Hyperbolic Embedding and Routing for
Dynamic Graphs
Andrej Cvetkovski
Department of Computer Science
Boston University
Email: andrej@cs.bu.edu
Mark Crovella
Department of Computer Science
Boston University
Email: crovella@cs.bu.edu
Abstract—We propose an embedding and routing scheme for
arbitrary network connectivity graphs, based on greedy routing
and utilizing virtual node coordinates. In dynamic multihop
packet-switching communication networks, routing elements can
join or leave during network operation or exhibit intermittent
failures. We present an algorithm for online greedy graph
embedding in the hyperbolic plane that enables incremental
embedding of network nodes as they join the network, without
disturbing the global embedding. Even a single link or node
removal may invalidate the greedy routing success guarantees in
network embeddings based on an embedded spanning tree sub-
graph. As an alternative to frequent reembedding of temporally
dynamic network graphs in order to retain the greedy embedding
property, we propose a simple but robust generalization of greedy
distance routing called Gravity–Pressure (GP) routing. Our rout-
ing method always succeeds in finding a route to the destination
provided that a path exists, even if a significant fraction of links or
nodes is removed subsequent to the embedding. GP routing does
not require precomputation or maintenance of special spanning
subgraphs and, as demonstrated by our numerical evaluation, is
particularly suitable for operation in tandem with our proposed
algorithm for online graph embedding.
I. INTRODUCTION
The applicability of the greedy geometric packet routing
paradigm to large internetworks was studied in details by Finn
[1] as an alternative to the classical routing table approach.
According to this type of addressing and routing scheme,
each router in the network is assigned a coordinate denoting
its location – a process referred to as network embedding –
and the shortest path geometric distances between the nodes
can be calculated based on these coordinates. Each packet is
then forwarded toward the destination, choosing intermediate
routing elements that provide most progress toward the desti-
nation. This makes the straight-line distance from the packet
to the destination node a monotonically decreasing function
throughout the journey, thus eventually reaching zero – when
the packet arrives at the destination.
The simplest deterministic greedy routing rule picks as a
next hop a directly connected neighboring node that would
decrease the distance of the packet to the destination the
most. The notable advantages of greedy routing – its small
computational complexity, the small memory requirement per
node, and the use of local information only (each node finds
a next hop based on the coordinates of its neighbors), make
this scheme suitable for distributed operation in large scale
networks.
Greedy geometric routing using the actual physical coordi-
nates of the nodes has been studied due to its simplicity and
scalability ([1]–[3]), and regained popularity in the research
community recently with the proliferation of GPS-capable
communication devices ([4], [5]). Greedy routing based on
node locations and Euclidean distances has been shown to
have a high rate of success, but fails when a packet reaches
a node that is closer to the destination than all of its direct
neighbors even if a path to the destination exists (e.g. [6]).
Changing the actual physical locations of nodes to address
this issue and improve the success rate of a greedy routing
scheme is perhaps a rather impractical idea, but the observation
that the distance-to-destination function of each path in an
embedded graph is determined by the embedding motivated
the quest for network embeddings that would support a 100%
successful greedy routing, even if nodes have to be assigned
artificial, virtual coordinates that do not necessarily represent
locations in physical space ([4], [5]). Limiting this quest to
true metric spaces, R. Kleinberg, in his groundbreaking work
[7] presented a constructive proof that every finite, connected,
undirected graph has a greedy embedding in two-dimensional
hyperbolic space – that is, an embedding that allows greedy
routing for all source-destination pairs.
The embedding in [7] uses the hyperbolic plane as an
underlying space, and the corresponding distance function is
the standard hyperbolic distance (see e.g. [8]). The generic
algorithm of [7] can find a greedy embedding of an infinite
d-regular tree for any integer d ≥ 3. To embed an actual graph
G, first a spanning tree T of G is chosen to serve as a minimal,
loop-free subgraph that spans all the vertices and provides a
unique simple path between any two of them. Subsequently,
the maximal degree d of T is determined. Finally, the nodes
of T are identified with the embedded nodes of the d-regular
tree, as obtained by the embedding algorithm. This completes
the greedy embedding of the tree T . It is easy to see that this
embedding is also a greedy embedding of the graph G.
Although the procedure described in [7] is aimed at ad-
hoc wireless networks and sensornets, there seem to be some
obstacles to successful application of this type of embedding
to communication networks whose topology can change over
time. In [7], the entire embedding is a function of a local
topological property of the graph, namely the maximum degree
of the chosen spanning tree T . Since newly added nodes
increase the node degree locally, their embedding is not always
possible without changing the coordinates of all nodes in
the network. Further, the greediness of the embedding in
[7] depends critically on the connectivity provided by the
underlying embedded spanning tree. This implies that local
changes in connectivity caused by nodes leaving the network
or failing links, can invalidate the greedy property of the entire
embedding. Such properties are undesirable of embedding
algorithms intended for distributed operation.
Our goal in this paper is to address the above issues by (i)
constructing a greedy graph embedding that supports addition
of an arbitrary number of nodes to the embedding in an online
fashion while requiring no changes to the previously assigned
node coordinates in order to retain the greedy property; and
(ii) constructing a greedy routing procedure that guarantees
delivery even in the presence of disturbances of the greedy
property of the embedding caused by nodes and/or links failing
unexpectedly or exhibiting intermittent periods of downtime or
standby-time.
Toward this end, in this paper we present an algorithm for
online calculation of a greedy embedding in the hyperbolic
plane for a given arbitrary, connected graph with edges repre-
senting the two-way connectivity in a communication network.
Our algorithm supports incremental embedding of network
nodes as they join the network during network operation time,
without affecting the rest of the embedding.
In greedy embeddings with guarantees based on the exis-
tence of “at least one” greedy next hop at each node, even
a single node or link failure may invalidate the greediness
of the embedding, thus causing the need to re-embed the
entire network if the greedy property is to be reestablished.
As an alternative to frequent reembedding of network graphs
due to intermittent node or link failures, or nodes leaving
the network, in this paper we propose a simple but robust
greedy routing method called Gravity–Pressure (GP) routing.
Our routing algorithm can be viewed as a generalization of
the simplest greedy distance routing, and always succeeds in
finding a route to the destination if a path in the network
exists. For its operation, GP routing does not require pre-
computation or maintenance of special spanning subgraphs.
Since no assumptions are made about the type of the network
coordinates, GP routing can be applied to embedded networks
using physical coordinates as well as virtual coordinates in
Euclidean or hyperbolic space. However, as elaborated in
Section IV, GP routing is particularly suitable for application
in graphs embedded using the online embedding procedure
described in this paper. Intuitively, if the fraction of nodes that
left the network after the initial greedy embedding is small, the
remaining network embedding is still “almost greedy”. Thus,
if the two proposed algorithms are used together, most of the
time GP routing will function as plain greedy distance routing,
making only positive progress toward the destination.
The rest of this paper is organized as follows. In Section
II-A we formulate a sufficient condition for a graph embedding
to be greedy. Based on this formulation, we present our online
embedding algorithm in Section II-B and further discuss its
construction and properties in Section II-C. Section III-A
offers an intuitive overview of the Gravity–Pressure routing,
and a precise algorithm statement is given in Section III-B.
Section IV presents a brief experimental evaluation of the
overall proposed routing and addressing scheme. Concluding
remarks are given in Sections V and VI.
II. O
NLINE GREEDY EMBEDDING
A. Preliminaries
We start by considering graph embeddings in a d-
dimensional Euclidean or hyperbolic space.
Definition 1: Given a connected finite graph G with vertex
set V ,anembedding of G in R
d
resp. H
d
is a mapping C (G) :
V → R
d
resp. C (G) : V → H
d
that assigns to each vertex v ∈V
a virtual coordinate C (v).
Definition 2: For two points v,w ∈ R
d
resp. H
d
,theEu-
clidean resp. hyperbolic bisector of the Euclidean resp. hy-
perbolic line segment determined by v and w is the locus of
points in R
d
resp. H
d
equidistant from v and w in terms of
Euclidean resp. hyperbolic distance.
In R
d
, the bisector is the Euclidean hyperplane perpendic-
ular to the segment [v, w] at its midpoint. In H
d
, the bisector
is the hyperbolic hyperplane perpendicular at the segment’s
midpoint to the hyperbolic line segment joining v and w.
Lemma 1: Let X be either R
d
or H
d
and
ρ
be the corre-
sponding distance function. Let v and w be different points in
X and let b be the bisector of the segment joining v and w.
Then for all u ∈ X it holds that
ρ
(v, u) <
ρ
(w, u) if and only if
v and u are in the same half-space with respect to the bisector
b.
Proof. Follows from the triangle inequality applied to the
triangle determined by v, u and x, where x is the intersection of
b and the segment joining u and w. Namely,
ρ
(v, u) <
ρ
(v, x)+
ρ
(u,x). From the definition of the bisector,
ρ
(v, x)=
ρ
(w, x),
and from the definition of x,
ρ
(u,x)+
ρ
(x,w)=
ρ
(u,w).
Combining these yields
ρ
(v, u) <
ρ
(w, u).
Likewise, for all u ∈ X it holds that
ρ
(w, u) <
ρ
(v, u) if and
only if w and u are in the same half-space with respect to b.
Definition 3: For a graph G (V,E) and its embedding C (G)
in R
d
resp. H
d
,lete ∈ E be an edge connecting the vertices
u and v.Anembedded edge of G is the Euclidean resp.
hyperbolic line segment C (e)=C (u, v) joining the points C (u)
and C (v) in R
d
resp. H
d
.
Lemma 2 (Greedy Embedding): For a graph G with embed-
ding C(G),letT be a spanning tree of G. For each edge
e ∈ T ,letb(e) be the perpendicular bisector of the embedded
edge C (e). Then a sufficient condition for C to be a greedy
embedding of G is that for each e ∈ T , b (e) intersects no
embedded edges of T other than C (e).
Proof. Consider any edge (u,v) ∈ T . To prove the lemma,
it suffices to show that u has a greedy route to any node s for
which the path in the tree from u has next hop v. Consider the
bisector b of (u,v). Since b intersects no other edges of T , s
must be in the half-space of v with respect to b (cf. Lemma
1). Therefore
ρ
(v, s) <
ρ
(u,s),sou has a greedy next hop to
s, namely v. Applying this argument to each edge on the path
from u to s confirms that the route in T (and thus in G)fromu
to s has a monotonically decreasing distance to the destination
i.e. is a greedy route. Consequently, C is a greedy embedding
of G.
The rest of this section concentrates on two-dimensional hy-
perbolic space and systematizes the concepts from hyperbolic
geometry that will be used in the subsequent presentation.
The Poincar
´
e Disk model will be used throughout for
visualization purposes. That is, we will use complex numbers
from the set D =
{
z ∈ C |
|
z
|
< 1
}
to represent the virtual co-
ordinates of the embedded vertices in the hyperbolic plane. An
introductory characterization of the more common hyperbolic
geometry models and their elementary geometric objects can
be found e.g. in [8].
As a distance function for the greedy embeddings consid-
ered in this section, we use the standard hyperbolic distance
ρ
for the Poincar
´
e Disk model: ∀z
1
,z
2
∈ D,
cosh
ρ
(z
1
,z
2
)=
2
|
z
1
− z
2
|
2
(1 −
|
z
1
|
2
)(1 −
|
z
2
|
2
)
+ 1. (1)
The existence of a closed-form expression for the hyperbolic
distance in D makes the choice of this model suitable for the
implementation of greedy embedding algorithms. The element
of hyperbolic length
2
|
dz
|
1 −
|
z
|
2
(2)
associated with this distance has circular symmetry: all points
on a Euclidean circle in D centered at the origin have same
distortion of the Euclidean element of length
|
dz
|
.
The Euclidean circle
∂
D =
{
z ∈ C |
|
z
|
= 1
}
represents the
boundary at infinity of this model. We also refer to this circle
as the horizon and to its points as the points at infinity of D
or ideal points.
In hyperbolic geometry, the path that realizes the hyperbolic
distance between two points (i.e. the shortest path) is the
hyperbolic line or geodesic. In the Poincar
´
e Disk model, paths
realizing (1) are represented by arcs of Euclidean circles in D
that are perpendicular to
∂
D. Two distinct points on
∂
D thus
determine a hyperbolic line in D. For a hyperbolic line in D
determined by two ideal points, of interest in this work are the
center and the radius of the Euclidean circle in the Riemannian
sphere
C = C∪
{
∞
}
containing the line. It is easy to show that
given two ideal points a = e
i
α
and b = e
i
β
, the center of the
Euclidean circle in
C containing the hyperbolic line whose
endpoints at infinity are a and b, and the corresponding radius
are given by
c = 1/m
∗
, R
2
= 1/
|
m
|
2
− 1(3)
where m =(a + b)/2 is the midpoint of the Euclidean chord
joining a and b, and m
∗
is the complex conjugate of m.
Two hyperbolic lines disjoint in D are said to be parallel.
Specifically, parallel hyperbolic lines in D contained in disjoint
Euclidean circles in
C are termed ultraparallel, to be distin-
guished from parallel hyperbolic lines that share an endpoint
at infinity.
B. Online Greedy Embedding
In this section, we present our algorithm for computation
of a greedy embedding of a given graph. The algorithm
takes as input a connected graph G =(V,E) specified by
a set of vertices V and the connections between them E =
{
(u,v) | u,v ∈ V
}
. The graph G serves as an abstraction of a
communication network – the nodes in the network correspond
to the vertices of the graph and two vertices in the graph are
connected if and only if the corresponding nodes can exchange
data bidirectionally. The neighbors of a vertex v ∈ V are the
directly connected vertices: N
v
=
{
u | (u, v) ∈ E
}
.
As noted in Sec. I, a greedy embedding of a tree graph
T which spans a given connected graph G is also a greedy
embedding of the graph G. Thus as a first step, the network
constructs a spanning tree T of the graph G. Any type of
a spanning tree can be used. A type of tree suitable for
distributed construction is a minimal-depth tree. To form a
minimal-depth tree, first the network nodes elect a root node.
Subsequently, each node n elects from N
n
its parent node to be
the node that has the smallest distance in hops to the root node.
Except for the root node r, each node in G is thus assumed to
have identified a parent node for itself. The parent of a node
n is referred to as p
n
.
Fig. 1 contains a precise statement of the online embedding
algorithm.
Procedure Online Embedding C (G)
1) Initialize by assigning to the root node r of the tree: (i)
a virtual coordinate C (r) in the hyperbolic plane; and
(ii) the angles
α
r
=
π
and
β
r
= 2
π
corresponding to the
ideal points a
r
= e
i
α
r
and b
r
= e
i
β
r
.
2) For each node n ∈ G:
a) Its parent p
n
: (i) sends C (p
n
),
α
n
=
α
p
n
and
β
n
=
(
α
p
n
+
β
p
n
)/2ton; and (ii) updates
α
p
n
:=
β
n
.
b) Node n: (i) calculates c and R according to (3) with
a
n
= e
i
α
n
and b
n
= e
i
β
n
and its own coordinate
C (n)=
R
2
(C (p
n
))
∗
− c
∗
+ c (4)
and (ii) updates
α
n
:=(
α
n
+
β
n
)/2.
Fig. 1. Online embedding procedure
In the initialization step of the algorithm, the values of
α
r
and
β
r
determine the possible choices for the root location
C (r). C (r) is chosen from the interior of the hyperbolic
triangle OAB defined by the geodesic G
1
, its bisector OB, and
the ray OA defined by
β
r
as shown in Fig. 2. With this choice
of initial conditions, the assignment of virtual coordinates to
the vertices of the spanning tree T (and thus the graph G)
obtained using the procedure in Fig. 1, corresponds to a greedy
embedding. We formalize this claim in the following
O
α
r
β
r
(α
r
+β
r
)/2
B
A
r
G
1
Fig. 2. Positioning of the root node for a greedy embedding
Proposition 1 (Correctness): If C (r) is an interior point of
the hyperbolic triangle OAB as in Fig. 2, then the embedding
C (G) obtained with the online embedding algorithm for an
arbitrary graph G with a spanning tree T is a greedy embed-
ding.
Proof. According to the greedy embedding lemma from Sec.
II-A, it suffices to show that no bisector of an edge e ∈ T
embedded in the hyperbolic plane intersects other edges of
the embedded tree. We begin by observing several properties
of the online embedding procedure above.
For a node n ∈ T ,letG
n
be the hyperbolic line in D
associated with n, whose endpoints at infinity are a
n
and
b
n
as in step 2a of the algorithm, and denote by H
n
the
corresponding region of D bounded by G
n
and containing
the point C (n). The virtual coordinate of the node n obtained
via (4) is the reflection of the location of the parent node
C (p
n
) in the hyperbolic line G
n
. Therefore, the hyperbolic
line segment joining C (p
n
) and C (n) is the embedded edge
C (p
n
,n) of T and that G
n
is its perpendicular bisector. To
see this, pick an isometry transform on D that maps the
endpoints of the segment of the embedded edge to a point p on
the imaginary axis in D and its complex conjugate p
∗
while
mapping the intersection of G
n
and C (p
n
,n) to the origin.
Since the isometries on D are conformal, it is easy to see
that under the chosen transform, the image of the Euclidean
circle in
C containing G
n
maps to the extended real axis
R = R ∪
{
∞
}
. From the symmetry of the hyperbolic length
element (2),
R is the perpendicular hyperbolic bisector of the
hyperbolic line segment joining p and p
∗
and consequently,
G
n
is the bisector of the embedded edge C (p
n
,n) as desired.
It remains to show that for any node n ∈ T , G
n
intersects
no embedded edges of T other than C (p
n
,n). We observe
that a point in D, its reflection in a hyperbolic line, and the
center of the Euclidean circle containing the hyperbolic line
are collinear in the Euclidean sense. Therefore a point p in D
and its reflection from a hyperbolic line G
n
always lie in the
same half of the subspace H
n
with respect to the Euclidean
bisector b(G
n
) of the arc in D containing G
n
. Since a node n
and the hyperbolic line G
c
associated with a child node of n
c are by construction contained in opposite halves of H
n
with
respect to b(G
n
), it follows that the embedded edge C (p
n
,n)
and G
c
are disjoint. Finally, note that by construction, for any
node n ∈ T , the hyperbolic line containing the embedded edge
C (p
n
,n) is ultraparallel to the hyperbolic line associated with
any sibling of n. Thus any embedded edge C (p
n
,n) is disjoint
with the hyperbolic bisector of any other embedded edge of
the tree T . Consequently, the embedding C (G) is a greedy
embedding.
When a new node, say n, joins an already embedded graph,
it can obtain a virtual coordinate simply by identifying a parent
node for itself, say p
n
, and executing step 2) of the algorithm
in Fig. 1. Note that this method does not require changes to
the virtual coordinates of the existing nodes when a new node
enters the graph. This is possible since our algorithm allows
allocation of disjoint subspaces of the hyperbolic plane in an
online fashion. By construction, the number of child-nodes any
node can have is not limited, and there is always free space
to be allocated for a newly added node.
Figure 3 illustrates an example of a graph embedded in the
Poincar
´
e Disk according to our online embedding procedure.
The figure shows the embedded edges of a spanning tree of
the graph; for clarity, the non-tree edges are not shown.
r
Fig. 3. Example of a greedy embedding of an irregular spanning tree in the
Poincar
´
e disk model
C. Remarks
All steps of the presented algorithm are suitable for dis-
tributed and asynchronous computation. Communication takes
place only between a node joining the embedded graph and its
parent node, which is elected from the immediate topographic
neighborhood in the graph.
The online embedding algorithm presented in Sec. III-B
generates node coordinates without use of any information
about physical locations of the nodes. The only initial virtual
coordinate needed is the root coordinate, which can easily be
chosen by the elected root node.
The region of allowable virtual coordinates for the root
node in the initialization of the algorithm can be derived from
the requirement in the proof of Proposition 1 that the point
in D representing any embedded node n and the associated
hyperbolic line of any child-node of n be at the opposite
sides of the Euclidean bisector of the region H
n
associated
with n. This requirement ensures that no hyperbolic bisector
associated with an embedded node intersects the embedded
edge of its parent node. The allowable region for the root node
is obtained as the intersection of the allowable regions with
respect to the associated geodesics of all possible child-nodes
of the root node:
J =
∞
n=1
J
n
.
It is easy to show that the region J corresponds to the
hyperbolic triangle OAB whose vertices are the origin O,the
ideal point A with coordinate −e
i
β
r
and the midpoint B of
the arc containing the geodesic G
1
associated with the first
child-node of the root node. (See Fig. 2.) It can be shown
that for this triangle to have a non-zero area, it is sufficient to
choose values of
α
r
and
β
r
that satisfy
β
r
−
α
r
< 4
π
/3. Fig.
3 illustrates the case when
β
r
−
α
r
=
π
.
Satisfying the condition of the greedy embedding lemma in
Sec. II-A is not the only way to achieve a greedy embedding,
but is sufficient. We remark that the construction implied by
the lemma is possible in the hyperbolic plane owing to the
fact that parallelism is a less restrictive quality in hyperbolic
space than in Euclidean space. More specifically, parallelism
is not a transitive relation in hyperbolic space and allows
every embedded edge to be parallel to the bisectors of all
other embedded edges. This is not possible in Euclidean
space without violating the condition of the greedy embedding
lemma, but is easily done in hyperbolic space. The online
embedding algorithm can thus embed an irregular tree directly
rather than identifying a regular tree as a superset of nodes to
be embedded in the hyperbolic plane.
III. T
HE GRAVITY–PRESSURE
GREEDY ROUTING ALGORITHM
A. Overview
The choice of a spanning tree as a subgraph type to be used
in the graph embedding procedure described in Sec. II is based
on the fact that spanning trees have simple enough structure to
allow incremental embedding, yet they contain a path between
any two nodes in the original graph. Adding a new node to
an existing spanning tree amounts to adding a single edge to
the already embedded spanning subgraph, and the condition
of Lemma 2 can be easily satisfied.
However, every spanning tree provides exactly one path for
each pair of nodes in the graph; removal of any graph edge
that is a non-leaf tree edge in the embedded subtree, partitions
the spanning tree into two unconnected subgraphs. Similarly,
removal of any node from the original graph other than leaf
nodes in the tree, partitions the spanning tree into a forest of
d subtrees, where d is the node degree of the removed node,
and thus disturbs the connectivity property of the tree. It is
easy to construct examples of graphs where partitioning of
the embedded spanning tree violates the greedy property of
the embedding. In fact, we have produced a number of such
embedded graphs for the purposes of Sec. IV of this paper.
To cope with greedy routing failures caused by local max-
ima of the packet progress toward the destination, one could
reinitiate the network embedding procedure on demand, or use
more sophisticated routing schemes that would either be able
to avoid such local maxima, or to continue the routing after a
data packet had reached a dead end. For the latter approach,
numerous advanced routing and route discovery procedures
have been proposed in the recent literature on location-based
routing (see e.g. [9]). These procedures can be roughly divided
into proactive, reactive, and hybrid, based on whether they
precompute auxiliary data structures for possible use in finding
a non-greedy route if a greedy route to the destination does
not exist.
In real network environments, link and node failures are
expected to happen often. Recent experimental studies have
shown that most failures are temporary, and in fact short-
lived (e.g. [10]). In such conditions, repeating the embedding
procedure to regain the greedy property, or precomputing data
structures every time a network element or link becomes
unavailable, may be unjustified from the standpoints of ef-
ficiency and conservation of resources. Instead, we propose
a simple generalization of the greedy distance routing rule
that does not require proactive computation or maintenance
of special data structures for its operation, and as such, is
suitable for application in temporally dynamic graphs. Our
routing method, called Gravity–Pressure (GP) routing, always
succeeds in finding a route to the destination, if a path in the
network exists.
In the rest of this section we provide an intuitive overview of
the GP routing procedure. A precise statement of the routing
algorithm is postponed to Section III-B. We will discuss some
of the advantages and disadvantages of GP routing when used
in conjunction with the greedy embedding algorithm of Sec.
II in more detail in Section V.
GP routing normally forwards packets to the neighbor that
provides most progress toward the destination. By analogy
with a liquid flowing through a system of pipes in gravitational
field of spherical symmetry toward the center located at the
destination node, we refer to this routing mode as the gravity
routing mode. The packet may occasionally reach a local
minimum, or a “valley”. In that case, GP forwards the packet
to a next hop that provides the least negative progress with
respect to the location of the destination. To deal with the
possibility of the packet entering a loop and periodically
returning to the same local lowermost point, we introduce the
