Effective resistance of Gromov-hyperbolic graphs: Application to asymptotic sensor network problems

Abstract

The technique of effective resistance has seen growing popularity in problems ranging from escape probability of random walks on graphs to asymptotic space localization in sensor networks. The results obtained thus far deal with such problems on Euclidean lattices, on which their asymptotic nature already reveals that the crucial issue is the large scale behavior of such lattices. Here we investigate how such results have to be amended on a class of graphs, referred to as Gromov hyperbolic, which behave in the large scale as negatively curved Riemannian manifolds. It is argued that Gromov hyperbolic graphs occur quite naturally in many situations. Among the results developed here, we will mention the nonvanishing probability of escape of a random walk to a Cantor set Gromov boundary and the facts that the space localization error of sensors networked in a Gromov hyperbolic fashion grows linearly with the distance to a sensor whose geographical position is known, but would become uniformly bounded in an idealized situation in which the geographical locations of the nodes at the Gromov boundary are known.

Full text

Page 1

Effective Resistance of Gromov-Hyperbolic Graphs:
Application to Asymptotic Sensor Network Problems
Edmond A. Jonckheere & Mingji Lou and Jo˜ ao Hespanha & Prabir Barooah
Abstract—The technique of effective resistance has seen
growing popularity in problems ranging from escape probabil-
ity of random walks on graphs to asymptotic space localization
in sensor networks. The results obtained thus far deal with
such problems on Euclidean lattices, on which their asymptotic
nature already reveals that the crucial issue is the large
scale behavior of such lattices. Here we investigate how such
results have to be amended on a class of graphs, referred
to as Gromov hyperbolic, which behave in the large scale
as negatively curved Riemannian manifolds. It is argued that
Gromov hyperbolic graphs occur quite naturally in many
situations. Among the results developed here, we will mention
the nonvanishing probability of escape of a random walk to
a Cantor set Gromov boundary and the facts that the space
localization error of sensors networked in a Gromov hyperbolic
fashion grows linearly with the distance to a sensor whose
geographical position is known, but would become uniformly
bounded in an idealized situation in which the geographical
locations of the nodes at the Gromov boundary are known
I. INTRODUCTION
The effective resistance between two nodes in a graph
is the voltage drop that would be observed in a electrical
network obtained by placing one resistor at each edge of the
graph, when a unit current is injected in one of the nodes
and removed from the other. In this paper, we extend the
result of Doyle and Snell [12] on the effective resistance
of a regular tree of degree bounded but greater than 2 and
its ramifications in various asymptotic space localization and
coordination problems for autonomous agents as initiated by
Barooah and Hespanha [5], [3], [4].
A tree is regular if the degree of its nodes, except for the
root, is constant. Under this degree condition, the effective
resistance from the root of the tree to “infinity” is finite. The
latter has the important consequence that a random walk on
the tree has a finite probability of escaping to infinity. In the
context of computer security, it follows that a worm is more
likely to escape to infinity and therefore cause more damage
than one that is recurrent (see Remark 1). In the context of
localization based on relative measurements, this means that
if only the absolute positions of the far-away leafs of a large
tree are available, all nodes in the tree can still be accurately
localized just based on noisy relative measurements between
adjacent nodes in the graph [4]. The reader is referred to
[5] for other connections between effective resistance and
distributed control problems.
These authors are with the Dept. of Elec. Eng., Univ. of South. California,
Los Angeles, CA 90089, jonckhee@usc.edu.
These authors are with the Univ. of California, Santa Barbara, CA 93106,
hespanha@ece.ucsb.edu
The natural generalization of the concept of tree is that
of Gromov hyperbolic graph [14]. Intuitively, a Gromov
hyperbolic graph is a graph that has the property that it looks
like a tree when viewed at a distance, where the concept of
“viewing at a distance” is formalized in large-scale geometry,
also referred to as coarse geometry or asymptotic metric
theory [15]. The importance of Gromov hyperbolic graphs
is that scale free, and hence Internet, graphs have that
property [18], [23], [19]. Since Gromov hyperbolic graphs
are a generalization of trees, the natural question is, “What
is the effective resistance of a Gromov hyperbolic graph?”
The importance of effective resistance of Gromov hyperbolic
graphs stems from the above mentioned fact that effective
resistances arise naturally in a variety of propagation and
localization problems.
Before extending the result of Doyle and Snell, a technical
question is whether such concept as “effective resistance
between the root and infinity” can be defined for those
hyperbolic graphs, in which the tree structure might not be
completely obvious. A “root” of a hyperbolic graph is a so-
called quasi-pole [10]. A quasi-pole Ω is a compact subset
of an infinite graph G such that for every vertex v of G
there exists a geodesic ray emanating from Ω and passing
within a bounded distance of v. One of the premises of coarse
geometry is that finite subsets, like Ω, can be coalesced
to points. As such, Ω can be coalesced to ω ∈ Ω, which
becomes the “root” of the graph. Once a root is chosen, the
next issue is to define “infinity” so as to be able to define
the effective resistance between the quasi-pole and infinity.
In the context of the coarse geometry of Gromov hyperbolic
graphs, “infinity” is the Gromov boundary at infinity of the
graphs, ∂∞G. The latter is defined as the equivalence class of
infinite geodesic rays emanating from the quasi-pole under
the equivalence relation that two rays are equivalent if their
Hausdorff distance remains finite [9, III.H.3].
While the Gromov boundary ∂∞G of a regular tree can
be easily visualized, this is not so when all that is known
about G is that it is Gromov hyperbolic, since it could
be anything ranging from N to the circle S1, passing by
the Cantor set, the Menger curve, etc. (See [21] for a
survey.) The difficulty is that the topology of ∂∞G dictates
such application-relevant issues as the Cheeger isoperimetric
constant and the exponential growth of balls [10]. As far
as the present paper is concerned, in order to have finite
effective resistance between Ω and ∂∞G, ∂∞G must have
the cardinality of the continuum.
Since in this paper we use coarse geometric techniques
in which we do not care about distortion as long as it is

Page 2

uniformly bounded, the exact value of the effective resistance
R(ω,∂∞G) is irrelevant. Is relevant, however, the question
of whether the effective resistance is vanishing, finite (0 <
R(ω,∂∞G) < ∞), or infinite. In fact, the latter are coarse
geometry invariants.
Due to space limitations, most proofs have been omitted,
but they are available in [20].
II. MATHEMATICAL SET-UP
A. Graphs
A graph G is defined by its vertex set and its edge set,
(V,E). The edge with end vertices u,v will be written uv.
All graphs here have bounded local geometry, i.e., the degree
of their nodes is uniformly bounded. As such, an infinite
graph has infinitely many vertices. Edges are equipped with
a length function, ? : E → R+, with the property that
?min:= inf
e∈E?(e) > 0,?max:= sup
e∈E?(e) < ∞
(1)
Upon identification of every edge with a homeomorph of the
unit interval [0,1], the length function is easily extended to
Lebesgue-measurable subsets of the edges.
A path from x to y is a continuous map p : [a,b] → G
such that p(a) = x, p(b) = y. The length of a path joining
x to y is the sum of the lengths of the (subsets of) edges
traversed by the path. The distance d(x,y) between x,y is
the infimum of the lengths of all paths joining x to y. The
latter will sometimes be referred to as length distance. With
this length distance, (G,d) becomes a metric space.
An isometric embedding f : G → H of the graph G =
(VG,EG) into the graph H = (VH,EH) is a map (usually
induced by a vertex transformation VG→ VH) that preserves
the metric, viz., dH(f(x),f(y)) = dG(x,y).
A geodesic γ = [xy] from x to y in G is an isometric
embedding γ : [0,?(γ)] → G such that γ(0) = x, γ(?(γ)) =
y. Because of the bounded local geometry, every pair of
points x,y can be joined by a geodesic [xy] such that
?([xy]) = d(x,y). In such a situation, (G,d) is said to be a
geodesic metric space.
B. Gromov hyperbolicity
A geodesic triangle ?uvw on vertices u,v,w is defined as
[uv]∪[vw]∪[wu]. A graph (G,d), or an arbitrary geodesic
metric space (X,d) for that matter, is Gromov hyperbolic
if there exists a δs < ∞ such that, ∀?uvw, we have
[vw] ⊂ Nδs([uv]) ∪ Nδs([uw]), where Nδs([uv]) denotes
the δs-neighborhood of [uv] for the metric d. An equivalent
characterization is the existence of a constant δi < ∞
such that, ∀?uvw, for points i(v,w) ∈ [vw], i(u,w) ∈
[uw], i(u,v) ∈ [uv] such that d(u,i(u,v)) = d(u,i(u,w)),
d(v,i(v,w)) = d(v,i(v,u)), d(w,i(w,v)) = d(w,i(w,u)),
we have diam{i(u,v),i(v,w),i(w,u)} ≤ δi. It can be
shown that δs < ∞ iff δi < ∞ (see [9, Chap. III.H,
Prop. 1.17]). Yet a third equivalent definition states that, in
any quadrilateral, the largest sum of the lengths of pairs of
opposite diagonals cannot exceed the medium sum by more
than 2δq< ∞ (see [9, p. 411]).
The most extreme example of a Gromov hyperbolic graph
is the tree, as δi= δs= δq= 0.
The following theorem formulates an important property
of Gromov hyperbolic spaces, a property that is sometimes
even taken to be the definition of Gromov-hyperbolicity [10,
Def. 1.1]:
Theorem 1: Let (X,d) be a δs-Gromov hyperbolic space.
Let γ1, γ2be two arc length parameterized geodesics such
that γ1(0) = γ2(0) = v0. Assume that, for some R > 0,
d(γ1(R),γ2(R)) > 3δs. Then any path p joining γ1(R+r),
γ2(r + R) and outside the ball BR+r(v0) is such that
?(p) ≥ δs2
r
δs−2
This theorem has the (negative) consequence that, if an
outage affects a ball BR+r(v0), a message scheduled to
transit along [uv] ? v0will have to make a detour of
exponential length δs2
shown in [14], this bound can be refined to δs2
r
δs−2to circumvent the outage. As
r+R
δs−1.
C. Quasi-isometry
A (λ,?)-quasi-isometric embedding of the graph G =
(VG,EG) into the graph H = (VH,EH) is a (not necessarily
continuous) function f : G → H (usually induced by a
vertex transformation VG→ VH) such that
1
λdG(x,y) − ? ≤ dH(f(x),f(y)) ≤ λdG(x,y) + ?,
∀x,y ∈ G. Such an f is quasi-injective in the sense that
f(x) = f(y)
⇒
f is said to be a quasi-isometry if, in addition, there exists
a constant c such that
dG(x,y) ≤ ?λ,
∀x,y ∈ G.
dH(f(G),z) ≤ c,
∀z ∈ H
which means that f is quasi-surjective. In this case, f has at
least one quasi-inverse f−q: H → G in the sense that there
exists a constant d for which
dG(f−q◦ f(x),x) ≤ d,
dH(f ◦ f−q(z),z) ≤ d,
∀x ∈ G,
∀z ∈ H,
(2)
(3)
and this quasi-inverse is both quasi-injective and quasi-
surjective. Quasi-isometry is an equivalence relation.
D. Gromov boundary
A geodesic ray r emanating from ω ∈ V is an isometric
embedding r : [0,∞) → G such that r(0) = ω. The same
ray will sometimes be written as r(0)/r(∞). Two rays r,r?
are said to be equivalent iff sups∈[0,∞)d(r(s),r?(s)) < ∞;
equivalently, if dH(r,r?) < ∞, where dH denotes the
Hausdorff distance. The set of equivalence class of rays
is the Gromov boundary, ∂∞G. The equivalence class of
the ray r is written r∞. Given two rays, ru
parameterized by the arc length s, we define
∞,rw
∞∈ ∂∞G,
ρ∞(ru
∞,rw
∞) := lim inf
s→∞e−?(s−1
2dG(ru(s),rw(s)))
The above is a semimetric [6, I.1.5] on ∂∞G. The idea
behind this definition is that, in a tree-like graph, two rays

Page 3

ru,rwemanating from the same quasi-pole ω will first follow
the same path and then separate at some vertex a(ru,rw),
the “common ancestor” of the points ru(∞),rw(∞). Then
ρ∞(ru
ρ∞ is as follows: Write the hyperbolic law of cosines in
?ωru(s)rw(s) as embedded in a surface of small negative
sectional curvature κ = −?2:
cosh?dG(ru(s),rw(s)) =
cosh?d(ω,ru(s))cosh?d(ω,rw(s))
− sinh?d(ω,ru(s))sinh?d(ω,rw(s))cos ˆ ω
where ˆ ω is a short for ∠ru(s)ωrw(s). Since the arc
length parameterization of geodesic rays is an isometry,
d(ω,ru(s)) = d(ω,rw(s)) = s. Then, for a distance d
such that ?d ? 1, cosh?d ≈ e?dand, for a small angle ˆ ω,
cos ˆ ω ≈ 1 −ˆ ω2
law of cosines yields
e?dG(ru(s),rw(s))= 1 + e2?sˆ ω
∞,rw
∞) = e−?d(ω,a(ru,rw)). Another intuition behind
2. With these approximations, the hyperbolic
2
In light of the above, ρ∞turns up to be the angle metric
ρ∞(ru
Now we make the semimetric ρ∞a metric by enforcing the
triangle inequality
∞,rw
∞) = lim inf
s→∞
∠ru(s)ˆ ωrw(s)
√2
d∞(ru
∞,rw
∞) := inf
ru=a0,a1,...,an=rw
n−1
?
i=0
ρ∞(ai,ai+1)
For ? < log2/δG, d∞is a visual metric in the sense that
1
4ρ∞(ru
(See [16, Sec. 2], [9, III.H.3.19].) This distance in turn
defines a topology on ∂∞G.
The Gromov boundary might have varying cardinalities,
and even Gromov boundaries with the same cardinality
might be quite different. An example of card(∂∞G) =
ℵ0 is provided by Figure 1. It should also be obvious
that the cardinality of the Gromov boundary of a binary
tree T2 is 2ℵ0, which is the cardinality of the continuum
c. Another example of card(∂∞G) = c is provided by
G = Cay(π1(Mn)), the Cayley graph of the fundamental
group of a compact Riemannian manifold Mnof curvature
uniformly bounded from above by κ < 0 (see [11, V.58]).
The significant difference between the continua ∂∞T2 and
∂∞Cay(π1(Mn)) is that the former is a Cantor set while the
latter is Sn−1.
Other examples of Gromov boundaries that are circles
include the Poincar´ e disk or any of its tessellation by the
image of a fundamental domain [22, 4.10].
Theorem 2: Under a quasi-isometry f : G → H, G =
(VG,EG) is Gromov hyperbolic iff H = (VH,EH) is
Gromov hyperbolic; furthermore, f induces a (quasi-M¨ obius)
homeomorphism f∞: ∂∞G → ∂∞H.
It transpires form the above that, for a Gromov hyperbolic
graph to be quasi-isometric to a tree, it is necessary that its
Gromov boundary be a Cantor set.
∞,rw
∞) ≤ d∞(ru
∞,rw
∞) ≤ ρ∞(ru
∞,rw
∞)
0
0 a
1
0
0
1
aa =
0
2 a
0
3 a
2
0
1
1
aa =
3
0
2
1
aa =
4
0
3
1
aa =
1
2 a
1
4 a
2
3 a
4
1a
3
2 a
0
5 a
2
2 a
1
3 a
0
4 a
G
∞
∂
Fig. 1.A tree, the Gromov boundary of which has cardinality ℵ0.
E. Isoperimetric behavior and exponential growth of balls
Given an infinite graph G, the combinatorial Cheeger
isoperimetric constant is defined as
h(G) = inf
S⊆VG
|∂S|
|S|
where the infimum is over all finite subsets, |S| denotes the
cardinality, and
∂S = {v ∈ V \ S : v ∈ N(s) for some s ∈ S}.
Observe that ∂S can be interpreted as the boundary of S and
therefore if h(G) > 0 the above means that the “volume”
|S| of a subset of vertices is bounded by a linear function
of the “area” |∂S| of its boundary.
The exponential growth of balls is clearly related to the
condition h(G) > 0. Indeed, if a subset S of vertices is
“infected” at time t = 0, the number of vertices infected prior
to or at time t ≥ 0 is bounded from below by |S|(1+h(G))t.
It turns out that a Gromov hyperbolic graph need not have
h(G) > 0. For the latter to hold, an extra condition on the
boundary at infinity is needed:
Theorem 3: Let G be a infinite, bounded geometry, com-
plete, Gromov hyperbolic graph with a quasi-pole. If, in
addition, the infimum of the diameters of the connected
components of ∂∞G is > 0, then h(G) > 0, which implies
that λmin(L(G)) > 0.
There is a slight refinement of the preceding result:
Theorem 4: λmin(L(G)) ≥h2(G)
III. INSTANCES OF COARSE GEOMETRY
Here we make the preceding concepts a little more pallat-
able by linking them with known engineering concepts.
4
.
A. Instances of hyperbolic graphs
Gromov hyperbolic graphs are an idealization of situations
that readily occur in practice. Typically, we are given a
set of agents {vi} =: V , which attempt to form an ad
hoc network, and which in doing so define a distance
function d : V × V
d(vi,vj) is the communication delay between viand vj.
As such, (V,d) becomes a metric space. The question is
to determine the manifold Mnand its Riemannian met-
ric g such that the agents can be thought of as oper-
ating on the manifold; precisely, the question is whether
→ R+. Typically, the distance

Page 4

there exists an isometric embedding (V,d) ?→ (Mn,g).
Specifically, there exists an isometric embedding (V,d) ?→
(Hn,g), where Hndenotes the standard n-dimensional Rie-
mannian manifold of constant sectional curvature κ < 0,
iff det{cosh?d(vi,vj)√−κ?}{1,...,k}×{1,...,k} = (−1)k+1.
situations that naturally lead to negatively curved networks.
a) Growth/preferential attachment graphs: They have
been shown to be scaled Gromov hyperbolic [18], [19].
b) High power transceivers in a wireless sensor net-
work: They have a tendency to network in a negatively
curved graph, while the low power transceivers rather net-
work in a positively curved graph [2].
c) Geometric optics propagation reflecting on convex
obstacles: Assume that in the two-dimensional Euclidean
space R2, we define a set of transceivers {vi} along with
convex obstacles. Assume that, at least for some pairs of
agents, communication can only be established by radio
waves reflecting on convex obstacles. It can be shown that
the embedding is in a negatively curved surface of a genus
equal to the number of convex obstacles.
d) Delaunay triangulation of nonuniformly distributed
vertices: Assume a set of agents {vi} are nonuniformly
distributed in R2. The Delaunay triangulation (V,E) of a
set of Gauss distributed agents is shown in Fig. 2 and is
negatively curved in the following sense. Define the clus-
tering coefficient as cluster(vi) =
(See [6], [7].) Practically speaking, we cite a few specific
#{(vivj,vivk):vjvk∈E}
(deg(vi)
2
2, the curvature is locally
)
∈
[0,1]. When 0 ≤ cluster(vi) ≤
negative [2], [13]. The distribution of cluster is shown in
Fig. 3, indicating negative curvature.
1
4647484950 515253
47
48
49
50
51
52
53
54
Fig. 2.Delaunay triangulation of nonuniformly distributed agents.
0.10.20.30.4 0.50.60.70.80.91
0
50
100
150
200
250
300
350
Clustering Cofficient
Number of Nodes
Fig. 3.Clustering coefficient distribution of Delaunay triangulation.
B. Instances of quasi-isometries
The examples are plentiful, the most obvious being that a
Euclidean lattice is quasi-isometric to the Euclidean space of
the same dimension. Precisely, the natural embedding Zn→
Rnis a (λ = 1,? = 1)-quasi-isometry. More generally, we
have the following:
Theorem 5: The drawing function f : G → Enof a
graph G that is both sparse and dense in the n-dimensional
Euclidean space Enis a (λ,?)-quasi-isometry for finite
constants λ,? > 0.
IV. EFFECTIVE RESISTANCE: A GENERALIZED RAYLEIGH
MONOTONICITY PRINCIPLE
The graph G = (V,E) can be made into an electrical
network by replacing every edge by a resistor of resistance
Re= ?(e). If j(vi) denotes the current injected at node vi,
and v(vi) denotes the voltage at node virelative to some
ground potential, then j = Lv, where L is the Laplacian
operator defined as (Lv)i =
N(vi) := {v ∈ V
resistance of the edge vivj. If ?(e) = Re = 1, ∀e ∈ E,
then L = diag(deg(vi)) − A, where A is the adjacency
matrix. It is easily verified that L ≥ 0. To define the effective
resistance at a driving point vertex vi, a subset V0of short-
circuited vertices, connected to the graph, is declared at
ground potential, v(V0) = 0. If V0 is finite and at finite
distance from some driving point vertex vi?∈ V0, the effective
resistance is defined as R(vi,V0) =?L−1
corresponding to V0, easily seen to be invertible. If the graph
is infinite, viewing L : ?2(N) → ?2(N) as a bounded Hilbert
space operator requires v(∂∞G) = 0, so that the 0 reference
potential is naturally set across ∂∞G. By the same token,
boundedness of L requires i(vi) = 0, ∀vi∈ ∂∞G. Assuming
for a moment that λ1(L) > 0, where λ1(L) = inf spec(L),
then?L−1?
crucial issue:
Theorem 6: R(vi,∂∞G) <
λ1(L(G)) > 0.
One can also show that effective resistance satisfies the
following triangle inequality
?
vj∈N(vi)
(vi−vj)
Re(vi,vj), where
: vvi∈ E} and Re(vi,vj) is the
0
?
ii, where L0is the
operator obtained after removing the rows and columns of L
iiis the effective resistance R(vi,∂∞G) between
viand ∂∞G. The latter consideration makes λ1(L) > 0 a
¯R < ∞, ∀vi∈ VG iff
RG(u,w) ≤ RG(u,v) + RG(v,w),
Hence effective resistance is a distance, which is different
from the length distance, unless G is a tree.
The following monotonicity result can be viewed as a gen-
eralization of Rayleigh’s Monotonicity Law [12] to “quasi-
embeddings”:
Theorem 7 (Rayleigh Monotonicity Principle): Consider
two uniformly-bounded degree graphs G = (VG,EG),
H = (VH,EH) for which there exists a quasi-injective
function f : G → H and constants λ,? < ∞ such that
dH(f(x),f(y)) ≤ λdG(x,y) + ?,
∀u,v,w ∈ VG.
∀x,y ∈ G.
(4)

Page 5

Then there exists a finite constant α > 0 such that
RH(f(x),f(y)) ≤ αRG(x,y),
If, in addition, f is quasi-surjective then we also have that
∀x,y ∈ VG.
(5)
RH(z,w) ≤ αRG(f−q(z),f−q(w)) + β, ∀z,w ∈ VH, (6)
where f−q(·) is a quasi-inverse of f and β a finite constant.
When f is actually surjective, then (6) holds with β = 0.
In the expressions above, RG and RH denote effective
resistances in electrical networks constructed from the graphs
G and H, respectively.
The following result can be obtained by applying Theo-
rem 7 to a quasi-isometry f and to its quasi-inverse f−q.
Corollary 1: Suppose that f is a (λ,?)-quasi-isometry be-
tween two uniformly-bounded degree graphs G = (VG,EG),
H = (VH,EH). Then there exist finite constants α,β > 0
such that
1
αRG(x,y) − β ≤ RH(f(x),f(y)) ≤ αRG(x,y),
∀x,y ∈ VG, where RGand RHdenote effective resistances
in electrical networks constructed from the graphs G and H,
respectively. When f is actually injective, then (7) holds with
β = 0.
(7)
V. TWO ASYMPTOTIC PROBLEMS
Here we formulate two asymptotic problems amenable to
effective resistance methods:
1) The escape probability of a random walk on a Gromov-
hyperbolic graph, closely related to the effective resis-
tance between the root and the Gromov boundary of
the graph.
2) The asymptotically space localization error at large
distance from the reference sensor, closely related to
the asymptotic effective resistance between two points
in the limit of large distance.
A. Escape probability of random walks
The exponential growth of balls is relevant to a propaga-
tion scheme in which every node infects every neighboring
node. Other processes include random walks on a graph—
that is, processes through which an infected node infects only
one of its neighbors chosen at random. More specifically, if
the walker is at vi, the probability of jumping to vj∈ N(vi)
is
p(vi,vj) =
1
Re(vi,vj)
?
v∈N(vi)
1
Re(v,vj)
The problem is to determine whether there is a nonvanishing
probability of the walk reaching the Gromov boundary at
infinity. The latter is called escape probability. A slight
generalization of [12, Sec. 1.3.4] yields
pesc(ω,∂∞G) =
1
R(ω,∂∞G)
?
vi∈N(ω)
1
Re(ω,vi)
where ω is a vertex of a quasi-pole. In other words, the
probability of escape from a vertex of a quasi-pole to the
boundary at infinity is proportional to the reciprocal of the
effective resistance between ω and the Gromov boundary.
As already said, the mere fact that the graph is Gromov
hyperbolic with a quasi-pole does not provide enough data
to determine an exact value for R(ω,∂∞G), but it is enough
to determine whether or not R(ω,∂∞G) is finite.
Remark 1. The difference between a randomly walking
worm escaping to infinity and an exponentially growing
percolating worm is a matter of the gap between λ1(L(G))
and h2(G)/4, as specified by Theorem 4.
B. Space localization
Assume that autonomous agents distributed in the Euclid-
ean space E2or E3communicate via the graph G. Assume
that a subset of vertices V0has known geographical position
and that the position of the other agents has to be computed
from noisy measurement of the distances dG(vi,vj), vivj∈
E, and azimuth data. The following is proved in [4]:
Theorem 8: If V0= {v0}, then the unbiased least square
estimation error of the geographical position x(vi) of viis
given by
E(||x(vi) − ˆ x(vi)||2) = RG(vi,v0).
If G is Gromov hyperbolic, and under the idealized situation
where V0= ∂∞G, we have
E(||x(vi) − ˆ x(vi)||2) = R(vi,∂∞G).
VI. RESULTS
The solution to the two problems defined in the previous
section relies on a generic theorem, stating that a Gromov
hyperbolic graph is quasi-isometric to a tree. Several such
results are already known (see, e.g., [8]), but here we need
to set-up the quasi-isometry in a specific way that easily
translates to effective resistance and geographic estimation
errors.
Theorem 9: A planar Gromov hyperbolic graph G with
a quasi-pole Ω and a Cantor Gromov boundary is (quasi-
surjectively) quasi-isometric to a tree T. Furthermore, if
?(e) = 1, ∀e ∈ E, the quasi-isometry can be taken such
that
dG(u,v) − δi≤ dT(u,v) ≤ λdG(u,v),
with λ = 7δi+ 2.
We note that, in [8], it is shown that there exists a quasi-
isometric embedding from a binary tree to a Gromov hy-
perbolic graph with Cantor Gromov boundary. The problem
is that this embedding is not in general quasi-surjective and
hence cannot be used to construct a quasi-isometry between
the graph and the binary tree.
∀u,v ∈ V
A. Escape probability
Theorem 10: Let G be a bounded geometry, Gromov hy-
perbolic graph with a quasi-pole Ω and a Gromov boundary
that is a Cantor set. Then 0 < R(ω,∂∞G) < ∞, i.e., there
is a nonvanishing probability of escape from the quasi-pole
to the boundary at infinity ∂∞G.