Opinion Maximization in Social Networks

0.44 MB·English

by  Aristides Gionis

Opinion maximization in social networks
Aristides Gionis∗ Evimaria Terzi† Panayiotis Tsaparas‡
Abstract terfere with the opinion-formation process by influenc-
Theprocessofopinionformationthroughsynthesisand ing the opinions of appropriately selected individuals,
contrast of different viewpoints has been the subject such that the overall favorable opinion about the spe-
of many studies in economics and social sciences. To- cific information item is strengthened.
3 day, this process manifests itself also in online social The idea of leveraging social influence for market-
1 networks and social media. The key characteristic of ingcampaignshasbeenstudiedextensivelyindatamin-
0 successful promotion campaigns is that they take into ing. Introduced by the work of Domingos and Richard-
2 considerationsuchopinion-formationdynamicsinorder son [20], and Kempe et al. [14] the problem of influence

n to create a overall favorable opinion about a specific in- maximization asks to identify the most influential indi-
a formationitem,suchasaperson,aproduct,oranidea. viduals, whose adoption of a product or an action will
J
Inthispaper,weadoptawell-establishedmodelfor spread maximally in the social network. This line of
0
3 social-opinion dynamics and formalize the campaign- workemploysprobabilisticpropagationmodels,suchas
design problem as the problem of identifying a set the independent-cascade or the linear-threshold model,

] of target individuals whose positive opinion about an whichspecifyhowactionsspreadamongtheindividuals
I
S information item will maximize the overall positive of the social network. At a high level, such propagation
. opinion for the item in the social network. We call models distinguish the individuals to either active or
s
c this problem Campaign. We study the complexity inactive, and assume that active individuals influence
[ of the Campaign problem, and design algorithms for their neighbors to become active according to certain

1 solving it. Our experiments on real data demonstrate probabilistic rules, so that activity spreads in the net-
v the efficiency and practical utility of our algorithms. work in a cascading manner.
5 Inthispaper,weareinterestedintheprocessofhow
5 1 Introduction individuals form opinions, rather than how they adopt
4 products or actions. We find that models such as the
7 Individuals who participate in social networks form
independent cascade and the linear threshold are not
. their opinions through synthesis and contrast of dif-
1 appropriate for modeling the process of forming opin-
0 ferent viewpoints they encounter in their social circles.
ions in social networks. First, opinions cannot be ac-
3 Suchprocessesmanifestthemselvesmorestronglyinon-
curately modeled by binary states, such as being either
1 line social networks and social media where opinions
: active or inactive, but they can take continuous values
v and ideas propagate faster through virtual connections
in an interval. Second, and perhaps more importantly,
i between social-network individuals. Opinion dynam-
X the formation of opinions does not resemble a discrete
ics have been of considerable interest to marketing and
r cascading process; it better resembles a social game in
a opinion-formationagencies,whichareinterestedinrais-
which individuals constantly influence each other until
ing public awareness on important issues (e.g., health,
convergence to an equilibrium.
social justice), or increasing the popularity of person or
Accordingly,weadopttheopinion-formationmodel
an item (e.g., a presidential candidate, or a product).
of Friedkin and Johnsen [11], which assumes that each
Today, social-networking platforms and social me-
node i of a social network G = (V,E) has an internal
diatakeupasignificantamountofanypromotioncam-
opinions andanexpressedopinionz . Whiletheinter-
paignbudget. Itisnotuncommontoseeactivistgroups, i i
nalopinionsofindividualsarefixedandnotamenableto
political parties, or corporations launching campaigns
external influences, the expressed opinions can change
primarily via Facebook or Twitter. Such campaigns in-
due to the influence from their neighbors. More specif-
ically, people change their expressed opinion z so that
i
∗Aalto University, Finland. This work was done while the they minimize their social cost, which is defined as the
authorwaswithYahoo!Research. disagreement between their expressed opinion and the
†Boston University, USA. Supported by NSF awards: CNS-
expressed opinion of their social neighbors, and the di-
1017529,III-1218437andgiftsfromMicrosoftandGoogle.
‡UniversityofIoannina,Greece. vergence between the expressed opinion and their true
internalbelief. Itcanbeshownthattheprocessofcom- sociology and computer science.
puting expressed opinion values by repeated averaging In his original work in 1974, DeGrout [6] was the
leads to a Nash equilibrium of the game where the util- first to define a model where individuals organized in
ities are the node social costs [3]. a social network G = (V,E) have a starting opinion
Armed with this opinion-formation model, we in- (represented by a real value) which they update by
troduce and study a problem similar to the influence- repeatedlyadoptingtheaverageopinionoftheirfriends.
maximizationproblem,introducedbyKempeetal.[14]. This model and its variants, has been subject of recent
We consider a social network of individuals, each hold- studies in social sciences and economics [2, 7, 11, 12,
ing an opinion value about an information item. We 13]. Most of these studies focus on identifying the
view an opinion as a numeric value between zero (neg- conditionsunderwhichsuchrepeated-averagingmodels
ative) and one (positive). We then ask to identify k converge to a consensus. In our work, we also adopt a
individuals, so that once their expressed opinions be- repeated averaging model. However, our focus is not
comes 1, the opinions of the rest of the nodes — at the characterization or the reachability of consensus.
an equilibrium state — are, on average, as positive as In fact, we do not assume consensus is ever reached.
possible. We call this problem Campaign. Rather, we assume that the individuals participatingin
Note that there is a fundamental qualitative differ- the network reach a stable point, where everyone has
ence between our framework and existing work on in- crystallized a personal opinion. As has been recently
fluence maximization: In contrast to other models used notedbysocialscientistDavideKrackhardt[16],studies
ininfluence-maximizationliterature,ourworkviewsthe of such non-consensus states are much more realistic
nodesasrationalindividualswhowishtooptimizetheir since consensus states are rarely reached.
own objectives. Thus, we assume that the opinions of Key to our paper is the work by Bindel et al. [3].
individuals get formed through best-response dynamics In fact, we adopt the same opinion-dynamics model
of a social game, which in turn is inspired by classi- as in [3], where individuals selfishly form opinions
calopinion-dynamicsmodelsstudiedextensivelyineco- to minimize their personal cost. However, Bindel et
nomics [2, 6, 7, 12, 13]. al. focus on quantifying the social cost of this lack
Furthermore, from a technical point of view, the of central coordination between individuals, i.e., the
maximization problem that results from our framework price of anarchy, and they consider a network-design
requires a completely different toolbox of techniques problem with the objective of reducing the social cost
than the standard influence-maximization problem. To atequilibrium. Ourworkontheotherhand, focuseson
this end, we exploit an interesting connection between designingapromotioncampaignsothat,atequilibrium,
our problem and absorbing random walks in order to the overall positive inclination towards a particular
establishourtechnicalresults,includingthecomplexity opinion is maximized.
and the approximability of the problem. Interestingly, Recently, there has been a lot of work in the
as with the influence-maximization problem, we show computer-science literature on identifying a set of in-
that the objective function of the Campaign problem dividuals to advertise an item (e.g., a product or an
issubmodular,andthus,itcanbeapproximatedwithin idea) so that the spread of the item in the network is
factor (1− 1) using a greedy algorithm. maximized. Different assumptions about how informa-
e
In addition, motivated by the properties of the tion items propagate in the network has led to a rich
greedy algorithm, we propose scalable heuristics that literature of targeted-advertisement methods (e.g., see
can handle large datasets. Our experiments show that [5, 9, 14, 20]). Although at a high level our work has
in practice the heuristics have performance comparable the same goal as all of these methods, there are also
to the greedy algorithm, and that are scalable to very important differences, as we have already discussed in
large graphs. Finally, we discuss two natural variants the introduction.
of the Campaign problem. Our discussion reveals a
surprising property of the opinion formation process on 3 Problem definition
undirected graphs: the average opinion of the network 3.1 Preliminaries. We consider a social graph G =
dependsonlyontheinternalopinionsoftheindividuals (V,E) with n nodes and m edges. The nodes of the
and not on the network structure. graph represent people and the edges represent social
affinity between them. We refer to the members of
2 Related Work the social graph by letters such as i and j, and we
Tothebestofourknowledge,wearethefirsttoformally write (i,j) to denote the edges of the graph. With
defineandstudytheCampaignproblem. However,our each edge (i,j) we associate a weight w ≥ 0, which
ij
work is related to a lot of existing work in economics, expresses the strength of the social affinity or influence
from person i to person j. We write N(i) to denote Equation (3.1). That is, the stationary vector of opin-
the social neighborhood of person i, that is, N(i)={j | ions z is such that no node i has an incentive to change
(i,j) ∈ E}. Unless explicitly mentioned, we do not their opinion to improve their cost c .
i
make an assumption whether the graph G is directed
or undirected; most of our results and our algorithms 3.2 Problem definition. The goal of a promotion
carryoverforbothtypesofgraphs. Thedirected-graph campaign is to improve the overall opinion about a
model is more natural as in many real-world situations product, person, or idea in a social network. Given an
the influence w from person i to person j is not equal opinion vector z, we define the overall opinion g(z) as
ij
to w .
ji
Following the framework of Bindel et al. [3] we as- (cid:88)n
g(z)= z ,
sume that person i has a persistent internal opinion s , i
i
whichremainsunchangedfromexternalinfluences. Per- i=1
son i has also an expressed opinion z , which depends
i which is also proportional to the average expressed
on their internal opinion s , as well as on the expressed
i opinion oftheindividualsinG. Thegoalofacampaign
opinions of their social neighborhood N(i). The under-
is to maximize g(z). Following the paradigm of Kempe
lying assumption is that individuals form opinions that
et al. [14], we assume that such a campaign relies on
combine their internal predisposition with the opinions
selecting a set of target nodes T, which are going to be
of those in their social circle.
convinced to change their expressed opinions to 1. For
Wemodeltheinternalandexternalopinionss and
i therestofthediscussion, wewilluseg(z |T)todenote
z as real values in the interval [0,1]. The convention
i the overall opinion in the network, when vector z is the
is that 0 denotes a negative opinion, and 1 a positive
Nash-equilibrium vector obtained under the constraint
opinion. Thevaluesin-betweencapturedifferentshades
that the expressed opinions of all nodes in T are fixed
of positive and negative opinions. Given a set of to 1. Given this notation, we can define the Campaign
expressed opinion values for all the people in the social
problem as follows.
graph, represented by an opinion vector z = (z : i ∈
i
V), and the vector of internal opinions s=(s :i∈V),
i Problem 1. (Campaign) Given a graph G = (V,E)
we consider that the personal cost for individual i is
and an integer k, identify a set T of k nodes such that
(cid:88) fixing the expressed opinions of the nodes in T to 1,
(3.1) c (z)=(s −z )2+ w (z −z )2.
i i i ij i j maximizes the overall opinion g(z |T).
j∈N(i)
This cost models the fact that the expressed opinion zi Weemphasizethatfixingzi =1foralli∈T means
of an individual i is a “compromise” between their own thatEquation(3.2)isonlyappliedforthezj’ssuchthat
internalbeliefsi andtheopinionsoftheirneighbors. As j ∈/ T, while for the nodes i∈T the values zi remain 1.
theindividualiformsanopinionz ,theirinternalopin- The definition of the Campaign problem reflects
i
ion s and the opinions z of their neighbors j ∈ N(i) our belief of what constitutes a feasible and effec-
i j
mayhavedifferentimportance. Therelativeimportance tive campaign strategy. Expressed opinions are more
of those opinions is captured by the weights w . amenable to change, and have stronger effect on the
ij
Now assume that, as a result of social influence overall opinion in the social network. Thus, it is rea-
and conflict resolution, every individual i is selfishly sonable for a campaign to target these opinions. We
minimizingtheirowncostc (z). Iftheinternalopinions note that other campaign strategies are also possible,
i
are persistent and cannot change (an assumption that resulting in different problem definitions. For example,
we carry throughout), minimizing the cost c implies one can define the problem where the campaign aims
i
changing the expressed opinion z to the weighted at changing the fundamental beliefs of people by alter-
i
averageamongtheinternalopinionsi andtheexpressed ing their internal opinions si. It is also conceivable to
opinions of the neighbors of i. In other words, ask whether it is possible to improve the overall opin-
ion g(z) by introducing a number of new edges in the
(cid:80)
si+ j∈N(i)wijzj social graph, e.g., via a link-suggestion application. We
(3.2) z = .
i 1+(cid:80) w discuss both of these variants at the end of the paper.
j∈N(i) ij
It turns out that from the algorithmic point of view,
Infact,itcanbeshownthatifeverypersoniiteratively boththeseproblemsarerelativelysimple. Forinstance,
updates their expressed opinion using Equation (3.2), we can show that for undirected graphs, surprisingly, it
then the iterations converge to a unique Nash Equilib- is not possible to improve the overall opinion g(z) by
rium for the game with utilities of players expressed by introducing new edges in the graph.
3.3 Background. We now show the connection be- Assume that each absorbing node j ∈ B is associ-
tweencomputingtheNash-equilibriumopinionvectorz ated with a fixed value b . If a random walk starting
j
and a random walk on a graph with absorbing nodes. fromtransientnodei∈U getsabsorbedinanabsorbing
This connection is essential in the analysis of the Cam- node j ∈B, then we assign to node i the value b . The
j
paign problem. probability of the random walk starting from node i to
Absorbing random walks: Let H = (X,R) be a be absorbed in j is QUB(i,j). Therefore, the expected
(cid:80)
graph with a set of N nodes X, and a set of edges value of i is fi = j∈BQUB(i,j)bj. If fU is the vector
R. The graph is also associated with the following withtheexpectedvaluesforalli∈U,andfB keepsthe
three N ×N matrices: (i) the weight matrix W with values {bj} for all j ∈B, then we have that
entries W(i,j) denoting the weight of the edges; (ii)
(3.3) f =Q f .
the degree matrix D, which is a diagonal matrix such U UB B
that D(i,i) = (cid:80)N W(i,j); (iii) the transition matrix
j=1 A fundamental observation, which highlights the
P = D−1W, which is a row-stochastic matrix; P(i,j)
connection between our work and random walks with
expressestheprobabilityofmovingfromnodeitonode
absorbing states, is that the expected value f of node
i
j in a random walk on the graph H.
i ∈ U can be computed by repeatedly averaging the
In such a random walk on the graph H, we say
values of the neighbors of i in the graph H. Therefore,
that a node b ∈ X is an absorbing node, if the random
thecomputationoftheNash-Equilibriumopinionvector
walk can only transition into that node, but not out
z canbedoneusingEquation(3.3)onanappropriately
of it (and thus, the random walk is absorbed in node
constructedgraphH. WediscusstheconstructionofH
b). Let B ⊆ X denote the set of all absorbing nodes
below. Moredetailsonabsorbingwalkscanbefoundin
of the random walk. The set of the remaining nodes
the excellent monograph of Doyle and Snell [8].
U =X\B arenon-absorbing,ortransient nodes. Given
The augmented graph. We will now show how the
this partition of the states in X, the transition matrix
theory of absorbing random walks described above can
of this random walk can be written as follows:
be leveraged for solving the Campaign problem. This
(cid:18) (cid:19)
P P connection is achieved by performing a random walk
P = UB UU .
I O with absorbing states on an augmented graph H =
(X,R), whose construction we describe below.
In the above equation, I is an (N −|U|)×(N −|U|) GivenasocialnetworkG=(V,E)whereeveryedge
identitymatrixandO amatrixwithallitsentriesequal (i,j)∈Eisassociatedwithweightw ,weconstructthe
ij
to 0; PUU is the |U| × |U| sub-matrix of P with the augmented graph H =(X,R) of G as follows:
transition probabilities between transient states; and
P isthe|U|×|B|sub-matrixofP withthetransition (i) thesetofverticesX ofH isdefinedasX =V ∪V,
UB
probabilities from transient to absorbing states. where V is a set of n new nodes such that for each
An important quantity of an absorbing random node i∈V there is a copy σ(i)∈V;
walk is the expected number of visits to a transient
(ii) the set of edges R of H includes all the edges E of
state j when starting from a transient state i before
G,plusanewsetofedgesbetweeneachnodei∈V
being absorbed. The probability of transitioning from
and its copy σ(i) ∈ V. That is, R = E ∪E, and
i to j in exactly (cid:96) steps is the (i,j)-entry of the matrix
E ={(i,σ(i))|i∈V};
(P )(cid:96). Therefore, the probability that a random walk
UU
starting from state i ends in j without being absorbed (iii) theweightsofallthenewedges(i,σ(i))∈Rareset
is given by the (i,j) entry of the |U|×|U| matrix to 1, i.e., W(i,σ(i)) = 1. For i,j ∈ V, the weight
of the edge (i,j) ∈ R is equal to the weight of the
∞
F =(cid:88)(P )(cid:96) =(1−P )−1, corresponding edge in G, i.e., W(i,j)=wij.
UU UU
(cid:96)=0 Our main observation is that we can compute the
opinion vector z that corresponds to the Nash equi-
which is known as the fundamental matrix of the
librium defined by Equation (3.1) by performing an
absorbing random walk. Finally, the matrix
absorbingrandomwalkonthegraphH. Inthisrandom
Q =F P walk, we set B = V and U = V, that is, we make
UB UB
all copy nodes in V to be absorbing. We also set
is an |U|×|B| matrix, with Q (i,j) being the proba- f = s, that is, we assign value s to each absorbing
UB B i
bilitythatarandomwalkwhichstartsattransientstate node σ(i). The Nash-equilibrium opinion-vector z can
i ends up being absorbed at state j ∈B. be computed using Equation (3.3), that is, z = Q s.
UB
(cid:80)
The opinion z = Q (i,j)s is the expected the less competition there is for x (i.e., the smaller the
i j∈B UB j
internal opinion value at the node of absorption for a size of B), the more mass of the random walk will be
random walk that starts from node i ∈ V. Given the absorbed in x, and the larger the increase of g(z | T).
vector z we can compute the overall opinion g(z). Hence g(z |T) is submodular.
TheCampaignproblemcanbenaturallydefinedin
this setting. Selecting a set of nodes T is equivalent to 5 Algorithms
addingthenodesinT intothesetofabsorbingnodesB, 5.1 Estimating the Nash-equilibrium vector z.
andassigningthemvalue1. Thatis,wehaveB =V ∪T A central component in all the algorithms presented in
and U = V \T. For the vector fB, we have fσ(i) = si thissectionistheestimationofthetheopinionfunction
for all σ(i) ∈ V, and fj = 1 for all j ∈ T. We use g(z |T). In Section 3.3, we have already discussed that
Equation (3.3) to compute vector z and using this z, this can be done by evaluating Equation (3.3). For ap-
we can then compute the overall opinion g(z | T). propriatelydefinedsetsU andB, thisrequirescomput-
Hence, the Campaign problem becomes the problem ing the matrix Q = (1−P )−1P . Hence, this
UB UU UB
of selecting a set of k nodes T ⊆ V to make absorbing calculation involves a matrix inversion, which is very
with value 1, such that g(z |T) is maximized. inefficient. The reason is that despite the fact that the
social graph is typically sparse, matrix inversion does
4 Problem complexity not preserve sparseness. Thus, it may be too expensive
In this section, we establish the complexity of the to even store the matrix QUB.
Campaign problem by showing that it is an NP-hard Instead, we resort to the power-iteration method
problem. We also discuss properties of the objective implied by Equation (3.2): at each iteration we update
function g(z | T), which give rise to a constant-factor the opinion zi of a node i ∈ V by averaging the
approximation algorithm for the Campaign problem. opinions of its neighbors j ∈ N(i) and its own internal
opinion s . During the iterations we do not update the
i
Theorem 4.1. Problem Campaign is NP-hard. values of opinions that are fixed. This power-iteration
method is known to converge to the equilibrium vector
The proof of the theorem appears in the Ap-
z, and it is highly scalable, since it only involves
pendix A. The proof relies on a reduction from
multiplication of a sparse matrix with a vector. For
the Vertex Cover on Regular Graphs problem
a graph with n nodes, m edges, and thus, average
(VCRG) [10]. degree d = 2m, the algorithm requires O(nd) = O(m)
Since the Campaign problem is NP-hard, we are n
operationsperiterations. Therefore,theoverallrunning
content with algorithms that approximate the optimal
timeisO(mI),whereI isthetotalnumberofiterations.
solution in polynomial time. Fortunately, we can show
In our experiments we found the the method converges
thatthefunctiong(z |T)ismonotoneandsubmodular,
in around 50-100 iterations, depending on the dataset.
and thus a simple greedy heuristic yields a constant-
factor approximation to the optimal solution.
5.2 Algorithms for the Campaign problem.
Our algorithms for the Campaign problem, include a
Theorem 4.2. The function g(z |T) is monotone and
constant-factorapproximationalgorithmaswellassev-
submodular.
eral efficient and effective heuristics.
Proof. We only give here a proof sketch. A detailed The Greedy algorithm. It is known that the greedy
proof is given in Appendix B. Recall that g(z | T) = algorithm is a (1 − 1)-approximation algorithm for
(cid:80) z . In the absorbing random walk interpretation maximizing a submoduelar function h : Y → R subject
i∈V i
of the opinion formation process, we have shown that to cardinality constraints, i.e., finding a set A⊆Y that
the expressed opinion of node i is the expected opinion maximizes h(A) such that |A| ≤ k [19]. Consequently,
valueatthepointofabsorbtionforarandomwalkthat the Greedy algorithm constructs the set T by adding
(cid:80)
starts from node i. That is, z = P(b | i)f , one node in each iteration. In the t-th iteration the
i b∈B b
where B is the set of absorbing nodes, P(b | i) is the algorithm extends the set T(t−1) by adding the node i
probability of the random walk starting from node i that maximizes g(z |T(t)) when setting z =1.
i
to be absorbed at node b, and f the opinion value at The computational bottleneck of Greedy is due to
b
node b. When we add a node x to T, and hence to the fact that we need to compute the Nash-equilibrium
the set B, some of the probability mass of the random opinion vector z that results from setting z =1 for all
t
walk will be absorbed at x. Since x has the maximum t ∈ T ∪ {j}, and we need to do such a computation
possibleopinionvalue,f =1,iffollowsthatz canonly for all candidate nodes j ∈ V \ T. Overall, for a
x i
increase, and thus g(z | T) is monotone. Furthermore, solution T of size |T| = k Greedy needs to perform
Karate club
degree 51015 llllllllllllllllllllllllllllllll degree to free neighbors 051015 llllllllllllllllllllllllllllllll internal opinion 0.00.20.40.60.8 llllllllllllllllllllllllllllllll
0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30
k k k
Figure 1: Measures of graph nodes plotted in order selected by the Greedy algorithm.
O(nk) computations of finding the optimal vector z. already selected by Greedy. Finally, in the third panel
As we saw, each of these computations is performed ofFigure1weseetheinternalopinions ofnodesinthe
i
by a power-iteration in time O(mI), yielding an overall order selected by the Greedy. We see that the Greedy
running time O(nmkI). Such a running time is super- tends to select first nodes with low internal opinion.
quadratic and therefore the algorithm is not scalable to There are a few exceptions of nodes with high internal
very large datasets. opinion s selected at the initial steps of greedy. Such
i
One way to speedup the algorithm is by storing, nodes are nodes with high degree, connected to many
for each node that it is not yet selected, its marginal nodes with small values of s .
i
improvement on the score, at the last time it was com- Armed with intuition from this analysis we now
puted. This speedup, which is commonly used in op- proceed to describe our heuristics.
timization problems with submodular functions [17], is
The Degree algorithm. This algorithm simply sorts
not adequate to make the greedy algorithm applicable
the nodes of G = (V,E) in decreasing order of their in
for large data, at least for the version of the algorithm
degree and forms the set of target nodes T by picking
described here. The reason is that in the very first iter-
the top-k nodes of the ranking. The running time of
ationthereisnopruningandthereforeweneedtomake
Degree is O(nlogn), i.e., the time required for sorting.
O(n) power-iteration computations, yielding again a
quadraticalgorithm. Toovercome thesescalability lim- The FreeDegree algorithm. This algorithm is a
itations, we present a number of scalable heuristics. “greedy” variant of Degree; FreeDegree forms the set
T iteratively by choosing at every iteration the node
5.3 Designing the heuristics. To characterize the with the highest free degree. The free degree of a node
nodes selected by Greedy we execute the algorithm on is the sum of the weights of the edges that are incident
small datasets, and we compute a number of measures to it and are are connected to nodes not already in T.
for each node selected by Greedy. In particular, for WhenthesetT consistsofk nodes,therunningtimeof
each node we compute measures such as its degree, the FreeDegree is O(kn).
average degree of its neighbors, the maximum degree of The RWR algorithm. As we saw in Figure 1, a
its neighbors, the value of its internal opinion si, the good choice for nodes to be added in the solution
average value of si over its neighbors, and so on. Three are not only the nodes of high degree but also the
of the features with the most clear signal are shown nodes of small value of internal opinion s . The RWR
i
in Figure 1 for the karate club dataset (described in algorithm combines both of these features: selecting
detail in Section 6). We obtain similar behavior on all nodes with high degree and with small s . This is done
i
the datasets we tried. byperformingarandomwalkwithrestart(RWR),where
In Figure 1 we plot measures of nodes in the order the probability of restarting at a node i is proportional
selected by the Greedy. A good measure would be to r = s − s , where s = max s , and
i max i max i∈V i
one that is monotonic with respect to this order. In orderingthenodesaccordingtotheresultingstationary
the first panel, we show the degree of a node in the distribution. The intuition is that a random walk
selectionorderof Greedy,andweseethatGreedytends favors high-degree nodes, and using the specific restart
to select first high degree nodes. As shown in the probabilities favors nodes with low value of s .
i
secondpanel,thisdependenceisevenmoreclearforthe For the restart probability, we use the parameter
free degree, i.e., the number of neighbors that are not α = 0.15, which has been established as a standard
parameter of the PageRank algorithm [4]. Making 6.2 Bibliographic datasets: A “data mining”
one RWR computation can be achieved by the power- campaign. We also evaluate our algorithms on two
iteration method, which similarly to computing the large social networks, derived from bibliographic data.
optimal vector z, has running time O(mI). Therefore, The first dataset, bibsonomy, is extracted from
theoverallrunningtimeofthealgorithmforselectinga bibsonomy [1], a social-bookmarking and publication-
set T of size k is O(mkI). sharing system. From the available data, we extract a
social graph of 45329 nodes representing authors and
The Min-S and Min-Z algorithms. The Min-S
149895 edges representing co-authorship relations. For
algorithm simply selects the k nodes with the smallest
each author we also keep the set of tags that have been
value s . This heuristic is motivated by the observation
i
used for the papers of that author.
that the Greedy algorithm tends to select nodes with
The second dataset, dblp, is also a co-authorship
small value s . For completeness, we also experiment
i
graph among computer scientists extracted from the
with the Min-Z algorithm, which greedily selects and
dblp site.2 The dataset is a large graph containing
add in the solution set T the node that at the current
635585 nodes and 1423716 edges. Again, for each
iteration has the smallest value of expressed opinion z .
i
author we keep the set of terms they have been used
in the titles of the papers they have co-authored.
6 Experimental evaluation
Inthisexperiment,wegeneratetheinternalopinion
The objective of our experiments is to compare the
vectors by identifying keywords related to data mining
proposed heuristics against Greedy, the algorithm with
(e.g., we picked data, mining, social, networks, graph,
the approximation guarantee, and demonstrate their
clustering, learning, and community). For each author
scalability.
i ∈ V we then set s to be the fraction of the above
i
keywords present in his set of terms. This setting
6.1 Small networks. Weexperimentwithanumber
corresponds to a hypothetical scenario of designing a
of publicly available small networks.1 We evaluate our
campaign to promote the “data mining” topic among
algorithms by reporting the value of the objective func-
all computer-science researchers.
tion g(z) as a function of the solution set size |T| = k.
The results of the heuristics for the two datasets
Our results for three small networks are shown in Fig-
are shown in Figure 3. For the bibsonomy dataset
ure 2. The datasets shown in the figure are the follow-
(left), there is a clear distinction between the three
ing: (i) karate club: a social network of friendships
heuristics;RWRclearlyperformsbetterthanbothDegree
between34membersofakarateclubataUSuniversity
and FreeDegree. This superior performance of RWR
in the 1970s [21]; (ii) les miserables: co-appearance
is expected as this algorithm takes into account both
network of characters in the novel Les Miserables [15];
the degrees and the values of the internal opinions s .
and(iii)dolphins: anundirectedsocialnetworkoffre- i
Also the better performance of FreeDegree compared
quentassociationsbetween62dolphins[18]. Forthisset
with the performance of Degree is consistent with the
of experiments we set the internal opinions s to be a
i results obtained for smaller networks. On the other
uniformly-sampled value in [0,1].
hand, on the dblp dataset (right part of Figure 3) the
We see that the difference between all the algo-
behavior of the three heuristics is more surprisingly, as
rithmsisrelativelysmallbuttheirrelativeperformance
all three perform almost identical. Finally, for both
is consistent. The Greedy algorithm achieves the best
datasets, the difference of the three best heuristics
results,whilethethreeheuristics,Degree,FreeDegree,
Degree, FreeDegree, and RWR with the other two
and RWR come close together. Min-S and Min-Z have
heuristics, Min-S and Min-Z is more pronounced. In
the poorest performance, even though for larger val-
fact, the performance of Min-S and Min-Z is very poor.
ues of k they improve and slightly outperform some of
We investigate the difference on the relative perfor-
the heuristics. Between the two, Min-S performs best,
manceofthebestthreeheuristics,Degree,FreeDegree,
outperforming Min-Z, especially for small values of k.
and RWR, on the two datasets by plotting the degrees of
Both of those trends are expected: the three heuris-
the nodes versus their s value. This is shown in Fig-
ticsDegree,FreeDegree,andRWR,arebettermotivated i
ure 4. Recall that our intuition for selecting nodes that
than Min-S, which in turn, is better motivated than
istochoosenodesthathavelargedegreeandsmallvalue
Min-Z.
of s . Figure 4 demonstrates that in the dblp dataset,
We obtain similar results for other small networks, i
large degree correlates well with small s values, while
although we do not provide the plots for lack of space. i
this is not the case for the bibsonomy dataset. There-
1www-personal.umich.edu/˜mejn/netdata/ 2www.informatik.uni-trier.de/˜ley/db/
Karate club Les miserables Dolphins
g(z)02530 l ll ll ll lll llgfrreellee.ddlleygrlleell g(z)06070 llllllllllllllllllllllllllllgfrllreelleell.ddlleyllgllrllellell g(z)405060 ll ll ll ll ll ll ll lll ll llgfllrrelleeell.ddlleyllgrllellell
2 RWR 5 RWR RWR
l l
5 l degree 0 l degree 30 l degree
1 l min.s 4 l min.s l min.s
l
min.z min.z min.z
10 ll 30 ll 20 ll
0 5 10 15 20 25 30 0 10 20 30 40 50 60 70 0 10 20 30 40 50
k k k
Figure 2: Performance of the algorithms on small networks.
Bibsonomy −− data mining DBLP −− data mining
l
z) 0010000 ll RfdmmreeWiinnge..Rr.szdeeegrleel l l l l l l l l l l l z) 5000060000 ll RfdmmreeWiinnge..Rr.szdeeegreel l l l l l l l l l l
g( 60 l l g( 000 l l
0
l 4 l
02000 ll l l l l l l l l l l l l l l l l 30000 ll ll l l l l l l l l l l l l l l l
0 100 200 300 400 0 100 200 300 400
k k
Figure 3: Performance of the algorithms on the bibsonomy and dblp networks for the topic “data mining”.
7 Problem variants
The Campaign problem we studied in this paper fo-
cusesoncampaignsthataimtoaltertheexpressedopin-
ions of individuals. However, other campaign strategies
arealsopossible. Forexampleonecouldaimataltering
the internal opinions of individuals such that the over-
all opinion is improved as much as possible. Formally,
the goal would be to select a set of nodes S, which are
going to be convinced to change their internal opinions
s to 1, such that the resulting overall opinion g(z | S)
i
Figure 4: Scatter plot of degrees vs. internal opinion is maximized. We call this problem the i-Campaign
values s in the two datasets, bibsonomy and dblp. problem. The difference between the Campaign and i-
i
Campaignproblemsisthatintheformerweareasking
to fix the expressed opinions z for k individuals, while
i
in the latter we are asking to fix the internal opinions
s . Even though the difference is seemingly small, the
i
problems are computationally very different.
fore,fordblpallthreeheuristicspickhigh-degreenodes,
Algorithmically, theproblemofselectingS individ-
which makes their performance almost identical.
uals to change their internal opinions, so that we max- References
imize g(z | S) is much simpler. In fact, we can show
that for an undirected social graph G = (V,E), where
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A Proof of Theorem 4.1 in node σ is strictly greater than zero. Therefore, the
(cid:96)
Proof. We prove the theorem by reducing an instance probability of being absorbed in some node not in T is
oftheVertex Cover on Regular Graphsproblem strictly greater than d+11, and thus zj < d+d1. It follows
(VCRG)[10]toaninstanceofthedecisionversionofthe that g(z |T)<(n−k) d .
d+1
Campaign problem. We remind that a graph is called
regular if all its nodes have the same degree. B Proof of Lemma 4.2
Given a regular graph GVC = (VVC,EVC) and an Proof. Recallthatg(z |T)=(cid:80)i∈V zi,wherethevalues
integerK theVCRGproblemaskswhetherthereexists of vector z are computed using Equation (3.3). As we
a set of nodes Y ⊆ VVC such that |Y| ≤ K and Y is have already described, we can view the computation
a vertex cover (i.e., for every (i,j) ∈ EVC it is i ∈ Y of the vector z as performing a random walk with
or j ∈Y). absorbing nodes on the augmented graph H = (V ∪
An instance of the decision version of the Cam- V,E∪E). AssumethatB isthesetofabsorbingnodes,
paign problem consists of a social graph G = (V,E), and that each b ∈ B is associated with value f . Let
b
internal opinions s, an integer k and a number θ. The P (b | i) be the probability that a random walk that
B
solution to the decision version is “yes” iff there exists starts from i gets absorbed at node b, when the set of
a set T ⊆V such that |T|≤k and g(z |T)≥θ. absorbing nodes is B. The expressed opinion of node
Given an instance of the VCRG problem, we will i∈/ B is
construct an instance of the decision version of Cam- z = (cid:88)P (b|i)f .
i B b
paign by setting G = (V,E) to be equal to G =
VC b∈B
(V ,E ), s = 0 for every i ∈ V, k = K and
VC VC i Since this value depends on the set B, we will write
θ =(n−k) d . Then,weshowthatT ⊆V isasolution
d+1 z(i | B) to denote the value of z when the set of
to the Campaign problem with value g(z | T) ≥ θ if i
absorbing nodes is B.
and only if T is a vertex cover of the input instance of
Initially, the set of absorbing nodes is B = V and
VCRG.
f =s for all b∈B. When we select a subset of nodes
Inordertoshowthis, weusetheabsorbingrandom b b
T ⊆V suchthattheirexpressedopinionsarefixedto1,
walk interpretation of the Campaign problem. Recall
wehavethatB =V∪T,f =s forallb∈V andf =1
that by the definition of the Campaign problem, every b b b
for all b∈T. Since the set of nodes V is always part of
node i ∈ T becomes an absorbing node with value
the set B, and the parameter that we are interested in
z =1. For every other node j ∈/ T we compute a value
i for this proof is the target set of nodes T, we will use
z , which is the expected value at the absorption point
j z(i | T) to denote z(i | B) where B = V ∪T. We thus
ofarandomwalkthatstartsfromj. Sinces =0forall
i have
i ∈ V and z = 1 for all i ∈ T, the value z represents (cid:88)
i j g(z |T)= z(i|T).
the probability that the random walk starting from j
i∈V
will be absorbed in some node in T.
NotethatthesummationisoverallnodesinV including
Now suppose that T is a vertex cover for G .
VC
the nodes in T. If i ∈ T, then P (i | i) = 1, and
Then, for every non-absorbing vertex j ∈ V \ T, for B
P (j | i) = 0 for all j (cid:54)= i. Therefore, z(i | T) = 1 for
each one of the d edges (j,i)∈E incident on j, it must B
all i∈T.
be that i ∈ T; otherwise edge (j,i) is not covered, and
We now make the following key observation. Let j
T is not a vertex cover. Therefore, in the augmented
be a non-absorbing node in V \T. We have that
graph H, node j is connected to d nodes in T, and to
nodeσj,allofthemabsorbing. Arandomwalkstarting (cid:88)(cid:0) (cid:1)
z(i|T)= P (b|i)+P (j |i)P (b|j) f .
from j will be absorbed and converge in a single step. B∪{j} B∪{j} B b
The probability of it being absorbed in a node in T is b∈B
zj = d+d1. There are n−k nodes in V \T, therefore, Theequationabovefollowsfromtheobservationthatwe
g(z |T)=(n−k) d . can express the probability of a random walk starting
d+1
If T is not a vertex cover for GVC, then there is an from i to be absorbed in some node b ∈ B as the sum
edge (j,(cid:96))∈E, such that j,(cid:96)(cid:54)∈T. As we noted before, of two terms: (i) the probability P (b | i) that the
B∪{j}
zj is the probability of being absorbed in some node in random walk is absorbed in b, while avoiding passing
T. The transition probability of the edge (j,σj) is d+11, through j (thus we add j in the absorbing set); (ii)
therefore, node j has probability at least 1 of being the probability P (j | i) that the random walk
d+1 B∪{j}
absorbed in σ . Since there is a path from j to σ with is absorbed in j while avoiding the nodes in B (that
j (cid:96)
non-zeroprobability,theprobabilityofjbeingabsorbed is, the probability of all paths that go from i to j of
arbitrarylength,withoutpassingthroughjorB),times