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Modeling Opportunistic Social Networks with Decayed ... · Modeling Opportunistic Social Networks...
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Modeling Opportunistic Social Networks with Decayed
Aggregation Graph
Peiyan Yuan1, 2
and Yali Wang1
1School of Computer and Information Engineering, Henan Normal University, Xinxiang, China
2Engineering Laboratory of Intellectual Business and Internet of Things Technologies, Henan Province
Email: [email protected], [email protected]
Abstract—The emerging mobile applications including
contact-based forwarding and online worm containment have
put a heavy burden on opportunistic social networks (OSNs),
the first goal is to develop an effective model that can reveal the
hidden, important social features underlying the OSNs. This
problem is especially challenging because of the time-varying
network topology. Traditional time expanded graph caches each
snapshot of networks, resulting in low computation efficiency
and high storage overhead. By aggregating past contact events
between two nodes to a simple boolean indicator, the binary
graph model can alleviate this issue. However, it only provides
a coarse-grained level of identifying the relationship between
nodes. It neglects the differences in contact events. Intuitively,
recent contact events are generally more important than old
ones, and in computing an aggregation, we should assign bigger
weights to them. In addition, since each contact has its own
duration, we should take this factor into account as well.
Motivated by these observations, we propose DAG, a decayed
aggregation graph for modeling OSNs at a fine-grained level.
By implementing DAG in different real scenarios, we show that
DAG is efficient in characterizing the relationship between
nodes and in improving the performance of mobile applications.
We simultaneously prove that DAG achieves approximate space
complexity as the binary graph.
Index Terms—Opportunistic social networks, decayed
aggregation graph, worm diffusion, data forwarding
I. INTRODUCTION
The pervasive deployment of portable devices (such as
smart phones) has resulted in a demand for opportunistic
social networks to support emerging mobile applications
such as contact-based forwarding and online worm
containment [1]. However, the success of these mobile
applications is restrained by the lack of effective model
that can reveal the hidden, important social relationships
underlying the OSNs [2]. Indeed, detecting the interesting
social features is of considerable advantage in mobile
applications. Consider the contact-based forwarding
problem in OSNs [3]. Due to the intermittently connected
links between nodes, an apparent challenge for any
Manuscript received January 1, 2015; revised March 27, 2015. This work was supported by the National Natural Science
Foundation of China under Grant No. U1404602, the Science and
Technology Foundation of Henan Educational Committee under Grant No.14A520031 and the Dr. Startup Project of Henan Normal University
under Grant No. qd14136. Corresponding author email: [email protected].
forwarding algorithm is to deliver the packet as quickly
as possible, without introducing many replicas of packets.
Since people have different social relationships (e.g.,
friend, acquaintance or stranger [4], there are also
different similarity metrics in the underlying network as a
reflection. A desired forwarding method is therefore to
select relays having high similarity scores with the
destination, instead of purely forwarding replicas to each
encountered node.
Another great advantage of exploiting social features
can be found in the online worm containment [5]-[6].
Suppose that a popular mobile application is infected
with malicious codes, the first thing is to prevent worms
from spreading massively by immunizing central nodes.
This example, again, highlights the big role of social
features in mobile applications. Unfortunately, it is very
challenging to identify them in OSNs, this is mainly
because of the time-varying network topology.
An ideal solution to the above problems would try to
duplicate each snapshot of the network as the traditional
time expanded graph [7]-[8] or its variant [9] did.
Nonetheless, this method suffers from some major
disadvantages: a) the huge storage overhead due to the
replication of the network, and b) the low computation
efficiency because of the increased problem size. To this
end, the authors of [10]-[13] tend to aggregate node
contacts by using a binary graph, where a link forms if
two nodes have at least a contact in the past time window
W. The binary graph achieves a lightweight overhead, but
it can only answer simple queries such as “Did Alice
meet up Bob in the past W ?”. In other words, it identifies
node relationship at a coarse-grained level, in the form of
aggregating past contact events between two nodes to a
boolean indicator. It obviously neglects the differences in
contact events. Intuitively, recent contact events are
generally more important than old ones, and in computing
an aggregate, we should assign bigger weights to them. In
addition, since each contact event has its own duration,
we should consider this factor as well.
Motivated by the observations, we propose DAG, a
decayed aggregation graph to model OSNs at a fine-
grained level. For any two nodes, we take the duration of
each contact and its decayed age into account, and
formulate their relationship strength as a decayed sum
problem (a decayed age a is equal to iT e , where T is the
current time and ei is the end moment of the ith contact,
respectively). We summarize our contributions as follows:
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doi:10.12720/jcm.10.3.213-220
©2015 Journal of Communications
We propose DAG, a decayed aggregation graph for
not only tracing the relationship between nodes over
time but also revealing the social features underlying
the OSNs at a fine-grained level.
We prove that DAG achieves approximate space
complexity compared to the binary graph.
We validate DAG on different real opportunistic
scenarios. The experimental results show that DAG is
effective in characterizing the relationship between
nodes and in improving the performance of mobile
applications.
The remainder of this paper is organized as follows.
Section II briefly reviews the related work. Section III
presents the motivation. Section IV gives our solution.
We have a comparative analysis with binary graph in
Section V, followed by a short conclusion in Section VI.
(a) GTW (t=3000s) (b) STW (t=3000s, W=3000s)
(c) DAG (t=3000s, λ=1) (d) GTW (t=6000s)
(e) STW (t=6000s, W=3000s) (f) DAG (t=6000s, λ=1)
Fig. 1. Adjacency matrixes of three models at different time instances. Black → 0 and white → 1. We here take the data-set KAIST as a sample. For more information of the real data-sets, please refer to the section V.
II. RELATED WORK
Modeling the time-varying network topology has
attracted much attention since its introduction. Based on
the underlying principles, we generally classify the
existing models in two categories including time
expanded graph and the binary graph.
Time expanded graph [7]-[9] can accurately identify
the social relationship between nodes by caching each
snapshot of the original network. New edges connecting a
node and its copy are added at the next snapshot in
addition to the edges having been existed in the last
snapshot. Therefore, the network size increases quickly
with the elapsed time. Furthermore, the method suffers
from the low computation efficiency as well, because of
the increased problem size. To this end, recent works
[10]-[13] tend to aggregate node contacts by using a time
window. For example, the authors of [10] employed
growing time window to compute the social features over
a social graph, where an edge emerges if there exists at
least one contact between two nodes at any time in the
past. A sliding time window is used in [11] and [12],
where time is split into a sequence of time frames with
same size W and only contacts in the last time frame form
edges of graph. In addition, metrics such as most recent
contacts and most frequent contacts were taken in [13].
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Compared to the time expanded graph, the proposed
binary graph indeed alleviates the storage overhead.
However, this method neglects the duration of each
contact and its decayed age. It only characterizes the
time-varying network at a coarse-grained level. Detailed
comparison between DAG and the binary graph is
presented in section IV.
III. MOTIVATION
We take the binary graph with growing time window
(GTW) and that with sliding time window (STW) as two
examples to analyze the possible problem caused by the
traditional solutions. Let N denote number of nodes in the
network. Let uv N NR r t
and uv N N
S s t
denote the
adjacency matrixes of GTW and STW, respectively. We
have 1uvr if node u and node v have at least a contact at
any time in the past, otherwise 0uvr , and 1uvs if they
have at least a contact in the past time window W,
otherwise 0uvs .
Apparently, the matrixes will get more and more
identical with the contacts aggregation as shown in Fig. 1
(a), Fig. 1 (d), Fig. 1 (b) and Fig. 1 (e). As a consequence,
heterogeneity of nodes cannot be well reflected, which in
turn degenerates the networking performance (please
refer to the section V). Hence, we conclude that the
binary graph only characterizes the OSNs at a coarse
level. On the contrary, if we model OSNs by a decayed
aggregation graph (please refer to the next section), we
achieve a fine-grained level of characterizing relationship
between nodes (Fig. 1(c) and Fig. 1 (f)).
IV. DECAYED GRAPH
A. DAG
We model an opportunistic social network as a
Decayed Aggregation Graph (DAG) G=(V, E), where V
denotes set of nodes in the network and E denotes set of
edges [14]. Let uvh t denote the relationship strength
between nodes u and v and ( ) uv N NH t h
denote G’s
adjacency matrix at moment t. We discuss how to
compute uvh t by aggregating their past contact events
as follows.
Fig. 2. Decayed aggregation graph and its components.
Let n denote the number of contact events between two
nodes (in general, n N ). We consider the contact series
1 2, ,..., nC C C C . Each element iC is a tuple ,i is e as
shown in the upper subfigure of Fig. 2, where ,i is e and
id denotes the start time, end time and duration of the ith
contact, respectively. As discussed above, the aggregate
will be time-decayed. That is, older contact events will be
decayed faster than more recent ones. The exact model of
decay is in general specified by a user through a decayed
function [15] [16].
Definition 1(Decayed Function): We call a function
0f x x , a decayed function, if it satisfies (a)
0 ( ) 1f x and (b) if 1 2x x , then 1 2f x f x .
At current time T, the decay of ei is defined as iT e .
At the same time, the intensity of each contact event
should be considered as well, since each contact has
different durations. The differentiated intensity is denoted
by a weighted function.
Definition 2 (Weighted Function): A function
0w x x is called a weighted function, if it holds (a)
0w x and (b) if 1 2x x , then 1 2w x w x .
For each contact iC , its weight is defined as
i ie s .
Problem 1(Decayed Sum Problem): Given the contact
series C, functions f x and w x , the first goal is to
estimate the decayed sum at current time T.
1
n
i ii
uvh T w x f x
(1)
where xi denotes the ith contact, i i iw x e s and
i if x f T e .
Since the inter-contact time between nodes always
follows an exponential decay [17], we further set
iT e
if T e e
. Hence, the Eq. (1) can be
reformulated as
( )
1
i
nT e
uv i i
i
h T e s e
(2)
The bottom subfigure of Fig. 2 shows the decayed
aggregation graph and its components, where the left
subfigure presents the weight function i iw e s and
decayed function if t e (the shadow part, ie t T )
of the ith contact, and the right part denotes the value of
uvh t . We next analyze the space complexity of DAG.
B. Space Complexity of DAG
Obviously, exact tracking of uvh t needs ( )n
storage bits. Considering scalability issue, we should
further reduce the storage overhead while keeping the
same calculation precision.
Let i ig t e s if t equals to ei, otherwise, 0g t ,
we obtain the following lemma 1.
Lemma 1: In a continuous interval [0,T], the Eq. (2) is
equivalent to the following Eq. (3)
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GGREGATIONA
©2015 Journal of Communications
( )T t
uv t Th T g t e
(3)
Proof: Let us split the interval [0, T] into two disjoined
parts D1 and D2, where `
1
( [ , ])n
i i i i
i
D d d s e
and 1 2 [0, ]D D T .
We have
1 2
( ) ( ) ( )( ) ( ) ( )T t T t T t
t T t D t D
g t e g t e g t e
For any 2t D , since
it e (note that 1ie D and
1 2D D ), we have g(t)=0. Hence,
1
( ) ( )
( )
1
( )
1
( ) ( )
( )
( )
= ( )
i
i
T t T t
t T t D
nT t
i t d
nT e
i ii
uv
g t e g t e
g t e
e sd e
h T
Theorem 1: At each time slot 0,1,2,...,s T , the value
of ( )uvh T can be maintained easily using
( ) ( ) ( 1)uv uvh T g T e h T (4)
Proof: From the Lemma 1, we know
( )
( ) ( )
1
( 1 1)
1
( 1 )
1
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( 1)
T s
uv
s T
T T T s
s T
T s
s T
T s
s T
uv
h T g s e
g T e g s e
g T g s e
g T e g s e
g T e h T
Theorem 2: The space complexity of DAG is 2(2 )O N .
Proof: From the Theorem 1, we know that each node
only requires carrying one counter to exactly track the
relationship between it and any other nodes. Since there
are total N nodes, we need ( ( 1))O N N storage bits, plus
the 2( )O N storage bits required by the adjacency
matrix H , we get the conclusion.
TABLE I: SPACE COMPLEXITY OF TEG, GTW, STW AND DAG
Four models TEG[8] GTW STW DAG
Space
complexity 2( )O nN 2( )O N 2(2 )O N 2(2 )O N
C. Discussion of the Space Complexity
Table I lists the space complexity of time expanded
graph (TEG), GTW, STW and DAG. We can see that
DAG has the same space complexity as that of STW, and
both of them have a slightly higher space complexity than
GTW, as the latter does not need to store additional
contact information. Note that for DAG, we can achieve
the approximate space complexity even we exploit other
popular decayed functions such as polynomial decay and
poly-exponential decay. Please refer to [15] for more
information about the impact of decayed function on
storage overhead.
By taking two important social features, centrality and
similarity of nodes as samples, we next observe which of
the DAG, GTW and STW can best extract such
knowledge from the underlying OSNs. In addition, we
also evaluate the impact of similarity metrics revealed by
the three models on opportunistic forwarding, and that of
centrality on worm diffusion speed, two interesting
mobile applications.
V. PERFORMANCE EVALUATION
A. Data-Sets
We use two real data-sets, referred to as North
Carolina State Fair and KAIST. Both of them have been
applied into different scenarios (e.g., message deletion
mechanism in [18] and the localization of mobile
networks in [19]).
In KAIST, 34 students who live in a campus dormitory
carried the GPS devices (GPS 60CSx) from 2006-09-26
to 2007-10-03 and altogether 92 daily traces were
gathered. In Statefair, 19 traces were gathered from 18
volunteers who visited a local state fair that includes
many street arcades, small street food stands and
showcases. Each participant in this site spent less than
three hours. More than one thousand people daily were
attracted for two weeks due to the popular event. Besides,
the site is completely outdoor. The detailed statistics of
the two data-sets can be found in [20].
B. Evaluating Centrality
We first explore the impact of the three models on
evaluating centrality metric. Node centrality reflects the
relative importance of a node in the network. There exist
a lot of methods to measure the centrality. In this paper,
we employ the betweenness measure [21], one of the
most popular methods [22], to evaluate centrality.
Betweenness centrality reflects the controlling capability
of a node to other nodes, which measures the extent to
which a node falls on the shortest paths between two
other nodes. The higher the betweenness centrality of a
node is, the bigger the ability it has to facilitate
communication to other nodes within the network is.
Betweenness centrality of a node i is computed as:
1 1
( )N Nuv
ij k uv
l iB
l
(5)
where luv is the total number of shortest path between
nodes u and v, and luv(i) is the number of those paths that
include node i .
Currently, the exact calculation of Eq. (5) is to flood a
large number of messages in the network, and count the
number of times a node acts as a relay for other nodes on
all the shortest paths. We call this method Flooding and
take it as a benchmark. We next approximately estimate
Bi by simulating the message propagation process [23].
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For a concise presentation, let uvx denote the contact
relationship between nodes u and v measured by the
above three models (i.e., uv uvx h in DAG, uv uvx r in
GTW or uv uvx s in STW). For any randomly generated
message m with source node s and destination node d,
when node s encounters node v, node v is selected as a
relay with a normalized probability
1
( )
( )v N
i
vP
i
(6)
where the term ( )i denote the degree of node i.
(a) Flooding versus DAG
(b) Flooding versus GTW
(c) Flooding versus STW
Fig. 3. Fast Fourier Transform (FFT) of centrality distribution at KAIST.
Node v takes the same infection strategy if it is
selected. The above process is repeated until the message
m reaches the destination node d. Thus, a path from s to d
is collected. Using this method, we can observe which
node plays a big role in the messages propagation process
by computing its appearance frequency in the shortest
paths we collect.
We present our results in Fig. 3 and Fig. 4, where the
centrality distributions of nodes are presented by the Fast
Fourier Transform (FFT). We can see that DAG achieves
the most similar distribution pattern with the flooding
pattern.
Table II and Table III further list the similarity scores
between Flooding measure and one of the three measures
under different similarity functions: Person coefficient, L2
norm and cosine angle. It is clear to see that DAG shows
the best performance compared to GTW and STW, for
example, at KAIST, it improves the accuracy of two
models (GTW and STW) for estimating the centrality of
nodes by 60% and 40% at KAIST, and even by 60% and
100% at Statefair with the L2 norm. The reason behind
this is that it exploits two factors (decayed and weighted)
to differentiate the contact events when modeling the
relationship between two nodes, instead of purely
recording whether there happens a contact event between
them or not.
(a) Flooding versus DAG
(b) Flooding versus GTW
(c) Flooding versus STW
Fig. 4. Fast Fourier Transform (FFT) of centrality distribution at Statefair.
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TABLE II: SIMILARITY BETWEEN FLOODING AND ONE OF THE THREE
MODELS
(at KAIST) Person L2 Cosine
Flooding vs DAG 0.8664 1.0390 0.9639
Flooding vs GTW 0.5810 1.6633 0.8653
Flooding vs STW 0.8066 1.4797 0.9486
TABLE III: SIMILARITY BETWEEN FLOODING AND ONE OF THE THREE
MODELS
(at Statefair) Person L2 Cosine
Flooding vs DAG 0.8606 0.4543 0.9737
Flooding vs GTW 0.5263 0.7238 0.9324
Flooding vs STW 0.3250 0.9048 0.8984
We now analyze the impact of centrality metric on
worm diffusion speed. To do so, we first calculate the
worm spreading time for each trace including all nodes.
We then repeat the same experiment by removing the
central nodes. We define a central node as a node that
belongs to the top 10% nodes with the highest importance
in the network, the remaining nodes are called non-
central nodes (similar selection/definition has been used
in [24]). Table IV presents the results. The first
phenomenon is that the central nodes play a big role in
worm spreading, removing the central nodes obviously
slows down the diffusion process. The second and
expected phenomenon is that the central nodes selected
from DAG model are responsible for most of the worm
spreading in opportunistic networks. For example,
removing them increases the diffusion time almost by
70% at Statefair, whereas, removing the nodes evaluated
with the other two models only increase the diffusion
time by 27% and 32%, respectively. These results verify
the effectiveness of DAG model.
TABLE IV: WORM DIFFUSION SPEED BY REMOVING THE CENTRAL
NODES EVALUATED WITH THE THREE MODELS
(at) Including
ALL DAG GTW STW
Statefair (s) 130.0 220.0 165.0 172.0
KAIST (s) 311.0 433.0 331.0 383.0
C. Evaluating Similarity
We then explore the impact of the three models (DAG,
GTW and STW) on evaluating similarity between nodes.
Similarity reflects the associations between nodes in the
network. Sociologists have observed the phenomenon
long before, which is called “clustering” in physics, that
if two people have one or more common friends, they can
also be friends with high probability. In general, we use
the min-max function to measure the number of common
neighbors between two nodes.
Let uvSim denote their similarity score, we have
1
min( , )
max( , )
Nui vi
uv
i ui vi
x xSim
x x
(7)
By using the similarity metric, we present another
mobile application where the similarity between nodes
has a heavy influence on the forwarding algorithms in
opportunistic networks [25]-[27]. We employ this
application to observe which of three models best
characterizes the social relationship between nodes, and
at the same time, we also evaluate the impact of similarity
metrics revealed by the three models on the performance
of opportunistic forwarding.
During the forwarding process, if a node u carrying
message m meets up node v, node u forwards the message
to node v and deletes the message from its buffer (i.e., a
single-copy forwarding strategy) if ud vdSim Sim ,where
d denotes the destination of m . The intuition behind this
forwarding strategy is that nodes having high similarity
scores with the destination will deliver the message more
quickly than those with low metrics. We test this intuition
by utilizing the aforementioned two real data-sets. For
each data-set, total 1000 messages are generated and the
emulation results are averaged for consistency.
(a) Statefair
(b) KAIST
Fig. 5. Packet delivery ratio (PDR).
Fig. 5 and Table V show the results on packet delivery
ratio (PDR) and mean delivery delay, two important
performance criteria for opportunistic social networks.
The criterion of packet delivery ratio represents the
delivery reliability in the network in terms of the number
of successfully received messages over that the total sent
messages, and the criterion of mean delivery delay
denotes the delivery efficiency, that is, how much time
we should spend in delivering a message. We can see
that DAG considerably quickens the delivery speed.
Compared to GTW and STW, it improves the delivery
delay by 65% and 52% at Statefair, and by 75% and 28%
at KAIST, respectively, while achieving better results on
packet delivery ratio as well.
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TABLE V: MEAN DELIVERY DELAY UNDER THE THREE MODELS
(at) DAG GTW STW
Statefair (s) 565.0 934.0 856.0
KAIST (S) 724.0 1266.0 931.0
VI. CONCLUSION AND FUTURE WORK
In this paper, we propose DAG, a decayed aggregation
graph, to model opportunistic social network. Using this
model, we can not only trace the relationship between
nodes over time but also reveal the social features
underlying the OSNs at a fine-grained level. We also
prove that DAG has the same space complexity as the
binary graph with sliding time window. We validate
DAG on two real opportunistic scenarios. The experiment
results show that DAG is effective in characterizing the
relationship between nodes and in improving the
performance of mobile applications. The significant
topics for future work include validating DAG with other
important social features such as the overlapping
community structure, and implementing it into other
crucial applications including recommendation system etc.
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Peiyan Yuan is an associate professor of
Computer Science at the Henan Normal
University since 2014, where he received his
B.S. degree at Computer Science in 2001.
After that, he got his M.S. (2007) and Ph.D.
(2014) degree at Computer Science from
Wuhan University of Technology and Beijing
University of Posts and Telecommunications,
respectively. His research interests include
future networks and distributed systems ranging from content-centric
networks, mobile opportunistic networks to online social networks and
crowd sensing applications etc. He is a member of the CCF and ACM.
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Journal of Communications Vol. 10, No. 3, March 2015
©2015 Journal of Communications
Yali Wang received her B.S. degree in
computer science from Henan Normal
University, China, and M.S. degree from
Soochow University, China. She is currently a
Ph.D. candidate in computer science and
technology at the State Key Laboratory of
Networking and Switching Technology,
BUPT. Her research interests are in the areas
of Network architecture and Network services.
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Journal of Communications Vol. 10, No. 3, March 2015
©2015 Journal of Communications