Social diversity for reducing the impact of information cascades on social learning

So cial diversit y fo r reducing the impact of info rmation cascades on so cial lea rning F ernando Rosas 1 , 2 , Kw ang-Cheng Chen 3 and Deniz G¨ und¨ uz 2 1 Centre of Complexit y Science and Depa rtment of Mathematics, Imp erial College London, UK 2 Depa rtment of Electrical and Electronic Engineering, Imp erial College London, UK 3 Depa rtment of Electrical Engineering, Universit y of South Florida, USA Abstract Collective b ehavio r in online so cial media and netw o rks is known to be capable of generating non-intuitive dynamics asso ciated with cro wd wisdom and herd b ehaviour. Even though these topics have b een well-studied in so cial science, the explosive growth of Internet computing and e-commerce mak es urgent to understand their effects within the digital so ciet y . In this w o rk w e explo re ho w the sto chasticit y intro duced by so cial diversit y can help agents involved in a inference process to improve their collective p erfo rmance. Our results show ho w so cial diversit y can reduce the undesirable effects of info rmation cascades, in which rational agents choose to igno re p ersonal knowledge in order to follow a p redominant so cial b ehaviour. Situations where so cial diversit y is never desirable are also distinguished, and consequences of these findings for engineering and so cial scenarios a re discussed. 1 In tro duction The high interconnectedness enabled by communication technologies and online media is p rogres- sively increasing the complexit y of our aggregated so cial behaviour [1]. In fact, these complex dynamics w ere dramatically illustrated by the failure of our prediction to ols in the fo recast of recent p olitical events, including the Brexit referendum and the latest US p residential election. A k ey op en challenge is to clarify how the la rge amount of information that is constantly exchanged among individuals affects their decisions. F ascinating dynamics tak e place when so cial agents engage in sequencial decision-making. F or example, most p eople no wada ys use the Internet to check other p eople’s recommendations prio r to mak e decisions, which enable more info rmed decisions thanks to the inclusion of evidence from pre- vious exp eriences. Subsequent decisions are, how ever, heavily influenced b y earlier agents, allowing misinfo rmation o r fake news to be reinfo rced and sp read across the so cial netw o rk. These non-trivial so cial learning dynamics are known to pla y a critical role in a numb er of key s o cial phenomena, e.g., in the adoption o r rejection of new technology , and in the fo rmation of p olitical opinions [2, 3]. Mo reover, so cial lea rning also plays a k ey role in the context of e-commerce and digital so ciety , e.g., in recommendation systems of online sto res where users access opinions of p revious customers while cho osing their products [4, 5]. This is also the case in the emergence of viral media contents in va rious Internet p o rtals, which are based on sequential actions of like or dislike. A deep understanding of so cial lea rning dynamics is crucial for enabling robust platfo rm design against fake news and data falsification, which is an urgent need in our mo dern net w ork ed so ciet y . As a matter of fact, digital misinformation w as listed b y the W o rld Economic F orum (WEF) as one of the main threats to our mo dern so ciety [6]. 1 So cial lea rning have b een thoughtfully studied since the 90’s b y researchers from economics and so cial sciences [7 – 9] (for mo dern reviews see [2, 3]). These studies have shown that so cial learning is driven by t wo comp eting mechanisms. In one hand, the well-kno wn cro wd wisdom improve the decision-making capabilities of agents within large netw o rks, as more information b ecomes available to latter agents in the decision sequence. The accumulation of so cial exp erience can, on the other hand, overload agents and generate info rmation cascades , which pushes them to igno re their p rivate kno wledge and to adopt a predominant so cial b ehaviour. Interestingly , it has b een shown that the combination of these t w o mechanisms can serve to p rovide net wo rk resilience against data falsification attacks [10, 11], p ointing out p romising possibilities fo r the design of resilient so cial lea rning platforms. Motivated by the b enefits that diversit y can provide in biological and so cial systems [12, 13], in this wo rk w e study how so cial diversity affects the lea rning rate in a so cial lea rning scenario. Fo r this, w e consider a net wo rk of rational agents that have diverse p references and p rio r info rmation, having some simila rities to the wo rks reported in [14, 15]. Using a communication theoretical interp retation of this scenario, we sho w that so cial diversit y is equivalent to additive noise in a communication channel —which one w ould exp ect to be detrimental fo r the lea rning process. Surp risingly , our findings sho w that so cial diversity can help to avoid info rmation cascades, intro ducing imp ortant imp rovements in the asymptotic learning p erfo rmance. The rest of this a rticle is structured as follo ws. Sections 2 intro duces the considered so cial lea rning scena rio, and develops our definition of so cial diversity . Section 3 defines information cascades, and characterize theoretically their b ehaviour with resp ect to so cial diversity . Section 4 p resents numerical evaluations that verify the theoretical results, and finally Section 5 summarizes our main conclusions. Notation : upp ercase letters X are used to denote random variables and lo wercase x realiza- tions of them, while boldface letters X and x represent vectors. Also, P w { X = x | Y = y } : = P { X = x | Y = y , W = w } is used as a shorthand notation. 2 So cial learning mo del 2.1 Preliminaries and basic assumptions Let us consider a so cial netw o rk comp osed b y N agents, who a re engaged in a decision-making p ro cess. In this process each agent need to mak e a decision b etw een tw o options 1 , which could co rresp ond to a choice b etw een tw o restaurants, tw o brands, or tw o p olitical pa rties. It is assumed that decisions o ccur sequentially , and a re lab eled acco rding to the o rder in which they tak e place. The decision of the n -th agent, denoted as X n ∈ { 0 , 1 } , is based on t w o sources of info rmation (see Figure 1): a private signal S n ∈ S , which is a continuous or discrete random va riable that rep resents p ersonal information that the n -th agent p ossesses, and so cial information given b y the decisions of the p revious agents, denoted b y X n − 1 : = ( X 1 , . . . , X n − 1 ) ∈ { 0 , 1 } n − 1 . All the agents a re assumed to have equivalent observation capabilities, and therefore the p rivate signals S n a re identically distributed. These signals are affected by environment conditions, which fo r simplicit y are represented by a bina ry variable W . F or the sake of tractabilit y , we follow the existent literature in assuming that the private signals S n a re conditionally indep endent given W , leaving other cases for future w ork. The corresponding conditional p robabilit y distributions of S n given the event { W = w } are denoted b y µ w . We further assume that no realization of S n is capable of completely determining W , which is equivalent to the measure theoretic notion of absolute continuit y b et ween µ 0 and µ 1 [16]. As a consequence of this assumption, the log-likelihoo d ratio 1 Although generalizations for more than tw o options are p ossible, we focus in the case of binary decisions for simplifying the presentation. 2 Figure 1: A social learning scenario, where an agen t needs to make a decision ( π n ) based on p ersonal information coming from a priv ate signal ( S n ) and so cial information ( X n − 1 ) coming from a so cial net work. of these tw o distributions µ 1 and µ 0 is well-defined and given b y the logarithm of the co rresp onding Radon-Nik o dym derivative Λ S ( s ) = log dµ 1 dµ 0 ( s ) 2 . A strategy is a rule fo r generating a decision X n based on S n = s and X n − 1 , i.e. a collection of deterministic o r random functions π n : S × { 0 , 1 } n − 1 → { 0 , 1 } such that X n = π n ( S n , X n − 1 ) . 2.2 Ba y esian strategy , agen ts’ preferences and prior information Let us assume that the p references of the n -th agent are enco ded in an utility function u n ( x, w ) , which determines the pa yoff that the agent receives when making the decision X n = x under the condition { W = w } . We consider rational agents that follow a Ba yesian strategy , which seeks to maximize their average pay off given b y E  u ( π n ( S n , X n − 1 ) , W )  , with E {·} b eing the exp ected value op erator. It has b een shown that the Ba yesian strategy for the n -th agent can b e expressed succinctly as [4] P  W = 1 | S n , X n − 1  P  W = 0 | S n , X n − 1  X n =0 ≶ X n =1 e ν n , (1) where ν n = log u n (0 , 0) − u n (0 , 1) u n (1 , 1) − u n (1 , 0) reflects the effect of the cost function. F or considering agents with diverse p references, w e assume that ν i a re indep endent and identically distributed (i.i.d.) random va riables. Let us further consider the case where the agents have no absolute kno wledge ab out the p rior distribution of W . Note that b ecause W is binary , its distribution is completely determined by the value of P { W = 1 } . Follo wing the framewo rk of Bay esian inference [17], let us consider θ n ∈ [0 , 1] to b e a collection of i.i.d. random va riables following a distribution f θ ( θ ) that reflects the state of kno wledge of the agents ab out P { W = 1 } . In pa rticular, if the agent has complete kno wledge then f θ ( θ ) is a delta centered in the true value of P { W = 1 } and hence θ n = P { W = 1 } for all n , while if agents has no info rmation then f θ ( θ ) corresponds to an uniform distribution over [0 , 1] . Noting that X n − 1 dep ends only on ( S 1 , . . . , S n − 1 ) , and therefo re is conditionally indep endent of S n , a direct application of the Bay es rule on P  W = 1 | S n , X n − 1  and P  W = 0 | S n , X n − 1  2 When S n tak es a discrete num b er of v alues then dµ 1 dµ 0 ( s ) = P { S n = s | W =1 } P { S n = s | W =0 } , while if S n is a contin uous random v ariable with conditional p.d.f. p ( s | w ) then dµ 1 dµ 0 ( s ) = p ( s | w =1) p ( s | w =0) . 3 sho ws that (1) can b e re-written as Λ S ( S n ) + Λ X n − 1 ( X n − 1 ) X n =0 ≶ X n =1 ν n + log θ n 1 − θ n , (2) where Λ S ( S n ) and Λ X n − 1 ( X n − 1 ) are the log-likelihoo d ratios of S n and X n − 1 , resp ectively . Note that an efficient metho d for computing τ n ( X n − 1 ) has b een rep orted in [10]. 2.3 Comm unication theoretic in terpretation By using an adequate decision labeling, one can consider the event { X n = W } to b e mo re desirable than { X n 6 = W } , or equivalently , that u n (1 , 1) ≥ u n (1 , 0) and u n (0 , 0) ≥ u n (0 , 1) . The Bay esian strategy is, hence, to cho ose X n as simila r to W as p ossible using the information provided b y S n and X n − 1 . Therefo re, the decisions π n ( S n , X n − 1 ) = X n can b e considered to b e noisy estimations of W . T o further explore this p ersp ective, let us re-formulate (2) as Λ S ( S n ) + ξ n X n =0 ≶ X n =1 τ n ( X n − 1 ) , (3) where ξ n := log(1 − θ n ) /θ n − ν n and τ n ( X n − 1 ) := − Λ X n − 1 ( X n − 1 ) . The ab ove can b e understo o d as a classic signal deco der within communication theory [4, Section IV], where Λ S ( S n ) is the decision signal and ξ n is additive noise. Mo reover, τ n ( X n − 1 is a decision threshold that establishes the deco ding rule based on a Vono roi tessellation that divides R in t wo semi-op en intervals given by ( −∞ , τ n ( X n − 1 )) and ( τ n ( X n − 1 ) , ∞ ) . 3 Av oiding information cascades via noise 3.1 Lo cal and global information cascades In general, the decision π n ( S n , X n − 1 ) is made based in complementary evidence provided by b oth X n − 1 and S n . The n -th agent is said to fall into a lo cal information cascade when the information convey ed by S n is not included in the decision-making process due to a dominant influence of X n − 1 . The term “lo cal” is used to emphasize that this event is related to the data fusion taking place at an individual agent. The notion of lo cal information cascade is fo rmalized in the follo wing definition, which is based on the notion of conditional mutual information [18], denoted as I ( · ; ·|· ) . Definition 1. The so cial information x n − 1 c ∈ { 0 , 1 } n − 1 gener ates a local information cascade for the n -th agent if I ( π n ; S n | X n − 1 = x n − 1 c ) = 0 . The ab ove condition summarizes t wo p ossibilities: either π n ( s, x n − 1 c ) is constant fo r all values of s ∈ S , o r there is still variabilit y but this variabilit y is conditionally indep endent of S n (e.g. in the case of sto chastic strategies —not considered in this wo rk). In b oth cases, the ab ove definition highlights the fact that the decision π n contains no unique info rmation 3 coming from S n when a lo cal cascade takes place . Next w e define global info rmation cascades , which a re avalanches of lo cal information cascades that affect all the agents after their ignition. Definition 2. The so cial information ve ctor x n − 1 c ∈ { 0 , 1 } n − 1 triggers a global information cascade if I ( π m : S m | X n − 1 = x n − 1 c ) = 0 holds for al l m ≥ n . The relationship b et ween lo cal and global info rmation cascades is explored in the next section (c.f. Prop osition 1). 3 F or a rigorous definition of unique information in Mark o v chains c.f. [19]. 4 3.2 The effect of so cial div ersit y o v er information cascades Let us first intro duce F w ( z ) = P w { Λ S ( S n ) + ξ n ≤ z } as a sho rthand notation for the cumulative distribution function of Λ S ( S n ) + ξ n conditioned on the event { W = w } . Note that, thank to the fact that Λ S ( S 1 ) and ξ 1 a re indep endent random va riables, one can compute F w ( · ) as the convolution of their densit y functions. Lemma 1. The c onditional statistics of π n given X n − 1 ar e define d by P w  π n = 0 | X n − 1 = x n − 1  = F w ( τ n ( x n − 1 )) . (4) Pr o of. A direct calculation shows that P w { π 1 ( S 1 ) = 0 } = P w { Λ S ( S 1 ) + ξ 1 < 0 } = F w (0) . F ollowing a similar rationale, one can find that P w  π n = 0 | X n − 1 = x n − 1  = Z S P w  π n ( s, x n − 1 ) = 0 | S n = s  µ w ( s )d s = Z S 1  π n ( s, x n − 1 ) = 0  µ w ( s )d s = P w  Λ S ( s ) + ξ n < τ n ( x n − 1 )  = F w ( τ n ( x n − 1 )) . Ab o v e, the first equality is a consequence of the fact that S n is conditionally indep endent of X n − 1 giv en W = w , while the second equalit y is a consequence that π n is a deterministic function of X n − 1 and S n , and hence b ecomes conditionally indep endent of W . Next, using Lemma 1, one can show that τ n is an effective summa ry of the information provided b y X n − 1 that is relevant fo r generating the decision π n . Lemma 2. The variables X n − 1 → τ n → π n form a Markov Chain, i.e. τ n is a sufficient statistic of X n − 1 for pr e dicting the de cision π n . Pr o of. Using (4), one can find that P w  π n = 0 | τ n , X n − 1  = F w ( τ n ) = P w { π n = 0 | τ n } , (5) and therefore the conditional indep endency of π n and X n − 1 giv en τ n is clear. W e no w p resent a p rop osition that cla rifies the relationship betw een lo cal and global information cascades. This result extends [4, Theorem 1] to the current scenario. Prop osition 1. Each lo c al information c asc ade triggers a glob al information c asc ade over the so cial network. Pr o of. Letus first note that τ n +1 ( X n ) − τ n ( X n − 1 ) =Λ X n − 1 ( X n − 1 ) − Λ X n ( X n ) = − Λ X n | X n − 1 ( X n | X n − 1 ) , (6) where the conditional log-lik eliho o d is given by Λ X n | X n − 1 ( X n | X n − 1 ) = log P 1  X n | X n − 1  P 0  X n | X n − 1  . 5 Let us consider x n − 1 c ∈ { 0 , 1 } n − 1 suc h that it pro duce a lo cal cascade in the n -th no de. As Bay esian strategies are deterministic, lo cal information cascades corresp onds to the ev ents where π n is fully determined by X n − 1 , i.e. when the probability of the ev ent { π n = 0 | X n − 1 = x n − 1 c } = { X n = 0 | X n − 1 = x n − 1 c } is either 0 or 1. This, in turn, implies that Λ X n | X n − 1 ( X n | x n − 1 c ) = 0 almost surely , and therefore, conditioned on the ev ent { X n − 1 = x n − 1 c } one has that τ m ( X m ) = τ n ( x n − 1 c ) for all m ≥ n. (7) Finally , by using (4), one can show that P w  π m = 0 | X m , X n − 1 = x n − 1 c  = F w ( τ m ( X m )) = F w ( x n − 1 c ) (8) Therefore, P w { π m = 0 | X n − 1 = x n − 1 c , X n , . . . , X m } is also either zero or one, sho wing that the m -th agen t also is affected b y a lo cal information cascade. Let us now intro duce U s = ess sup Λ S ( S n ) , U ξ n = ess sup ξ n , L s = ess inf Λ S ( S n ) and L ξ n = ess inf ξ n as shorthand notations for the essential sup ermum and infimum of Λ S ( S n ) and ξ n 4 . In pa rticular, U s and L s co rresp ond to the signals within S that most strongly supp o rt the hyp othesis { W = 1 } and { W = 0 } , resp ectively . If one of these quantities diverge, this implies that there a re signals s ∈ S that p rovide overwhelming evidence in favour of one of the comp eting hyp otheses. On the other hand, if U s and L s a re b oth finite then the agents a re said to have b ounded beliefs [3]. Simila rly , when b oth U ξ and L ξ a re finite we say agents have b ounded diversit y , which implies that the diversity among p riors and cost functions is not to o high. Using this notions, we present the main result of this w o rk. Theorem 1. L o c al information c asc ades c annot take plac e when agents have either unb ounde d b eliefs or unb ounde d diversity. Pr o of. Note first that, due to the indep endency b etw een Λ S ( S n ) and ξ n , one has that U total : = ess sup { Λ S ( S n ) + ξ n } = U s + U ξ , (9) L total : = ess inf { Λ S ( S n ) + ξ n } = L s + L ξ . (10) F rom this, U total and L total are un b ounded if and only the agents ha ve unbounded b eliefs or un b ounded div ersity . F rom Lemma 1, it is clear that π n is fully determined by x n − 1 ∈ { 0 , 1 } n − 1 if and only if τ n ( x n − 1 ) is suc h that F w ( τ n ( x n − 1 )) is zero or 1 for w ∈ { 0 , 1 } . Because of the definition of F w , this happ ens whenever τ n ( X n − 1 ) / ∈ [ L total , U total ], pro ving the Prop osition. Info rmation cascades a re kno wn to degrade the learning process, p reventing the error rate of the lea rning process P { π n 6 = W } from converging to zero when the so cial netw o rk grows [4]. Therefore, Theo rem 1 reveals a non-intuitive value of social diversity , as it can safegua rd so cial learning from info rmation cascades. In this w ay , so cial diversit y can guarantee p erfect so cial learning to happ en asymptotically , even when agents have b ounded b eliefs and a re hence prone to herd b ehaviour [4]. Ho wever, this b enefit usually comes at the p rice of a slo w er convergence, which can b e detrimental fo r the first agents of the decision sequence. This trade-off is explored in the next section. 4 The essen tial suprem um is the smallest upp er bound of a random v ariable that holds almost surely , b eing the natural measure-theoretic extension of the notion of suprem um [20]. 6 4 Pro of of concept F or illustrating the findings p resented in Section 3, this section p resents results of simulations of a so cial net w ork follo wing the mo del presented in Section 2. We considered t wo scenarios: one where S n a re binary variables that follow a binna ry symmetric channel with P { S n 6 = w | W = w } = 1 / 4 , and other where S n given { W = w } are Gaussian variables N ( µ w , σ 2 ) with µ w = ( − 1) 1 − w and σ 2 = 4 . These tw o signal mo dels were cho osen b ecause it is kno wn that agent following binary signals a re strongly affected by information cascades, while agents follo wing Gaussian signals are not affected b y them (for further details ab out these scenarios c.f. [4, Section VI]). F or simpicit y , the so cial diversit y has b een mo deled considering ξ n to b e i.i.d. follo wing a Gaussian distribution N (0 , σ 2 ξ ) , and hence σ 2 ξ quantifies the “diverstiy strength” of the so cial net wo rk. Each scena rio was simulated 10 5 realizations, and the statistics of the lea rning error rate, defined as P { π n 6 = W } w ere computed afterw ards. In agreement with Theorem 1, results confirm that so cial learning p ro cesses can b e b enefited b y so cial diversit y . Figure 2a sho ws how the results of a collective inference ca rried out by agents driven by binary private signals achieve b etter p erformance asymptotically . How ever, for some values of so cial diversity the lea rning rate can b e rather slow, making social lea rning not useful for small so cial net wo rks. In all the studied cases it was seen that so cial diversity degrades the p erfo rmance of the first agents in the decision sequence; ho wever an adequate level of diversity can intro duce a fast learning rate. In contrast, as illustrated in Figure 2a for agents follo wing Gaussian signals, so cial diversity was found to b e alwa ys detrimental in cases where agents have unb ounded beliefs. This confirms the fact that the b enefits of so cial diversit y is to avoid info rmation cascades, which a re the main cause of p o or p erformance of so cial learning in large netw o rks [4]. 0 50 100 150 200 250 300 350 400 0 . 7 0 . 72 0 . 74 0 . 76 0 . 78 0 . 8 0 . 82 0 . 84 0 . 86 0 . 88 0 . 9 Agent Learning rate – P { π n = W } σ 2 ξ = 0 . 01 σ 2 ξ = 0 . 1 σ 2 ξ = 0 . 5 σ 2 ξ = 0 . 7 (a) Binary priv ate signals 0 50 100 150 200 250 300 350 400 0 . 7 0 . 75 0 . 8 0 . 85 0 . 9 0 . 95 1 Agent Learning rate – P { π n = W } σ 2 ξ = 0 . 01 σ 2 ξ = 0 . 1 σ 2 ξ = 0 . 5 σ 2 ξ = 0 . 7 (b) Gaussian priv ate signals Figure 2: So cial learning rate for agents following binary or Gaussian priv ate signals, under v arious lev els of so cial diversit y ( σ 2 ξ ). So cial net works that follo w binary signals are vulnerable to information cascades, and hence a non-zero so cial diversit y improv e their asymptotic learning rate. In contrast, so cial net works that follo w Gaussian signals are inm une to information cascades, and hence so cial div ersity ha ve a purely detrimen tal effect. The different effect that so cial diversit y has over agents lo cated at different p ositions in the inference process is further illustrated b y Figure 3. W e found that, fo r each agent, there exists an optimal level of so cial diversity that reduces the effect of info rmation cascades without intro ducing to o much noise. Agents lo cated in the first places of the decision sequence are alw ays affected negatively b y so cial diversit y , and hence for them is optimal to have σ 2 ξ = 0 . 7 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 76 0 . 78 0 . 8 0 . 82 0 . 84 0 . 86 0 . 88 0 . 9 Social diversit y σ 2 ξ Learning rate – P { π n = W } 10-th agent 100-th agent 400-th agent 800-th agent Figure 3: An optimal level of so cial div ersity exist that can improv e the so cial learning p erfor- mance of agents lo cated late in the inference pro cess. How ev er desirable, this b etter p erformance of late agen ts comes at the exp ense of a detrimenta l effect to the first agents. 5 Conclusion This pap er aims to undestand how so cial lea rning i s affected when it is pursued by a diverse p opu- lation. Our scena rio considered rational agents with heterogeneous preferences, as encoded b y their utilit y functions, and diverse prio r info rmation ab out the ta rget va riable. A communication theo retic analysis sho wed that this kind of so cial diversity is equivalent to additive noise in a communication channel. Ho wever, it was found that an unb ounded so cial diversit y prevent information cascades and, hence, intro duces imp ortant imp rovements into the asymptotic so cial learning rate that can b e achieved by a p opulation. So cial learning is, therefore, one of those rare cases where noise can imp rove the overall p erfo rmance. T o understand ho w can noise be b eneficial, let us point out that rational so cial agents maximize their individual p erfo rmance while ignoring the consequences of their actions on the aggregated b ehaviour. This selfish qualit y of the agent’s b ehavior makes their actions lo cally optimal while b eing globaly sub optimal. In this context, the heterogeneit y intro duced b y so cial diversit y makes the decisions of each agent less informative to others. This generates a reduced so cial pressure that, in turn, p revents info rmation cascades and herd b ehaviour, intro ducing great imp rovements in the asymptotic so cial learning p erformance. It is to b e noted that the b enefits of so cial diversity a re only exp erienced by agents that are p rone to info rmation cascades. Therefo re, so cial diversity is not b eneficial, e.g., for agents with unb ounded b eliefs. How ever, in most applications agent’s b eliefs a re b ounded, either b ecause their signals information content is limited or b ecause the signals themselves are b ounded. The latter is the case in most engineering applications, e.g. in the scenario studied in [10]. Finally , it is important to remark that so cial diversity p rovides b enefits to the latter agents in the decision sequence, while degrading the performance of the first agents. Therefo re, so cial diversit y might in general b e detrimental for the p erformance of so cial learning in small netw orks. Ac kno wledgemen ts F ernando Rosas is supp orted by the Europ ean Union’s H2020 resea rch and innovation programme, under the Ma rie Sk lo do wsk a-Curie grant agreement No. 702981. 8 References [1] Y. Ba r-Y am, “Complexity rising: From human b eings to human civilization, a complexity p rofile,” Encyclop edia of Life Supp ort Systems (EOLSS), UNESCO, EOLSS Publishers, Oxford, UK , 2002. [2] D. Easley and J. Kleinb erg, “Netw orks, cro wds, and ma rkets,” Cambridge Universit y Press , vol. 1, no. 2.1, pp. 2–1, 2010. [3] D. Acemoglu, M. A. Dahleh, I. Lob el, and A. Ozdaglar, “Bay esian lea rning in so cial net w o rks,” The Review of Economic Studies , vol. 78, no. 4, pp. 1201–1236, 2011. [4] F. Rosas, J.-H. Hsiao, and K.-C. Chen, “A technological p ersp ective on info rmation cascades via so cial learning,” IEEE Access , vol. 5, pp. 22 605–22 633, 2017. [5] J. Hsiao and K. C. Chen, “Steering information cascades in a so cial system by selective rewiring and incentive seeding,” in to b e included in 2016 IEEE International Conference on Communi- cations (ICC) , 2016. [6] L. How ell, “Digital wildfires in a hyp erconnected wo rld,” WEF Rep ort , 2013. [7] A. V. Banerjee, “A simple mo del of herd b ehavior,” The Quarterly Journal of Economics , pp. 797–817, 1992. [8] S. Bikhchandani, D. Hirshleifer, and I. Welch, “A theo ry of fads, fashion, custom, and cultural change as info rmational cascades,” Journal of p olitical Economy , pp. 992–1026, 1992. [9] ——, “Lea rning from the b ehavio r of others: Conformit y , fads, and informational cascades,” The Journal of Economic P ersp ectives , vol. 12, no. 3, pp. 151–170, 1998. [10] F. Rosas and K.-C. Chen, “Social learning against data falsification in sensor net wo rks,” in International W orkshop on Complex Netw o rks and their Applications . Sp ringer, 2017, pp. 704–716. [11] F. Rosas, K.-C. Chen, and D. Gunduz, “So cial learning for resilient data fusion against data falsification attacks,” a rXiv p reprint arXiv:1804.00356 , 2018. [12] J. Mathiesen, N. Mitarai, K. Snepp en, and A. T rusina, “Ecosystems with mutually exclusive interactions self-organize to a state of high diversity ,” Physical review letters , vol. 107, no. 18, p. 188101, 2011. [13] F. C. Santos, M. D. Santos, and J. M. P acheco, “So cial diversity p romotes the emergence of co op eration in public go o ds games,” Nature , vol. 454, no. 7201, p. 213, 2008. [14] L. Smith and P . Sørensen, “P athological outcomes of observational lea rning,” Econometrica , vol. 68, no. 2, pp. 371–398, 2000. [15] V. Bala and S. Go y al, “Conformism and diversit y under so cial lea rning,” Economic theory , vol. 17, no. 1, pp. 101–120, 2001. [16] M. Lo eve, Probabilit y Theory I . Springer, 1978. [17] A. Gelman, J. B. Ca rlin, H. S. Stern, D. B. Dunson, A. Vehta ri, and D. B. Rubin, Bay esian data analysis . CRC press Bo ca Raton, FL, 2014, vol. 2. [18] T. M. Cover and J. A. Thomas, Elements of info rmation theo ry . John Wiley & Sons, 2012. 9 [19] F. Rosas, V. Ntranos, C. J. Ellison, S. P ollin, and M. Verhelst, “Understanding interdep endency through complex info rmation sha ring,” Entropy , vol. 18, no. 2, p. 38, 2016. [20] J. Dieudonne, T reatise on Analysis . Asso ciated Press, New Y o rk, 1976, vol. I I. 10