 0 \\
      \vdots & \ddots & \vdots \\
      0 & \dots & \cond_{x_nt} \\
    \end{array}\right]
    - \frac{1}{\cond(t)}
    \left[\begin{array}{rrrc}
      \cond_{x_1t} \\
      \vdots \\
      \cond_{x_nt}
    \end{array}\right]
    \left[\begin{array}{rrrc}
      \cond_{x_1t} & \dots & \cond_{x_nt}
    \end{array}\right],
  \end{align*}
  whence the new conductance from $x_i$ to $x_j$ is given by $\cond_{x_it}\cond_{tx_j}/\cond(t)$. It is interesting to note that the operator being subtracted corresponds to the projection to the rank-one subspace spanned by the probabilities of leaving $t$:
  \linenopax
  \begin{align*}
    \frac{1}{\cond(t)}
    \left[\begin{array}{r}
      \cond_{x_1t} \\
      \vdots \\
      \cond_{x_nt}
    \end{array}\right]
    \left[\begin{array}{rrr}
      \cond_{x_1t} & \dots & \cond_{x_nt}
    \end{array}\right]
    = \cond(t) |v\ra \la v|,
  \end{align*}
  using Dirac's ket-bra notation for the projection to a rank-1 subspace spanned by $v$ where
  \linenopax
  \begin{align*}
    v = \left[\begin{array}{rrr}
      p(t,x_1)  & \dots & p(t,x_n)
    \end{array}\right].
  \end{align*}
In fact, $|v\ra \la v | = \Prob_X$, in the notation of \eqref{eqn:Schur-complement-as-sum}. In general, the trace construction (Schur complement) has the effect of probabilistically projecting away the complement of the subnetwork.

    In Remark~\ref{rem:Resistance-distance-via-network-reduction} we described how the effective resistance can be interpreted as the correct resistance for a single edge which replaces a subnetwork. The following corollary of Theorem~\ref{thm:edges-of-H^S} formalizes this interpretation by exploiting the fact that the Schur complement construction is viable for arbitrary subsets of vertices; see Remark~\ref{rem:trace-valid-for-subsets}. In this case, one takes the trace of the (typically disconnected) subset $\{x,y\} \ci \verts$; note that $\left[\begin{smallmatrix} 1 & -1 \\ -1 & 1 \end{smallmatrix}\right]$ is the Laplacian of the trivial 2-vertex network when the edge between them has unit conductance. The following result is also an extension of \cite[(2.1.4)]{Kig01} to infinite networks.
\end{remark}

\begin{cor}\label{thm:resistance-as-2-vert-network}
  Let $\verts[H] = \{x,y\}$ be any two vertices of a transient network \Graph. Then the trace resistance can be computed via
  \linenopax
  \begin{equation}\label{eqn:resistance-as-2-vert-network}
    \trc(\Lap,H)
    = \frac1{R^\trc(x,y)} \left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right]
    = A - B^T D^{-1} B.
  \end{equation}
  \begin{proof}
    Take $H=\{x,y\}$ in Theorem~\ref{thm:edges-of-H^S}. As discussed in Remark~\ref{rem:trace-valid-for-subsets}, it is not necessary to have $x \nbr y$. Note that in this case, $\left(\Prob_{B^T} \sum_n \Prob_D^n \Prob_B\right)(x,y)$ corresponds all paths from $x$ to $y$ that consist of more than one step:
  \linenopax
  \begin{align}
    \prob[x \to y]\evald{H^\complm}
    &= \Prob_A(x,y) + \left(\Prob_{B^T} \sum_{n=0}^\iy \Prob_D^n \Prob_B\right)(x,y)
        = p(x,y)

    + \sum_{|\cpath| \geq 2} \prob(\cpath).
    \label{eqn:sum-over-paths-decomposition}
    \qedhere
  \end{align}
  \end{proof}
\end{cor}

\version{}{\marginpar{Alternative idea}Use Kigami's approach (\energy, Theorem~\ref{thm:resistance-as-2-vert-network}) and take limits.}

\begin{cor}\label{thm:resistance-as-path-integral}
  The trace resistance $R^\trc(x,y)$ is given by
  \linenopax
  \begin{equation}\label{eqn:resistance-as-path-integral}
    R^\trc(x,y) = \frac1{\cond(x) \prob[x \to y]}
  \end{equation}
  \begin{proof}
    Again, take $\verts[H] = \{x,y\}$.  Then
    \linenopax
    \begin{align*}
      R^\trc(x,y)^{-1}
      = \cond_{xy}^{H^S}
      &= \cond_{xy} + \cond(x) \prob[x \to y]\evald{H^\complm} \\
      &= \cond(x) \left(p(x,y) + \prob[x \to y]\evald{H^\complm}\right) \\
      &= \cond(x) \prob[x \to y],
    \end{align*}
    where Corollary~\ref{thm:resistance-as-2-vert-network} gives the first equality and Theorem~\ref{thm:edges-of-H^S} gives the second.
  \end{proof}
\end{cor}

\begin{remark}[Effective resistance as ``path integral'']
  \label{rem:resistance-as-path-integral}
  Corollary~\ref{thm:resistance-as-path-integral} may also be obtained by the more elegant (and much shorter) approach of \cite[\S2.2]{Lyons:ProbOnTrees}, where it is stated as follows: the mean number of times a random walk visits $a$ before reaching $b$ is $\prob[a \to b]^{-1} = \cond(a) R(a,b)$. We give the present proof to highlight and explain the underlying role of the Schur complement with respect to network reduction; see Remarks~\ref{rem:trace-valid-for-subsets}--\ref{rem:Resistance-distance-via-Schur-complement}.
  A key point of the present approach is to emphasize the expression of effective resistance $R(a,b)$ in terms of a \emph{sum over all possible paths from $a$ to $b$}.   By Remark~\ref{rem:wired-vs-free-as-Kirchhoff-conditions+}, it is apparent that this ``path-integral'' interpretation makes $R^\trc$ much more closely related to $R^F$ than to $R^W$, as seen by the following results.
\end{remark}

\begin{cor}[{\cite[Prop.~2.1.11]{Kig01}}]
  \label{thm:trace-invariance-of-resistance}
  Let $H_2 \ci H_1$ be finite subnetworks of a transient network \Graph. Then for $a,b \in \verts[H_2]$, one has $R_{H_1}^S(a,b) = R_{H_2}^S(a,b)$.

\end{cor}

\begin{cor}\label{thm:trace-resistance-is-free-resistance}
  On any network, $R^\trc(a,b) = R^F(a,b)$.
  \begin{proof}
    By Corollary~\ref{thm:trace-invariance-of-resistance}, it is clear that $R_{G_k^\trc}(a,b) = R_{G_{k+1}^\trc}(a,b)$ for all $k$. Meanwhile, any path from $a$ to $b$ will lie in $G_k$ for sufficiently large $k$, so it is clear by Theorem~\ref{thm:resistance-as-path-integral}, the sequence $\{R^F_{G_k}(a,b)\}_{k=0}^\iy$ is monotonically decreasing with limit $R^F(a,b) = R^\trc(a,b)$.
  \end{proof}
\end{cor}

\begin{remark}
  \version{}{\marginpar{Work out, and insert here, the corresponding formula for $R^W$}}
  Writing $[x \to y \,|\, \cpath \ci H]$ to indicate a restriction to paths from $x$ to $y$ that lie entirely in $H$, as in Remark~\ref{rem:notation-restriction-to-Gk}, one has
  \linenopax
  \begin{align*}
    R_{G_k^\trc}(x,y)
    &= \frac1{\cond(x) \left(\prob[x \to y \,|\, \cpath \ci G_k] + \prob[x \to y  \,|\, \cpath \nsubseteq G_k]\right)}  \\
    &\leq \frac1{\cond(x) \prob[x \to y \,|\, \cpath \ci G_k]}
    = R_{G_k^F}(x,y).
  \end{align*}

  Essentially, Corollary~\ref{thm:trace-invariance-of-resistance} is an expression of the first equality and Corollary~\ref{thm:trace-resistance-is-free-resistance} is a consequence of the inequality and how it tends to an equality as $k \to \iy$.
\end{remark}

\begin{comment}
\begin{lemma}\label{thm:renormalized-Schur}
  Suppose that $M$ is diagonal, with block decomposition $M = \left[\begin{smallmatrix} M_1 & 0 \\ 0 & M_2\end{smallmatrix}\right]$ corresponding to \eqref{eqn:Lap-block-decomp}. Then $(M\Lap)_H = M_1 \trc(\Lap,H)$.
  \begin{proof}
    One computes $(M\Lap)_H$ directly:
    \linenopax
    \begin{align*}
      \left(\left[\begin{smallmatrix} M_1 A & M_1 B^T \\ M_2 B & M_2 D\end{smallmatrix}\right]\right)_H
      &= M_1 A - (M_1 B^T)(M_2 D)^{-1})(M_2 B)
      = M_1 A - M_1 B^T D^{-1} B
      = M_1 \trc(\Lap,H).
      \qedhere
    \end{align*}
  \end{proof}
\end{lemma}
\end{comment}

\subsection{The shorted operator}
\label{sec:shorted-operator}

It is worth noting that the operator $D$ defined in \eqref{eqn:action-of-Lap-quadrants} is always invertible as in the discussion following \eqref{eqn:Schur-complement-as-sum}.
However, the Schur complement construction is valid more generally. As is pointed out in \cite{Butler}, the \emph{shorted operator} generalizes the Schur complement construction to positive operators on a (typically infinite-dimensional) Hilbert space \sH; see \cite{Anderson,AndersonTrapp,Krein}. In general, let $T=T^\ad$ be a positive operator so $\la \gf, T\gf\ra \geq 0$ for all $\gf \in \sH$, and let $S$ be a closed subspace of \sH. Partition $T$ analogously to \eqref{eqn:action-of-Lap-quadrants}, so that $A:S \to S$, $B:S \to S^\complm$, $B^T:S^\complm \to S$, and $D:S^\complm \to S^\complm$.

\begin{theorem}[{\cite{AndersonTrapp}}]
  With respect to the usual ordering of self-adjoint operators, there exists a unique operator $\sS{h}(T)$ such that
  \linenopax
  \begin{align*}
    \sS{h}(T) = \sup_L\left\{L \geq 0 \suth \left[\begin{array}{cc} L & 0 \\ 0 & 0
  \end{array}\right] \leq T\right\},
  \end{align*}
  and it is given by
  \linenopax
  \begin{align*}
    \sS{h}(T) = \lim_{\ge \to 0^+} \left(A - B^T (D + \ge)^{-1} B\right).
  \end{align*}
  In particular, the shorted operator coincides with the Schur complement, whenever the latter exists.
\end{theorem}

There is another characterization of the shorted operator due to \cite{Butler}.

\begin{theorem}[{\cite{Butler}}]
  Suppose $\{\gy_n\} \ci \sH$ is a sequence satisfying $\la\gy_n, D\gy_n\ra \leq M$ for some $M \in \bR$, and $\lim_{n \to \iy} T \left[\begin{smallmatrix} \gf \\ \gy_n \end{smallmatrix}\right] = \left[\begin{smallmatrix} \gq \\ 0 \end{smallmatrix}\right]$. Then $\sS{h}(T) \gf = \lim_{n \to \iy} \left(A \gf + B^T\gy_n\right)$.
\end{theorem}

\section{Projections in Hilbert space and the conditioning of the random walk}
\label{sec:probabilistic-interp-of-v_x-and-f_x}

  In Remark~\ref{rem:wired-vs-free-as-boundary-conditions}, we gave an operator-theoretic account of the difference between $R^F$ and $R^W$. The foregoing probabilistic discussions might lead one to wonder if there is a probabilistic counterpart. An alternative approach is given in \cite[App.~B]{Kig03}.

  On a finite network, it is well-known that
  \begin{align}\label{eqn:vx-as-prob-on-finite}
    v_x = R(o,x) u_x,
  \end{align}
  where $u_x(y)$ is the probability that a random walker (RW) started at $y$ reaches $x$ before $o$:
  \linenopax
  \begin{align}\label{eqn:u_x-defined-probabilistically}
    u_x(y) := \prob_y[\gt_x < \gt_o].
  \end{align}
  Here again, $\gt_x$ denotes the hitting time of $x$ as in Definition~\ref{def:hitting-time}. Note that \eqref{eqn:def:R(x,y)-energy} gives $u_x = \frac{v_x}{\energy(v_x)}$. The relationship \eqref{eqn:vx-as-prob-on-finite} is discussed in \cite{DoSn84,LevPerWil08,Lyons:ProbOnTrees}.

  Theorem~\ref{thm:fx-as-prob} is a wired extension of \eqref{eqn:vx-as-prob-on-finite} to transient networks.
  The corresponding free version appears in Conjecture~\ref{thm:v_x-as-prob-on-infinite-network}.

\begin{theorem}\label{thm:fx-as-prob}
  On a transient network, let $f_x$ be the representative of $\Pfin v_x$ specified by $f_x(o)=0$. Then for $x \neq o$, $f_x$ is computed probabilistically by
  \linenopax
  \begin{align}\label{eqn:prob-expr-for-fx}
    f_x(y) = R^W(o,x) \left(\prob_y[\gt_x < \gt_o]
    \right. \\ & \hstr[5] \left.
    + \prob_y^{G}[\gt_o = \gt_x = \iy] \prob_x^{G}[\gt_o = \iy] \lim_{k \to \iy} \tfrac{\cond(x)}{\cond(\iy_k)} \prob_{\iy_k}^{G_k^W}[\gt_{\iy_k} < \gt_{\{x,o\}}]\right).

  \end{align}

  \begin{proof}
    Fix $x,y$ and an exhaustion $\{G_k\}_{k=1}^\iy$, and suppose without loss of generality that $o,x,y \in G_1$.

    Since $v_x=f_x$ on any finite network, the identity \eqref{eqn:vx-as-prob-on-finite} gives $f_x^{(k)} = R_{G_k^W}(o,x) \check u_x^{(k)}$, where $f_x^{(k)}$ is the unique solution to $\Lap v = \gd_x - \gd_o$ on the finite (wired) subnetwork $G_k^W$, and
    \linenopax
    \begin{align*}
      \check u_x^{(k)}(y) := \prob_y^{G_k^W}[\gt_x < \gt_o].
    \end{align*}
    where the superscript indicates the network in which the random walk travels.
    As in the previous case, we just need to check the limit of $\check u_x^{(k)}$, for which, we have
    \begin{align}\label{eqn:fx-as-prob-derivation-1}
       \check u_x^{(k)}(y)
       &= \prob_y^{G_k^W}[\gt_x < \gt_o \text{ \& } \gt_x < \gt_{\iy_k}] + \prob_y^{G_k^W}[\gt_x < \gt_o \text{ \& } \gt_x > \gt_{\iy_k}]
    \end{align}

    The first probability in \eqref{eqn:fx-as-prob-derivation-1} is
    \begin{align*}
      \prob_y^{G_k^W}[\gt_x < \gt_o \text{ \& } \gt_x < \gt_{\iy_k}]
      &= \prob_y^{G}[\gt_x < \gt_o \text{ \& } \gt_x < \gt_{G_k^\complm}] \\
      \limas{k}&\hstr[2.2] \prob_y^{G}[\gt_x < \gt_o \text{ \& } \gt_x < \iy]
      = \prob_y^{G}[\gt_x < \gt_o],
    \end{align*}
    where the last equality follows because $\gt_x < \gt_o$ implies $\gt_x < \iy$.

    The latter probability in \eqref{eqn:fx-as-prob-derivation-1} measures the set of paths which travel from $y$ to $\iy_k$ without hitting $x$ or $o$, and then on to $x$ without passing through $o$, and hence can be rewritten
    \begin{align*}
      \prob_y^{G_k^W}&[\gt_{\iy_k} < \gt_x < \gt_o]
      = \prob_y^{G_k^W}[\gt_{\iy_k} < \gt_{\{o,x\}}] \prob_{\iy_k}^{G_k^W}[\gt_x < \gt_o] \\
      &= \prob_y^{G_k^W}[\gt_{\iy_k} < \gt_{\{o,x\}}] \left(\prob_{\iy_k}^{G_k^W}[\gt_{\iy_k} < \gt_{\{x,o\}}]\prob_{\iy_k}^{G_k^W}[\gt_x < \gt_{\{\iy_k,o\}}] + \prob_{\iy_k}^{G_k^W}[\gt_x < \gt_{\{\iy_k,o\}}]\right),
    \end{align*}
    since a walk starting at $\iy_k$ may or may not return to $\iy_k$ before reaching $x$.

    First, consider only those walks which do not loop back through $\iy_k$ (i.e., multiply out the above expression and take the second term) to observe
    \begin{align}
      \prob_y^{G_k^W}[\gt_{\iy_k} < \gt_{\{o,x\}}] &\prob_{\iy_k}^{G_k^W}[\gt_x < \gt_{\{\iy_k,o\}}]
      = \prob_y^{G_k^W}[\gt_{\iy_k} < \gt_{\{o,x\}}]\prob_x^{G_k^W}[\gt_{\iy_k} < \gt_o] \tfrac{\cond(x)}{\cond(\iy_k)} \label{eqn:fx-as-prob-derivation-3} \\
      &= \left(1 - \prob_y^{G_k^W}[\gt_{\{o,x\}} < \gt_{\iy_k}]\right) \left(1 - \prob_x^{G_k^W}[\gt_o < \gt_{\iy_k}]\right) \tfrac{\cond(x)}{\cond(\iy_k)} \notag \\

      &= \left(1 - \prob_y^{G}[\gt_{\{o,x\}} < \gt_{G_k^\complm}]\right) \left(1 - \prob_x^{G}[\gt_o < \gt_{G_k^\complm}]\right) \tfrac{\cond(x)}{\cond(\iy_k)} \notag \\
      \limas{k}
      &\hstr[2.2] \left(1 - \prob_y^{G}[\gt_{\{o,x\}} < \iy]\right) \left(1 - \prob_x^{G}[\gt_o < \iy]\right) \lim_{k \to \iy} \tfrac{\cond(x)}{\cond(\iy_k)} \notag \\
      &= \prob_y^{G}[\gt_o = \gt_x = \iy] \prob_x^{G}[\gt_o = \iy] \lim_{k \to \iy} \tfrac{\cond(x)}{\cond(\iy_k)}.
      \label{eqn:fx-as-prob-derivation-2}
    \end{align}
    Note that \eqref{eqn:fx-as-prob-derivation-3} comes by reversibility of the walk, and the way probability is computed for paths from $\iy_k$ to $x$ which avoid $o$ and $\iy_k$. Since the network is transient, $\sum_{k=1}^\iy{\cond(\iy_k)^{-1}}$ is summable by Nash-William's criterion and so $\lim_{k \to \iy} \tfrac{\cond(x)}{\cond(\iy_k)} = 0$ causes \eqref{eqn:fx-as-prob-derivation-2} to vanish.

    Now for walks which do loop back through $\iy_k$, the same arguments as above yield
    \begin{align*}
      \prob_y^{G_k^W}[\gt_{\iy_k} < \gt_{\{o,x\}}] &\prob_{\iy_k}^{G_k^W}[\gt_{\iy_k} < \gt_{\{x,o\}}]\prob_{\iy_k}^{G_k^W}[\gt_x < \gt_{\{\iy_k,o\}}] \\
      &\limas{k} \hstr[1] \prob_y^{G}[\gt_o = \gt_x = \iy] \prob_x^{G}[\gt_o = \iy] \lim_{k \to \iy} \tfrac{\cond(x)}{\cond(\iy_k)} \prob_{\iy_k}^{G_k^W}[\gt_{\iy_k} < \gt_{\{x,o\}}],
    \end{align*}
    and the conclusion follows.
  \end{proof}
\end{theorem}

The following conjecture expresses a free extension of \eqref{eqn:vx-as-prob-on-finite} to infinite networks. We offer an erroneous ``proof'' in the hopes that it may inspire the reader to find a correct proof. The error is discussed in Remark~\ref{rem:error-in-prob-expr-for-vx}, just below. In the statement of Conjecture~\ref{thm:v_x-as-prob-on-infinite-network}, we use the notation
\begin{align}\label{eqn:bounded-traj}
  |\cpath| < \iy
\end{align}
to denote the event that the walk is bounded, i.e., that the trajectory is contained in a finite subnetwork of \Graph.

\begin{conj}\label{thm:v_x-as-prob-on-infinite-network}
  On an infinite resistance network, let $v_x$ be the representative of an element of the energy kernel specified by $v_x(o)=0$. Then for $x \neq o$, $v_x$ is computed probabilistically by

  \linenopax
  \begin{align}\label{eqn:prob-expr-for-vx}
    v_x(y) = R^F(o,x) \prob_y[\gt_x < \gt_o \,|\, |\cpath| < \iy],
  \end{align}
  that is, the walk is conditioned to lie entirely in some finite subnetwork as in \eqref{eqn:bounded-traj}.
  \begin{proof}[``Proof.'']
    Fix $x,y$ and suppose without loss of generality that $o,x,y \in G_1$. One can  write \eqref{eqn:vx-as-prob-on-finite} on $G_k$ as $v_x^{(k)} = R_{G_k^F}(o,x) u_x^{(k)}$. In other words, $v_x^{(k)}$ is the unique solution to $\Lap v = \gd_x - \gd_o$ on the finite subnetwork $G_k^F$.

    Since $R^F(x,y) = \lim_{k \to \iy} R_{G_k^F}(x,y)$ by \eqref{eqn:def:free-resistance}, it only remains to check the limit of $u_x^{(k)}$. Using a superscript to indicate the network in which the random walk travels, we have
    \linenopax
    \begin{align}\label{eqn:prob-expr-for-vx:error}
      \lim_{k \to \iy} u_x^{(k)}(y)
      &= \lim_{k \to \iy} \prob_y^{G_k^F} [\gt_x < \gt_o]
       = \lim_{k \to \iy} \prob_y^{G} [\gt_x < \gt_o \,|\, \cpath \ci G_k^F].

    \end{align}
    Here again, the notation $[\cpath \ci H]$ denotes the event that the random walk never leaves the subnetwork $H$, i.e., $\gt_{H^\complm} = \iy$. The events $[\cpath \ci G_k^F]$ are nested and increasing, so the limit is the union, and \eqref{eqn:prob-expr-for-vx} follows.
    Note that $G_k^F$ is recurrent, so $\cpath \ci G_k^F$ implies $\gt_x < \iy$.
  \end{proof}
\end{conj}

\begin{remark}\label{rem:error-in-prob-expr-for-vx}
  As indicated, the argument outlined above is incomplete due to the second equality of \eqref{eqn:prob-expr-for-vx:error}. While the set of paths from $y$ to $x$ in $G_k^F$ is the same as the set of paths from $y$ to $x$ in $G$ which lie in $G_k$, the probability of a given path may differ when computed in network or the other. This happens precisely when \cpath passes through a boundary point: the transition probability away from a point in $\bd G_k$ is strictly larger in $G_k^F$ than it is in $G_k$.
\end{remark}

\section{Comparison of resistance metric to other metrics}
\label{sec:Comparison-to-other-metrics}

\subsection{Comparison to geodesic metric}
\label{sec:Comparison-to-shortest-path-metric}

On a Riemannian manifold $(\gW,g)$, the geodesic distance is
  \linenopax
  \begin{equation*}
    \distmin(x,y)
    := \inf_{\gg} \left\{\int_0^1 g(\gg'(t),\gg'(t))^{1/2} \,dt \suth \gg(0)=x, \gg(1)=y, \gg \in C^1\right\}.
  \end{equation*}

\begin{defn}\label{def:geodesic-metric}
  On $(\Graph, \cond)$, the \emph{geodesic distance} from $x$ to $y$ is
  \linenopax
  \begin{equation}\label{eqn:def:geodesic-metric}
    \distmin(x,y)
    := \inf \{r(\cpath) \suth \cpath \in \Paths(x,y)\},
  \end{equation}
  where $r(\cpath) := \left(\sum_{(x,y) \in \cpath} \cond_{xy}^{-1} \right)$ (for resistors in series, the total resistance is the sum).
\end{defn}
  \glossary{name={$\distmin(x,y)$},description={geodesic distance, shortest-path metric},sort=d,format=textbf}
  \glossary{name={$r(\cpath)$},description={total resistance of a path},sort=r,format=textbf}

\begin{remark}\label{rem:shortest-path-metric}
  Definition~\ref{def:geodesic-metric} differs from the definition of shortest path metric found in the literature on general graph theory; without weights on the edges one usually defines the shortest path metric simply as the minimal number of edges in a path from $x$ to $y$. (This corresponds to taking $\cond \equiv 1$.) Such shortest paths always exist. According to Definition~\ref{def:geodesic-metric}, shortest paths may not exist (cf.~Example~\ref{exm:one-sided-ladder-model}). Of course, even when they do exist, they are not always unique.

It should be observed that effective resistance is not a geodesic metric, in the usual sense of metric geometry;  it does not correspond to a length structure in the sense of \cite[\S2]{Burago}.
\end{remark}

\begin{lemma}\label{thm:shortest-path-bounds-resistance-distance}
  The effective resistance is bounded above by the geodesic distance. More precisely, $R^F(x,y) \leq \distmin(x,y)$ with equality if and only if \Graph is a tree.
  \begin{proof}
    If there is a second path, then some positive amount of current will pass along it (i.e., there is a positive probability of getting to $y$ via this route). To make this precise, let $v = v_x - v_y$ and let $\cpath = (x=x_0,x_1,\dots,x_n=y)$ be any path from $x$ to $y$:
    \linenopax
    \begin{align*}
      R^F(x,y)^2
      = |v(x)-v(y)|^2
      \leq r(\cpath) \energy(v),
    \end{align*}
    by the exact same computation as in the proof of Lemma~\ref{thm:L_x-is-bounded}, but with $u=v$. The desired inequality then follows by dividing both sides by $\energy(v) = R^F(x,y)$.

    The other claim follows by observing that trees are characterized by the property of having exactly one path \cpath between any $x$ and $y$ in \verts. By \eqref{eqn:def:R^F(x,y)-diss}, $R^F(x,y)$ can be found by computing the dissipation of the unit current which runs entirely along \cpath from $x$ to $y$. This means that $\curr(x_{i-1},x_i)=1$ on \cpath, and $\curr = 0$ elsewhere, so
    \linenopax
    \begin{align*}
      R^F(x,y)
      &= \diss(\curr)
      = \sum_{i=1}^n \frac1{\cond_{x_{i-1} x_i}} \curr(x_{i-1},x_i)^2
      = \sum_{i=1}^n \frac1{\cond_{x_{i-1} x_i}}
      = r(\cpath).
      \qedhere
    \end{align*}
  \end{proof}
\end{lemma}

This type of inequality is explicitly calculated in Example~\ref{exm:finite-cyclic-model}.

\begin{remark}\label{rem:shortcut-through-infinity}
  It is clear from the end of the proof of Lemma~\ref{thm:shortest-path-bounds-resistance-distance} that on a tree, $v_x-v_y$ is locally constant on the complement of the unique path from $x$ to $y$. However, this may not hold for $f_x-f_y$, where $f_x = \Pfin v_x$; see Example~\ref{exm:binary-tree:reproducing-kernel}. This is an example of how the wired resistance can ``cheat'' by considering currents which take a shortcut through infinity; compare \eqref{eqn:def:R^F(x,y)-diss} to \eqref{eqn:def:R^W(x,y)-diss}.
\end{remark}

\subsection{Comparison to Connes' metric}
\label{sec:Comparison-to-Connes-metric}

  The formulation of $R(x,y)$ given in \eqref{eqn:def:R(x,y)-Lap} may evoke Connes' maxim that a metric can be thought of as the inverse of a Dirac operator; cf.~\cite{Con94}. This does not appear to have a literal incarnation in the current context, but we do have the inequality of Lemma~\ref{thm:Connes'-metric} in the case when $\cond = \one$. In this formulation, $v \in \HE$ is considered as a multiplication operator defined on $u$ by
  \linenopax
  \begin{equation}\label{eqn:v-as-operator}
    (vu)(x) := v(x) u(x), \q \forall x \in \verts,
  \end{equation}
  and both $v$ and \Lap are considered as operators on $\ell^2(\verts \cap \dom \energy$. We use the commutator notation $[v,\Lap] := v\Lap-\Lap v$, and $\| [v,\Lap] \|$ is understood as the usual operator norm on $\ell^2$.

\begin{lemma}\label{thm:Connes'-metric}
  If $\cond = \one$, then for all $x,y \in \verts$ one has
  \linenopax
  \begin{align}\label{eqn:Connes'-metric}
    R(x,y) \leq \sup \{|v(x)-v(y)|^2 \suth \| [v,\Lap] \| \leq \sqrt2, v \in \dom \energy\}.
  \end{align}
  \begin{proof}
    We will compare \eqref{eqn:Connes'-metric} to \eqref{eqn:def:R(x,y)-sup}. Writing $M_v$ for multiplication by $v$, it is straightforward to compute from the definitions
    \linenopax
    \begin{align*}
      (M_v \Lap - \Lap M_v)u(x)
      &= \sum_{y \nbr x} (v(y)-v(x))u(y),
    \end{align*}
    so that the Schwarz inequality gives
    \linenopax
    \begin{align*}
      \|[M_v,\Lap]u\|_2^2
      &= \sum_{x \in \verts} \left|\sum_{y \nbr x} (v(y)-v(x))u(y)\right|^2 \\
      &\leq \sum_{x \in \verts} \left(\sum_{y \nbr x} |v(y)-v(x)|^2\right) \left(\sum_{y \nbr x} |u(y)|^2\right).
    \end{align*}
    By extending the sum of $|u(x)|^2$ to all $x \in \verts$ (an admittedly crude estimate), this gives $\|[v,\Lap]u\|_2^2 \leq 2 \|u\|_2^2 \energy(v)$, and hence $\|[v,\Lap]\|^2 \leq 2 \energy(v)$
  \end{proof}
\end{lemma}

\section{Generalized resistance metrics}
\label{sec:Generalized resistance metrics}

In this section, we describe a notion of effective resistance between probability measures, of which $R(x,y)$ (or $R^F$ and $R^W$) is a special case. This concept is closely related to the notion of total variation of measures, and hence is related to mixing times of Markov chains; cf. \cite[\S4.1]{LevPerWil08}. When the Markov chain is taken to be random walk on an ERN, the state space is just the vertices of \Graph.

\begin{defn}\label{def:resistance-distance-of-measures}
  Let \gm and \gn be two probability measures on \verts. Then the total variation distance between them is
  \linenopax
  \begin{align}\label{eqn:def:resistance-distance-of-measures}
    \dist_{\TV}(\gm,\gn)
    := 2 \sup _{A \ci \verts} |\gm(A)-\gn(A)|.
  \end{align}
\end{defn}

\begin{prop}[{\cite[Prop.~4.5]{LevPerWil08}}]
  \label{thm:resistance-distance-of-measures-as-norm}
  Let \gm and \gn be two probability measures on the state space \gW of a (discrete) Markov chain. Then the total variation distance between them is
  \linenopax
  \begin{align}\label{eqn:resistance-distance-of-measures-as-norm}
    \dist_{\TV}(\gm,\gn)
    = \sup\left\{ \left|\sum_{x \in \gW} u(x) \gm(x) - \sum_{x \in \gW} u(x) \gn(x)\right| \suth \|u\|_\iy \leq 1 \right\}.
  \end{align}
  Here, $\|u\|_\iy := \sup_{x \in \verts}|u(x)|$.
\end{prop}
  \glossary{name={$\dist_{\TV}(\gm,\gn)$},description={total variation metric},sort=d,format=textbf}

\subsection{Effective resistance between measures}
\label{sec:Effective-resistance-between-measures}

If we think of \gm as a linear functional acting on the space of bounded functions, then it is clear that \eqref{eqn:resistance-distance-of-measures-as-norm} expresses $\dist_{\TV}(\gm,\gn)$ as the operator norm $\|\gm-\gn\|$. That is, it expresses the pairing between $\gm \in \ell^1$ and $u \in \ell^\iy$. We can therefore extend $R^F$ directly (see \eqref{eqn:def:R^F(x,y)-S}--\eqref{eqn:def:R^F(x,y)-sup} and Remark~\ref{rem:R(x,y) = |L{xy}|}).

\begin{defn}
  The free resistance between two probability measures is
  \linenopax
  \begin{align}
    \dist_{R^F}(\gm,\gn)
    := \sup\left\{ \left|\sum_{x \in \verts} u(x) \gm(x) - \sum_{x \in \verts} u(x) \gn(x)\right|^2 \suth \|u\|_\energy \leq 1 \right\}.
  \end{align}
\end{defn}
  \glossary{name={$\dist_{R^F}(\gm,\gn)$},description={free resistance between two probability measures},sort=d,format=textbf}

It is clear from this definition (and Remark~\ref{rem:R(x,y) = |L{xy}|}) that $R^F(x,y) = \dist_{R^F}(\gd_x,\gd_y)$. This extension of $R^F$ to measures was motivated by a question of Marc Rieffel in \cite{Rieffel99}.

\subsection{Total variation spaces}
\label{sec:Total-variation-spaces}

\begin{defn}\label{def:TV-form}
  Since $\dom \energy$ is a Banach space, we may define a new pairing via the bilinear form
  \linenopax
  \begin{align}\label{eqn:def:TV-form}
    \la u, \gm\ra_{\TV}
    := \sum_{x \in \verts} u(x) \gm(x),
  \end{align}
  where \gm is an element of
  \linenopax
  \begin{align}\label{eqn:def:TV}
    \TV
    := \{\gm:\verts \to \bR \suth \exists k_\gm \text{ s.t.}
      |\la u, \gm\ra_{\TV}| \leq k_\gm \cdot \energy(u)^{1/2},
      \forall u \in \dom \energy\}.
  \end{align}
  \glossary{name={$\TV$},description={total variation space},sort=T,format=textbf}
  Then $\TV= \dom \la u, \cdot\ra_\TV$ is the dual of $\dom \energy$ with respect to the total variation topology induced by \eqref{eqn:def:TV-form}. Also, the norm in \TV is given by
  \linenopax
  \begin{align}\label{eqn:def:TV-norm}
    \|\gm\|_\TV
    := \inf \{k \suth |\la u, \gm\ra_{\TV}| \leq k \cdot \energy(u)^{1/2}, \forall u \in \dom \energy\}.
  \end{align}
\end{defn}

\begin{remark}
  Since \TV is a Banach space which is the dual of a normed space, the unit ball
  \linenopax
  \begin{align}\label{eqn:unit-ball-in-TV}
    \{\gm \in \TV \suth \|\gm\|_{\TV} \leq 1\}
  \end{align}
  is compact in the weak-$\star$ topology, by Alaoglu's theorem.
\end{remark}

\begin{lemma}\label{thm:Lap-maps-Fin-to-TV}
  The Laplacian \Lap maps \HE into \TV with $\|\Lap v\|_\TV \leq \|\Pfin v\|_\energy$.
  \begin{proof}
    For $u,v \in \HE$, write $v=f+h$ with $f=\Pfin v$ and $h=\Phar v$, so that
    \linenopax
    \begin{align*}
      \left|\sum_{x \in \verts} \cj{u}(x) \Lap v(x)\right|
      &\leq \left|\sum_{x \in \verts} \cj{u}(x) \Lap f(x)\right|
        + \cancel{\left|\sum_{x \in \verts} \cj{u}(x) \Lap h(x)\right|}
      = |\la u, f\ra_\energy|
      \leq \|u\|_\energy \|f\|_\energy,
    \end{align*}
    by Theorem~\ref{thm:E(u,v)=<u,Lapv>-on-Fin} followed by the Schwarz inequality. The mapping is contractive relative to the respective norms because $\|v\|_\energy$ is an element of the set on the right-hand side of \eqref{eqn:def:TV-norm}, and hence at least as big as the infimum, whence $\|\Lap v\|_\TV \leq \|f\|_\energy \leq \|v\|_\energy$.
  \end{proof}
\end{lemma}

\section{Remarks and references}
\label{sec:Remarks-and-References-resistance-metrics}

A key reference for this chapter is \cite{Kig03}; the relationship between the free and wired resistance can be elegantly phrased in terms of resistance forms, as we describe in the following remark. Additionally, the role of the trace resistance is apparent in Kigami's work \cite{Kig01,Kig03, Kigami95, Kigami94, Kigami93}. However, the potential-theoretic approach can be intimidating to the uninitiated, and we hope that our treatment of effective resistance from the first principles of Hilbert space theory will provide a gentle introduction, as well as new insights. As an example of this, we feel that the probabilistic proof of Theorem~\ref{thm:edges-of-H^S} (to which we are indebted to Jun Kigami) offers insight as to \emph{why} the operation of Schur complement should correspond to taking the trace.

After PowersÕ papers in the 70s (starting with \cite{Pow76b}), there has been an explosive in the interest in metric geometry, potential theory \cite{Brelot},spectral theory \cite{Chu04}, and harmonic analysis \cite{Car73a} on infinite graphs. As illustrated in \cite{Kig03} a good deal of the motivation arose from a parallel research track dealing with analysis of fractals. In addition, some of the early work was motivated by problems in statistical mechanics (see e.g., \cite{Rue69} and \cite{Rue04}, on thermodynamic formalism).

\begin{remark}[Comparison with resistance forms]
  \label{rem:comparison-to-resistance-forms}
  In \cite[Def.~2.8]{Kig03}, a \emph{resistance form} is defined as follows: let $X$ be a set and let \energy be a symmetric quadratic form on $\ell(X)$, the space of all functions on $X$, and let \sF denote the domain of \energy. Then $(\energy, \sF)$ is a resistance form iff:
  \begin{enumerate}[(RF1)]
    \item \sF is a linear subspace of $\ell(X)$ containing the constant functions and \energy is nonnegative on \sF with $\energy(u) = 0$ iff $u$ is constant.
    \item $\sF/\sim$ is a Hilbert with inner product \energy, where $\sim$ is the equivalence relation defined on \sF by $u \sim v$ iff $u-v$ is constant.
    \item For any finite subset $V \ci U$ and for any $v \in \ell(V)$, there is $u \in \sF$ such that $u \evald{V} = v$.
    \item For any $p,q \in X$, the number
      \begin{align}\label{eqn:form-resistance}
        R_{\energy,\sF}(p,q) := \sup \left\{ \tfrac{|u(p)-u(q)|^2}{\energy(u)}
          \suth u \in \sF, \energy(u) > 0\right\}
      \end{align}
      is finite. Then $R_{\energy,\sF}$ is called the \emph{effective resistance associated to the form} $(\energy,\sF)$.
    \item If $u \in \sF$, then $\cj{u}$ defined by $\cj{u}(x) := \min\{1,\max\{0,u(x)\}\}$ (the unit normal contraction of $u$, in the language of Dirichlet forms) is also in \sF.
  \end{enumerate}
  Upon comparison of \eqref{eqn:def:R^F(x,y)-R}--\eqref{eqn:def:R^F(x,y)-S} to \eqref{eqn:def:R^W(x,y)-R}--\eqref{eqn:def:R^W(x,y)-sup}, one can see that $R^F$ is the effective resistance associated to the resistance form $(\energy,\HE)$, and that $R^W$ is the effective resistance associated to the resistance form $(\energy,\Fin)$. We are grateful to Jun Kigami for pointing this out to us. Note that the wired resistance is not related to the ``shorted resistance form'' of \cite[\S3]{Kig03} (see Prop.~3.6 in particular). See also Remark~\ref{rem:Rharm-has-no-form}.
\end{remark}

The reader will also find \cite{Soardi94} to be an good reference for effective resistance. While the sources \cite{Soardi94, DoSn84, Lyons:ProbOnTrees, LevPerWil08, Peres99} do not especially emphasize the metric aspect of effective resistance, they provide an exceptional description of the relationship between effective resistance and random walks. The books \cite{Kig01} and \cite{Str06} are also useful for understanding connections between effective resistance and the energy form and Laplace operator, on graphs and on self-similar fractals.

For different formulations effective resistance appearing in the literature, see \cite{Pow76b} and \cite[\S8]{Peres99} for \eqref{eqn:def:R(x,y)-Lap}, \cite{DoSn84} for \eqref{eqn:def:R(x,y)-energy}, \cite{DoSn84,Pow76b} for \eqref{eqn:def:R(x,y)-diss}, \cite{Kig03,Kig01,Str06} for \eqref{eqn:def:R(x,y)-R}--\eqref{eqn:def:R(x,y)-S}.

For investigations of the ``limit current'' (corresponding to free resistance) and ``minimal current'' (corresponding to wired resistance), one should consult \cite{Soardi94} and the earlier sources \cite{Flanders71, Thomassen90, Zemanian76}.

The role of effective resistance in combinatorics (dimer configurations, percolation on finite sets, etc.) is discussed in \cite{BeKo05, Rieffel99, KeWi09}. The role of Schur complement in the trace of a resistance form appears in \cite{Kig03}, and less specifically also in \cite{Metz, KeWi09, Butler, AndersonTrapp, Anderson}, where it is sometimes called the \emph{shorted operator}. Also see \cite{KndZuLiRo} for the role of Schur complement in regard to Dirichlet-to-Neumann maps.


text:
\begin{document}
\input{symbols}
\maketitle

\begin{abstract}
We design a physics-aware auto-encoder to specifically reduce the
  dimensionality of solutions arising from convection-dominated nonlinear
  physical systems.  Although existing nonlinear manifold learning methods seem
  to be compelling tools to reduce the dimensionality of data characterized by
  a large Kolmogorov n-width, they typically lack a straightforward mapping
  from the latent space to the high-dimensional physical space.  Moreover, the
  realized latent variables are often hard to interpret.  Therefore, many of
  these methods are often dismissed in the reduced order modeling of dynamical
  systems governed by the partial differential equations (PDEs).  Accordingly,
  we propose an auto-encoder type nonlinear dimensionality reduction algorithm.
  The unsupervised learning problem trains a diffeomorphic spatio-temporal
  grid, that registers the output sequence of the PDEs on a non-uniform
  parameter/time-varying grid, such that the Kolmogorov n-width of the mapped
  data on the learned grid is minimized.  We demonstrate the efficacy and
  interpretability of our approach to separate convection/advection from
  diffusion/scaling on various manufactured and physical systems.
\end{abstract}

\section{Introduction}

\input{introduction}

\section{Preliminaries}

\input{preliminaries}

\section{Physics-aware registration}

\input{method}

\section{Experiments}

\input{experiments}

\section{Conclusion}
\input{conclusion}

\medskip

\small
\bibliographystyle{unsrtnat}
\bibliography{NeurIPS20_PhysicsAware}{}

\end{document}

config: arxiv