Role of Conformity in Opinion Dynamics in SocialNetworks

Abhimanyu DasMicrosoft ResearchMountain View, CA

abhidas@microsoft.com

Sreenivas GollapudiMicrosoft ResearchMountain View, CA

sreenig@microsoft.comArindam Khan

School of Computer ScienceGeorgia Tech, Atlanta, GAakhan67@gatech.edu

Renato Paes LemeGoogle Research

New York, NYrenatoppl@google.com

ABSTRACTSocial networks serve as important platforms for users to ex-press, exchange and form opinions on various topics. Sev-eral opinion dynamics models have been proposed to char-acterize how a user iteratively updates her expressed opinionbased on her innate opinion and the opinion of her neigh-bors. The extent to how much a user is influenced by herneighboring opinions, as opposed to her own innate opin-ion, is governed by a measure of her “conformity’ parameter.Characterizing this degree of conformity for users of a socialnetwork is critical for several applications such as debiasingonline surveys and finding social influencers. In this paper,we address the problem of estimating these conformity val-ues for users, using only the expressed opinions and the so-cial graph. We pose this problem in a constrained optimiza-tion framework and design efficient algorithms, which wevalidate on both synthetic and real-world Twitter data. Us-ing these estimated conformity values, we then address theproblem of identifying the smallest subset of users in a so-cial graph that, when seeded initially with some non-neutralopinions, can accurately explain the current opinion valuesof users in the entire social graph. We call this problem seedrecovery. Using ideas from compressed sensing, we analyzeand design algorithms for both conformity estimation andseed recovery, and validate them on real and synthetic data.

1. INTRODUCTIONThe widespread use of online social networks has a

very direct bearing on how users form and express opin-ions on various issues such as politics, technology, con-sumer products, healthcare etc. Though users are in-creasingly spending more time on these networks, notall of them adopt and propagate ideas and opinions ina homogeneous way. Understanding how users in anonline social network shape and influence each other’sopinions is important in the context of viral marketing,behavioral targeting and information dissemination of

users in the network.There has been a plethora of work on modeling opin-

ion dynamics in social networks [7, 10, 29, 21] in thesociology and economics literature. Specifically, theseworks model how an individual user updates her opinionin the context of information learned from her neighborsin her social network, and then use the model to charac-terize the evolution of opinions in the network in termsof equilibria, convergence time, and the emergence of aconsensus or polarization.

More recent models [11, 1, 20] explicitly incorporatethe notion of an innate opinion – i.e., an endogenousopinion of each user about a certain topic – as opposedto her expressed opinion, which is a result of a socialprocess. These models describe how the opinion of eachuser in a social network evolves (i.e. how she changesher opinion over time) based on the following: (i) herinnate opinion, which is an immutable property of theuser, (ii) the expressed opinion of users in her socialcircle and (iii) internal parameters of the model. In thecase of the Friedkin and Johnsen [11] model, the modelparameters correspond to a real quantity for each userwhich measures her conformity, i.e., the likelihood ofthis user adopting the opinions of her neighbors in thenetwork.

While the expressed opinion of users are readily ob-servable in the social network (e.g. from a user’s tweetsor Facebook posts), both the innate opinion and confor-mity parameter of users are hidden and can only be in-ferred by reverse engineering the opinion dynamics pro-cess. In particular, identifying these endogenous con-formity values of users are a critical component towardenabling several interesting applications that leverageopinion dynamics processes. Salient examples includeefficient sampling of innate user opinions [4], seedingopinions to maximize opinion adoption [12], and iden-tifying candidates for viral marketing or targeting.

Inferring the conformity parameter for each user by

1

simply analyzing the history of opinions emitted by theuser is not practical largely due to lack of availabilityof a reasonable amount of historical opinion formationdata for each user. Therefore, we ask the following ques-tion, that we call ConformityExtraction : Can wereliably estimate conformity parameters of users in asocial network from a single snapshot of the stationarystate of the opinion dynamic?

Mathematically this is an under-determined problemsince it involves n constraints (the fact that the dynamicis in a stationary state for each user) and 2n variables(the innate opinion and conformity parameter for eachuser). We overcome this problem by relying on twonatural assumptions, viz.,

• homophily : we assume that if two users are closein the social network, then their innate opinionsare likely to be also close.

• access to a coarse conformity distribution: we as-sume that we have access to a coarse-grained esti-mate of the distribution of conformity parametersin the entire network.

We next define a novel problem, which we call See-dRecovery , that is fundamentally different from thewell-studied problem of influence maximization [23, 28].While the motivating application for both the problemsis finding influencers, the influencers from SeedRecov-ery are a small set of users whose early adoption (inthe past) of an opinion has been critical in shaping thecurrent snapshot of opinions in the network. On theother hand, the users computed via influence maximiza-tion are individual who, when currently seeded with theopinion, will be critical in shaping (or maximizing) thefuture spread of the opinion in the network, agnostic toany historical opinion dynamics.

We note that the study of SeedRecovery is enabledby our estimates of the conformity parameters. Moreformally, we are given the current state of expressedopinions in the social network and we want to identifyif there is a sparse set of innate opinions in the networkthat can explain the current expressed opinions. Thisenables us to distinguish opinions that might have beenseeded, i.e., are the result of a small number of plantedopinions (say by means of a viral market campaign)from opinions that arise naturally. It also helps capturea measure of the “heterogeneity” or “richness” of theopinion dynamics process in a social network. In fact,we identify this problem as a special case of the sparserecovery problem that is commonly studied in the com-pressed sensing and signal processing literature [2, 8,32] and for which greedy strategies are known to workwell1. We apply a similar Greedy algorithm for the

1We note here that unlike the influence maximization prob-lem, the sparse recovery objective is not submodular

SeedRecovery problem, and validate it for a specialcase analytically and more general, using experimentalanalysis on synthetic and Twitter data.

Contributions of this studyOur contributions in the study are three-fold:

• We address the ConformityExtraction prob-lem of recovering the conformity values of users ina social network, using only the stationary snap-shot of the opinion dynamics, along with assump-tions on homophily and a coarse-grained empiricaldistribution of user conformity.

• We formulate the problem of SeedRecovery ina social network, show that it is related to thewell-studied sparse recovery problem, and proposea Greedy algorithm for recovering the seed set ofusers for the opinion dynamics.

• We perform extensive experiments on both syn-thetic graphs and a large set of real-world Twit-ter data to validate the performance of our al-gorithms for both ConformityExtraction andSeedRecovery .

2. RELATED WORKThe broad research area of opinion formation is quite

classical, and we refer the interested reader to [21] fora survey. The earliest work in this domain comes fromthe sociology and statistics literature [29, 7, 10].

Several models for opinion formation and consensushave been studied in the sociology community. One no-table example is the work by DeGroot [7] which studieshow consensus is formed and reached when individualopinions are updated using the average of the neigh-borhood opinions in a network. The work of Friedkinand Johnsen (FJ) [11], is perhaps the first study to ex-tend the DeGroot model to include both disagreementand consensus, by associating with each node an innateopinion in addition to her expressed opinion. In theirmodel, a user adheres to her initial opinion with a cer-tain weight αi, while she is socially influenced by othersin her network with a weight 1− αi.

On the subject of conformity, recent work [31] fo-cused on computing conformity parameters under threedifferent notions of individual, peer, and group confor-mities. We differ from this line of work in that our focusis on leveraging the underlying opinion dynamics in asocial network to estimate user specific conformity pa-rameters. Further, estimating conformity values in thecontext of opinion dynamics also allows us to identifysparse seeded opinions (SeedRecovery ).

There has been a large body of work on modeling theadoption or spread of ideas, rumors or content amongonline users. Well known models in this domain include

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Threshold [18], Cascade models [14], and conformity-aware cascade models [25] that specify how a node adoptsa particular idea or product based on the adoption pat-tern prevalent in its neighborhood. Subsequently, sev-eral papers studied, both theoretically [23] and empiri-cally [13, 16, 27], the phenomenon of diffusion of ideasor content in a social network and the related problemof identifying influential nodes to seed, in order to max-imize adoption rates. Several of these papers are mainlyconcerned with binary-valued propagation of an idea orproducts where a user decides to either adopt or notadopt the idea, instead of a more continuous opiniondynamics model where a user opinion is influenced byher neighboring opinions to varying degrees. Howevera recent paper by Terzi et al. [12] considers influencemaximization in the context of opinion adoption basedon the Friedkin-Johnsen model. They pose the prob-lem of selecting a small set of nodes and seeding themwith a single positive opinion to maximize the adoptionof the overall positive opinion in the network. Theyshow that the resulting problem is submodular and canhence be maximized efficiently. However, they assumeknowledge of the user conformity values in the opiniondynamics model and do not address the problem of howto estimate these parameters.

As mentioned previously, the SeedRecovery prob-lem that we introduce is fundamentally different fromthe above influence maximization problems, since thegoal is not to seed users with products or opinions tomaximize adoption, but rather to understand if the cur-rent state of expressed opinions in the social networkcan be explained (from the opinion dynamics process)using the opinions of a small set of seed nodes, and ifso, to recover these seed nodes. Several results in thesparse recovery literature have shown [2, 8, 32, 5] thatgreedy and L1-relaxation techniques can recover the lin-ear combination efficiently as long as the matrix formedby taking the vectors xi as columns is well-conditionedand k is sufficiently small. However, we are not awareof any prior application of sparse recovery techniquesfor opinion formation problems.

In other related work, Das et al [4] addresses the prob-lem of sampling users in a social graph to estimate theaverage innate opinion of users, using only the expressedopinion of the sampled nodes. However, they too re-quire knowledge of the per-user conformity values fortheir sampling algorithms.

For the cascade-based models for diffusion of an ideaor product-adoption, the problem of estimating the adop-tion probabilities of a user has been studied in [17],[9] and [30]. Most of these papers use a probabilisticmodel, along with historical data of user adoption ac-tivity, to estimate adoption probabilities for each edgein the social graph. However, for the case of social opin-ion dynamics, to the best of our knowledge, we are not

aware of any related work for estimating the user con-formity parameters for applications that use opinion dy-namics models.

3. OPINION MODELWe consider a (possibly directed) social network graph

G = (V,E) with nodes V = {1, 2, . . . , n}, correspondingto individuals and edges E corresponding to social in-teractions. We will say that (i, j) ∈ E if i is influencedby j. We will denote by Ni = {j; (i, j) ∈ E}, the setof neighbors of node i and di = |Ni|, the out-degree ofnode i.

It will be convenient to express the graph in terms ofits adjacency matrix A, which corresponds to an n× nmatrix such that Aij = 1 if (i, j) ∈ E and Aij = 0otherwise. We will also use I to represent the n × nidentity matrix, 1 to represent the vector in Rn withall components 1 and given a vector v ∈ Rn, we willrepresent by dg(v) the matrix with the components ofv in the diagonal and zero elsewhere.

We are interested in studying opinion formation pro-cesses in social networks. We will distinguish betweenan agent’s innate opinion, which reflects the agent’s in-terval belief, and the agent’s expressed opinion, which isthe opinion an agent chooses to express in the networkas a result of a social influence process. Here we encodethe opinions as a single real quantity. Let yti ∈ R be theopinion expressed by node i on time t. We express byzi the innate opinion of agent i.

The classic model due to Friedkin and Johnsen [11]proposes a dynamic governing the opinion formationprocess. In their model, each agent is associated withconformity parameter αi in the [0, 1] range, which mea-sures how strong her innate opinions are, and how likelywill she be influenced by her neighborhood opinions.An αi value close to 1 implies that the individual ishighly opinionated, and her expressed opinion is similarto her innate opinion. While a value close to 0 impliesthat the individual has a very weak innate opinion andconsequently her expressed opinion is largely governedby the opinions of neighbors around her. According totheir model, in every timestep, each agent updates herexpressed opinion to a convex combination between herinnate opinion and the average of expressed opinions ofher neighbors in the previous timestep. The weight ofeach term in the

yt+1i = αi · zi + (1− αi) ·

1

di

∑j∈Ni

ytj (1)

In matrix form, we can re-write it as:

yt+1 = dg(α) · z + (I− dg(α)) · dg(d)−1Ayt (2)

It has been shown in [4] that the above opinion for-mation dynamics converge to an equilibrium that de-pends only on the innate opinions and the structure of

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the network and not on the original opinions, as long asαi > 0 for all i ∈ V , i.e., each individual holds an innateopinion that has some impact on what they express.

The equilibrium can be obtained as the unique fixedpoint of equation (2), i.e.:

y = (I− (I− dg(α)) · dg(d)−1A)−1dg(α) · z (3)

We will denote by F the matrix governing the Friedkin-Johnsen dynamic, i.e., F = (I−(I−dg(α))·dg(d)−1A)−1dg(α).Also, given any matrix M of size m × n and a subsetS ⊆ [n], we denote by MS the matrix of size m × |S|corresponding to the columns of M with indices in S.Similarly, given a vector x ∈ Rn, we will denote by xSthe vector in RS corresponding to the components of xwith indices on S.

4. CONFORMITY EXTRACTIONBased on the Friedkin-Johnsen opinion dynamics model

described in the previous section, we now address theproblem of estimating the conformity parameter αi of auser in a social network. Since we expect that these per-user conformity parameters depend on the topic that isbeing opined, in the remainder of this section we formu-late the conformity extraction problem in the context ofa particular topic. We then present a Linear Program-ming based approach for this problem.

We only assume knowledge of the current steady stateexpressed opinions of users in the social network. Inpractice, these opinions cam be obtained from opinionmining [26] of recent content posted by the user onthe social network, for example, her recent tweets onTwitter or posts on Facebook. We are also given thedirected social graph among these users, which couldcorrespond to, say, the Twitter follow graph or Face-book friend graph.

Using these two pieces of information, we would liketo estimate each user’s αi value. Clearly, as seen fromEquation 1, if we knew the user’s innate opinion zi,we could directly calculate her αi based on how far isher expressed opinion compared to her innate opinionand the mean of her neighboring expressed opinions.However, in practice, it is very hard to glean informa-tion about an individual user’s innate opinions. Hence,the main technical challenge is that we have an under-determined system of equations where we have two un-knowns per user: αi and zi while the opinion dynamicsmodel only gives us one equation per user.

To overcome the underdeterminacy of this system, wedevise two score functions and pick the candidate (α, z)pair that optimizes a combination of such scores:

Homophily score: We assume that users that are’close’ in the social network are also likely to have innateopinions that are close. This leads to the following score

function on the innate opinions:

H(z) =∑i,j∈V

|zi − zj | · (1−Dij/n)

where Dij is the distance from node i to node j in thesocial graph.

Conformity distribution score: We bucket therange of αi into three categories: low B1 = [0, 1/3],medium B2 = (1/3, 2/3] and high B3 = (2/3, 1] andassume that we have a coarse-grained estimate of theempirical distribution of the conformity values acrossbuckets: let λj be the estimate of what fraction of theαi values fall in bucket j. Then, given a vector α of con-formity parameters, the log-likelihood that this vectorwas generated by a distribution with bucket estimates(λ1, λ2, λ3) is given by:

L(α) = β1(α) · log λ1 + β2(α) · log λ2 + β3(α) · log λ3

where βi(α) =∑nj=1 βij and βij = 1 if αi ∈ Bj and

zero otherwise.Note that the assumption about knowledge of the

coarse-grained conformity distribution across the threebuckets is much weaker than an assumption about theexact distribution of the αi values. Furthermore, inpractice this is not an unrealistic assumption, since onecan manually go through a small set of users who havetweeted or created a post about the topic, peruse theirpast postings and use human judgment to infer whatfraction of them are likely to be highly conforming,moderately conforming, or stubborn.

4.1 Mathematical Program for Identifying Con-formity

Based on the score functions identified, we proposeidentifying conformity via the following MathematicalProgram:

Minimizeα,z H(z)− c · L(α)

s.t

y = dg(α) · z + (I− dg(α))dg(d)−1Ay

αi ∈ (0, 1), ∀i ∈ V

where c is a regularization constant that adjust thetrade-off between the score functions, and that is setin our experiments using cross validation. By rearrang-ing the update rule of the Friedkin-Johnsen model anddenoting the average opinion of neighbors of node i bymi = 1

di

∑j∈Ni

yi, we can write:

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Minimizeα,z H(z)− c · L(α)

s.t

αi(zi −mi) = yi −mi, ∀i ∈ Vαi ∈ (0, 1), ∀i ∈ V

Observe that this is not a Linear Program since bothterms αi and zi are variables and the product αi · ziappears in the constraints. Now we describe how tochange the program to get rid of the product. We notethat the objective function doesn’t depend on αi di-rectly, but rather on βij which are indicators of theevent αi ∈ Bj . Therefore, we propose to substitute theconstraint αi(zi −mi) = yi −mi by

βij = 1⇒ yi −mi

zi −mi∈ Bj

and enforce integrality constraints on βij as well as∑3j=1 βij = 1. We note that this doesn’t affect the

value of the mathematical program, since for any solu-tion of the program on (z, β), it is possible to recover asolution (z, α) with the same objective simply by takingαi = (yi −mi)/(zi −mi).

Now, we re-write the newly introduced constraints ina more amenable form. First, we enforce the constraintthat the ratio (yi −mi)/(zi −mi) is non-negative. Giventhis constraint, we can rephrase βij = 1⇒ bj ≤

yi−mi

zi−mi≤

bj where Bj = [bj , bj ] as follows:

bj · sgn(yi −mi) · (zi −mi) ≥ βij |yi −mi|

bj ·sgn(yi−mi) · (zi−mi) ≤ βij |yi −mi|+K · (1−βij)

where sgn(x) = 1 if x ≥ 0 and sgn(x) = −1 other-wise and K = c1(sgn(yi −mi) · (zi −mi)) where c1 isa constant to be chosen later. To see that those areequivalent, notice that if βij = 1, this gives exactlybj ≤

yi−mi

zi−mi≤ bj . If βij = 0, then both constraints are

trivially satisfied for c1 ≥ 2/3.

This leads to the following Integer Program:

Minimize H(z)− c ·∑3j=1(

∑i βij) log λi

s.t

(yi −mi)(zi − yi) ≥ 0,∀ibj · sgn(yi −mi) · (zi −mi) ≥ βij |yi −mi|,∀i, jbj · sgn(yi −mi) · (zi −mi) ≤ βij |yi −mi|+K · (1− βij),∀i, j∑k

j=1 βij = 1,∀i ∈ Vβij ∈ {0, 1},∀i, j

Relaxation and rounding. We refer to the so-lution of this integer program as the IPRecovery .

However, since solving an integer program is not feasi-ble for large instances, we begin with an LP relaxation,LPRecovery , and offer a simple rounding techniquefor the solution of LPRecovery to obtain the final so-lution. Our approach consists of relaxing the integralityconstraints to βij ∈ [0, 1], solving the resulting LinearProgram, obtaining z and βij and then recovering thevalues of αi by setting:

αi =yi −mi

zi −mi

Note that in the LP objective, there are no αi’s. Weassign node i to bucket Bj based on its αi value. Thisin a way, is a rounding of βij based on only αi val-ues. In the analysis, we show that original βij valuesfrom the LP and the rounded βij ’s from αi’s are quiteclose. Therefore, the objective value resulting from therounding is also close to the objective value of IPRe-covery since the solution to LPRecovery is a lowerbound to the objective function. Further, we also showempirically (see Section 6) our approach performs com-parably to IPRecovery .

Analysis. In the above linear program, we show thatthe parameters βij and the αi derived using the equa-tion αi = (yi −mi)/(zi −mi), are closely related.

Lemma 1. αi ∈ [0, 1].

Proof. From the constraint (yi −mi)(zi − yi) ≥ 0,sgn(yi −mi) = sgn(zi − yi) = sgn(zi −mi).Thus yi lies between zi and mi. Hence, |yi − mi| ≤|zi −mi|. Thus αi = (yi −mi)/(zi −mi) ≤ 1.On the other hand, from the same constraint we get,

(yi −mi)(zi − yi)(zi −mi)2

≥ 0

⇒ (yi −mi)/(zi −mi) ≥ 0

⇒ |yi −mi|/|zi −mi| ≥ 0

⇒ αi ≥ 0

Lemma 2. If βi1 > c2 then αi <1

3c2.

Proof. From bj ·sgn(yi−mi)·(zi−mi) ≥ βij |yi −mi|,13 |zi −mi| ≥ βi1 |yi −mi| > c2|yi −mi|⇒ 1

3c2> |yi−mi||zi−mi| = αi.

Corollary 1. If βi1 > 1/2 then αi <23 .

Lemma 3. If βi3 > c3 then αi > c1 − 3c1−23c3

.

Proof. From bj ·sgn(yi−mi)·(zi−mi) ≤ βij |yi −mi|+K · (1− βij),⇒ 2

3 ≤ βi3 · αi + c1(1− βi3).⇒ (c1 − 2

3 ) ≥ βi3(c1 − αi) > c3(c1 − αi)⇒ αi > c1 − 3c1−2

3c3.

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Corollary 2. If βi3 > 1/2 and c1 = 1 then αi >13 .

Theorem 1. If βij > 1/2 for some j ∈ {1, 2, 3} andderived α ∈ Bk, then |j − k| ≤ 1.

Proof. For j = 2, the theorem is trivially true. Forj = 1, 3 the proof follows from Corollary 1 and Corollary2.

In words, Theorem 1 says that if β has reasonableweight (¿1/2) on some bucket Bj , then the derivedbucket Bk from α is very close to Bj .

5. SEED RECOVERYIn the previous section, we discussed how to infer the

conformity parameters αi for users in a social network.The knowledge of those parameters enables various in-teresting applications. In this section, we discuss onesuch application called the SeedRecovery problem.

Consider the scenario where yi represents the opin-ion about a certain product. By means of a marketingcampaign, a company might try to influence the generalopinions on this product by planting few nodes withvery high innate opinions about such products. Thiscould be done, for example, by paying celebrities totweet about certain products or events. The problemof how to choose a few nodes to seed an opinion on anetwork has been extensively studied [23, 12]. In thecontext of SeedRecovery , we ask the following ques-tion: given the expressed opinions in a network, howlikely it is that those were seeded by a small number ofnodes ?

We assume that opinions are normalized in such away that zi = 0 represents a default neutral opinion,i.e., the node hasn’t heard about that particular prod-uct or has neither positive nor negative innate opinionsabout it. Now, given a certain observed expressed opin-ions y, is there a vector z with a small number of innatenon-neutral opinions that could have produced y as anoutcome of the Friedkin-Johnsen dynamic ? In otherwords, for a given k, can we estimate:

errork = minz‖y − Fz‖22 s.t. ‖z‖0 = k

This can be cast as an instance of the sparse recoveryproblem from compressed sensing. The two main ap-proaches to solve this problem are convex relaxation[2, 8] and greedy algorithms [32, 5]. We take the lat-ter route and apply the well-known Forward Regressionalgorithm that was analyzed in [5]. Davis et al. [6]showed that the problem of minimizing errork admitsno multiplicative approximation, by showing that it isNP-hard to check for a given instance if errork = 0. Asa result, approximation guarantees for this problem areusually given in terms of the squared multiple correla-tion or R2 objective (R2 = 1−errork/‖y‖22), which is awell known measure of goodness-of-fit in statistics [22].

We note that since 0 ≤ errork ≤ ‖y‖22, the R2-objective is in the [0, 1] range, where R2 = 1 corre-sponds to a solution of errork = 0. We will say that asolution z is an α-approximation if R2(z) ≥ α · R2(z′)for any z′ ∈ Rn with ‖z′‖0 = k.

Now we describe the Forward Regression algorithmand derive its approximation guarantees for the See-dRecovery problem. First, we note that the problemcan be re-written as:

errork = minS:|S|=k

[minzS‖y − FSzS‖22

]For a fixed S ⊆ [n], it is a classic result in Linear Alge-bra (see [15] for example) that ‖y−FSzS‖22 is minimizedby the vector zS = (FTSFS)−1FTSy, therefore the errorcan be written as:

errork = minS:|S|=k

‖y − FS(FTSFS)−1FTSy‖22

= minS:|S|=k

‖y‖22 − yTFS(FTSFS)−1FTSy

since FS(FTSFS)−1FTSy and y − FS(FTSFS)−1FTSy areperpendicular vectors. This transformation allows usto write the problem in terms of the R2 objective as:

R2 = maxS:|S|=k

f(S) where f(S) = yTFS(FTSFS)−1FTS y

and y = y/‖y‖2. The Forward Regression algorithmbuilds a family of sets incrementally by adding the ele-ment that provides the maximum increase in value forf . The algorithm is initialized with S0 = ∅ and foreach k = 1 . . . , n, we define Sk+1 = Sk ∪ {i} for somei ∈ argmaxj /∈Sk

f(Sk ∪ {j}).We will provide a two-fold validation for the Forward

Regression algorithm for the Seed Recovery problem:the first is theoretical. We will use a result due to Dasand Kempe [5] to give a theoretical approximation guar-antee for this problem for a special case. The ForwardRegression algorithm notoriously performs much betterin practice than its theoretical bounds [5], however thetheoretical guarantee is useful to highlight the depen-dency of the algorithm on parameters of the instance.In particular, we will show that the approximation guar-antee improves for higher values of αi. In Section 7,we also perform experimental validation of this algo-rithm: we construct synthetic instances of the problemfor which the innate opinions form a sparse vector andevaluate the outcome of the Forward Regression algo-rithm against the ground truth.

The theoretical guarantee on the approximation ofthe Forward Regression algorithm can be obtained fromspectral properties of the matrix F:

Theorem 2 (Das and Kempe [5]). For each k =

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1, 2, . . . , n,

f(Sk) ≥[1− exp

(−λkmin(FTF)

maxi ‖Fi‖22

)]· maxS:|S|=k

f(S)

where F i is the i-th column of matrix F and λkmin(FTF)is the smallest k-sparse eigenvalue of the matrix FTF,i.e., λk,min(FTF) = minx∈Rn\0,‖x‖0=k ‖Fx‖22/‖x‖22

In what follows, we provide a lower bound on theexponent λkmin(FTF)/maxi ‖Fi‖22 for the special caseof regular undirected graphs and uniform αi values, i.e.,we will assume that αi is the same for all i and that thegraph is undirected and all nodes have the same degree.Since αi is the same and di is the same for all nodes,we drop the subscript i for the rest of the section. TheFriedkin-Johnsen matrix F in this case is symmetric andcan be written as:

F = α ·(

I− 1− αd·A)−1

= α

∞∑k=0

(1− α)k(

1

dA

)kusing the matrix identity (I−M)−1 =

∑∞k=0 Mk. From

this we can observe that the entries of F are non-negative.We use this fact to show the following result:

Lemma 4. If F is the Friedkin-Johnsen matrix as-sociated with an undirected regular graph with degree dand uniform values αi = α(> 0), then all columns ofthe matrix F have their norm bounded by 1.

Proof. Let 1 be the vector with all components equalto 1. Then if A is the matrix associated with the reg-ular graph, then clearly 1

dA1 = 1. Therefore F1 =

α∑∞k=0(1−α)k( 1

dA)k1 = 1α∑∞k=0(1−α)k = 1. There-

fore, for all rows i,∑j Fij = 1. Since the entries are

non-negative, it means that all entries are in the [0, 1]range. Finally, we use the symmetry of F to see that:

‖Fi‖22 =∑j

F2ij ≤

∑j

Fij = 1

The remaining term in Theorem 2 is λkmin(FTF), whichwe bound by the smallest eigenvalue of FTF:

λkmin(FTF) ≥ λmin(FTF) = minx 6=0

‖Fx‖22‖x‖22

In the subsequent proof, we use the concept of the op-erator norm of a matrix. Given a square matrix M, wedefine its operator norm ‖M‖2 = maxx∈Rn\{0} ‖Mx‖2/‖x‖2and use the following matrix inequalities:

‖M1 + M2‖2 ≤ ‖M1‖2 + ‖M2‖2 (4)

‖M1 ·M2‖2 ≤ ‖M1‖2 · ‖M2‖2 (5)

We refer to [15] for an extensive exposition on matrixnorms and spectral properties of matrices.

Also, given the adjacency matrix A of a regular graphof degree d. We will use the fact [3] that the operatornorm of the adjacency matrix is at most the maximumdegree of its vertices, hence,

‖1

dA‖2 ≤ 1. (6)

Lemma 5. If F is the Friedkin-Johnsen matrix asso-ciated with an undirected regular graph d and uniformvalues αi = α(> 0), then

λmin(FTF) ≥ α2

4

Proof. Since α > 0 then F is invertible, therefore:

λmin(FTF) = minx 6=0

‖Fx‖22‖x‖22

= miny 6=0

‖y‖22‖F−1y‖22

=

[maxy 6=0

‖F−1y‖2‖y‖2

]−2= ‖F−1‖−22

By the definition of F we have that:

F−1 = α−1 · (I− (1− α) · 1

dA)

Therefore:

‖F−1‖2 ≤ α−1(‖I‖2 + (1− α)‖ 1dA‖2) ≤ α−1(1 + 1− α) ≤ 2α−1

Here the first inequality follows from the matrix inequal-ities 4 and 5 and the second inequality follows from in-

equality 6. Thus, λmin(FTF) ≥ α2

4

Thus from Theorem 2, Lemma 4 and 5, we get thefollowing theorem:

Theorem 3. For each k = 1, 2, . . . , n,

f(Sk) ≥[1− exp

(−α2/4

)]· maxS:|S|=k

f(S).

6. CONFORMITY EXTRACTION EXPER-IMENTS

In this section, we present the results for our exper-iments for the ConformityExtraction problem onboth real world Twitter data and synthetic data.

6.1 Conformity in Synthetic DataWe first conduct synthetic experiments (where we

have complete access to ground truth and can there-fore obtain fine-grained validation) to show that our al-gorithms described in Section 4 can extract conformityvalues with high accuracy. Toward this end, we gen-erated graphs having regular, random and power lawdegree distributions. The number of nodes in the graphwas varied from 100 to 1000. The degree in the regulargraph case was set to 20 while the maximum degree inthe power-law and random graph was set to 100.

Every node was assigned one of 10 innate opinions in{0, 1, 2, . . . , 9}. To capture a homophily effect on the

7

innate opinions, we imposed a Lipschitz constraint onthe assignment of innate opinions to nodes, such thatfor 85% of the graph edges, the difference in opinionsbetween the nodes forming the edge is at most 1.

For assigning α values to nodes, we used three dif-ferent distributions: α values distributed uniformly in[0, 1], α values taken from a power-law distribution withmost of the values close to 0, and a bimodal Gaussiandistribution with peaks close to 0 and 0.5. Finally, wenote that all results are averaged over 10 runs.

Using the α distribution and innate opinions of nodesin the graph, we ran 5000 rounds of the Friedkin-Johnsenopinion dynamics, and considered the final opinions ofthe nodes as their expressed opinions.

Based on these expressed opinions and the graph struc-ture, we then estimated the α values of all nodes in thegraph, using our LPRecovery and IPRecovery al-gorithms for ConformityExtraction described inSection 4. For all our experiments, we used a commer-cial optimization solver [19] to run LPRecovery andIPRecovery . To measure the effectiveness of our al-gorithms, we compared the estimated α values withthe ground-truth α parameters. We first categorizedthe nodes into three buckets based on their groundtruth α values: low, medium, and high corresponding tothe ranges [0, 1/3], (1/3, 2/3], and (2/3, 1] respectively.Then, we plotted the performance of our algorithms(separately for each bucket and also for for the overallset of nodes) using two metrics: 1) a fine-grained mea-sure corresponding to the absolute error between theground truth α and the estimated α; 2) a coarse-grainedmeasure corresponding to the accuracy or percentage ofnodes for which we estimated their buckets correctly.

Figure 1 illustrates the results for random graphs us-ing the LPRecovery algorithm. We varied the num-ber of nodes in {100, 250, 500, 750, 1000} and ran theexperiments with the three different types of α distribu-tions. In almost all the cases, the absolute error in esti-mating the conformity parameters was less then 0.12. Ingeneral, we observe that the accuracy of our algorithmis slightly better for the sparse α distribution wheremost of the nodes have low α values, than with otherdistributions. We also note that the estimation errorfor the α values is lower for large graphs than for smallgraphs. This is likely due to the fact that as we increasethe number of nodes in the random graph, the expecteddegree of a node increases, which then strengthens thehomophily assumptions used by the linear program toprune its feasible solution space.

The results for the accuracy measure are qualitativelysimilar. As seen from the figure, we recover at least 80%of the values in each bucket. Again, in the natural caseof sparse α distribution, this number increases to 90%.

We observe that we obtained qualitatively similar re-sults for the case of regular and power-law graphs and

for the case where we used the IPRecovery algorithminstead of LPRecovery . The results are omitted dueto lack of space.

6.2 Conformity in Twitter DataNext, we ran our algorithms for ConformityEx-

traction on real world social network data from Twit-ter. Our data set comprised of user opinions extractedfrom a large set of tweets corresponding to one of threetopics, namely: organic food, weight loss and electriccars. We considered all tweets related to these top-ics (using simple keyword-based classifiers) within a 6-month timeline from 12/1/2012 to 5/31/2013. Thetotal number of tweets in each topic varied between100000 and 2000000. We then ran each tweet througha commercial sentiment analyzer [24] to obtain senti-ment values ranging from−1 (corresponding to negativesentiment) to 1 (corresponding to positive sentiment),which was then smoothed into one of 10 opinion buck-ets. We treated the median of a user’s last three opinionvalues as her expressed opinion. Using the Twitter fol-low graph, we obtained the induced subgraph over thenodes (across all topics) with around 1 million nodesand 100 million edges. As before, our goal is to extractα values of the users using the expressed opinions andgraph structure.

One significant difference in this experiment is that,unlike the synthetic experiments, there is no explicitground truth α values available for the users. To over-come this problem, we identified, using tweet histories, asmall set of ground truth consisting of users with highα and low α. To obtain this ground truth for eachtopic, we first extract users with at least 5 tweets onthe topic, and have at least 5 neighbors in the Twittergraph who have also tweeted about the topic. We cate-gorize a user’s tweet into a positive, negative or neutralopinion based on the sentiment value. We then definea stubborn user (high α bucket) as one whose opinionsdiffers from her majority neighboring opinion for ε frac-tion of her tweets. Similarly a user is conforming (lowα bucket) if her final opinion is different from her initialopinion and whose set of opinions is the same as the fi-nal majority opinion in her neighborhood for γ fractionof her tweets. We ignored the nodes that do not satisfythese conditions in our analysis. For our experimentswe set ε to be 0.7 and γ to be 0.3.

Note that the above method to compute the groundtruth cannot be used as a general algorithm to extractconformity values for all users since only a small setof users satisfy the aforementioned criteria to reliablymeasure their conformity. This necessitates the designof algorithms such as the ones proposed in this paper.

For each of the topics, we ran our IPRecovery andLPRecovery algorithms using the expressed opinionsand the induced graph structure on all Twitter users

8

0

0.05

0.1

0.15

0.2

0.25

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Ab

solu

te E

rro

r

Number of nodes

Random Graph, Sparse α

overall α small α medium α large α

(a) Absolute Error for sparse α

0

0.02

0.04

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0.1

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0.18

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Ab

solu

te E

rro

r

Number of nodes

Random Graph, Uniform α

overall α

small α

medium α

large α

(b) Absolute Error for uniform α

0

0.02

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0.14

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Ab

solu

te E

rro

r

number of nodes

Random Graph, Bimodal α

overall α small α medium α large α

(c) Absolute Error for bimodal α

0

20

40

60

80

100

120

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Acc

ura

cy

number of nodes

Random Graph, Sparse α overall α small α medium α large α

(d) Accuracy for sparse α

0102030405060708090

100

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Acc

ura

cy

Number of nodes

Random Graph, Uniform α

overall α small α medium α large α

(e) Accuracy for uniform α

0

20

40

60

80

100

120

100 nodes 250 nodes 500 nodes 750 nodes 1000 nodes

Acc

ura

cy

Number of nodes

Random Graph, Bimodal αoverall α

small α

medium α

large α

(f) Accuracy for bimodal α

Figure 1: Effectiveness of LPRecovery on random graphs

Topic Algorithm Percentageof stub-born nodesrecovered

Percentageof conform-ing nodesrecovered

Electric Car LPRecovery 75 65Electric Car IPRecovery 77 82.5Organic Food LPRecovery 87.5 66.7Organic Food IPRecovery 89 59Weight Loss LPRecovery 55 65Weight Loss IPRecovery 55 61

Table 1: Conformity Extraction on Twitter

who have tweeted about the topic at least 5 times andhave at least 5 neighbors, and estimated their confor-mity parameter α. This yielded around 2000 to 15000users for each topic. From among these users, we thenextracted a smaller ground truth set of stubborn usersand conforming users. We considered the estimated αvalue for each user in the ground truth set, and useda threshold of 0.66 and 0.33 on their estimated α topredict whether these users are stubborn, conforming,or neither. We then measure the recall with respectto the stubborn user set (i.e., what fraction of groundtruth stubborn users were correctly recovered by our al-gorithm) and similarly with respect to the conforminguser set. Table 1 summarizes the results.

As the table shows, our algorithms perform well inrecovering the stubborn users from the ground truthset for many instances. In particular, for the topic oforganic food, our algorithms recover almost 90% of thestubborn users, but recover a lesser (60%) fraction ofthe conforming user. On the other hand, for weight loss,the performance of the algorithms in recovering the con-forming users (65%) is better than that for the stubbornusers (55%). This correlated with the observed skew

toward stubborn users for organic food and conformingusers for weight loss (see Figure 3).

The table above also validates empirically that theLPRecovery algorithm is a good approximation tothe (much slower) IPRecovery algorithm, since thegap in performance between the two algorithms for mostcases is less than 10%.

6.3 Validation of homophily in TwitterThe assumption of homophily in the innate opinions

of users in a social networks is crucial in our LPRe-covery and IPRecovery algorithms, since it helpsus solve an under-determined system of equations. Tovalidate this assumption, we set out to observe the dif-ference between the innate values of neighbors in theTwitter follow graph for the three topics. For the ho-mophily experiments, we define a user’s innate opinionon a particular topic to be the average of the sentimentvalues associated with the first three tweets posted bythe user in the 6 month time period. Recall that theuser’s opinion values as extracted from our sentimentanalyser are bucketed into {0, 1, 2, . . . , 9}. Table 2 re-ports the average difference in opinions between everypair of users connected by an edge in the Twitter graph,as well as the total fraction of edges in the graph forwhich this difference in opinions is less than 1. As seenin the table for our topics of interest, we observe that formore than 64% of the users, the difference in opinionsacross an edge is less than 1. Furthermore the averagedifference between a pair of neighboring users is lessthan 1 for all the topics.

7. SEED RECOVERY EXPERIMENTSIn the next set of experiments, we used the α values

9

Topics Avg gap inneighboringopinions

Fraction ofedges withgap < 1

Electric Car 0.75 64%Weight Loss 0.79 69%Organic Food 0.60 83%

Table 2: Homophily in Twitter data

from our previous ConformityExtraction experi-ments to address the SeedRecovery problem for bothsynthetic as well as Twitter graphs. Thus, we would liketo compute a small set of nodes with a given innateopinion (and assuming neutral innate opinion on allother nodes) that can best explain the current expressedopinions in the network resulting from Friedkin-Johnsendynamics. As described in Section 5, we measure thediscrepancy in the predicted expressed opinions usingthis seed set versus the ground truth expressed opinionacross all the nodes in the graph. As mentioned ear-lier, for measuring this discrepancy, we use the squaredmultiple correlation (R2) metric, which lies in [0, 1] andis essentially equal to 1 − L2Error, where L2Error isthe normalized L2 norm error between the predicted ex-pressed opinion vector and the ground truth expressedopinion vector. Our goal is to recover a seed set of knodes in the graph that can maximize the R2 measure.

We use the Greedy algorithm (defined in Section5) to recover the best seed sets of size k (we vary kfrom 1 to 20) for both synthetic and Twitter data. Wealso report the characteristics of the computed seed setas we vary its size k. We compare the performance ofthe Greedy algorithm against two natural baselinesthat have been used in similar problems ( [23]): select-ing nodes with the highest α-values and selecting nodeswith the highest degrees in the graph.

7.1 Synthetic DataSimilar to Section 6, we generated synthetic graphs of

1000 nodes with regular, random and power law degreedistributions, and assigned innate opinions in a similarmanner as earlier. We also used the same three dis-tributions of α values as earlier: uniform, bimodal andsparse. We only report the result of random graphs (theregular and power-law graphs had qualitatively simi-lar results). In Figure 2(a), we plot the R2 metric asa function of k for the various α distributions for theGreedy algorithm.

First, as expected, the R2 increases as the size of theseed set increases. More interestingly, even for as lowas 6 seed nodes, we get an R2 value of close to 0.92indicating a very good agreement between the origi-nal and predicted expressed opinions. For comparison,we also plot the corresponding R2 values using the twobaseline algorithms (selecting nodes according to the α

values and degrees respectively) for each of the α dis-tributions. Clearly, our Greedy algorithm performssignificantly better than both the baselines. Secondly,in terms of the characteristics of the seed set selectedby Greedy , we observe that the algorithm starts offby initially selecting high-degree, stubborn nodes, andthen moves to nodes with lower α and degree values(Figures 2(b) and 2(d)) as the size of the seed set in-creases. We also measured the difference in the α valuesbetween a node in the seed set and its average neighbor-ing α. Figure 2(c) indicates that our algorithm favorsselecting seed nodes that have high α but whose neigh-bors have low α values. These observations agree withour intuitive expectation that the most likely seed nodesare ones that are stubborn and have a large number ofconforming neighbors. Interestingly, similar behaviourhas also been observed previously in [4] in the contextof selecting nodes to best estimate the average innateopinion in the network.

In a more direct experiment (Figures 2(e) and 2(f)),we “planted” 10 seed nodes in a 1000 node randomgraph with α = 0.95 for all the 10 nodes. Further,these 10 nodes were initialized with non-neutral innateopinions while those of the remaining nodes had neu-tral opinions. As before, the α values of the remain-ing nodes were drawn from three different distributions– sparse, bimodal, and uniform. The goal of this ex-periment was to validate if our algorithm can indeed”recover” the planted seed nodes, purely based on theexpressed opinions and alpha values of all the nodes.Note that the graph also contained several (based onthe specific α distribution) stubborn nodes that werenot seeds, and hence it is not sufficient to simply picknodes with large α values as the seeds. This is corrob-orated by Figure 2(e) which shows that the R2 value ofthe seed set obtained by the Greedy algorithm outper-forms the α-based algorithm. The α-based algorithmin turn outperforms the degree-based algorithm, due tothe fact that the seed nodes in this case have high alphavalues.

In particular, Figure 2(e) shows that the Greedy al-gorithm finds exactly the right set of 10 seed nodes.This is because for any size ≥ 10, the R2 value is 1.0,implying that the selected seed set actually contains allthe 10 seed nodes! This observation is further corrobo-rated by Figure 2(f) where we see that the average valueof α for the seed set of size 10 is precisely 0.95 whichis indeed the α value for each of the planted stubbornnode.

7.2 Twitter DataNext, we repeated the SeedRecovery experiments

using Twitter data for various topics. The dataset andresulting social graph are exactly the same as in Sec-tion 6.2. As earlier, we used the α values generated

10

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

R2

Number of Seed Nodes

Sparse (greedy) Bimodal (Greedy) Uniform (Greedy)sparse (alpha) bimodal (alpha) uniform (alpha)sparse (degree) bimodal (degree) uniform (degree)

(a) Squared multiple correlation R2

0

0.2

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1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ave

rage

α

Number of Seed Nodes

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(b) Average α of the seed set

0

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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rage

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iffe

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ce

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(c) Avg difference in α values of theseed set node and its neighborhood

0

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120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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rage

de

gre

e

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Bimodal (Greedy)

Uniform (Greedy)

(d) Average degree of seed set for apower-law graph

0

0.2

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1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15R

2Number of Seed Nodes

Sparse (greedy) Bimodal (Greedy) Uniform (Greedy)

sparse (alpha) bimodal (alpha) uniform (alpha)

sparse (degree) bimodal (degree) uniform (degree)

(e) Squared multiple correlation R2 ofthe recovered seed set

0

0.1

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0.9

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Ave

rage

α

Number of Seed Nodes

Sparse (greedy)Bimodal (Greedy)Uniform (Greedy)

(f) Average α of the recovered seedset

Figure 2: Characteristics of the seed set for synthetic random graphs, with uniform, bimodal, andpower-law α-distributions. Figures 2(e) and 2(f) show results for the “planted” seed set experiment.

from the conformity extraction experiments describedin Section 6.2. Figure 3 shows the distribution of theseα values for different topics. The α distributions for allthe topics resemble either sparse or bimodal distribu-tions, and we remind the reader that we covered bothof them in our simulations.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 - 0.33 0.33 - 0.66 0.66 - 1

α value

Electric Car Organic Food Weight Loss

Figure 3: Distribution of α across the threebuckets

The results are summarized in Figure 4. Qualita-tively, even for this data, we observe similar results tothe simulations. The R2 of the selected seed set is muchlarger than the α-based and degree-based baseline algo-rithms, for all the topics. Similarly, based on the plotsshowing the average α value, average degree, and av-erage neighborhood difference in α values for the seedset, the Greedy algorithm shows a clear preference forselecting seed nodes that are moderately stubborn andhave a large number of conforming neighbors. (Notethat just selecting seed nodes based on high alpha valuesalone does not suffice, as shown by the correspondingbaseline performance in 4(a)).

However, we do see interesting differences in the R2

plots between various topics. For example, while thesquared multiple correlation for Electric cars is around0.6 for k = 20, it is only around 0.2 for the case ofOrganic food or Weight loss. This suggests that for theTwitter data, the equilibrium opinions for Electric carsis much more likely to have been generated by a smallnumber of seed nodes, as compared to those for organicfood or weight loss. We therefore surmise that the R2

measure of the seed set for a topic might provide insightsinto a notion of how “heterogenous” or “diverse” is theopinion dynamics for that topic in a social network.

8. CONCLUSIONSThe notion of conformity plays a central role in shap-

ing of users opinions in online social networks. In thisstudy, we proposed algorithms for estimating confor-mity of users using only the expressed opinions of usersresulting from the underlying opinion dynamics and thesocial graph. Under some natural conditions, we showusing both simulations and Twitter data that our algo-rithms perform well on extracting the conformity valuesof the users.

Further, we propose efficient algorithms to recoverthe smallest set of source nodes in the graph that bestexplain the current distribution of opinions in the entiregraph. We refer to this problem as seed recovery andwe believe this and similar problems have many appli-cations in running effective marketing campaigns, un-derstanding information flow in social networks etc. Asbefore, we validate our algorithms for this problem us-ing both simulations and Twitter data. An interesting

11

0

0.2

0.4

0.6

0.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

R2

Number of Seed Nodes

Electric Car (Greedy) Electric Car (alpha)Electric Car (degree) Organic Food (Greedy)Organic Food (alpha) Organic Food (degree)Weight Loss (Greedy) Weight Loss (alpha)Weight Loss (degree)

(a) Squared multiple corre-lation R2

0

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1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Ave

rage

α

Number of Seed Nodes

Electric Car (Greedy) Organic Food (Greedy)

Weight Loss (Greedy)

(b) Average α of the seedset

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Ave

rage

α-d

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ren

ce

Number of Seed Nodes

Electric Car (Greedy) Organic Food (Greedy)

Weight Loss (Greedy)

(c) Average difference in αvalues of the seed set nodeand its neighborhood

0

5

10

15

20

25

30

35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Ave

rage

de

gre

e

Number of Seed Nodes

Electric Car (Greedy)Organic Food (Greedy)Weight Loss (Greedy)

(d) Average degree of seedset

Figure 4: Characteristics of seed set for the three different topics on Twitter

open question is to generalize the conformity extractionproblem to other well-studied opinion dynamics mod-els.

9. REFERENCES

[1] D. Bindel, J. M. Kleinberg, and S. Oren. How badis forming your own opinion? In FOCS, pages57–66, 2011.

[2] E. J. Candes, J. Romberg, and T. Tao. Stablesignal recovery from incomplete and inaccuratemeasurements. Communications on Pure andApplied Mathematics, 59:1207–1223, 2005.

[3] F. Chung. Spectral Graph Theory. Number no. 92in CBMS Regional Conference Series. AmericanMathematical Society, 1997.

[4] A. Das, S. Gollapudi, R. Panigrahy, and M. Salek.Debiasing social wisdom. In Proceedings of the19th ACM SIGKDD, pages 500–508. ACM, 2013.

[5] A. Das and D. Kempe. Submodular meetsspectral: Greedy algorithms for subset selection,sparse approximation and dictionary selection. InICML, pages 1057–1064, 2011.

[6] G. Davis, S. Mallat, and M. Avellaneda. Adaptivegreedy approximations. ConstructiveApproximation, 13(1):57–98, Mar. 1997.

[7] M. H. DeGroot. Reaching a consensus. Journal ofthe American Statistical Association,69(345):118–121, 1974.

[8] D. Donoho. For most large underdeterminedsystems of linear equations, the minimal 11-normnear-solution approximates the sparsestnear-solution. Communications on Pure andApplied Mathematics, 59:1207–1223, 2005.

[9] X. Fang, P. J.-H. Hu, Z. L. Li, and W. Tsai.Predicting adoption probabilities in socialnetworks. Information Systems Research,24(1):128–145, 2013.

[10] J. Fowler and N. Christakis. Cooperative behaviorcascades in human social networks. Proc. Nat.Acad. Sci., 107(12):5334–8, 2010.

[11] N. E. Friedkin and E. C. Johnsen. Social influenceand opinions. Journal of Mathematical Sociology,15(3-4):193–206, 1990.

[12] A. Gionis, E. Terzi, and P. Tsaparas. Opinionmaximization in social networks. arXiv preprintarXiv:1301.7455, 2013.

[13] S. Goel, D. J. Watts, and D. G. Goldstein. Thestructure of online diffusion networks. In Proc. ofthe 13th ACM EC, pages 623–638. ACM, 2012.

[14] J. Goldenberg, B. Libai, and E. Muller. Talk ofthe network: A complex systems look at theunderlying process of word-of-mouth. Marketingletters, 12(3):211–223, 2001.

[15] G. H. Golub and C. F. Van Loan. Matrixcomputations (3rd ed.). Johns Hopkins UniversityPress, Baltimore, MD, USA, 1996.

[16] M. Gomez Rodriguez, J. Leskovec, andA. Krause. Inferring networks of diffusion andinfluence. In Proc. of the 16th ACM SIGKDD,pages 1019–1028. ACM, 2010.

[17] A. Goyal, F. Bonchi, and L. V. Lakshmanan.Learning influence probabilities in socialnetworks. In Proceedings of the third ACMinternational conference on Web search and datamining, pages 241–250. ACM, 2010.

[18] M. Granovetter. Threshold models of collectivebehavior. American journal of sociology, pages1420–1443, 1978.

[19] Gurobi. Optimizer -http://www.gurobi.com/products/gurobi-optimizer/gurobi-overview.

[20] R. Hegselmann and U. Krause. Opinion dynamicsand bounded confidence models, analysis, andsimulation. Journal of Artificial Societies andSocial Simulation, 5(3), 2002.

[21] M. O. Jackson. Social and Economic Networks.Princeton University Press, Princeton, NJ, USA,2008.

[22] R. A. Johnson and D. W. Wichern. Appliedmultivariate statistical analysis, volume 5.Prentice hall Upper Saddle River, NJ, 2002.

[23] D. Kempe, J. Kleinberg, and E. Tardos.Maximizing the spread of influence through asocial network. In Proc. of the ninth ACMSIGKDD, pages 137–146. ACM, 2003.

[24] Lexalytics. Salience engine -http://www.lexalytics.com/technical-

12

info/salience-engine-for-text-analysis.[25] H. Li, S. S. Bhowmick, and A. Sun. Cinema:

conformity-aware greedy algorithm for influencemaximization in online social networks. InProceedings of the 16th International Conferenceon Extending Database Technology, pages323–334. ACM, 2013.

[26] B. Liu. Sentiment analysis and opinion mining.Synthesis Lectures on Human LanguageTechnologies, 5(1), 2012.

[27] S. A. Myers, C. Zhu, and J. Leskovec. Informationdiffusion and external influence in networks. InProc. of the 18th ACM SIGKDD, pages 33–41.ACM, 2012.

[28] M. Richardson and P. Domingos. Miningknowledge-sharing sites for viral marketing. InProceedings of the eighth ACM SIGKDD, pages61–70. ACM, 2002.

[29] T. C. Schelling. Models of segregation. AmericanEconomic Review, 59(2):488–493, 1969.

[30] J. Tang, J. Sun, C. Wang, and Z. Yang. Socialinfluence analysis in large-scale networks. InProceedings of the 15th ACM SIGKDD, pages807–816. ACM, 2009.

[31] J. Tang, S. Wu, and J. Sun. Confluence:Conformity influence in large social networks. InProceedings of the 19th ACM SIGKDDinternational conference on Knowledge discoveryand data mining, pages 347–355. ACM, 2013.

[32] T. Zhang. On the consistency of feature selectionusing greedy least squares regression. Journal ofMachine Learning Research, 10:555–568, 2009.

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