Diversity Improves Speed and Accuracy in Social Networks
Bhargav Karamched, Megan Stickler, William Ott, Benjamin Lindner, Zachary Kilpatrick, Kresimir Josic
DDiversity Improves Speed and Accuracy in Social Networks
Bhargav Karamched, ∗ Megan Stickler, William Ott, BenjaminLindner,
2, 3
Zachary P. Kilpatrick, and Kreˇsimir Josi´c
1, 5 Department of Mathematics, University of Houston, Houston, Texas 77004, USA Physics Department, Humboldt University, Berlin, Germany Bernstein Center for Computational Neuroscience, Berlin, Germany Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA Department of Biology and Biochemistry, University of Houston, Houston, Texas 77004, USA † (Dated: July 14, 2020)How does temporally structured private and social information shape collective decisions? To ad-dress this question we consider a network of rational agents who independently accumulate privateevidence that triggers a decision upon reaching a threshold. When seen by the whole network, thefirst agent’s choice initiates a wave of new decisions; later decisions have less impact. In heteroge-neous networks, first decisions are made quickly by impulsive individuals who need little evidenceto make a choice, but, even when wrong, can reveal the correct options to nearly everyone else.We conclude that groups comprised of diverse individuals can make more efficient decisions thanhomogenous ones. A central question in biology, sociology, and economicsis how the exchange of information shapes group de-cisions [1–7]. Humans and other animals observe thechoices of their peers to guide their own decisions [8–11]: Argentinian ants form trails by following their peers[12], African wild dogs depart a congregation in responseto their neighbor’s sneezes [13], and pedestrians look toeach other to decide when to cross a road [14].How do individuals combine private evidence and so-cial information to make decisions? What should theydo when they observe choices at odds with their ownbeliefs? To address these questions we propose an ana-lytically tractable model of collective, rational decision-making. Agents in the network accumulate private ev-idence according to the widely-adopted drift-diffusionmodel (DDM) [15–17]. They do not share private in-formation, but observe each other’s choices [6, 18]. Adecision reveals the evidence an agent accumulated andmay trigger decisions by undecided observers.We show that in a group of identical agents, a wrongfirst decision leads approximately half the network astray.However, in heterogeneous networks a wrong first choiceis usually made by a hasty, uninformed agent and onlyconvinces others who are similarly quick to decide. Morecautious agents can observe the decisions of these earlyadopters, and make the right choice. We conclude that indiverse groups decisions by unreliable agents, even whenwrong, can reveal the better option.
Model Description.
We consider an all-to-all network,or clique , of agents, each deciding between two options(Fig. 1). To do so, agents continuously accumulate pri-vate evidence and social evidence from other agents. Theagents do not share their private observations, but knowthe statistics of the observations each agent makes, and ∗ [email protected] † [email protected] observe the choices of all agents in the network. A deci-sion, once made, is final and cannot be undone.Before giving precise definitions, an example providessome intuition: A group of people is deciding which one oftwo new products to buy. To do so, they study the prod-ucts’ specifications, examine their performance, and readreviews, making a sequence of observations of varying re-liability. They also observe which product their friendschoose to buy. To decide, each person combines theirprivate observations (product reviews) with social infor-mation (purchase decisions of friends). They do not ex-change information before making a purchase, but knowthe type of information their friends gather, and thus thestatistics of how beliefs evolve [6, 19]. Once a purchaseis made, the product cannot be returned. a. Evolution of observers’ beliefs. We assume N agents (observers) accumulate noisy private observationsand optimally combine them with information obtainedfrom observing the decisions of their neighbors to choosebetween two hypotheses, H + or H − . Either hypothe-sis is a priori equally likely to be correct. Each agent, i, makes decisions based on their belief , y i ( t ) , whichequals the log likelihood ratio (LLR) between the hy-potheses given all available evidence. After making asequence of private observations, ξ ( i )1: t , the belief is there-fore y i ( t ) = log[ P ( H + | ξ ( i )1: t ) / P ( H − | ξ ( i )1: t )]. If private obser-vations are rapid and uncorrelated in time and betweenagents, each agent’s belief approximately evolves as dy i = ± αdt + √ αdW i , (1)where the sign of the drift equals that of the correct hy-potheses and W i ( t ) are independent, standard Wienerprocesses [17, 20]. Each observer starts with no evidenceor bias, so y i (0) = 0. We assume henceforth that H + iscorrect, and that the drift in Eq. (1) is α = 1. When H − is correct and α (cid:54) = 1 the analysis is similar.Each agent, i, sets a threshold, θ i , and chooses H + ( H − ) at time T i if y i ( T i ) ≥ θ i ( y i ( T i ) ≤ − θ i ), and a r X i v : . [ phy s i c s . s o c - ph ] J u l c li q u e f r a c t i o n i n f i r s t w a v e s i z e o f f i r s t w a v e f i r s t d e c i s i o n t i m e R + abc de theorysimulations FIG. 1. (Color online)
Waves of collective decisions. (a) The first in a clique of identical agents gathers sufficientprivate evidence but decides incorrectly (red). (b) The firstdecision convinces a small wave of agents to agree. Since thiswave is small, it reveals to the undecided (blue) agents thatthe first decision was likely wrong. (c) The difference betweendecided and undecided agents leads a second wave of agentsto choose correctly (green). (d) The first wave increases withnetwork size (red), but comprises a smaller fraction of thepopulation (blue; See Eq. (5)). (e) The time to the first de-cision decreases with network size (red), allowing each agentless time to accumulate private information (See Eq. (4)). Theamount of information provided by a first wave decision alsodecreases ( R + , blue; See Eq. (6)). We used θ = 0 . y i ( t ) ∈ ( − θ i , θ i ) for 0 ≤ t < T i . An agent’s decision isobserved by all other agents and cannot be undone.An agent that observes a decision will know whetherthe decider chose H + or H − , but may not know thethreshold of the decider. We will consider omniscient agents who know each other’s thresholds, and the case of consensus bias where agents assume all others have thesame thresholds they do. b. Belief updates due to a decision. Without loss ofgenerality, we assume the belief of agent i = 1 is the firstto reach threshold at time t = T (See Fig. 1a). The prob-ability that this decision is correct is (1+exp( − θ )) − [17].Until the first decision, beliefs of all agents, y i ( t ) , evolve independently according to Eq. (1). Upon ob-serving the first decision, omniscient agents update theirbelief by the amount of evidence independently accumu-lated by the first decider, y i ( T ) → y i ( T ) ± θ [18]. A pos-itive ( H + ) first decision ( y ( T ) = θ ) and update causesany belief that satisfies y i ( T − ) ∈ [ θ i − θ , θ i ) just be-fore the first choice, to cross the positive threshold, θ i ,evoking a positive decision by agent i . Similarly, agentswhose belief satisfies y i ( T − ) ∈ ( − θ i , θ − θ i ] would fol-low a negative first decision. Agents subject to consensusbias update their belief as y i ( T ) → y i ( T ) ± θ i . For sim- plicity we assume agents exchange all social informationbefore accumulating further private information.Multiple waves of decisions can now follow: The firstchoice is followed by a wave of a agreeing agents (SeeFig. 1b). Each of the N − a − Homogeneous populations.
To answer this question,first suppose all agents have identical thresholds, θ i = θ ,for all i , so that the cases of omniscience and consen-sus bias are equivalent. Observing that agent i (cid:54) = 1follows a positive first decision tells other agents that y i ( T − ) ∈ [0 , θ ). An agent’s belief equals the LLR ofthe conditional probabilities of the two options, given allavailable information. Therefore observing first wave de-cision of agent i leads to an increment in belief equalto [18]LLR( y i ( T ) ∈ (0 , θ )) def = log (cid:18) P ( y i ( T ) ∈ [0 , θ ) | H + ) P ( y i ( T ) ∈ [0 , θ ) | H − ) (cid:19) = log (cid:32) (cid:82) θ p + ( x, T ) dx (cid:82) θ p − ( x, T ) dx (cid:33) ≡ R + ( T ) . Here p ± ( x, t )∆ x = P ( y i ( t ) ∈ ( x, x + ∆ x ) | H ± ) + o (∆ x )is the conditional probability density for the belief ofagent i at time t . Since thresholds are symmetric, (cid:82) θ p − ( x, t ) dx = (cid:82) − θ p + ( x, t ) dx , so observing an agent j who remains undecided after the first decision reveals y j ( T ) ∈ ( − θ, y j ( T ) ∈ ( − θ, ≡ R − ( T ) = − R + ( T ).We assume that agents know the statistics of pri-vate observations and can therefore compute p + ( x, t ) and p − ( x, t ). Agents thus know that beliefs evolve accordingto Eq. (1) and that the belief distribution prior to anydecision satisfies: ∂ t p ± = ∓ ∂ x p ± + ∂ xx p ± , p ± ( ± θ, t ) = 0 , (2)if H ± is correct, with p ± ( x,
0) = δ ( x ). Agents do notknow which hypothesis is correct, and compute the beliefupdate, R + ( T ) , using the belief distributions, p ± ( x, t ).If the first agent chooses H + , agents undecided afterthe first wave combine the information from all observeddecisions and indecisions. Since private measurementsare independent, information obtained from agents in thefirst wave is additive, and the belief increment is c +1 def = a R + ( T ) + ( N − a − R − ( T )= (2 a − N + 2) R + ( T ) . (3)If the first decision sways more than half the network, a > N/ − , the weight of new evidence favors thechoice of the first agent. Conversely, observing that themajority of agents remain undecided provides evidenceagainst the first agent’s decision. All undecided agentsincrement their belief by c +1 , leading to a second wave of a decisions, all of equal sign (See Fig. 1c). Agents in thefirst wave agree with the first decision, while agents in thesecond wave agree only if the sign of c +1 matches the firstdecision. Observers still undecided after this second waveupdate their LLR by a new increment c +2 , and waves ofdecisions follow until either all agents make a choice, orno new agent makes a decision after some belief update, c + k , k ≥
2. Undecided agents then continue to accumulateprivate information. Whether the first decision is right orwrong, we will show that in large populations the secondwave of decisions encompasses the entire population.If the first agent wrongly chooses H − , the computa-tions are similar: Observing a decision in the first waveprovides a belief increment R − ( t ) = − R + ( t ), and ob-serving an undecided agent provides an increment R + ( t ) , giving a total increment c − = (2 a − N + 2) R − ( T ). Fur-ther decision waves follow equivalently. a. Decisions in large groups. As N grows, the timeto the first decision, T, approaches 0, and we can ap-proximate the solution to Eq. (2) using the method ofimages [22, 23] . Using the resulting lifetime distributionand extreme value theory we find that the expected firstdecision time is [24–28]: E [ T ] ≈ θ N . (4)The mean time decreases logarithmically with N , al-lowing each agent less time to gather private information(Fig. 1e). When the first decision time, T , is small theremaining beliefs are distributed almost symmetricallyaround the origin, and approximately half the populationagrees with the first decision, whether right or wrong. In-deed, we find that, E [ a | y ( T ) = ± θ ] ≈ N − (cid:16) ± θ √ π ln N (cid:17) , (5)where the last term is positive (negative) if the first agentcorrectly (incorrectly) chooses H + ( H − ). Thus, slightlymore than half of a large clique immediately follows a cor-rect first decision, and slightly less than half the cliquefollows a wrong first choice (Fig. 1d shows the mean num-ber and fraction following a correct first decision).The number of agents in excess of half the populationfollowing a correct first decision scales as N (ln N ) − / .But as the population grows agents in the first wave ac-cumulate less private information prior to their choice.We find that for large N , the expected social informa-tion communicated by each decision is E [ R + ( T )] ≈ E [ (cid:112) T /π ] ≈ θ/ √ π ln N . (6)Thus, as the population increases the size of the firstwave, a , grows (Fig. 1d), but each first wave decisionprovides less information (Fig. 1e). However, the loga-rithmic decrease in the revealed information, R ± ( T ), isoutweighed by the nearly linear growth in the numberof agents, a : Using Eqs. (3), (4) and (6), we find theexpected belief update, ˆ c ± ≡ E [ c ± ], of undecided agents θ min s e c o n d w a v e i n c r e m e n t θ min a cc / t o t clique size10 s e c o n d w a v e i n c r e m e n t θ =0.3 θ =0.5 θ =1 10 clique size0.81.0 p r o b a b ili t y w h o l e c li q u e d e c i d e s t h r e s h o l d a b
020 0.5 1.0 θ min f i r s t d e c i s i o n t i m e γ =0.05 γ =0.3 γ =0.7 c d e theorysimulations FIG. 2. (Color online)
Decision statistics for large ho-mogeneous and dichotomous cliques. (a) Belief incre-ment ˆ c ± of agents in the second wave in homogeneous cliques.(b) Simulations of the probability the full homogeneous net-work decides after the second wave as a function of size, N .Chebyshev’s Inequality provides an upper bound on cliquesize by which the probability is reached. Inset: Threshold θ at which ˆ c ± = 2 θ as clique size N is varied. (c) First de-cision time for dichotomous threshold cliques for various γ .(d) Fraction of accurate deciders in dichotomous thresholdcliques under consensus bias. (e) Belief increment of agentsin the second wave in dichotomous threshold cliques underconsensus bias. Clique size N = 15000 in panels (c–e). in the second wave grows nearly linearly in N ,ˆ c ± ≈ θ N π ln N . (7)The expected update per agent in the second wave fol-lowing a correct or incorrect decision, ˆ c +1 or ˆ c − , is pos-itive: If the first decision is correct, more than half thenetwork is in the first wave, and both (2 E [ a ] − N − R + ( T ) are positive in Eq. (3). Both of these termsare negative when the first decision is wrong. Thus thesecond wave is self-correcting : When the network is suf-ficiently large, ˆ c ± > θ (see Fig. 2a), and we expectall undecided agents to make the correct choice in thesecond wave, regardless of the choice of the first agent.As Eq. (3) approximates the average belief update, wecannot use it to estimate the probability that the entireclique will decide. However, we can use Chebyshev’s In-equality to show that when N ≥ π ( θ (1 − x )) − theentire network decides by the second wave with proba-bility at least x (See Fig 2b).In sum, the first choice triggers a wave of a decisions inagreement with the first decider, whether right or wrong.In large networks, all remaining agents decide correctly inthe second wave, regardless of the first agent’s decision. Heterogeneous Populations.
A population of decisionmakers is rarely homogeneous. Some people are quick tomake decisions based on little evidence. Others requiresubstantial information before making a choice [29, 30].Some have access to high quality information, while oth-ers rely on poor evidence. How does such diversity im- θ min γ . . . . time → d e c / t o t θ min . γ a cc / t o t . . . ad ea cb cb undecidedcorrectincorrect agent with threshold θ min agent with threshold θ max FIG. 3. (Color online)
Balancing hasty and deliberatedecisions in cliques with dichotomous threshold dis-tributions. (a) With too few low threshold agents, the re-maining agents do not receive sufficient information to decideafter the first wave; (b) With many low threshold agents, awrong first decision can sway much of the network; (c) Withthe right number of low threshold agents, a small numberof hasty agents follows a wrong first decision, but the differ-ence between agreeing and disagreeing low threshold agentsis just sufficient for the rest of the clique to choose correctly.(d) Fraction of the clique choosing accurately for a dichoto-mous threshold clique. (e) Fraction of the clique deciding bythe end of the second wave. Isoclines indicate time to firstdecision. Clique size N = 15000 in (b) and (c). pact decisions of the collective?Here we focus on diversity in the amount of evidenceagents require to make a choice by assuming agents’ deci-sion thresholds are distributed on an interval [ θ min , θ max ].Agents with a low threshold are more likely to decidefirst, but also to make a wrong choice [31]. The ensuingexchange of social information depends on assumptionsagents make about each other. While collective deci-sions in heterogeneous networks under consensus bias aresimilar to those in homogeneous populations, omniscientagents can leverage quick, unreliable decisions to improvethe response of the population. a. Dichotomous threshold distribution. The casewhen all agents have either a high or a low thresholdis tractable and sheds light on more general examples.Before a decision the belief of each agent evolves accord-ing to Eq. (1) on a symmetric interval with absorbingboundaries at − θ i < < θ i . We assume that γN agentsshare threshold θ min and (1 − γ ) N share threshold θ max for some 0 < θ min < θ max and γ ∈ (0 , E [ T ] ≈ θ γN ) which breaksdown when 0 < γ (cid:28) − θ min )) − [17]. The social network is homogeneous from the perspec-tive of an observer under consensus bias. We thus againexpect about half of the network to follow the first choice,whether right or wrong. Indeed, the expected size of thefirst wave is given by an expression similar to Eq. (5), E [ a ] ≈ N − (cid:16) ± θ min √ π ln γN (cid:17) . The expected belief incre-ment in the second wave is ˆ c ± ≈ θ γN π ln γN which is analo-gous to Eq. (7), and is governed primarily by the timingof the first choice (See Fig. 2e). In large populationsdecisions happen quickly, before the belief distributionscan interact with the boundaries. Therefore the expectedbelief increment, ˆ c ± , is approximately independent of theobserver’s threshold : Following the first wave low andhigh threshold agents make the same update.As in homogeneous networks, the size of the expectedupdate, ˆ c ± , grows with population size. When the up-date exceeds 2 θ max , we expect all agents to decide by thesecond wave. If the first decision is correct, the entireclique follows. A wrong first choice is followed by the firstwave constituting about half the network (see Fig. 2d),while the second wave decides correctly. Under consensusbias, dichotomous cliques behave as if they were homoge-nous with threshold θ min : Uninformed agents govern de-cisions, leading to fast, inaccurate choices.In contrast, omniscient agents correctly weigh evi-dence revealed by a hasty first decider. We expect abouthalf of the low threshold agents, γN/
2, to decide in thefirst wave. Indeed, we find E [ a ] ≈ γN − (cid:16) ± θ min √ π ln γN (cid:17) .The evidence revealed by a single low threshold agentis unlikely to sway high threshold agents. However, ifthe subpopulation of low threshold agents is sufficientlylarge, the difference between those convinced and uncon-vinced by the first choice can trigger a correct decisionin the rest of the population (See Fig. 3b,c).Thus, in a network of omniscient agents, hasty ob-servers again govern the speed of the first decision andmostly comprise the first wave. The remaining agents canthen observe the choices of the early adopters to make theright decision. The fraction of wrong decisions can thusbe smaller than in homogeneous networks.In finite populations this argument requires γ and θ min be large enough for the first wave to convince the re-mainder of the population (Fig. 3a), but small enoughto buffer the majority of the population from followingan incorrect first choice (Fig. 3b). We thus expect thatthe population makes the best decisions at intermediatevalues of γ and θ min (star in Fig. 3d). A balance be-tween these cases is reached when ˆ c − = 2 θ max , whichcorresponds to a fraction of low threshold agents givenby γ ≈ πθ max N ln Nθ . Almost all agents decide by the second wave (Fig. 3e).Thus a finite population with dichotomous thresholdscan sacrifice a small fraction of early adopters so most a cc / t o t homogeneous simsbest dichotomous simsuniform sims0.05 0.50 1.00 θ min a cc / t o t θ min θ max a b homogeneous simshomogeneous sims FIG. 4. (Color online)
Improved accuracy for fixed deci-sion speed. (a) Mean fraction of the entire clique choosingaccurately after two waves for different threshold distributionsin an omniscient population. For the dichotomous case, γ ischosen to maximize accuracy for each θ min value. (b) Overa range of possible first decision times, heterogeneous cliquesgive better accuracy than homogeneous ones with omniscientsocial updating. Here, N = 15000 and θ max = 1. of the population makes a fast, correct choice. Agents inheterogenous networks can thus decide more quickly, andoutperform agents in homogeneous networks in recover-ing from a wrong first choice (Figs. 3c and 4). b. Different threshold distributions. We next consider N agents with thresholds, θ i , following different distri-butions supported on the interval [ θ min , θ max ]. The ex-pected time to the first decision is again governed by thesmallest threshold, θ min , and under consensus biased thesize of the first wave is E [ a ] ≈ N − (cid:104) ± θ min √ π ln N (cid:105) , withthe sign determined by the first decision. In either case,the increment to the undecided agents after the first waveis given by Eq. (7) with θ min replacing θ . For sufficientlylarge N , ˆ c > θ max . Therefore, under consensus biasthe clique again behaves as a homogeneous clique withthreshold θ min .The omniscient case is more complicated. Numericalsimulations show the trends observed in the dichotomous case persist for a large class of threshold distributions:Hasty agents decide first, and deliberate agents decidebased on which early adopters followed the first choice(See Figs. 4), leading to faster and more accurate choicesthan in homogeneous networks. Conclusion.
Our model of collective decision makingis analytically tractable and shows how diverse popula-tions can make quicker, more accurate decisions than ho-mogeneous ones. Previous models of collective decision-making have either ignored temporal aspects of evidenceaccumulation [6, 19, 32] or did not describe rationalagents [33, 34]. Our model incorporates both aspectsand can serve as a baseline to understand when and howdecisions depart from rationality [35].There are a number of ways to include more realisticfeatures in the model: Observations can be correlated,rather than conditionally independent , resulting incorrelated noise in Eqs. (1) and (2) [36]. Agents couldaccumulate evidence at different rates, resulting in eachhaving different drift and diffusion coefficients [17, 20].The framework we provide can be extended to thesecases to probe how different conditions influence deci-sions.
ACKNOWLEDGMENTS
We would like to thank Thibaud Taillefumier for help-ful comments. This work was supported by NSF/NIHCRCNS grant R01MH115557. BK and KJ were sup-ported by NSF grant DMS-1662305. KJ was also sup-ported by NSF NeuroNex grant DBI-1707400. ZPK wasalso supported by NSF grant DMS-1853630. WO wassupported by NSF grant DMS-1816315 and NIH grantR01GM117138. [1] I. D. Couzin, Trends in cognitive sciences , 36 (2009).[2] W. Edwards, Psychological Bulletin , 380 (1954).[3] C. D. Frith and U. Frith, Annual review of psychology , 287 (2012).[4] A. P´erez-Escudero and G. G. D. Polavieja, PLoS Com-putational Biology (2011).[5] S. Arganda, A. P´erez-Escudero, and G. G. D. Polavieja,Proceedings of the National Academy of Sciences ,20508 (2012).[6] R. P. Mann, Proceedings of the National Academy ofSciences , E10387 (2018).[7] R. P. Mann, Proceedings of the National Academy ofSciences (2020).[8] A. J. W. Ward, D. J. T. Sumpter, I. D. Couzin, P. J. B.Hart, and J. Krause, Proceedings of the NationalAcademy of Sciences , 6948 (2008).[9] A. Ward, J. Krause, and D. Sumpter, PloS one (2012).[10] G. E. Gall, A. Strandburg-Peshkin, T. Clutton-Brock,and M. B. Manser, Animal Behavior , 91 (2017).[11] A. Strandburg-Peshkin, D. R. Farine, I. D. Couzin, andM. C. Crofoot, Science , 1358 (2015).[12] A. Perna, B. Granovskiy, S. Garnier, S. C. Nicolis, M. Lab´edan, G. Theraulaz, V. Fourcassi´e, and D. J. T.Sumpter, PLoS Computational Biology , e1002592(2012).[13] R. H. Walker, A. J. King, J. W. McNutt, and N. R.Jordan, Prceedings of the Royal Society B (2017).[14] J. Faria, S. Krause, and J. Krause, Behavioral Ecology , 1236 (2010).[15] R. Ratcliff, Psychological review , 59 (1978).[16] J. I. Gold and M. N. Shadlen, Neuron , 299 (2002).[17] R. Bogacz, E. Brown, J. Moehlis, P. Holmes, and J. D.Cohen, Psychological Review , 700 (2006).[18] B. R. Karamched, S. Stolarczyk, Z. P. Kilpatrick, andK. Josi´c, SIAM Journal on Applied Dynamical Systems in press (2020).[19] S. b. Goyal, Connections: An Introduction to the Eco-nomics of Networks (Princeton University Press, 2012)pp. 1–289, arXiv:9809069v1 [arXiv:gr-qc].[20] A. Veliz-Cuba, Z. P. Kilpatrick, and K. Josi´c, SIAMReview , 264 (2016).[21] Supplementary Material contains details about the sim-ulations and calculations.[22] D. R. Cox and H. D. Miller, The theory of stochastic processes (Chapman and Hall, 1965).[23] J. Drugowitsch, Scientific Reports , 20490 (2016).[24] G. H. Weiss, K. E. Shuler, and K. Lindenberg, J. Stat.Phys. , 255 (1983).[25] S. B. Yuste and K. Lindenberg, J. Stat. Phys. , 501(1996).[26] Z. Schuss, K. Basnayake, and D. Holcman, Physics ofLife Reviews (2019).[27] K. Basnayake, Z. Schuss, and D. Holcman, Journal ofNonlinear Science , 461 (2019).[28] S. D. Lawley and J. B. Madrid, Journal of NonlinearScience , 1 (2020).[29] T. Postmes, R. Spears, and S. Cihangir, Journal of Per- sonality and Social Psychology , 918 (2001).[30] N. R. Franks, A. Dornhaus, J. P. Fitzsimmons, andM. Stevens, Prceedings of the Royal Society B , 2457(2003).[31] V. Srivastava and N. E. Leonard, IEEE Transactions onControl of Network Systems , 121 (2014).[32] M. Mueller-Frank, Theoretical Economics , 1 (2013).[33] D. J. Watts, Proceedings of the National Academy ofSciences , 5766 (2002).[34] R. J. Caginalp and B. Doiron, SIAM Journal on AppliedDynamical Systems , 1543 (2017).[35] W. S. Geisler, The visual neurosciences , 12 (2003).[36] R. Moreno-Bote, Neural computation22