Probabilistic Broadcast for Flooding in Wireless
Mobile Ad hoc Networks
Yoav Sasson David Cavin Andr´e Schiper
{yoav.sasson,david.cavin,andre.schiper}@epfl.ch
´
Ecole Polytechnique F´ed´erale de Lausanne (EPFL)
1015 Lausanne, Switzerland
Abstract— Although far from optimal, flooding is an indis-
pensable message dissemination technique for network-wide
broadcast within mobile ad hoc networks (MANETs). As such, the
plain flooding algorithm provokes a high number of unnecessary
packet rebroadcasts, causing contention, packet collisions and
ultimately wasting precious limited bandwidth. We explore the
phase transition phenomenon observed in percolation theory and
random graphs as a basis for defining probabilistic flooding
algorithms.
By considering ideal and realistic models, we acquire a better
understanding of the factors that determine phase transition, the
consequences of the passage to realistic MANET conditions and
to what extent we may benefit from probabilistic flooding in real
MANET networks.
I. INTRODUCTION
Mobile ad hoc networks (MANETs) are self-organizing mo-
bile wireless networks that do not rely on a preexisting infras-
tructure to communicate. Nodes of such networks have limited
transmission range, and packets may need to traverse multiple
other nodes before reaching their destination. Research in
MANETs was initiated 20 years ago by DARPA for packet
radio projects [13], but has regained popularity nowadays due
to the widespread availability of portable wireless devices such
as cell phones, PDAs and WiFi / Bluetooth enabled laptops.
Because of the ever-changing topology of MANETs, broad-
casting [19] is a fundamental communication primitive, essen-
tial to ad hoc routing algorithms (e.g., [20], [5]) for route
discovery. The usual approach for broadcasting is through
flooding. Flooding is well suited for MANETs as it requires no
topological knowledge. It consists in each node rebroadcasting
a message to its neighbors upon receiving it for the first time.
Although straightforward, flooding is far from optimal and
generates a high number of redundant messages, wasting
valuable limited resources such as bandwidth and energy
supplies. Besides research mentioned in Section II, more effort
has been devoted to defining MAC and routing algorithms
adapted to MANETs, than to flooding. Since flooding is a
The work presented in this paper was supported by the National Compe-
tence Center in Research on Mobile Information and Communication Systems
(NCCR-MICS), a center supported by the Swiss National Science Foundation
under grant 5005-67322.
low-level primitive, optimizing it will drastically improve the
overall performance of MANETs.
One direction to optimize flooding is to take a probabilistic
approach. In order to flood, a node in the network broad-
casts a message with probability p and takes no action with
probability 1
p. In our paper we explore the possibility of
applying a phenomenon well studied in percolation theory and
random graphs, phase transition, as a basis for selecting p.
Above a certain threshold for p, in graphs of a certain size for
random graphs and lattices of a certain density for percolation,
an infinite spanning cluster abruptly appears instead of a set
of finite clusters. An infinite spanning cluster is a unbounded
connected component, which if transposed to a MANET would
translate in the very high probability of the existence of a
multi-hop path between any two nodes within the network.
To the best of our knowledge, besides [12], previous pub-
lications having studied probabilistic broadcast for flooding
in MANETs [6], [16] have not done so within the context
of phase transition. This paper contributes in a first stage to
a better understanding of the various factors that influence
phase transition in ideal MANET environments (no packet
collisions). By opposition to traditional theoretical phase tran-
sition analysis and simulation, we specifically consider factors
that would typically intervene within probabilistic algorithms
deployed on MANETs. In a second stage, we illustrate the
consequences of considering realistic effects such as packet
collisions and node mobility. To the contrary of [12], we
concentrate on pure flooding in order to understand the varia-
tions in performance due solely to the parameters simulating
realistic MANET environments. Our results therefore provide
a general understanding of the behavior to be expected from
probabilistic flooding.
The remainder of the paper is organized as follows. Sec-
tion II gives an overview of other works that seek to reduce the
overhead of flooding in MANETs. In Section III we introduce
the phase transition phenomenon, known results, and how it
may benefit flooding in MANETs. In Section IV we present
two models for which we study the phase transition behavior.
Section V contains simulations and results of our algorithms.
Finally, we conclude and describe future work in Section VI.
II. RELATED WORK
In this section we examine related work which directly or
indirectly aims at reducing the number of broadcast packets
generated by the flooding algorithm.
The high number of redundant broadcast packets due to
flooding in MANETs has been referred to as the Broadcast
Storm Problem in [16]. The paper proposes several schemes,
of which one probabilistic, in order to reduce the number
of broadcast packets while maintaining high reliability. More
recently, [6] provides a comparative study of broadcasting
algorithms, including probability based methods. Given the
scenarios and results in these two papers, it is difficult to make
any statement regarding phase transition (Section III).
As for deterministic studies of the problem, [18] and [15]
make use of local topology knowledge in order to avoid
unnecessary rebroadcasts by comparing the added coverage
between the rebroadcast of a destination node to that of the
source node at each hop. [18] enhances the algorithm by tak-
ing into consideration statistical information about broadcast
duplicates, whereas [15] enhances the algorithm by taking
into account two-hop coverage. [21] restricts flooding to a
subset of nodes (”multipoint relays”) by selecting for each
node a minimum number of one-hop neighbors covering all
second-hop neighbors. [17]
1
and more recently [23] (with
an optimized approach) explore the idea of superimposing
a communications graph — a cluster — over the network
so that only particular nodes rebroadcast the packets. Albeit
reducing the number of rebroadcast packets, constructing and
maintaining the clusters introduce a new source of overhead
in a mobile network.
Other fields such as percolation theory and random graphs
have recently been a source of inspiration for designing
solutions within MANETs. Both are based on a probabilistic
model and exhibit an interesting phenomenon called phase
transition. They will be presented in more detail in Section III.
Phase transition has been applied to reduce traffic for mul-
ticast in wired networks [2], to study optimum power ranges
for connectivity [3], [11] and for enhancing connectivity in
hybrid MANET/Wired networks [7]. Only recently however,
and in parallel with our research, have characteristics from
these fields been applied to reduce flooding in MANETs:
[14] points out that the phase transition phenomenon also
occurs in MANETs and may be taken advantage for the
elaboration of probabilistic algorithms such as flooding
and routing within such networks.
[12] studies a gossip-based approach to flooding. Through
simulations the authors show that for large networks, a
simple gossiping uses up to 35% fewer messages than
flooding, and that the performance of AODV routing [20]
relying on gossip-based flooding is improved even in
small networks of 150 nodes.
This paper is based on the same inspiration as [14] and [12],
yet we obtain different results and gain a better understanding
of the phase transition behavior.
1
The goal of this paper is primarily to provide reliable broadcast delivery.
III. THE PHASE TRANSITION PHENOMENON
A phase transition is a phenomenon where a system un-
dergoes a sudden change of state: small changes of a given
parameter in the system induces a great shift in the system’s
global behavior. This abrupt transition occurs at a specific
value p
c
called the critical point or critical threshold. Below
p
c
the system is said to be in a subcritical phase — the global
behavior is non-existent. Above p
c
the system is in a super-
critical phase and the global property may be almost surely
observed. Figure 1 illustrates the phase transition probability
θ given the probability p of a problem specific parameter λ.
L denotes the size of the system considered.
It would be extremely cost-efficient to observe phase transi-
tion in a probabilistic flooding algorithm within all or known
subsets of MANET topologies. The implication within such
cases would be that there exists a certain probability threshold
p
c
1 at which the flooded message will almost surely reach
all nodes within multihop broadcast reach. Broadcasting with
a probability p
p
c
will not provide any significant improve-
ment. We now present two areas of research where phase
transition applies in order to extract models for MANETS in
Section IV and study their phase transition properties.
A. Percolation Theory
Percolation theory studies the flow of fluid in random
media and has been generally credited as being introduced
in 1957 by Broadbent and Hammersley [4]. Two main two-
dimensional lattice square percolation models are studied, site
percolation and bond percolation. In the bond percolation
model (Figure 2(a)), each edge of the lattice is said to be open
with probability p and closed with probability 1
p. The fluid
flows through the open edges of the lattice. The site percolation
model on the other hand considers the lattice squares or sites to
be the relevant entities (Figure 2(b)): A lattice site is open with
probability p and closed with probability 1
p, and the fluid
flows from open site to open site across the lattice. Figure 2(b)
illustrates an example of site percolation with p
0
55.
Phase transition in percolation models is observed as the
change of state between having a finite number of clusters
and having one infinite cluster. A cluster is a set of connected
entities (edges for bond percolation and sites for site perco-
lation). A cluster that reaches from one side of the lattice to
the other is said to be an infinite cluster. Percolation theory
Fig. 1. Phase Transition
studies the existence and value p
c
for which phase transition
occurs, as well as cluster numbers, sizes and structures.
Percolation theory has numerous interesting applications to
cases which involve some form of propagation or connectivity
such as the spreading of infectious diseases with regard
to population density or the spreading of forest fires. The
question is whether results may also be derived for message
propagation within real-world MANETs.
Great effort within percolation theory has been devoted to
finding the exact value of p
c
at which the phase transition
occurs. Unfortunately, p
c
is not universal but specific to each
lattice geometry. Besides a few distinct cases, there is no
general analytical formula to obtain p
c
, which is usually
computed case by case through Monte Carlo simulations.
B. Random Graphs
Another predominant area of research for phase transition
is Random Graphs. A random graph G is a graph where
the number of nodes, edges and connections between them
are determined in some random manner. The phase transition
property has been well studied in the context of random
graphs. Erd˝os and R´enyi [8] have shown that the probability
of a random graph being connected tends to 1 if the number
of edges E is greater than p
c
E
N logN
2
. Although the
results of Erd˝os and R´enyi are for large values of N, Frank and
Martel have shown by simulation in [9] that phase transition
occurs also in graphs of moderate size (between 30 to 480
vertices). In other words, we may view p
c
E
as a critical
value for the number of edges above which a phase transition
will occur, resulting in a quick convergence for obtaining a
connected graph. As such, we are not able to use random
graphs to represent MANETs: In random graphs, an edge may
connect any two vertices’s in the euclidean plane. In MANETs
however, communication links connect nodes that are within
communication range only. In Section IV-B we describe
the Fixed Radius Model which is an ideal representation of
MANET topologies. It remains a question whether results as
in [8], [9] may be observed in the fixed radius model.
C. Discussion
Phase transition properties depend greatly on the graph
geometry. There is no general theoretical result that enables
(a) Bond Percolation
(open edges are in
bold)
(b) Site Percolation (open
sites are in black)
Fig. 2. Bond and Site Percolation
us to determine the critical threshold p
c
at which the tran-
sition will take place, if at all. Therefore p
c
will have to
somehow be approximated. Furthermore, the few results we
have from percolation theory are for infinite lattices (L
∞).
As we take smaller configurations (L
∞), the transition
from the subcritical to the supercritical state becomes less
abrupt. The dashed tail of the graph in Figure 1 illustrates
that in non infinite lattice configurations, the probability of
percolation taking place becomes linear. We must therefore
determine graph characteristics for which border effects are
not significant.
IV. APPLYING PHASE TRANSITION TO FLOODING
Similarly to wired networks, we may model a MANET by a
graph. Let G
V
E
be an undirected graph. A vertex v
i
V
represents a mobile node, and an edge e
i j
E means that the
nodes i and j are within communication range of each other.
Within this paper we assume that all mobile nodes possess the
same constant transmission range, and do not consider other
properties such as energy levels or consumption.
Given a broadcast source node S, let G
B
be the connected
subgraph of G representing all nodes that will receive the
broadcast message by flooding (S
G
B
) (Figure 3). Since the
message reaches all the nodes in the graph, G
B
may be thought
of as an infinite open cluster as defined in Section III-A.
An efficient probabilistic algorithm will remove edges from
G
B
while still remaining above G
B
’s percolation threshold
p
c
, thus maintaining the infinite open cluster. By remaining
in the supercritical phase, we expect to observe a significant
reduction of message traffic due to flooding while minimizing
the loss of reachability.
We must however ultimately take into consideration that
real-world MANETs differ from mathematical graphs on
several points. The differences that impact phase transition
properties are:
1) Typical real-world MANETs as we see them are not
infinite but may be composed of a few tens to a few
thousand nodes. Border effects may therefore eventually
impact the system’s behavior.
2) Nodes may join or leave the network for various rea-
sons, constantly modifying the network’s density over
time. This directly affects the network’s phase transition
properties.
3) Packet loss: Packets within a MANET are lost due
to packet collisions and contention as well as node
Fig. 3. G
B
Subgraph
mobility. In percolation theory there is no loss of ”fluid”.
We now consider two models as a basis for studying
the above points. The first model is quite simplistic but is
nonetheless useful for extracting best-case results for a specific
MANET topology, as we show that it may be reduced to a
well studied percolation case with known theoretical results.
Through the second and more realistic model any MANET
topology may be represented.
A. Square Grid Model
We define the square grid model as follows. Consider a
m
m square grid with nodes placed at each intersection
as illustrated in Figure 4(a). Each node communicates with
is direct vertical and horizontal neighbors, such that each
node has exactly four neighbors. We broadcast one message
from a single source positioned at the center of the grid.
Using the regular algorithm for flooding in order to achieve
our broadcast, a total of m
2
messages will be transmitted
(Algorithm 1).
Let’s now consider a probabilistic approach. Instead of
systematically rebroadcasting a message upon receiving it for
the first time, we slightly modify Algorithm 1 in order to
rebroadcast the message with a probability p (Algorithm 2).
The exception is the source that broadcasts always (p
1) to
initiate the flooding. With Algorithm 2 and besides the non-
probabilistic broadcast source, our case becomes equivalent to
the site percolation on the plane square lattice as described
in Section III-A. Indeed, since p is constant throughout the
flooding operation, and has the same value at all nodes, it
is like saying that we initially decide to remove links from
the graph with probability p, and then executing a regular
non-probabilistic flooding operation. The threshold value for
percolation in such a case is known to be p
c
0
59 [22]. We
furthermore note that there has been no loss of generality by
assuming that all sites are populated, as flooding with proba-
bility p
f
on a grid of occupation probability p
o
is equivalent to
site percolation on the square lattice of occupation probability
p
f
p
o
. By choosing p
p
c
for Algorithm 2, we expect to
observe an infinite open cluster, translating in our flooding
reaching nearly all nodes in the graph.
Algorithm 1 flood(m)
1: upon reception of message m at node n:
2: if message m received for the first time then
3: broadcast(m)
this is the basic local broadcast primitive
to nodes within range only
4: end if
Algorithm 2 p-flood(m,p)
1: upon reception of message m at node n:
2: if message m received for the first time then
3: broadcast(m) with probability p
this is the basic local
broadcast primitive to nodes within range only
4: end if
(a) Square Grid Model
(b) Fixed Radius Model
Fig. 4. Models
B. Fixed Radius Model
The previous model is useful for reducing a particular
MANET configuration to a well studied percolation model
in order to compare results. Unfortunately, the model only
enables us to consider particular graphs of maximum node
degree 4. A general model adapted to MANETs may be
defined as follows. Let R be the nodes’ communication range.
The nodes are randomly placed on an m
n area according to
a probability distribution such as Poisson. A link l
i j
connecting
nodes i and j is added to the graph if the Euclidean distance
between the nodes is less than R. We have thus obtained
fixed radius random graph as described in [14] and illustrated
in Figure 4(b). We must however note an that probabilistic
flooding in such a model implies that a node may choose
not to broadcast a message to all its neighbors within range
with probability 1
p, resulting in the ”fluid” not flowing
in any of the links attached to the node using percolation
terminology. In random graph models, edges are added or
removed independently.
V. SIMULATION AND RESULTS
Given the two models presented in Section IV, we are
interested in analyzing the phase transition properties of prob-
abilistic flooding as defined in Algorithm 2. Throughout the
cases, we define the success rate SR as the ratio of distinct
packets received at each node by the total number of distinct
packets broadcast in the network, averaged across all nodes.
A. Probabilistic Flooding with Ideal Network Conditions
The motivation behind our first series of simulations is to
obtain best case results. We have written a discrete event
simulator in Java to simulate the Square Grid Model described
in Section IV-A. Mobility is not considered, and the wireless
medium is collision-free. We measure the success rate of
probabilistic flooding for a single packet broadcast at the
center of 3
3, 5
5, 10
10 and 50
50 size square lattices.
The center broadcasts with probability p
1, and we consider
lattices of average node degrees 4 and 8. Figure 5 presents the
results for simulations averaged over 10 and 300 runs.
We conclude from the results in Figure 5 that there are three
factors that affect the phase transition properties in our chosen
scenarios: network size, average node degree, and the number
of simulation runs over which the success rate is averaged.
Network Size: In all four graphs phase transition becomes
apparent as of 100 nodes (10
10 lattices). The success rate
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Success Rate
p
Success Rate - 4 Neighbors - 300 Runs
3x3
5x5
10x10
50x50
(a) 4 Neighbors
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Success Rate
p
Success Rate - 8 Neighbors - 300 Runs
3x3
5x5
10x10
50x50
(b) 8 Neighbors
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Success Rate
p
Success Rate - 4 Neighbors - 10 Runs
3x3
5x5
10x10
50x50
(c) 4 Neighbors
0
0.2
0.4
0.6
0.8
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Success Rate
p
Success Rate - 8 Neighbors - 10 Runs
3x3
5x5
10x10
50x50
(d) 8 Neighbors
Fig. 5. Ideal network (no collisions): success rate for probabilistic flooding in n
n square lattice configurations with no collisions as a function of the
broadcast probability p
graphs tend to become linear as the number of nodes in the
network decreases, due to boundary effects.
Average Node Degree: Higher success rate values are ob-
tained for lower values of p when the average node degree
is of 8 instead of 4. A potentially interesting and exploitable
result is that success rates of over 90% are achieved as ”early”
as p
0
65 for small networks in absence of phase transition
(linear success rate curves).
Number of Simulation Runs: The success rate average
curves become less robust to the number of simulation runs
as the number of nodes considered decreases. We observe
nonetheless in Figure 5(d) that in this case likewise a high
average node degree compensates for a small number of
simulation runs, even for small networks.
The main result of this series of simulations is that for higher
average node degrees, probabilistic flooding may be used to
significantly reduce the amount of broadcast packets even for
small size networks and in absence of phase transition. This
result is obtained in an ideal case of a perfectly symmetrical
topology, no packet collisions and an absence of node mobility.
The question is whether and how is the success rate and
phase transition affected by network conditions of realistic
MANETs.
B. Probabilistic Flooding with Realistic Network Conditions
We now examine node distribution and topology corre-
sponding to the Fixed Radius Model described in Section IV-
B. We have used the ns2 network simulator [1] to simulate var-
ious scenarios for probabilistic flooding. We have considered
small to medium-sized networks of 9, 25 and 100 nodes with
little to no mobility using the 802.11b MAC layer [10] in DCF
mode. Due to the large number of simulations conducted and
ns2’s limited scalability, simulation duration for potentially
significant larger networks would have been prohibitive. Node
transmission ranges of 150 and 250 meters and simulation
areas of 0
25km
2
and 1km
2
were chosen to vary network
density. Note that the 802.11b MAC layer specification uses
CSMA/CA and enforces RTS/CTS/ACK control frames for
unicast communication only. Collision control for broadcast
is limited to basic collision avoidance carrier sensing and
broadcast is therefore extremely prone to packet collisions.
A straightforward tweak to reduce collisions is to have nodes
wait for a random small amount of time before rebroadcasting
(JITTER). We had
N broadcast sources emit a maximum
of one hundred 64 byte packets at constant bit rate with
an interval of 0.05 second, where N is the total number of
nodes in the network. The radio model is ns2’s default, which
simulates Lucent’s WaveLAN wireless card with a 2Mb/sec
bit rate. Simulation duration is of 30 seconds. Figures 6 and 7
present the success rate for the various scenarios while varying
the probabilistic flooding probability p. In order to evaluate the
MANET connectivity, we have displayed the upper bound for
success rate when relevant. This upper bound was obtained
by running the simulations with regular flooding (p
1) over
a collision-free ideal MAC layer. The ideal success rate is of
1
0 in the highly dense networks represented by Figure 7, and
has therefore been omitted from the plots.
