Download Scalable Analysis and Design of Ad Hoc Networks Via Random

Scalable Analysis and Design of Ad Hoc Networks Via
Random Graph Theory∗
András Faragó
Department of Computer Science
Erik Jonsson School of Engineering and Computer Science
The University of Texas at Dallas
P.O.B. 830688, Richardson, TX 75083-0688, U.S.A.
[email protected]
ABSTRACT
for many networking functions, such as routing, medium access, distributed control, etc.
We lay down the foundations of a new approach for the
scalable analysis of ad hoc networks, with special regard
to the properties of the random network topology. The
proposed methodology is rooted in the theory of random
graphs, but we significantly extend the conventional random graph model, as in its original form it would be too
“sterile” to capture realistic ad hoc networks. We analyze
some fundamental properties of the proposed new, general
model and demonstrate that it is capable of solving analysis/design problems which would otherwise be difficult to
capture.
The scalable analysis, design and control of large, random ad hoc networks is a great challenge that calls for
new methodologies. The goal of this paper is to present results that lay down the foundations of a new approach that
can help in the scalable analysis of ad hoc networks. The
proposed methodology is rooted in the theory of random
graphs. The conventional random graph model, however, is
too “sterile” for realistic ad hoc networks. Our main objective in this paper is to introduce and analyze a much more
general and refined random graph model that can help in
analyzing/designing realistic networks.
Categories and Subject Descriptors
2.
C.2.1 [Computer Communication Networks]: Network
Architecture and Design—Network Topology
REFERENCE SCENARIO
To illustrate and motivate our approach, let us consider
the following reference scenario. Consider an ad hoc network
in which many nodes are positioned randomly in a certain
area. Let A and B be two given nodes in the network at
random locations x and y, respectively, and at a distance
of r = |x − y| from each other. Assume that the presence
or abscence of a direct radio link between them depends on
the (random) distance r. To capture this dependence at a
general level, let us say that the probability of having a link
between A and B is f (r), where 0 ≤ f (r) ≤ 1 is a given
function. For example, if we take
1 if r ≤ r0
P(A–B link exists) = f (r) =
0 if r > r0
General Terms
Design, Theory
Keywords
Ad hoc network, random graph model, random network
topology
1. INTRODUCTION
As it is well known, in ad hoc networks the radio stations communicate directly with each other without fixed
infrastructure. The nodes are typically mobile, they may be
positioned at random locations and usually not all of them
are within range and line of sight of each other. This situation typically results in a random, frequently changing,
irregular network topology, which creates a major challenge
then the link exists if and only if A and B are within a certain transmission radius r0 . In this case the node locations
unambigously determine which links exist, due to the 0-1
valued probability. It is also possible, however, that the nature of radio propagation is taken into account in a more
refined way. For example, we may use
nα o
P(A–B link exists) = f (r) = min β , 1 ,
r
∗Research supported in part by NSF Grant ANI-0105985
where α, β > 0 are constants, obtained from a radio wave
propagation model (the minimum operation is needed to
make sure the probability cannot exceed 1). In this case
even for known node locations the network topology is still
random, because only the probability of link existence is determined by the distance.
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Let us further assume some additonal factors:
43
• The nodes may become unavailable with a certain probability. The event that a node becomes unavailable is
not necessarily independent of the (random) location
of the node, since in certain areas the danger of becoming unavailable may be higher than in others. For example, in a battlefield scenario, certain locations may
have higher risk of attack or jamming. Or, in any
application, certain areas may have higher risk of congestion.
3. PREVIOUS ANALYTICAL MODELS
Since the scalability of the modeling apparatus is a key
requirement for large ad hoc networks, therefore, we review
only those approaches that can be considered scalable, i.e.,
they do not involve exponentially growing computation time
when the network grows large1 .
The natural first idea to capture situations like our reference scenario would be the conventional random graph
model, denoted by Gn,p . This is a random graph on n vertices, such that the probability that any two given nodes
are connected by an edge is p, independently for each pair
of nodes. The theory of random graphs was initiated by
the Hungarian mathematicians P. Erdős and A. Rényi in
the 50’s and since then a large number of results have been
achieved. For deep and comprehensive overviews see the
books [4, 11]. Another interesting and important feature
is that many notoriously difficult algorithmic problems become much easier for random graphs [7]. A well known
example is to decide whether the graph contains a clique
(fully connected subgraph) on k vertices. This is a classic
NP-complete problem [8] for deterministic graphs. Moreover, the corresponding optimization version (the maximum
clique problem) where we wish to compute the maximum
clique size ω(G) in the graph G, cannot even be approximated by a polynomial time algorithm within an error factor of n1− for any fixed > 0, under standard complexity
theoretic assumptions [9]. Thus, this problem clearly defies
scalability, as with growing n (=number of nodes) no fast
algorithm can be expected to guarantee even an acceptable
approximation for the worst case. On the other hand, in
the random graph Gn,p the maximum size of a complete
subgraph can be easily computed, as it is known [4] that
asymptotically
• Random fading can influence the ability of receiving a
transmission.
• Shadowing can prevent reception, due to the presence
of obstacles or uneven terrain.
Remark: We can also refine the approach by defining several degrees of availability, depending on reliability, security
trust level, traffic load or other factors. For example, in a
Bluetooth scatternet, if a unit acts as a bridge between two
piconets, then it has to periodically switch between the piconets, as it can only transmit/receive in one piconet at a
time, so it has a lower degree of availability in each piconet.
Thus, in the reference scenario we have a large number of
randomly positioned nodes with random links among them.
The links depend on the random node positions, but this
dependence may be probabilistic (as in the case of the second example function above). Additionally, nodes may randomly become unavailable, or fall to a lower degree of availability, and correct reception can be prevented by fading or
shadowing. The occurence of these disturbances is not necessarily independent of the random network topology. Now,
with such a situation in mind, we can ask natural (but difficult!) questions about the network. A few examples: What
is the probability that the available nodes form a connected
network? A k-connected network? What is the expected
value of the network diameter? What is the diameter if
only those paths are accepted in which each node has a prescribed high degree of availability? What is the probability
that the maximum number of neighbors of any node is not
more than some number k? (This may be important for the
medium access protocol). What is the expected size of the
largest clique (fully connected subnetwork), that represents
a most congested (densest) part in the network topology?
What is the expected minimal number of clusters that cover
the entire network, where a cluster is defined as a subset
of nodes that spans a subnetwork with prescribed properties (e.g., fully connected)? Combinations of the above: e.g.,
what is the probability that the network diameter is at most
d and every node has at most k neighbors?
ω(Gn,p ) ∼ 2 log1/p n
and the probability distribution of ω(Gn,p ) sharply concentrates on two consequtive integers. For a finite graph this
holds almost surely, which means that the fraction of all
graphs that deviate from the formula tends to 0 as n grows.
Thus, for a random graph, one can practically determine
the desired quantity with vanishing error probability, despite
its worst-case algorithmic intractability. Theoretical analysis shows that, interestingly, such algorithmic advantages in
fact occur in much more cases than one would expect [6].
The applicability of such results, however, critically depends on how adequately Gn,p can model a realistic network. Unfortunately, Gn,p is a too simplistic and too uniform model to capture a realistic ad hoc network, since for
a given number of nodes the model only has one parameter
(p) to describe the entire network and it totally ignores the
geometry of the configuration and correlations among links.
For example, if the nodes are randomly located in a large
area and the condition for having a link between two nodes
is that they are at most at a (relatively small) distance r0 ,
then obvious dependencies arise: if A is connected to B and
B is connected to C, then the A–C distance can be at most
2r0 , which results in a much higher probability of having a
link between A and C, than the unconditional probability of
having a link between them. Such natural geometric considerations are fully ignored by the Gn,p model. Nevertheless,
All the above analysis questions can be directly turned
into design problems. For example: how large transmission
radius can guarantee that a given network property holds at
least with a prescribed probability?
It would not be very hard to continue the list of questions
in a similar spirit. It is much more difficult, however, to
answer any of them. In the next section we briefly overview
the existing analysis aproaches to attack such problems and
then, in Section 4, our new, more general model will be
introduced.
1
Traditional Markov models very quickly lead to an exponential explosion of state space, practically excluding scalability, so we do not consider them here.
44
below. In what follows n will always denote the number of
nodes.
in certain special situations Gn,p can be still successfully
applied. An example is when all the nodes are located in a
small area and the existence of links depends only on independent random obstacles and not on distance. This special
situtation may occur, e.g., in indoor wireless systems. Under such conditions Jacquet and Laouiti [10] analyzed ad
hoc routing algorithms in the random graph model.
• Node variables. Each node i is associated with a
random variable ξi , i = 1, . . . , n. In full generality,
these random variables may be arbitrary and can take
values from any set. Thus, ξi may be binary, integer,
real, complex, scalar, vector, matrix etc., with arbitrary distribution.
Other papers take into account the geometric aspects of
the ad hoc network topology. For example, McDonald and
Znati [14] capture link availability using a mobility model,
involving geometric considerations. They assume, however,
that the links along a path behave independently. Basagni
et al. [2] apply a more refined approach to account for the
probability of path availability, expressing it via conditional
link existence probabilities that are derived from the geometry of the random node locations. While the approach
can correctly describe the existence probability of a particular path, it cannot capture global properties of the network
topology, such as network diameter etc. In an earlier paper,
Philips et al. [16] use a different approach, they model the
random node positions by a 2-dimensional Poisson process.
The condition for link existence is that the distance between
two nodes is at most a given transmission radius. They obtain results on the transmission radius needed for global
connectivity. The approach does not seem to generalize,
however, to the analysis of more complex graph properties.
Bettstetter [3] follows a related approach to determine the
minimum node degree in a random network topology. The
result also applies to k-connectivity, as it is known [15] that
for a geometric random graph (in at least two dimensions)
k-connectivity is essentially equivalent to the property that
the minimum degree is ≥ k. In the mathematical literature
there are also other sporadic investigations on geometrically
induced random graphs, but only for special cases, such as
the analysis of Appel and Russo [1] for uniformly distributed
points in the unit square with the distance generated by the
maximum norm.
Random graph related models have been proposed not
only for ad hoc networks, but also to model the Internet
topology and the hyperlink structure of the World Wide
Web, see, e.g., [17, 5, 12]. These models do not seem to
apply to ad hoc networks.
To the author’s best knowledge, no model (with nontrivial results) has been proposed that could capture situations
like the reference ad hoc network scenario outlined in Section 2. The difficulty of this scenario is that it combines
quite different effects, such as the random geometry of the
network, the features of radio propagation, as well as random and possibly location dependent node/link availability
etc., so it does not easily lend itself for transparent analytical modeling. In the next section we introduce a proposed
new model, with the intent to capture the different effects
in a unified framework.
4.
• Link function. A function f is given that maps any
two node variables into a real number in the interval
[0, 1]. The value of the function is interpreted as the
probability of having a link between two nodes. That
is, nodes i, j will be connected by a random link that
exists with probability f (ξi , ξj ).
Since the model is determined by the random variables
ξ = (ξ1 , . . . , ξn ) and by the function f, therefore, we use
the notation Gξ,f for the arising RVM random graph. In
this paper we only consider undirected graphs, so it is assumed that f (ξi , ξj ) = f (ξj , ξi ) holds. It is straightforward,
however, to define a similar model for directed graphs.
Before analyzing some fundamental properties of the model,
let us show through a few examples that by the appropriate
choice of ξ and f many important cases can be captured in
the unified framework.
4.1
Ordinary Random graphs
The classic Gn,p model arises as a direct special case: if
each ξi is a degenerated random variable that can take only
a single value with probability 1 and f (ξi , ξj ) = p, then we
get Gn,p .
4.2
Ad Hoc Network Topology with Transmission Range
Let ξi be a planar vector that represents the random location of node i (may not be uniformly distributed). Choose
f according to
1 if |ξi − ξj | ≤ r0
f (ξi , ξj ) =
0 if |ξi − ξj | > r0
where r0 is the transmission range.
4.3
Ad Hoc Network Topology with Radio
Propagation Model
Let ξi be again a planar vector that represents the random
location of node i. Choose f such that it reflects the probability of link existence on the basis of a radio propagation
model, for example,
nα o
f (ξi , ξj ) = min β , 1 ,
r
with appropriate constants α, β > 0.
THE RANDOM VERTEX MODEL
4.4
In our model most of the randomness is hidden in the vertices of the graph, this motivates the name Random Vertex
Model (RVM). If a random graph is generated by this model,
for brevity we call it an RVM random graph. In what follows
we use the terms vertex and node interchangeably. Similarly,
the terms edge and link will be interchangeable.
The model is built from two key components, as described
Obstacles, Unavailability and Other Factors
Assume that in the area where the nodes are located there
are certain obstacles that are not transparent to radio waves.
Let u denote a vector, representing potential node locations.
If u, v are two such vectors (locations), then let L(u, v) = 1
if no obstacle separates u and v, and set L(u, v) = 0 if
45
5.
they are separated by an obstacle. Further, let Ai be the
indicator variable of node i’s availability, that is,
1 if node i is available
Ai =
0 if node i is unavailable.
Naturally, the decisive factor regarding the usability of the
above introduced modeling approach is whether it is feasible to achieve meaningful analytical results in the proposed
framework. In this section we prove that it is possible.
Let the variable for node i contain the location and the
availability indicator, that is ξi = (ui, Ai). Then, using the
transmission radius based link function, we can define
L(ui , uj )Ai Aj if |ui − uj | ≤ r0
f (ξi , ξj ) =
0
if |ui − uj | > r0
For the detailed exposition let us introduce first a few definitions. The intended meaning of the first definition is to
introduce a geometric flavor and to exclude certain degenerated cases. Specifically, Definition 1 requires three properties (see below). The first condition introduces a geometric
character in the model. The second condition intends to
capture the property that if two nodes get closer to each
other, this can only increase the probability of having a link
between them and if they conincide in space then they are
surely connected. These properties naturally hold in many
ad hoc network models. (Often the normed vector space is
simply the Euclidean plane). The third condition is a regularity condition for the probability distribution that is only
needed for the proof of Theorem 1. This condition does not
imply any significant restriction, since any probability distribution can be approximated with vanishing error by one
that satisfies the condition. Note that in the definition it is
not required that the node variables are independent.
Note that even if we require that the ξi are independent
random variables, the components of the same node variable ξi need not be independent, so Ai is not necessarily
independent of ui, allowing location dependent availability.
One can read from the function f (ξi , ξj ) that the (i, j) link
exists if the two nodes are within the transmission range,
not separated by an obstacle and both nodes are available.
It is possible, of course, to use here the propagation model
based link function instead of the distance based one. Also,
one may include other factors in the model, such as fading, interference, available bandwidth, queueing delay etc.,
whenever they can be expressed by their influence on the link
availability, which is then reflected by the value of f (ξi , ξj ).
4.5
ANALYSIS
Definition 1 An RVM random graph Gξ,f is called geometric if the following hold:
Arbitrary Random Graphs
• The ξi take their values from a normed linear vector
space and the link fucntion f is continuous.
As a general comment, it is worth mentioning that an
RVM random graph, in full generality, can mimic any possible random graph model on a given number of vertices, as it
allows any probability distribution over the set of all graphs
on a given number of vertices. To see this let G1 , . . . , GN
be an enumeration of all graphs on n vertices, each assigned
with an arbitrary probability p(Gk ), k = 1, . . . , N,
P
p(G
k ) = 1. Let ξ1 = ξ2 = . . . = ξn ∈ {1, . . . , N } with
k
Pr(ξi = k) = p(Gk ) and set
8
if ξi = k and there is an edge
< 1
between i, j in Gk
f (ξi , ξj ) =
:
0
otherwise
• The norm ||.|| in this space has the property that the
link function tends monotonely to 1 when x, y approach
each other in terms of the distance generated by the
norm. Formally, ||x − y|| ≤ ||u − v|| implies f (x, y) ≥
f (u, v) and ||x − y|| → 0 implies f (x, y) → 1.
• The probability distribution of ξ has a density function
hξ (x1 , . . . , xn ) with a convex support, i.e., it is positive
in a convex domain and 0 outside.
Often in ad hoc networks the nodes are distributed randomly over a certain area so that any two nodes can have
a link between each other with the same probability. This
important special case is reflected by the next definition.
Definition 2 Gξ,f is called homogeneous if E f (ξi, ξj ) is
the same for every i, j (i = j).
It follows directly from the definition that Pr(Gξ,f = Gk ) =
p(Gk ), so we generated a random graph from the prescribed
arbitrary probability distribution over all graphs on n vertices.
Of course, the above constructed RVM random graph has
strongly dependent vertex variables with an arbitrary probabillity distribution, which makes the analysis difficult. In
ad hoc network models, however, it is often enough to restrict ourselves to more special node variables. They usually
have some geometric character and/or are often independent. Note that even if the node variables are independent,
it does not mean that the different components within the
same node variable also have to be independent (for example, the location and the availability of the same node can
depend on each other). Moreover, the edges are usually not
independent even when the node variables are. For example, if ξi , ξj , ξk are three independent node variables, then
the existence of links (i, j) and (j, k) can both depend on the
value of ξj and, therefore, are generally not independent.
The following definition introduces another important concept in our treatment, the independent core of a generalized
random graph.
Definition 3 Let Gξ,f be an RVM random graph on n nodes
and assume that Gξ,f is homogeneous. The independent core
of Gξ,f is defined as the (conventional) random graph Gn,p ,
where p is given by the expected value of the link existence
probability:
(1)
p = E f (ξi, ξj ) .
Note that the value of p in (1) does not depend on i, j, by the
assumption of homogeneity. The meaning of the definition
is that while in Gξ,f the links are generally not independent,
the associated independent core Gn,p keeps only the average link existence probability, but ignores all dependences.
A key result will be that, under certain conditions, if the
46
therefore, if we sum up these probabilities for all Gi , then
we get precisely the expected number of the graphs with
property Q that occur in Gξ,f , as each Gi contributes to
(i)
this expectation by 1 with probability P(e1 ,(i) . . . , eki ) and
independent core has a property, then the generalized random graph Gξ,f will also have the property. Using this, we
will be able to reduce the analysis of the complex geometric
Gξ,f with dependent links to the analysis of the independent
core, for which we can use the rich treasury of results that
exist on Gn,p .
Now we can present the analytical results. The first theorem states that the links in the geometric Gξ,f model are
always positively correlated. This will be a key ingredient
in comparing the properties of Gξ,f and Gn,p . Note that we
do not require the node variables to be independent, which
makes it possible to account for rather messy situations, too.
For the ease of notation, let P(e1 , . . . , ek ) denote the probability that the edges e1 , . . . , ek are all included in the graph.
Similarly, P(ei ) means the probability that edge ei is in the
graph.
(i)
by 0 with probability 1 − P(e1 ,(i) . . . , eki ). Thus,
r
X
(i)
E N (Q, Gξ,f ) =
P(e1 ,(i) . . . , eki )
i=1
holds. Similarly, we have
P(Gi ⊆ Gn,p ) =
P(e1 , . . . , ek ) ≥
(i)
P(ej )
j=1
for the independent core Gn,p , implying
ki
r Y
X
(i)
E N (Q, Gn,p ) =
P(ej ).
Theorem 1 Let Gξ,f be a geometric RVM random graph.
Then the edges of Gξ,f are always positively correlated, that
is,
k
Y
ki
Y
i=1 j=1
Consequently, by making a term by term comparison using
(2), we obtain
E N (Q, Gξ,f ) ≥ E N (Q, Gn,p ) ,
P(ei )
i=1
holds for any choice of the edges e1 , . . . , ek , even if the node
variables are not independent.
completing the proof.
Proof: See Appendix A.
The message of Theorem 2 is that whenever the independent link probability p is such that the independent core
has a property, then the original, much more complex, geometric graph will also behave similarly, in the sense that
the subgraphs with the property are at least as abundant on
the average in Gξ,f as in Gn,p . Note that this holds for any
property Q.
✷
Now we can address the issue of how to use the rich treasury of results that exist for the classic Gn,p model, in order
to analyze the more complex (but also more realistic) Gξ,f
model. Informally, one can argue as follows. Consider a homogeneous geometric RVM random graph Gξ,f and let Gn,p
be its independent core (Definition 3). Since the edges of
Gξ,f are positively correlated (Theorem 1), therefore, whenever a given subgraph G occurs in Gn,p (i.e., a given subset
of edges is present), the probability that G occurs in Gξ,f
can only be higher. Thus, we can expect that the presence
of any given subgraph is more abundant in Gξ,f . This informal argument is made precise in the following theorem.
To state it concisely, we introduce the following notation.
For any given graph property Q and any graph G let us
denote the number of subgraphs in G that have property Q
by N (Q, G).
On the other hand, the fact that the expected number of
subgraphs with property Q is at least as large in Gξ,f as
in its independent core does not necessarily mean that the
same holds for the probability of having a subgraph with
property Q. The reason is that this probability can be low
despite a high expected number. For example, let M be a
large number. Assume that with probability 1/M Gξ,f has
M 2 subgraphs with property Q and no such subgraph with
probability 1−1/M. Then the expected number of subgraphs
with property Q is M , which is a large number, while the
probability of having such a subgraph at all is only 1/M, a
small number.
Theorem 2 Let Gξ,f be a homogeneous geometric RVM
random graph. Then for any graph property Q
E N (Q, Gξ,f ) ≥ E N (Q, Gn,p )
Under further assumptions, however, one can also prove
that not just expected number, but also the probability of
having the property in Gξ,f will be at least as large as in
its independent core. This can be concluded when the considered graph parameter shows a concentration of measure,
a frequent phenomenon in random graphs. When this happens, it allows the direct usage of many random graph theorems which claim that the (ordinary) random graph has a
property almost surely if the edge probability satisfies certain conditions. For space limitations we omit the details of
this refinement here.
holds, where Gn,p is the independent core of Gξ,f .
Proof: Let G1 , . . . , Gr be the possible subgraphs with property Q that can occur in Gξ,f with positive probability.
Each Gi is defined by the set of edges it contains, let it
(i)
(i)
be {e1 , . . . , eki } for Gi . (Note that different occurences of
the same graph are counted separately.) By Theorem 1 we
have for Gξ,f
(i)
P(e1 ,(i) . . . , eki ) ≥
ki
Y
(i)
P(ej ).
✷
6. A CASE STUDY
(2)
Consider an ad hoc network that is modeled as follows.
The nodes are distributed independently and uniformly at
random over a convex region in the plane with density ρ
nodes per unit area, on the average. Two nodes can directly
j=1
Now note that since
(i)
P(Gi ⊆ Gξ,f ) = P(e1 ,(i) . . . , eki ),
47
then we satisfy limA→∞ g(A) = ∞, so we obtain that a
choice of
communicate if their distance is at most r, the transmission
radius.
Assume that for reliable connectivity and for routing purposes we prefer a network topology in which many different spanning trees exist. For this reason we would like to
choose the transmission radius such that as the covered domain grows (its area tends to infinity), the expected number
of spanning trees in the network topology also tends to infinity. We would like to design the needed transmission radius
r to satisfy the above requirement under the condition that
the node density remains constant.
Let us define a geometric, homogeneous RVM random
graph Gξ,f for this problem in the natural way, by taking
a node variable ξi for the random position (coordinates) of
the node and using the following link function
1 if |ξi − ξj | ≤ r
f (ξi , ξj ) =
0 if |ξi − ξj | > r
r = log A
suffices to have E N (T, Gn,p ) → ∞. Now we can use the
fact that by Theorem 2
E N (T, Gξ,f ) ≥ E N (T, Gn,p )
holds, which yields that E N (T, Gξ,f ) → ∞ also holds.
Thus, the result is that if the transmission radius grows proportionally with the logarithm of the area, then it is guaranteed that the expected number of spanning trees will tend
to infinity, as required.
The moral of this example is that by using the new mathematical tools, in combination with results from traditional
random graph theory, we are capable of easily solving a problem that would be otherwise difficult to solve directly. Note
that even though a simple model is considered in this example, the geometric random graph that describes the network
topology has dependent links, that would not allow the direct use of the conventional random graph machinery.
Let us now consider the independent core Gn,p of Gξ,f .
First we compute the value of p. Let A be the area of the covered convex domain. Assume that its shape remains nondegenerated when the area grows, i.e., the ratio of the radius of
the smallest circumscribed circle versus the largest inscribed
circle remains bounded. Elementary geometric calculations
show that the probability of having a link between any two
given nodes with different location is the ratio of the area
covered by the transmission of a single node versus the entire
area:
7. CONCLUSION
We have presented the foundations of a random graph
modeling approach for the scalable analysis/design of the
topological properties of large ad hoc networks. We have analyzed some fundamental properties of the methodology and
demonstrated that it can be used to solve analysis/design
problems that would otherwise be infeasible to solve directly
with traditional random graph theory.
r2 π
,
(3)
A
apart from vanishing side effects at the region boundary,
which become negligible when A → ∞.
Let T denote the graph property of being a spanning tree.
It is known from the theory of (conventional) random graphs
[4] that if
pn
→∞
(4)
log n
p=
8. REFERENCES
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Degree of a Graph on Uniform Points in [0, 1]d ”, Adv.
Appl. Prob., 29(1997), pp. 567-581.
[2] S. Basagni, I. Chlamtac, A. Faragó, V.R. Syrotiuk and
R. Talebi, ”Route Selection in Mobile Multimedia Ad
Hoc Networks”, Sixth IEEE International Workshop
on Mobile Multimedia Communications
(MOMUC’99), San Diego, CA, Nov. 15–17, 1999.
holds, then it implies
E N (T, Gn,p ) → ∞,
i.e., the expected number of spanning trees in Gn,p tends to
infinity with n. The condition (4) can be expressed as
[3] C. Bettstetter, “On the Minimum Node Degree and
Connectivity of a Wireless Multihop Network”,
MOBIHOC’02, Lausanne, Switzerland, June 9-11,
2002, pp. 80-91.
log n
g(n),
(5)
n
where g(n) is any function with limn→∞ g(n) = ∞.
Now using (3) and (5) we can conclude that a choice of
r
A log n
g(n)
r=
nπ
is sufficient for E N (T, Gn,p ) → ∞. Let us take into account now the relationship n = ρA and that for constant ρ,
g(ρA) → ∞ implies g(A) → ∞. Then we obtain that
s
log A + log ρ
g(A)
r=
ρπ
is sufficient for E N (T, Gn,p ) → ∞. If we choose
p=
g(A) =
[4] B. Bollobás, Random Graphs, Academic Press,
London, UK, 1985.
[5] W. Aiello, F. Chung and L. Lu, “A Random Graph
Model for Massive Graphs”, 32nd ACM Symp. on
Theory of Computing (STOC’2000).
[6] A. Faragó, “Almost Surely Almost Exact
Optimization in Random Graphs”, Technical Report,
Dept. of Comp. Sci., The University of Texas at
Dallas, Oct 2000.
[7] A. Frieze and C. McDiarmid, “Algorithmic Theory of
Random Graphs”, Random Structures and Algorithms,
10(1997), pp. 5-42.
[8] M. R. Garey and D. S. Johnson, Computers and
Intractability: A Guide to the Theory of
NP-Completeness, Freeman, San Francisco, CA, 1979.
ρπ log2 A
log A + log ρ
48
APPENDIX
A. PROOF OF THEOREM 1
[9] J. Håstad, “Clique is hard to approximate within
n1− ”, in Proceedings of the 37th Annual Symposium
on Foundations of Computer Science, Burlington, VT,
14–16 October 1996, pp. 627–636.
Let us express P(e1 , . . . , ek ) using conditional probability:
P(e1 , . . . , ek ) = P(e1 |e2 , . . . , ek )P(e2 , . . . , ek )
[10] P. Jacquet and A. Laouiti, “Analysis of Mobile Ad
Hoc Network Routing Protocols in Random Graph
Models”, MOBICOM’2000 (See also INRIA Research
Report #3835).
If we can show that
[11] S. Janson, T. Luczak and A. Rucinski, Random
Graphs, Wiley-Interscience, 2000.
then it will imply P(e1 , . . . , ek ) ≥ P (e1 )P(e2 , . . . , ek ) by (6),
which then directly yields
P(e1 |e2 , . . . , ek ) ≥ P (e1 ),
[12] J. Kleinberg, “Authoritative Sources in a Hyperlinked
Environment” 9th ACM-SIAM Symposium on
Discrete Algorithms (SODA’98), 1998.
P(e1 , . . . , ek ) ≥
k
Y
(6)
(7)
P(ei )
i=1
[13] J. Kleinberg “The Small-World Phenomenon: An
Algorithmic Perspective”, Cornell Computer Science
Technical Report 99-1776, October 1999.
by induction, noting that the initial step (when k = 1) trivially holds.
To prove (7) let us first note that we can generate Gξ,f in
the following way. Draw the node variables ξ = (ξ1 , . . . , ξn )
according to their distribution. For each potential edge also
draw a random number ηl , l = 1, . . . , m, where m = n(n −
1)/2 is the number of all possible edges. The ηl random
variables are independent and uniformly distributed in the
[0, 1] interval. Then the edge indexed by l between nodes i, j
will be put in the graph if and only if ηl ≤ f (ξi, ξj ) holds.
In this way the edge will exist with probability f (ξi , ξj ), as
desired.
Let x = (x1 , . . . , xn ) be a realization of the node variables
ξ = (ξ1 , . . . , ξn ), such that the probability density function
hξ (x) is positive at x. Let us assume, without loss of generality, that one of the end nodes of e1 is indexed by 1. We
proceed by creating a special partition of all realizations,
called star decomposition, so that each class will have desirable properties.
Let us say that another realization y = (y1 , . . . , yn ) is
similar to x, denoted by x ∼ y, if y can be obtained from x
by shrinking or magnifying the configuration with center x1
(making use that the node variables take their values from a
linear vector space). Formally, x ∼ y if there exists a λ ≥ 0
such that
[14] A.B. McDonald and T. Znati, ”A Path Availability
Model for Wireless Ad-Hoc Networks”, IEEE
Wrireless Communications and Networking
Conference (WCNC’99), New Orleans, LA, Sep.
21-24, 1999.
[15] M.D. Penrose, “On k-connectivity for a Geometric
Random Graph”, Random Structures and Algorithms,
15(1999/2), pp. 145-164.
[16] T.K. Philips, S.S. Panwar and A.N. Tantawi,
“Connectivity Properties of a Packet Radio Model”,
IEEE Trans. Inf. Theory, 35(1989), pp. 1044-1046.
[17] E.W. Zegura, K.L. Clavert and M.J. Donahoo, “A
Quantitative Comparison of Graph-Based Models for
Internet Topology”, IEEE/ACM Trans. Networking,
5(1997), pp. 770-783.
yi = x1 + λ(xi − x1 ), i = 1, . . . , n
(8)
holds. (Note that it implies x1 = y1 ).
Now let zl be a realization of the ηl random variable, l =
1, . . . , m and set z = (z1 , . . . , zm ). Without loss of generality
we can assume zl < 1, since this occurs with probability 1.
Consider the set of all possible joint realizations (x, z) of
ξ = (ξ1 , . . . , ξn ) and η = (η1 , . . . , ηn ). For any possible value
of (x, z), such that hξ (x) > 0, define the following set, called
the star of x :
Sx,z = {(y, z) | x ∼ y}.
Visually, this can be imagined as fixing a realization of the
node positions and then taking all possible linear shrinkings
and magnifications with center x1 . The value of z remains
fixed in a such a star and also x1 remains fixed, being the
center of shrinking or magnification. A key property is that,
by the definition of similarity, for any x, x and z, the sets
Sx,z and Sx ,z are either equal or disjoint. Thus, in this way
we defined a partition of all possible realizations. Now we
are going to prove (7) by conditioning on such a star Sx,z ,
which makes the arising probabilities dependent on the star,
49
and then taking the expectation over all stars. Formally, (7)
will be proven by showing
E P(e1 | e2 , . . . , ek , (ξ, η) ∈ Sx,z ) ≥
E P (e1 | (ξ, η) ∈ Sx,z ) ,
(9)
where the expectation is taken over all stars Sx,z . (Since the
distibution of (ξ, η) is continuous, the probabilities in (9)
are interpreted with respect to the conditional probability
density function, conditioned on (ξ, η) ∈ Sx,z .) To prove (9)
it is enough to show that for any fixed Sx,z
P e1 | e2 , . . . , ek , (ξ, η) ∈ Sx,z ≥ P e1 | (ξ, η) ∈ Sx,z (10)
holds, whenever hξ (x) > 0. Thus, let us fix a star Sx,z , hξ (x) >
0. Note that we can run over all elements of the star by running the parameter λ in (8). In other words, the distribution
of ξ generates a distribution Fx,z (λ) on Sx,z for any z. If we
denote by e1 (λ) the indicator of the event that edge e1 exists
for a given λ, then we can write
Z ∞
e1 (λ)dFx,z (λ)
P e1 | (ξ, η) ∈ Sx,z =
0
Since by assumption Gξ,f is geometric, therefore, if e1 (λ) =
1 holds for some λ, then e1 (λ ) = 1 should also hold for
any 0 ≤ λ < λ by the construction. This implies that
the integrand is 1 in an interval, 0 outside, and the interval
starts at λ = 0. Denoting by λ1 (x, z) the upper limit, we
obtain
Z λ1 (x,z)
dFx,z (λ).
(11)
P e1 | (ξ, η) ∈ Sx,z =
0
Consider now the probability P e2 , . . . , ek , | (ξ, η) ∈ Sx,z .
Due to the assumption that Gξ,f is geometric and hξ (x) > 0,
this probability must be positive, since by sufficiently contracting the star, the value of f (xi, xj ) can be forced arbitrarily close to 1 (this contraction can occur with positive
probability due to the last condition in Definition 1). Using
again that Gξ,f is geometric, we obtain that
Z λ2 (x,z)
dFx,z (λ)
P e2 , . . . , ek , | (ξ, η) ∈ Sx,z =
0
holds with some λ2 (x, z) > 0. (The visual meaning of λ2 is
that how much shrinking of the star is needed to force each
of the edges e2 , . . . , ek to exist.)
Now, if λ1 (x, z) ≥ λ2 (x, z), then we have that the existence of e2 , . . . , ek implies the existence of e1 (conditioned
on Sx,z ), yielding P e1 | e2 , . . . , ek , (ξ, η) ∈ Sx,z = 1. Otherwise, if λ1 (x, z) < λ2 (x, z), then we have
R λ1 (x,z)
dFx,z (λ)
P e1 | e2 , . . . , ek , (ξ, η) ∈ Sx,z = R0λ (x,z)
2
dFx,z (λ)
0
Z
≥
λ1 (x,z)
0
dFx,z (λ) = P e1 | (ξ, η) ∈ Sx,z ,
where the inequality follows from the fact that the integral
in the denominator is ≤ 1, being a probability. Thus, in any
case we obtain
P e1 | e2 , . . . , ek , (ξ, η) ∈ Sx,z ≥ P e1 | (ξ, η) ∈ Sx,z ,
which completes the proof.
✷
50