Journal of Ambient Intelligence and Humanized Computing
https://doi.org/10.1007/s12652-019-01286-2
ORIGINAL RESEARCH
A fuzzy logic approach to influence maximization in social networks
Yacine Atif1
· Kanna Al‑Falahi2 · Tshering Wangchuk3 · Birgitta Lindström1
Received: 3 January 2019 / Accepted: 26 March 2019
© The Author(s) 2019
Abstract
Within a community, social relationships are paramount to profile individuals’ conduct. For instance, an individual within a
social network might be compelled to embrace a behaviour that his/her companion has recently adopted. Such social attitude
is labelled social influence, which assesses the extent by which an individual’s social neighbourhood adopt that individual’s
behaviour. We suggest an original approach to influence maximization using a fuzzy-logic based model, which combines
influence-weights associated with historical logs of the social network users, and their favourable location in the network.
Our approach uses a two-phases process to maximise influence diffusion. First, we harness the complexity of the problem
by partitioning the network into significantly-enriched community-structures, which we then use as modules to locate the
most influential nodes across the entire network. These key users are determined relatively to a fuzzy-logic based technique
that identifies the most influential users, out of which the seed-set candidates to diffuse a behaviour or an innovation are
extracted following the allocated budget for the influence campaign. This way to deal with influence propagation in social
networks, is different from previous models, which do not compare structural and behavioural attributes among members
of the network. The performance results show the validity of the proposed partitioning-approach of a social network into
communities, and its contribution to “activate” a higher number of nodes overall. Our experimental study involves both
empirical and real contemporary social-networks, whereby a smaller seed set of key users, is shown to scale influence to the
high-end compared to some renowned techniques, which employ a larger seed set of key users and yet they influence less
nodes in the social network.
Keywords Social networks · Community detection · Influence propagation · Fuzzy logic
1 Introduction
The impact of online social networks (OSNs) has undeniably
affected a sizeable proportion of the world population who
in way or the other, tend to use YouTube, Facebook, Twitter, Flickr, MySpace and LinkedIn, etc. The impact of social
networks on individuals’ behaviour throughout various
* Yacine Atif
Yacine.Atif@his.se
Kanna Al-Falahi
k.alfalahi@uaeu.ac.ae
Tshering Wangchuk
tshering_wangchuk@rim.edu.bt
1
School of Informatics, University of Skövde, Skövde,
Sweden
2
College of Information Technology, United Arab Emirates
University, Al-Ain, UAE
3
Royal Institute of Management, Thimphu, Bhutan
stages of their life has been extensive, and in different circumstances. Subsequently, social networks have become a
prime venue for propagating influences using different techniques, and disseminating information of all kinds. This phenomenon is facilitated by social connections which spread
information from one individual to another at a faster pace,
particularly when critical events arise. For example, tweets
(i.e. Twitter posts) have considerable increased in volume
during the severe 2011 Tsunami in Japan (Acar and Muraki
2011), during which individuals around the devastated areas
posted tweets to alert followers about their situation. Similarly, and in the same year, the political unrest in Egypt and
Tunisia were driven by bloggers posting their exasperation
against their respective government practices, on social networks. The extraordinary observation during these events,
is that massive physical-protestations took place on streets
following virtual frustration expressions to get rid of dictatorships. This illustrates the influential power of social
networks, whereby actions are not just embraced online,
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Y. Atif et al.
but individuals do engage into translating them across the
physical world also.
Social networks have long been a nature of the humanity
life, where distinguished individuals of a community could
drive members of their community into embracing a faith or
adopting a behaviour or change their life conditions (Khousa
and Atif 2018). This natural humanity trait, has subsequently
been digitalised in OSNs. However the propagation pace in
OSNs is much faster than in real-life networks, and connections between OSN users scale much higher and quicker in
OSNs leading to increasingly new relationships and new
community memberships. Like in real-life social circles,
people expand and benefit from their social relationships
in OSNs as well. Traditionally, influencers tend to be those
who accumulate many relationships. However, social ties
develop also when the same action cascades over OSNs,
which generates a propagation wave that may reach many
more individuals than direct connections. This is how bloggers “like” a particular topic, and spread the induced message that spawns growing discussions, across contemporary
social networks (Dumenco 2011). Similarly, a product or a
service may be embraced by many individuals across intermediate connections, as well. Subsequently, the concept of
interest graphs evolved, where nodes designating individuals express their mutual interests for a content node (Solis
2011). Hence, there are two types of node, and two types
of links as well, that link people to content of their interests, and content to content to express content relationships.
This graph concept supports brand evolution by intersecting
interest graphs to build larger communities that are used
to spread influence in a targeted advertisement campaign
for example. This approach has later evolved further into a
major marketing trend in contemporary OSNs (Solis 2011).
OSNs are investigated using concepts from graph theory
(Newth 2006) in the context of computer science and social
science (Scott 2000) disciplines, leading to an interdisciplinary social network analysis (SNA) area. Therefore, an interdisciplinary approach is required to investigate and evaluate
OSNs. A graph representation of OSNs include nodes or
vertices that represent OSN individuals, and links or edges
that represent social ties, such as friendship. This representation is frequently used in the evolving SNA field (Hanneman
and Riddle 2005) to identify the triggers and distribution
patterns of influence propagation in social networks. The
influential power of a user augments with the relationships
he or she develops with peers who have varying degrees of
influence power, in the social network. SNA attributes of a
node centrality such as degree, closeness and betweenness
centralities (Hanneman and Riddle 2005) were used earlier to quantify nodes’ power in the network. The influence
power of a node grows with the node’s degree, closeness or
betweenness values. A graph-theory structure of a social network sample is depicted in Fig. 1. The graph representation
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Fig. 1 Social network diagram (Source: http://www.fmsasg.co/
SocialNetworkAnalysis)
displays groups of nodes with some central ones that can
trigger a great influential campaign when disseminating their
social influence. However, centrality attributes alone reflect
only structural aspects of a node. We argue in this paper that
the influence assets of a node are also driven by the dynamic
impacts across links and connections originating from that
(influencer) node within OSN. This impact occurs when
users along connecting paths to the influencer embrace the
influencer’s advocated behaviour or embrace the influencer’s
action, labelled throughout this paper as “common-action”.
We propose an approach by which a set of k nodes in a
social network are discovered based on both their structural
and historical-actions attributes to maximize influence propagation. The proposed approach employs three successive
processes that work in tandem. Initially, “artificial” communities are built whereby “similar” users are assembled
together, based on a judicious similarity function. Next, for
each of these synthetic communities, we identify key users
using a computational intelligence technique that employs
a fuzzy-logic based function to discriminate nodes based on
both their structural-centrality and influence-weight attributes. To measure influence weights, we suggest to crawl
action-logs across nodes to figure out instances of common-actions adoption. Obviously, these measurements are
subject to dynamic changes, depending on the accumulated
behaviours of users from past activity logs in the social network. Finally, we rank these key users based on their influence power, by simulating an influence diffusion process to
determine the seed-set of candidate influence propagation
nodes. The rationale to pre-process the original social network through the identification of virtual communities, is
inspired from the fact that members of the same community
tend to think the same and hence they facilitate the propagation of incoming influences from peer members of their own
community. The final step in the above three-steps process
predicts the most influential members from candidate key
A fuzzy logic approach to influence maximization in social networks
nodes based on the available marketing budget to shortlist
agents for promoting products or services and recommending them around social networks.
The rest of this paper is organized as follows: Sect. 2 provides a background and related works about the main areas
that are relevant to our proposed work. Section 3 depicts
our proposed community-aware social influence diffusion
approach. Section 4 demonstrates the efficiency of our proposed approach through an experimental analysis, based on
data sets from simulated and existing social networks. We
wrap-up the suggested developments discussed in this paper
within Sect. 5, where we also reveal some of our ongoing
extensions.
2 Problem and background
Members of a social network are expected to build connections with other members of the network. To model members
and inherent connections, a graph G (N, E), represents the
set of members N = {1, 2, … , n} which is implemented as an
n × n adjacency matrix. The influence weights: 0 ≤ Eij ≤ 1
are the adjacency matrix entries. Thereby the graph G(N, E)
is said to be a weighted graph. When Eij ≠ Eji , the graph is
said to be directed and undirected otherwise.
An influence occurs when a user u of a social network
represented by the graph G embraces a behaviour, that was
previously embraced by another user v, in which case v is
said to have activated u or u is said to have been activated.
Other nodes of G that do not perform the action or embrace
the behaviour of v are said to remain inactive. Subsequently,
influence maximization consists in discovering a subset of
key users U in the social network modelled by the graph
G, where |U| = k , who activate as many users in the social
network as possible. Identifying U is the core problem of
influence maximization (Goyal et al. 2010). The discovery
of key users is subject to a multi-criteria decision-making
dilemma due to the contribution of both nodes’ influence
weights and their topological attributes within G. This
dilemma motivates the rationale of our proposed computational intelligence technique based on a fuzzy-logic model
to maximize influence, which forms the main contribution of
this paper. To understand further this rationale, we introduce
some relevant fuzzy-logic concepts and use them to illustrate
our proposed approach.
2.1 Fuzzy logic
Fuzzy logic is a prominent development in computational
intelligence (Zadeh 1965). This theory tolerates logical
assertions to carry a progressive extent of values that lie
within the interval [0, 1] as an alternative to true/false assertions (Hellmann 2001). The approximation in the reasoning
processing led to several applications in contrast to its crisp
counterpart, and appears more natural to mimic human,
rather than machine reasoning (Zadeh 1984) to evaluate realworld considerations. We adopted fuzzy-logic to break the
dilemma induced when selecting highly influential nodes,
thereafter labelled “key nodes”, out of which we pick the
seed-set of nodes to use for an actual influential propagation
instance, in order to meet marketing budget limitations. The
joint criteria used to determine “key nodes” membership is
found to be effectively addressed using a fuzzy-logic based
membership function.
Fuzzy-logic sets are characterised with partial membership features, unlike crisp set counterparts (i.e. either an element is a member of set or not), and thus they adapt better
to natural membership expressions used in real-world situations (Baig et al. 2013). A membership function is used
to evaluate the extent of membership and which is context
dependent to meet the realistic real-world features (Rahman
and Ratrout 2009). The membership function computes the
actual membership extent within the interval [0, 1] to assert
a statement with a certain context-related degree, that contrasts with traditional logical assertions with exclusively true
or false propositions (Rojas 1996).
2.2 Illustrative scenario
To illustrate the application of fuzzy-logic to our proposed
approach to select “key nodes”, consider the following
example that is adapted from (Wolfram 2014). Consider the
problem of discovering the most influential users in a social
network made up of the following nodes = {1, 2, 3, 4, 5}. The
goal is to determine nodes that combine centrality location
and influence-weight on other nodes attributes. A first fuzzy
set is developed to represent the centrality degree as follows:
Centrality (C) = {{1, 0.4}, {2, 0.6}, {3, 0.8}, {4, 0.4}, {5, 0.5}}.
Note that Node 3 is deemed most central given its membership value or grade. However, Nodes 1 and 4 are the
least central nodes in the network. The other fuzzy set
Influence Weight (IW) = {{1, 0.1}, {2, 0.9}, {3, 0.7}, {4, 1},
{5, 0.2}} represents the influence power of each node. Note
that the influence weight here represents an averaged value
of influence that a node has over all nodes in the network.
A high value indicates a high influence power. Therefore,
Node 4 is deemed to be the most influential user in the entire
network.
Next, we engage into a decision-making process to
identify the set of nodes that are most influential considering both of the above constraints, i.e. favourable location vs. influence weight. The natural answer to these
joint criteria is a fuzzy intersection of both membership
sets to identify nodes that optimise both features, simultaneously. A fuzzy intersection considers members with
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lowest grades in each of the fuzzy sets Centrality (C)
and InfluenceWeight (IW) (Rojas 1996), thus resulting in
C ∩ IW = {{1, 0.1}, {2, 0.6}, {3, 0.7}, {4, 0.4}, {5, 0.2}} .
This decision-making process is illustrated in Fig. 2. The
maximum grade value grants highest influence feature to
a user. It appears that Node 3 is elected as the one with the
highest grade and thus is deemed to be the most influential
in the network considering both constraints.
There were fuzzy-logic considerations to model social
networks. However, they were limited to studying common
network attributes such as degree, clustering and betweenness (Kundu and Pal 2015). This approach focused on fuzzy
relationships among social network users. Similarly, distinguished relationships among actors in social networks were
modelled using “fuzzy graphs” in Nair and Sarasamma
(2007). Our approach advocates fuzzy logic to select key
nodes in our community-driven influence propagation
approach.
3 Related works
The process of identifying key users in a large network such
as those found in contemporary social networks, can be
harnessed by decomposing the network into communities.
Intuitively, the influence propagation process is expected
to spread faster among members of the same community
with shared interests. Subsequently, community detection
is combined with influence maximization in this paper, and
thus our review of existing works encompasses both areas.
3.1 Community detection
Identifying members within their circle of common interests
has been a vector to direct marketing campaigns according
Fig. 2 Fuzzy decision plot
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to the interests of the social circle members. However, this
identification process requires the discovery of social network members with shared interests. One of the prominent
works found in the literature discovers these clusters of
social network members by hierarchically dividing the network through eliminating iteratively network edges (Newman and Girvan 2004). This process leads to a division of
the network into dense clusters of users, thereby leading
to community structures. The candidate edges for removal
are those with high-betweenness value. This value that is
associated to a candidate edge, quantifies the length of the
shortest-path between any two nodes, when that path passes
through the candidate edge. However, as edges are taken
out from the network, all betweenness-values need to be
recomputed since the paths based on which the previous
computation was made may have changed. A desired threshold is used to evaluate the quality of the detected communities in each iteration to decide whether to stop the network
division process. Nevertheless, this technique is seldom
employed for identifying communities due to its complexity and incurred computational costs. Instead, the opposite
agglomerative alternative is mostly used. A hierarchical
clustering approach built upon the above technique discovers and takes out edges iteratively from the network based
on a centrality value (Fortunato et al. 2004). The authors
show the effectiveness of their approach, despite the O (n4 )
complexity of the proposed algorithm.
A quality-driven division approach has also been proposed using a metric called modularity (Newman 2004).
Labelled Q, the modularity is a function that sizes the significance degree of detected communities. This approach is
distinguished by its simplicity and viable worst-case computational complexity of O (n2 ). Subsequently, this approach
has been deemed attractive and employed in several applications. Nevertheless, the modularity is upper-bounded by a
A fuzzy logic approach to influence maximization in social networks
threshold, which is a function of the network cardinality, and
communities with modularity values lower than that threshold could not be detected. To overcome these bounds, a metric that measures communities’ density has been suggested
Li et al. (2008). This approach employs both vertices and
edges, while preserving the iterative division process of the
network to detect communities. Yet, this proposed procedure
is NP-hard. An alternative resolution of the upper-bounds
problem involves a variation in the modularity function formula (Arenas et al. 2008). Later, it was shown that such
variation of the modularity function exhibits also boundary
issues when combining smaller clusters and dividing larger
ones (Lancichinetti and Fortunato 2011).
As mentioned earlier, an alternative approach to divisive
techniques was the agglomerative one (Clauset et al. 2004a)
suggested by the same authors, and labelled following their
initials CNM. This method proceeds from the bottom of the
dendrogram that hierarchically displays the relationships
between nodes, and move up in a greedy way, while assembling clusters of the network. Although analogous to (Newman 2004), this approach has better complexity performance
of O (nlog2 n) in worst-cases.
3.2 Influence maximization
The influence maximization problem has been extensively
investigated in the literature, particularly in quantifying the
expectation of a node to influence other nodes. But existing
approaches face a limited capacity to maximize the activated
social network nodes to the higher end, while minimizing
the seed-set size k of selected nodes used to propagate influence. Constant probabilistic values are assigned to nodes in
the static approaches to model influence propagation within
social networks based on time-independent observations
(Goyal et al. 2010). However, these methods assume a static
propagation of influence that do not evolve over time, given
the constant probabilistic values. This means, they do not
address the development of influence probabilities following users’ activities in the social network. The Bernoulli
probability distribution was used in the above static modelling approaches to represent social network users attempting
to activate neighbouring peers. The influence propagation
models using static approaches are simple to use, but the
natural evolution of social networks limits their applicability.
The induced constant probabilities assumption oversimplifies influence measurements to accommodate contemporary
social networks.
Dynamic approaches to represent influence propagation
such as the Snapshot approach (Kossinets and Watts 2006;
Backstrom et al. 2006; Shi et al. 2009) does consider the
evolution of probabilistic values to reflect the nodes’ evolving influence power over time. As the name implies, this
approach considers successive snapshots of the network over
time to infer its evolution. This approach has been extensively used given its capacity to pick up the dynamics of
social network data for analytical purposes, including the
evaluation of influence state among nodes across successive
timestamps. However, consecutive snapshots increase substantially the size of data to analyse. Alternatively, ordinaltime approaches limit the observation sequences to activation occurrence instants (Cosley et al. 2010). That is, when
there is a change in the network induced by an influencerelated activity, a snapshot of the network is retrieved, which
lowers the size of data to analyse. Nevertheless, timestamped
snapshots of an entire social web structure are complex to
collect, which reduces the implementation efficiency of
related approaches, that aim at evaluating activation patterns
across influence-propagation processes.
Alternative approaches to model influence that appear
to be less sensitive to the above drawbacks have been proposed. The landmark Linear Threshold Model (or LTM) and
Independent Cascade Model (thereafter labelled ICM or IC)
fall in this category. LTM (Domingos and Richardson 2001;
Kempe et al. 2003; Richardson and Domingos 2002) accumulates the influence weight contribution from each node
towards a common neighbour. When the resulting accumulated value exceeds a threshold, the common neighbour is
activated. Edge weights reflect the influence power a node
may have over his neighbours. ICM (Kempe et al. 2003),
advocates a binary states of nodes whereby each node has
a single chance to be activated or not during an influencediffusion that cascades over neighbouring nodes. Activated
nodes will have the same chance to activate their neighbours, recursively. This process is similar to viral spreading across ties in conventional social networks where users
incite peers to watch the same movie, or embrace a certain
political opinion. Subsequently, a cascade is enacted which
diffuses the influence over the network structure. Activation
occurs at a given node based on some probabilistic value,
which evolves according to the interaction intensity between
nodes. These approaches speed up the influence propagation
process, particularly when the seed-set of highly influential
users is pre-established.
However, the above LTM as well as IC models do not
consider the mutual relationships among node actions. This
observation called for alternative approaches that consider
users’ actions towards a common context. Topical graphs
mine users’ activities using a machine learning approach to
infer influence probabilities following users’ interest in particular topics (Tang et al. 2009). A related subsequent investigation found that similar users tend to influence each other
(Sun and Tang 2011). This relationship between similarity
based on social ties and influence activation supports further
our influence-propagation approach and our rationale for our
proposed community-driven influence propagation. However, the effectiveness of influence propagation is enhanced
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by decreasing the seed set of highly-influential nodes (Hosseini-Pozveh et al. 2017) and harness the complexity by
using a modular propagation approach, such as those that
employ communities (Wang et al. 2010), like we do. But
these approaches are developed for specific purposes and
do not incorporate computational intelligence techniques to
optimise the seed-set selection as we do with our proposed
fuzzy-logic based influence propagation model.
4 Community based influence propagation
algorithm
In this section, we reveal our approach to influence-maximization which includes a community-enrichment preprocessing step to scale-up the diffusion process and the number
of activated users within a social network. In doing so, we
join together two of our previous works, namely an original
approach to identify communities (AlFalahi et al. 2013) and
a technique to evaluate influence weights (AlFalahi et al.
2014). The combination of these works generates a new
approach whereby the previous techniques are employed in
tandem, to obtain a set of users who can incite neighbouring peers to embrace an advocated behaviour. The analysis
results shown later in the experiments section, reveal the
effectiveness of this new approach to maximize influence
on synthetic and existing social networks data.
Our suggested approach consists actually, of three successive steps, namely (1) Communities identification via
an enhanced Similarity-CNM network algorithm, (2) Key
Users discovery within each detected community, and (3)
Seed set identification from ranking key users, to effectively
drive the diffusion process over neighbouring peers. Figure 3
illustrates our proposed framework and Table 1 provides an
explanatory reference of the symbols used throughout this
paper.
4.1 Similarity‑CNM
Given an input social network, our proposed approach starts
by discovering communities. This essential step of our
approach employs a similarity function to support behavioural embracement among similar peers. This preprocessing
step ensures that the search space for key users is reduced
into modular communities and facilitates further the subsequent diffusion process. Our inspiration that is supported
also by previous investigations (Wang et al. 2010), is that
users with high similarity-attributes are more susceptible to
embrace common attitudes. Thus, the community structures
which first assemble similar users into modular communities facilitate the process of key-users discovery. These key
users are the first seed-set candidates to propagate influence, that are further ranked to extract a subset that meets
13
Fig. 3 Proposed framework
some budgeting resources allocated to a given marketing
campaign.
An improved version of the CNM algorithm (AlFalahi
et al. 2013) is shown in Algorithm 1 depicted next. Named
Similarity-CNM, this approach detects communities
through an improved version of the existing CNM landmark approach (Clauset et al. 2004a). Based on the performance results revealed later in this paper, the quality of the
communities from the improved CNM-Similarity version
outperforms the original CNM. Subsequently, the influence
modelling steps follows the community detection one, using
a network of modular virtual-communities instead of the
original plain network. This community-enriched network is
deemed to supply additional information that guide further
the spread of influence across the entire network. The discovery of communities is preceded by enriching the network
with synthetic links that join similar nodes together, in order
to obtain denser community structures. The preprocessing
step incurs a computational complexity of O (n2 ). However,
this preprocessing step is carried out offline to alleviate
this additional computational cost throughout the influence
maximisation process.
A fuzzy logic approach to influence maximization in social networks
Table 1 Legend of symbols
Symbol
Legend
G
N
n
E
Di
Eij
ni
cnij
C
Q
Similarity_Threshold
CentralityWeighti
Threshold_IU
Degree Centralities
Central Users
Threshold_S
InfluenceWeightij
Influence Weights Avgi
Intersectioni
Social network graph
Nodes set in a social network
The number of nodes in the network, i.e. |N|
Edges set in a social network
Degree of Node i, i.e the number of ties that Node i has in the network
Adjacency matrix entry corresponding to Nodes i and j, i.e. = 1 if there is a link between i and j, and 0, otherwise.
Number of nodes adjacent to Node i, i.e. number of i’s neighbours.
Number of common adjacent nodes to Nodes i and j
A community in the social network
Modularity value of a social network with respect to community structures, and determined by Eq. (7)
Parameter used to determine similar nodes with respect to the value delivered by Eq. (1)
Level of Node i centrality in the network, and determined by Eq. (2)
Parameter used to determine the number of important users in a network with respect to their centrality weight values
Vector of centrality weight values of nodes i, ∀i ∈ N
The first ThresholdI U nodes with highest centrality weight
Parameter to set the desired size of influential-nodes seed set
Influence weight of Node i over a particular Node j, given Eij = 1 and determined by Eq. (3)
Average of influence weights that Node i has over all nodes j, given Eij = 1, and determined by Eq. (4)
A fuzzy intersection value, between CentralityWeighti and InfluenceWeightsAvgi membership grades determined by
Eq. (5)
A set of nodes with both favourable location and influence history determined by Eq. 6
Key Users
The preprocessing step results in an identical network from
the original one, with added links that connect similar users,
thereafter labelled similarity-network. A directed unweighted
graph representation of the network is considered, where
users’ similarity represent weight annotations over the network
edges. These similarity weights provide indicators about users’
relationships to construct well-structured communities. The
discovery and evaluation of community structures from the
similarity-network uses the standard CNM algorithm (Clauset
et al. 2004b). As stated earlier, this algorithm clusters network
nodes iteratively following an agglomerative approach that
moves up through the hierarchical network structure, while
joining clusters together. The modularity is computed on the
way up, to evaluate the clusters’ quality Q, which represents
the variation of links within clusters and a presumed number of links. Good structures rise with such variations (Newman 2004). Initially a small set of nodes is built up without
any links, and hence with a poor modularity. The community
structure is iteratively enriched with edges while merging cluster pairs, which raises modularity values. The prospects of
building sparse communities (Fortunato 2010) is reduced by
supplying CNM with enriched similarity-network. We refer
to this combined algorithm and enriched input as SimilarityCNM approach. The algorithmic steps to obtain the virtual
similarity-network G′, given an input network G, are revealed
in Algorithm 1. The employed similarity function is shown
in Eq. (1):
Similarity (i, j) =
Eij + cnij
n i + nj
.
(1)
Equation (1), makes use of adjacency values Eij that represent the size of common-neighbours cnij between the nodes
i and j, using their respective degree ni and nj . The proposed
pre-processing step results in more inclusive communities
and speeds-up the community-detection process. The objective is to improve the structure of detected communities with
high-modularity values. The discovered virtual communities
are used to find candidate key users. This process begins by
identifying users with highest centrality values within each
community. Then, the nodes with highest influence weights
are obtained.
The complexity of the algorithm is of order O (n2 ) , but
since the process is performed offline as a preprocessing
step to virtually transform the input network, the complexity doesn’t really affect the performance of the framework.
The next step of the framework is to establish communities
in the enriched network G′ by applying some contemporary
community detection algorithms, such as CNM (Clauset
et al. 2004b).
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We show later in the experimental analysis section, that
the community quality is highly optimizied when the proposed pre-processing step of Algorithm 1 is applied. Next,
we show how this virtual division of the network into wellstructured communities contributes to elicit nodes used to
propagate influence.
4.2 Key users
Key users are discovered initially from the the communities generated in the previous Similarity-CNM algorithm
step. They represent seed-set candidates to propagate influence. They are distinguished by their favourable position
in the network which is quantified through structural centrality values, and their influence-weight which is quantified from historical logs data. As stated earlier, to break the
dilemma of dealing with dual-criteria simultaneously, we
employ fuzzy-logic theory (Kahraman 2008) to select key
users that optimise both criteria. In doing so, we identify the
attributes involved in the key-users membership-function,
as well as the associated weights to reflect the importance
of some attributes over others (Peneva and Ivan 2008). The
attributes here are the centrality and the influence power of
users in the network, whereas the weights are importance
parameters associated with each of these two criteria. Following the definition of criteria and associated weights, key
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users are elicited using the fuzzy-logic process shown in
Algorithm 2, which we elaborate further next.
4.2.1 Central users fuzzy set
The structural attribute of key users reflect their favourable
position, such as the ones with high-degree values, or those
bridging two or more clusters, who have the capacity of carrying influence across cluster users. These are examples of
key user structural attributes, which are some interpretation
of user centrality. We adopt Degree Centrality to measure
structural attribute values. Central users fuzzy-set is determined with these values derived from the corresponding
membership function, which we discuss next.
Initially, and for each detected community, a Degree Centrality is computed. To do this, the out-degree and in-degree
are of each user node are determined and then cumulated.
To recognise central users, we employ a centralityThreshold, whereby user nodes with degree exceeding centralityThreshold, are deemed structurally central, and will be
carried forward to the next stage. Based on this approach, the
degree centrality for all users is computed using the following membership function to determine central users:
CentralityWeighti =
Di
.
|E|
(2)
A fuzzy logic approach to influence maximization in social networks
In Eq. (2), Di represents Node’s i degree, which cumulates
the in-degree and out-degree of Node i. The overall number
of links in the network is formulated by |E|. Structural centrality values are thus determined by Eq. (2) which define
Central Users fuzzy-set. These values fall within the interval
[0.1], to reflect the centrality extent of each network node.
Those nodes with close to 1 centrality value, indicate a
high-centrality position. Central Users fuzzy-set values are
employed in the selection process of key users, as discussed
next.
4.2.2 Influence weights fuzzy set
In addition to the structural attribute, key users are also
determined based on their capacity to spread influence
across the network. The computation of this influence
capacity determines the influence-weight value of a user,
which is formulated using a Common Actions version of
Jaccard coefficient (AlFalahi et al. 2014). As an illustration of this computation, suppose a user node A triggers
a behaviour at timestamp T1, and at a later stage User B
embraces that behaviour at timestamp T2. This sequence of
events indicates that an activation instance occurred when
User B adopts the behaviour initiated by User A. To calculate
the Common Actions Jaccard coefficient, we enumerate the
actions that a user adopted, and that were previously triggered by a neighbouring user in the network. The real-world
experimental data we used shows that an action is triggered
by a single source, and thus this consideration is assumed
throughout our proposed influence-propagation algorithm.
Equation (3) shows the actual formulation of the common
actions Jaccard coefficient.
InfluenceWeightij =
Aij
.
Ai + Aj − Aij
(3)
In Eq. (3), Ai represents the number of actions accomplished
by Node i, Aj represents the number of actions accomplished
by Node j and Aij represents the number of common actions,
13
Y. Atif et al.
that represent those actions accomplished by Node i and
subsequently, accomplished by Node j, as well.
The computation of influence weights, is followed by
averaging them to distinguish those nodes that are deemed
highly active, based on historical logs data. This process
reveals the Influence Weights fuzzy set membership function, which is formulated as follows:
∑n
InfluenceWeightsij
j=1
(4)
.
InfluenceWeightsAvgi =
n
Equation (4) shows that the cumulated value of influence
weights that a specific Node i has over each Node j across
the network, is normalized by the size of the social network
n.
4.2.3 Fuzzy decision making
After computing both central and influential user fuzzy
sets for every user, we decide on key users using a fuzzyset intersection operation for every Node i, considering the corresponding fuzzy sets CentralityWeighti and
InfluenceWeightsAvgi , using the following formulation:
Intersectioni = min (CentralityWeighti , InfluenceWeightsAvgi ).
(5)
Equation (5) shows that the intersection between the fuzzy
sets picks the smallest of degree centrality and influence
weight values. Subsequently, Eq. (6) shows that, ultimately
the key users are those which maximise their intersecting
structural and influence fuzzy-membership sets. The rationale of this approach is to address deficiencies each node may
have in either its structural or influence power dimensions.
This is why, the fuzzy-intersection considers the minimum
of both values, so that users with less deficiency in either
attribute get picked. This results in a set of user nodes with a
single associated value, that is the least deficient, in terms of
13
influence or structural shortcomings. Subsequently, the key
nodes are determined based on the maximum of these single
valuations of each node, as formulated by Eq. (6):
KeyUsers = argmaxi∈N (Intersectioni ).
(6)
In Eq. (6), N represents the social network user nodes set.
4.2.4 Seed set
At last, the seed set which represents the actual set of users
to drive influence propagations is selected as the top k
nodes of key users, where k is a parameter that depends
on the allocated budget to a given marketing campaign to
account for cost involving in recruiting seed set users to
promote a given product or a service or spread a desirable
behavioural campaign, such as stop-smoking. Hence, we
need to rank the key users in order to be able to pick the
top k ones. For that, Algorithm 3 is employed to evaluate
the influence spread for each user using the IC model.
For each run of the algorithm, we account the number
of activations that a candidate key user scores. The computational cost of this approach is similar to that of IC
model, however the input key user nodes are judiciously
picked in our case using our proposed fuzzy-logic based
selection process. Our approach also contrasts with LTM
which does not consider action logs data, like we advocate. In addition, LTM is NP-Hard, calling for heuristic
approaches to harness the problem. Instead, we harness the
problem through the gradual three modular steps process
that are: (1) detecting virtual communities using correlations between user actions, (2) identifying key users in
each of these communities, and (3) finding the seed-set
(among those key users) to propagate influence across the
entire social network.
A fuzzy logic approach to influence maximization in social networks
5 Experiments and performance analysis
This section describes the experiments we conducted to evaluate the community-based influence propagation approach
we introduced in this paper. As mentioned in Step 1 of Algorithm 2, we propose to use a similarity based preprocessing
step to enrich the input social network using Algorithm 1,
before applying Step 2 which detects communities in the
enhanced social network. The resulting Similarity-CNM task
is poised to detect better community structure as explained
further in Sect. 4.1. Hence, we propose to first reveal the
outcomes of this pre-processing step, whereby community
quality is measured using Modularity as evaluation metric.
Subsequently, we implemented the remaining steps of Algorithm 1 to generate the key-users which accumulate both
a favorable location in the network and a good account of
influence (S). And finally, we run the second experiment to
assess the propagation extent of influence propagated by the
highest key-users, which form the actual seed-sets. Throughout both experiments, we hypothesize that the similarity
based preprocessing step on a social network G is effective
with respect to the quality of communities, which are used to
select candidates for the further influence-propagation step.
In doing so, we hypothesize also that the fuzzy-logic based
combination of favourable location within those communities, and the prior activity history elects highly influential
candidates across the entire network. We implemented the
proposed algorithms in this paper using Python and related
iGraph and Networkx libraries. A Mac OS X version 10.14.2
(Mojave) platform powered with an i7 processor of 2.50GHz
and a RAM of 16 GB was used to implemented the algorithms presented in this paper in order to evaluate their
performance.
5.1 Datasets
We used both artificially-generated social networks and
actual network data. To evaluate the similarity-based community detection algorithm, we used LFR benchmark networks as dataset (Lancichinetti 2008). This benchmark was
used in several researches dealing with community-detection
in social networks (Cao et al. 2015; Hafez et al. 2014; Chen
et al. 2016; Emmons et al. 2016; Orman et al. 2012). Simulated networks are employed in the community-detection
experiment to overcome the difficulty to evaluate communities in real-world networks due to an absence of community
ground-truths (Cao et al. 2015), and to assess communityquality under varying degrees of structural parameters.
Nevertheless, LFR Benchmark networks do simulate networks that are very close to real-world social networks’
data (Bródka et al. 2010), and this benchmark is becoming
a de-facto standard network-generator for evaluating the
performance of different community-detection algorithms
(Largeron et al. 2015). We generated a network of 10,000
nodes using LFR benchmark for the first experiment. The
most important parameter used to vary the structure of the
network is known as the mixing parameter 𝜇 , which represents the fraction of intra-community edges incident to
each node. Its value ranges from 0 to 1, where 0 results in
graphs that have high community structure, and 1 results
in graphs that have low community structure. The mixing
parameter generates this connection based on (1 − 𝜇 ) for
intra-community edges and (𝜇) for inter-community edges.
Thus, values between 0 and 0.5 yield proper community
structures, and values between 0.5 and 1 yield loose community structures. The other parameters are 𝜏1 and 𝜏2, respectively the “power law exponent of degree distribution” and
“power law exponent for the community size distribution”
(Lancichinetti 2008; Lancichinetti and Fortunato 2011), and
which are respectively set to 2 and 1.5 in our experiment.
Further parameters are the average and maximum node
degree set to 10 and 50 respectively in our experiment, and
the community-size set between 20 and 60. These values are
consistent with those proposed by LFR benchmark providers (Lancichinetti 2008; Lancichinetti and Fortunato 2011).
For the second batch of experiments however, related to
influence-propagation reach, we employed real-world data
sets from Flickr social network. This social-network is distinguished by photo sharing activities. Users of Flickr post
photographs or include them into blogs and other users may
“like” the posted photographs as an instance of an activation.
This dataset is graciously made available by some published
works (Cha et al. 2009), and consists of over 2.5 million
nodes with over 33 million links. Due to computational
constraints and as part of our preliminary experiments, we
extracted two subgraph samples of 500 and 5000 nodes, randomly to observe the results of our experiments across real
networks and assess the scalability properties of the obtained
results. The targeted indicators from this second batch of
experiments relates to the performance of the fuzzy-logic
based intersection between favourable and influential nodes,
that are poised to optimize diffusion across social networks.
The extracted subgraphs preserve the original links which
amount to 26,223 edges for the 500 nodes network and
242,600 edges for the 5000 nodes network.
5.2 Candidate algorithms
In the context of community detection, we propose to illustrate the performance of the similarity-based algorithm
against a series of known community-detection algorithms,
including pioneering CNM (Newman 2004), as well as InfoMap (Rosvall and Bergstrom 2008), Louvain (Blondel et al.
2008), and Multilevel (Rotta and Noack 2011) algorithms.
CNM is modularity-based and very fast. Infomap is a search
13
Y. Atif et al.
not exploit action correlations among users of the social network whereas our proposed Algorithm 3 integrates influence
weights based on common actions weights. In addition, IC
propagates influence over the input network, whereas we
consider our enriched similarity-network. To assess the
gain in activated nodes following influence propagation, we
apply IC model with randomly selected seed-nodes against
the seed-set users generated from 3. With this comparison,
we evaluate the value of the fuzzy-intersection introduced
in this paper to select the most appropriate nodes for influence diffusion.
5.3 Performance metrics
A practical metric used to evaluate the fabric structure of
communities is the modularity, which is denoted Q and
which evaluate the partitions in a social network. This evaluation is based on the variance of the amount of edges linking nodes within the same cluster from an expected amount
of edges in an arbitrary network (that is typically unstructured). Better communities are detected when this difference
is large. According to (Clauset et al. 2004a), the value of Q
above 0.3 is considered as a significant community structure.
This value is derived from the following formula:
∑
Q=
(eij − a2i ),
(7)
i
Fig. 4 Modularity of community structure when applying similarity
network
algorithm for minimizing a map equation over possible network partitions. Louvain is a greedy optimization approach
that maximizes the modularity of a partition in the network
in two steps. Initially, “small” communities are established
through a local optimisation of the modularity value, and
then community nodes are aggregated to construct a new
network. The process iterates over these two steps until a
maximum value of the modularity is reached. The multilevel
refinement method is a multistep approach, which repeatedly
prioritises the process of joining pairs of clusters that do not
decrease the modularity. The priority criterion is a parameter
of the algorithm.
Subsequently, and in the context of influence diffusion,
we conducted experiments to measure the performance
gain of our proposed approach compared to the original IC
approach (Kempe et al. 2003), by setting the input set of triggering nodes A0. The members of this set should be carefully
selected to maximise influence propagation. IC model does
13
where eij is the proportion of edges linking vertices in com∑
munities i and j, and ai = i eij is the proportion of edgeendpoints that connect vertices in community i. Q value
ranges between [− 1, 1] , and measures the density of vertices within the same community to that of nodes belonging
to a different community. The larger the modularity score,
the better is the partitioning of nodes into communities. A
low-score means there is less community structure and highscore means communities are very well partitioned (structured). The similarity threshold of Algorithm 1 was set to
0.005 following observed experimental instances of Q for
various threshold settings.
As for the influence-maximization evaluation, we made
use of activated nodes size as a performance metric. We
determine the number of activated nodes as it is the paramount element to contrast the performance of influence
propagation for both the IC model and our proposed fuzzylogic based model. Nodes activation can be explained as the
embracement of a certain action by a node, which is triggered by another node. This is the main goal for influence
propagation, whereby algorithms strive to scale-up nodes
activation that conveys the adoption of a marketed product or
a targeted behaviour. The influence threshold of Algorithm 3
was set to 0.1.
A fuzzy logic approach to influence maximization in social networks
Fig. 5 Community-detection modularities over 10,000 node networks with various topologies
5.4 Performance results
Our experiment results are organized in two stages. First, we
show the gain in modularity of communities obtained when
applying our pre-processing step of Algorithm 1. Then, we
reveal the gain in activated nodes obtained when our proposed fuzzy-logic based approach to elicit key-users shown
in Algorithm 2 is employed, and subsequent diffusion of
Algorithm 3 is carried out to generate the seed set of highlyinfluential users.
5.4.1 Similarity‑based preprocessing
Figure 4a shows the performance when our proposed
enriched synthetic-network is used for community-detection by a range of landmark algorithms. The original
LFR-generated network of 1000 nodes delivers a lower
community structure than the enriched similarity network
obtained by adding edges between similar nodes as dictated
by Algorithm 1. The same 20% margin-gain in modularity
is observed in a higher-scale network of 10,000 nodes as
shown in Fig. 4b. Another observation is that the similarity
network creates better opportunity for community clusters,
when the network size increases as the modularity rises
between the 1000-nodes network to the 10,000-nodes network, whereas it appears stationary in the original network
cases. In both experiments, the mixing parameter 𝜇 was set
to 0.3.
From the above experiment, we observe the value of the
judicious similarity-function based on common-neighbors
employed to support community-detection algorithms detect
better communities. We use the synthetic similarity network
13
Y. Atif et al.
Fig. 7 Activated nodes with varying diffusion steps using 10 seed
nodes in a 5000-nodes Flickr network
Fig. 6 Activated Flick network nodes with varying seed-set sizes
to detect virtual-communities via various community detection algorithms, for the purpose of using the enhanced community-structure to find the important nodes within each of
these virtual communities. Later, in subsequent experiments,
we observe the value of the proposed fuzzy-logic based
approach to calculate the influence weight for each node
within each community, based on both centrality measure
and common actions history. Using the community structure
scales-down the complexity of finding important nodes.
The enhanced community-structure brought about by
the similarity-network spans various network topologies,
as illustrated further in Fig. 5, where similarity-based community detection algorithms outperform original algorithms
across a range of mixing parameter values. The performance
of detected community-structures degrades as the value of
mixing parameter 𝜇 rises. Each point in the graphs represents an instance of modularity for a given LFR-generated
network of 10,000 nodes shaped through the indicated
mixing parameter in the x-axis. A low mixing-parameter
13
is conducive to dense community-structures, since the fraction of neighboring nodes outside any community (i.e. 𝜇 )
is low, and hence higher modularity is inferred given the
tight community structures of the sample network. On the
other hand, a high mixing-parameter is conducive to loose
community-structures since the fraction of nodes outside any
community is high, and hence a low modularity is inferred.
However, for each community-detection algorithm, the degradation is moderate when applying our proposed preprocessing approach to the original network.
Furthermore, for each community-detection algorithm,
the modularity does not fall below the 0.3 threshold, which
is the minimum value mentioned in Clauset et al. (2004a), to
indicate a significant community structure. It is noteworthy
that InfoMap case shown in Fig. 5c, the similarity-network
shows the steepest loss in modularity. This is because Infomap community-detection algorithm is distinguished from
the other community-detection algorithms as it relies on a
map equation, whereas the others are modularity-maximisation approaches. The employed map equation in Infomap
partitions the network following some patterns within the
network, to build communities. Hence, for our subsequent
step of influence maximization, we use a candidate from
modularity-based approaches, namely CNM.
5.4.2 Social‑influence propagation
Through our second batch of experiments, we evaluate the
influence-propagation reach when employing the fuzzylogic based approach discussed in Sect. 4.2, after preprocessing the social network using the approach presented in
Sect. 4.1. As stated earlier, real-world data sets from Flickr
social network are used in these experiments. Two subgraph
samples of 500 nodes with 26,223 edges, and 5000 nodes
A fuzzy logic approach to influence maximization in social networks
with 242,600 edges, have been extracted from Flickr data to
evaluate the diffusion spread and the scalability performance
of candidate algorithms. The output is measured in terms
of the number of activated nodes. This performance metric
estimates the diffusion along the network, given an input of
judiciously selected seed-set users, which in our proposed
approach are inferred from our fuzzy-logic based technique.
The results produced from these experiments batch reveal
interesting tradeoffs between our proposed method and IC
model, whereby a larger number of nodes are activated by
our proposed community-based influence-diffusion model
(discussed in Algorithm 2). and yet involving a smaller
seed-set (compared to IC model). This is illustrated by Fig. 6
which reports the influence diffusion results for two samplesize Flick networks. First, Fig. 6a shows the results for a
snapshot of 500 nodes, where our proposed social-influence
based propagation which uses the fuzzy-logic discrimination to identify seed-set nodes quickly reaches a high range
of nodes activation while using few seed nodes. Indeed, 10
nodes activate about 350 nodes in our social-based propagation, while original IC model which chooses random seedset nodes activates about 30 nodes, only. As the seed-set
threshold increases. the social-influence approach scales
up the range of activated nodes to the higher-end, reaching about about 400 nodes for an initial 50 seed nodes. The
results show the pursuit of nodes activation to cover almost
the entire 500 nodes of the sample Flickr social network.
They also show that IC model requires 50 seed nodes to activate the third of what our proposed social-influence propagation achieve with less merely 10 nodes. The activation
gain is about 90% attributed to our proposed social-influence
approach compared to IC benchmark. These results are the
outcome of 10 diffusion steps, where the judicious combination of seed nodes’ location and historical influence brought
by the fuzzy intersection of Eq. (5) enable faster influence
propagation to the high-end of the social network. The
results show that exploiting correlations exposed by centrality attribute and historical action logs, the influence propagation process scales higher the activation process. The triggering users are more successful in persuading neighbours or
neighbours-of-neighbours to embrace the propagated action.
In another experiment, the size of the Flickr sample network is extended to 5000 nodes. The results are illustrated
in Fig. 6b and in this case we report the results of just one
diffusion step, and limiting the threshold of “important
users” selected for their centrality value within their communities to 10. This is the centralityThreshold mentioned
in Sect. 4.2.1, which reflects the topological eligibility of
seed-set nodes. Figure 6b shows that in just one diffusion
step, the number of activated nodes rises quickly to over
600 users, in an influence campaign driven by just 10 seed
users. By contrast, IC model activates about 100 users
with the same number of seed nodes. However, the gap
between the two models grows when increasing the seedset size to reach 1000 more activated nodes by the socialbased propagation over original IC model. Hence, using
IC approach, 400 nodes are activated by the 50 nodes of
the seed set used to diffuse influence in the social network
as shown in Fig. 6c, whereas this number climbs to over
1400 in the social propagation model. This is an important
outcome, considering investment decisions made by businesses to promote a product using our approach, as they
could persuade less number of initial people to promote
their product to expand the outcome of a marketing campaign. This approach induces substantial marketing savings to provide free samples to those influence-inceptive
individuals forming the seed-set. In addition, businesses
raise their income, as those inceptive-individuals have the
capacity to entice a large number of social-network users
to adopt the product at a later stage.
We also noticed some more interesting results when
analyzing the performance with varying diffusion steps
as illustrated in Fig. 7. The social-based propagation is
able to reach almost the entire Flickr social network of
5000 nodes after just five diffusion steps using only 10
seed-set nodes, and outperforming the original IC propagation by an increasing margin as the diffusion step rises.
This means that 0.2% of individuals in a social network
are able to convince the entire network population. This
influence scalability is attributed to the fuzzy-set intersection approach we employed to determine key users. The
combined value obtained from both CentralityWeightn and
InfluenceWeightsAvgn criteria to decide on key user node
n, compensates the defects in either centrality or influence
weight to reach neighbouring nodes. The impacts of this
result contribute to an efficient seed set of potential candidates for influence propagation, by its reduced size and
higher influence (reachability).
6 Conclusion
In the presented work within this paper, we address the
prominent social-network problem pertaining to influencemaximization, for which we contribute a computationalintelligence approach to expand the influence diffusion rates
in contemporary social networks. The experimental analysis
results reveal the potential benefits of using a communityenrichment preprocessing step before applying influencediffusion algorithms. We also suggested a new method to
find “key nodes” in social networks using a computational
intelligence approach that adapts fuzzy-logic theory to key
users selection. This technique discovers the most influential nodes as seed set for influence propagation, by combining multiple criteria such as nodes’ location and influence
weights in the social network. The propagation of influence
13
Y. Atif et al.
in social networks involves naturally some vagueness, given
dynamic nodes’ relationships and location in the network.
The proposed fuzzy-logic approach is suggested to overcome this typical vagueness in social networks, by combining both of these node properties to assert key nodes that are
candidate seed set members for diffusing influence. These
correlations have practical implications across a range of
business, political or social campaigns that aim at generating revenues while minimizing costs, or adopting desired
behaviours across a society with fewer interventions.
Future directions to extend the influence maximization
algorithm presented in this paper are numerous to investigate further efficiency and scalability opportunities. We are
also working on applying the proposed approach to other
real-word datasets such as YouTube and come up with new
insights about robustness in finding the most influential seed
set. We are also exploring the effectiveness of employing
various centrality attributes for a better precision of the
obtained results, such as betweenness, closeness, etc.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativeco
mmons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate
credit to the original author(s) and the source, provide a link to the
Creative Commons license, and indicate if changes were made.
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