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Positive Inuence Dominating Set Generation in
Social Networks
Akshaye Dhawan
Ursinus College $1)/,-%)/-! /
Mahew Rink
Ursinus College
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Positive Influence Dominating Set Generation in Social Networks
Akshaye Dhawan, Matthew Rink
Department of Mathematics and Computer Science
Ursinus College
Collegeville, PA
adhawan@ursinus.edu, marink@ursinus.edu
Abstract
Current algorithms in the Positive Influence Dominating
Set (PIDS) problem domain are focused on a specific
type of PIDS, the Total Positive Influence Dominating
Set (TPIDS). We have developed an algorithm specifically
targeted towards the non-total type of PIDS. In addition
to our new algorithm, we adapted two existing TPIDS
algorithms to generate PIDS. We ran simulations for
all three algorithms, and our new algorithm consistently
generates smaller PIDS than both existing algorithms, with
our algorithm generating PIDS approximately 5% smaller
than the better of the two existing algorithms.
Index Terms—Greedy Algorithms; Social Networks; Dom-
inating Sets;
I. Introduction
Social networks have grown immensely in size and popu-
larity in the last decade. The explosion of these networks
has led to a great deal of research pursuing algorithms
[1] that provide insight into the structure and properties
of these networks. Social networks are often represented
as mathematical graphs with users of the network being
modeled as nodes in a graph and the relationships between
users being modeled as edges between nodes.
An important algorithmic problem in these network
is that of modeling the spread of information within a
group. Social science research has shown that individuals
are strongly influenced by the ideas of those around
them. Various studies in the context of drinking [2] and
smoking [3] have shown that the likelihood of an individual
to participate in these activities increases as their peers
engage in these activities.
Within the context of social networks, recent work [4]
[5] [6] has looked at modeling the spread of positive
behavior in a social network. For example, a campaign to
eat healthier or to not binge drink can only gain traction
if a key number of users propagate this message to others
using their influence. This problem has been defined as the
Positive Influence Dominating Set problem (PIDS).
PIDS vs. TPIDS A Positive Influence Dominating Set
(PIDS) is a subset P of a graph G = (V, E) such that
for each node u ∈ V , if less than some fraction of the
neighbors of u are not in P , then u ∈ P . In the literature
on social networks this fraction is fixed at 0.5 [4].
Figure 1a shows an example of a PIDS on a graph. The
nodes in the set P are shown in red. As can be seen from
the figure, every gray node has at least half its neighbors
in the set P .
A Total Positive Influence Dominating Set (TPID) is
similar to a PIDS, with a single alteration. While in a
PIDS P , the nodes in P are exempt from the requirement
of needing half their neighbors to be in P , in a TPIDS this
is not true. Therefore, in a TPIDS T , every node in V has
half of its neighbors in T , including the nodes in T itself.
Note the difference between Figure 1a and Figure 1b.
The difference in definition between TPIDS and PIDS
is simple, but it is sufficiently different that algorithms that
are suited to creating one may not be suited for creating
the other. In this paper we tackle the problem of computing
a PIDS.
The remainder of this paper is as follows. Section II
discusses the literature on Positive Influence Dominating
Sets in social networks. In Section III we present our
algorithm for computing a PIDS that we call AltGreedy.
In Section IV we present simulation results on PIDS
generation that compare our work to algorithms in the
literature. Finally, we conclude and discuss future work
in Section V.
(a) PIDS on the graph
(b) TPIDS on the graph
Fig. 1: Example of PIDS and TPIDS on a power-law graph
II. Related Work
While the topic of dominating sets has several years
of research behind it in particular for backbone formation
in Wireless Sensor Networks [7] [8] [9], the problem
of Positive Influence Dominating Sets (PIDS) in social
networks has been studied relatively recently. The problem
was introduced in [4] within the context of looking for a
solution to social issues. An example the paper cites is the
issue of binge drinking on college campuses. Intervention
programs may not have the budget to advertise to or
take in all binge drinkers. It is then desirable to select
a subset of the binge drinkers such that if a binge drinker
is not directly participating in an intervention program,
at least half of their binge drinking friends are. For
the problem application of binge drinking intervention,
Wang et. al. [4] categorized nodes as positive, neutral,
or negative. They collapsed neutral and negative into a
single negative category, and then used the two remaining
categories (positive and negative) as a starting point for
their PIDS computation. Nodes that are initially positive
(in the notation of the paper, the nodes in C) are able to
influence the other nodes they are connected to. The subset
P that is output by the algorithm proposed in [4] is not
a PIDS by itself, but P ∪ C will be the complete PIDS
of the graph. The strategy to select a PIDS used by Wang
et al. in [4] is to repeatedly select a 1-dominating set of
the nodes V − P ∪ C that have yet to be dominated, and
then dominate the nodes in P ∪ C by selecting the highest
degree neighbors of those nodes. The notion of the initial
positive compartment C is one that is generally not seen
beyond this paper. Wang et al. do mention that a graph
with C = ∅ is one where every node is initially negative,
which is the assumption most PIDS research has used. It
is important to note that in this paper ([4]), the definition
provided for PIDS is what is later called a TPIDS by [6],
and so for the purposes of the current work we use the
definitions in [6], which is what the definition for PIDS
and TPIDS presented in Section I follow.
In [10], a followup paper to [4], the authors present a
greedy algorithm to generate a TPIDS. Their algorithm is
based on a selection function f () defined as:
f(P ) =
P
u∈V
min(h(u), n
P
(u))
The strategy for this algorithm, henceforth referred to
as WangGreedy [10] is to greedily choose the node that
maximizes their evaluation function f. This function sums
the number of neighbors that each node has in the TPIDS-
in-progress P (given by the function n
P
(u)), with the
stipulation that each node can only contribute up to half
its neighbors to the sum. The function h(u) computes half
the neighbors of u.
We now take a look at how WangGreedy’s evaluation
function f behaves when generating TPIDS in comparison
to generating PIDS. When generating a TPIDS, the node
that maximizes f is simply the node that touches the
most unsatisfied nodes. To see this, consider first that the
TPIDS-in-progress P is empty initially. Then the node that
maximizes f will be the one with the highest degree, since
at first f is 0 because no nodes have any neighbors in
P and therefore all nodes are unsatisfied. Now consider
the case where P already contains some nodes. Then the
node u that maximizes f will also be the one that touches
the most unsatisfied nodes, since nodes that are already
satisfied by P cannot contribute any more to the sum in
f because of the min limitation, therefore the only way f
will increase is if u touches unmet nodes.
We adapt WangGreedy to generate PIDS by altering its
evaluation function:
f(P ) =
X
u∈V −P
min(h(u), n
P
(u))
We make these changes because it accurately represents
the notion that in comparison to a TPIDS, a PIDS does not
require the nodes in P to also have half their neighbors in
P . However, when adapted to the requirements for PIDS,
WangGreedy no longer chooses the nodes that touch the
most unmet nodes. This is because the evaluation function
f no longer considers nodes in P .
The authors of [11] take a very different greedy ap-
proach to generating a TPIDS. Their approach is to value
a node u based on how needy u’s neighbors are. This
has the effect of valuing nodes that might not necessarily
connect to a large number of unsatisfied nodes, but perhaps
instead connect to a few very needy nodes. Need is tracked
by checking if a node has at least half its neighbors in P
or if n
P
(u) <
l
deg (u)
2
m
. We refer to this algorithm as
RaeiGreedy in our comparison.
III. AltGreedy: A Greedy Algorithm for Pos-
itive Influence Dominating Set Generation
In this section, we present our algorithm for construct-
ing a Positive Influence Dominating Set (PIDS) which we
call AltGreedy. Since both WangGreedy [10] and Raei-
Greedy [11] were designed with Total Positive Influence
Dominating Sets (TPIDS) in mind, we seek to develop an
algorithm specifically targeted towards generating a PIDS
efficiently. This is because the PIDS problem is a more
accurate representation of spreading a positive message
since nodes participating in spreading the message are
already considered to be influenced and thus do not need
half their neighbors to repeat the message back to them.
This is also more closely in line with the traditional
definition of a dominating set whereby nodes in the set
do not need to be dominated themselves.
A. Notation
Here we define the notation necessary for our algorithm.
Let G = (V, E) be a graph G with nodes V and edges
E. The nodes V typically represent users on the social
network and the edges E represent relationships. For the
purpose of this research the graph G is undirected. Let P
be our PIDS on G. For some node u ∈ V , we define the
following notation.
Notation Definition
d(u) degree of node u
h(u)
l
d(u)
2
m
n(u) the set of the neighbors of u
n
P
(u) n(u) ∩ P
B. Algorithm: AltGreedy
We first define the following functions s() and g() on
a node u :
s(u) =
(
1 if n
P
(u) ≥ h(u) or u ∈ P
0 otherwise
g(u) = s(u) +
X
w∈n(u)
1 − s(w)
Our first function, s(u), represents whether a node is
satisfied under the definition of a PIDS. A node is said to
be satisfied when at least half its neighbors are in P .
Our second function, g(u), tallies the number of unsat-
isfied neighbors a given node u touches. To this tally, we
add 1 if u itself is unsatisfied. We do this because as noted
in the definition of PIDS, a node can be satisfied either by
having half of its neighbors in our PIDS P , or by being
in P itself. Thus we increment the tally by 1 to indicate
that selecting this node comes with the added benefit of
satisfying itself instantly.
Our algorithm which we call AltGreedy can be written
as follows.
1: P ← ∅
2: compute g(u) value for all u ∈ V
3: while P is not a PIDS do
4: select u ∈ V − P to maximize g(u)
5: and set P ← P ∪ {u}
6: revise g(w) values for all w ∈ V − P
7: end while
8: return P
Our algorithm’s strategy for generating a PIDS is to be
greedy and select the node that is connected to the most
unsatisfied nodes, and add that node to the PIDS under-
construction represented by the set P . To accomplish this,
after creating our empty set P that will become our PIDS
at termination, we compute g values for all the nodes in
our graph V . Our loop simply chooses the node with the
maximum g value, adds that to P , and recomputes the g
values for the remaining nodes that are not in P (denoted
in Line 6 by the nodes in V − P ). This is very similar in
flow to the algorithm presented by Raei et al. [11], but we
are substituting RaeiGreedy’s cover-degree criterion with
our g function as the basis of selecting nodes to add in V .
We chose to use the strategy of selecting the node that
is connected to the most unsatisfied nodes, taking into
account whether the node in question is unsatisfied or
not, because we are specifically targeting PIDS generation
with our algorithm. Both algorithms WangGreedy [10]
and RaeiGreedy [11] are aiming to construct a TPIDS.
Section IV examines the performance of those algorithms
when compared to AltGreedy via several experiments. For
fairness we test the algorithms to generate both a PIDS
and a TPIDS.
Complexity Analysis: Since the AltGreedy algorithm is
structured much like that of RaeiGreedy [11], we are able
to reference their time complexity proof to show that our
algorithm is O(n
2
).
The loop in line 3 will run at most n times (where n is
the number of nodes in the graph), since we can only add
as many nodes to our PIDS P as we have altogether, and
each iteration of the loop adds a node from V to P .
The first thing we do in the loop is to check if P is a
PIDS or not, and only continue if the latter is true (this
also takes place on line 3). In the worst case, we need
to evaluate every node to check if it satisfies the PIDS
conditions, so the complexity of this step is O(n).
Lines 4 and 6 of our algorithm both take O(n), since
we will have to choose from and revise first n nodes, then
n − 1 nodes, etc. Line 5 takes constant time (O(1)).
Therefore, our algorithm’s time complexity is O(n
2
).
IV. Simulation Results
In order to evaluate performance, all three algorithms -
WangGreedy, RaeiGreedy and AltGreedy were tested by
having them generate both a Positive Influence Dominating
Set (PIDS) and a Total Positive Influence Dominating Set
(TPIDS) on random scale-free graphs. The simulations
were designed to allow us to compare our algorithms with
those in [11]. Both WangGreedy and RaeiGreedy were
modified in the PIDS experiments to stop when a PIDS
has been achieved.
A. SNAP Library
We used the Stanford Network Analysis Platform
(SNAP) [12] to implement and perform our simulations.
The C + + SNAP library allowed us to easily generate
scale-free networks using their implementation of the
Barabasi-Albert model [13]. Using SNAP allowed us to
quickly implement the algorithms in question and scale
to larger graphs easily. Scale-free networks are a type of
graph that follow a power-law degree distribution. This
means that there are few nodes with relatively high degree
(which can be viewed as hubs), and many nodes with
low degree. [13] and [1] showed that social networks tend
to form scale-free networks. Thus for our experimental
work, we model social networks by randomly generating
graphs using the Barabasi-Albert model of preferential
attachment.
B. Simulations
Using the SNAP library to generate graphs with power-
law degree distribution (via the Barabasi-Albert algorithm),
we ran several simulations to compare the PIDS and
TPIDS generated by the three algorithms - WangGreedy,
RaeiGreedy, and AltGreedy.
1) PIDS Generation: PIDS were generated on graphs
of node sizes 500-1000 (increments of 100), all with an
average degree of 10. A total of 1000 random graphs were
generated using the Barabasi-Albert model for each data
point in this experiment. The average size of the PIDS
generated by the algorithms during this simulation are
displayed in Figure 2. As can be seen from the figure
, AltGreedy out performs WangGreedy and RaeiGreedy
by generating a PIDS that is about 10% smaller than the
other algorithms. Figure 2b shows what percentage of the
total nodes in the graph participate in the PIDS. Again, as
expected AltGreedy uses 2 −5% fewer nodes in the PIDS.
In addition to these results, we ran the RaeiGreedy
(since this represents the state-of-the-art in the literature
and performs better than WangGreedy) and AltGreedy
algorithms on data sets of larger graphs of with 5000 to
25000 nodes in increments of 5000, all with an average
degree 10. The results for this can be seen in Figure 3. The
trend seen in smaller graphs is preserved with AltGreedy
generating a smaller PIDS and using fewer nodes than
RaeiGreedy.
2) TPIDS Generation: We also ran simulations for the
Total Positive Influence Dominating Set (TPIDS) gener-
ation. To construct a TPIDS, the function s described in
Section III-B, can also be adapted to test satisfaction under
TPIDS, by simply removing the u ∈ P qualification, since
in a TPIDS every node needs to have at least half of its
neighbors in our TPIDS subset. As expected, on average
the size of the TPIDS generated were larger than that of the
PIDS for the same graph. Interestingly, TPIDS and PIDS
exhibit opposite behaviors for certain tests. To replicate
an experiment from [11], we used all three algorithms
to generate both TPIDS (as in the original experiment)
and also PIDS for graphs of a fixed size 200, while
incrementing the average degree by 2.
For TPIDS, the data corresponds to the results published
in [11], in that the higher the average degree, the smaller
the average TPIDS. However, the opposite behavior is
exhibited when we generate a PIDS, i.e., a higher average
degree means we select more nodes. This behavior makes
sense, because when generating a TPIDS (T ), not only do
the nodes not in T need to have half of their neighbors in
T , but so do the nodes in T itself. When there is a low
