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Novel Multidimensional Models of OpinionDynamics in Social Networks

Sergey E. Parsegov, Anton V. Proskurnikov,Member, IEEE, Roberto Tempo,Fellow, IEEEand Noah E. Friedkin

Abstract—Unlike many complex networks studied in the lit-erature, social networks rarely exhibit unanimous behavior, orconsensus. This requires a development of mathematical modelsthat are sufficiently simple to be examined and capture, atthe same time, the complex behavior of real social groups,where opinions and actions related to them may form clustersof different size. One such model, proposed by Friedkin andJohnsen, extends the idea of conventional consensus algorithm(also referred to as the iterative opinion pooling) to take intoaccount the actors’ prejudices, caused by some exogenous factorsand leading to disagreement in the final opinions.

In this paper, we offer a novel multidimensional extension,describing the evolution of the agents’ opinions on severaltopics.Unlike the existing models, these topics are interdependent, andhence the opinions being formed on these topics are also mutuallydependent. We rigorous examine stability properties of thepro-posed model, in particular, convergence of the agents’ opinions.Although our model assumes synchronous communication amongthe agents, we show that the same final opinions may be reached“on average” via asynchronous gossip-based protocols.

I. I NTRODUCTION

A social network is an important and attractive case studyin the theory of complex networks and multi-agent systems.Unlike many natural and man-made complex networks, whosecooperative behavior is motivated by the attainment of someglobal coordination among the agents, e.g.consensus, opinionsof social actors usually disagree and may form irregularfactions (clusters). We use the term “opinion” to broadly referto individuals’ displayed cognitive orientations to objects (e.g.,topics or issues); the term includes displayed attitudes (signedorientations) and beliefs (subjective certainties). A challengingproblem is to develop a model of opinion dynamics, admittingmathematically rigorous analysis, and yet sufficiently instruc-tive to capture the main properties of real social networks.

S.E. Parsegov is with V.A. Trapeznikov Institute for Control Sciencesof Russian Academy of Sciences (ICS RAS), Moscow, Russia, e-mail:s.e.parsegov@gmail.com

A.V. Proskurnikov is with Engineering and Technology Institute (EN-TEG) at the University of Groningen, The Netherlands, and also withInstitute for Problems of Mechanical Engineering of Russian Academy ofSciences (IPME RAS) and ITMO University, St. Petersburg, Russia, e-mail:anton.p.1982@ieee.org

N.E. Friedkin is with the University of Santa-Barbara, CA, USA e-mail:friedkin@soc.ucsb.edu

R. Tempo is with CNR-IEIIT, Politecnico di Torino, Italy, e-mail:roberto.tempo@polito.it

Partial funding is provided by the European Research Council (grantERCStG-307207), CNR International Joint Lab COOPS, RFBR (grant 14-08-01015), Russian Federation President’s Grant MD-6325.2016.8 and St.Petersburg State University, grant 6.38.230.2015. Theorem 1 was obtainedunder sole support of Russian Science Fund (RSF) grant 14-29-00142.Theorem 2 was obtained under sole support of RSF grant 16-11-00063.Theorem 3 was obtained under sole support of RSF grant grant 16-19-00057.

The backbone of many mathematical models, explaining theclustering of continuous opinions, is the idea ofhomophilyorbiased assimilation[1]: a social actor readily adopts opinionsof like-minded individuals (under the assumption that its smalldifferences of opinion with others are not evaluated as im-portant), accepting the more deviant opinions with discretion.This principle is prominently manifested by variousboundedconfidencemodels, where the agents completely ignore theopinions outside their confidence intervals [2]–[5]. Thesemodels demonstrate clustering of opinions, however, theirrigorous mathematical analysis remains a non-trivial problem;it is very difficult, for instance, to predict the structure ofopinion clusters for a given initial condition. Another possibleexplanation of opinion disagreement isantagonismamongsome pairs of agents, naturally described bynegative ties[6].Special dynamics of this type, leading to opinion polarization,were addressed in [7]–[11]. It should be noticed, however,that experimental evidence securing the postulate of ubiquitousnegative interpersonal influences (also known asboomerangeffects) seems to be currently unavailable. Since the firstdefinition of boomerang effects [12], the empirical literaturehas concentrated on the special conditions under which theseeffects might arise. This literature provides no assertionthatboomerang effects, sometimes observed in dyad systems, arenon-ignorable in multi-agent networks of social influence.

It is known that even a network with positive and linear cou-plings may exhibit persistent disagreement and clustering, ifits nodes are heterogeneous, e.g. some agents are “informed”,that is, influenced by some external signals [13], [14]. Oneof the first models of opinion dynamics, employing sucha heterogeneity, was suggested in [15]–[17] and henceforthis referred to as the Friedkin-Johnsen (FJ) model. The FJmodel promotes and extends the DeGroot iterative poolingscheme [18], taking its origins in French’s “theory of socialpower” [19], [20]. Unlike the DeGroot scheme, where eachactor updates its opinion based on its own and neighbors’opinions, in the FJ model actors can also factor their initialopinions, or prejudices, into every iteration of opinion. Inother words, some of the agents arestubborn in the sensethat they never forget their prejudices, and thus remain persis-tently influenced by exogenous conditions under which thoseprejudices were formed [15], [16]. In the recent papers [21],[22] a sufficient condition for stability of the FJ model wasobtained. Furthermore, although the original FJ model is basedon synchronous communication, in [21], [22] its “lazy” versionwas proposed. This version is based on asynchronous gossipinfluence and provides the same steady opinionon average, nomatter if one considers the probabilistic average (that is,the

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expectation) or time-average (the solution Cesaro mean).TheFJ model and its gossip modification are intimately related tothe PageRank computation algorithms [22]–[28]. In specialcases, the FJ model has been given a game-theoretic [29]and an electric interpretation [30]. Similar dynamics arise inLeontief economic models [31] and some protocols for multi-agent coordination [32]. Further extensions of the FJ modelare discussed in the recent papers [33], [34].

Whereas many of the aforementioned models of opiniondynamics deal with scalar opinions, we deal with influencethat may modify opinions on several topics, which makes itnatural to considervector-valuedopinions [4], [35]–[37]; eachopinion vector in such a model is constituted byd > 1 topic-specific scalar opinions. A corresponding multidimensionalextension has been also suggested for the FJ model [17], [34].However, these extensions assumed that opinions’ dimensionsareindependent, that is, agents’ attitudes to each specific topicevolve as if the other dimensions did not exist. In contrast,ifeach opinion vector is constituted by an agent’s opinions onseveralinterdependentissues, then the dynamics of the topic-specific opinions are entangled. It has long been recognizedthat such interdependence may exist and is important. A setof interdependent positions on multiple issues is referredtoas schemain psychology,ideology in political science, andculture in sociology and social anthropology; scientists moreoften use the termsparadigmand doctrine. Converse in hisseminal paper [38] defined abelief systemas a “configurationof ideas and attitudes in which elements are bound togetherby some form of constraints of functional interdependence”.All these closely related concepts share the common idea ofan interdependent set of cognitive orientations to objects.

The main contribution of this paper is a novel multidi-mensional extension of the FJ model, which describes thedynamics of vector-valued opinions, representing individuals’positions on several interdependent issues. This extension,describing the evolution of abelief system, cannot be obtainedby a replication of the scalar FJ model on each issue. Forboth classical and extended FJ models we obtain necessaryand sufficient conditions of stability and convergence. We alsodevelop a randomized asynchronous protocol, which providesconvergence to the same steady opinion vector as the originaldeterministic dynamics on average. This paper significantlyextends results of the paper [39], which deals with a specialcase of the FJ model [15], [21], [22] satisfying the “couplingcondition”. This condition restricts the agent’s susceptibilityto neighbors’ opinions to coincide with its self-weight.

The paper is organized as follows. Section II introducessome concepts and notation to be used throughout the paper.In Section III we introduce the scalar FJ model and relatedconcepts; its stability and convergence properties are studiedin Section IV. A novel multidimensional model of opiniondynamics is presented in Section V. Section VI offers anasynchronous randomized model of opinion dynamics, thatis equivalent to the deterministic model on average. Weillustrate the results by numerical experiments in SectionVII.In Section VIII we discuss approaches to the estimation of themulti-issues dependencies from experimental data. Proofsarecollected in Section IX. Section X concludes the paper.

II. PRELIMINARIES AND NOTATION

Given two integersm and n ≥ m, let m : n denote theset {m,m + 1, . . . , n}. Given a finite setV , its cardinalityis denoted by|V |. We denote matrices with capital lettersA = (aij), using lower case letters for vectors and scalarentries. The symbol1n denotes the column vector of ones(1, 1, . . . , 1)⊤ ∈ Rn, andIn is the identity matrix of sizen.

Given a square matrixA = (aij)ni,j=1, let diagA =

diag(a11, a22, . . . , ann) ∈ Rn×n stand for its main diagonalandρ(A) be itsspectral radius. The matrixA is Schur stableif ρ(A) < 1. The matrix A is row-stochasticif aij ≥ 0and

∑n

j=1 aij = 1 ∀i. Given a pair of matricesA ∈ Rm×n,B ∈ Rp×q, their Kronecker product [40], [41] is defined by

A⊗B =

a11B a12B · · · a1nBa21B a22B · · · a2nB

.... . .

...am1B am2B · · · amnB

∈ Rmp×nq.

A (directed)graph is a pairG = (V , E), whereV stands forthe finite set ofnodesor verticesandE ⊆ V ×V is the set ofarcs or edges. A sequencei = i0 7→ i1 7→ . . . 7→ ir = i′ iscalled awalk from i to i′; the nodei′ is reachablefrom thenode i if at least one walk leads fromi to i′. The graph isstrongly connectedif each node is reachable from any othernode. Unless otherwise stated, we assume that nodes of eachgraph are indexed from1 to n = |V|, so thatV = 1 : n.

III. T HE FJ AND DEGROOT MODELS

Consider a community ofn socialactors(or agents) indexed1 throughn, and letx = (x1, . . . , xn)

⊤ stand for the columnvector of their scalaropinionsxi ∈ R. The Friedkin-Johnsen(FJ) model of opinions evolution [15]–[17] is determined bytwo matrices, that is a row-stochastic matrix ofinterpersonalinfluencesW ∈ Rn×n and a diagonal matrix of actors’susceptibilitiesto the social influence0 ≤ Λ ≤ In (we followthe notations from [21], [22]). At each stagek = 0, 1, . . . ofthe influence process the agents’ opinions evolve as follows

x(k + 1) = ΛWx(k) + (I − Λ)u, x(0) = u. (1)

The valuesui = xi(0) are referred to as the agentsprejudices.The model (1) naturally extends DeGroot’s iterative scheme

of opinion pooling[18] whereΛ = In. Similar to DeGroot’smodel, it assumes anaveraging(convex combination) mecha-nism of information integration. Each agenti allocates weightsto the displayed opinions of others under the constraint of anongoing allocation of weight to the agent’s initial opinion.The natural and intensively investigated special case of thismodel assumes the “coupling condition”λii = 1 − wii ∀i(that is,Λ = I − diagW ). Under this assumption, the self-weight wii plays a special role, considered to be a measureof stubbornessor closure of the ith agent to interpersonalinfluence. If wii = 1 and thuswij = 0 ∀j 6= i, then it ismaximally stubborn and completely ignores opinions of itsneighbors. Conversely, ifwii = 0 (and thus its susceptibilityis maximalλii = 1), then the agent is completely open tointerpersonal influence, attaches no weight to its own opinion(and thus forgets its initial conditions), relying fully onothers’

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opinions. The susceptibility of theith agentλii = 1 − wii

varies between0 and1, where the extremal values correspondrespectively to maximally stubborn and open-minded agents.From its inception, the usefulness of this special case has beenempirically assessed with different measures of opinion andalternative measurement models of the interpersonal influencematrix W [15]–[17], [42].

In this section, we consider dynamics of (1) in the generalcase, where the diagonal susceptibility matrix0 ≤ Λ ≤ Inmay differ fromI − diagW . In the case wherewii = 1 andhencewij = 0 as i 6= j, one hasxi(1) = xi(0) = ui andthen, via induction onk, one easily getsxi(k) = ui for anyk = 0, 1, . . ., no matter howλii is chosen. On the other hand,if λii = 0, thenxi(k) = ui independent of the weightswij .Henceforth we assume, without loss of generality, that for anyi ∈ 1 : n one either haveλii = 0 andwii = 1 (entailing thatxi(k) ≡ ui) or λii < 1 andwii < 1.

It is convenient to associate the matrixW to the graphG[W ] = (V , E [W ]). The set of nodesV = 1 : n of thisgraph is in one-to-one correspondence with the agents andthe arcs stand for the inter-personal influences (orties), that is(i, j) ∈ E [W ] if and only if wij > 0. A positive self-influenceweight wii > 0 corresponds to the self-loop(i, i). We callG = G[W ] the interaction graphof the social network.

Example 1: Consider a social network ofn = 4 actors, ad-dressed in [15] and having interpersonal influences as follows

W =

0.220 0.120 0.360 0.3000.147 0.215 0.344 0.2940 0 1 0

0.090 0.178 0.446 0.286

. (2)

Fig. 1 illustrates the corresponding interaction graph.

3

1

2

4

Fig. 1: Interaction graphG[W ], corresponding to matrix (2)

In this section we are primarily interested inconvergenceof the FJ model to a stationary point (if such a point exists).

Definition 1: (Convergence).The FJ model (1) isconver-gent, if for any vectoru ∈ Rn the sequencex(k) has a limit

x′ = limk→∞

x(k) =⇒ x′ = ΛWx′ + (I − Λ)u. (3)

It should be noticed that the limit valuex′ = x′(u) ingeneraldependson the initial conditionx(0) = u. A specialsituation where any solution converges tothe sameequilibriumis the exponential stabilityof the linear system (1), whichmeans thatΛW is a Schur stable matrix:ρ(ΛW ) < 1. A

stable FJ model is convergent, and the only stationary pointis

x′ =

∞∑

k=0

(ΛW )k(I − Λ)u = (I − ΛW )−1(I − Λ)u. (4)

As will be shown, the class of convergent FJ models is in factwider than that of stable ones. This is not surprising since,forinstance, the classical DeGroot model [18] whereΛ = In isnever stable, yet converges to a consensus value (x′

1 = . . . =x′n) wheneverW is stochastic indecomposable aperiodic (SIA)

[43], for instance,Wm is positive for somem > 0 (i.e. Wis primitive) [18]. In fact, any unstable FJ model contains asubgroup of agents whose opinions obey the DeGroot model,being independent on the remaining network. To formulate thecorresponding results, we introduce the following definition.

Definition 2: (Stubborness and oblivion).We call theithagentstubborn if λii < 1 and totally stubbornif λii = 0.An agent that is neither stubborn nor influenced by a stubbornagent (connected to some stubborn agent by a walk in theinteraction graphG[W ]) is calledoblivious.

Example 2: Consider the FJ model (1), whereW is from(2) andΛ = I − diagW . It should be noticed that this modelwas reconstructed from real data, obtained in experiments witha small group of individuals, following the method proposedin [15]. Figure 2 illustrates the graph of the coupling matrixΛW and the constant “input” (prejudice)u. In this model theagent3 (drawn in red) is totally stubborn, and the three agents1, 2 and4 are stubborn. Hence, there are no oblivious agents inthis model. As will be shown in the next section (Theorem 1),the absence of oblivious agents implies stability.

3

1

2

4

u1

u2

u3

u4

Fig. 2: The structure of couplings among the agents and “in-puts” for the FJ model withW from (2) andΛ = I −diagW

The prejudicesui are considered to be formed by someexogenous conditions [15], and the agent’s stubborness canbe considered as their ongoing influence. A totally stubbornagent remains affected by those external “cues” and ignoresthe others’ opinions, so its opinion is unchangedxi(k) ≡ ui.Stubborn agents, being not completely “open-minded”, neverforget their prejudices and factor them into every iteration ofopinion. Non-stubborn agents are not “anchored” to their ownprejudices yet be influenced by the others’ prejudices via com-munication (such an influence corresponds to a walk inG[W ]from the agent to some stubborn individual), such individualscan be considered as “implicitly stubborn”. Unlike them, foran oblivious agent the prejudice does not affect any stage ofthe opinion iteration, except for the first one. The dynamicsof

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oblivious agents thus depend on the “prehistory” of the socialnetwork only through the initial conditionx(0) = u.

After renumbering the agents, we assume that stubbornagents and agents, influenced by them, are numbered1 throughn′ ≤ n and oblivious agents (if they exist) have indices fromn′ + 1 to n. For the oblivious agenti we haveλii = 1 andwij = 0 ∀j ≤ n′. Indeed, ifwij > 0 for somej ≤ n′, thenthe ith agent is connected by a walk to some stubborn agentvia agentj and hence is not oblivious. The matricesW,Λ andvectorsx(k) are therefore decomposed as follows

W =

[W 11 W 12

0 W 22

]

,Λ =

[Λ11 00 I

]

, x(k) =

[x1(k)x2(k)

]

, (5)

wherex1 ∈ Rn′

andW 11 andΛ11 have dimensionsn′ × n′.If n′ = n then x2(k), W 12 andW 22 are absent, otherwisethe oblivious agents obey the conventional DeGroot dynamicsx2(k + 1) = W 22x2(k), being independent on the remainingagents. If the FJ model is convergent, then the limitW 22

∗ =limk→∞

(W 22)k obviously exists, in other words, the matrixW 22

is regular in the sense of [44, Ch.XIII,§7].Definition 3: (Regularity) A matrix A ∈ Rd×d is called

regular [44] if a limit A∗ = limk→∞

Ak exists. A regularrow-

stochastic matrixA is calledfully regular [44] or SIA [43] ifall rows ofA∗ are identical, i.e.A∗ = 1dv

⊤, wherev ∈ Rd.In the literature, the regularity is usually defined for non-

negative matrices [44], but in this paper we use this term forageneral matrix. In the Appendix we examine some propertiesof stochasticregular matrices, which play an important rolein the convergence properties of the FJ model.

IV. STABILITY AND CONVERGENCE OF THEFJ MODEL

The main contribution of this section is the followingcriterion for the convergence of the FJ model, which employsthe decomposition (5).

Theorem 1: (Stability and convergence) The matrixΛ11W 11 is Schur stable. The system (1) is stable if and onlyif there are no oblivious agents, that is,ΛW = Λ11W 11. TheFJ model with oblivious agents is convergent if and only ifW 22 is regular, i.e. the limitW 22

∗ = limk→∞

(W 22)k exists. In

this case, the limiting opinionx′ = limk→∞

x(k) is given by

x′ =

[(I − Λ11W 11)−1 0

0 I

] [I − Λ11 Λ11W 12W 22

∗

0 W 22∗

]

u.

(6)An important consequence of Theorem 1 is the stability of

the FJ model, whose interaction graph is strongly connected(or, equivalently, the matrixW is irreducible [44]).

Corollary 1: If the interaction graphG[W ] is stronglyconnected andΛ 6= I (i.e. at least one stubborn agent exists),then the FJ model (1) is stable.

Proof: The strong connectivity implies that each agentis either stubborn or connected by a walk to any of stubbornagents; hence, there are no oblivious agents.

Theorem 1 also implies that the FJ model is featured by thefollowing property. For a general system with constant input

x(k + 1) = Ax(k) +Bu, (7)

the regularity of the matrixA is a necessary and sufficientcondition for convergence ifBu = 0, sincex(k) = Akx(0) →A∗x(0). For Bu 6= 0, regularity is not sufficient for theexistence of a limit lim

k→∞x(k): a trivial counterexample is

A = I. Iterating the equation (7) with regularA, one obtains

x(k) = Akx(0)+

k∑

j=1

AjBu −−−−→k→∞

A∗x(0)+

∞∑

k=0

AkBu, (8)

where the convergence takes place if and only if the series inthe right-hand side converge. The convergence criterion fromTheorem 1 implies that for the FJ model (1) withA = ΛWandB = I −Λ the regularity ofA is necessary and sufficientfor convergence [34]; for any convergent FJ model (8) holds.

Corollary 2: The FJ model (1) is convergent if and only ifA = ΛW is regular. If this holds, the limit of powersA∗ is

A∗ = limk→∞

(ΛW )k =

[0 (I − Λ11W 11)−1Λ11W 12W 22

∗

0 W 22∗

]

,

(9)and the series from (8) (withB = I − Λ) converge to

∞∑

k=0

(ΛW )k(I − Λ)u =

[(I − Λ11W 11)−1(I − Λ11)u1

0

]

.

(10)Due to (8), the final opinionx′ from (6) decomposes into

x′ = A∗u+

∞∑

k=0

(ΛW )k(I − Λ)u. (11)

Proof: Theorem 1 implies that the matrixA = ΛW isdecomposed as follows

A =

[Λ11W 11 Λ11W 12

0 W 22

]

,

where the submatrixΛ11W 11 is Schur stable. It is obvious thatA is not regular unlessW 22 is regular, sinceAk contains theright-bottom block(W 22)k. A straightforward computationshows that ifW 22 is regular, then (9) and (10) hold, inparticular,A is regular as well.

Note that the first equality in (4) in generalfails for unstableyet convergent FJ model, even though the series (10) convergesto a stationary point of the system (1) (the second equality in(4) makes no sense asI − ΛW is not invertible). Unlike thestable case, in the presence of oblivious agents the FJ modelhas multiple stationary points for the same vector of prejudicesu; the opinionsx(k) and the series (10) converge todistinctstationary points unlessW 22

∗ u2 = 0.As shown in the Appendix, for a regular row-stochastic

matrix A the limit A∗ equals to

A∗ = limk→∞

Ak = limα→1

(I − αA)−1(1− α). (12)

Theorem 1, combined with (12), entails the following im-portant approximation result. Along with the FJ model (1),consider the following “stubborn” approximation

xα(k + 1) = αΛWxα(k) + (I − αΛ)u, xα(0) = u, (13)

where α ∈ (0; 1). HenceαΛ < I, which implies that allagents in the model (13) are stubborn, the model (13) is stable,

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converging to the stationary opinionxα(k) −−−−→k→∞

x′α = (I −

αΛW )−1(I − αΛ)u. It is obvious thatxα(k) −−−→α→1

x(k) for

anyk = 1, 2, . . ., a question arises if such a convergence takesplace fork = ∞, that is,x′

α → x′ asα → 1. A straightforwardcomputation, using (12) forA = W 22 and (6), shows that thisis the case whenever the original model (1) is convergent.Moreover, the convergence is uniform inu, provided thatuvaries in some compact set. In this sense anyconvergentFJmodel can be approximated with the models, where all ofthe agents are stubborn (Λ < In). The proof of (12) in theAppendix allows to get explicit estimates for‖x′

α − x′‖ that,however, do not appear useful for the subsequent analysis.

V. A MULTIDIMENSIONAL EXTENSION OF THE FJ MODEL

In this section, we propose an extension of the FJ model,dealing with vector opinionsx1(k), . . . , xn(k) ∈ Rm. Theelements of each vectorxi(k) = (x1

i (k), . . . , xmi (k)) stand

for the opinions of theith agent onm different issues.

A. Opinions on independent issues

In the simplest situation where agents communicate onmcompletely unrelated issues, it is natural to assume that theparticular issuesxj

1(k), xj2(k), . . . , x

jn(k) satisfy the FJ model

(1) for anyj = 1, . . . ,m, and therefore

xi(k+1) = λii

n∑

j=1

wijxj(k)+(1−λii)ui, ui := xi(0). (14)

Example 3: Consider the FJ model (14) withW from (2)andΛ = I − diagW . Unlike Example 2, now the opinionsxj(k) are two-dimensional, that is, m = 2 and xj(k) =(x1

j (k), x2j (k))

⊤ represent the opinions on two independenttopics (a) and (b). The structure of the system, consistingof two copies of the usual FJ model (1), is illustrated byFigure 3. Since the topic-specific opinionsx1

j (k), x2j (k) evolve

independently, their limits can be calculated independently,applying (4) toui = (xi

1(0), xi2(0), x

i3(0), x

i4(0))

⊤, i = 1, 2.For instance, choosing the initial condition

x(0) = u = [ 25, 25,︸ ︷︷ ︸

u1=x1(0)

25, 15,︸ ︷︷ ︸

u2=x2(0)

75,−50,︸ ︷︷ ︸

u3=x3(0)

85, 5︸︷︷︸

u4=x4(0)

]⊤, (15)

the vector of steady agents’ opinion is

x′ = [60,−19.3, 60,−21.5, 75,−50, 75,−23.2]⊤. (16)

B. Interdependent issues: a belief system’s dynamics

Dealing with opinions oninterdependenttopics, the opin-ions being formed on one topic are influenced by the opinionsheld on some of the other topics, so that the topic-specificopinions are entangled. Consider a group of people discussingtwo related topics, e.g. fish (as a part of diet) in general andsalmon. Salmon is nested in fish. A person disliking fish alsodislikes salmon. If the influence process changes individuals’attitudes toward fish, say promoting fish as a healthy part ofa diet, then the door is opened for influences on salmon as apart of this diet. If, on the other hand, the influence process

3a

1a

2a

4a

u1a

u2a

u3a

u4a

3b

1b

2b

4b

u1b

u2b

u3b

u4b

Fig. 3: The structure of the two-dimensional model (14)

changes individuals’ attitudes against fish, say warning thatfish are now contaminated by toxic chemicals, then the dooris closed for influences on salmon as part of this diet.

Adjusting his/her position on one of the interdependentissues, an individual might have to adjust the positions onseveral related issues simultaneously in order to maintainthebelief system’s consistency. Contradictions and other inconsis-tencies between beliefs, attitudes and ideas trigger tensions anddiscomfort (“cognitive dissonance”) that can be resolved bya within-individual (introspective) process. This introspectiveprocess, studied in cognitive dissonance and cognitive con-sistency theory, is thought to be an automatic process of thehuman brain, enabling an individual to develop a “coherent”system of attitudes and beliefs [45], [46].

To the best of the authors’ knowledge, no model describinghow networks of interpersonal influences may generate beliefsystems is available in the literature. In this section, we makethe first step towards filling this gap and propose a model,based on the classical FJ model, that takes issues interde-pendencies into account. We modify the multidimensional FJmodel (14) (withxj(k) ∈ Rm) as follows

xi(k + 1) = λiiC

n∑

j=1

wijxj(k) + (1− λii)ui. (17)

The model (17) inherits the structure of the usual FJ dynamics,including the matrix of social influencesW and the matrix ofagents’ susceptibilitiesΛ. On each stage of opinion iterationthe agenti calculates an “average” opinion, being the weightedsum

∑

j wijxj(k) of its own and its neighbors’ opinions;along with the agent’s prejudiceui it determines the updatedopinion xi(k + 1). The crucial difference with the FJ modelis the presence of additional introspective transformation, ad-justing and mixing the averaged topic-specific opinions. Thistransformation is described by a constant “coupling matrix”C ∈ Rm×m, henceforth called the matrix ofmulti-issuesdependence structure(MiDS). In the caseC = Im the model(17) reduces to the usual FJ model (14).

To clarify the role of the MiDS matrixC, consider forthe moment a network with star-shape topology where allthe agents follow a totally stubborn leader, i.e. there existsj ∈ {1, 2, . . . , n} such thatλjj = 0 andwij = 1 = λii forany i 6= j, so thatxi(k + 1) = Cuj . The opinion changes inthis system are movements of the opinions of the followers

6

toward the initial opinions of the leader, and these movementsare strictly based on the direct influences of the leader. Theentries of the MiDS matrix govern the relative contributionsof the leader’s issue-specific opinions to the formation of thefollowers’ opinions. Sincexp

i (k+1) =∑m

q=1 cpquqj , thencpq

is a contribution of theqth issue of the leader’s opinion to thepth issue of the follower’s one. In general, instead of a simpleleader-follower network we have a group of agents, commu-nicating onm different issues in accordance with the matrixof interpersonal influencesW . During such communications,the ith agent calculates the average

∑

j wijxj(k) of its ownopinion and those displayed by the neighbors. The weightcpqmeasures the effect of theqth issue of this averaged opinionto thepth issue of the updated opinionxi(k + 1).

Notice that the origins and roles of matricesW andC inthe multidimensional model (17) are very different. The matrixW is a property of the social network, describing its topologyand social influence structure, which is henceforth assumedto be known (the measurement models for the structuralmatricesΛ,W are discussed in [15]–[17]). At the same time,C expresses the interrelations between different topics ofinterest. It seems reasonable that the MiDS matrix shouldbe independent of the social network itself, depending onintrospective processes, forming an individual’s belief system.

We proceed with examples, which show that introducingthe MiDS matrix C can substantially change the opiniondynamics. These examples deal with the social network ofn = 4 actors from [15], having the influence matrix (2) andthe susceptibility matrixΛ = I − diagW . Unlike Example 3,the agents discussinterdependenttopics.

Example 4: Let the agents discuss two topics, (a) and (b),say the attitudes towards fish (as a part of diet) in general andsalmon. We start from the initial condition (15), which meansthat agents1 and 2 have modest positive liking for fish andsalmon; the third (totally stubborn) agent has a strong likingfor fish, but dislikes salmon; the agent4 has a strong likingfor fish and a weak positive liking for salmon. Neglectingthe issues interdependence (C = I2), the final opinion wascalculated in Example 3 and is given in (16).

We now introduce a MiDS matrix, taking into account thedependencies between the topics

C1 =

[0.8 0.20.3 0.7

]

. (18)

As will be shown in Theorem 2, the opinions converge to

x′C1

= [39.2, 12, 39, 10.1, 75,−50, 56, 5.3]⊤. (19)

Hence, introducing the MiDS matrixC1 from (18), withits dominant main diagonal, imposes a substantial drag inopinions of the “open-minded” agents1 and2. In both casestheir attitudes toward fish become more positive and thosetoward salmon become less positive, compared to the initialvalues (15). However, in the case of dependent issues theirattitudes toward salmon do not become negative as they didin the case of independence. As for the agent4, its attitudetowards salmon under the MiDS matrix (18) becomes evenmore positive, compared to the initial value (15), whereas forC = I2 this attitude becomes strongly negative.

The difference in behavior of the systems (14) and (17) iscaused by the presence of additional ties (couplings) betweenthe topic-specific opinions, imposed by the MiDS matrixC,drawn in Fig. 4 in green. For simplicity, we show only three ofthese extra ties; there are also ties between the topic-specificopinions 1b and 2a, 3a, 4a; 2a and 1b, 3b, 4b etc.

3a

1a

2a

4a

u1a

u2a

u3a

u4a

3b

1b

2b

4b

u1b

u2b

u3b

u4b

Fig. 4: The structure of the two-dimensional FJ model (17)with C from (18): emergent extra couplings between topic-specific opinions

In Example 4 the additional ties are positive, bringing thetopic-specific opinions closer to each other. The requirementof consistency of a belief systems may also imply thenegativecouplings between different topics if the multidimensionalopinion contains attitudes to a pair of contrary issues.

Example 5: Consider Example 4, replacing the stochasticMiDS matrix (18) with the non-positive matrix

C2 =

[0.8 −0.2−0.3 0.7

]

. (20)

The topics (a) and (b), discussed by the agents, are interrelatedbut opposite, e.g. the agents discuss their attitudes to vege-tarian and all-meat diets. As will be shown, the system (17)remains stable. Starting from the same initial opinion (15), theagents’ opinions converge to the final value

x′C2

= [52.3,−30.9, 52.1,−33.3, 75,−50, 68.4,−33.2]⊤.

Similar to the case of uncoupled topics (Example 3), the agentsconverge on positive attitude to vegetarian diets and negativeattitude to all-meat diet, following thus the prejudice of thetotally stubborn individual3. However, the final opinions infact substantially differ. In the case of decoupled topic-specificopinions (C = I2) the agents get stronger positive attitudes tovegetarian diets than in the case of negative coupling (C = C2)while their negative attitudes against all-meat diets are weaker.

Remark 1: (Restrictions on the MiDS matrix) In manyapplications the topic-specific opinions may vary only in somepredefined interval. For instance, treating them ascertaintiesof belief [47] or subjective probabilities [18], [37], the modelof opinion evolution should imply that the topic-specificopinions belong to[0, 1]. Similarly, the agentsattitudes, i.e.signed orientations towards some issues [34], are often scaledto the interval [−1, 1]. Such a limitation on the opinionscan make it natural to choose the MiDS matrixC from a

7

special class. For instance, choosingC row-stochastic, themodel (17) inherits important property of the FJ model [34]:if uj

i = xji (0) ∈ [a, b] ∀i, j then xj

i (k) ∈ [a, b] ∀i, j for anyk ≥ 0 (here [a, b] ⊂ R is a given interval,i ∈ 1 : n andj ∈ 1 : m). This is easily derived from (17) via inductionon k: if xi(k) ∈ [a, b]m for k = 0, . . . , s and anyi, thenCxi(k) ∈ [a, b]m and hencexi(s+ 1) ∈ [a, b]m.

In the case of speciala and b the assumption of row-stochasticity can be further relaxed. For instance, to keepthe vector opinionsxi(k) in [0, 1]m wheneverui ∈ [0, 1]m,the matrixC should be chosensubstochastic: cij ≥ 0 and∑m

j=1 cij ≤ 1 for any m. To provide the invariance of thehypercube[−1, 1]m, the matrixC should satisfy the condition

‖C‖∞ = maxi

m∑

j=1

|cij | ≤ 1. (21)

More generally, via induction onk the following invarianceproperty can be proved: ifD is a convex set andCx ∈ D forany x ∈ D, thenD is an invariant set for the dynamics (17):if ui = xi(0) ∈ D thenxi(k) ∈ D for any k.

In the next subsection, the problems of stability and con-vergence of the model (17) are addressed.

C. Convergence of the multidimensional FJ model

Similar to (15), the stack vectors of opinionsx(k) =(x1(k)

⊤, . . . , xn(k)⊤)⊤ and prejudicesu = (u⊤

1 , . . . , u⊤n )

⊤ =x(0) can be constructed. The dynamics (17) now becomes

x(k + 1) = [(ΛW )⊗ C]x(k) + [(In − Λ)⊗ Im], (22)

which is a convenient representation of (17) in the matrix form.We begin with stability analysis of the model (22). In the

case whenC is row-stochastic the stability conditions remainthe same as for the initial model (1). However, the model (22)remains stable for many non-stochastic matrices, includingthose with exponentially unstable eigenvalues.

Theorem 2: (Stability) The model (22) is stable (i.e.ΛW ⊗ C is Schur stable) if and only ifρ(ΛW )ρ(C) < 1.If this holds, then the vector of ultimate opinions is

x′C := lim

k→∞x(k) = (Imn−ΛW⊗C)−1[(In−Λ)⊗Im]u. (23)

If ρ(C) = 1, the stability of (22) is equivalent to the stabilityof the FJ model (1), i.e. to the absence of oblivious agents.

Remark 2: Theorem 2, in particular, guarantees stabilitywhen the original FJ model (1) is stable (ρ(ΛW ) < 1)and C is row-stochastic or, more generally, satisfies (21).These conditions, however, are not necessary for stability; thesystem (22) remains stable wheneverρ(C) < ρ(ΛW )−1.

In the case where some agents are oblivious, the conver-gence of the model (22) is not possible unlessC is regular,that is, the limitC∗ = lim

k→∞Ck exists (in particular,ρ(C) ≤

1). As in Theorem 1, we assume that oblivious agents areindexedn′+1 throughn and consider the decomposition (5).

Theorem 3: (Convergence)Let n′ < n. The model (22)is convergent if and only ifC is regular and eitherC∗ = 0 or

W 22 is regular. If this holds thenx(k) −−−−→k→∞

x′C , where

x′C =

[(I − Λ11W 11 ⊗ C)−1 0

0 I

]

Pu,

P =

[(I − Λ11)⊗ Im (Λ11W 12W 22

∗ )⊗ CC∗

0 W 22∗ ⊗ C∗

]

.

(24)

By definition, in the case whereC∗ = 0 but the limitlimk→∞(W 22)k does not exist we putW∗ = 0.

Remark 3: (Extensions) In the model (22) we do notassume the interdependencies between the initial topic-specificopinions; one may also consider a more general case whenxi(0) = Dui and hencex(0) = [In ⊗ D]u, where D isa constantm × m matrix. This affects neither stability norconvergence conditions, and formulas (23), (24) forx′

C remainvalid, replacingP in the latter equation with

P =

[(I − Λ11)⊗ Im (Λ11W 12W 22

∗ )⊗ CC∗D0 W 22

∗ ⊗ C∗D

]

.

VI. OPINION DYNAMICS UNDER GOSSIP-BASED

COMMUNICATION

A considerable restriction of the model (22), inheritedfrom the original Friedkin-Johnsen model, is thesimultaneouscommunication between the agents. That is, on each step theactors simultaneously communicate to all of their neighbors.This type of communication can hardly be implemented in alarge-scale social network, since, as was mentioned in [15],“it is obvious that interpersonal influences do not occur inthe simultaneous way and there are complex sequences ofinterpersonal influences in a group”. A more realistic opiniondynamics can be based on asynchronousgossip-based[48],[49] communication, assuming that only two agents interactduring each step. An asynchronous version of the FJ model(1) was proposed in [21], [22].

The idea of the model from [21], [22] is as follows. On eachstep an arc is randomly sampled with the uniform distributionfrom the interaction graphG[W ] = (V , E). If this arc is(i, j),then theith agent updates its opinion in accordance with

xi(k+1) = hi ((1 − γij)xi(k) + γijxj(k))+(1−hi)ui. (25)

Hence, the new opinion of the agent is a weighted average ofhis/her previous opinion, the prejudice and the neighbor’spre-vious opinion. The opinions of other agents remain unchanged

xl(k + 1) = xl(k) ∀l 6= i. (26)

The coefficienthi ∈ [0, 1] is a measure of the agent“obstinacy”. If an arc(i, i) is sampled, then

xi(k + 1) = hixi(k) + (1− hi)ui. (27)

The smaller ishi, the more stubborn is the agent, forhi = 0it becomes totally stubborn. Conversely, forhi = 1 the agentis “open-minded” and forgets its prejudice. The coefficientγij ∈ [0, 1] expresses how strong is the influence of thejthagent on theith one. Since the arc(i, j) exists if and only ifwij > 0, one may assume thatγij = 0 wheneverwij = 0.

It was shown in [21], [22] that, forstableFJ model withΛ = I − diagW , under proper choice of the coefficientshi

8

andγij , the expectationEx(k) converges to the same steadyvalue x′ as the Friedkin-Johnsen model and, moreover, theprocess isergodicin both mean-square and almost sure sense.In other words, both probabilistic averages (expectations) andtime averages (referred to as theCesaro or Polyak averages)of the random opinions converge to the final opinion in theFJ model. It should be noticed that opinions themselves arenot convergent(see numerical simulations below) but oscillatearound their expected values. In this section, we extend thegossip algorithm from [21], [22] to the case whereΛ 6= I −diagW and the opinions are multidimensional.

Let G[W ] = (V , E) be the interaction graph of the net-work. Given two matricesΓ1,Γ2 such thatγ1

ij , γ2ij ≥ 0 and

γ1ij + γ2

ij ≤ 1, we consider the following multidimensionalextension of the algorithm (25), (26). On each step an arc isuniformly sampled in the setE . If this arc is(i, j), then agenti communicates to agentj and updates its opinion as follows

xi(k+ 1) = (1− γ1ij − γ2

ij)xi(k) + γ1ijCxj(k) + γ2

ijui. (28)

Hence during each interaction the agent’s opinion is aver-aged with its ownprejudiceand modified neighbors’ opinionCxj(k). The other opinions remain unchanged as in (26).

The following theorem shows that under the assumption ofthe stability of the original FJ model (22) and proper choiceofΓ1,Γ2 the model (28), (26) inherits the asymptotic propertiesof the deterministic model (22).

Theorem 4: (Ergodicity) Assume thatρ(ΛW ) < 1, i.e.there are no oblivious agents, andC is row-stochastic. LetΓ1 = ΛW and Γ2 = (I − Λ)W . Then, the limit x∗ =limk→∞

Ex(k) exists and equals to the final opinion (23) of the FJ

model (22), i.e.x∗ = x′C . The random processx(k) is almost

sure ergodic, which means thatx(k) → x∗ with probability1, andLp-ergodicso thatE‖x(k)− x∗‖

p −−−−→k→∞

0, where

x(k) :=1

k + 1

k∑

l=0

x(l). (29)

Both equalityx∗ = x′C and ergodicity remain valid, replacing

Γ2 = (I−Λ)W with any matrix, such that0 ≤ γ2ij ≤ 1−γ1

ij,∑n

j=1 γ2ij = 1− λii andγ2

ij = 0 as (i, j) 6∈ E .As a corollary, we obtain the result from [21], [22], stating

the equivalence on average between the asynchronous opiniondynamics (25), (26) and the scalar FJ model (1).

Corollary 3: Let di be theout-branchdegree of theithnode, i.e. the cardinality of the set{j : (i, j) ∈ E}. Considerthe algorithm (25), (26), wherexi ∈ R, (1−hi)di = 1−λii ∀i,γij ∈ [0, 1] and hiγij = λiiwij wheneveri 6= j. Then, thelimit x∗ = lim

k→∞Ex(k) exists and equals to the steady-state

opinion (4) of the FJ model (1):x∗ = x′. The random processx(k) is almost sure and mean-square ergodic.

Proof: The algorithm (25), (26) can be considered as aspecial case of (28), (26), whereC = 1, γ1

ij = hiγij andγ2ij =

1 − hi. Since the valuesγ1ii have no effect on the dynamics

(28) with C = 1, one can, changingγ1ii if necessary, assume

thatΓ1 = ΛW . The claim now follows from Theorem 4 since1− γ2

ij = hi ≥ γ1ij and

∑

j γ2ij = (1 − hi)di = 1− λii.

Hence, the gossip algorithm, proposed in [21], [22] is onlyone element of a broad class of protocols (28) (withC = 1),satisfying assumptions of Theorem 4.

Remark 4: (Random opinions) Whereas the Cesaro-Polyak averagesx(k) do converge to their average valuex∗,the random opinionsx(k) themselvesdo not, exhibiting non-decaying oscillations aroundx∗, see [21] and the numericalsimulations in Section VII. As implied by [22, Theorem 1],x(k) converges in probability to a random vectorx∞, whosedistribution is the unique invariant distribution of the dynamics(28), (26) and is determined by the triple(Λ,W,C).

VII. N UMERICAL EXPERIMENTS

In this section, we give numerical tests, which illustrate theconvergence of the “synchronous” multidimensional FJ modeland its “lazy” gossip version.

We start with the opinion dynamics of a social networkwith n = 4 actors, the matrix of interpersonal influencesW from (2) and susceptibility matrixΛ = I − diagW . InFig. 5 we illustrate the dynamics of opinions in Examples 3-5: Fig. 5a shows the case of independent issuesC = I2,Fig. 5b illustrates the model (17) with the stochastic matrix Cfrom (18), and Fig. 5c demonstrates the dynamics under theMiDS matrix (20). As was discussed in Examples 4 and 5, theinterdependencies between the topics lead to substantial dragsin the opinions of the agents 1,2 and 4, compared with thecase of independent topics.

It is useful to compare the final opinion of the models justconsidered with the DeGroot-like dynamics1 (Fig. 6) where theinitial opinions and matricesC are the same, however,Λ = In.In the case of independent issuesC = I2 all the opinions areattracted by the stubborn agent’s opinion (Fig. 6a)

limk→∞

x(k) = [75,−50, 75,−50, 75,−50, 75,−50]⊤.

In the case of positive ties between topics (Fig. 6b) we have

limk→∞

x(k) = [25, 25, 25, 25, 25, 25, 25, 25]⊤.

In fact, the stubborn agent3 constantly averages the issues ofits opinions so that they reach agreement, all other issues arealso attracted to this consensus value. In the case of negativelycoupled topics (Fig. 6c) the topic-specific opinionspolarize

limk→∞

x(k) = [65,−65, 65,−65, 65,−65, 65,−65]⊤.

The next simulation (Fig. 7) deals with the randomizedgossip-based counterparts of the models from Examples (3)and (4). The Cesaro-Polyak averages (Figs. 7a and 7b)x(k) ofthe opinions under the gossip-based protocol from Theorem 4converge to the same limits as the deterministic model (22)(blue circles). However, the random opinionsx(k) themselvesdo not converge and exhibit oscillations (Fig. 7c).

1In the DeGroot model [18] the components of the opinion vectors xi(k)are independent. This corresponds to the case whereC = Im. One canconsider a generalized DeGroot’s model as well, which is a special case of (22)with Λ = In but C 6= Im. This implies the issues interdependency, whichcan make all topic-specific opinions (that is, opinions on different issues)converge to the same consensus value or polarize, as shown inFig. 6.

9

step, k0 1 2 3 4 5 6 7

x i(k)

-40

-20

0

20

40

60

80

(a) Example 3: topics are independent

step, k0 1 2 3 4 5 6 7

x i(k)

-40

-20

0

20

40

60

80

(b) Example 4: topics are positively coupled

step, k0 1 2 3 4 5 6 7

x i(k)

-40

-20

0

20

40

60

80

(c) Example 5: topics are negatively coupled

Fig. 5: Dynamics of opinions in Examples 3-5.

The last example deals with the group ofn = 51 agents,consisting of a totally stubborn “leader” andN = 10 groups,each containing5 agents (Fig. 8). In each subgroup a “localleader” or “representative” exists, who is the only subgroupmember influenced from outside. The leader of the firstsubgroup is influenced by the totally stubborn agent, and theleader of theith subgroup (i = 2, . . . , N ) is influenced bythat of the (i − 1)th subgroup. All other members in eachsubgroup are influenced by the local leader and by each other,as shown in Fig. 8. Notice that each agent has a non-zero self-weight, but we intentionally do not draw self-loops around thenodes in order to make the network structure more clear. Wesimulated the dynamics of the network, assuming that the first

step, k0 1 2 3 4 5 6 7

x i(k)

-40

-20

0

20

40

60

80

(a) DeGroot’s model: independent issues

step, k0 1 2 3 4 5 6 7

x i(k)

-40

-20

0

20

40

60

80

(b) Extended DeGroot’s model: MiDS matrix (18)

step, k0 1 2 3 4 5 6 7

x i(k)

-60

-40

-20

0

20

40

60

80

(c) Extended DeGroot’s model: MiDS matrix (20)

Fig. 6: Opinion dynamics in the extended DeGroot model

local leader has the self-weight0.1 (and assigns the weight0.9 to the opinion of the totally stubborn agent), and the otherlocal leaders have self-weights0.5 (assigning the weight0.5 tothe leaders of predecessing subgroups). All the weights insidethe subgroups are chosen randomly in a way thatW is row-stochastic (we do not provide this matrix here due to spacelimitations). We assume thatΛ = I − diagW and choose theMiDS matrix as follows

C =

[0.9 0.10.1 0.9

]

.

The initial conditions for the totally stubborn agent arex1(0) = [100,−100]⊤, the other initial conditions are ran-domly distributed in[−10, 10]. The dynamics of opinions in

10

0 200 400 600 800 1000

−40

−20

0

20

40

60

80

step, k

aver

aged

xi(k

)

(a) Convergence of the Cesaro-Polyak averages, theMiDS matrix C = I2

0 200 400 600 800 1000

−40

−20

0

20

40

60

80

step, k

aver

aged

xi(k

)

(b) Convergence of the Cesaro-Polyak averages, theMiDS matrix (18)

0 200 400 600 800 1000

−40

−20

0

20

40

60

80

step, k

x i(k)

(c) The dynamics of opinions with the MiDS ma-trix (18) show no convergence

Fig. 7: Gossip-based counterparts of Examples 3 and 4

the deterministic model and averaged opinions in the gossipmodel are shown respectively in Figs. 9a and 9b. One can seethat several clusters of opinions emerge, and the gossip-basedprotocol is equivalent to the deterministic model on average,in spite of rather slow convergence.

VIII. E STIMATION OF THE M IDS MATRIX

Identification of structures and dynamics in social networks,using experimental data, is an emerging area, which is activelystudied by different research communities, working on statis-tics [50], physics [51] and signal processing [52]. We assume

Fig. 8: Hierarchical structure withn = 51 agents

that the structure of social influence, that is, the matricesW and Λ are known. A procedure for their experimentalidentification was discussed in [15], [17]. In this section,wefocus on the estimation of the MiDS matrixC. As discussedin [52], identification of social network is an important stepfrom network analysis to network control and “sensing” (thatis, observing or filtering).

To estimateC, an experiment can be performed where agroup of individuals with given matricesΛ andW commu-nicates onm interdependent issues. The agents are asked torecord their initial opinions, constituting the vectoru = x(0),after which they start to communicate. The agents interact inpairs and can be separated from each other; the matrixWdetermines the interaction topology of the network, that is,which pairs of agent are able to interact. Two natural types ofmethods, allowing to estimateC, can be referred to as “finite-horizon” and “infinite horizon” identification procedures.

A. Finite-horizon identification procedure

In the experiment of the first kind the agents are asked toaccomplishT ≥ 1 full rounds of conversations and recordtheir opinionsxj(1), . . . , xj(T ) after each of these rounds,which can be grouped into stack vectorsx(1), . . . , x(T ). Aftercollection of this data,C can be estimated as the matrixbest fitting the equations (22) for0 ≤ k < T . Givenx(0) = u, x(1), . . . , x(T ), consider the optimization problem

minimizeT∑

j=1

‖εj‖22 subject to

εj = x(j)− (ΛW ⊗ C x(j − 1) + (I − Λ)⊗ I u) .

(30)

As was discussed in Remark 1, to provide the model“feasibility”, that is, belonging of the opinions to a givenset, it

11

0 10 20 30 40 50

−100

−80

−60

−40

−20

0

20

40

60

80

100

step, k

x i(k)

(a)

0 2 4 6 8 10

x 104

−100

−80

−60

−40

−20

0

20

40

60

80

100

step, k

aver

aged

xi(k

)

(b)

Fig. 9: Opinion dynamics ofn = 51 agents: (a) the determin-istic model; (b) averaged opinions in the gossip-based model

can be natural to restrict the MiDS matrix to some known setC ∈ C, which may e.g. consist of all row-stochastic matrices

m∑

j=1

cij = 1 ∀i, cij ≥ 0 ∀i, j, (31)

or all matrices, satisfying (21). In both of these examples,the problem (30) remains a convex QP problem, adding theconstraintC ∈ C. More generally, one may replace (30) withthe following convex optimizationproblem

minimize f(ε1, . . . , εT ) subject to

εj = x(j)− (ΛW ⊗ C x(j − 1) + (I − Λ)⊗ I u) ,

C ∈ C.

(32)

Here f is a convex andpositive definitefunction (that is,f(ε1, . . . , εT ) ≥ 0 and the inequality is strict unlessε1 =. . . = εT = 0) and C stands for a closed convex set. In thecase wheref(ε1, . . . , εT ) =

∑T

j=1 |εj | or f(ε1, . . . , εT ) =maxj |εj | the problem (32) becomes a standard linear pro-gramming problem.

Remark 5: In the case whereC is a (non-empty)compactset, the minimum in (32) exists wheneverf is continuous.Moreover, if f stands for astrictly convexnorm [53, Section8.1] on RT then the minimum exists even for unboundedclosed convex setsC since the set of all possible vectors{(ε1, . . . , εT )} is closed and convex, and the optimizationproblem (32) boils down to the projection on this set. For

instance, the minimum in the unconstrained optimization prob-lem (30) is always achieved.

B. Infinite-horizon identification procedure

The experiment of the second kind is applicable only tostable models, which inevitably implies a restriction on theMiDS matrix C. As in the previous section, we suppose thatC ∈ C, whereC is convex closed set of matrices. We supposealso that all elements ofC satisfy Theorem 2, that is,ρ(C) <ρ(ΛW )−1. As discussed in Remark 2, the setC may consiste.g. of all stochastic matrices (31) or matrices, satisfying (21).

The agents are not required to trace the history of their opin-ions, and their interactions are not limited to any prescribednumber of rounds. Instead, similar to the experiments from[15], the agents interact until their opinions stabilize (“agentscommunicate until consensus or deadlock is reached”[15]).In this sense, one may assume that the agents know the finalopinionx′. The matrixC should satisfy the equation

x′ = ΛW ⊗ C x′ + (I − Λ)⊗ I u, (33)

obtained as a limit of (22) ask → ∞. To find a matrixC bestfitting (33) we solve an optimization problem similar to (30)

minimize ‖ε‖22 subject to

ε = x′ − (ΛW ⊗ C x′ + (In − Λ)⊗ Im u) ,

C ∈ C.

(34)

In the case whereC stands for the polyhedron of matrices(e.g. we are confined to row-stochastic MiDS matricesC), theproblem (34) boils down to a convex QP problem; replacing‖ · ‖22 with ‖ · ‖1 or ‖ · ‖∞ norm, one gets a LP problem. Thesolution in the optimization problem (34) exists for any closedand convex setC for the reasons explained in Remark 5.

Both types of experiments thus lead to convex optimizationproblems. The advantage of the “finite-horizon” experimentisits independence of the system convergence. Also, allocatingsome fixed time for each dyadic interaction (and hence for theround of interactions), the data collection can be accomplishedin known time (linearly depending onT ). In many applica-tions, one is primarily interested in the opinion dynamics ona finite interval. This approach, however, requires to storethe whole trajectory of the system, collecting thus a largeamount of data (growing asnT ) and leads to a larger convexoptimization problem. The loss of data from one of the agentsin general requires to restart the experiment. On the other hand,the “infinite-horizon” experiment is applicable only to stablemodels, and one cannot predict how fast the convergence willbe. This experiment does not require agents to trace theirhistory and thus reduces the size of the optimization problem.

C. Transformation of the equality constraints

Both optimization problems (30) and (34) are featured bynon-standard linear constraints, involving Kronecker products.To avoid Kronecker operations, we transform the constraintsinto a standard formAx = b, whereA is a matrix andx, bare vectors. To this end, we perform a vectorization operation.Given a matrixM , its vectorizationvecM is a column vector

12

obtained by stacking the columns ofM , one on top of oneanother [40], e.g.vec ( 1 0

2 1 ) = [1, 2, 0, 1]⊤.Lemma 1: [40] For any three matricesA,B, C such that

the productABC is defined, one has

vecABC = (C⊤ ⊗A) vecB. (35)

In particular, forA ∈ Rm×l andB ∈ Rl×n one obtains

vecAB = (In ⊗A) vecB = (B⊤ ⊗ Im) vecA. (36)

The constraints in (30), (34) can be simplified. Consider firstthe constraint in (34). Letx′

i be the final opinion of theithagent andX ′ = [x′

1, . . . , x′n] be the matrix constituted by

them, hencex′ = vecX ′. Applying (36) forA = C andB =X ′ entails that[In ⊗C]x′ = [(X ′⊤ ⊗ Im] vecC, thus[ΛW ⊗C]x′ = [ΛW ⊗ Im][In ⊗ C]x′ = [ΛW (X ′)⊤ ⊗ Im] vecC.Denotingc = vecC, the constraint in (34) shapes into

ε+ [ΛWX ′⊤ ⊗ Im]c = x′ − [(In − Λ)⊗ Im]u, (37)

where the matrixΛWX⊤ ⊗ Im and the right-hand side areknown. The constraints in (30) can be rewritten as

εj+[ΛWX(j−1)⊤⊗Im]c = x(j)− [(In−Λ)⊗Im]u. (38)

HereX(j) is the matrix[x1(j), . . . , xn(j)] and hencex(j) =vecX(j).

D. Numerical examples

To illustrate the identification procedures, we consider twonumerical examples.

Example 6: Consider a social network with the matrixWfrom (2), Λ = I − diagW and the prejudice vector (15).Unlike Example 4,C is unknown and is to be found from the“infinite-horizon” experiment. Suppose that agents were askedto compute the final opinion, obtaining

x′ = [35, 11, 35, 10, 75,−50, 53, 5]⊤.

Solving the problem (34), one gets the minimal residual‖ε‖2 = 0.9322, which gives the MiDS matrix

C =

[0.7562 0.24380.3032 0.6968

]

.

In accordance with (23), this matrixC corresponds to thesteady opinion

x′C = [35.316, 11.443, 35.092, 9.483, 75,−50, 52.386, 4.915]⊤.

Example 7: For Λ, W andu from the previous example,agents were asked to conductT = 3 full rounds of conversa-tion (“finite horizon” experiment), obtaining the opinions

x(1) = [42.80, 14.05, 43.59, 12.51, 75,−50, 61.49, 7.18]⊤

x(2) = [41.31, 13.37, 41.45, 11.43, 75,−50, 55.48, 6.45]⊤

x(3) = [41.74, 12.30, 40.41, 10.84, 75,−50, 58.99, 6.02]⊤.

We are interested in finding a row-stochastic matrixC bestfitting the data. Solving the corresponding QP problem (30)with additional constraints (31), the MiDS matrix is

C =

[0.8181 0.18190.2983 0.7017

]

.

The model (17) with this matrixC gives the opinions

x(1) = [43.12, 14.66, 42.54, 12.37, 75,−50, 59.90, 7.17]⊤

x(2) = [41.93, 13.26, 41.73, 11.35, 75,−50, 58.37, 6.26]⊤

x(3) = [41.30, 12.69, 41.12, 10.79, 75,−50, 57.90, 5.83]⊤.

Remark 6: Examples 6 and 7, demonstrating the estima-tion procedures, are constructed as follows. We get the model(22) with W from (2),Λ = I − diagW andC from (18) andslightly change its final value (19) (Example 6) and trajectory(Example 7). Due to these perturbations, the estimated MiDSmatrix does not exactly coincide with (18) yet is close to it.

IX. PROOFS

We start with the proof of Theorem 1, which requires someadditional techniques.

Definition 4: (Substochasticity) A non-negative matrixA = (aij) is row-substochastic, if

∑

j aij ≤ 1 ∀i. Given sucha matrix sizedn × n, we call a subset of indicesJ ⊆ 1 : nstochasticif the corresponding submatrix(aij)i,j∈J is row-stochastic, i.e.

∑

j∈J

aij = 1 ∀i ∈ J .

The Gerschgorin Disk Theorem [44] implies that any sub-stochastic matrixA has ρ(A) ≤ 1. Our aim is to identifythe class of substochastic matrices withρ(A) = 1. As it willbe shown, such matrices are either row-stochastic or containa row-stochastic submatrix, i.e. has a non-empty stochasticsubset of indices.

Lemma 2: Any square substochastic matrixA withρ(A) = 1 admits a non-empty stochastic subset of indices.The union of two stochastic subsets is stochastic again, so thatthemaximalstochastic subsetJ∗ exists. Making a permutationof indices such thatJ∗ = (n′ + 1) : n, where0 ≤ n′ < n, thematrix A is decomposed into upper triangular form

A =

(A11 A12

0 A22

)

, (39)

whereA11 is a Schur stablen′ × n′-matrix (ρ(A11) < 1) andA22 is row-stochastic.

Proof: Thanks to the Perron-Frobenius Theorem [41],[44], ρ(A) = 1 is an eigenvalue ofA, corresponding to a non-negative eigenvectorv ∈ Rn (heren stands for the dimensionof A). Without loss of generality, assume thatmaxi vi = 1.Then we either havevi = 1n and henceA is row-stochastic (sothe claim is obvious), or there exists a non-empty setJ ( 1 : nof such indicesi that vi = 1. We are going to show thatJ isstochastic. Sincevi = 1 for i ∈ Jc = 1 : n \ J , one has

1 =∑

j∈Jc

aijvj +∑

j∈J

aij ≤ 1∀i ∈ J.

Since vj < 1 as j ∈ Jc, the equality holds only ifaij =0 ∀i ∈ J, j 6∈ J and

∑

j∈J aij = 1, i.e. J is a stochastic set.This proves the first claim of Lemma 2.

Given a stochastic subsetJ , it is obvious thataij = 0when i ∈ J and j 6∈ J , since otherwise one would have∑

j∈1:n

aij > 1. This implies that given two stochastic subsets

J1, J2 and choosingi ∈ J1, one has∑

j∈J1∪J2

aij =∑

j∈J1

aij +

13

∑

j∈J2∩Jc1

aij = 1. The same holds fori ∈ J2, which proves

stochasticity of the setJ1 ∪ J2. This proves the second claimof Lemma 2 and the existence of the maximal stochastic subsetJ∗, which, after a permutation of indices, becomes as followsJ∗ = (n′ + 1) : n. Recalling thataij = 0 ∀i ∈ J∗, j ∈ Jc

∗ , oneshows that the matrix is decomposed as (39), whereA22 isrow-stochastic. It remains to show thatρ(A11) < 1. Assume,on the contrary, thatρ(A11) = 1. Applying the first claimof Lemma 2 toA11, one proves the existence of anotherstochastic subsetJ ′ ⊆ 1 : n′, which contradicts the maximalityof J∗. This contradiction shows thatA11 is Schur stable.

Returning to the FJ model (1), it is easily shown now thatthe maximal stochastic subset of indices of the matrixΛWconsists of indices ofobliviousagents.

Lemma 3: Given a FJ model (1) with the matrixΛ diag-onal (where0 ≤ λii ≤ 1) and the matrixW row-stochastic,the maximal stochastic set of indicesJ∗ for the matrixΛW isconstituted by the indices of oblivious agents. In other words,j ∈ J∗ if and only if thejth agent is oblivious.

Proof: Notice, first, that the setJ∗ consists of obliviousagents. Indeed,1 =

∑

j∈J∗

λiiwij ≤ λii ≤ 1 for any i ∈ J∗,and hence none of agents fromJ∗ is stubborn. Sinceaij =0 ∀i ∈ J∗, j ∈ Jc

∗ (see the proof of Lemma 2), the agentsfrom J∗ are also unaffected by stubborn agents, being thusoblivious. Consider the setJ of all oblivious agents, which,as has been just proved, comprisesJ∗: J ⊇ J∗. By definition,λjj = 1 ∀j ∈ J . Furthermore, no walk in the graph fromJto Jc exists, and hencewij = 0 as i ∈ J, j 6∈ Jc, so that∑

j∈J wij = 1 ∀i ∈ J . Therefore, indices of oblivious agentsconstitute a stochastic setJ , and henceJ ⊆ J∗. ThereforeJ = J∗, which finishes the proof.

We are now ready to prove Theorem 1.Proof of Theorem 1:Applying Lemma 2 to the matrix

A = ΛW , we prove that agents can be re-indexed in a waythatA is decomposed as (39), whereA11 = Λ11W 11 is Schurstable andA22 is row-stochastic (ifA is Schur stable, thenA = A11 andA22 andA12 are absent). Lemma 3 shows thatindices1 : n′ correspond to stubborn agents and agents theyinfluence, whereas indices(n′ + 1) : n denumerate obliviousagents that are, in particular, not stubborn and henceλjj = 1as j > n′ so thatA22 = W 22. This proves the first claim ofTheorem 1, concerning the Schur stability ofΛ11W 11.

By noticing thatx2(k) = (W 22)kx2(0), one shows thatconvergence of the FJ model is possible only whenW 22 isregular, i.e.(W 22)k → W 22

∗ and hencex2(k) → W 22∗ u2. If

this holds, one immediately obtains (6) since

x1(k + 1) = Λ11W 11x1(k) + Λ11W 12x2(k) + (I − Λ11)u1

andΛ11W 11 is Schur stable.The proof of Theorem 2 follows from the well-known

property of the Kronecker product.Lemma 4: [40, Theorem 13.12] The spectrum of the

matrix A⊗B consists of all productsλiµj , whereλ1, . . . , λn

are eigenvalues ofA andµ1, . . . , µm are those ofB.Proof of Theorem 2:Lemma 4 entails thatρ(ΛW⊗C) =

ρ(ΛW )ρ(C), hence the system (22) is stable if and only if

ρ(ΛW )ρ(C) < 1. In particular, ifρ(C) = 1 then the system(22) is stable if and only ifρ(ΛW ) < 1.

The proof of Theorem 3 is similar to that of Theorem 1.After renumbering the agents, one can assume that obliv-ious agents are indexedn′ + 1 through n and considerthe corresponding submatricesW 11,W 12,W 22,Λ11, used inTheorem 1. Then the matrixΛW⊗C can also be decomposed

ΛW ⊗ C =

(Λ11W 11 ⊗ C Λ11W 12 ⊗ C

0 W 22 ⊗ C

)

, (40)

where the matricesΛ11W 11 ⊗ C has dimensionsmn′ ×mn′ and m(n − n′) × m(n − n′) respectively. We con-sider the corresponding subdivision of the vectorsx(k) =[x1(k)⊤, x2(k)⊤]⊤ and u = [(u1)⊤, (u2)⊤]⊤.

Proof of Theorem 3: Since the opinion dynamics ofoblivious agents is given byx2(k + 1) = W 22 ⊗ Cx2(k),the convergence implies regularity of the matrixW 22 ⊗ C.The regularity ofW 22 ⊗ C entails thatC is regular. Indeed,consider a left eigenvectorz of W 22 at 1 (that is,zW = z)and denotev = z ⊗ Im. Since vT (W 22 ⊗ C)k = z ⊗ Ck

has a limit ask → ∞ and z 6= 0, C is regular and, inparticular,ρ(C) ≤ 1. Obviously, the limitC∗ = limk→∞ Ck

is zero if and only ifC is Schur stable, i.e.ρ(C) < 1. Ifρ(C) = 1 then λ = 1 the only possible eigenvalue on theunit circle {λ : |λ| = 1}. Consider a right eigenvectorzof C (possibly, complex), corresponding to this eigenvalue.Denotingv = In⊗ z, the matrix(W 22⊗C)kv = (W 22)k ⊗ vhas a limit ask → ∞, and thusW 22 is regular. Thenecessity part is proved. To prove sufficiency, notice thatx2(k) → W 22

∗ ⊗ C∗u2 as k → ∞ (where we putW∗ = 0

whenC∗ = 0, and thus (24) follows from the equation

x1(k + 1) = [Λ11W 11 ⊗ C]x1(k) + [Λ11W 12 ⊗ C]x2(k)+

+[I − Λ11]⊗ Im u1,

whereΛ11W 11 ⊗ C is Schur stable.To proceed with the proof of Theorem 4, we need some

extra notation. As for the scalar opinion case in [21], [22] thegossip-based protocol (28), (26) shapes into

x(k + 1) = P (k)x(k) +B(k)u, (41)

where P (k), B(k) are independent identically distributed(i.i.d.) random matrices. If arc(i, j) is sampled, thenP (k) =A(i,j) andB(k) = B(i,j), where by definition

P (i,j) =(Imn − (γ1

ij + γ2ij)eie

⊤i ⊗ Im + γ1

ijeie⊤j ⊗ C

),

B(i,j) = γ2ijeie

⊤i ⊗ Im.

Denotingα := |E|−1 ∈ (0, 1] and noticing thatEP (k) =α∑

(i,j)∈EP (i,j) andEB(k) = α

∑

(i,j)∈EB(i,j), the follow-

ing equalities are easily obtained

EP (k) = Imn − α [Imn − ΛW ⊗ C]

EB(k) = α(In − Λ)⊗ Im.(42)

Proof of Theorem 4:As implied by equations (41) and(42), the opinion dynamics obeys the equation

x(k + 1) = P (k)x(k) + v(k), (43)

14

where the matricesP (k) and vectorsv(k) are i.i.d. and theirfinite first moments are given by the following

EP (k) = (1−α)I +αΛW ⊗C, Ev(k) = α(In −Λ)⊗ Im u,

whereα ∈ (0; 1]. SinceP (k) are non-negative, Theorem 1from [22] is applicable to (43), entailing that the processx(k)is almost sure ergodic andEx(k) → x∗ ask → ∞, where

x∗ = [I − ΛW ⊗ C]−1[(In − Λ)⊗ Im]u = x′C .

To prove theLp-ergodicity, notice thatx(k) and x(k) remainbounded due to Remark 1, and henceE‖x(k) − x∗‖

p → 0thanks to the Dominated Convergence Theorem [54].

Remark 7: (Convergence rate)For the case ofp = 2(mean-square ergodicity) there is an elegant estimate for theconvergence rate [21], [27]:E‖x(k) − x∗‖

2 ≤ χ/(k + 1),whereχ depends on the spectral radiusρ(ΛW ) and the vectorof prejudicesu. An analogous estimate can be proved for themultidimensional gossip algorithm (28), (26).

Remark 8: (Relaxation of the stochasticity condition)Ascan be seen from the proof, Theorem 4 retains its validityfor substochastic matrices, since they are non-negative andprovide boundedness of the solutions. Furthermore, the proofof almost sureergodicity does not rely on the solutions’boundedness and hence is preserved wheneverC is non-negative andρ(C)ρ(ΛW ) < 1. A closer examination of theproof of [22, Theorem 1] shows that it can be extended to thecase whereP (k) with negative entries. The non-negativity ofC can thus be relaxed, however, this relaxation is beyond thescope of this paper.

X. CONCLUSION

In this paper, we propose a novel model of opinion dynamicsin a social network with static topology. Our model is asignificant extension of the classical Friedkin-Johnsen model[15] to the case where agents’ opinions on two or moreinterdependent topics are being influenced. The extension isnatural if the agent are communicating on several “logically”related topics. In the sociological literature, an interdependentset of attitudes and beliefs on multiple issues is referredto as an ideological or belief system [38]. A specificationof the interpersonal influence mechanisms and networks thatcontribute to the formation of ideological-belief systemshasremained an open problem.

We establish necessary and sufficient conditions for thestability of our model and its convergence, which meansthat opinions converge to finite limit values for any initialconditions. We also address the problem of identification ofthe multi-issue interdependence structure. Although our modelrequires the agents to communicate synchronously, we showthat the same final opinions can be reached by use of adecentralized and asynchronous gossip-based protocol.

Several potential topics of future research are concernedwith experimental validation of our models for large setsof data and investigation of their system-theoretic propertiessuch as e.g. robustness and controllability. An important openproblem is the stability of an extension of the model (17),where the agents’ MiDS matrices are heterogeneous. A nu-merical analysis of a system with two different MiDS matrices

is available in our recent paper [55], further developing thetheory of logically constrained belief systems formation.

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APPENDIX

STOCHASTIC REGULAR MATRICES

We state a spectral criterion for regularity.

Lemma 5: [44, Ch.XIII, §7]. A row-stochastic squarematrix A is regular if and only ifdet(λI −A) 6= 0 wheneverλ 6= 1 and|λ| = 1; in other words, all eigenvalues ofA exceptfor 1 lie strictly inside the unit circle. A regular matrix is fullyregular if and only if1 is a simple eigenvalue, i.e.1d is the onlyeigenvector at1 up to rescaling:Az = z ⇒ z = c1d, c ∈ R.

In the case of irreducible [44] matrixA regularity and fullregularity are both equivalent to the property calledprimitivity,i.e. strict positivity of the matrixAm for some m ≥ 0which implies that all states of the irreducible Markov chain,generated byA, are aperiodic [44]. Lemma 5 also gives ageometric interpretation of the matrixA∗. Let the spectrumof A be λ1 = 1, λ2, . . . , λd, where |λj | < 1 as j > 1.Then Rd can be decomposed into a direct sum of invariant

root subspacesRd =d⊕

j=1

Lj, corresponding to the eigenvalues

λj . Moreover, the algebraic and geometric multiplicities ofλ1 = 1 always coincide [44, Ch.XIII,§6], so L1 consists ofeigenvectors. Therefore, the restrictionsAj = A|Lj

of A ontoLj are Schur stable forj > 1, whereasA1 is the identityoperator. Considering a decomposition of an arbitrary vectorv =

∑

j vj , wherevj ∈ Lj , one hasAkv1 = v1 andAkvj → 0ask → ∞ for anyj > 1. Therefore, the operatorA∗ : v 7→ v1is simply theprojector onto the subspaceL1.

As a consequence, we now can easily obtain the equality(12). Indeed, taking a decompositionv = v1 + . . . + vd, oneeasily notices that(I − αA)−1v1 = (1 − α)−1v1 and (I −αA)−1vi → (I − Ai)

−1vi as α → 1 for any i > 1. Hencelimα→1

(I − αA)−1(1− α)v = v1 = A∗v, which proves (12).

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