An Influence Network Model to Study Discrepancies in Expressed And

An Inﬂuence Network Model to Study Discrepancies in Expressed and Private Opinions ?

Mengbin Ye a, Yuzhen Qin b, Alain Govaert b, Brian D.O. Anderson a,c,d, Ming Cao b

aResearch School of Engineering, the Australian National University, Canberra, A.C.T. 2601, Australia bFaculty of Mathematics and Natural Sciences, University of Groningen, The Netherlands

cSchool of Automation, Hangzhou Dianzi University, Hangzhou 310018, China

dData61-CSIRO, Canberra, A.C.T. 2601, Australia

Abstract

In many social situations, a discrepancy arises between an individual’s private and expressed opinions on a given topic. Motivated by Solomon Asch’s seminal experiments on social conformity and other related socio-psychological works, we propose a novel opinion dynamics model to study how such a discrepancy can arise in general social networks of interpersonal inﬂuence. Each individual in the network has both a private and an expressed opinion: an individual’s private opinion evolves under social inﬂuence from the expressed opinions of the individual’s neighbours, while the individual determines his or her expressed opinion under a pressure to conform to the average expressed opinion of his or her neighbours, termed the local public opinion. General conditions on the network that guarantee exponentially fast convergence of the opinions to a limit are obtained. Further analysis of the limit yields several semi-quantitative conclusions, which have insightful social interpretations, including the establishing of conditions that ensure every individual in the network has such a discrepancy. Last, we show the generality and validity of the model by using it to explain and predict the results of Solomon Asch’s seminal experiments.

Key words: opinion dynamics; social network analysis; networked systems; agent-based model; social conformity

1 Introduction Discrepancies in private and expressed opinions of individuals can arise in many situations, with a variety of consequential phenomena. Over one third of jurors The study of dynamic models of opinion evolution on in criminal trials would have privately voted against social networks has recently become of interest to the the final decision of their jury [4]. Large differences be- systems and control community. Most models are agent- tween a population’s private and expressed opinions can based, in which the opinion(s) of each individual (agent) create discontent and tension, a factor associated with evolve via interaction and communication with neigh- the Arab Spring movement [5] and the fall of the So- arXiv:1806.11236v3 [cs.SI] 22 Feb 2019 bouring individuals. This paper aims to develop a novel viet Union [6]. Access to the public action of individu- opinion dynamics model as a general theoretical frame- als, without being able to observe their thoughts, can work to study how discrepancies arise in individuals’ pri- create informational cascades where all subsequent in- vate and expressed opinions, and thus bridge the current dividuals select the wrong action [7]. Other phenom- gap between socio-psychological studies on conformity ena linked to such discrepancies include pluralistic igno- and dynamic models of interpersonal influence. Inter- rance, where individuals privately reject a view but be- ested readers are referred to [1,2,3] for surveys on the lieve the majority of other individuals accept it [8], the many works on opinion dynamics models. “spiral of silence” [9,10], and enforcement of unpopular social norms [11,12]. Whether occurring in a jury panel, ? This paper was not presented at any IFAC meeting. Cor- a company boardroom or in the general population for responding author: M. Ye. a sensitive political issue, the potential societal ramifi- Email addresses: [email protected] (Mengbin Ye), cations of large and persistent discrepancies in private [email protected] (Yuzhen Qin), [email protected] (Alain and expressed opinions are clear, and serve as a key mo- Govaert), [email protected] (Brian D.O. tivator for our investigations. Anderson), [email protected] (Ming Cao).

Preprint submitted to Automatica 25 February 2019 1.1 Existing Work continuous interval, but is extremely complex and non- linear. The properties of the models in [11,29] have only been partially characterised by simulation-based analy- Conformity: Empirical Data and Static Models. One sis, which is computationally expensive if detailed anal- common reason such discrepancies arise is a pressure on ysis is desired. an individual to conform in a group situation; formal study of such phenomena goes back over six decades. In 1951, Solomon E. Asch’s seminal paper [13] showed We seek to expand from [11,29] to build an ABM of lower an individual’s public support for an indisputable fact complexity that is still powerful enough to capture how could be distorted due to the pressure to conform to a discrepancies in expressed and private opinions might unanimous group of others opposing this fact. Asch’s evolve in social networks, and to allow study by theoret- work was among the many studies examining the ef- ical analysis, as opposed to only by simulation. Impor- fects of pressures to conform to the group standard or tantly also, a minimal number of parameters per agent opinion, using both controlled laboratory experiments makes data fitting and parameter estimation in experi- and data gathered from field studies. Many of the lab mental investigations a tractable process, as highlighted experiments focus on Asch-like studies, perhaps with by the successful validations of the Friedkin–Johnsen various modifications. A meta-analysis of 125 such model [30,31,32], whereas experiments for more compli- studies was presented in [14]. Pluralistic ignorance is cated models are rare. often associated with pressures to conform to social norms [8,15,16]. With a focus on the seminal Asch ex- 1.2 Contributions of This Paper periments, a number of static models were proposed to describe a single individual conforming to a unanimous In this paper, we aim to bridge the gap between the liter- majority [17,18,19], with obvious common limitations ature on conformity and the opinion dynamics models, in generalisation to dynamics on social networks. by proposing a model where each individual (agent) has both a private and an expressed opinion. Inspired by the Opinion Dynamics Models. Agent-based models Friedkin–Johnsen model, we propose that an individ- (ABMs) have proved to be both versatile and powerful, ual’s private opinion evolves under social influence ex- with simple agent-level dynamics leading to interesting erted by the individual’s network neighbours’ expressed emergent network-level social phenomena. The seminal opinions, but each individual remains attached to his or French–DeGroot model [20,21] showed that a network her initial opinion with a level of stubbornness. Then, of individuals can reach a consensus of opinions via and motivated by existing works on the pressures to con- weighted averaging of their opinions, a mechanism mod- form in a group situation, we propose that each individ- elling “social influence”. Indeed, the term “influence ual has some resilience to this pressure, and each individ- network” arose to reflect the social influence exerted via ual expresses an opinion altered from his or her private the interpersonal network. Since then, the roles of ho- opinion to be closer to the average expressed opinion. mophily [22,23], bias assimilation [24], social distancing [25], and antagonistic interactions [26,27] in generating Rigorous analysis of the model is given, leading to a clustering, polarisation, and disagreement of opinions in number of semi-quantitative conclusions with insightful the social network have also been studied. Individuals social interpretations. We show that for strongly con- who remain somewhat attached to their initial opin- nected networks and almost all parameter values for ions were introduced in the Friedkin–Johnsen model stubbornness and resilience, individuals’ opinions con- [28] to explain the persistent disagreements observed in verge exponentially fast to a steady-state of persistent real communities. However, a key assumption in most disagreement. We identify that the combination of (i) existing ABMs (including those above), is that each stubbornness, (ii) resilience, and (iii) connectivity of the individual has a single opinion for a given topic. These network generically leads to every individual having a models are unable to capture phenomena in which an discrepancy between his or her limiting expressed and individual holds, for the same topic, a private opinion private opinions. We give a method for underbounding different to the opinion he or she expresses. the disagreement among the limiting private opinions given limited knowledge of the network, and show that a A few complex ABMs do exist in which each agent has change in an individual’s resilience to the pressure has a both an expressed opinion and a private opinion for the propagating effect on every other individual’s expressed same topic. The work [11] studies norm enforcement and opinion. Last, we apply our model to the seminal experi- assumes that each agent has two binary variables rep- ments on conformity by Asch [13]. Asch recorded 3 differ- resenting private and public acceptance or rejection of ent types of responses among test individuals who must a norm. We are motivated to consider opinions as con- choose between expressing support for an indisputable tinuous variables to better capture discrepancies in ex- fact and siding with a unanimous majority claiming the pressed and private opinions, since an individual’s opin- fact to be false. We identify stubbornness and resilience ion may range in its intensity. The model in [29] does as- parameter ranges for all 3 responses; this capturing of sume the expressed and private opinions take values in a all 3 responses is a first among ABMs, and underlines

2 our model’s strength as a general framework for study- We are now ready to propose the agent-based model. For ing the evolution of expressed and private opinions. a population of n individuals, let yi(t) ∈ R andy î(t) ∈ R, i = 1, . . . , n, represent, at time t = 0, 1,..., individual i’s Our work extends from (i) the static models of confor- private and expressed opinions on a given topic, respec- mity, by generalising to opinion dynamics on arbitrary tively. In general, yi(t) andy î(t) are not the same, and networks, and (ii) the dynamic agent-based models, by we regard yi as individual i’s true opinion. Individual i introducing mechanisms inspired by socio-psychological may refrain from expressing yi(t) for many reasons, e.g. literature to model the expressed and private opinions of political correctness when discussing a sensitive topic. each individual separately. The result is a general mod- For instance, preference falsification [35] occurs when an elling framework, which is shown to be consistent with individual falsifies his or her view due to social pressure empirical data, and may be used to further the study (be it imaginary or real), or deliberately, e.g. by a politi- of phenomena involving discrepancies in private and ex- cian seeking to garner votes. In our model, an individual pressed opinions in social networks. falsifies his or her opinion due to a pressure to conform to the group average opinion. The terms “opinion”, “belief”, and “attitude” all appear in the literature, with The rest of the paper is structured as follows. The model various related definitions Our model is general enough is presented in Section 2, with theoretical results detailed to cover all these terms, since in all such instances, one in Section 3. Section 4 applies the model to Asch’s ex- can scale y (t), yˆ (t) to be in some real interval [a, b], periments, with concluding remarks given in Section 5. i i where a and b represent the two extreme positions on the topic. For consistency, we will only use “opinion” unless explicitly stated otherwise. 2 A Novel Model of Opinion Evolution Under Pressure to Conform The individuals discuss their expressed opinionsy î(t) over a network described by a graph G[W ], and as a Before introducing the model, we define some notation, result, their private and expressed opinions, yi(t) and and introduce graphs, which are used to model the net- yî(t) evolve in a process qualitatively described in Fig. 1. work of interpersonal influence. Formally, individual i’s private opinion evolves as

n Notations: The n-column vector of all ones and zeros X yi(t+1) = λiwiiyi(t)+λi wijyˆj(t)+(1−λi)yi(0) (1) is given by 1n and 0n respectively. The n × n identity n×m j6=i matrix is given by In. For a matrix A ∈ R (respectively a vector a ∈ Rn), we denote the (i, j)th element th as aij (respectively the i element as ai). A matrix A and expressed opiniony î(t) is determined according to is said to be nonnegative, denoted by A ≥ 0 (respectively positive, denoted by A > 0) if all of its entries yî(t) = φiyi(t) + (1 − φi)ˆyi,lavg(t − 1). (2) aij are nonnegative (respectively positive). A nonnegative matrix A is said to be row-stochastic (respectively In Eq. (1), the influence weight that individual i ac- Pn row-substochastic) if for all i, there holds j=1 aij = 1 cords to individual j’s expressed opiniony ˆj(t) is cap- Pn Pn tured by w ≥ 0, satisfying Pn w = 1 for all i ∈ I. (respectively j=1 aij ≤ 1 and ∃k : j=1 akj < 1). ij j=1 ij The term wii ≥ 0 represents the self-confidence (if any) 1 Graphs: Given any nonnegative not necessarily symmet- of individual i in i’s own private opinion . The con- n×n stant λ ∈ [0, 1] represents individual i’s susceptibility ric A ∈ R , we can associate with it a graph G[A] = i to interpersonal influence changing i’s private opinion (V, E[A], A). Here, V = {v1, . . . , vn} is the set of nodes, (1 − λi is thus i’s stubbornness regarding initial opin- with index set I = {1, . . . , n}. An edge eij = (vi, vj) is in the set of ordered edges E[A] ⊆ V × V if and only if ion yi(0)). Individual i is maximally or minimally sus- a > 0. The edge e is said to be incoming with respect ceptible if λi = 1 or λi = 0, respectively. In Eq. (2), ji ij the quantityy ˆ (t) = P m yˆ (t) is specific to in- to j and outgoing with respect to i. We allow self-loops, i,lavg j∈Ni ij i i.e. eii is allowed to be in E. The neighbour set of vi is de- dividual i, and includes only the expressedy ˆj(t) of i’s noted by Ni = {vj ∈ V :(vj, vi) ∈ E}. A directed path neighbours. We assume that the weight mij ≥ 0 satisfies w > 0 ⇔ m > 0 and P m = 1; the ma- is a sequence of edges of the form (vp1 , vp2 ), (vp2 , vp3 ), ..., ij ij j∈Ni ij where vpi ∈ V, epj pk ∈ E. A graph G[A] is strongly con- trix M = {mij} is therefore row-stochastic and G[M] nected if and only if there is a path from every node to has the same connectivity properties as G[W ]. A nat- −1 every other node [33], or equivalently, if and only if A ural choice is mij = |Ni| for all j : eji ∈ E[W ], is irreducible [33]. A cycle is a directed path that starts and ends at the same vertex, and contains no repeated 1 In most situations, one can assume wii > 0, and models vertex except the initial (also the final) vertex, and a di- for studying the dynamics of wii exist [36,37]. Presence of rected graph is aperiodic if there exists no integer k > 1 wii > 0 can also ensure convergence of the opinions, e.g. in that divides the length of every cycle of the graph [34]. the DeGroot model [1].

3 while a reasonable alternative is mij = wij, ∀i, j ∈ I. Thus,y î,lavg(t) represents the group standard or norm as viewed by individual i at time t, and is termed the local public opinion as perceived by individual i. The constant φi ∈ [0, 1] encodes individual i’s resilience to pressures to conform to the local public opinion (maximally 1, and minimally 0), or resilience for short. The initial expressed opinion is set to bey î(0) = yi(0), which means Eq. (1) comes into effect for t = 1. As it turns out, under mild assumptions on λi, the final opinion values are dependent on yi(0) but independent ofy î(0); one could also Fig. 1. The discussion process. Each individual i, at time step t, expresses opiniony ˆ (t) and learns of others’ expressed select other initialisations fory î(0) with the final opin- i ions unchanged (though the transient would change). opinionsy ˆj (t), j 6= i. Next, the privately held opinion yi(t+1) evolves according to Eq. (1). After this, individual i then determines the newy î(t + 1) to be expressed in the next Sociology literature indicates that the pressure to con- round of discussion, according to Eq. (2). form causes an individual to express an opinion that is in the direction of the perceived group standard [13,38,10], sociology literature, in which an individual determines which in our model isy î,lavg(t). Some pressures of con- his or her expressed opinion under a pressure to conform. formity may derive from unspoken traditions [39], or a fear or being different [13], and others arise because of a desire to be in the group, driven by e.g. monetary incen- 2.1 The Networked System Dynamics tives, status or rewards [40]. Thus, Eq. (2) aims to capture individual i expressing an opinion equal to i’s pri- We now obtain a matrix form equation for the dynam- vate opinion modified or altered due to normative pres- ics of all individuals’ opinions on the network. Let y = sure (proportional to 1 − φi) to be closer to the pub- [y , y , . . . , y ]> and yˆ = [ˆy , yˆ ,..., yˆ ]> be the stacked lic opinion as perceived by individual i, which exerts a 1 2 n 1 2 n vectors of private and expressed opinions yi andy î of the “force” (1 − φi)ˆyi,lavg(t − 1). Heterogeneous φi captures n individuals in the influence network, respectively. The the fact that some individuals are less inhibited/reserved influence matrix W can be decomposed as W = W +W than others when expressing their opinions. In addition, f c pressures are exerted (or perceived to be exerted), dif- where Wf is a diagonal matrix with diagonal entries ferentially for individuals, e.g. due to status [41,38]. w˜ii = wii. The matrix Wc has entries wbij = wij for all j 6= i and wbii = 0 for all i. Define Λ = diag(λi) and Φ = diag(φi). Substitutingy ˆj(t) from Eq. (2) into Remark 1 Use of a local public opinion yî,lavg(t) en- P Eq. (1), and recalling thaty î,lavg = mijyˆj, yields sures the model’s scalability to large networks, but in j∈Nj small networks, e.g. a boardroom of 10 people, one could n replace yî,lavg(t) with the global public opinion yâvg(t) = n X 1 P yi(t + 1)=λiwiiyi(t) + λi wijφjyj(t)+ (1 − λi)yi(0) n j=1 yˆj(t) since it is likely to be discernible to every individual. It turns out that all but one of the high-level j6=i n theoretical conclusions, including convergence, do not de- X X pend on the choice of weights of the local public opinion, + λi wij(1 − φj) mjkyˆk(t − 1). nor on whether a local or global public opinion is used. j6=i k∈Nj However, preliminary observations show that the distri- (3) bution of the final opinion values can vary significantly depending on the aforementioned choices, and we leave From Eq. (3) and Eq. (2), one obtains characterisation of the difference to future investigations. " # " # " # y(t + 1) y(t) (In − Λ) y(0) Remark 2 A key feature in our model, departing from = P + , (4) most existing models, is the associating of two states yi, yî yˆ(t) yˆ(t − 1) 0n for each individual and the restriction that only other yˆj (and no y ) may be available to individual i. Importantly, j where P consists of the following block matrices note that yî(t) evolves dynamically via Eq. (2); yî(t) is not simply an output variable. However, notice that setting " # " # φi = 1 for all i recovers the Friedkin–Johnsen model, Λ(Wf + WcΦ) ΛWc(In − Φ)M P 11 P 12 while φ = λ = 1 for all i, recovers the DeGroot model = (5) i i Φ (I − Φ) M P P [21]. One may also notice the time-shift in Eq. (2) of n 21 22 yî,lavg(t−1), which ensures that Eq. (2) is consistent with the qualitative process described in Fig. 1. Thus, Eq. (2) As stated above, we set the initialisation as yˆ(0) = y(0), aims to capture a natural manner, widely supported in the yielding y(1) = (ΛW + In − Λ)y(0).

4 3 Analysis of the Opinion Dynamical System though the transient will differ. The row-stochasticity of R and S implies that the final private and expressed We now investigate the evolution of yi(t) andy î(t), ac- opinions are a convex combination of the initial private cording to Eq. (1) and Eq. (2), for the n individuals inter- opinions. Additionally, R, S > 0 means every individ- acting on the influence network G[W ]. In order to place ual i’s initial yi(0) has an influence on every individual ∗ ∗ the focus on social interpretations, we first present the j’s final opinions yj andy ˆj , a reflection of the strongly theoretical statements, and then discuss conclusions. All connected network. The following corollary establishes the proofs are deferred to the Appendix, since the key a condition for consensus of opinions, though one notes focus of this section is to secure conclusions via analysis that part of the hypothesis for Theorem 1 is discarded. of Eq. (4) regarding the discrepancies between expressed and private opinions that form over time. Throughout Corollary 1 (Consensus of Opinions) Suppose that this section, we make the following assumption on the φi ∈ (0, 1), and λi = 1, for all i ∈ I. Suppose further that social network. G[W ] is strongly connected and aperiodic, and W is row- stochastic. Then, for the system Eq. (4), limt→∞ y(t) = Assumption 1 The network G[W ] is strongly con- limt→∞ yˆ(t) = α1n for some α ∈ R, exponentially fast. nected and aperiodic, and W is row-stochastic. Further- more, there holds λi, φi ∈ (0, 1), ∀ i ∈ I. 3.2 Discrepancies and Persistent Disagreement It should be noted that for the purpose of convergence analysis, almost certainly one could relax the assumption This section establishes how disagreement among the to include graphs which are not strongly connected, and opinions at steady state may arise. In the following theo- for φ , λ ∈ [0, 1], which we leave for future work. i i rem, let zmax , maxi=1,...,n zi and zmin , mini=1,...,n zi n Pn denote the largest and smallest element of z ∈ R . Notice that because j=1 wij = 1 and λi ∈ [0, 1], Eq. (1) indicates that yi(t + 1) is a convex combination of yi(0), Theorem 2 Suppose that the hypotheses in Theorem 1 yi(t), andy ˆj(t), j ∈ Ni. Similarly,y î(t) is a convex com- hold. If y(0) 6= α1n for some α ∈ R, then the final bination of yi(t) andy î,lavg(t − 1). It follows that opinions obey the following inequalities

S = {yi, yî : min yk(0) ≤ yi, yî ≤ max yj(0), i ∈ I} (6) ∗ ∗ k∈I j∈I y(0)max > ymax > yˆmax (9a) ∗ ∗ y(0)min < y < yˆ (9b) is a positive invariant set of the system Eq. (4), which min min is a desirable property. If yi(0) ∈ [a, b], where a, b ∈ R ∗ ∗ and yˆmin 6=y ˆmax. Moreover, given a network G[W ] represent the two extremes of the opinion spectrum, and > and parameter vectors φ = [φ1, . . . , φn] and λ = S is a positive invariant set of Eq. (4), then the opinions > are always well defined. [λ1, . . . , λn] , the set of initial conditions y(0) for which precisely m > 0 individuals ij ∈ {i1, . . . , im} ⊆ I have y∗ =y ˆ∗ , i.e. m |{i ∈ I : y∗ =y ˆ∗}|, lies in a subspace 3.1 Convergence ij ij , i i of Rn with dimension n − m. The main convergence theorem, and a subsequent corollary for consensus, are now presented. This result shows that for generic initial conditions there is a persistent disagreement of final opinions at the Theorem 1 (Exponential Convergence) Consider steady-state. This is a consequence of individuals not a network G[W ] where each individual i’s opinions yi(t) being maximally susceptible to influence, λi < 1 ∀ i ∈ I. and yî(t) evolve according to Eq. (1) and Eq. (2), respec- One of the key conclusions of this paper is that for ∗ ∗ tively. Suppose Assumption 1 holds. Then, the system any individual i in the network, yi 6=y î for generic Eq. (4) converges exponentially fast to the limit initial conditions, which is a subtle but significant difference from Eq. (9). More precisely, the presence of both ∗ lim y(t) , y = Ry(0) (7) stubbornness and pressure to conform, and the strong t→∞ connectedness of the network creates a discrepancy be- ∗ lim yˆ(t) yˆ = Sy∗, (8) tween the private and expressed opinions of an individ- t→∞ , ual. Without stubbornness (λi = 1, ∀ i), a consensus of −1 opinions is reached, and without a pressure to conform where R = (In − (P 11 + P 12S)) (In − Λ) and S = −1 (φ = 1), an individual has the same private and ex- (In − P 22) P 21 are positive and row-stochastic, with i pressed opinions. Without strong connectedness, some P ij defined in Eq. (5). individuals will not be influenced to change opinions. The above shows that the final private and expressed ∗ ∗ opinions depend on y(0), while yˆ(0) are forgotten ex- One further consequence of Eq. (9) is that ymax −ymin > ∗ ∗ ponentially fast; one could initialise yˆ(0) arbitrarily, yˆmax − yˆmin, which implies that the level of agreement is

5 greater among the final expressed opinions when com- be available. While one cannot expect to know every φi, pared to the final private opinions. In other words, indi- we argue that φmax and φmin might be obtained, if not viduals are more willing to agree with others when they accurately then approximately. If the global public opin- are expressing their opinions in a social network due to iony âvg acts on all individuals, then Corollary 2 gives a pressure to conform. Moreover, the extreme final ex- a method for computing a lower bound on the level of pressed opinions are upper and lower bounded by the private disagreement given some limited knowledge. final private opinions, which are in turn upper and lower bounded by the extreme initial private opinions, show- It is obvious that if κ(φ) is small (if φmax is small and the ing the effects of interpersonal influence and a pressure ratio φmin/φmax is close to 1), then even strong agree- to conform. ∗ ∗ ment among the expressed opinions (a smally ˆmax −yˆmin) does not preclude significant disagreement in the final Remark 3 Theorem 2 states that generically, there will private opinions of the individuals. This might occur in be no two individuals who have the same final private e.g., an authoritarian government. The tightness of the opinions, and no individual will have the same final pri- bound Eq. (10) depends on the ratio φmin/φmax; the vate and expressed opinion. Let the parameters defin- closer the ratio is to one (i.e. as the “force” of the pres- ing the system (W , φ and λ) be given and suppose that sure to conform felt by each individual becomes more one runs p experiments with yi(0) sampled independently uniform), the tighter the bound. from a distribution (uniform, normal, beta, etc.) over a non-degenerate interval 2 . If q is the number of those ex- ∗ ∗ 3.4 An Individual’s Resilience Affects Everyone periments which result in yi =y î for some i ∈ I, then limp→∞ q/p = 0. From yet another perspective, the set ∗ ∗ An interesting result is now presented, that shows how of y(0) for which yi =y î for some i ∈ I belongs in a subspace of Rn that has a Lebesgue measure of zero. Sim- individual i’s resilience φi is propagated through the net- ∗ ∗ work. ilarly, yi = yj for i 6= j generically.

3.3 Estimating Disagreement in the Private Opinions Corollary 3 Suppose that the hypotheses in Theorem 1 hold. Then, the matrix S in Eq. (8) has partial derivative ∂(S) with strictly positive entries in the ith column and We now give a quantitative method for underbounding ∂φi the disagreement in the steady-state private opinions with all other entries strictly negative. for a special case of the model, where we replace the local public opiniony î,lavg with the global public opinion Recall below Theorem 1 that individual k’s final ex- −1 Pn ∗ yâvg = n yî in Eq. (2) for all individuals. pressed opiniony ˆk is a convex combination of all indi- j=1 ∗ viduals’ final private opinions yj , with convex weights skj, j = 1, . . . , n. Intuitively, increasing φk makes indi- Corollary 2 Suppose that, for all i ∈ I, yî,lavg(t − 1) in −1 Pn vidual k more resilient to the pressure to conform, and Eq. (2) is replaced with yâvg = n yî. Let κ(φ) = ∂s j=1 this is confirmed by the above; ∂skk > 0 and kj < 0 φmin ∂φk ∂φk 1 − φ (1 − φmax) ∈ (0, 1) and φmax = maxi∈I φi, ∗ ∗ max for any j 6= k and thusy ˆk → yk as φk → 1. φmin = mini∈I φi. Suppose further that the hypotheses in Theorem 1 hold. Then, More importantly, the above result yields a surprising th ∗ ∗ and nontrivial fact; every entry of the k column of yˆmax − yˆmin ∗ ∗ ∂(S) ∂(S) ≤ ymax − ymin. (10) is strictly positive, and all other entries of are κ(φ) ∂φk ∂φk strictly negative. In context, any change in individual k’s resilience directly impacts every other individual’s final expressed opinion due to the network of interpersonal For the purposes of monitoring the level of unvoiced dis- influences. In particular, as φk increases (decreases), an content in a network (e.g. to prevent drastic and un- ∗ individual j’s final expressed opiniony ˆj becomes closer foreseen actions or violence [5,6,29]), it is of interest to ∗ to (further from) the final private opinion yk of individ- obtain more knowledge about the level of disagreement ∂s ∂s ∗ ∗ ual k, since jk > 0 (decreasing, since jk < 0). among the private opinions: ymax −ymin. A fundamental ∂φk ∂φk issue is that such information is by definition unlikely to be obtainable (except in certain situations like the post- 3.5 Simulations experimental interviews conducted by Asch in his experiments, see Section 4). On the other hand, one expects that the level of expressed disagreementy ˆ∗ −yˆ∗ may Two simulations are now presented to illustrate the the- max min oretical results. A 3-regular network 3 G[W ] with n = 18 2 A statistical distribution is degenerate if for some k0 the 3 cumulative distribution function F (x, k0) = 0 if x < k0 and A k-regular graph is one which every node vi has k neigh- F (x, k0) = 1 if x ≥ k0. bours, i.e. |Ni| = k ∀ i ∈ I.

6 is generated. Self-loops are added to each node (to en- 1 sure G[W ] is aperiodic), and the influence weights wij Private Opinions are obtained as follows. The value of each w is drawn Expressed Opinions ij 0.8 randomly from a uniform distribution in the interval (0, 1) if (vj, vi) ∈ E, and once all wij are determined, the weights are normalised by dividing all entries in row i 0.6 Pn by j=1 wij. This ensures that W is row-stochastic and 0.4 nonnegative. For i 6= j, it is not required that wij = wji (which would result in an undirected graph), but for simplicity and convenience the simulations impose 4 that 0.2 wij > 0 ⇔ wji > 0. The values of yi(0), φi, and λi, are selected from beta distributions, which have two param- 0 0 1 2 3 4 5 6 7 8 910 eters α and β. For α, β > 1, a beta distribution of the Time step, t variable x is unimodal and satisfies x ∈ (0, 1), which is precisely what is required to satisfy Assumption 1 re- Fig. 2. Temporal evolution of opinions for 18 individuals in garding φi, λi. The beta distribution parameters are (i) an influence network. The green and dotted blue lines rep- α = 2, β = 2 for yi(0), (ii) α = 2, β = 2 for φi, and (iii) resent the expressed and private opinions of the individuals, α = 2, β = 8 for λi. In the simulation, we use the global respectively. public opinion model (see Remark 1) to also showcase Corollary 2. 1 Private Opinions Expressed Opinions The temporal evolution of opinions is shown in Fig. 2. 0.8 Several of the results detailed in this section can be observed. In particular, it is clear that Eq. (8) holds. That 0.6 is, there is no consensus of the limiting expressed or private opinions. Moreover, the disagreement among the ∗ ∗ 0.4 final expressed opinions,y ˆmax − yˆmin, is strictly smaller than the disagreement among the final private opinions, ∗ ∗ 0.2 ymax − ymin. Separate to this, the final private opinions enclose the final expressed opinions from above and be- 0 low. For the given simulation, the largest and smallest 0 5 10 15 resilience values are φmax = 0.9437 and φmin = 0.1994, Time step, t respectively. This implies that κ(φ) = 0.9881. One can ∗ ∗ also obtain thaty ˆmax − yˆmin = 0.1613. From Eq. (10), Fig. 3. Temporal evolution of opinions for 18 individuals in ∗ ∗ an influence network. The green and dotted blue lines rep- this indicates that ymax − ymin ≥ 0.163. The simulation result is consistent with the lower bound, in that resent the expressed and private opinions of the individuals, y∗ − y∗ = 0.3455. Also, the bound is not tight, since respectively. The lack of stubbornness, λi = 1, ∀ i, means max min that all opinions reach a consensus. φmin/φmax is far from 1 (see Section 3.3).

individual’s susceptibility λi and resilience φi that de- For the same G[W ], with the same initial conditions termine the individual’s reaction to a unanimous major- yi(0) and resilience φi, a second simulation is run with ity’s pressure to conform, and thus give an agent-based λ1 = 1, ∀i ∈ I. As shown in Fig. 3, the opinions converge model explanation of the recorded observations. In or- ∗ ∗ to a consensus y = yˆ = α1n, for some α ∈ R, which der for the reader to fully appreciate and understand the illustrates Corollary 1. results, a brief overview of the experiments and its results are now given, and the reader is referred to [13] for full details on the results. In summary, the experiments 4 Application to Asch’s Experiments studied an individual’s response to “two contradictory and irreconcilable forces” [13] of (i) a clear and indis- We now use the model to revisit Solomon E. Asch’s sem- putable fact, and (ii) a unanimous majority of the others inal experiments on conformity [13]. There are at least who take positions opposing this fact. two objectives. For one, successfully capturing Asch’s empirical data constitutes a form of soft validation for In the experiment, eight individuals are instructed to the model. Second, we aim to identify the values of the judge a series of line lengths. Of the eight individuals, one is in fact the test subject, and the other seven “con- 5 4 Such an assumption is not needed for the theoretical re- federates” have been told a priori about what they sults, but is a simple way to ensure that all directed graphs generated using the MATLAB package are strongly con- 5 These other individuals have become referred to as “connected. federates” in later literature.

7 been developed). The qualitative observations made in this section are invariant to the weights wij, and focus is instead placed on examining Asch’s experimental results from the perspective of our model. In the following Section 4.2, the impact of W (and in particular the weight w11), and parameters φ1 and λ1, are shown using analytic calculations. Fig. 4. Example of the Asch experiment. The individuals openly discuss their individual beliefs as to which one of 4.1 Types of Individuals A, B, C has the same length as the green line. Clearly A is equal in length to the green line. The test individual is the Asch observed three broad types of individuals. In par- red node. The confederates (seven blue nodes) unanimously ticular, he divided the test individuals as: (i) indepen- express belief in the same wrong answer, e.g. B. dent individuals, (ii) yielding individuals with distortion of judgment, and (iii) yielding individuals with distor- should do. An example of the line length judging experi- tion of action. The assigned values for the parameters ment is shown in Fig. 4. There are three lines of unequal λ and φ for each type of individual are summarised in length, and the group has open discussions concerning 1 1 Table. 1. Values of φ1, λ1 in this neighbourhood gener- which one of the lines A, B, C is equal in length to the ate responses that are qualitatively the same at a high green line. Each individual is required to independently level; the differences lie in the exact values of the final declare his choice, and the confederates (blue individ- opinions. uals) unanimously select the same wrong answer, e.g. B. The reactions of the test individual (red node) are Independent individuals can be divided further into dif- then recorded, followed by a post-experiment interview 6 ferent subgroups depending on the reasoning behind to evaluate the test individual’s private belief . their independence, but this will not be considered because we focus only on the final outcome or observed In order to apply our model, and with Fig. 4 as an il- result and not the reasons for independence. Asch iden- lustrative example, we frame yi, yî ∈ [0, 1] to be indi- tified an independent individual as someone who was vidual i’s belief in the statement “the green line is of strongly confident that A was correct. This individual the same length as line A.” Specifically, yi = 1 (respec- did not change his expressed belief, i.e. did not yield to tively yi = 0) implies individual i is maximally certain the confederates’ unanimous declaration that A was in- the statement is true (respectively, maximally certain correct, despite the confederates insistently questioning the statement is false). Asch found close to 100% of indi- the individual. Asch’s descriptions indicate that the test viduals in control groups had yi(0) = 1. Without loss of individual is extremely stubborn (i.e. closed to influence) generality, we therefore denote the test individual as in- and confident his belief is correct, and is resilient to the dividual 1 and set y1(0) =y ˆ(0) = 1. Confederates are set group pressure. It is then obvious that one would as- to have yi(0) =y î(0) = 0, for i = 2, . . . , n, with λi = 0 sign to such individuals values of λ1 close to zero and φ1 and φi = 1. That is, they consistently express maximal close to one. With the framing of the experiments given certainty that “the green line of the same length as line above, our model would be said to accurately capture an A” is a false statement. independent individual if test individual 1 with parameter values of λ1 close to zero and φ1 close to one, has ∗ ∗ It should be noted that in the experiments, Asch never final beliefsy ˆ1 , y1 ≈ 1. assigned values of susceptibility λi, and resilience φi to the individuals because the quantitatively measured Asch also identified yielding individuals, who could be data by Asch was the number of incorrect answers over divided into two groups. Those who experienced a distor- 12 iterations per group, and the behaviour of the indi- tion of judgment/perception either (i) lacked confidence, vidual being tested. However, based on his written de- assumed the group was correct and thus concluded A scription of individuals (including excerpts of the inter- was incorrect, or (ii) did not realise he had been influ- views), it was clear to the authors of this paper what enced by the group at all and changed his private belief the approximate range of values of the parameters λi, φi to be certain that A was incorrect. This indicates that should be for each type of individual. (Some of these the individual is open to influence (i.e. not stubborn in descriptions and excerpts will be provided immediately y1(0) = 1) and is highly affected by the group pressure below). Also, the experiments did not attempt to de- (i.e. not resilient). One concludes that for such individ- termine the influence matrix W (at the time, influence uals λ1 is likely to be close to one, and φ1 to be close to network theory in the sense of DeGroot etc. had not yet zero. As shown in the sequel, it turns out that the value of φ1 plays only a minor role for such an individual be- 6 cause he is already extremely susceptible to influence. In this section, we refer to yi, yî as beliefs, as the variables represent individual i’s certainty on an issue that is provably For our model to accurately capture such an individual, true or false. As noted in Section 2, our model is general then for λ1 close to one, and φ1 close to zero, one expects ∗ ∗ enough to cover both subjective and intellective topics. y1 , yˆ1 ≈ 0.

8 Table 1 From this, one concludes that the test subject’s final pri- Types of test individuals and their susceptibility and re- vate belief is dependent on his level of stubbornness in silience parameters believing that A is the correct answer, i.e. λ1, and on λ1 φ1 his self-weight w11, i.e. how much he trusts his own be- ∗ Independent low high lief relative to the others in the group. Interestingly, y1 does not depend on individual 1’s resilience φ , though Yielding, judgment distortion high any 1 it must be noted that this is a special case when the Yielding, action distortion low low other individuals are all confederates. In general net- ∗ works beyond the Asch framework, y1 will depend not Other yielding individuals experienced a distortion of ac- only on φ1, but also the other φi. For simplicity, consider tion. This type of individual, on being interviewed (and a natural selection of wii = 1 − λi [28]. As a result, one ∗ before being informed of the true nature of the experi- obtains that y1 = (1 − λ1)/(1 − λ1(1 − λ1)). Examina- ment) stated that he remained privately certain that A tion of the function f(λ1) = (1 − λ1)/(1 − λ1(1 − λ1)), was the correct answer, but suppressed his observations for λ1 ∈ [0, 1], reveals how the test subject’s final pri- as to not publicly generate friction with the group. Such vate belief changes as a function of his openness to influ- an individual has full awareness of the difference between ence; the function f(λ1) is plotted in Fig. 5. Notice that the truth and the majority’s position. This individual is f(λ1) = (1−λ1)/(1−λ1(1−λ1)) ≥ 1−λ1 for λ1 ∈ [0, 1] closed to influence (i.e. stubborn) but not resilient, and with equality if and only if λ1 = {0, 1}. This implies ∗ it is predicted that such individuals will have λ1 and φ1 that the test individual’s final y1 will always be greater both close to zero. If our model were to accurately cap- than his stubbornness 1 − λ1, except if he has λ1 = 0 ture such an individual, then the final beliefs would be (maximally stubborn) or λ1 = 1 (maximally open to in- ∗ ∗ expected to be y1 ≈ 1 andy ˆ1 ≈ 0. fluence).

4.2 Theoretical Analysis Next, consider the final expressed belief, which is given This section will present theoretical calculations of asy ˆ∗ = nφ1 y∗. The relative closeness ofy ˆ∗ to y∗, 1 n−1+φ1 1 1 1 Asch’s experiments in the framework of the our model, ∗ ∗ as measured byy ˆ1 /y1 , is determined by n and φ1. De- showing how y1, yˆ1 vary with W , λ1 ∈ [0, 1] and nφ1 fine g(φ1, n) = . The function g(φ1, n) is plotted n−1+φ1 φ1 ∈ [0, 1]. Analysis will be conducted for n ≥ 2, to investigate the effects of the majority size on the in Fig. 6. Observe that g(φ1, n) ≥ φ1 for any n, for all belief evolution. We make the mild assumption that φ1 ∈ [0, 1], and with equality if and only if φ1 = {0, 1}. This implies that the test individual’s final expressed w11 ∈ (0, 1), which implies that individual 1 considers his/her own private belief during the discussions. belief will always be closer to his final private belief than his resilience level. Most interestingly, observe that g(φ1, n) → φ1 from above, as n → ∞, but the differ- Because λi = 0 and φi = 1 for all i = 2, . . . , n, one ence between g(φ1, n) and φ1 when going from n = 2 to concludes from Eq. (1) and Eq. (2) that yi(t) =y î(t) = 0 > n = 2 × 2 = 4 is much greater than the differences go- for all t. With y(0) = [1, 0,..., 0] , it follows that test ing from n = 4 to n = 4 × 2 = 8. This may explain the individual 1’s belief evolves as observation in [13] that increasing the majority size did " # " # " # not produce a correspondingly larger distortion effect y1(t + 1) y1(t) 1 − λ1 beyond majorities of three to four individuals, at least = V + . (11) yˆ (t) yˆ (t − 1) 0 for test individuals with low λ1. That is, an increase in 1 1 n does not produce a matching increase in distortion of the final expressed opinion from the final private opin- where ∗ ∗ " # ion, represented asy ˆ1 /y1 = g(φ1, n) → 1 as n → ∞. λ1w11 0 V = . (12) 1 φ1 n (1 − φ1) From the fact that n ≥ 2, λ ∈ [0, 1], w ∈ (0, 1), and Also of note is that for individuals with λ1 close to one, 1 11 ∗ ∗ φ ∈ [0, 1], it follows that V has eigenvalues inside the y1 is already close to zero, and boundsy ˆ1 from above. 1 ∗ ∗ unit circle and thus the system in Eq. (11) converges The magnitude of the difference, |y1 − yˆ1 |, only changes to limit exponentially fast. Straightforward calculations slightly as φ1 is varied, which indicates that for individ- show that this limit is given by uals who yielded with distortion of judgment, the value of φ1 plays only a minor role in the determining the absolute (as opposed to relative) difference between ex- ∗ 1 − λ1 lim y1(t) y1 = (13) pressed and private beliefs. This is in contrast to indi- t→∞ , 1 − λ w 1 11 viduals with low susceptibility, where the behaviour of ∗ nφ1 ∗ an individual can vary significantly by varying φ from lim yˆ1(t) , yˆ1 = y1 . (14) 1 t→∞ n − 1 + φ1 1 to 0.

9 1 which might be seen as a concession towards the major- f( ) 1 ity belief. There is also a small shift away from maximal 1- 1 ∗ 0.8 certainty of yi = 1, with y1 ≈ 0.93; in [13], one independent test individual stated 0.6 I would follow my own view, though part of my reason 0.4 would tell me that I might be wrong.

0.2 Figure 7b shows the belief evolution of a yielding test individual who, under group pressure, exhibits distor- 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 tion of judgment/perception. The figure shows that both 1 ∗ ∗ y1 andy ˆ1 are heavily influenced by the group pressure, and thus individual 1 is no longer privately certain that Fig. 5. The function f(λ1) and 1 − λ1 plotted A is the correct answer. In other words, this individ- against λ1. The analytical calculations show that ∗ ual is highly susceptible to interpersonal influence, and y1 = f(λ1), and thus the red line represents individual 1’s final private belief as a function of even his private view becomes affected by the majority. his susceptibility to influence. Of great interest is the evolution of beliefs observed in Fig. 7c, which involves an experiment with a yielding 1 test individual exhibiting distortion of action. Accord- ing to Asch, Individual 1 0.8

0.6 yields because of an overmastering need to not appear diﬀerent or inferior to others, because of an inability

0.4 to tolerate the appearance of defectiveness in the eyes

g( ,2) 1 of the group ˜[13]. g( ,4) 0.2 1 g( ,8) 1 ∗ 1 In other words v1’s expressed belief y1 is heavily dis- 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 torted by the pressure to conform to the majority. How-

1 ever, this individual is still able to “conclude that they ∗ [themselves] are not wrong” [13], i.e. yi ≈ 0.93. Fig. 6. The function g(φ1, n), with n = 2, 4, 8, plotted against φ1. The analytical calculations Other simulations with values of λ1, φ1 in the neigh- ∗ ∗ show thaty ˆ1 = g(φ1, n)y1 , and thus the plot bourhood of those used also display similar behaviour shows how the test individual’s final expressed as shown in Fig. 7a to 7b, indicating a robust ability opinion is changed from his final private opinion for our model to capture Asch’s experiments is an in- by his resilience φ1, and by n. trinsic property of the model, and rather than resulting 4.3 Simulations from careful reverse engineering. All three types of individual behaviours can be predicted by our model using pairs of parameters λ , φ , providing a measure of valida- The Asch experiments are simulated using the proposed i i tion for our model. At the same time, we have provided model. An arbitrary W is generated with weights w ij an agent-based model explanation of the empirical find- sampled randomly from a uniform distribution and nor- n ings of Asch’s experiments; it might now be possible to malised to ensure P w = 1. The other parameters j=1 ij analyse the many subsequent works derived from Asch are described in the third paragraph of Section 4. In the can be analysed common framework, whereas existing following plots of Fig. 7a, 7b and 7c, the values of λ1 static models of conformity are tied to specific empiri- and φ1 are given. The red lines correspond to test indi- cal data (see the Introduction). The Friedkin–Johnsen vidual 1, with the solid line showing private belief y1(t) model has also been applied to the Asch experiments and the dotted line showing expressed beliefy ˆ1(t). The [30], but (unsurprisingly) was not able to capture all of blue line represents the confederates k = 2,..., 8, who the types of individuals reported because the Friedkin– have yk(t) =y ˆk(t) = 0 for all t. Johnsen model does not assume that each individual has a separate private and expressed belief. Figure 7a shows the evolution of beliefs when the test individual is independent. It can be seen that both the 4.4 Threshold Variant and Asch’s Second Experiments private and expressed beliefs of v1 are largely unaffected by the confederates’ unanimous expressed belief and the ∗ ∗ pressure exerted by the group. Note thaty ˆ1 < y1 , which The simulations above assumed that the individuals ex- is also reported in [13]; despite expressing his belief that press a continuous real-valued beliefsy î(t), whereas it is A is the correct answer, one independent test individual perhaps more appropriate to sety î(t) as a binary vari- stated “You’re probably right, but you may be wrong!”, able, withy î(t) = 1 andy î(t) = 0 denoting individual i

10 (a) An independent individual, with λ1 = (b) A yielding individual with distortion (c) A yielding individual with distortion 0.1, φ1 = 0.9. of judgment, with λ1 = 0.9, φ1 = 0.1. of action, with λ1 = 0.1, φ1 = 0.1. Fig. 7. Fig. 7a, 7c, and 7b show the evolution of beliefs for all three types of reactions recorded by Asch, as they appear in our model. The red solid and dotted line denote the private and expressed belief, respectively, of the test individual 1 (i.e. y1(t) andy ˆ1(t)). The blue line is the belief of the unanimous confederate group, who express a belief ofy ˆi(t) = 0.

∗ ∗ 1 picking A and not picking A as the correct answer. The is a small interval region τ1 ∈ φ1y1 , φ1y1 + (1 − φ1) n ∗ proposed model can be modified to accommodate situa- of width (1 − φ1)/n wherey ˆ1 depends on the initial tions where the expressed variable denotes an action, or conditiony ˆ1(0). decision by replacing Eq. (2) with 4.4.2 Asch’s Second Experiments yî(t) = σi (φiyi(t) + (1 − φi)ˆyi,lavg(t − 1)) , (15) Asch conducted several variations to the original experi- where σi(x) : [0, 1] → {0, 1} is a threshold function sat- ments, as reported in [13,42]. In one particular variation, isfying σi(x) = 0 if x ∈ [0, τi] and σi(x) = 1 if x ∈ (τi, 1], one confederate also told a priori to select the correct for some threshold value τi ∈ (0, 1). Applying the thresh- answer; the frequency of individuals showing distortion old variant of the model with τi = 0.5 yields no quali- of action or distortion of judgment decreased dramati- tative difference for the simulations in Section 4.3. That cally. We now frame this variation of the experiment in is, pairs of parameter values λ1, φ1 which in the original our model’s framework, and call it Asch’s Second Exper- model were associated with an independent, distortion iment for convenience. The parameter matrix W , and of action, or distortion of judgment individual (Table 1) parameters λi and φi, i = 1,..., 8 are unchanged from were almost always also associated with the same type the first experiment described in Section 4. The setup of individual in the threshold model. of individual 1 is also the same. However, different from Section 4, the n − 1 confederates’ beliefs are now set 4.4.1 Calculations to be y2(0) =y ˆ2(0) = 1, and yi(0) =y î(0) = 0 for i = 3, . . . , n. It should be noted that theoretical calculations of the final private and expressed beliefs of individ- Because of the highly specialised setup for the Asch ex- ual 1 can also be completed, following the same method periments, it turns out that one can theoretically cal- as in Section 4.4.1. culate the final beliefs of test individual 1 even under the threshold model. This would not be the case for the threshold model in general scenarios. In fact, it is unclear 4.4.3 Simulations if the threshold model will always converge in a general setting, especially if individuals update synchronously. We now provide simulations for Asch’s Second Experi- ment, using both the original model proposed in Eq. (2), We perform calculations for Asch’s experiments (Sec- and the threshold model in Eq. (15). tion 4). First, we remark that the private opinion dynam- Case 1: The behaviour of individuals with high φ and ics y (t) of test individual 1 is unchanged in the thresh- 1 1 low λ (independent individuals in Asch’s First Experi- old model when compared to the original model, since i ment) are the same, qualitatively, when comparing the the expressed beliefs of all of individual 1’s neighbours ∗ 1−λ1 original model and the threshold model. We omit the are stationary. Thus, limt→∞ y1(t) y = as , 1 1−λ1w11 simulation results for such individuals. in the original model calculations in Section 4.2. Case 2: Next, we simulate a test individual that has low One can then consider y1(t) as an input to Eq. (15). It φ1 and low λi (in Asch’s First Experiment, these indi- follows thaty ˆ1(t) converges. In particular, and assuming viduals were said to show distortion of action). Fig. 8a ∗ global public opinion is used, then limt→∞ yˆ1(t) , yˆ1 = and 8b show a test individual with λ1 = 0.1, φ1 = 0.1, ∗ 1 ∗ ∗ 1 if φ1y1 +(1−φ1) n ≥ τ1 andy ˆ1 = 0 if φ1y1 < τ1. There for the original and threshold model, respectively.

11 Case 3: Last, we simulate a test individual that has low References φ1 and high λi (in the original Asch setup, these individuals were said to show distortion of judgment). Fig. 9a [1] A. V. Proskurnikov and R. Tempo, “A tutorial on modeling and 9b show a test individual with λ1 = 0.9, φ1 = 0.1, for and analysis of dynamic social networks. Part I,” Annual the original and threshold model, respectively. Finally, Reviews in Control, vol. 43, pp. 65–79, 2017. Fig. 10 shows Case 4, which simulates a test individual [2] ——, “A tutorial on modeling and analysis of dynamic social with the same parameter set of λ1 = 0.9, φ1 = 0.1, but networks. Part II,” Annual Reviews in Control, vol. 45, pp. with the threshold changed from τ1 = 0.5 to τ1 = 0.6. 166–190, 2018. [3] A. Flache, M. Mäs, T. Feliciani, E. Chattoe-Brown, G. Deffuant, S. Huet, and J. Lorenz, “Models of Social Whether the original model or the threshold model is Influence: Towards the Next Frontiers,” Journal of Artificial used, it can been seen that introduction of an actor (con- Societies & Social Simulation, vol. 20, no. 4, 2017. federate) telling the truth has a major impact on the [4] N. L. Waters and V. P. Hans, “A Jury of One: Opinion belief evolution of the test individual in Case 2 and 3 Formation, Conformity, and Dissent on Juries,” Journal of (compare Fig. 7c with Fig. 8a and 8b, and Fig. 7b with Empirical Legal Studies, vol. 6, no. 3, pp. 513–540, 2009. Fig. 9a, 9b and 10). The impact is significantly more [5] J. Goodwin, “Why We Were Surprised (Again) by the Arab pronounced under the threshold model, such that a test Spring,” Swiss Political Science Review, vol. 17, no. 4, pp. individual with λ1 = 0.9, φ1 = 0.1 and τi = 0.5 (Case 452–456, 2011. 3) will still pick the correct answer when another ac- [6] T. Kuran, “Sparks and prairie fires: A theory of unanticipated tor tells the truth. When the threshold is adjusted to political revolution,” Public Choice, vol. 61, no. 1, pp. 41–74, τi = 0.6 (Case 4), the test individual picks the wrong 1989. answer along with the confederates. [7] S. Bikhchandani, D. Hirshleifer, and I. Welch, “A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades,” Journal of Political Economy, vol. 100, no. 5, pp. 992–1026, 1992. 5 Conclusions [8] F. H. Allport, Social Psychology. Boston: Houghton Mifflin Company, 1924. We have proposed a novel agent-based model of opin- [9] E. Noelle-Neumann, The Spiral of Silence: Public Opinion, Our Social Skin. University of Chicago Press, 1993. ion evolution on interpersonal influence networks, where each individual has separate expressed and private opin- [10] D. G. Taylor, “Pluralistic Ignorance and the Spiral of Silence: A Formal Analysis,” Public Opinion Quarterly, vol. 46, no. 3, ions that evolve in a coupled manner. Conditions on the pp. 311–335, 1982. network and the values of susceptibility and resilience for the individuals were established for ensuring that the [11] D. Centola, R. Willer, and M. Macy, “The Emperors Dilemma: A Computational Model of Self-Enforcing Norms,” opinions converged exponentially fast to a steady-state American Journal of Sociology, vol. 110, no. 4, pp. 1009– of persistent disagreement. Further analysis of the fi- 1040, 2005. nal opinion values yielded semi-quantitative conclusions [12] R. Willer, K. Kuwabara, and M. W. Macy, “The False that led to insightful social interpretations, including the Enforcement of Unpopular Norms,” American Journal of conditions that lead to a discrepancy between the ex- Sociology, vol. 115, no. 2, pp. 451–490, 2009. pressed and private opinions of an individual. We then [13] S. E. Asch, Groups, Leadership, and Men. Carnegie Press: used the model to study Asch’s experiments [13], show- Pittsburgh, 1951, ch. Effects of Group Pressure Upon the ing that all 3 types of reactions from the test individual Modification and Distortion of Judgments, pp. 222–236. could be captured within our framework. A number of [14] R. Bond, “Group Size and Conformity,” Group Processes & interesting future directions can be considered. Prelim- Intergroup Relations, vol. 8, no. 4, pp. 331–354, 2005. inary simulations show that our model can also capture [15] D. A. Prentice and D. T. Miller, “Pluralistic Ignorance pluralistic ignorance, with network structure and place- and Alcohol Use on Campus: Some Consequences of ment of extremist nodes having a significant effect on Misperceiving the Social Norm,” Journal of personality and the observed phenomena. Clearly the threshold model social psychology, vol. 64, no. 2, pp. 243–256, 1993. in Section 4.4 requires further study, and one could also [16] H. J. O’Gorman, “Pluralistic Ignorance and White Estimates consider the model in a continuous-time setting, or with of White Support for Racial Segregation,” Public Opinion asynchronous updating, or both. Quarterly, vol. 39, no. 3, pp. 313–330, 1975. [17] S. Tanford and S. Penrod, “Social Influence Model: A Formal Integration of Research on Majority and Minority Influence Processes,” Psychological Bulletin, vol. 95, no. 2, p. 189, 1984. Acknowledgements [18] B. Mullen, “Operationalizing the Effect of the Group on the Individual: A Self-Attention Perspective,” Journal of Experimental Social Psychology, vol. 19, no. 4, pp. 295–322, The authors would like to thank Julien Hendricks for 1983. his helpful discussion on the proof of Theorem 2, and [19] G. Stasser and J. H. 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12 1 1

0.8 0.8 Unanimous Group Test Individual Private Belief 0.6 0.6 Test Individual Expressed Belief

0.4 0.4

0.2 Unanimous Group 0.2 Test Individual Private Belief Test Individual Expressed Belief 0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Time Step, t Time Step, t

(a) Asch’s second experiment: An individual (b) Asch’s second experiment: An individual with λ1 = 0.1, φ1 = 0.1, original model. with λ1 = 0.1, φ1 = 0.1, threshold model. Fig. 8. Fig. 8a and 8b show the evolution of beliefs, for two diﬀerent models, of the variation of the Asch experiment where a second actor supports the truth. The red dashed and dotted line denote the private and expressed belief, respectively, of the test individual 1 (i.e. y1(t) andy ˆ1(t)). The blue line is the belief of the unanimous confederate group, who express a belief of yˆi(t) = 0.

1 1 Unanimous Group Test Individual Private Belief 0.8 Test Individual Expressed Belief 0.8

0.6 0.6

Unanimous Group 0.4 0.4 Test Individual Private Belief Test Individual Expressed Belief 0.2 0.2

0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Time Step, t Time Step, t

(a) Asch’s second experiment: An individual (b) Asch’s second experiment: An individual with λ1 = 0.9, φ1 = 0.1, original model. with λ1 = 0.9, φ1 = 0.1, threshold model. Fig. 9. Fig. 9a and 9b show the evolution of beliefs, for two diﬀerent models, of the variation of the Asch experiment where a second actor supports the truth. The red dashed and dotted line denote the private and expressed belief, respectively, of the test individual 1 (i.e. y1(t) andy ˆ1(t)). The blue line is the belief of the unanimous confederate group, who express a belief of yˆi(t) = 0.

1 American Statistical Association, vol. 69, no. 345, pp. 118– Unanimous Group 121, 1974. Test Individual Private Belief 0.8 Test Individual Expressed Belief [22] R. Hegselmann and U. Krause, “Opinion dynamics and bounded conﬁdence models, analysis, and simulation,”

0.6 Journal of Artiﬁcial Societies and Social Simulation, vol. 5, no. 3, 2002.

0.4 [23] W. Su, G. Chen, and Y. Hong, “Noise leads to quasi-consensus of hegselmann–krause opinion dynamics,” 0.2 Automatica, vol. 85, pp. 448 – 454, 2017. [24] P. Dandekar, A. Goel, and D. T. Lee, “Biased assimilation, 0 homophily, and the dynamics of polarization,” Proceedings 2 4 6 8 10 12 14 of the National Academy of Sciences, vol. 110, no. 15, pp. Time Step, t 5791–5796, 2013. Fig. 10. Asch’s second experiment: An individual with [25] M. Mäs,A. Flache, and J. A. Kitts, “Cultural Integration and Differentiation in Groups and Organizations,” in Perspectives λ1 = 0.9, φ1 = 0.1, threshold model, with τi = 0.6 on Culture and Agent-based Simulations. Springer, 2014, pp. 71–90. [20] J. R. P. French Jr, “A Formal Theory of Social Power,” [26] C. Altafini, “Consensus Problems on Networks with Psychological Review, vol. 63, no. 3, pp. 181–194, 1956. Antagonistic Interactions,” IEEE Transactions on Automatic [21] M. H. DeGroot, “Reaching a Consensus,” Journal of the Control, vol. 58, no. 4, pp. 935–946, 2013.

13 [27] A. Proskurnikov, A. Matveev, and M. Cao, “Opinion matrix A ≥ 0 is primitive if there exists k ∈ N such that dynamics in social networks with hostile camps: Consensus Ak > 0 [34, Definition 1.12]. A graph G[A] is strongly vs. polarization,” IEEE Transaction on Automatic Control, connected and aperiodic if and only if A is primitive, i.e. vol. 61, no. 6, pp. 1524–1536, 2016. k ∃k ∈ N such that A is a positive matrix [34, Proposition [28] N. E. Friedkin and E. C. Johnsen, “Social Influence and th n Opinions,” Journal of Mathematical Sociology, vol. 15, no. 1.35]. We denote the i canonical base unit vector of R n×n 3-4, pp. 193–206, 1990. as ei. The spectral radius of a matrix A ∈ R is given [29] P. Duggins, “A Psychologically-Motivated Model of Opinion by ρ(A). Change with Applications to American Politics,” Journal of Artificial Societies and Social Simulation, vol. 20, no. 1, pp. Lemma 1 If A ∈ Rn×n is row-substochastic and irre- 1–13, 2017. ducible, then ρ(A) < 1. [30] N. E. Friedkin and E. C. Johnsen, Social Influence Network Theory: A Sociological Examination of Small Group Dynamics. Cambridge University Press, 2011, vol. 33. Proof: This lemma is an immediate consequence of [43, [31] N. E. Friedkin and F. Bullo, “How truth wins in opinion Lemma 2.8]. 2 dynamics along issue sequences,” Proceedings of the National Academy of Sciences, vol. 114, no. 43, pp. 11 380–11 385, 2017. A.1 Performance Function and Ergodicity Coefficient [32] N. E. Friedkin, P. Jia, and F. Bullo, “A Theory of the Evolution of Social Power: Natural Trajectories of In order to analyse the disagreement among the opinions Interpersonal Influence Systems along Issue Sequences,” at steady state, we introduce a performance function and Sociological Science, vol. 3, pp. 444–472, 2016. n a coefficient of ergodicity. For a vector x ∈ R , define [33] C. D. Godsil and G. Royle, Algebraic Graph Theory. the performance function V (x): Rn 7→ R as Springer: New York, 2001, vol. 207. [34] F. Bullo, J. Cortes, and S. Martinez, Distributed Control of V (x) = max xi − min xj, (A.1) Robotic Networks. Princeton University Press, 2009. i∈{1,...,n} j∈{1,...,n} [35] T. Kuran, Private Truths, Public Lies: The Social Consequences of Preference Falsification. Harvard University Press, 1997. In context, V (y) measures the “level of disagreement” in the vector of opinions y(t), and consensus of opinions, [36] P. Jia, A. MirTabatabaei, N. E. Friedkin, and F. Bullo, “Opinion Dynamics and the Evolution of Social Power in i.e. y(t) = α1n, α ∈ R, is reached if and only if V (y(t)) = Influence Networks,” SIAM Review, vol. 57, no. 3, pp. 367– 0. Next consider the following coefficient of ergodicity, 397, 2015. τ(A) for a row-stochastic matrix A ∈ Rn×n, defined [44] [37] M. Ye, J. Liu, B. D. O. Anderson, C. Yu, and T. Ba¸sar, as “Evolution of Social Power in Social Networks with Dynamic n Topology,” IEEE Transaction on Automatic Control, vol. 63, X no. 11, pp. 3793–3808, Nov. 2019. τ (A) = 1 − min min{ais, ajs}. (A.2) i,j∈{1,...,n} [38] R. L. Gorden, “Interaction Between Attitude and the s=1 Definition of the Situation in the Expression of Opinion,” American Sociological Review, vol. 17, no. 1, pp. 50–58, 1952. This coefficient of ergodicity satisfies 0 ≤ τ(A) ≤ 1, and [39] F. Merei, “Group Leadership and Institutionalization,” > τ(A) = 0 if and only if A = 1nz for some z ≥ 0. Human Relations, vol. 2, no. 1, pp. 23–39, 1949. Importantly, there holds τ(A) < 1 if A > 0. Also, there [40] L. Festinger, “Informal Social Communication,” holds V (Ax) ≤ τ(A)V (x) (see [44]) Psychological Review, vol. 57, no. 5, p. 271, 1950. [41] S. Schachter, “Deviation, Rejection, and Communication,” The Journal of Abnormal and Social Psychology, vol. 46, A.2 Supporting Lemmas no. 2, pp. 190–207, 1951. [42] S. E. Asch, “Opinions and Social Pressure,” Scientific American, vol. 193, no. 5, pp. 31–35, 1955. Two lemmas are introduced to establish several proper- −1 [43] R. S. Varga, Matrix Iterative Analysis. Springer Science & ties of P and (I2n − P ) , which will be used to help Business Media, 2009, vol. 27. prove the main results. [44] E. Seneta, Non-negative Matrices and Markov Chains. Springer Science & Business Media, 2006. Lemma 2 Suppose that Assumption 1 holds. Then, P [45] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and given in Eq. (5) is nonnegative, the graph G[P ] is strongly Formulas. Princeton University Press, 2009. connected and aperiodic, and there holds ρ(P ) < 1. [46] W. J. Rugh, Linear System Theory, 2nd ed. Prentice Hall, Upper Saddle River, New Jersey, 1996. Lemma 3 Suppose that Assumption 1 holds. With P given in Eq. (5), define Q as A Preliminaries " # " # Q Q In − P 11 −P 12 In this section, we record some definitions, and notations Q = 11 12 = . to be used in the proofs of the main results. A square Q21 Q22 −P 21 In − P 22

14 −1 Then, Q11, Q22 are nonsingular, and Q > 0 is connected and aperiodic. Since G1 and G2 are both, separately, strongly connected, then if there exists 1) an " # edge from any node in V1 to any node V2, and 2) an edge −1 AB Q = , (A.3) from any node in V2 to any node in V1, one can conclude CD that the graph G[P ] is strongly connected. It suﬃces to show that E12 6= ∅ and E21 6= ∅. Since P 21 = Φ −1 −1 has strictly positive diagonal entries, this proves that where A = (Q11 − Q12Q22 Q21) , D = (Q22 − −1 −1 −1 −1 E12 6= ∅. From the fact that In − Φ has strictly positive Q21Q11 Q12) , B = −Q11 Q12D, C = −Q22 Q21A. −1 diagonal entries, and because Wc is irreducible, it fol- Moreover, R = A(In − Λ) and S = −Q22 Q21 are invertible, positive row-stochastic matrices. lows that P 12 = ΛWc (In − Φ) M 6= 0n×n. This shows that E21 6= ∅.

B Proofs It has therefore been proved that G[P ] is strongly connected and aperiodic, which also proves that P is B.1 Proof of Lemma 2 irreducible. Since λi < 1 ∀ i, P is row-substochastic, Lemma 1 establishes that ρ(P ) < 1. This completes the First, we verify that P ≥ 0 by using the fact that W , proof. Λ, In − Φ, M are all nonnegative. Next, observe that B.2 Proof of Lemma 3 " #" # " # Λ(Wf + WcΦ) ΛWc (In − Φ) M 1n Λ1n = Φ (I − Φ) M 1 1 Lemma 2 showed that G[P ] is strongly connected and n n n aperiodic, which implies that P is primitive. Since −1 −1 Q = (I2n − P ) ans ρ(P ) < 1, the Neumann series because M and W = W + W are row-stochastic. −1 P∞ k f c yields Q = k=0 P > 0. Next, it will be shown −1 Q11, Q22 and D = Q11 − Q12Q22 Q21 are all invert- Notice that the graph G[P ] = (V, E[P ], P ) has ible, which will allow Q−1 to be expressed in the form 2n nodes, with V = {1,..., 2n}. The node subset of Eq. (A.3) by use of [45, Proposition 2.8.7, pg. 108– V1 = {v1, . . . , vn} contains node vi which is associ- 109]. Under Assumption 1, G1[P 11] and G2[P 22] are ated with individual i’s private opinion yi, i ∈ I. The both strongly connected and aperiodic; Lemma 1 states node subset V2 = {vn+1, . . . , v2n} contains node vn+i that ρ(P ), ρ(P ) < 1. Since Q = I − P and which is associated with individual i’s expressed opin- 11 22 11 n 11 Q22 = In − P 22, the same method as above can be iony ˆi, i ∈ I. Deﬁne the following two subgraphs; used to prove that Q , Q are invertible, and satisfy G = (V , E[P ], P ) and G = (V , E[P ], P ). The 11 22 1 1 11 11 2 2 22 22 Q−1, Q−1 > 0. edge set of G[P ] can be divided as follows 11 22

" # " # In order to prove that D is invertible, we ﬁrst establish P 11 0n×n 0n×n P 12 −1 −1 E = E , E = E , some properties of S = −Q22 Q21. Since Q22 > 0, it fol- 11 12 lows from the fact that Φ = diag(φ ) is a positive diago- 0n×n 0n×n 0n×n 0n×n i −1 " # " # nal matrix, that S = Q22 Φ > 0. To prove that S is row- 0n×n 0n×n 0n×n 0n×n stochastic, ﬁrst note that det(S) = det(Q−1) det(Φ) 6= E = E , E = E , 22 21 22 0 (we have φ ∈ (0, 1), ∀ i ⇒ det(Φ) 6= 0). Since P 21 0n×n 0n×n P 22 i (AB)−1 = B−1A−1, observe that

In other words, E11 contains only edges between nodes −1 −1 −1 in V1 and E22 contains only edges between nodes in V2. S = Φ − Φ (In − Φ)M . (B.1) The edge set E12 contains only edges from nodes in V2 to nodes in V1, while the edge set E21 contains only edges −1 From Eq. (B.1), verify that S 1n = 1n, which implies from nodes in V1 to nodes in V2. Clearly E[P ] = E11 ∪ −1 SS 1n = S1n ⇔ S1n = 1n, i.e. S is row-stochastic. E12 ∪E21 ∪E22. It will now be shown that G[P ] is strongly connected and aperiodic, implies that P is primitive. We now turn to proving that D is invertible. Notice that Since the diagonal entries of Λ, Φ are strictly posi- S, −Q12 = P 12, and Λ(Wf + WcΦ) are all nonnegative. We write D = In − U where U = P 11 + P 12S ≥ 0. tive, it is obvious that P 11 = Λ(Wf + WcΦ) ∼ W . Because G[W ] is strongly connected and aperiodic, it Observe that U1n = P 111n + ΛWc (In − Φ) 1n = follows that G1 is strongly connected and aperiodic. Λ1n because (Wc + Wf)1n = 1n. In other words, the th Similarly, the edges of G2 are E[P 22]. Because In − Φ i row of U sums to λi < 1 (see Assumption 1), which has strictly positive diagonal entries, one concludes that implies that kUk∞ < 1 ⇒ ρ(U) < 1. Because it was P 22 = (In − Φ) M ∼ G[M] ∼ G[W ], i.e. G2 is strongly shown in the proof of Lemma 2 that G[P 11] is strongly

15 −1 connected and aperiodic, it is straightforward to show y(0) = βR 1n = β1n. In other words, a consensus of ∗ that G[U] is also strongly connected and aperiodic. It the final private opinions, y = β1n, occurs if and only follows that U is primitive, which implies that D−1 > 0 if the initial private opinions are at a consensus. Recall- ∞ from the Neumann series D−1 = P U k. Thus, R = ing the theorem hypothesis that y(0) 6= α1n, for some k=0 ∗ −1 α ∈ R, it follows that y is not at a consensus. Thus, D (In − Λ) > 0, because In − Λ is a positive diagonal ∗ ∗ matrix. Finally, one can verify that R is row-stochastic ymin 6= ymax as claimed. with the following computation: D1n = (In − U)1n = −1 −1 (In − Λ)1n ⇒ R1n = D (In − Λ)1n = D D1n = Next, the inequalities Eq. (9a) and Eq. (9b) are proved. 1n. This completes the proof. Since R, S > 0 are row-stochastic, τ(R), τ(S) < 1. Be- > n cause R is invertible, R 6= 1nz for some z ∈ R . This B.3 Proof of Theorem 1 and Corollary 1 means that τ(R) > 0 (see below Eq. (A.2)). Similarly, one can prove that τ(S) > 0. In the above paragraph, Proof of Theorem 1: Lemma 2 established that the time- it was shown that if there is no consensus of the ini- invariant matrix P satisfies ρ(P ) < 1. Standard lin- tial private opinions, then V (y∗ = Ry(0)) > 0. By re- ear systems theory [46] is used to conclude that the lin- calling that V (Ax) ≤ τ(A)V (x) (see Appendix A.1) ear, time-invariant system Eq. (4), with constant input and the above facts, we conclude that 0 < V (y∗ = > >> Ry(0)) < V (y(0)), which establishes the left hand in- ((In − Λ)y(0)) , 0n , converges exponentially fast to equality of Eq. (9a) and Eq. (9b). Following steps similar to the above, but which are omitted, one can show ∗ ∗ ∗ " # " ∗# " # that 0 < V (yˆ = Sy ) < V (y ), which establishes the limt→∞ y(t) y (In − Λ)y(0) = (I − P )−1 right hand inequality of Eq. (9a) and Eq. (9b), and also , ∗ 2n ∗ ∗ limt→∞ yˆ(t) yˆ 0n establishes thaty ˆmin 6=y ˆmax. " # −1 (In − Λ)y(0) = Q . Last, it remains to prove that for generic initial con- 0n ∗ ∗ ∗ ∗ ∗ ditions, yi 6=y î . Observe thaty î = yi ⇒ yâvg = (B.2) > ∗ ∗ ∗ 1n yˆ /n. Thus,y î = yi for m specific individuals if −1 and only if there are m independent equations satisfy- Having calculated the form of Q in Eq. (A.3), it is 1 > ∗ ∗ ∗ ∗ ing (ei − n 1n) y = 0. This implies thaty ˆ must lie straightforward to verify that y = Ry(0) and yˆ = n ∗ in an n − m-dimensional subspace of R , denoted as D. SRy(0) = Sy . Here, the definitions of R and S are From Theorem 1, one has y∗ = RSy(0). It follows that given in Lemma 3, which also proved their positivity and ∗ ∗ yî = yi for m specific individuals only if y(0) belongs to row-stochasticity. This completes the proof. 2 the inverse image (by RS) of D, and the inverse image has dimension n − m because R, S are invertible. This Proof of Corollary 1: The assumption that Λ = In completes the proof. 2 implies that P is nonnegative and row-stochastic. The proof of Lemma 2 established that G[P ] is strongly connected and aperiodic, and this remains unchanged when Λ = In. Standard results on the DeGroot model [1] then imply that consensus is achieved exponentially fast, i.e. B.5 Proof of Corollary 2 limt→∞ y(t) = yˆ(t) = α1n for some α ∈ R. 2 Recall the definition of V in Appendix A.1. From The- B.4 Proof of Theorem 2 orem 1, one has that V (yˆ∗) = V (Sy∗) ≤ τ(S)V (y∗), which implies that there holds V (yˆ∗)/τ(S) ≤ V (y∗). If y(0) = α1n, for some α ∈ R (i.e. the initial private ∗ ∗ Thus, Eq. (10) can be proved by showing that τ(S) ≤ opinions are at a consensus), then y = yˆ = α1n be- κ(φ). Note that since global public opiniony âvg is used, cause R and S are row-stochastic. In what follows, it will −1 > −1 M in Eq. (5) becomes M = n 1n1n . Recall that Q22 be proved that if the initial private opinions are not at −1 P∞ a consensus, then there is disagreement at steady state. can be expressed as Q22 = k=0 P 22. Since P 22 = −1 > n (In − Φ) 1n1n and Q21 = −Φ, we obtain S = > k ∗ ∗ ∗ P∞ 1n1n First, we establish ymin 6= ymax. Note that V (y ) = 0 Φ + H where H (In − Φ) Φ > 0. ∗ , k=1 n if and only if y = β1n, for some β ∈ R. Next, observe ∗ that y = β1n if and only if Ry(0) = β1n, for some β ∈ R. Note that R is invertible, because it is the prod- Let a = mini,j aij denote the smallest element of a ma- uct of two invertible matrices (see Lemma 3). Moreover, trix A, and observe that s = h because S = Φ + H has th because R is row-stochastic, there holds R1n = 1n ⇔ the same offdiagonal entries as H, and the i diagonal −1 −1 −1 R R1n = R 1n ⇔ R 1n = 1n. Thus, premulti- entry of S is greater than that of H by φi > 0. Since −1 plying by R on both sides of Ry(0) = β1n yields S > 0, Eq. (A.2) yields τ(S) ≤ 1 − ns ≤ 1 − nh. We

16 now analyse H. For any A ∈ Rn×n, there holds we obtain

n−1 (I − Φ) 1 1>A ∂S(φ) ∂S−1(φ) n n n = −S(φ) S(φ). (B.5)  Pn Pn  ∂φi ∂φi (1 − φ1) j=1 a1j ··· (1 − φ1) j=1 anj 1    . .. .  Below, the argument φ will be dropped from S(φ) and =  . . .  . n   −1 ∂Φ−1 Pn Pn S (φ) when there is no confusion. Note that ∂φ = (1 − φn) a1j ··· (1 − φn) anj i j=1 j=1 −2 > −φi eiei . Using Eq. (B.1) and Eq. (B.5), one obtains ∂S(φ) −2 > > > th th = φ Sei e − m S, where m is the i row By recursion, we obtain that the (i, j) entry of ∂φi i i i i > of M. It suffices to prove the corollary claim, if it can be 1n1n k (1−φi) [(In − Φ) ] is given by k γk, where > > n n shown that the row vector ei − mi S has a strictly th n n n positive i entry and all other entries are strictly nega- h X X X i tive. This is because S > 0 ⇒ Se > 0. We achieve this γ = ··· (1 − φ )(1 − φ ) ··· (1 − φ ) i k p1 p2 pk−1 by showing that p1=1 p2=1 pk−1=1 | {z } > > k-1 summation terms (ei − mi )Sei > 0 (B.6) > > Pn Pn (ei − mi )Sej < 0 , ∀ j 6= i. (B.7) This is obtained by recursively using i=1 j=1 aibj = Pn Pn Pn Pn i=1 ai j=1 bj = i=1 ai j=1 bj . Next, define Observe the following useful quantity: 1 1> Zk = [(I − Φ) n n ]kΦ. From the above, one can show n n > −1 > −1 −1 th k ei S = ei Φ − Φ (In − Φ)M that the (i, j) element of Z is given by zij(k) = 1 = φ−1e> − (φ−1 − 1)m>. (B.8) nk (1 − φi)φjγk. It follows that the smallest element of i i i i Zk, denoted by z(k), is bounded as follows Postmultiplying by S on both sides of Eq. (B.8) yields > −1 > −1 > 1 ei = φi ei S −(φi −1)mi S. Rearranging this yields z(k) ≥ (1 − φ )φ γ . (B.3) nk max min k > > > ei S = φiei + (1 − φi)mi S (B.9) Observe that 1 − φ ≥ 1 − φ , ∀ i ⇒ Pn 1 − φ ≥ > −1 > > i max a=1 a mi S = (1 − φi) ei S − φiei . (B.10) n(1 − φmax). It follows that By using the equality of Eq. (B.9) for substitution, ob- 1 z(k) ≥ φ (1 − φ )k. (B.4) serve that the left hand side of Eq. (B.7) is n min max > > P∞ k P∞ (ei S − mi S)ej Since H = k=1 Z , there holds h ≥ k=1 z(k) ≥ > > > > −1 = φiei + (1 − φi)mi S − mi S ej = −φimi Sej, φmin(1 − φmax)(nφmax) . We can obtain this by noting that for any r ∈ (−1, 1), the geometric series is ∞ k 1 ∞ k 1 > > P P because e ej = 0 for any j 6= i. Note that m Sej > 0 k=0 r = 1−r ⇔ k=1 r = 1−r − 1, and 0 < 1 − i i > > φmax < 1. From τ(S), τ(H) ≤ 1 − nh, and the above because M being irreducible implies mi 6= 0n . Thus, φ > arguments, we obtain τ(S) ≤ 1 − nh = 1 − min (1 − −φimi Sej/n < 0, which proves Eq. (B.7). Substituting φmax φ ) = κ(φ) as in the corollary statement. Since 0 < the equality in Eq. (B.10), observe that the left hand max side of Eq. (B.6) is φmin/φmax < 1 and 0 < 1 − φmax < 1, one has 0 < κ(φ) < 1 and thus τ(S) ≤ κ(φ) holds ∀ φi ∈ (0, 1). > > > 1 > > (ei S − mi S)ei = ei Sei − ei Sei − φiei ei Key to the proof is that the coefficient of ergodicity for 1 − φi S is bounded from above as τ(S) ≤ κ(φ). The tightness φi > = 1 − ei Sei > 0. (B.11) of τ(S) ≤ κ(φ) depends on φmin/φmax: this can be con- 1 − φi cluded by examining the proof, and noting that the key inequalities in Eq. (B.3) and Eq. (B.4) involve φmin and The inequality is obtained by observing that 1) φi ∈ > φmax. If φmin/φmax = 1, then τ(S) = κ(φ). (0, 1) ⇒ φi/(1 − φi) > 0, and 2) 1 − ei Sei > 0 because > 0 < ei Sei = sii < 1. This proves Eq. (B.6). 2 B.6 Proof of Corollary 3

First, verify that S is invertible, and continuously dif- ferentiable, for all φi ∈ (0, 1). From [45, Fact 10.11.20]