1. Annual Conference2014 - IET-Sri Lanka
MODELING SCOCIAL NETWORKS IN MOBILE
TELECOMMUNICATIONS NETWORKS VIA
INFLUENCE DIFFUSION
Asoka Korale, Ph.D., MBA, CEng., MIET
Abstract: The identification of leaders and
communities of followers in Social Networks is
vital for implementing innovative business
strategies and marketing campaigns that can be
put in to effect by exploiting the hierarchical
relationships between individuals and their
consumption preferences.
This paper presents a novel approach for
analyzing the Social Network that arises as a
result of the hierarchical nature of communication
that occurs between the members of a particular
segment of the subscribers of a Mobile
Telecommunications Network. We identify leaders
and communities comprised of followers by
processing the way in which calls are initiated and
terminated between the subscribers. The call
pattern is translated in to a call graph using the
inherent properties of the interactions between
individual subscribers of the mobile network and
appropriately represented in an Adjacency matrix.
A community detection algorithm [2] is then
employed to identify those individuals with
significant influence in the network. The leaders
so identified are also validated using existing
notions about what constitutes a leader of a
subscriber segment in the context of a Mobile
Telecommunication Network.
Results from subscribers belonging to a Corporate
Telecommunications Network so analyzed are
presented identifying the leaders and the
communities formed around them. The
connectivity properties of the identified leaders are
then evaluated using existing methods to validate
the results of the new approach. A significant
innovation of this approach is the estimation of the
degree of membership of a given individual in a
particular community made possible by employing
this influence diffusion technique [2]. This leads
to the discovery of overlapping communities
populated by individuals who may display different
degrees of membership in more than one
community at a given time.
Business strategies a mobile network operator may
deploy by exploiting the underlying social network
discovered by this modeling to enhance revenues
through product promotions, advertisement
diffusion, churn reduction, and recommender
systems are also presented.
1. INTRODUCTION
It is often the case that networks exhibit hierarchical
properties and the analysis of such networks to
identify those nodes which lie at the root of the
hierarchy is important for many scientific and
business applications. It also the case that many of
these networks exhibit modularity properties where
the network may display natural partitions or
clustering of nodes which can be described and
understood as a process of community formation.
Thus under these conditions it would be possible to
partition the network so as to isolate the different
communities with the aim of identifying groups of

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nodes that belong to each of the community. This
partitioning is usually done in a way that the there is
a high density of links between members of a
particular community while the links between
communities are minimized.
Graph partitioning [6] and Spectral partitioning [7,
8] are the main techniques currently used for
dividing networks in to clusters or groups. Graph
partitioning algorithms however require as input the
number of partitions that need to be created and
thus do not automatically determine the optimum
way of segmenting a network.
Thus the partitions so estimated have analogies to
communities in social networks and the
communities are usually headed by a leaders that
are often the nodes near the root of the hierarchy. In
this way one is able to identify communities made
up of followers and leaders who have high
importance or influence.
In this paper we examine several methods for
modeling the Social Networks arising in Mobile
Telecommunications Networks by modeling the
mobile phone call pattern between subscribers. We
present results for a novel method for the analysis
of such Social Networks through the use an
algorithm [2] that displays superior performance to
the existing techniques employing modularity
function [3, 4], graph and spectral partitioning. This
algorithm performs automatic community detection
and does not suffer from the drawbacks of the
modularity approach related to resolution limit
which may give results that may not be in line with
the most intuitive partition of the network [5].
The algorithm also presents a method for estimating
the degree to which a node belongs to a particular
community by a membership vector with elements
associated with each detected community. In this
way it is possible to identify not only leaders of a
particular community but also their respective
followers who comprise the members of the
community. A novel feature is then the detection of
nodes that belong to more than one community to a
certain degree at a given time as expressed by the
membership function. Thus we are also able to
identify nodes or individuals that act as bridges
between communities.
2. SOCIAL NETWORK ANALYSIS
Social Network Analysis (SNA) is derived from the
study of the interactions between individuals and
groups and has its origins in the science of
Sociology, Sociometry and Psychology. In the
graphical representation of such a social network
the nodes are referred to as actors and the edges
which represent the interactions between the nodes
as ties or links.
While there is not much ambiguity in the definition
of the nodes in social networks the definition of the
edges however depends on the particular type of
problem being studied. As such the physical
meanings of the edges can be taken to signify
friendships between individuals, professional
relationships as found within certain organizations
for instance, communication patterns between
individuals in certain situations and even the
exchange of goods/money.
Other representations of social networks take the
form of “Affiliation Networks” (also referred to as
Bipartite graphs) where two types of vertices are
present with the links between the vertices
indicating membership of an individual in a
particular event / group.
An ego centric network on the other hand is the
study of the network surrounding a particular
individual, and thus does not encompass the entire
social network or depict the connections between all
actors. This is suitable when the network is very
large comprising of a vast number of nodes and the
aim is to picture the immediate neighborhood of the
ego or the individual under study and those around
it known in the literature as alters.
3. MODELING THE SOCIAL NETWORK
ARISING BETWEEN SUBSCRIBERS OF A
MOBILE TELECOMMUNICATION
NETWORK
The interactions between subscribers in a Mobile
Telecommunications Networks can be modeled as a
social network by utilizing the calling patterns
between the individual subscribers to construct a
“call graph” or a network of connections between
subscribers. The nodes in the graph are then the
subscribers or MSISDNs (Mobile Numbers
representing the subscribers) and the links are
indicative of calls between the nodes.
We believe that in such a scheme an edge can be
established between two nodes if there has been a
call between the respective MSISDNs. Often it is
the case that an edge is established between two
nodes only if there has been a reciprocal call (i.e.

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both MSISDNs have called one another). This is to
eliminate spurious links that may be formed by
external agents and weak and “unimportant”
relationships between subscribers. In such a
scheme the call graph may be a simple unweighted
and undirected one where a link is only established
between nodes that have made a reciprocal call or
exchanged a Short Message Service (SMS) message
between them. In other configurations the graph
may be a directed with the direction of the link
indicating the direction of the originating call or
SMS.
Unweighted call graphs of the nature described
above don’t account for the strength of the
relationship as expressed by the tie strength
between two nodes. The model can account for this
aspect by assigning a weight to each edge
depending on the number of calls exchanged
between two nodes or on the total amount of call
time (minutes of use) between the subscribers. The
total rupee value of the calls between two nodes can
also considered in determining the tie strength,
noting that this is a variable that may change with
the time of day and day of week in markets that
have differentiated pricing depending on the time
the call is made.
There is greater importance attached to incoming
calls as opposed to those outgoing in the social
networks of telecommunications. This follows from
an analogy taken from the World Wide Web, where
the importance of a web page is dependent on the
number of other web pages pointing to it. Thus web
pages that have many links to it are considered of a
higher rank than those that merely have many links
pointing to other pages. The Google page rank
algorithm utilizes this criterion and is based on the
premise that while one can create a page with a
large number of links to other pages it is the number
of references by other pages to the web page in
question that is indicative of its importance and
should determine its place in the network hierarchy
[10].
Considering the above inferences and arguments, in
the analysis presented in this paper we modeled the
Social Network of connections between the
subscribers of the Mobile Telecommunications
Network as a weighted undirected graph where a
connection is established between two subscribers
only if there are calls of a certain minimum duration
and if there is at least one call in the reverse
direction between the subscribers.
4. MATHEMATICS OF NETWORK
ANALYSIS [1]
Network theory is a vast discipline and only the
most important details necessary for identification
of leaders and community detection are presented
here. The adjacency Matrix is a common
mathematical representation of a network and in the
case of a directed graph takes the form below.
When the edges are weighted the value of the
weight will apply to the corresponding matrix entry
instead of unity
1ijA , if there is an edge from j to i
0ijA , otherwise
Centrality is a family of measures that seek to
quantify the importance of a particular node in a
network. As such there are many such centrality
measures of which Degree centrality, Katz
centrality, Page Rank and Eigen vector centrality
are perhaps most encountered. Katz and Page Rank
are closely related to and is a variant of the Eigen
Vector centrality used in this paper.
Eigen Vector centrality is based on the premise that
a node’s importance lies not only in the number of
connections it has but also on the connections of its
neighbors. So a node’s importance is measured by
the number of connections it has and this is
increased by having connections to other nodes that
are also important. Eigen vector centrality thus
assigns a score to each node that is the sum of the
scores of its neighbors, and mathematically this is
expressed as

j
jiji xAx'
, and when expressed as a vector
)1()(  nAxnx which implies that by iterating
)0()( xAnx n

It can be shown that in the limit for large n, x
approaches the principal Eigen vector of A
(corresponding to the largest Eigen value). An
important observation is that the Eigen values and
Eigen vectors of a matrix provide global
information about its structure. Thus the elements of
the principal Eigen vector also provide an indication
of the importance of each corresponding node.
Influence Diffusion, Spectral Clustering and
Spectral Graph Partitioning are a few of the many
techniques available for community detection.
Figure 2, depicts the results for the Spectral Graph
Partitioning of the “Highland Tribes” test network

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depicted in Figure 1, obtained by thresholding the
Eigen vector corresponding to the second smallest
Eigen value of the Laplacian matrix.
Figure 1: Highland Tribes Social Network
The Laplacian L, is defined as ADL  where
D is a diagonal matrix composed of the sum of the
corresponding columns of the Adjacency matrix

i
iji AD .
Figure 2: Detected Communities
The three communities and two components of the
social network are partitioned via the thresholds,
Figure 2.
Degree centrality measures the number of incoming
and outgoing connections at a particular node and is
a powerful measure of the importance of a node.
In-Degree 
j
iji ADin
Out-Degree 
i
iji ADout
5. ALGORITHM FOR LEADER
IDENTIFICATION AND COMMUNITY
DETECTION [2]
The algorithm implemented in the following
analysis is based on that which was derived by
Smilkov etal [2] and only the most salient points are
reproduced here.
An influence matrix '
A is defined using the
adjacency matrix A as

k
k
jijiji CAA'
where },min{ jkki
k
ji AAC  the
transitive link weight from node i to j and its
defined only for neighboring nodes.
To determine the overall influence at a node *
ix the
average influence of its neighbors are calculated via
an iterative procedure
)()1( txTtx  where


k
kj
ij
ij
A
A
T '
'
The identification of leaders is via the set of
neighbors with the largest influence on node i. The
neighborhood is defined }max|{ kikjii TTj 
and that node is denoted a leader if jjiiij xTxT 
for all ij  .
A significant innovation introduced by this
algorithm is the calculation of a membership
function that allows fuzzy boundaries between
communities, where one actor can belong to more
than one community to a different degree. Thus
each node has a membership vector where its
elements sum to one, each element indicating the
degree of membership of that node in a particular
community.
Considering consensus dynamics the membership
vector is determined via



j
jji
j
ji
i tyA
A
ty )(
1
)1( , at each iteration a
weighted average membership figure is calculated.
Thus each node will have a membership vector
giving its degree of participation in each community
where the sumof this vector is unity.
6. RESULTS FROM A CORPORATE
TELECOMMUNICATION NETWORK TO
VALIDATE TECHNIQUE
A well defined network of subscribers chosen from
a corporate network was chosen to illustrate and
validate the technique for community detection,
identifying leaders and their degree of membership
in a particular community. It is shown that the
leaders so identified belong almost exclusively to a

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single community and their followers exhibit a high
degree of membership in their leader’s community
while they may also have a lower degree of
involvement in other communities.
The In-Degree is a commonly used and direct
measure of the importance of a particular node
(subscriber) or actor in this type of communication
network with high degree measure indicating a
greater level of importance bestowed upon that
node. In the following analysis this In-Degree
measure is used to validate those individuals
identified as leaders by demonstrating that they
indeed display a high value when compared with
the other subscribers in the network.
The Adjacency matrix for the calls between the
members of the corporate mobile network was
constructed in such a way that only reciprocated
calls of total duration greater than 6 minutes for all
calls over the period were considered in establishing
a link between two nodes (subscribers). The
network was modeled as a weighted undirected
graph with link weight taken to be the total amount
of minutes of use between a particular pair of nodes.
The initial call records contained approximately
4000 incoming and outgoing calls between
members of the mobile network having 55 members
captured over a period of one week. The data was
parsed to meet the above eligibility criteria as it is
believed that meaningful relations exist between
members that have reasonably long calls and also
only when there is at least one call in the reverse
direction that meets the total cumulative minimum
duration criteria. Thus the link weight was taken to
be the sum of the total number of incoming and
outgoing minutes between two subscribers (nodes).
Figure 3: Degree of Membership
The degree of membership result (Figure 3)
indicates that most nodes in this analysis mostly
belong to one community and that the overlap of
communities is not so prevalent. A less stringent
requirement for the data parsing condition it was
observed resulted in a greater number of nodes that
exhibited greater overlap between several
communities. Thus when a link is established
between nodes for lower cumulative call duration
than the six minutes chosen, a greater degree of
overlap between communities was observed. Hence
this parameter is one that can tune the degree of
overlap by indirectly selecting the strength of the
relationship between a pair of subscribers.
Figure 4: Communities and Leaders
Figure 4, presents the leaders (nodes 3, 9, 23 and
36) and the corresponding number of followers
given by community size in their respective
communities. The red hatches indicate those nodes
that have not attached to a particular community or
leader, but are counted as part of community 1, for
sake of presentation. Figure 4, also indicates that
leaders with node numbers 3, 9, 23, and 36 belongs
to the communities 1, 2, 3 and 4 respectively.
It is also observed that the leaders (nodes) so
identified belong exclusively to one community by
having a membership value of unity (Figure 3).
Thus nodes 3, 9, 23, and 36 belong only to one
community and are not members of any other
community even to a lower degree. This also
validates their leadership position in the network. It
also is observed that there are a few nodes that have
membership values less than unity thus belonging
simultaneously to more than one community. It is
also observed that this degree of participation is
clustered mainly at the lower and upper regions of
the graph indicating that those nodes that have
multiple memberships still strongly associate with
one cluster (community) than the other.
0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
MSIDN or Cx No
DegreeofMembershipinaCommunity
Degree of Membership in Communities
Community 1
Community 2
Community 3
Community 4
0 5 10 15 20 25 30 35 40 45 50 55
1
2
3
4
5
Communities
Leader Community Size
3
9
23
36
39
9
3
4
MSISDN or Cx No
CommunityID

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Figure 5: In-Degree at each node
Figure 5 depicts the In-Degree measure of each of
the nodes in the network participating in the
respective communities. There is strong correlation
between In-Degree and selection of a node as a
leader. Thus it is observed that nodes 3, 9, 23 and
36 do indeed exhibit a high In-Degree relative to the
other subscribers which is commensurate with their
elevation to the status of leader of a community. In-
Degree alone however does not make a leader as
it’s the connectivity of the neighboring nodes that
ultimately counts. There are a number of nodes with
zero In-Degree that are the result of the original
filtering to obtain only those relationships that are
meaningful and eliminate spurious links. Thus we
are able to validate the results of the Social Network
modeling of the Mobile Telecommunication
Network via the influence diffusion analysis
technique by utilizing the more traditional approach
of In-Degree.
7.0 BUSINESS AND MARKET STRATEGIES
FOR NETWORK OPERATORS
Increasingly businesses are aware of the value of
the “buzz” created by campaigns and watch social
media for signs of chatter that can help predict the
success or failure of a new product or initiative.
Social media then is an indicator as well as a
method for spreading the word or a particular
message across its communities. The ability to
identify those individuals within the network with
“social power” is key to implementing these
strategies.
The identification of corporate leaders such as
senior managers and CEOs for inclusion in ones
customer base is a less subtle form of the same
strategy that has the benefit of attracting those
subordinate connections froma particular institution
and is a well practiced approach. Thus diffusing an
advertisement, product idea or message across the
customer base and then to the wider public is more
efficiently done by targeting the message to these
key individuals in the social hierarchy in a network.
In this way the message is more likely to filter to
the larger communities lead by the “influential”
individuals. This is far more efficient and cost
effective than a blanket advertizing campaign or
broadcasting the message to a randomsample of the
base.
The increasing synergies that arise due to the
convergence between the mobile
telecommunications and the internet worlds give a
promoter the opportunity of spreading a message
between the two environments. Thus internet
products may be advertized through mobile and
vice versa. Additionally the social networks arising
from social media such as Facebook and Twitter
can readily be exploited by the social networks in
telecommunications networks provided care is
taken to match the “interest profile” of a potential
customer. In general those individuals that form
social networks do have many commonalities that
can take the form of common interests,
demographics, and lineage among other attributes.
Thus it may be construed that a mobile user who
accesses the internet and browsers news sites may
be a candidate for a subscription to a certain type of
online news magazine.
Further, the analysis of the attributes of the
members of a particular community formed around
a leader may lead to the discovery of common traits
that can exploited by a shrewd marketer. Thus the
identification of a leader of a community combined
with the knowledge of some overall characteristics
of the followers will enable the creation of a well
tailored message targeting the leader but aimed to
be also appealing to the followers of that segment
and the community at large.
Sentiment analysis of twitter feeds and postings on
social media plays an increasingly larger role in a
company’s communications strategy providing
almost immediate feedback on an organization’s
initiatives and corporate image. As influencing the
opinions of existing and potential new customers is
important to the success ofa business in the internet
age, the corporate image usually built on
fundamentals such as service quality, customer
service and reliability benefits from managing and
influencing the conversations taking place on social
media. In this regard well designed messages
directed to “influential” individuals in the social
hierarchy can help burnish a company’s image,
0 10 20 30 40 50 60
0
1
2
3
4
5
6
7
8
9
In Degree
MSIDN or Cx No
Degree

7. Annual Conference2014 - IET-Sri Lanka
create awareness and push products, raising the
profile of a company.
Product bundling and opportunities for cross sell
and up sell are all made more effective through the
utilization of social networks, so much so that social
media marketing is a vital part of many innovative
organizations. These promotional initiatives can be
considered as being the result of a chain process and
when triggered by individuals with influence and
social power can have the effect of rippling through
the network triggering other individuals to perform
similar actions. Churn events is a prime example of
this phenomena where the loss of an important
customer can result in the loss of a whole host of
other “followers” [9]. Thus the retention of key
individuals in the network hierarchy is crucial to a
business as it lowers churn rates, reduces negative
vibes and rationalizes customer acquisition costs.
Additionally segmentation of the customer base and
can also be made more effective when based on
communities combined with other profiling
attributes of interest.
Product and service recommendations derived from
Recommender systems are a very common part of
the internet experience. Essentially the
recommender algorithm determines the
combinations of services and products a customer
has purchased and makes a recommendation on a
product not already in the portfolio of the consumer
based on the consumption pattern of products of
other customers who have a very similar profile to
the customer. This profile will take in to account a
number of attributes such as demographics, income
and spending, and product preferences. Social
networks enhance this process as recommendations
from friends and those with social influence go
further than mere advertisements or ratings. Thus
for example a hotel / service recommendation from
a friend, colleague or someone with influence will
make a significant difference in influencing a
choice than a mere banner / pop up advertisement or
ratings service [11].
Finally, the single view of the customer envisaged
by business to arrive at a complete understanding of
the needs and worth of a customer will also depend
on that person’s position in the social hierarchy. As
examined earlier, the importance of a customer or
subscriber cannot be determined solely on revenue
contributions but also should take in to account that
person’s position in the social network, which is a
measure of the ability to act as an attractor for other
customers, products, services and the spread of
influence.
8. CONCLUSION
This paper presented a novel technique for the
analysis of the Social Networks that arise in Mobile
Telecommunications Networks. In this technique
we modeled the call pattern between the subscribers
of the mobile network as a weighted undirected
graph using an influence diffusion algorithm to
estimate the key actors and their properties in the
network. Several schemes for modeling the mobile
phone call patterns between the subscribers in the
network as call graphs were studied and the impact
on the resulting social network analyzed.
Leaders and followers of communities formed
among the mobile subscribers of a corporate
network were identified via this technique and
validated using In-Degree measure. Their more
significant attributes such as connectivity, centrality
were also analyzed to understand the role they play
in the communities. The degree of membership in a
particular community was presented as a measure to
determine those subscribers that may act as links
between two or more communities of subscribers
instead of merely belonging exclusively to one
community and in that respect play an important
role.
In conclusion the strategies a mobile operator
should adopt to enhance its competitive position in
the industry through the deeper understanding ofthe
interactions and relationships of its key subscribers
and the communities they represent were discussed.
The modeling thus illustrates the significance,
relevance and power of social network analysis in
mobile telecommunications.
9. ACKNOWLEDGEMENTS
N.S. Ekanayake, BSc. (Hons) for providing
business intelligence and detailed call data records
from a live corporate mobile telecommunication
network.
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[3] M. E. J Newman and M. Girvan. Physical
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