Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM REVIEW
c
!
2008 Society for Industrial and Applied Mathematics
Vol. 50, No. 1, pp. 37–66
Minimizing Effective Resistance
of a Graph
∗
Arpita Ghosh
†
Stephen Boyd
†
Amin Saberi
‡
Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance
seen between the nodes of a resistor network with branch conductances given by the edge
weights. The effective resistance comes up in many applications and fields in addition
to electrical network analysis, including, for example, Markov chains and continuous-time
averaging networks. In this paper we study the problem of allocating edge weights on
a given graph in order to minimize the total effective resistance, i.e., the sum of the
resistances between all pairs of nodes. We show that this is a convex optimization problem
and can be solved efficiently either numerically or, in some cases, analytically. We show
that optimal allocation of the edge weights can reduce the total effective resistance of the
graph (compared to uniform weights) by a factor that grows unboundedly with the size of
the graph. We show that among all graphs with n nodes, the path has the largest value
of optimal total effective resistance and the complete graph has the least.
Key words. weighted Laplacian eigenvalues, electrical network, weighted graph
AMS subject classifications. 05C50, 05C12, 90C25, 90C35, 90C46
DOI. 10.1137/050645452
1. Introduction. Let N be a network with n nodes and m edges, i.e., an undi-
rected graph (V, E) with |V | = n, |E| = m, and nonnegative weights on the edges. We
call the weight on edge l its conductance and denote it by g
l
. The effective resistance
between a pair of nodes i and j, denoted R
ij
, is the electrical resistance measured
across nodes i and j when the network represents an electrical circuit with each edge
(or branch, in the terminology of electrical circuits) a resistor with (electrical) conduc-
tance g
l
. In other words, R
ij
is the potential difference that appears across terminals
i and j when a unit current source is applied between them. We will give a formal,
precise definition of effective resistance later; for now we simply note that it is a mea-
sure of how “close” the nodes i and j are: R
ij
is small when there are many paths
between nodes i and j with high conductance edges, and R
ij
is large when there are
few paths, with lower conductance, between nodes i and j. Indeed, the resistance R
ij
is sometimes referred to as the resistance distance between nodes i and j.
∗
Received by the editors November 16, 2005; accepted for publication (in revised form) August
8, 2006; published electronically February 1, 2008. This work was supported in part by a Stanford
Graduate Fellowship, the MARCO Focus Center for Circuit and System Solutions (www.c2s2.org)
under contract 2003-CT-888, by AFOSR grant AF F49620-01-1-0365, by NSF grant ECS-0423905,
and by DARPA/MIT grant 5710001848.
http://www.siam.org/journals/sirev/50-1/64545.html
†
Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9510 (arpitag@
stanford.edu, boyd@stanford.edu).
‡
Department of Management Science and Engineering, Stanford University, Stanford, CA 94305-
9510 (saberi@stanford.edu).
37
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
38 ARPITA GHOSH, STEPHEN BOYD, AND AMIN SABERI
We define the total effective resistance, R
tot
, as the sum of the effective resistance
between all distinct pairs of nodes,
R
tot
=
1
2
n
!
i,j=1
R
ij
=
!
i<j
R
ij
.(1)
The total effective resistance is evidently a quantitative scalar measure of how well
“connected” the network is, or how “large” the network is, in terms of resistance
distance. The total effective resistance comes up in a number of contexts. In an
electrical network, R
tot
is related to the average power dissipation of the circuit with
a random current excitation. The total effective resistance arises in Markov chains
as well: R
tot
is, up to a scale factor, the average commute time (or average hitting
time) of a Markov chain on the graph, with weights given by the edge conductances
g. In this context, a network with small total effective resistance corresponds to a
Markov chain with small hitting or commute times between nodes, and a large total
effective resistance corresponds to a Markov chain with large hitting or commute times
between at least some pairs of no des. We will see several other applications where
the total effective resistance arises, including averaging networks, experiment design,
and Euclidean distance embeddings.
In this paper we address the problem of allocating a fixed total conductance
among the edges so as to minimize the total effective resistance of the graph. We can
assume without loss of generality that the total conductance to be allo cated is 1, so
we have the optimization problem
minimize R
tot
subject to 1
T
g =1,g≥ 0.
(2)
Here, the optimization variable is g ∈ R
m
, the vector of edge conductances, and
the problem data is the graph (topology) (V, E). The symbol 1 denotes the vector
with all entries 1, and the inequality symbol ≥ between vectors means componentwise
inequality. We refer to problem (2) as the effective resistance minimization problem
(ERMP).
We will give several interpretations of this problem. In the context of electrical
networks, the ERMP corresponds to allocating conductance to the branches of a
circuit so as to achieve low resistance between the nodes; in a Markov chain context,
the ERMP is the problem of selecting the weights on the edges to minimize the average
commute (or hitting) time between nodes. When R
ij
are interpreted as distances, the
ERMP is the problem of allocating conductance to a graph to make the graph “small,”
in the sense of average distance between nodes.
We denote the optimal solution of the ERMP (which we will show always exists
and is unique) as g
!
and the corresponding optimal total effective resistance as R
!
tot
.
In [15], Fiedler introduced the general idea of optimizing some function, say, Φ, of a
weighted graph over all nonnegative edge weights that add to m, i.e., with average
edge weight 1. He refers to the optimal value of this problem as the absolute Φ of the
graph. For example, if Φ is the second smallest eigenvalue of the associated Laplacian,
which is called the algebraic connectivity of a (weighted) graph, then the absolute
algebraic connectivity (of a graph) is the maximum value of the second eigenvalue,
optimized over all weights on the edges that sum to m. Our ERMP is thus a special
case of Fiedler’s construction: (1/m)R
!
tot
is what Fielder would call the absolute total
effective resistance of the graph.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MINIMIZING EFFECTIVE RESISTANCE OF A GRAPH 39
In this paper, we will show that problem (2) is a convex optimization problem,
which can be formulated as a semidefinite program (SDP) [7]. This has several im-
plications, both practical and theoretical. One practical consequence is that we can
solve the ERMP efficiently. On the theoretical side, convexity of the ERMP allows
us to form necessary and sufficient optimality conditions and several associated dual
problems (with zero duality gap). Feasible points in these dual problems give us lower
bounds on R
!
tot
; in fact, we obtain a lower bound on R
!
tot
given any feasible allocation
of conductances. This gives us an easily computable upper bound on the suboptimal-
ity, i.e., a duality gap, given a conductance allocation g. We use this duality gap in a
simple interior-point algorithm for solving the ERMP.
We describe several families of graphs for which the solution to (2) can be found
analytically, by exploiting symmetry or other structure. These include trees, wheels,
and the barbell graph K
n
−K
n
. For the barbell graph, we show that the ratio of R
!
tot
to R
tot
obtained with uniform edge weights converges to zero as the size of the graph
increases. Thus, the total effective resistance of a graph, with optimized edge weights,
can be unboundedly better (i.e., smaller) than the total effective resistance of a graph
with uniform allocation of weights to the edges.
This paper is organized as follows. In section 2, we give a formal definition of the
effective resistance, derive a number of formulas and expressions for R
ij
, R
tot
, and the
first and second derivatives of R
tot
, and establish several important properties, such
as convexity of R
ij
and R
tot
as a function of the edge conductances. In section 3, we
give several interpretations of R
ij
, R
tot
, and the ERMP.
We study the ERMP in section 4, giving the SDP formulation, (necessary and
sufficient) optimality conditions, two dual problems, and a simple but effective custom
interior-point method for solving it. In section 5, we study some families of graphs for
which the ERMP can be solved analytically. In section 6, we show that of all graphs
on n nodes, the optimal value of the ERMP is smallest for the complete graph and
largest for the path. We give some numerical examples in section 7 and describe some
extensions in section 8.
1.1. Related Problems. The ERMP is related to several other convex optimiza-
tion problems that involve choice of some weights on the edges of a graph. One such
problem (already mentioned above) is to assign nonnegative weights, which add to
1, to the edges of a graph so as to maximize the second smallest eigenvalue of the
Laplacian:
maximize λ
2
(L)
subject to 1
T
g =1,g≥ 0.
(3)
Here L denotes the Laplacian of the weighted graph. This problem has been studied
in different contexts. The eigenvalue λ
2
(L) is related to the mixing rate of the Markov
process with edge transition rates given by the edge weights. In [28], the weights g are
optimized to obtain the fastest mixing Markov process on the given graph. Problem
(3) has also been studied in the context of algebraic connectivity [14]. The algebraic
connectivity is the second smallest eigenvalue of the Laplacian matrix L of a graph
(with unit edge weights) and is a measure of how well connected the graph is. Fiedler
defined the absolute algebraic connectivity of a graph as the maximum value of λ
2
(L)
over all nonnegative edge weights that add up to m, i.e., 1/m times the optimal value
of (3). The problem of finding the absolute algebraic connectivity of a graph was
discussed in [15, 16], and an analytical solution was presented for tree graphs.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
40 ARPITA GHOSH, STEPHEN BOYD, AND AMIN SABERI
Other convex problems involving edge weights on graphs include the problem of
finding the fastest mixing Markov chain on a given graph [6, 19, 28], the problem of
finding the edge weights (which can be negative) that give the fastest convergence
in an averaging network [30], and the problem of finding edge weights that give the
smallest least mean-square (LMS) consensus error [31]. Convex optimization can also
be used to obtain bounds on various quantities over a family of graphs; see [18].
In [8], Boyd et al. considered the sizing of the wires in the power supply network of
an integrated circuit, with unknown load currents modeled stochastically. This turns
out to be closely related to our ERMP, with the wire segment widths proportional to
the edge weights.
Some papers on various aspects of resistance distance include [22, 32, 2, 23, 24].
2. The Effective Resistance.
2.1. Definition. Suppose edge l connects nodes i and j. We define a
l
∈ R
n
as
(a
l
)
i
= 1, (a
l
)
j
= −1, and all other entries 0. The conductance matrix (or weighted
Laplacian) of the network is defined as
G =
m
!
l=1
g
l
a
l
a
T
l
= A diag(g)A
T
,
where diag(g) ∈ R
m×m
is the diagonal matrix formed from g and A ∈ R
n×m
is the
incidence matrix of the graph,
A =[a
1
···a
m
].
Since g
l
≥ 0, G is positive semidefinite, which we write as G $ 0. (The symbol $
denotes denotes matrix inequality between symmetric matrices: P $ Q means that
P − Q is positive semidefinite.) The matrix G satisfies G1 = 0, since a
T
l
1 = 0 for
each edge l. Thus, G has smallest eigenvalue 0, corresponding to the eigenvector 1.
Throughout this paper we make the following assumption about the edge weights:
The subgraph of edges with positive edge weights is connected.(4)
(If this is not the case, the effective resistance between any pair of nodes not connected
by a path of edges with positive conductance is infinite, and many of our formulas are
no longer valid.)
With this assumption, all other eigenvalues of G are positive. We denote the
eigenvalues of G as
0 <λ
2
≤ ··· ≤ λ
n
.
The nullspace of G is one-dimensional, the line along 1; its range has codimension 1,
and is given by 1
⊥
(i.e., all vectors v with 1
T
v = 0).
Let G
(k)
be the submatrix obtained by deleting the kth row and column of G.
Our assumption (4) implies that each G
(k)
is nonsingular (see, e.g., [10]). We will
refer to G
(k)
as the reduced conductance matrix (obtained by grounding node k).
Now we can define the effective resistance R
ij
between a pair of nodes i and j.
Let v be a solution of the equation
Gv = e
i
− e
j
,
where e
i
denotes the ith unit vector, with 1 in the ith position and 0 elsewhere. This
equation has a solution since e
i
− e
j
is in the range of G. We define R
ij
as
R
ij
= v
i
− v
j
.
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
MINIMIZING EFFECTIVE RESISTANCE OF A GRAPH 41
This is well defined; all solutions of Gv = e
i
− e
j
give the same value of v
i
− v
j
.
(This follows since the difference of any two solutions has the form α1 for some α ∈ R.)
We define the effective resistance matrix R ∈ R
n×n
as the matrix with i, j entry R
ij
.
The effective resistance matrix is evidently symmetric and has diagonal entries zero,
since R
ii
= 0.
2.2. Effective Resistance in an Electrical Network. The term effective resis-
tance (as well as several other terms used here) comes from electrical network analy-
sis. We consider an electrical network with conductance g
l
on branch (or edge) l. Let
v ∈ R
n
denote the vector of node potentials, and suppose a current J
i
is injected into
node i. The sum of the currents injected into the network must be zero in order for
Kirchhoff’s current law to hold, i.e., we must have 1
T
J = 0. The injected currents
and node potentials are related by Gv = J. There are many solutions of this equation,
but all differ by a constant vector. Thus, the potential difference between a pair of
nodes is always well defined.
One way to fix the node potentials is to assign a potential zero to some node, say,
the kth node. This corresponds to grounding the kth node. When this is done, the
circuit equations are given by G
(k)
v
(k)
= J
(k)
, where G
(k)
is the reduced conductance
matrix, v
(k)
is the reduced potential vector, obtained by deleting the kth entry of
v (which is zero), and J
(k)
is the reduced current vector, obtained by deleting the
kth entry of J. In this formulation, J
(k)
has no restrictions; alternatively, we can
say that J
k
is implicitly defined as J
k
= −1
T
J
(k)
. From our assumption (4), G
(k)
is nonsingular, so there is a unique reduced potential vector v
(k)
for any vector of
injected currents J
(k)
.
Now consider the specific case when the external current is J = e
i
− e
j
, which
corresponds to a one ampere current source connected from node j to node i. Any
solution v of Gv = e
i
− e
j
is a valid vector of node potentials; all of these differ
by a constant. The difference v
i
− v
j
is the same for all valid node potentials and
is the voltage developed across terminals i and j. This voltage is R
ij
, the effective
resistance between nodes i and j. (The effective resistance between two nodes of a
circuit is defined as the ratio of voltage across the nodes to the current flow injected
into them.)
The effective resistance R
ij
is the total power dissipated in the resistor network
when J = e
i
− e
j
, i.e., a one ampere current source is applied between nodes i and
j. This can be shown directly or by a power conservation argument. The voltage
developed across nodes i and j is R
ij
(by definition), so the power supplied by the
current source, which is current times voltage, is R
ij
. The power supplied by the
external current source must equal the total power dissipated in the resistors of the
network, so the latter is also R
ij
.
2.3. Effective Resistance for a Tree. We first consider the case when N is a
tree. In this case the effective resistance between nodes i and j can be expressed as
R
ij
=
!
1
g
l
,
where the sum is over edges that lie on the (unique) path between i and j. Therefore,
the total effective resistance for a tree is given by
R
tot
=
m
!
l=1
b
l
g
l
,(5)
