shaping opinion dynamics in social...

Shaping Opinion Dynamics in Social NetworksAbir De∗

IIT [email protected]

Sourangshu BhattacharyaIIT Kharagpur

[email protected]

Niloy GangulyIIT Kharagpur

[email protected]

ABSTRACT

A networked opinion diffusion process that usually involves ex-tensive spontaneous discussions between connected users, is of-ten propelled by external sources of news or feeds recommendedto them. In many applications like marketing design, or productlaunch, etc., corporations often post curated news or feeds on socialmedia in order to steer the users’ opinions in a desired way. Wecall such scenarios as opinion shaping or opinion control wherebya few select users called control users post opinionated messagesto drive the others’ opinions to reach a given state. In this paper,we propose SmartShape, an opinion control package that jointlyselects the control users, as well as computes the optimum rate ofcontrol messages, thereby driving the networked opinion dynamicsto the desired direction. Furthermore, our proposal also includes arobust shaping suit which makes our control framework resilientto stochastic fluctuations of opinion dynamics, originating fromseveral sources of randomness. Experiments on several syntheticand real datasets gathered from Twitter, show that SmartShapecan accurately determine the quality of a set of control users aswell as shape the opinion dynamics more effectively than severalbaselines.

ACM Reference Format:

Abir De, Sourangshu Bhattacharya, and Niloy Ganguly. 2018. Shaping Opin-ionDynamics in Social Networks. In Proc. of the 17th International Conferenceon Autonomous Agents and Multiagent Systems (AAMAS 2018), Stockholm,Sweden, July 10–15, 2018, IFAAMAS, 9 pages.

1 INTRODUCTION

It is widely accepted that opinions on social networks evolve acrosstime, often influenced by network neighbors that are representedby connecting edges. Modeling such a networked opinion dynami-cal process has been researched in many years, as evidenced by aplethora of works [1–8]. However, in contrast to these works thatconsider spontaneous opinion exchange between users, in practice,news or informations are often curated by professional journalists,market design agencies, etc. to alter the opinion of the users in adesired way. One can think of these scenarios as different opinionshaping or opinion control tasks, in which feeds or news are postedon the wall of a few people called as control users, in order to steerthe opinions of others to a given state. In this paper, our goal is toidentify appropriate control users, and devise an efficient opinionshaping mechanism, in order to curate the overall opinion dynam-ics in a favorable manner.Limitations of prior works: Opinion shaping has been studied

∗Now affiliated to MPI for Software Systems, Germany.

Proc. of the 17th International Conference on Autonomous Agents and Multiagent Systems(AAMAS 2018), M. Dastani, G. Sukthankar, E. André, S. Koenig (eds.), July 10–15, 2018,Stockholm, Sweden. © 2018 International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved. . . . $15.00

in different guises mostly by the control theorists [9–13]. Neverthe-less, the traditional control approaches towards opinion shapingshare several limitations. For example, they emphasize consensus-control, where the opinions of some pre-specified nodes are curatedfor steering others’ opinion to reach consensus. Such a setting haslimited applicability in most practical scenarios, as an example,in political campaigns, polarization is often the main goal. Fur-thermore, most of them assume control opinions as continuoussignals, whereas in practice, the expressed opinions are discreteevents observed only through the messages or posts. Only veryrecently Wang et al. [14] attempts to overcome these limitations bymodeling control signals as discrete epochs, which, however, offersan approximate and computationally inefficient solution.

Barring the individual limitations of these existing approaches,they all have looked into the opinion control problem throughthe tinted glass of a naive assumption – an apriori and completespecification of the control users which, however in practice, isnot known beforehand. Appropriate control users selection is theunderlying premise of many information propagation models [15–18]. However, it has been ignored by the existing opinion shapingapproaches, despite its compelling practical significance. In fact,the approach in [14] assigns the control signals to each and everynode, which in essence means that each user is a control user whogoverns the opinions of others; consequently, their proposal fallswayside of any practical importance.

In practice, the utility of an opinion shaping strategy dependson two primary factors. (i) Effective selection of control users, and(ii) robustness of the shaping strategy. Control users are usuallythe influential nodes that are actuated through external feeds orposts which further disseminate across others and shape their opin-ions. However, since a user’s capacity to curate the overall opiniondynamics strongly depends on both the graph structure and theopinion dynamics, computing a direct measure of influence itselfis a challenging task. The traditional centrality scores may serveas proxies for influence, however they completely disregard thedynamics of opinion flow.

Besides control user selection, a robust shaping design is an-other critical need for effective opinion control. In general, opinionevolution follows a stochastic dynamics having multiple sourcesof randomness like the arrival timings, message sentiments etc.Therefore, the design of the shaping strategy must ensure that itsperformance is insensitive to such random opinion fluctuations.While the simplistic assumptions like deterministic dynamics, re-strictive loss or consensus control are often useful notions for ashaping process to be explainable and tractable, they often hinderthe control operation in a complex environment, which renders theshaping strategy fragile and practically ineffective.Proposed approach: At the very outset, our opinion control ap-proach uses an underlying self exciting point process model calledHawkes process [8, 19–24] which allows users’ actual or organic

Session 34: Engineering Multiagent Systems 2 AAMAS 2018, July 10-15, 2018, Stockholm, Sweden

1336

opinions to be modulated over time, by both organic and controlopinions from their neighbors, expressed as sentiment messages.Using this underlying dynamics, we design SmartShape, a novelopinion shaping framework that employs a set of control users whoregulate others’ opinions to fulfill a desired objective, using someadditional (control) messages. The objectives of SmartShape aretwofolds; (i) optimal selection of control users, and (ii) computationof optimum rates for the control messages. To this aim, the proposedcontrol suit casts the opinion shaping task as a constrained mixedinteger programming (MIP) problem, where the binary variablesformulate the indicator function of the control users, the objectivespecifies the precise control task, and the constraints capture vari-ous traits of the opinion diffusion model. In contrast to the existingworks [9–13, 25] that consider the objective functions having fixedfunctional forms, our proposal only assumes convex shaping objec-tives. In a consequence, it can encompass a wide variety of opinionshaping tasks, which enhances its practical utility. Furthermore,such convexity allows the MIP to be relaxed into a convex problemthat in turn plummets the computational burden of MIP, therebyproviding a quick yet accurate solution. SmartShape can maintainits performance in the presence of random opinion fluctuations, aswell as random posting rates. Furthermore, we derive an expres-sion for the opinion covariance at the steady-state, which enablescarrying out robust opinion shaping by introducing an additionalconvex constraint to the utility maximization problems.

We experiment on both synthetic and real data gathered fromTwitter to evaluate the utility of our shaping strategies, whichshow that our framework can accurately determine the qualityof a set of control users for a wide variety of opinion shapingtasks. Furthermore through detailed analysis, we point out thatthe influential nodes are a heterogeneous mix of users with highyet diverse centrality values. Our approach appropriately identifiesthese key players from different centrality measures, which rendersthe shaping process quick and effective.

2 MODELING OPINION DYNAMICS

Any opinion shaping strategy has to be developed on top of anopinion dynamical model. A good opinion diffusion model needsto have the following two qualities: (i) It should comply with real-ity, by demonstrating substantial predictive prowess against othermodels, and (ii) it should capture various practical facets of opiniondynamics, like consensus, polarization, convergence etc., in a uni-fied way. Several date-driven models e.g. Biased Voter [6], AsLM [7],SLANT [8], etc. fulfil the above two criteria. Among these data-driven models, we choose SLANT [8] as the workhorse of ouropinion shaping proposal. SLANT which is equipped with a pointprocess (Hawkes) based probabilistic machinery, can accuratelyencapsulate the complex stochastic message dynamics. As a result,it offers a substantial performance boost beyond its competitors. Inthe following, we revisit the formulation of SLANT [8].

2.1 SLANT

In a nutshell, SLANT is driven by three intuitive ideas: (i) Thehistory of messages until time t influences the arrival process of theevents after time t , ii) users’ opinions are hidden or latent until theydecide to share it with their friends (or neighbors); and, iii) users

may update their opinions about a particular topic by learning fromthe opinions shared by their friends,Setup: Given a directed social network G = (V, E), we denoteN (u) as the set of users followed by a user u and each post e ase := (u,m, t ), where the triplet means that the user u ∈ V posted anorganic message with sentiment (expressed opinion)m at time t . WedenoteHu (t ) = {ei = (ui ,mi , ti )|ui = u and ti < t} as a collection ofall messages posted by user u until time t and H (t ) := ∪u ∈VHu (t )as the entire history of messages posted by any user until t . Fromnow onwards, we would write H (t ) as Ht to lighten the notations.Organic dynamics of opinions and messages. At the outset,we represent the message times by a set of counting processes. Inparticular, we denote the set of counting processes as a vector N (t ),in which the u-th entry, Nu (t ) ∈ {0} ∪ Z+, counts the number ofmessages user u posted until time t [26–32]. Then, we characterizethe message rate of user u using the conditional intensity functionλ∗u (t ) which is associated with the conditional probability of ob-serving an organic message event by user u in infinitesimal timeinterval [t , t + dt )], given the history H (t ) of posts until time t :

P( u posts a message in [t , t + dt )|H (t )) = λ∗u (t ) dt ∀u ∈ V

i.e. E[dN (t )|H (t )] = λ∗(t )dt (1)

where dN (t ) := ( dNu (t ) )u ∈V counts the message per user in theinterval [t , t + dt ) and λ∗(t ) := ( λ∗u (t ) )u ∈V denotes the associateduser intensities, which may depend on the history H (t ) (conform-ing with (i)). The functional form of λ(t ) is often designed to capturethe phenomenon of interest. In this paper, we assume that the mes-sages follow a multivariate Hawkes process where the intensityreflects the bursty nature of posts in social networks due the mu-tual excitation process between message events [19, 21–24, 33–41].Here, the intensity depends on the entire history of message events∪v ∈{u∪N(u)}Hv (t ) before t :

λ∗u (t ) = µu + bu∑

ei ∈Hv (t )κ(t − ti ) (2)

where the first term, µu ⩾ 0, captures the posts by user u onher own initiative, and the second term, with bu ⩾ 0, reflects theinfluence of previous messages posted by the users she follows haveon her intensity.We represent users’ organic latent opinions as a multidimensionalstochastic process x∗(t ), where the u-th entry, x∗u (t ) ∈ R, representsthe opinion of user u at time t and the sign ∗ means that it maydepend on the history H (t ). Then, every time a user u posts amessage at time t , the message sentimentm reflects the expressedopinion, which is a random variable drawn from a distributionp(m |x∗u (t )). The dynamics of x∗u (t ) is given by,

x∗u (t ) = αu +∑

v ∈N(u)avu

∑ei ∈Hv (t )

miд(t − ti ) (3)

where the first term, αu ∈ R, models the original opinion a useru and the second term, with avu ∈ R, models updates in useru’s opinion due to the influence that previous messages of herneighbors. Such influence usually depend on node features andother factors [42]. Here, we take κ(t ) = e−ν t and д(t ) = e−ωt (whereν , ω ⩾ 0) denote an exponential triggering kernel, which modelsthe decay of influence over time. The organic opinion and message


1337

dynamics (Eqs. (2), (3)) can be equivalently written as,

x∗(t ) = α +∫ t

0д(t − s)Am(s) ⊙ dN (s) (4)

λ∗(t ) = µ +∫ t

0Bκ(t − s)dN (s), (5)

with A = (avu )u,v ∈V , B = diag(bu ) and x∗(t ) = (x∗u (t ))u ∈V .Sentiment distribution. In our work, the sentiment distribution isassumed to be normal. That is,m ∈ R, i.e.,p(m |xu (t )) = N (xu (t ),σu ).This fits well scenarios in which sentiment is extracted from textusing sentiment analysis [43].

3 CONTROLLED OPINION DYNAMICS

Our opinion shaping design seeks to identify an appropriate set ofcontrol usersI who efficiently governsx∗

Ic(t ), the opinions of other

users Ic = V\I using additional posts, at optimal rates. In order tosteer x∗

Ic(t ), we aim to shape limt→∞ EHt [x∗

Ic(t )] i.e. the steady

state behavior of the expected opinion dynamics. Furthermore, toprovide a robust shaping design that is resilient to the randomnessof opinion flow, we regulate the opinion variance i.e. Tr(ΓIc ), whereΓIc := limt→∞ EHt [(x∗

Ic(t ) − EHt x

∗Ic

(t ))(x∗Ic

(t ) − EHt x∗Ic

(t ))T ].So, prior to going into the formulation of shaping tasks, it is crucialto characterize these quantities, which we describe next. We beginwith formally introducing the control users, then derive the opin-ion dynamics under the actions of these control users. Finally, wecompute differential dynamics of the expected opinion trajectoryand the opinion covariance, followed by their stabilities, and thesteady state expressions for stable systems under the control ac-tions. These properties will be exploited in the subsequent sectionto formally devise the opinion shaping strategies.

3.1 Introducing control users

We denote the set of (non) control users as (Ic ) I ⊂ V . Also fortractability, we assume that these users post binary opinions, i.e.positive or negative messages (±1) with constant rates η±

I∈ R+.

For notational consistency, assume η± = η±V. To represent the

arrival process of such control messages, we introduce additionalcounting processesM±(t ) which regulate the rate of publicationsof the corresponding opinions ±1, with E[dM±(t )] = η±dt .One hot representation of I: We define the one-hot representa-tion of the control nodes as SSS = 1[u ∈ I]u ∈V . Using such repre-sentation, we can write:

x∗Ic

(t ) = ((1 −SSS) ⊙ x∗(t )){u |SSS(u)=0},

η±I

= (SSS ⊙ η±){u |SSS(u)̸=0} . (6)

Now, the problem of finding I becomes equivalent to obtaining SSS.Characterization of I: The theoretical characterizations of con-trol users I is beyond the scope of this work. However, Section 5.1provides a detailed empirical analysis which reveals that an optimalI is a set of users with high yet diverse centrality values.

3.2 Opinion dynamics under control actions

In the same sprit of the integral representation of uncontrolled opin-ion and message dynamics (Eq. (5)), we represent the dynamics oforganic opinion and messages in the presence of control processes

M± as follows:x∗Ic

(t ) =αIc +∫ t

0д(t − s)[A1mIc (s) ⊙ dNIc (s)

+ A2dM+(s) −A2dM

−(s)] (7)

λ∗Ic

(t ) = µIc +∫ t

0κ(t − s)B1dNIc (s). (8)

Here, A1 = [avu ]v,u ∈Ic , A2 = [avu ]v ∈Ic ,u ∈I , andB1 = diag(bu )u ∈Ic . Similarly one can define αIc , and µIc . Notethat in Eq. (8), the control processM± does not influence the mes-sage intensities λ∗

Ic(t ) of the ordinary users. This is due to the

diagonal property of B which ensures that the value of λ∗Ic

onlydepends its own previous events.SDE based representation: Given the triggering kernels to be ex-ponential, the resulting opinion and event dynamics under controlactions are Markovian, and therefore can be represented as jumpstochastic differential equations. This representation will be usedin subsequent sections for opinion shaping, where the messagesrepresented by the counting processM±(t ), will act as the controlsignals to regulate the dynamics of the organic opinions x∗(t ).

Proposition 3.1. Given the triggering kernel д(t ) = e−ωt andκ(t ) = e−ν t , the tuple (x∗(t ),λ∗(t )) following Eqs. (4)- (5), is a Markovprocess, whose dynamics are defined by the following marked jumpedstochastic differential equations (SDE):

dx∗Ic

(t ) = ω(αIc − x∗Ic

(t ))dt + [A1mIc (t ) ⊙ dNIc (t )+ A2dM

+(t ) −A2dM−(t )]

dλ∗Ic

(t ) = ν (µIc − λ∗Ic

(t ))dt + B1 dNIc (t ). (9)

The proposition can be easily proved by differentiating Eqs. (4)and (5) respectively. A formal proof is given in [44].Dynamics ofmean and covariance of opinion dynamics:Now,in the following theorems, we provide the trajectories of mean andcovariance of opinion dynamics under control actions.

Theorem 3.2 (Dynamics of E(x∗Ic

(t ))). Given the message inten-sity of each of the non-control users u ∈ Ic = V\I follows λ∗u (t ) =µu + buu

∑ei ∈Hu (t ) e

−ν (t−ti ), the expected opinion EHt [x∗Ic

(t )] ofthese non-control nodes Ic follows,

dEHt [x ∗Ic (t )]

dt=[−ω I + A1ΛIc (t )]EHt [x ∗

Ic (t )]

+ A2(η+I− η−

I) + ωαIc , (10)

with, Λ∗Ic

(t ) = diag(EHt [λ∗Ic

(t )]), and

EHt [λ∗(t )] =[e(B−ν I )t + ν (B − ν I )−1

(e(B−ν I )t − I

) ]µ,

The proof follows immediately from Proposition 3.1. A detailedproof-sketch is given in [44].

Theorem 3.3 (Dynamics of covariance matrix ΓIc (t )). Giventhe notations used in Theorem 3.2, the dynamics of opinion-covarianceof the non-control nodes ΓIc (t ) := EHt [(x∗

Ic(t )−EHt x

∗Ic

(t ))(x∗Ic (t )−

EHt x∗Ic

(t ))T ] is given by:

dΓIc (t )dt

= −2ωΓIc (t ) + ΓIc (t )ΛIc (t )AT1 + A1ΛIc (t )ΓIc (t )

+ A1ΓIc ii (t )ΛIc (t )AT1 + σ 2A1ΛIc (t )AT1 + A2 diag(η+I

+ η−I

)AT2+ A1 diag(EHt [x ∗(t )])2ΛIc (t )AT1 (11)


1338

The theorem can be proved by computing the differential ofΓIc (t ), and then using Ito calculus [45] on it. A detailed proof isgiven in [44].Stability: Stability is the central challenge for any control strategydesign. In the following lemmas proved in supplementary mate-rial [44], we investigate stability of mean and covariance dynamicsof opinion diffusion under control actions.

Lemma 3.4 (Stability of EHt (x∗Ic

(t ))). Given the message dy-namics follows Poisson process with λ∗(t ) = µ, then the expectedopinion dynamics is bounded i.e. EHt [x∗

Ic(t )] < ∞ if,

Re[ξ (A1ΛIc )] < ω, where ξ (X ) is the eigenvalue of X .

Lemma 3.5 (Stability of ΓIc (t )). Given the message dynamicsfollows Poisson process with λ∗(t ) = µ, then the covariance of opiniondynamics is bounded i.e. ΓIc (t ) < ∞ iff

ξ [(−ωI + A1ΛIc ) ⊗ I + I ⊗ (−ωI + A1ΛIc ) + (A1 ⊗ A1)Λ̂Ic ] < 0.

where Λ̂i2,i2 = λ∗(i), Λ := diag[µ], and ⊗ indicates the Kroneckerproduct.

Steady state behavior: Here, we discuss the asymptotic proper-ties (expectations and variance as t → ∞) of the opinion dynamicsin control environment.

Lemma 3.6 (Asymptotic for mean opinion). If the mean opiniondynamics is stable, then the steady state average opinion of the non-control nodes is given by

limt→∞

EHt [x ∗Ic (t )] =

(I −

A1ΛIc

ω

)−1 (αIc + A2

(η+I− η−

I

)), (12)

where ΛIc = diag[I − B1

ν

]−1ηIc and B1 = diag([bv ]v ∈Ic ). Eq. (15)

is equivalent to,

(A1Λ1 − ω I )xIc + A2(η+I− η−

I

)+ ωαIc = 0 (13)

where xIc = limt→∞ EHt [x ∗Ic (t )].

In terms of SSS, the one-hot representation of I, Eq. (13) becomes,

(AΛ − ω I )[(1 − SSS) ⊙ x ] + A[SSS ⊙(η+ − η−

)]

+ ω(1 − SSS) ⊙ α = 0 (14)

where x is the steady state organic opinion of all the users.

On steady-state for a stable system, the asymptotic value ofexpected opinion value does not change. So,

dEHt [x ∗Ic (t )]

dt = 0 ast → ∞ which, upon putting on Theorem 3.2, immediately provesthe lemma. Now we set about computing ΓIc (∞) i.e. the covarianceof opinions at the steady state.

Lemma 3.7 (Asymptotic for opinion covariance). Let thesteady state solution of the covariance matrix of the non-control nodesfor a stable system be ΓIc = limt→∞ ΓIc (t ). Then ΓIc satisfies thefollowing equation:

− 2ωΓIc + ΓIc ΛIcAT1 + A1ΛIc ΓIc + A1ΓIc iiΛIcA

T

= −σ 2A1ΛIcAT1 −A1 diag(xIc )2ΛIcA

T1

−A2 diag(η+I

+ η−I

)AT2 , (15)

In terms, SSS, the above equation becomes

− 2ωΓSSSSSS + ΓSSSSSSΛAT + AΛΓSSSSSS + AΓiiSSSSSS

ΛA

= −σ 2AΛAT −Adiag((1 −SSS)x )2ΛAT

−Adiag[SSS ⊙(η+ + η−

)]AT (16)

where ΓSSSSSS := (I − diag[SSS])Γ(I − diag[SSS]) and ΓiiSSSSSS

= diag(ΓSSSSSS ).

The proof of this lemma directly comes from Theorem 3.3 byputting dΓIc /dt → 0 as t → ∞. The steady state covariancematrix obtained by solving Eq (15) has a closed form solution, isgiven below.

Lemma 3.8. The closed form ΓIc is given by,

vec(ΓIc ) = [(−ωI + A1ΛIc ) ⊗ I + I ⊗ (−ωI + A1ΛIc )T

+(A1 ⊗ A1)Λ̂Ic ]−1× vec[σ 2A1ΛIcA

T1

+ A1 diag(EH(∞)\H(t0)[x∗Ic (∞)|Ht0 ])2ΛIcAT1 ] (17)

Here, vec(X ) denotes the vectorization of a matrix X .

If we vectorize the matrix equation (15) using the fact vec(AXB) =(BT ⊗ A)vec(X ), we directly have the closed form of vec(ΓIc ).

Eqs. (14) and (16) provide us the relationships between the selec-tion vector, the organic and control message intensities of the users,the initial opinions, the steady-state opinions, and the covariancematrix. In the next section, we exploit these relationships to designa diverse range of convex programs to determine the message inten-sities of the control users in order to achieve a target steady-stateopinion for the non-control ones, given their initial opinions.

4 EFFICIENT OPINION SHAPING

In this section, we develop the opinion control framework SmartShape,by exploiting the underlying dynamics of opinion flow in the pres-ence of control actions. In a nutshell, SmartShape encompasses adiverse range of opinion shaping tasks as mixed integer program-ming problems that are further relaxed into a set of convex utilitymaximization problems that can be efficiently solved. Given anopinion model i.e. the knowledge of all the parameters α , µ,A,B,and the exact shaping task, we compute the selection vectorSSS (thusI and Ic ) and the control rates η± that steer the steady-state opin-ions of the non-control users x by maximizing a utility functionU (.), a concave function of the opinions of non-control users. Thisutility function formally specifies the underlying shaping objec-tive. Additionally, we associate costs with the control posts by theselected users. If the cost c = (c1, c2, . . . , c |V |)T ≥ 0 is cost percontrol event, and C is the total budget, then we impose

cT(η+ + η−

)⊙ SSS ≤ C, and η± ≥ 0. (18)

Finally, with the above impositions (Eq. (18)), we present differentlevels of SmartShape-Basic.SmartShape-Basic. Utility maximization without covariance con-trol: The basic form of SmartShape does not aim to regularizethe variance, rather simply maximize a given utility function ofopinions of organic users Ic .maximizex , SSS, η±

U ((1 −SSS) ⊙ x ) so that, (14), and (18) (19)

However, this problem is MIP due to the presence ofSSS ∈ {0, 1} |V | .We adopt the linearization of binary variables method described


1339

in [46–48]. In particular the linearization technique approximateseach of (1 − S) ⊙ x , and S ⊙ η± by four constraints. A cheatsheetof such techniques may also found in this blog 1.

SmartShape-Basic:

maximize

z,x ,η±,ξ ±,SSSU (z ) (20)

subject to: (AΛ − ω I )z + A(ξ + − ξ −

)] + ω(1 − SSS) ⊙ α = 0

cT(ξ + + ξ −

)≤ C, ξ + ≥ 0 ξ − ≥ 0

% Linearization of (1 − S) ⊙ x % Linearization of S ⊙ η±

x ≤ z ≤ x c ⊙ ξ ± ≤ CSSS

x ⊙ (1 − SSS) ≤ z ≤ x ⊙ (1 − SSS) 0 ≤ ξ ± ≤ η±

x − SSS ⊙ x ≤ z ≤ x − SSS ⊙ x c ⊙ η± − (111 − SSS)C ≤ c ⊙ ξ ±

z ≤ x + SSS ⊙ x 0 ≤ SSS ≤ 1

Here, we add an L1 regularizer and tune the regularizer valueto control the number of control nodes. The last two sets of con-straints, approximate the non-convex product-terms ( (1 −SSS) ⊙ xand SSS ⊙ η± ). Clearly, as long as the utility function is concave,the opinion shaping problem defined by Eq. 20 is jointly convexin z,x ,η±, ξ±, and SSS. Hence, the global optimum can be found bymany well-known algorithms.SmartShape-Centrality.UtilityMaximizationwith a pre-selectedI: One can also assume I, the set of control nodes, is already cho-sen using some centralitymeasures like PageRank, degree, closenesscentralities, etc. Indeed we shall use such setting as a baseline tocompare with the proposed user selection strategy of SmartShape-Basic (see Section 5.1).

maximizexI c ,η±

U (xIc ) subject to, (12) and (18) (21)

SmartShape-Robust. Utility maximization with covariance con-trol: The problem formulation of SmartShape-Basic aims to steerthe average users’ opinions to a given steady-state. However, par-ticular realizations (or trajectories) of the model may converge toa steady-state opinion far from the steady-state average opinion,limt→∞ EHt [x∗

Ic(t )]. Here, we reformulate the above opinion shap-

ing problems to ensure that particular realizations of the modelare typically close to the average, by adding a regularizer on thesteady-state opinion covariance of controlled-nodes quantified as,

maximizex , SSS, η±, ΓSSSSSS

U ((1 −SSS) ⊙ x) − γ Tr(ΓSSSSSS )

subject to (14), (16), (18), and ΓSSSSSS ⪰ 0.(22)

Using the linearization method as Eq. (20), we obtain :SmartShape-Robust :

maximize

z,x ,η±,ξ ±,SSS ,ΓSSSSSSU (z ) − γ Tr(ΓSSSSSS ) (23)

subject to:(AΛ − ω I )z + A (ξ + − ξ −)] + ω(1 − SSS) ⊙ α = 02ωΓSSSSSS + ΓSSSSSSΛAT + AΛΓSSSSSS + Γii

SSSSSS

= −σ 2AΛAT −A diag(z )2ΛAT −A diag[(ξ + + ξ −)]ATcT (ξ + + ξ −) ≤ C, ξ ± ≥ 0, Γ ⪰ 0

% Linearization of (1 − S) ⊙ x % Linearization of S ⊙ η±

x ≤ z ≤ x c ⊙ ξ ± ≤ CSSS

x ⊙ (1 − SSS) ≤ z ≤ x ⊙ (1 − SSS) 0 ≤ ξ ± ≤ η±

x − SSS ⊙ x ≤ z ≤ x − SSS ⊙ x c ⊙ η± − (111 − SSS)C ≤ c ⊙ ξ ±

z ≤ x + SSS ⊙ x 0 ≤ SSS ≤ 1

1https://www.leandro-coelho.com/linearization-product-variables/

Since, the second equality constraint in (23) is not a convex con-straint with respect to z, the optimization problem, as it is, mayseem very difficult to solve efficiently. However, we can overcomethis difficulty by following Lemma 3.8, and looking into the explicitexpression of ΓSSSSSS . Note that the objective function only dependson the trace of ΓSSSSSS . To compute that, we add an additional variablet and reconstruct the second equality constraint as,

t ≥ −(vec(I ))TV −1 vec(σ 2AΛAT + Adiag(z)2ΛAT

+ Adiag(ξ+ + ξ−

)AT ). (24)

V = (−ωI + AΛ) ⊗ I + I ⊗ (−ωI + AΛ)T +n∑i=1

(A ⊗ AT )Λ̂

Such transformation (Eq. (24)) makes objective function U (z) −γ t . Note that, at the optimal condition, we have exact equality inEq. (24). Finally we have z,x ,η±, ξ±,SSS and t as the optimizationvariables. Note that, since all the elements in diag(z)2 are multipliedby positive constants, the inequality (24) is convex with respect toz and thus the problem is jointly convex in all variables.

4.1 Instances of utility functions

Minimax opinion shaping [MMOSH-1, MMOSH-2]: Supposewe aim to make the users with the most positive opinion as negativeas possible. Then, we choose

MMOSH-1: U (xIc ) = − maxu ∈[m]

xIc ,u . (25)

Alternatively, in order to make the users with the most negativeopinions as positive as possible, we write the utility function as

MMOSH-2: U (xIc ) = minu ∈[m]

xIc ,u . (26)

Average opinion shaping [AOSH-1, AOSH-2] Suppose we aimto make the average opinions over users to be as low as possible.Then, we can choose:

AOSH-1: U (xIc ) = −∑u ∈Ic

xu . (27)

On the other hand, we also take the following utility:

AOSH-2: U (xIc ) =∑u ∈Ic

xu . (28)

Top-k opinion shaping [Top-k−OSH-{1,2}] Suppose we aim tomake the k users with the most positive opinion as negative aspossible. Then, we need to introduce an extra variable yIc , add theadditional constraints

Top-k−OSH-1: yIc ,u ≥ max(xIc ,u , 0) ∀u ∈ Ic , (29)

With, Top-k−OSH-{1,2}: U (yIc ) = −k∑i=1

|y[i] |, (30)

wherey[i] denotes the ith largest component ofy. Similarly, supposeour goal is to make the k users with the most negative opinion aspositive as possible. Then, we introduce the extra variable yIc , addthe additional constraints

Top-k−OSH-2: yIc ,u ≤ min(xIc ,u , 0) ∀u ∈ Ic , (31)

and use the utility function given by Eq. 30.


1340

https://www.leandro-coelho.com/linearization-product-variables/

https://www.leandro-coelho.com/linearization-product-variables/

PageRank InDegree OutDegree Closeness Eigen-Centrality SmartShape-Basic

1 10 20 40 50−8

−6

−4

−2

0

2

−maxuE(

xu)

% of Control Nodes

1 10 20 40 50−8

−6

−4

−2

0

2

min

uE(

xu)

% of Control Nodes

1 10 20 40 500

0.01

0.02

0.03

0.04

0.05

−∑

uE(x

u)/|V

|

% of Control Nodes

1 10 20 40 500

0.005

0.01

0.015

∑uE(

xu)/|V

|

% of Control Nodes

1 10 20 40 50−20

−15

−10

−5

0

−||E

(x)+

|| lgst,k

% of Control Nodes

1 10 20 40 50−6

−4

−2

0

−||E

(x)+

|| lgst,k

% of Control Nodes

Uniform: ηηη± = η∗η∗η∗±

Weighted: ηηη±u = Suη∗η∗η∗±/S SmartShape-Basic: ηηη± = η∗η∗η∗±

1 10 20 40 50−60

−40

−20

0

20

−maxuE(

xu)

% of Control Nodes

(a) MMOSH-1

1 10 20 40 50−60

−40

−20

0

20

min

uE(

xu)

% of Control Nodes

(b) MMOSH-2

1 10 20 40 50−0.3

−0.2

−0.1

0

0.1

−∑

uE(x

u)/|V

|

% of Control Nodes

(c) AOSH-1

1 10 20 40 500

0.005

0.01

0.015

∑uE(

xu)/|V

|

% of Control Nodes

(d) AOSH-2

1 10 20 40 50−25

−20

−15

−10

−5

0

−||E

(x)+

|| lgst,k

% of Control Nodes

(e) Top-k-OSH-1

1 10 20 40 50−8

−6

−4

−2

0

−||E

(x)+

|| lgst,k

% of Control Nodes

(f) Top-k-OSH-1

Figure 1: Opinion shaping on Kronecker Hierarchical (first three columns– (a) to (c)) and Politics (last three columns– (d)

to (f)) networks. The top row indicates the impact of SmartShape-Basic from user selection perspective, as compared to

SmartShape-Centrality where the control users are chosen based on some centrality measures like PageRank, InDegree,

OutDegree, and Closeness. The bottom row indicates the efficacy of SmartShape-Basic from rate computation viewpoint,

against various heuristics like “Uniform” and “Weighted”.

Datasets |V| |E | |H (T )| E[m] std[m]Politics 548 5271 20026 0.0169 0.1780Movie 567 4886 14016 0.5969 0.1358Fight 848 10118 21526 -0.0123 0.2577Bollywood 1031 34952 46845 0.5101 0.2310US 533 20067 18704 -0.0186 0.7135

Table 1: Real datasets statistics

5 EXPERIMENTS

We provide a comprehensive evaluation of SmartShape using bothsynthetic data and real data gathered from Twitter, and assess towhich extent the chosen control nodes can steer the users’ opinionsto a given state in several types of networks with very differentstructures.Synthetic datasets: We evaluate the accuracy of our shapingstrategies on five types of Kronecker networks [49]: i) Homophilynetworks (parameter matrix [0.96, 0.3; 0.3, 0.96]), ii) heterophilynetworks ([0.3, 0.96; 0.96, 0.3]), iii) core-periphery networks([0.9, 0.5; 0.5, 0.3]), (iv) random networks ([0.7, 0.7; 0.7, 0.7]); and,(v) hierarchical networks ([0.9, 0.1; 0.1, 0.9]). For each network, themessage intensities are multivariate Hawkes, µ and B are drawnfrom a uniform distribution U (0, 1), and α and A are drawn froma Gaussian distribution N (µ = 0,σ = 1). Also, we use exponentialkernels with parameters ω = 100 and ν = 1.Real datasets: We use five real datasets [8] corresponding to vari-ous real world events, collected from Twitter, which are also summa-rized in Table 1. For each of these datasets, we have a social networkG = (V, E), and a collection of messagesH (T ) = {(ui ,mi , ti )} gath-ered during a time period [0,T ). Using this information as input,we obtain the optimal parameters α , µ,A and B by maximizing thecorresponding likelihood function w.r.t. α , µ, A, B [8, 50].∑ei ∈H(T )

logp(mi |x∗ui (ti )) +

∑ei ∈H(T )

log λ∗ui (ti ) −∑u ∈V

∫T0λ∗u (τ )dτ

Setup:With these parameters, we run SmartShape both in syn-thetic and real networks. We focus on six tasks (Eqs. (25) to (31)).For each of these tasks we find the top control nodes, and assess theperformance exhibited by these nodes of different numbers. In allexperiments, we set the total budget toC = 10, and assume all usersentail the same cost. More in detail, we solve SmartShape-Basic(Eq. 20) and SmartShape-Robust (Eq. 23).We adjust the regularizerover SSS to vary the number of control nodes over (1, 10, 20, 40, 50)%of |V|. The performance of the control strategy is measured usingthe U , the value of the underlying objective function. We comparethe efficacy of our user selection strategy, as well as the messagerate computation against several competitors.Baselines: To probe the utility of the proposed user selection strat-egy, we compare SmartShape-Basicwith SmartShape-Centrality,where for the latter we fix I with the top x% (x = 1, 10, 20, 40, 50)nodes based on several centralitymeasures e.g. PageRank, in-degree,out-degree, closeness, and eigencentrality. On the other hand, wecompare the efficacy of message rate computation with two differ-ent heuristics, e.g.Uniform, andWeighted. Here, “Uniform” indicatesthat the rate of each user is constant and equals to the average ofthe individual optimum rates, i.e. η±u = ∑

i ∈Ic η±i /|I

c |. “Weighted”assumes that the message rate of each user u is proportional to herselection score Su , subject to the same aggregated message rate i.e.η±u = Suη∗±u /

∑u Su .

5.1 Comparison with baselines

The primary goals of SmartShape are (i) optimal control userselection, and (ii) optimal message rate selection. We evaluate theproposed shaping strategies from these two perspectives.Utility of optimal selection of I. To evaluate the impact of in-fluential user selection strategy, we compare SmartShape-Basic(Eq. (20)) with SmartShape-Centrality (Eq. (21)), where for the


1341

Politics Movie Fight Bollywood US

1 10 20 40 50−1.5

−1

−0.5

0

0.5

1

−maxuE(

xu)

% of Control Nodes

1 10 20 40 50−0.6

−0.4

−0.2

0

0.2

0.4

min

uE(

xu)

% of Control Nodes

1 10 20 40 500

0.2

0.4

0.6

0.8

−∑

uE(

xu)/|V

|

% of Control Nodes

1 10 20 40 500

0.05

0.1

0.15

0.2

0.25

∑uE(

xu)/|V

|

% of Control Nodes

1 10 20 40 50−150

−100

−50

0

−||E

(x)+

|| lgst,k

% of Control Nodes

1 10 20 40 50−50

−40

−30

−20

−10

0

−||E

(x)+

|| lgst,k

% of Control Nodes

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(+

→−)

(a) MMOSH-1

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(−

→+)

(b) MMOSH-2

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(+

→−)

(c) AOSH-1

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(−

→+)

(d) AOSH-2

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(+

→−)

(e) Top-k-OSH-1

1 10 20 40 500

0.2

0.4

0.6

0.8

1

% of Control Nodes

f v(−

→+)

(f) Top-k-OSH-2

Figure 2: Performance variation of SmartShape-Basic with % of control nodes. The top row shows the variation of actual

objective U , and the bottom row shows variation of fv , % of users who switched their polarity in the desired direction.

latter, the control nodes are pre-selected using well-known mea-sures of centrality such as PageRank, degree, closeness or eigencen-trality. Top row of Figure 1 summarizes the results which indicatethat our node-selection strategy steers the objective more efficientlythan other centrality-measures. It shows that network-structure isnot the only criteria for control-node selection. Rather one shouldalso consider the utility function and the complex dynamics ofopinion evolution. Our proposal combines these three factors in aunified-way. As a result, it significantly outperforms the existingbaselines.Utility of computation of optimal message rate η∗±. To un-derstand the utility of SmartShape from control rate computationviewpoint, we compare SmartShape-Basic (Eq. (20)) with differentrate selection strategies e.g. Uniform and Weighted, as described inthe last subsection. The bottom row of Figure 1 summarizes theresults. We observe that SmartShape steers the objective moreeffectively than others, indicating that it appropriately incorporatesthe network structure, the utility function and the complex opiniondynamics, that are not effectively combined the other two heuristics.5.2 Characterization of control node set I∗

The control users should have a high influence over other usersto make the steering task quick and effective. Here, we aim tocharacterize the optimally selected user set I∗, through variousstructural properties (Figure 3). In order to do that, we find out theoverlap of I∗ with top important users selected using standard cen-trality measures. In particular, we measure the Jaccard coefficientbetween I∗ and ICentrality, |I∗ ∩ICentrality |/|I

∗ ∪ICentrality |, withfixed number of control users. Here, ICentrality is the set of controlusers chosen based on a centrality score e.g. PageRank, Degree,etc., with |I∗ |= ICentrality |. Figure 3 shows the variation of thisoverlap measure with the number of control nodes. For the barnamed “All”, the overlap is given by |I∗ ∩ IAll |/|I

∗ ∪ IAll | whereIAll = ∪CentralityICentrality. So, the bar “All” shows the overlap ofthe optimally chosen control users with any of the centrality basedcontrol user set. Figure 3 reveals three key observations. (i) I∗ hasa lot of overlaps with various centrality measures. So the controlusers play strong yet different structurally influential roles in thenetwork. (ii) The bar named as “All” has a high value, which in-dicates that I∗ is a heterogeneous blend of nodes from variouscentrality measures, that is accurately identified by our proposal.

(iii) In most cases, the out-degree centrality has the highest overlapwith I∗, indicating that it is the key measure of influence in thecontext of opinion shaping.

5.3 Performance variation with |I |Variation of actual objective. Figure 2 shows that the perfor-mance of SmartShape-Basic for all shaping problems becomesbetter, as the number of control nodes increases. The shaping frame-work works quite effectively for US dataset (consistently secondbest for all shaping tasks except MMOSH-1), because it has the highaverage degree among all real datasets, which makes the influenceof the control nodes accessible to a large number of nodes.Change of polarity. Intuitively, by minimizing the most positiveopinion across users (MMOSH-1) or the average opinions (ASOH-1) or the opinions of top-k most positive users (Top-k-OSH-1),one could expect (some) users to switch from an initial positiveopinion to a steady-state negative opinion. Similarly, by maximizingthe most negative opinion across users (MMOSH-2), the averageopinions of the users (AOSH-2) or the opinions of the top-k mostnegative users (Top-k-OSH-2), one could expect (some) users toswitch from an initial negative opinion to a steady-state positiveopinion. Figure 2 confirms this intuition, by showing the percentageof nodes that change their opinion polarity against control nodeset sizes, for all the real datasets.

5.4 Analysis of SmartShape-Robust

Finally, we solve SmartShape-Robust, the same activity shapingtasks with covariance control (Eq. 23) for different penalty values γ .Figures 4, and 5 summarize the results. Figure 5 shows the variationof objective with number of nodes, where it is evidently clear thatmore number of control nodes help to reduce the opinion variance.Figure 4 shows the variation of objective with regularizer (γ ). Itshows a clear trade-off between variance and objective value. Incase of MMOSH-1 and MMOSH-2 (Figures 4a and 4g), we observethe shaping performance is most robust, i.e. the performance doesnot change much with regularizer variation. In a few cases likeAOSH-2 (Figures 4d and 4d), we observe that Bollywood suffers ahigher variance. It is because, the natural activity level of the usersin Bollywood is very high (See |H (t )| in Table 1) which is injectingan inherent high variance (through a higher Λ) in the system.


1342

1 10 20 40 500

0.5

1

% of Control Nodes

Ove

rla

p

PageRankInDegreeOutDegreeClosenessEigenAll

(a) Politics

1 10 20 40 500

0.5

1

% of Control Nodes

Ove

rla

p


(b) Movie

1 10 20 40 500

0.5

1

% of Control Nodes

Ove

rla

p


(c) Bollywood

Figure 3: Characterization of I∗, the control users spotted by SmartShape-Basic, for three real datasets. The structural traits

of I∗are measured using their overlap with users of top centrality scores. The overlap is measured as Jaccard coefficient

between I∗and I

Centrality, where the latter is the control users chosen based on a centrality score e.g. PageRank, Degree, etc.

The bar “All” indicates the overlap of the optimal users with any of the centrality based user-set.


1e−4 1e−3 1e−2 1e−1−0.8

−0.6

−0.4

−0.2

0

−maxuE(

xu)

γγγ →1e−4 1e−3 1e−2 1e−1

−0.1

0

0.1

0.2

0.3

0.4

min

uE(

xu)

γγγ →1e−4 1e−3 1e−2 1e−1−5

0

5

10

15

20

−∑

uE(

xu)/|V

|

γγγ →1e−4 1e−3 1e−2 1e−10

10

20

30

40

∑uE(

xu)/|V

|γγγ →

1e−4 1e−3 1e−2 1e−1−20

−15

−10

−5

0

−||E

(x)+

|| lgst,k

γγγ →1e−4 1e−3 1e−2 1e−1

−0.5

−0.4

−0.3

−0.2

−0.1

0

−||E

(x)+

|| lgst,k

γγγ →

1e−4 1e−3 1e−2 1e−10

2

4

6

8

10

σ2

σ2

σ2(t

→∞)→

γγγ →(a) MMOSH-1

1e−4 1e−3 1e−2 1e−10

20

40

60

σ2

σ2

σ2(t

→∞)→

γγγ →(b) MMOSH-2

1e−4 1e−3 1e−2 1e−10

1

2

3

4

5x 10

4

σ2

σ2

σ2(t

→∞)→

γγγ →(c) AOSH-1

1e−4 1e−3 1e−2 1e−10

1

2

3

4x 10

4

σ2

σ2

σ2(t

→∞)→

γγγ →(d) AOSH-2

1e−4 1e−3 1e−2 1e−10

1

2

3

4x 10

4

σ2

σ2

σ2(t

→∞)→

γγγ →(e) Top-k-OSH-1

1e−4 1e−3 1e−2 1e−10

100

200

300

400

500

σ2

σ2

σ2(t

→∞)→

γγγ →(f) Top-k-OSH-2

Figure 4: Performance variation of SmartShape-Robust, opinion shaping with covariance control, with respect to the regu-

larizer γ . The top (bottom) row shows the variation of control objective (variance).


1 10 20 40 500

200

400

600

800

1000

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(a) MMOSH-1

1 10 20 40 500

200

400

600

800

1000

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(b) MMOSH-2

1 10 20 40 500

200

400

600

800

1000

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(c) AOSH-1

1 10 20 40 500

200

400

600

800

1000

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(d) AOSH-2

1 10 20 40 500

200

400

600

800

1000

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(e) Top-k-OSH-1

1 10 20 40 500

50

100

150

% of Control Nodes

σ2

σ2

σ2(t

→∞)→

(f) Top-k-OSH-2

Figure 5: Variation of opinion variance for SmartShape-Robust with respect to % of control nodes.

6 CONCLUSIONS

Our principal contribution in this paper lies in designing a flexibleopinion shaping framework, that employs the best set of k controlusers who can steer the opinions of others, using additional mes-sages. To this aim, we proposed SmartShape, an novel opinioncontrol suit which jointly selects an optimal set of influential users,as well as the rate of their messages in a unified way. SmartShapeencompasses a wide variety of opinion shaping tasks addressingdifferent shaping objectives in various practical scenarios. Further-more, it is resilient to stochastic opinion fluctuations, as well asrandommessage arrivals. Experiments on several synthetic and realdata gathered from Twitter showed that SmartShape can identify

an appropriate set of influential users who can shape the networkedopinion dynamics in a desired way. Furthermore, a detailed empiri-cal analysis revealed that these control users are a heterogeneouscollection of users having high centrality values, but from differentcentrality measures. We believe that, apart from many immediateapplications like marketing design, product campaign, etc., our pro-posal can also be used to understand the dynamics of fake newsthat are spread by many incentivized users, as well as to designcounter-measures against it.Acknowledgment: The authors acknowledge Manuel Gomez Ro-driguez and Isabel Valera for useful discussions. N. Ganguly and S.Bhattacharya acknowledge an Intel grant.


1343

REFERENCES

[1] P. Clifford and A. Sudbury. A model for spatial conflict. Biometrika, 60(3):581–588,1973.

[2] E. Yildiz, A. Ozdaglar, D. Acemoglu, A. Saberi, and A. Scaglione. Binary opiniondynamics with stubborn agents. ACMTransactions on Economics and Computation,1(4):19, 2013.

[3] Frank Schweitzer and Laxmidhar Behera. Nonlinear voter models: the transitionfrom invasion to coexistence. The European Physical Journal B-Condensed Matterand Complex Systems, 67(3):301–318, 2009.

[4] M. E. Yildiz, R. Pagliari, A. Ozdaglar, and A. Scaglione. Voting models in randomnetworks. In Information Theory and Applications Workshop, pages 1–7, 2010.

[5] M. H. DeGroot. Reaching a consensus. Journal of the American StatisticalAssociation, 69(345):118–121, 1974.

[6] A. Das, S. Gollapudi, and K. Munagala. Modeling opinion dynamics in socialnetworks. InWSDM, 2014.

[7] A. De, S. Bhattacharya, P. Bhattacharya, N. Ganguly, and S. Chakrabarti. Learninga linear influence model from transient opinion dynamics. In CIKM, 2014.

[8] Abir De, Isabel Valera, Niloy Ganguly, Sourangshu Bhattacharya, and ManuelGomez Rodriguez. Learning and forecasting opinion dynamics in social networks.In NIPS. 2016.

[9] Jan ChristianDittmer. Consensus formation under bounded confidence. NonlinearAnalysis: Theory, Methods & Applications, 47(7):4615–4621, 2001.

[10] Seyed Rasoul Etesami and Tamer Başar. Game-theoretic analysis of thehegselmann-krause model for opinion dynamics in finite dimensions. IEEETransactions on Automatic Control, 60(7):1886–1897, 2015.

[11] Vincent D Blondel, Julien M Hendrickx, and John N Tsitsiklis. On krause’s multi-agent consensus model with state-dependent connectivity. IEEE transactions onAutomatic Control, 54(11):2586–2597, 2009.

[12] Noah E Friedkin. The problem of social control and coordination of complexsystems in sociology: A look at the community cleavage problem. IEEE ControlSystems, 35(3):40–51, 2015.

[13] Irinel-Constantin Morărescu, Samuel Martin, Antoine Girard, and Aurélie Muller-Gueudin. Coordination in networks of linear impulsive agents. IEEE Transactionson Automatic Control, 61(9):2402–2415, 2016.

[14] Yichen Wang, Grady Williams, Evangelos Theodorou, and Le Song. Variationalpolicy for guiding point processes. arXiv preprint arXiv:1701.08585, 2017.

[15] Jure Leskovec, Lada A Adamic, and Bernardo A Huberman. The dynamics ofviral marketing. ACM Transactions on the Web (TWEB), 1(1):5, 2007.

[16] Ali Zarezade, Abir De, Hamid Rabiee, and Manuel Gomez-Rodriguez. Cheshire:An online algorithm for activity maximization in social networks. In arXivpreprint arXiv:1703.02059, 2017.

[17] James Archbold and Nathan Griffiths. Limiting concept spread in environmentswith interacting concepts. In Proceedings of the 16th Conference on AutonomousAgents and MultiAgent Systems, pages 1332–1340. International Foundation forAutonomous Agents and Multiagent Systems, 2017.

[18] Amulya Yadav, Hau Chan, Albert Xin Jiang, Haifeng Xu, Eric Rice, and MilindTambe. Using social networks to aid homeless shelters: Dynamic influencemaximization under uncertainty. In Proceedings of the 2016 International Confer-ence on Autonomous Agents & Multiagent Systems, pages 740–748. InternationalFoundation for Autonomous Agents and Multiagent Systems, 2016.

[19] Mehrdad Farajtabar, Jiachen Yang, Xiaojing Ye, Huan Xu, Rakshit Trivedi, EliasKhalil, Shuang Li, Le Song, and Hongyuan Zha. Fake news mitigation via pointprocess based intervention. arXiv preprint arXiv:1703.07823, 2017.

[20] Ali Zarezade, Abir De, Utkarsh Upadhyay, Hamid R. Rabiee, and Manuel Gomez-Rodriguez. Steering social activity: A stochastic optimal control point of view.JMLR, 2018.

[21] Marian-Andrei Rizoiu, Lexing Xie, Scott Sanner, Manuel Cebrian, Honglin Yu,and Pascal Van Hentenryck. Expecting to be hip: Hawkes intensity processesfor social media popularity. In Proceedings of the 26th International Conferenceon World Wide Web, pages 735–744. International World Wide Web ConferencesSteering Committee, 2017.

[22] Seyed Abbas Hosseini, Ali Khodadadi, Ali Arabzadeh, and Hamid R Rabiee. Hnp3:A hierarchical nonparametric point process for modeling content diffusion oversocial media. In Data Mining (ICDM), 2016 IEEE 16th International Conference on,pages 943–948. IEEE, 2016.

[23] M Moore and M Davenport. Learning network structure via hawkes processes.In Proc. Work. Signal Processing with Adaptive Sparse Structured Representations(SPARS), 2015.

[24] Dominik Thalmeier, Vicenç Gómez, and Hilbert J Kappen. Action selection ingrowing state spaces: Control of network structure growth. Journal of Physics A:Mathematical and Theoretical, 50(3):034006, 2016.

[25] Igor Douven and Alexander Riegler. Extending the hegselmann–krause model i.Logic Journal of IGPL, 18(2):323–335, 2009.

[26] Bidisha Samanta, Abir De, Abhijnan Chakraborty, and Niloy Ganguly. Lmpp:A large margin point process combining reinforcement and competition formodeling hashtag popularity.

[27] Emmanuel Bacry, Stéphane Gaïffas, Iacopo Mastromatteo, and Jean-FrançoisMuzy. Mean-field inference of hawkes point processes. Journal of Physics A:Mathematical and Theoretical, 49(17):174006, 2016.

[28] Nan Du, Hanjun Dai, Rakshit Trivedi, Utkarsh Upadhyay, Manuel Gomez, andLe Song. Recurrent temporal point process.

[29] Martin Jankowiak andManuel Gomez-Rodriguez. Uncovering the spatiotemporalpatterns of collective social activity. In Proceedings of the 2017 SIAM InternationalConference on Data Mining, pages 822–830. SIAM, 2017.

[30] Arun Sathanur, Mahantesh Halappanavar, Yi Shi, and Walin Sagduyu. Exploringthe role of intrinsic nodal activation on the spread of influence in complexnetworks. arXiv preprint arXiv:1707.05287, 2017.

[31] Yichen Wang, Xiaojing Ye, Haomin Zhou, Hongyuan Zha, and Le Song. Linkingmicro event history to macro prediction in point process models. In ArtificialIntelligence and Statistics, pages 1375–1384, 2017.

[32] Utkarsh Upadhyay, Isabel Valera, and Manuel Gomez-Rodriguez. Uncoveringthe dynamics of crowdlearning and the value of knowledge. In Proceedings ofthe Tenth ACM International Conference on Web Search and Data Mining, pages61–70. ACM, 2017.

[33] Ali Zarezade, Sina Jafarzadeh, and Hamid R Rabiee. Spatio-temporal modelingof check-ins in location-based social networks. arXiv preprint arXiv:1611.07710,2016.

[34] Young Lee, KarWai Lim, and Cheng Soon Ong. Hawkes processes with stochasticexcitations. In International Conference on Machine Learning, pages 79–88, 2016.

[35] Behzad Tabibian, Isabel Valera, Mehrdad Farajtabar, Le Song, Bernhard Schölkopf,and Manuel Gomez-Rodriguez. Distilling information reliability and sourcetrustworthiness from digital traces. In Proceedings of the 26th InternationalConference on World Wide Web, pages 847–855. International World Wide WebConferences Steering Committee, 2017.

[36] Ali Zarezade, Utkarsh Upadhyay, Hamid R Rabiee, and Manuel Gomez-Rodriguez.Redqueen: An online algorithm for smart broadcasting in social networks. InProceedings of the Tenth ACM International Conference on Web Search and DataMining, pages 51–60. ACM, 2017.

[37] Mehrdad Farajtabar, Manuel Gomez-Rodriguez, Yichen Wang, Shuang Li,Hongyuan Zha, and Le Song. Co-evolutionary dynamics of information dif-fusion and network structure. In Proceedings of the 24th International Conferenceon World Wide Web, pages 619–620. ACM, 2015.

[38] Edward Choi, Nan Du, Robert Chen, Le Song, and Jimeng Sun. Constructingdisease network and temporal progression model via context-sensitive hawkesprocess. In Data Mining (ICDM), 2015 IEEE International Conference on, pages721–726. IEEE, 2015.

[39] Julio Cesar Louzada Pinto, Tijani Chahed, and Eitan Altman. Trend detectionin social networks using hawkes processes. In Advances in Social NetworksAnalysis and Mining (ASONAM), 2015 IEEE/ACM International Conference on,pages 1441–1448. IEEE, 2015.

[40] Ali Zarezade, Ali Khodadadi, Mehrdad Farajtabar, Hamid R Rabiee, and HongyuanZha. Correlated cascades: Compete or cooperate. In AAAI, pages 238–244, 2017.

[41] Mohammad Reza Karimi, Erfan Tavakoli, Mehrdad Farajtabar, Le Song, andManuel Gomez-Rodriguez. Smart broadcasting: Do you want to be seen? arXivpreprint arXiv:1605.06855, 2016.

[42] Abir De, Sourangshu Bhattacharya, Sourav Sarkar, Niloy Ganguly, and SoumenChakrabarti. Discriminative link prediction using local, community, and globalsignals. TKDE, 2016.

[43] Aniko Hannak, Eric Anderson, Lisa Feldman Barrett, Sune Lehmann, Alan Mis-love, and Mirek Riedewald. Tweetin’in the rain: Exploring societal-scale effectsof weather on mood. In ICWSM, 2012.

[44] Supplementary material. http://www.cnergres.iitkgp.ac.in/suppmat.pdf .[45] F. B. Hanson. Applied stochastic processes and control for Jump-diffusions: modeling,

analysis, and computation, volume 13. Siam, 2007.[46] Fred Glover. Improved linear integer programming formulations of nonlinear

integer problems. Management Science, 22(4):455–460, 1975.[47] Francisco E Torres. Linearization of mixed-integer products. Mathematical

programming, 49(1):427–428, 1990.[48] Hanif D Sherali and Warren P Adams. A reformulation-linearization technique for

solving discrete and continuous nonconvex problems, volume 31. Springer Science& Business Media, 2013.

[49] Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg, Christos Faloutsos, andZoubin Ghahramani. Kronecker graphs: An approach to modeling networks.The Journal of Machine Learning Research, 11:985–1042, 2010.

[50] Abir De, Sourangshu Bhattacharya, and Niloy Ganguly. Demarcating endogenousand exogenous opinion diffusion process on social networks. InWWW, 2018.


1344

http://www.cnergres.iitkgp.ac.in/suppmat.pdf

shaping opinion dynamics in social...

Documents