Effect of locality of node mobility on epidemic broadcasting in DTNs

Abstract

In DTNs (Delay/Disruption-Tolerant Networks) composed of mobile nodes, when the node movement has spatial locality, the nodes repeatedly miss opportunities to forward messages to other nodes, thus lowering communication performance. In this paper, we analyze the effect of locality of node mobility on message dissemination speed in epidemic broadcasting. We represent the locality of node mobility using the positional distribution of nodes in a stationary state, and we present a method for deriving the ratio of infected nodes from the positional distribution. Based on the results of a numerical experiment where the positional distribution of nodes obeys a two-dimensional normal distribution, we show that the message dissemination speed is heavily restricted by the locality of node mobility. Moreover, we clarify that a heavy-tailed positional distribution leads to a low locality of node mobility and entails mostly unrestricted message dissemination speed.

Full text

Effect of Locality of Node Mobility on
Epidemic Broadcasting in DTNs
K. Watabe and H. Ohsaki
This is the post-print version of the following article:
•K. Watabe and H. Ohsaki, “Effect of locality of node mobility on
epidemic broadcasting in dtns,” in Proceedings of the 6th Joint IFIP
Wireless and Mobile Networking Conference (WMNC 2013), Dubai,
United Arab Emirates, Apr. 2013. doi:10.1109/WMNC.2013.6549017.
The final publication is available at
http://ieeexplore.ieee.org/document/6549017/ .
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Effect of Locality of Node Mobility
on Epidemic Broadcasting in DTNs
Kohei Watabe Hiroyuki Osaki
Graduate School of Information Science and Technology, Osaka University
Suita-shi, Osaka 565–0871, Japan
Email: {k-watabe, oosaki}@ist.osaka-u.ac.jp
Abstract—In DTNs (Delay/Disruption-Tolerant Networks)
composed of mobile nodes, when the node movement has
spatial locality, the nodes repeatedly miss opportunities to for-
ward messages to other nodes, thus lowering communication
performance. In this paper, we analyze the effect of locality
of node mobility on message dissemination speed in epidemic
broadcasting. We represent the locality of node mobility using
the positional distribution of nodes in a stationary state, and
we present a method for deriving the ratio of infected nodes
from the positional distribution. Based on the results of a
numerical experiment where the positional distribution of
nodes obeys a two-dimensional normal distribution, we show
that the message dissemination speed is heavily restricted
by the locality of node mobility. Moreover, we clarify that
a heavy-tailed positional distribution leads to a low locality
of node mobility and entails mostly unrestricted message
dissemination speed.
I. INTRODUCTION
DTNs (Delay/Disruption-Tolerant Networks), repre-
sented by MANET (Mobile Ad-Hoc NETwork), where
nodes are sparse, have attracted considerable attention.
DTNs realize communication under conditions where con-
tinuous end-to-end connection is not guaranteed, and they
are expected to realize promising applications in networks
in a disaster areas, military networks, inter-planetary net-
works, sensor networks, and so forth. Many algorithms for
achieving efficient communication in DTNs composed of
mobile devices have been proposed , and most of them
compensate for the lack of connectivity by implementing
store-and-carry message forwarding.
Among methods for one-to-all communication in DTNs
composed of mobile nodes, epidemic broadcasting allows
nodes carrying a message (infected nodes) to forward the
message when they enter the communication range of
other nodes. A message spreads among nodes forming
the network since all infected nodes forward messages
repeatedly. Various algorithms for epidemic broadcasting
are proposed in previous works (refer to [1] and references
therein), in which performance metrics such as message
delivery time, coverage and number of duplicate messages
are evaluated.
Generally, node mobility affects the communication
performance of DTNs, where nodes communicate through
epidemic broadcasting or other manner, and previous works
have conducted various evaluations regarding the effects
of node mobility. In [2, 1], message delivery time and
throughput are compared using various mobility models,
including ones imposing constraints on the mobility of
nodes, as in the case of vehicles on a road, in addition to
fundamental mobility models, such as the random waypoint
mobility model, the random direction mobility model and
the random walk mobility model. Moreover, [3, 4] eval-
uates the effect of the distribution of rectilinear moving
distance on message delivery time, buffer utilization and
throughput using a mobility model in which nodes repeat-
edly perform rectilinear motion and randomly change their
direction. The effect of heterogeneity of node mobility on
message delivery time is also explored in [5].
As shown in Fig. 1, when nodes are likely to move
around a specific area in the field, their trajectories do not
intersect, and messages are difficult to spread because the
frequency of contact between nodes is limited. However, in
performance evaluations in the previous works mentioned
above, the areas in which nodes can move are assumed
to be equal, and mobility models are limited to models
where the stationary positional distributions of nodes are
identical. In [2], where the movement of each node is
assumed to be restricted to a grid, the stationary positional
distribution is the same for all nodes since they move under
the same restriction. Moreover, in the mobility models in
[3, 4], nodes are uniformly distributed in the field at a
steady state. When nodes forming a DTN are represented
by humans, node movement is centered at a certain point
in the field since it is assumed that humans tend to move
around the base of their activity (e.g., their home or office).
If we consider this characteristic of human mobility, models
in which node movement distribution in the field is not
restricted do not necessarily reflect all the characteristics
of motion of mobile nodes. To our knowledge, thus far
there has been no research focusing on models with local
positional distributions of nodes for evaluating message
dissemination speed in epidemic broadcasting, although [6]
has derived a scaling law of asymptotic throughput for
MANET. Therefore, it is important to evaluate message
dissemination speed under environments in which node
movement is restricted and nodes are distributed locally
in the field.
In our study, we evaluate the effect of locality of
node mobility on message dissemination speed in epidemic
broadcasting. First, we express the intensity of locality of
node mobility through the shape of the positional distribu-
tion of nodes and clarify the relationship between positional
distribution and frequency of contact between nodes. In
this way, we derive a weighted adjacency matrix whose

Mobile nodeField
Low locality of node mobility

Track does not
intersect each other, !
and there is no
contact opportunities. 
High locality of node mobility
Fig. 1. Intensity of locality of node mobility and contact frequency.
elements are the frequencies of contact between each pair
of nodes, and map the problem of epidemic broadcasting
for mobile nodes to the problem of epidemics on a graph.
In this paper, we compare message dissemination speeds
for various intensities of the locality, where the positional
distribution of nodes follow a two-dimensional normal
distribution, by changing the standard deviation. Moreover,
we examine the effect of the tail behavior of the positional
distribution of nodes on message dissemination speed by
comparing the results obtained with a two-dimensional
normal distribution and a heavy-tailed two-dimensional
Cauchy distribution.
The rest of the paper is organized as follows. First,
in Section II, we present the analysis method used to
evaluate the effect of locality of node mobility on mes-
sage dissemination speed in DTNs employing epidemic
broadcasting. Next, in Section III, we present the results
of numerical experiments in which we compare message
dissemination speeds for different intensities of the locality
of node mobility. We conclude the paper and describe the
direction of future work in Section IV.
II. ANALYSIS
In the analytical model used in this paper, we express
the intensity of locality of node mobility using the shape of
the positional distribution of nodes. We assume that nodes
i(i=1,2,...,N)have stationary motion patterns and
therefore follow a stationary positional distribution. Note
that we assume that the field in which the nodes can move
is a two-dimensional space. The expression of locality of
node mobility using positional distribution allows us to
consider a wide variety of motion patterns. We quantify the
intensity of locality of node mobility as the expectation of
the distance Lfrom the mean/mode point to the current
node position. Larger E[L]indicates a wider range of
motion of nodes and weaker locality of node mobility.
Each node has a communication range r, and a node can
communicate only when it is within the communication
range of other nodes (in other words, only when nodes are
in contact). An infected node forwards a message if it is
in contact with a susceptible node. Note that we assume
that infected nodes forward the message instantly, and we
do not take message size into account.
To derive the message dissemination speed in epidemic
broadcasting, first we derive the frequency of contact
between nodes from the stationary positional distribution of
the nodes. In DTNs, contact frequency strongly affects the
performance of epidemic broadcasting since infected nodes
forward and disseminate messages only when they are in
contact with other nodes. Using the probability density
function (pdf) of the positional distribution of node i, we
can derive the total contact duration Ttotal per unit time
for nodes iand jas follows.
Ttotal =!∞
−∞ !∞
−∞
pi(x, y)!!
Rx,y
pj(z, w)dzdwdxdy
≃r2π!∞
−∞ !∞
−∞
pi(x, y)pj(x, y)dxdy. (1)
where Rx,y denotes the inside area of a circle with a center
(x, y)and a radius r, and the approximate equation holds
when ris sufficiently small. If we assume that contact
duration is independent of contact frequency per unit
time, we can derive the relation between the total contact
duration Ttotal per unit time and the contact frequency wi,j
for nodes iand jfrom the average contact duration Tcd.
wi,j =Ttotal
Tcd
.(2)
Samar et al. have investigated the duration of contact
between mobile nodes [7]. According to [7], we can obtain
the average duration of contact T′
cd(v)between a node with
velocity vand other nodes as follows by assuming that
the velocity of other nodes obeys a uniform distribution
U(0,v
max),
T′
cd(v)= r
2vmax !π
0
log "
"
"
"
"
"
vmax +#v2
max −v2sin2φ
v+vcos φ"
"
"
"
"
"
dφ.
If we assume that valso obeys a uniform distribution
U(0,v
max), we can obtain Tcd as follows.
Tcd =1
vmax !vmax
0
T′
cd(v)dv. (3)
As a result, we can derive the contact frequency for any
arbitrary node pair by substituting Eqs. (1) and (3) into
Eq. (2).
We can map the problem of epidemic broadcasting
for mobile nodes with a stationary positional distribution
pi(x, y)to the problem of epidemics on a graph Gas-
sociated with a weighted adjacency matrix with elements
wi,j since the rate at which an infected node iforwards a
message to susceptible node jis given as the frequency
of contact wi,j between nodes iand j. The weighted
adjacency matrix of graph Gis
A=⎛
⎜
⎜
⎝
0w2,1··· wN,1
w1,20wN,2
.
.
.....
.
.
w1,N w2,N ... 0
⎞
⎟
⎟
⎠
,
where Ndenotes the number of nodes.
If the contact frequency is wfor every node pair, the
following differential equation holds by letting I(t)denotes
the ratio of infected nodes at time t[8].
d
dtI(t)=wI(t)(1 −I(t)).(4)

Extending Eq. (4) by A, we can derive the following
difference equation regarding the probability πi(t)that a
node iis an infected node.
π(t+∆t)=(E−Π)Aπ(t)∆t+π(t),(5)
π(t)=⎛
⎜
⎜
⎝
π1(t)
π2(t)
.
.
.
πN(t)
⎞
⎟
⎟
⎠
,Π=⎛
⎜
⎜
⎝
π1(t)0··· 0
0π2(t) 0
.
.
.....
.
.
00... π
N(t)
⎞
⎟
⎟
⎠
.
The ratio of infected nodes at time tis derived from π(t)
as follows.
I(t)= 1
N
N
*
i=1
πi(t).(6)
Therefore, by solving numerically the difference equation
in Eq. (5), we can obtain the ratio of infected nodes I(t)
at time t, which indicates the message dissemination speed
of epidemic broadcasting.
III. NUMERICAL EXPERIMENT
Though our analysis method can apply to any posi-
tional distribution pi(x, y), we present numerical results in
which the positional distribution follows a two-dimensional
normal distribution as one of the simplest possible cases.
We change the intensity of locality of node mobility by
manipulating the standard deviation of the normal distri-
bution, and compare the resulting message dissemination
speed. We denote the pdf pi(x, y)of node ias
pi(x, y)= 1
√2πσ2e−(x−hx,i)2
2σ2·1
√2πσ2e−(y−hy,i)2
2σ2,(7)
where σand (hx,i,h
y,i)denote the standard deviation and
coordinates of the mean point of the normal distribution, re-
spectively. A smaller standard deviation σleads to stronger
locality of node mobility since the node movements are
confined to a smaller area, and in this case the expectation
E[L]of the distance Lfrom (hx,i,h
y,i)to the current node
position is smaller. When the positional distribution of node
iis given by Eq. (7), we can derive the total duration of
contact Ttotal between nodes iand jas follows.
Ttotal =r2
4σ2e−(hx,i−hx,j )2+(hy,i −hy,j )2
4σ2
We consider the ratio of infected nodes at time t
in a DTN composed of N(= n×n)nodes with the
condition that the mean points (hx,i,h
y,i)line up along
nlines and nrows separated by a distance dwhich
form a two-dimensional lattice as shown in Fig. 2. We
denote the node infected at time 0as node 0, and thereby
π(0) = (1,0,...,0).
We change the intensity of locality of node mobility
by setting the standard deviation σto 100 [m], 200 [m],
400 [m], 800 [m], 1600 [m], 3200 [m] and 6400 [m], and
calculate the time until 50% of the nodes have become
infected (50% delivery time) using Eqs. (5) and (6). We
show the results in Fig. 3. The remaining parameters are
n= 10,d= 400 [m], r= 50 [m] and vmax = 8000 [m/h].
Figure 3 presents the results of a Monte Carlo simulation
(hx,1, hy,1)(hx,2, hy, 2)(hx,n, hy, n)
(hx,n+1, hy,n+1)(hx,n+2, hy,n+2)
(hx,n(n-1)+1, hy,n(n-1)+1)(hx,nn, hy, nn )
d
d
nd
nd
(hx,n(n-1)+2, hy,n(n-1)+2)
(hx,2n, hy,2n )
Fig. 2. Location of the mean points (hx,i,h
y,i)of positional distribu-
tions of nodes
in which an infected node forwards a message randomly in
accordance with the rate of link weights on a graph with a
weighted adjacency matrix A. According to the figure, we
can confirm that the 50% delivery time is extremely long
in the case of σ= 100 [m]. The main reason for this is that
infected nodes not likely to move to areas in which other
nodes are distributed since the locality of node mobility is
too strong. Similarly, the 50% delivery time is also long
when σis larger than 1600 [m]. The reason for this is the
extremely weak locality of node mobility, as a result of
which each node covers a rather wide area, and thereby
many opportunities for establishing contact are lost.
Moreover, to investigate the effect of a heavy-tailed
positional distribution of nodes on message dissemina-
tion speed, we compare the results for a two-dimensional
Cauchy distribution with those for a two-dimensional
normal distribution. The tail behavior of the positional
distribution reflecting the motion patterns of nodes strongly
affects the expectation E[L]of the distance Lfrom a
mean/mode point to the current node position. The Cauchy
distribution is well known as a heavy-tailed distribution [9],
although its pdf is symmetric and bell-shaped, similarly to
the pdf of a normal distribution. Similarly to case of normal
distribution, we give the positional distribution pi(x, y)of
node ias follows.
pi(x, y)= γ
π((x−hx,i)2+γ2)
γ
π((y−hy,i)2+γ2).(8)
Note that γand (hx,i,h
y,i)denote the scale parameter and
the coordinates of the mode point, respectively. When the
positional distribution of node iis given by Eq. (8), we
derive the total duration of contact Ttotal per unit time
between nodes iand jas follows.
Ttotal =4r2γ2
π((hx,i −hx,j )2+4γ2)((hy ,i −hy,j )2+4γ2).
Figure. 4 shows a comparison of the 50% delivery
time for a Cauchy distribution (calculated by numerically
solving the difference equation in Eq. (5)) and that for
a Monte Carlo simulation. To perform direct compari-
son with the results for a normal distribution, we adjust
the scale parameter γsuch that xcand xn, satisfying
Fc(xc)=0.9and Fn(xn)=0.9, take the same value,
where Fc(x)and Fn(x)denote the cumulative distribution
functions of Cauchy and normal distributions. Therefore,
the area in which a node is distributed with probability 0.8
(in other words, excluding the portions of the two tails of
the distribution corresponding to a probability of 0.1each)
is the same for both distributions. Note that the horizontal

0!
20!
40!
60!
80!
100!
100!1000!10000!
50% delivery time
Standard deviation of position distribution of nodes
Results with Monte Carlo simulation!
Solution of difference equation!
Fig. 3. 50% delivery times for different two-dimensional normal distribu-
tions used for the positional distributions of nodes.
0!
10!
20!
30!
40!
50!
60!
100!1000!10000!
50% delivery time
Standard deviation of corrresponding normal distribution
Results with Monte Carlo simulation!
Solution of difference equation!
Fig. 4. 50% delivery times when the positional distributions of nodes follow
a two-dimensional Cauchy distribution.
axis represents the standard deviation of the corresponding
normal distribution, and other parameters are the same as
in the example of a normal distribution. By comparing
Fig. 3 with Fig. 4, we can confirm that although the 50%
delivery time is long in the results of normal distribution
when σ= 100 [m], the corresponding result for a Cauchy
distribution is nearly the same as the results for Cauchy
distribution that are corresponding to σ= 200 [m] and
σ= 400 [m]. The main reason for this result is that
compared with case of a normal distribution, in the case
of a Cauchy distribution there is a high probability that a
node contacts another node located far from it since the
latter distribution has a heavy tail and the locality of node
mobility is extremely weak.
The results for the examples mentioned above provide
helpful suggestions for designing DTNs. In our result when
σ= 100, heavy-tail of positional distribution reduces
the 50% delivery time to about 1/10. For a heavy-tailed
positional distribution of mobile nodes, we can expect that
messages can be disseminated within a reasonable period of
time, even if the density of nodes is very low and the nodes
are located far apart. For instance, regarding real-world
human movement patterns, the distribution of the distances
to destinations is heavy-tailed [4], which suggests that the
positional distribution of nodes is heavy-tailed. Moreover,
in scenarios where we can control the motion patterns
of nodes (e.g., in sensor networks), fast delivery can be
achieved by designing heavy-tailed positional distributions.
IV. CONCLUSION AND FUTURE WORK
We evaluated the effect of locality of node mobility
on message dissemination speed in epidemic broadcasting.
We derived the contact frequency between node pairs
from the stationary positional distributions of nodes, and
presented a method for deriving the ratio of infected nodes.
Through numerical experiments, we investigated the case
in which the positional distributions of nodes follow a
two-dimensional normal distribution, and showed that the
message dissemination speed is considerably lower when
the standard deviation of the normal distribution is small
and the locality of node mobility is strong. Moreover,
we compared the results for a two-dimensional normal
distribution and those for a heavy-tailed two-dimensional
Cauchy distribution, and showed that message dissemi-
nation speed is almost unrestricted when the stationary
positional distribution of nodes is heavy-tailed (entailing
weak locality of node mobility). These results suggest
that when the motion patterns of nodes follow a heavy-
tailed positional distribution, messages can be disseminated
within a reasonable period of time even if the density of
nodes is low and they are located far from each other.
Although in this paper we focused on a method for
deriving the ratio of infected nodes, a weighted adjacency
matrix Awhose elements are the frequencies of contact
between nodes allows us to conduct various analyses. Epi-
demics on a graph have been studied in detail in a number
of previous works. For instance, an analysis of epidemic
thresholds has been conducted using the eigenvalues of an
adjacency matrix . In future work, we plan to utilize these
results to analyze message dissemination in the case of
epidemic broadcasting in DTNs composed of mobile nodes.
ACKNOWLEDGMENT
This work was supported by JSPS Grant-in-Aid for
JSPS Fellows Grant Number 24 ·3184.
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