A General Model of Opinion Dynamics with Tunable Sensitivity
11 A General Model of Opinion Dynamics withTunable Sensitivity
Anastasia Bizyaeva, Alessio Franci, and Naomi Ehrich Leonard
Abstract —We present a model of continuous-time opiniondynamics for an arbitrary number of agents that communicateover a network and form real-valued opinions about an arbitrarynumber of options. The model generalizes linear and nonlinearmodels in the literature. Drawing from biology, physics, and socialpsychology, we introduce an attention parameter to modulate so-cial influence and a saturation function to bound inter-agent andintra-agent opinion exchanges. This yields simply parameterizeddynamics that exhibit the range of opinion formation behaviorspredicted by model-independent bifurcation theory but not ex-hibited by linear models or existing nonlinear models. Behaviorsinclude rapid and reliable formation of multistable consensusand dissensus states, even in homogeneous networks, as wellas ultra-sensitivity to inputs, robustness to uncertainty, flexibletransitions between consensus and dissensus, and opinion cas-cades. Augmenting the opinion dynamics with feedback dynamicsfor the attention parameter results in tunable thresholds thatgovern sensitivity and robustness. The model provides new meansfor systematic study of dynamics on natural and engineerednetworks, from information spread and political polarization tocollective decision making and dynamic task allocation.
Index Terms —multi-agent systems, distributed control, deci-sion making, social networks, consensus, dissensus, cascades,sensitivity, robustness, bifurcations, nonlinear dynamics
I. I
NTRODUCTION
Opinion dynamics of networked agents are the subject oflong-standing interdisciplinary interest, and there is a largeand growing literature on agent-based models created to studymechanisms that drive the formation of consensus and opinionclustering in groups. These models appear, for example, instudies of collective animal behavior and voting patterns inhuman social networks. In engineering, they are fundamentalto designing distributed coordination of autonomous agentsand dynamic allocation of tasks across a network.Agent-based models are typically used to investigate pa-rameter regimes and network structures for which opinionsin a group converge over time to a desired configuration.However, natural groups exhibit much more flexibility thancaptured with existing models. Remarkably, groups in naturecan rapidly switch between different opinion configurations
This research has been supported in part by NSF grant CMMI-1635056,ONR grant N00014-19-1-2556, ARO grant W911NF-18-1-0325, DGAPA-UNAM PAPIIT grant IN102420, Conacyt grant A1-S-10610, and by NSFGraduate Research Fellowship DGE-2039656. Any opinion, findings, andconclusions or recommendations expressed in this material are those of theauthors(s) and do not necessarily reflect the views of the NSF.A. Bizyaeva and N.E. Leonard are with the Department of Mechanical andAerospace Engineering, Princeton University, Princeton, NJ, 08544 USA; e-mail: [email protected], [email protected]. Franci is with Math Departmemnt of the National AutonomousUniversity of Mexico, 04510 Mexico City, Mexico. e-mail:[email protected] in response to changes in their environment, and they canbreak deadlock, i.e., choose among options with little, if any,evidence that one option is better than another. Understandingthe mechanisms that explain the temporal dynamics of opinionformation in groups and the ultra-sensitivity and robustnessneeded for groups to pick out meaningful information andto break deadlock in uncertain and changing environments isimportant in its own right. It is also pivotal to developing themeans to design provably adaptable yet robust control lawsfor robotic teams and other networked multi-agent systems.Motivated by these observations, we explore the followingquestions in this paper. How can a network of decisionmakers come rapidly and reliably to a coherent configurationof opinions, including consensus or dissensus, on multipleoptions in response to, or in the absence of, internal biasesor external inputs? How can a network reliably transitionfrom one configuration of opinions to another in responseto change? How can the sensitivity of the opinion formationprocess be tuned so that real signals are distinguished fromspurious signals? To investigate these questions, we present anagent-based dynamic model of the opinion formation processthat generalizes linear and existing nonlinear models. Thegeneral model is maximally rich in the behaviors it exhibitsyet tractable to analysis by virtue of the small number ofparameters needed to generate the full range of behaviors.In classical models of opinion formation [1]–[5], eachagent’s opinion is represented by a scalar and linear averagingmechanisms govern response to agent interactions, commonlydefined by a network graph. Updates to self-weights [6] andinter-agent weights [7] are used to study evolution of socialpower. Prominent nonlinear variations on averaging modelsinclude “bounded confidence” models, which assume that theopinion of an agent is primarily influenced by its like-mindedneighbors [8]–[11]. For multiple topics, models are extendedto vector-valued opinion states, e.g., [12]–[18]. Typically, eachtopic has two options and opinion formation on different topicsis interrelated but also independent, in the sense that agentshave unlimited voting capacity across topics.In [19]–[25], nonlinear models are derived and studied inwhich agents with scalar opinions respond to a saturatingfunction of the opinions of others, much like in a Hopfieldneural network. In [19], the functional form represents biasedassimilation from social psychology. The use of a saturatingfunction has inspired the present development and these mod-els provide a precursor to the general model defined here.Closely related to these are nonlinear models that leveragecoupled oscillator dynamics [26]–[28], biologically inspiredmean-field models [29], and the Ising model for spin systems a r X i v : . [ m a t h . O C ] O c t [30], [31]. Consensus is also sometimes studied on a nonlinearmanifold opinion space, as in [32], [33], although analytictractability of this approach is difficult in general [34].In almost all of this literature, with [19], [25] being anexception, clustered or fragmented configurations of opinionstypically arise only when an asymmetry or heterogeneity isimposed. For example, the emergence of opinion clusters istied to special conditions on connectivity of a network in [3],[5], [17]. In bounded confidence models, clustering of opinionsis usually related to values of the opinion threshold, as in[16]. This implicitly creates a heterogeneous network structureamong agents, since agent pairs with sufficiently differentopinions delete their communication link. Fragmentation andopinion clustering have also been connected to heterogeneousbiases or external information among agents [35].In contrast to these findings, model-independent theoryrecently developed in [36] shows that opinion dynamics on anall-to-all network of homogeneous unbiased agents evaluatingan arbitrary number of options can give rise to a fragmentedglobal state. In this state, which we refer to as a type ofgroup dissensus , the group is neutral about every option onaverage while individual agents are opinionated. Emergenceof dissensus on a network is shown to be a generic outcomeof opinion formation, alongside consensus, when the opinionformation process is nonlinear. That the inherent coexistenceof consensus and dissensus is not tied to asymmetry orheterogeneity is a key feature of the general model.The theory in [36] also predicts coexistence and simulta-neous stability of many different consensus and/or dissensusequibria. This multistability can be understood to be funda-mentally related to the type of flexible behavior observed innatural groups, since it enables a group to easily transitionfrom one opinion configuration to another. Bistability of twoconsensus outcomes and the sensitivity of collective opinionformation is explored for two options in [20], [22], [24].Recent work in [37] explores the role of bistability in opiniondynamics of a single agent recursively navigating a number ofoptions organized in a binary tree hierarchy. Other nonlinearmodels of opinion formation among multiple options include[31], [38]. However, existing models of multi-option opiniondynamics for multi-agent systems do not capture maximallygeneric multistability of group outcomes.In this paper, we present a nonlinear model of multi-optionopinion dynamics that captures consensus and dissensus aspredicted in [36] and the more general formation of agreementand disagreement. The model also captures the all impor-tant multistability of opinion configurations, including distinctconsensus and/or dissensus equilibria. The model realizes themodel-independent structural conditions of [36] through thenonlinear signal properties of saturated nonlinearities. Satu-rated nonlinearities appear in virtually every natural and arti-ficial signaling network due to bounds on action and sensing.Using their structure is thus a logical modeling choice, which,as we show, captures the most general (model-independent)properties of the opinion formation process. Our general modelprovides a systematic means to generate testable hypothesesabout opinion dynamics and decision-making processes acrossa wide range of natural groups, including human social net- works and animal groups. Likewise, the model provides a sys-tematic means to leverage the ultra-sensitivity and robustnesstypical of biological systems in the design of novel distributedcontrol approaches for technological multi-agent systems.Our major contributions are as follows. 1) We introducea new general model for the study of multi-agent, multi-option opinion dynamics. The model allows a network tobe defined on intra-agent and inter-agent opinion exchangesand includes a resistance term, a social term weighted by anattention parameter, which can also represent social effort orsusceptibility to social influence, and an input term, which canrepresent, e.g., external stimuli, bias, or persistent opinions.2) We show that the general model inherits the richness ofgeneric opinion-formation behaviors predicted by the model-independent theory developed in [36]. This includes rapidand reliable opinion formation and multistable consensus anddissensus, with flexible transitions between them, even in thesymmetric and homogeneous case of equal agents and equallyvaluable options. It also includes ultra-sensitivity to inputsnear the opinion forming bifurcation, and robustness of theopinion formation process to heterogeneity in all parameters,away from the bifurcation. Moreover, the behaviors are gov-erned by a small number of key parameters, rendering themodel analytically tractable. We prove the central role of theeigenstructure of the network graph adjacency matrix. 3) Weshow how the model specializes to a range of models in theliterature, including linear consensus, and how the necessity ofstructurally balanced networks for clustered opinion formationin linear models with signed network graphs breaks downin the nonlinear setting. The central role of the networkgraph Laplacian in opinion dynamics in the literature fallsout as the specialization of the central role of the networkgraph adjacency matrix in the general model. 4) We introducedistributed feedback dynamics to the agent attention param-eter. 5) We show how attention feedback design parametersallow tunable sensitivity of opinion formation to inputs androbustness to changes in inputs, and tunable opinion cascadeseven in response to a single agent receiving an input. 6) Weexamine tunable transitions between consensus and dissensususing feedback dynamics also on network weights.In Section II we make definitions. We present the generalmodel in Section III and prove when it reduces to dynamics ofagent clusters. In Section IV we prove that opinions form asa bifurcation and multistable consensus and dissensus emergegenerically under symmetry. We show how the general modelspecializes to models in the literature and prove implicationsin Section V. We propose feedback dynamics for the attentionparameter in Section VI and prove and illustrate tunable sensi-tivity, robustness, cascades, and transitions between consensusand dissensus. We conclude in Section VII.II. N OTATION AND D EFINITIONS
Consider a network of N a agents forming opinions about N o options. Let x ij ≥ be the magnitude of the absolute opinion of agent i about option j and suppose that every agenthas the same total voting capacity: x i + · · · + x iN o = r, r > . (1) Absolute opinion state of agent i is X i = ( x i , . . . , x iN o ) ∈ ∆ ,the ( N o − -dimensional simplex, and absolute opinion stateof the system is X = ( X , . . . , X N a ) ∈ V = ∆ × · · · × ∆ . Thesimplex product space V is often associated with models ofopinion dynamics, e.g., in [6], [35], [39]. At the neutral point O = ( O , . . . , O N a ) ∈ V , each agent has the same opinionabout each option: O i = (cid:16) rN o , . . . , rN o (cid:17) ∈ ∆ .In this paper we design and analyze opinion dynamics onthe linear space V = T O V ⊂ R N a N o (the tangent space to V at O ), where V = V a × · · · × V a and V a = { ( v , . . . , v N o ) ∈ R N o | (cid:80) N o l =1 v l = 0 } . As long as V is forward-invariant for thedynamics and the dynamics are bounded, they can be mappedback to the opinion dynamics on simplex space V with anaffine coordinate change (proved in Section III for our model).Let Z i = ( z i , . . . , z iN o ) ∈ V a be the opinion state of agent i and Z = ( Z , . . . , Z N a ) ∈ V the opinion state of the system .These represent relative opinions: z ij is the opinion of agent i about option j relative to the other options: z i + · · · + z iN o = 0 . (2)The neutral point X = O ∈ V corresponds to Z = ∈ V .Agent i is unopinionated if its opinion state is close to theneutral point: (cid:107) Z i (cid:107) ≤ ϑ , for ϑ ≥ small and (cid:107)· (cid:107) = (cid:107)· (cid:107) .When (cid:107) Z i (cid:107) > ϑ , agent i is opinionated . Agent i favors option j when it is opinionated and z ij ≥ z ip − ϑ for all p (cid:54) = j .Conversely, agent i disfavors option j if it is opinionated and z ij < z ip − ϑ for some p . An agent is conflicted among a setof options if it has near equal and favorable opinions aboutall options in the set relative to the options not in the set.Two agents agree if both are opinionated and share the samequalitative opinion state (e.g. favoring option j or conflictedamong a set of options). Two agents disagree if they havedifferent qualitative opinion states. If all agents are opinionatedand agree, the group is in an agreement state . If the group isin agreement and (cid:107) Z i − Z k (cid:107)≤ ϑ , for all i, k , the group is ina consensus state . If at least one pair of agents disagrees,the group is in a disagreement state . If the group is indisagreement and the average agent is unopinionated, i.e., (cid:107) N a (cid:80) N a i =1 Z i (cid:107)≤ ϑ , the group is in a dissensus state . The groupis in an unopinionated state if all agents are unopinionated.The system opinion state-space decomposes as V = W c ⊕ W d , where W c is the multi-option consensus space defined as W c = { ( Z , . . . , Z N a ) | Z i = ˜ Z ∈ V a , ∀ i } , (3)and W d is the multi-option dissensus space defined as W d = { ( Z , . . . , Z N a ) | Z + · · · + Z N a = } . (4)On the consensus space W c , agents have identical opinions. Onthe dissensus space W d , agent opinions are balanced over theoptions such that the average opinion of the group is neutral. Symmetry and equivariance:
Let Γ be a compact Lie groupacting on R n . Consider a dynamical system ˙ x = h ( x ) where x ∈ R n and h : R n → R n . Then ρ ∈ Γ is a symmetry of thesystem, equivalently h is ρ - equivariant , if ρ h ( x ) = h ( ρ x ) .If h is ρ -equivariant for all ρ ∈ Γ , then h is Γ - equivariant [40]. In other words Γ -equivariance means that elements ofthe symmetry group Γ send solutions to solutions. The compact Lie group associated with permutation sym-metries of n objects is the symmetric group on n symbols S n , which is the set of all bijections of Ω n := { , . . . , n } toitself (i.e., all permutations of ordered sets of n elements). Maximally symmetric opinion dynamics are ( S N o × S N a ) -equivariant, where elements of S N a permute the N a -elementset of agents and elements of S N o permute the N o -element setof options [36]. Thus, maximally symmetric opinion dynamicsare unchanged under any permutation of agents or options.A subgroup Γ n ⊂ S n is transitive if the orbit Γ n ( i ) = { γ ( i ) , γ ∈ Γ n } = Ω , for some (and thus all) i ∈ Ω . (Γ N o × Γ N a ) -equivariant opinion dynamics, with transitive Γ N a , are still highly symmetric since any pair of agents, whilenot necessarily interchangeable by arbitrary permutations, canbe mapped into each other by the symmetry group action.For example, if Γ N a = D N a , the (transitive) dihedral groupof order N a , symmetries correspond to N a rotations and N a reflections. Thus, D N a -equivariant opinion dynamics areunchanged if agents are permuted by a rotation or a reflection,e.g., if agents communicate over a network defined by a cycle.III. G ENERAL M ODEL OF O PINION D YNAMICS
The general model describes continuous-time opinion dy-namics for an arbitrary number N a of agents (each repre-senting an individual or a subgroup) that communicate over anetwork and form opinions about an arbitrary number N o ofoptions. For subsets of agents with similar features, we showin Section III-C a reduction to dynamics on clusters of agents.A key feature of the model is the application of a saturatingfunction to exchanges of opinions among agents. Dynamicsthat evolve according to saturating interactions appear inspatially localized and extended neuronal population modelsof thalamo-cortical dynamics [41], [42], in Hopfield neuralnetwork models [43]–[45], and in nonlinear leaky competingaccumulator models of noisy decision making [46], [47]. Inthe discrete-time model of opinion formation for two optionsof [19], [25], a saturating Hill function is used to represent“biased assimilation” of social information, a well-studiedconcept from social psychology.In the general model, the saturating function is the onlynonlinearity. Yet, it is sufficient to yield consensus and dis-sensus as a function of just a few parameters, even in thehomogeneous and maximally symmetric cases. This is not thecase for opinion dynamics with a linear averaging term (e.g.,linear variations of the DeGroot model [1]), even allowing forantagonistic interconnections as in [5]; see Section V. A. Model
The general model defines the rate of change of z ij , agent i ’s opinion about option j , as a function of three terms: aresistance term, a social term, and an input term.The resistance term, parametrized by d ij > , on its owndrives z ij exponentially in time to 0 (neutral opinion). In thegeneral model, a larger d ij implies a greater resistance of agent i to forming a non-neutral opinion about option j .The social term is defined as the product of an attention parameter u i ≥ and a saturating function of weighted sums 𝒛 𝒊𝒋 𝒛 𝒊𝒍 𝒛 𝒊𝒑 𝒛 𝒌𝒋 𝒛 𝒌𝒍 𝒛 𝒌𝒑 𝑨 𝒊𝒌 𝒋𝒋 𝑨 𝒊𝒌 𝒋𝒍 𝑨 𝒊𝒊𝒑𝒑 𝑨 𝒌𝒌 𝒍𝒑 Agent 𝒌 Agent 𝒊 Fig. 1: Illustration of the four classes of interactions. of agent opinions that are available to agent i and influence itsopinion of option j . The social term can also be interpretedas an activation term. The weights are encoded in adjacencytensor A ∈ R N a × N a × N o × N o , such that z ij is affected by z kl only if A jlik ∈ R is nonzero. The magnitude of A jlik determinesthe strength of influence of agent k ’s opinion about option l on agent i ’s opinion about option j , and the sign of A jlik determines whether this interaction is excitatory ( A jlik > ) or inhibitory ( A jlik < ). We reserve indices i, k to refer to agentsand indices j, l to refer to options.We distinguish four classes of interactions (see Fig. 1):1) Intra-agent, same-option coupling : A jjii Intra-agent, inter-option coupling : A jlii , j (cid:54) = l Inter-agent, same-option coupling : A jjik , i (cid:54) = k Inter-agent, inter-option coupling : A jlik , i (cid:54) = k , j (cid:54) = l .Because options are mutually exclusive, agents i and k are cooperative if min { A jjik , A jjki } > max { A jlik , A jlki } , for alloptions j (cid:54) = l , i.e., the opinion of agent i (agent k ) aboutoption j is more strongly excited by the opinion of agent k (agent i ) about the same option j than by its opinion about adistinct option l . Conversely, agents i and k are competitive if max { A jjik , A jjki } < min { A jlik , A jlki } .The attention parameter u i ≥ governs the strength ofthe social term relative to resistance and can have its owndynamics. We show in Section VI how u i can be used as adynamic feedback control parameter to tune sensitivity of theopinion dynamics to input. u i can model urgency in adoptinga non-neutral opinion by having it grow with increasing time-urgency (election day approaching), or spatial-urgency (targetgetting closer). u i can also be used to model social effort , excitability , or susceptibility of agent i to social influence.The input b ij ∈ R represents an input signal from theenvironment or a bias or predisposition that directly affectsagent i ’s opinion of option j . For example, the input b ij can be used to model the exogenous influence of agent i ’sinitial opinions, as in [2], where agents hold on to their initialopinions (sometimes called “stubborn” agents as in [48]).The general model of opinion dynamics is given, for everyagent i = 1 , . . . , N a and every option j ∈ , . . . , N o , by ˙ z ij = F ij ( Z ) − N o N o (cid:88) l =1 F il ( Z ) (5a) F ij ( Z ) = − d ij z ij (5b) + u i S (cid:32) N a (cid:88) k =1 A jjik z kj (cid:33) + N o (cid:88) l (cid:54) = jl =1 S (cid:32) N a (cid:88) k =1 A jlik z kl (cid:33) + b ij . The drift F ij : V → R , defined in (5b), is the sum of the threeinfluences on agent i ’s opinion of option j : resistance, social,and input. S q : R → [ − k q , k q ] with k q , k q ∈ R > for q ∈ { , } is a generic sigmoidal saturating function satisfyingconstraints S q (0) = 0 , S (cid:48) q (0) = 1 , S (cid:48)(cid:48) q (0) (cid:54) = 0 , S (cid:48)(cid:48)(cid:48) q (0) (cid:54) = 0 . S saturates same-option interactions, and S saturates inter-option interactions. S and S could be the same but aredistinguished in (5) for the most general statement of themodel. The rate of change of z ij is defined in (5a) to ensurethat z ij remains agent i ’s opinion of option j relative to itsopinions of the other options. Subtracting the average driftover options in (5a) models the mutual exclusivity of optionsand makes state space V forward invariant for (5), as provedin Lemma III.3 below.Let ˆ b i = N o (cid:80) N o l =1 b il be the average input to agent i andlet b ⊥ ij = b ij − ˆ b i be the relative input to agent i for option j . Lemma III.1.
The average input ˆ b i for each agent i has noeffect on the emergent dynamics of the network.Proof. Define G ij ( Z ) = F ij ( Z ) − b ij , G (cid:48) ij = G ij − (cid:80) N o p =1 G ip .Then ˙ z ij = G (cid:48) ij ( Z ) + b ij − N o (cid:80) N o p =1 b ip = G ij ( Z ) + b ⊥ ij .Lemma III.1 implies that only relative inputs affect theopinion dynamics. Without relative inputs, the system (5)always has the neutral point as an equilibrium. Lemma III.2. Z = is an equilibrium for (5) if and only ifthere are no relative inputs, i.e., b ⊥ ij = 0 for all i and all j . When relative inputs are small, i.e., they do not dominate thedynamics, the formation of opinions in the general model (5)is governed by the balance between the resistance term, whichresists opinion formation, and the social term, which promotesopinion formation. For illustrative purposes, consider the casein which u i = u ≥ for all i . Then for u small, resistancedominates and the system behaves linearly. The opinions z ij remain small and their relative magnitude is determined bythe small inputs b ⊥ ij . For u large, the social term dominatesand the system behaves nonlinearly. Opinions z ij form thatare much larger than, and potentially unrelated to, inputs b ⊥ ij ,even for very small initial conditions. B. Well-Definedness of Model
We show that the general model (5) is well defined byshowing in Lemma III.3 that V is forward invariant for (5)and in Theorem III.4 that solutions are bounded. This implies,as we show in Corollary III.4.1, that (5) can be mapped from V to the simplex product space V , so that the general model(5) describes opinion dynamics of agents with equal votingcapacity. Lemma III.3. V is forward invariant for (5) .Proof. For all i = 1 , . . . , N a , (cid:80) N o j =1 ˙ z ij = 0 , so if z i (0) + · · · + z iN o (0) = 0 , z i ( t ) + · · · + z iN o ( t ) = 0 for all t > . Let D i = − N o − N o d i N o d i . . . N o d iN o N o d i − N o − N o d i . . . N o d iN o ... ... . . . ... N o d i N o d i . . . − N o − N o d iN o . (6) Theorem III.4 (Boundedness) . Let ¯ U be a compact subsetof R . There exists R > such that, for all u, d ij , A jlik , b ij ∈ ¯ U , i, k = 1 , . . . , N a , j, l = 1 , . . . , N o , the set V ∩ {| z ij | ≤ R, i = 1 , . . . , N a , j = 1 , . . . , N o } is forward invariant for (5) .This implies that the solutions Z ( t ) of the dynamics (5) arebounded for all time t ≥ .Proof. By boundedness of S p ( · ) , there exists ˜ R > such that,for all u i , d ij , A jlik , b ij ∈ ¯ U , F ij ( Z ) = − d ij z ij + C ij ( Z ) , with | C ij ( Z ) | ≤ ˜ R . For all Z ∈ V , it holds that ddt (cid:107) Z (cid:107) = N a (cid:88) i =1 N o (cid:88) j =1 z ij ˙ z ij = N a (cid:88) i =1 N o (cid:88) j =1 z ij (cid:16) − d ij z ij + C ij ( Z )+ 1 N o N o (cid:88) l =1 ( d il z il − C il ( Z )) (cid:17) = Z T D Z + N a (cid:88) i =1 N o (cid:88) j =1 z ij (cid:32) C ij ( Z ) − N o N o (cid:88) l =1 C il ( Z ) (cid:33) ≤ Z T D Z + N a N o ˜ R (cid:107) Z (cid:107) where we have used (cid:80) N o j =1 z ij = 0 for all i . We compute Z T D Z = N a (cid:88) i =1 N o (cid:88) j =1 (cid:0) − d ij z ij (cid:1) + 1 N o N a (cid:88) i =1 N o (cid:88) l =1 d il z il N o (cid:88) j =1 z ij = N a (cid:88) i =1 N o (cid:88) j =1 − d ij z ij ≤ − min i,j { d ij }(cid:107) Z (cid:107) . Then, for all (cid:107) Z (cid:107) ≥ N a N o ˜ R min i,j { d ij } , it follows that ddt (cid:107) Z (cid:107) ≤−(cid:107) Z (cid:107) (cid:16) min i,j { d ij }(cid:107) Z (cid:107) − N a N o ˜ R (cid:17) ≤ . The result followsby [49, Theorem 4.18]. Corollary III.4.1.
Mapping to the Simplex Product V . Givena bounded set ¯ U ⊂ R , assume u, d ij , A jlik , b ij ∈ ¯ U , i, k =1 , . . . , N a , j, l = 1 , . . . , N o . Let r be defined by (1). Then, thevector field of (5) can be mapped from the forward invariantregion V ∩ {| z ij | ≤ R, i = 1 , . . . , N a , j = 1 , . . . , N o } tothe product of simplex V by the affine change of coordinates L : V ∩ {| z ij | ≤ R, i = 1 , . . . , N a , j = 1 , . . . , N o } → V Z (cid:55)→ rN o R Z + rN o .C. Clustering and Model Reduction Special network structure sometimes referred to as externalequitable partitions is often tied to synchronization and clus-tering in networked systems [50]–[53]. In this section we provethat such network structure constitutes a sufficient conditionfor the dynamics (5) of N a agents and N o options to reduceto the dynamics of N c clusters and N o options on a lowerdimensional attractive manifold. Each cluster p = 1 , . . . , N c represents N p of the N a agents forming opinions as a unit and (cid:80) N c p =1 N p = N a . Theorem III.5 (Model Reduction with Opinion Clusters) . Suppose there are N c clusters with N p agents in the p th clustersuch that (cid:80) N c p =1 N p = N a . Let I p be the set of indices foragents in the p th cluster. Assume for every p = 1 , . . . , N c : 1) u i = u p , d ij = d pj , b ij = ˆ b i + b ⊥ pj for i ∈ I p ; 2) A jjii = ¯ B jjpp , A jjik = B jjpp , A jlii = ¯ B jlpp , A jlik = B jlpp for i, k ∈ I p , and i (cid:54) = k ;3) A jjik = B jjps , A jlik = B jlps for i ∈ I p , k ∈ I s s = 1 , . . . , N c and s (cid:54) = p , with d pj > , u p ≥ , and B jlps , b ⊥ pj ∈ R for all p, s = 1 , . . . , N c , j, l = 1 , . . . , N o , p (cid:54) = s , j (cid:54) = l .Define bounded set K q ⊂ R > , q = 1 , , as the image of thederivative of the saturating function S (cid:48) q of (5) . If the followingcondition holds: sup κ ∈ K ,κ ∈ K (cid:110) − min j { d pj } + u p κ max j { ¯ B jjpp − B jjpp } + u p κ max j { B jlpp − ¯ B jlpp } (cid:111) < ∀ p = 1 , . . . , N c , (7) then every trajectory of (5) converges exponentially in time toan N c ( N o − -dimensional attracting manifold defined by E = { Z ∈ V | z ij = z kj ∀ i, k ∈ I p , p = 1 , . . . , N c } . (8) The dynamics on E reduce to (5) with N c agents with opinionstates ˆ z pj = z ij for any i ∈ I p , p = 1 , . . . , N c , with weights ˆ A jjpp = ¯ B jjpp + ( N p − B jjpp , ˆ A jjps = N s B jjps , (9a) ˆ A jlpp = ¯ B jlpp + ( N p − B jlpp , ˆ A jlps = N s B jlps . (9b) Proof.
Opinion dynamics (5) of agent i ∈ I p are defined by F ij ( Z ) = − d pj z ij + +ˆ b i + b ⊥ pj (10) + u p S ¯ B jjpp z ij + B jjpp (cid:88) k ∈I p z kj + N c (cid:88) s (cid:54) = ps =1 (cid:88) k ∈I s B jjps z kj + N o (cid:88) l (cid:54) = jl =1 S ¯ B jlpp z il + B jlpp (cid:88) k ∈I p z kl + N c (cid:88) s (cid:54) = ps =1 (cid:88) k ∈I s B jlps z kl . Let V T ( Z ) = (cid:80) N c p =1 V p ( Z ) , V p ( Z ) = (cid:80) i,k ∈I p (cid:80) N o j =1 ( z ij − z kj ) . Again let F ij ( Z ) = − d ij z ij + C ij ( Z ) . Then ˙ V p ( Z ) = − (cid:88) i ∈I p (cid:88) k ∈I p ( Z i − Z k ) T D p ( Z i − Z k )+ (cid:88) i ∈I p (cid:88) k ∈I p N o (cid:88) j =1 ( z ij − z kj )( C ij ( Z ) − C kj ( Z )) − N o (cid:88) i ∈I p (cid:88) k ∈I p N o (cid:88) j =1 N o (cid:88) l =1 ( z ij − z kj )( C il ( Z ) − C kl ( Z )) , (11)where D p is from (6). The third term in (11) is zero be-cause (cid:80) N o j =1 z ij = 0 on V . By the Mean Value Theo-rem, we can write C ij ( Z ) − C kj ( Z ) in the second term as u p (cid:0) κ ( ¯ B jjpp − B jjpp ) − κ ( ¯ B jlpp − B jlpp ) (cid:1) ( z ij − z kj ) , where κ ∈ K and κ ∈ K . Then we find that ˙ V p ( Z ) ≤ sup κ ∈ K ,κ ∈ K (cid:110) − min j { d pj } + u p κ max j { ¯ B jjpp − B jjpp } + u p κ max j { B jlpp − ¯ B jlpp } (cid:111) V p ( Z ) . When (7) is satisfied, using LaSalle’s invariance principle [49,Theorem 4.4] every trajectory of (5) converges exponentiallyin time to the largest invariant set of V T ( Z ) = 0 , which is E .Let ˆ z pj = z ij for any i ∈ I p . The dynamics (10) on E reduceto (5) with N a = N c and weights (9).Whenever conditions of Theorem III.5 are met, the groupof N a agents will converge to a clustered group opinion state.This can happen for a broad class of interaction networks butalso for an all-to-all network and interaction weights with thesame sign; see Section V for an illustration with two options.IV. O PINION F ORMATION : C
ONSENSUS AND D ISSENSUS
In this section, we show the following key results on opinionformation for dynamics (5) in the homogeneous regime whereagents have same parameter values and no relative input, andthe inter-agent network graph with adjacency matrix denoted ˜ A is unweighted. In Section VI we consider non-trivial inputs.1) Opinion formation can be modeled as a bifurcation, anintrinsically nonlinear dynamical phenomenon. Opinionsform rapidly since they are driven by nonlinearity ratherthan slow linear integration of evidence. Opinions canform even in the absence of input, as long as attention(urgency or susceptibility, etc.) is sufficiently high.2) The way opinions form (e.g., agreement or disagreement)at a bifurcation depends on the eigenstructure of ˜ A andwhether agents are cooperative or competitive.3) At the bifurcation, there are multiple stable solutions, andopinion formation breaks deadlock.4) Near the bifurcation, opinion formation is ultra-sensitiveto input.5) Away from the bifurcation, opinion formation is robust tosmall heterogeneity in parameter values and small inputs.6) For symmetric networks, (multistable) consensus and dissensus emerge generically.7) Consensus and dissensus can co-exist , revealing the possi-bility of easy transition between consensus and dissensus.We conclude with a generalization to clustered dynamics. A. Cooperative and Competitive Homogeneous Agents
The homogeneous regime of model (5) is defined as b ij = ˆ b ∈ R , d ij = d > , u i = u ≥ , A jjii = α ∈ R A jlii = β ∈ R , A jjik = γ ˜ a ik , A jlik = δ ˜ a ik (12)where γ, δ ∈ R and ˜ a ik ∈ { , } for all i, k = 1 , . . . , N a , i (cid:54) = k , and for all j, l = 1 , . . . , N o , j (cid:54) = l . The intra-agent networkis all-to-all for each agent (i.e., no clusters or hierarchies inthe option space ), whereas the inter-agent network is general This is the natural assumption, for instance, in the lowest-dimensional case N o = 2 , considered in most of the opinion-formation literature. with adjacency matrix (˜ a ik ) =: ˜ A ∈ R N a × N a . Agents havethe same resistance d , attention u , and input ˆ b for all options,i.e., relative inputs b ⊥ ij = 0 so all options have equal value.Parameters α , β , γ , δ determine qualitative properties ofopinion interactions. Parameter α determines sign and mag-nitude of opinion self-interaction. To avoid redundancy withresistance d , we assume α ≥ , i.e., either no self-coupling( α = 0 ) or self-reinforcing coupling ( α > ). Parameter β determines how different intra-agent opinions interact. Sincetotal voting capacity is fixed, it is natural to assume β < , i.e.,opinions are mutually exclusive . Parameters γ and δ determinewhether agents cooperate ( γ − δ > ) or compete ( γ − δ < . B. Opinion Formation Appears through a Bifurcation
The following theorem provides sufficient conditions un-der which opinion formation appears as a bifurcation fromthe neutral equilibrium Z = 0 and provides formulas tocompute the kernel along which the bifurcation appears. Let I N ∈ R N × N be the identity, N ∈ R N the vector of ones,and P o = ( I N o − N o N o TN o ) the projection onto ⊥ N o . Let R { v , . . . , v k } be the span of vectors v , . . . , v k ∈ R n . Theorem IV.1 (Opinion Formation as a Bifurcation) . Con-sider model (5) with parameters as in (12) . Let J be the Ja-cobian of the system evaluated at neutral equilibrium Z = .A. Cooperative agents.
Suppose that ( α − β ) > , ( γ − δ ) > .Then Z = is locally exponentially stable for < u < u c , u c = dα − β + λ max ( γ − δ ) (13) and λ max ≥ is the largest eigenvalue of ˜ A . Z = is un-stable for u > u c . At u = u c , an opinion-forming bifurcationappears along ker J = R { v a ⊗ v jo , j = 1 , . . . , N o } , where v a is an eigenvector of ˜ A corresponding to the eigenvalue λ max and v jo ∈ ⊥ N o , j ∈ { , . . . , N o } is a basis of V .B. Competitive agents.
Suppose that ( α − β ) > , ( γ − δ ) < .Then Z = is locally exponentially stable for < u < u d , u d = dα − β + λ min ( γ − δ ) (14) and λ min ≤ is the smallest eigenvalue of ˜ A . Z = is un-stable for u > u c . At u = u d , an opinion-forming bifurcationappears along ker J = R { v a ⊗ v jo , j = 1 , . . . , N o } , where v a is an eigenvector of ˜ A corresponding to the eigenvalue λ min and v jo ∈ ⊥ N o , j ∈ { , . . . , N o } is a basis of V .Proof. J = (cid:16) ( − d + u ( α − β )) I N a + u ( γ − δ ) ˜ A (cid:17) ⊗ P o . Thus, eigenvalues of J are of the form λ a λ o , where λ a isan eigenvalue of ( − d + u ( α − β )) I N a + u ( γ − δ ) ˜ A and λ o is an eigenvalue of P o restricted to V . Because the onlyeigenvalue of P o restricted to V is one, λ o = 1 , whereas λ a = − d + u ( α − β )) + uλ i ( γ − δ ) , where λ i , i = 1 , . . . , N a is an eigenvalue of ˜ A . Thus, for γ − δ > , λ a is largestfor λ i = λ max , whereas for γ − δ < , λ a is largest for λ i = λ min . It follows that, if γ − δ > , all eigenvalues of J are negative for u < u c , zero is an eigenvalue of J for u = u c (with multiplicity ( N o − N λ max , where N λ min isthe multiplicity of λ max ), and there exist positive eigenvalues for u > u c . Conversely, if γ − δ < , all eigenvalues of J are negative for u < u d , zero is an eigenvalue of J for u = u d (with multiplicity ( N o − N λ min , where N λ min isthe multiplicity of λ min ), and there exist positive eigenvaluesfor u > u d . The form of the eigenvectors of J correspondingto its zero eigenvalue for u = u c (resp., u = u d ) follows sincethe eigenvectors of the Kroenecker product of matrices is theKroenecker product of the eigenvectors.Theorem IV.1 reveals how opinions can form even withoutinput: opinions form when attention u is greater than threshold u c (if agents are cooperative) or u d (if agents are competitive). Remark IV.1 (Agreement and Disagreement Decision Makingfor General Connected Graphs) . By Theorem IV.1, the eigen-vectors associated to the maximum and minimum eigenvaluesof ˜ A predict how opinion formation appears in the cooperativeand competitive homogeneous regimes, respectively. In [54]we use this result to study how opinions distribute over thenetwork in the formation of agreement for cooperative agentsand disagreement for competitive agents. Remark IV.2 (Intra-agent Coupling) . The influence of hier-archies, clusters, or more general network structures, in theoption space can be studied with the same techniques as inthe proof of Theorem IV.1 by replacing the eigenvalue andeigenvectors of P o with those of the intra-agent Jacobianarising from the intra-agent adjacency matrix. Remark IV.3 (Multistability and Breaking Deadlock) . Thereis multistability of opinion configurations at the bifurcation.When agents cooperate and ker J is made of agreementvectors, if agreement in favor of one option is stable thenagreement in favor of each other option is stable, and likewisefor disagreement solutions. There is a deadlock when u < u c ( u < u d ) and breaking of deadlock when u > u c ( u > u d ). Remark IV.4 (Ultra-sensitivity) . At the bifurcation the lin-earization is singular, and the model is ultra-sensitive attransition from neutral to opinionated. Even infinitesimal per-turbations (e.g., tiny difference in option values) are sufficientto destroy multistability at the bifurcation by selecting a subsetof stable equilibria to which the model can converge (e.g.,those corresponding to higher-valued options), a phenomenonknown as forced-symmetry breaking and widely exploited innonlinear decision-making models [22], [29], [38].
Remark IV.5 (Robustness) . Generically, stable equilibriathat appear at the bifurcation are hyperbolic, and thus theyand their basin of attraction are robust to perturbations,a key property that ensures stability of opinion formationdespite (sufficiently small) changes in inputs, heterogeneityin parameters, perturbations in the communication network.Robustness bounds can be derived using methods like thoseused for Hopfield networks in [55]. Robust multistability ofequilibria gives the opinion-forming process hysteresis, andthus memory, between different opinion states: once an opinionis formed in favor of an option, a sufficiently large change inthe inputs, is necessary for a switch to a new opinion state. C. Consensus and Dissensus Generic for Symmetric Systems
When the inter-agent network has symmetry , we can useTheorem IV.1 to prove that consensus and dissensus solu-tions are generic. Model-independent results [36, Theorem 4.6and Remark 4.7] ensure that, in the presence of symmetry, ker J = W c or ker J = W d . I.e., if (5) is symmetric withrespect to a group Γ a that acts by swapping the agent indicestransitively, then generically ker J = W c or ker J = W d .In the homogeneous regime (12), agent symmetry of (5) isfully determined by ˜ A as proved in the following propositionfor the maximally symmetric case Γ a = S N a and the highlysymmetric case Γ a = D N a . The same result holds, with similarproof, for other agent symmetries, e.g., Γ a = Z N a , Γ a = A N a . Proposition IV.2.
Consider model (5) in the homogeneousregime defined by (12) . Then the following hold true:1) Model (5) is ( S N o × S N a ) -equivariant if and only if ˜ A isthe adjacency matrix of an all-to-all graph;2) If ˜ A is the adjacency matrix of an undirected cycle graph,then model (5) is ( S N o × D N a ) -equivariant.Proof. The proof of (1) follows analogously to that of [36,Theorem 2.5] with the additional coefficient d ij on the linearterms. It is omitted due to space constraints.To prove (2), it is sufficient to show equivariance of the dy-namics under the action of generators of S N o × D N a . Element σ ∈ S N o acts on V by permuting the elements of each agent’sopinion Z i . Generators of S N o are N o transpositions σ j whereeach σ j swaps elements j and j + 1 (or N o and when j = N o ). Let F i ( Z ) = ( F i ( Z ) , . . . , F iN o ( Z )) and observe that σ j F i ( Z ) = ( F i ( Z ) , . . . , F i ( j +1) ( Z ) , F ij ( Z ) , . . . , F iN o ( Z )) .Computing F i ( σ j Z ) , only F ij and F i ( j +1) are changed, with F ij ( σ j Z ) = − dz i ( j +1) + u (cid:16) S (cid:0) αz i ( j +1) + γz ( i − j +1) + γz ( i +1)( j +1) (cid:1) + N o (cid:88) l (cid:54) =( j +1) l =1 S (cid:0) βz il + δz ( i − j +1) + δz ( i +1)( j +1) (cid:1) (cid:17) + ˆ b and analogously for F i ( j +1) . Thus, σ j F i ( Z ) = F i ( σ j Z ) forall j = 1 ...N o , and for all i = 1 , . . . , N a , and the dynamicsare equivariant under the action of S N o . Element ρ ∈ D N a acts on V by permuting the order of the agent vectors Z i inthe total system vector Z = ( Z , . . . , Z N a ) . The generatorsof D N a are the reflection element ρ which reverses the orderof elements in Z , and a rotation ρ which cycles forward thevector by one element, mapping each element i to i + 1 (and N a to ). Let F ( Z ) = ( F ( Z ) , . . . , F N a ( Z )) and observe that ρ F ( Z ) = ( F N a ( Z ) , F N a − ( Z ) , . . . , F ( Z ) , F ( Z )) and ρ F ( Z ) = ( F N a ( Z ) , F ( Z ) , F ( Z ) , . . . , F N a − ( Z )) . Forcompactness we leave out the full expression for F ij ( ρ p Z ) .We conclude that ρ F ( Z ) = F ( ρ Z ) and ρ F ( Z ) = F ( ρ Z ) ,i.e., the dynamics are equivariant under the action of D N a . Remark IV.6.
More generally, the symmetry group of theopinion dynamics is determined by the automorphism groupof the multi-graph associated to ˜ A . The proof follows as forProposition IV.2. One can break part of the option symmetry by introducing structure in how opinions about different optionsinteract, at both the intra- and inter-agent level. Proposition IV.2 reveals that symmetry in the opinion net-work constrains the number of parameters that control opinionformation. Because Proposition IV.2 is necessary and sufficientin the maximally symmetric case S N o × S N a , it follows thatthe four parameters α, β, γ, δ are all and the only parametersthat can tune the intra- and inter-agent opinion interactionsof a maximally symmetric network. When the agent sym-metry group is smaller than S N a , e.g., D N a , the number ofparameters determining the inter-agent opinion dynamics canincrease but the sufficient condition in Proposition IV.2 showsthat there are again only two such parameters associated toany given inter-agent network graph with symmetry D N a ,e.g., an undirected N a -cycle. The same is true for other agentsymmetry groups, or when some option symmetry is lost.The next corollary follows from Theorem IV.1, Proposi-tion IV.2, and [36, Theorem 4.6 and Remark 4.7]. The highlysymmetric cases considered provide an opinion-formationanalogue of the classical all-to-all and balanced cases forconsensus and synchronization dynamics (e.g., [4], [56]–[58]). Corollary IV.2.1 (Consensus from Cooperation and Dissensusfrom Competition) . Consider model (5) in the homogeneousregime (12) with inter-agent adjacency matrix ˜ A and α − β > . Let u c and u d be defined by (13) and (14) . Cooperative Agents and Consensus . If agents are coopera-tive ( γ − δ > ) and ˜ A is either all-to-all or an undirectedcycle, then opinion formation appears as a bifurcation alongthe consensus space at u = u c with λ max = N a − for theall-to-all case and λ max = 2 for the cycle case. Competitive Agents and Dissensus . If agents are competitive( γ − δ < ) and ˜ A is either all-to-all or an undirected cycle,then opinion formation appears as a bifurcation along thedissensus space at u = u d with λ min = − for the all-to-allcase, λ min = − for the cycle case, when N a is even, and λ min = 2 cos( π ( N a − /N a ) , when N a is odd. Remark IV.7 (Stability of Consensus and Dissensus) . Con-sensus and dissensus solution branches predicted for thesymmetric networks in Corollary IV.2.1 are a consequence ofthe Equivariant Branching Lemma [40, Section 1.4], and aremade of hyperbolic equilibria. Their stability can be provedusing the tools in [59, Section XIII.4] and [40, Section 2.3].
Remark IV.8 (Mode Interaction and Coexistence of Consen-sus and Dissensus) . When γ = δ , there is mode interaction [60], and bifurcations along the consensus and dissensusspaces appear at the same critical value of u . This regimeis especially interesting because it allows for co-existence ofstable consensus and dissensus solutions, which can result inagents easily transitioning between consensus and dissensus inresponse to changing conditions. However, additional primarysolution branches not captured by equivariant analysis canappear in this regime; classification of these for higher di-mensional permutation symmetries remains an open problem. We generalize Theorem IV.1 to a network with N c clusters. (a) (b) Fig. 2: Simulations for N o = 2 options and N a = 8 agents (top)and N o = 3 options and N a = 12 agents (bottom). Opinions form(a) consensus when agents are cooperative: γ = 0 . , δ = − . ;(b) dissensus when agents are competitive: γ = − . , δ = 0 . .In each plot, α = 0 . , β = 0 . , d = 1 , u = 3 , ˆ b = 0 , andrandom initial conditions are the same. All parameters (includingcommunication weights α, β, γ, δ ) were perturbed with small randomadditive perturbations drawn from a normal distribution with (a)variance 0.01, (b) variance 0.001. Opinions z ij are mapped to thesimplex for 3 options using Corollary III.4.1 with r = 1 and R = 2 u . Proposition IV.3 (Symmetric Clustered Dynamics) . Supposein addition to satisfying assumptions of Theorem III.5, theparameters of (10) satisfy u p = u ≥ , d pj = d > , b pj = ˆ b ∈ R , and the interaction weights satisfy α := ¯ B jjpp + ( N p − B jjpp , γ := N s B jjps , (15a) β := ¯ B jlpp + ( N p − B jlpp , δ := N s B jlps , (15b) with α, β, δ, γ ∈ R , for all p, s = 1 , . . . , N c , j, l = 1 , . . . , N o , p (cid:54) = s , j (cid:54) = l . Every trajectory of the dynamics (10) convergesexponentially in time to a N c ( N o − -dimensional attractingmanifold E defined by (8), the reduced dynamics on which are ( S N o × S N c ) -equivariant. The associated clustered dissensusspace along which solution branches emerge is ˆ W d = { ( Z , . . . , Z N a ) | N c (cid:88) p =1 N p (cid:88) i ∈I p Z i = } . (16) Proof.
This follows directly from model reduction in TheoremIII.5 and Proposition IV.2.Fig. 3 illustrates consensus and dissensus formation in a3-cluster network with three options, and small perturbationsadded to all parameters including interactions weights.V. S
PECIALIZATION TO M ODELS IN THE L ITERATURE
In this section we show how the general model (5) for N o = 2 options specializes to well-studied nonlinear and linearconsensus models in the literature. We also show that networkconditions, e.g., structural balance, proved to be necessary forclustering in linear models, e.g., for bipartite consensus [5],are not necessary when there is a saturating nonlinearity. (a) (b) Fig. 3: Dynamics of a 3-cluster network with N = 2 , N = 3 , N = 4 . For all trajectories u = 4 , d = 1 , ˆ b = 0 , and the weightsin (15) satisfy ¯ B jjpp = B jjpp , ¯ B jlpp = B jlpp , α = 0 . , β = 0 . . (a)Consensus: γ = 0 . , δ = − . ; (b) Dissensus: γ = − . , δ = 0 . .Small random perturbations drawn from a normal distribution withvariance 0.01 were added to all of the model parameters. Mapping tosimplex was obtained from Corollary III.4.1 with r = 1 and R = 2 u . For N o = 2 , the opinion state of agent i is one-dimensional: Z i = ( z i , z i ) , with z i + z i = 0 . We define x i = z i = − z i as agent i ’s opinion. Then, opinion dynamics (5) reduce to ˙ x i = − d i x i + 12 u i (cid:32) S (cid:32) N a (cid:88) k =1 A ik x k (cid:33) − S (cid:32) − N a (cid:88) k =1 A ik x k (cid:33) + S (cid:32) − N a (cid:88) k =1 A ik x k (cid:33) − S (cid:32) N a (cid:88) k =1 A ik x k (cid:33)(cid:33) + b ⊥ i (17)where b i := b i = − b i , and d i = ( d i + d i ) . Let thenetwork opinion state be x = ( x , . . . , x N a ) ∈ R N a and vectorof relative inputs be b ⊥ = ( b ⊥ , . . . , b ⊥ N a ) ∈ R N a .We first show how (17) reduces to the nonlinear model ofconsensus studied in [20], [22], [24]. Theorem V.1.
For N a agents and N o = 2 options, let A jjii = A jlii = A jlik = 0 , A jjik = γ ik ∈ R , u i = u ≥ , and d ij = (cid:80) N a k =1 (cid:80) l =1 | A jlik | = (cid:80) N a k =1 | γ ik | := d i , for i, k = 1 , . . . , N a , i (cid:54) = k , j, l = 1 , , j (cid:54) = l . Then, opinion dynamics (5) reduce to ˙ x i = − d i x i + u ˆ S (cid:32) N a (cid:88) k =1 γ ik x k (cid:33) + b ⊥ i , (18) where i (cid:54) = k and ˆ S p ( x ) = (cid:0) S p ( x ) − S p ( − x ) (cid:1) , p ∈ { , } ,are odd sigmoids and x i := z i = − z i , b i := b i = − b i ,and d i = ( d i + d i ) . In particular, (5) specializes to thenonlinear consensus dynamics studied in [20], [22], [24].Proof. Since N o = 2 , (5) reduces to (17). By the assumptions,(17) further reduces to (18). For γ ik ≥ , dynamics (18) areequivalent to the nonlinear model [20], [22], [24].In the specialization of Theorem V.1, the self-reinforcingweights are set to zero and the resistance parameter d i is setequal to agent i ’s indegree. This reduces the relevance of theadjacency matrix in the general model to relevance of theLaplacian, as becomes apparent in the linearization of (18).We next show that the linearization of (18) is exactly the“Altafini” model of linear consensus with antagonistic, i.e.,signed, interconnections [5], [61], [62]. Proposition V.2.
The linearization of (18) about x = 0 with u = 1 and b ⊥ i = 0 is ˙ x = −L x , where L = D − A is the
Fig. 4: For small initial conditions, the model from [5] (left) approx-imates the response of the nonlinear model (18) (right) with N a = 3 , u = 1 , b ⊥ i = 0 , and ˆ S = tanh . As expected, clustered behavior isobserved in both models for the structurally balanced graph definedin [5, Example 1] and the same initial conditions.Fig. 5: The model from [5] (left) and nonlinear model (18) (right)with N a = 5 , u = 5 , b ⊥ i = 0 , ˆ S = tanh , same random initialconditions, and same adjacency matrix given by γ ik = − for i, k ∈I p , i (cid:54) = k , and γ ik = − for i ∈ I p , k ∈ I s , p (cid:54) = s for clusterswith indices I = { , } and I = { , , } . Because the adjacencymatrix is not structurally balanced, the Altafini model converges tothe neutral solution. The nonlinear model, however, converges to astable dissensus state, as predicted by Theorems III.5 and IV.1. signed graph Laplacian matrix, D = diag( d i ) ∈ R N a × N a is the associated degree matrix and the entries of adjacencymatrix A ∈ R N a × N a are the inter-agent weights γ ik .Proof. The proof follows by linearization of (18).In the Altafini model, a clustered disagreement outcome,called bipartite consensus , arises for the network only ifthe network graph is structurally balanced ( L has a zeroeigenvalue); see [5, Theorem 2]. Multi-option generalizationsof linear consensus also require structural balance for clusteredbehavior [17]. By Proposition V.2, this property is recoveredlocally in the nonlinear model (18), as illustrated in Fig. 4.However, in light of Theorems III.5 and IV.3, we have thefollowing corollary, illustrated in Fig. 5. Corollary V.2.1.
A structurally balanced network is not nec-essary for opinion clustering in the nonlinear model (5) . VI. D
YNAMIC F EEDBACK AND T UNABLE S ENSITIVITY
We have established that existence of consensus and dis-sensus equilibria and multistability of opinion formation out-comes arise from bifurcations of the general opinion dynamicmodel (5). In this section we explore how ultra-sensitivity toinputs b ⊥ ij , robustness to changes in inputs, cascade dynamics,and transitions between consensus and dissensus all arise asa consequence of opinion multistability . With the additionof dynamic state feedback for model parameters in (5), theopinion formation process can reliably amplify arbitrarilysmall relative inputs b ⊥ ij , reject small changes in input as unwanted disturbance, facilitate an opinion cascade even ifonly one agent gets an input, and enable groups to move easilybetween consensus and dissensus. The choice of feedbackdesign parameters determine implicit thresholds that make allof these behaviors tunable.In Section VI-A we propose a dynamic state feedback lawfor attention u i in (5). that provides tunable sensitivity androbustness (Section VI-B), tunable cascade dynamics (VI-C),and tunable consensus/dissensus transitions (VI-D). A. Dynamic State Feedback Law for Attention
Using feedback design like that in [63] and [22], weaugment the opinion dynamics (5) by introducing feedbackdynamics on the attention parameter u i for each agent i , inthe form of a saturated nonlinear leaky integrator: τ u ˙ u i = − u i + S u N o N o (cid:88) j =1 N a (cid:88) k =1 N o (cid:88) l =1 (cid:16) ¯ A jlik z kl (cid:17) = − u i + S u (cid:18) N o (cid:107) ¯ A i Z (cid:107) (cid:19) , (19)where ¯ A i ∈ R N o × N a N o Here, τ u > is the attention dynamicstime scale, which can be freely chosen. S u is a smoothsaturating function, satisfying S u (0) = 0 , S u ( y ) → u sat > as y → ∞ , S (cid:48) u ( y ) > for all y ∈ R , and S (cid:48)(cid:48) u ( y ) < for all y > y m , and y m > . We can always decompose S u as S u ( y ) = u f (cid:16) F (cid:0) g ( y − y m ) (cid:1) − F ( − gy m ) (cid:17) , (20) F : R → [0 , a smooth monotone sigmoid with F (cid:48) ( x ) > for all x ∈ R , F (cid:48) ( x ) = F (cid:48) ( − x ) , F (cid:48)(cid:48) ( x ) < for x > . Designparameter y m defines the half-activation point of S u , u f theupper bound of S u , and g the slope of S u .The weights ¯ A jlik can be real or binary { , } . ¯ A jlik canbe related to the corresponding interaction weights A jlik in(5), or independently defined. The latter case allows agentsto exert influence by exciting their neighbors to form anopinion without influencing what opinion they form. For allsimulations we define S u by (20) with F ( x ) = 1 / (1 + e − x ) ,Large inputs saturate to u sat = u f e gy m / (1 + e gy m ) . For gy m > , u f is a good approximation of u sat . Remark VI.1 (Well-Definedness of Opinion Dynamics withFeedback) . Observe that ˙ u i (0) < for u i (0) ≥ u sat and ˙ u i (0) > for u i (0) ≤ . It follows that the set { ≤ u i ≤ u sat } is attractive and forward invariant, and | ˙ u i | is uniformlybounded in { ≤ u i ≤ u sat } . Thus, boundedness of Z ( t ) inthe presence of attention feedback dynamics follows along thesame lines as the proof of Theorem III.4. So, the attentionfeedback dynamics do not modify invariance of the simplexproduct V and the coupled dynamics (5),(19) are well-defined.B. Tunable Sensitivity and Robustness of Opinion Formation We focus on the case N o = 2 and first consider uncoupledagents, i.e. γ ik = δ ik = 0 , with A ii = A ii = α i and A ii = A = β i , in which case dynamics (17) reduce to ˙ x i = − d i x i + u i ˆ S ( x i )+ b ⊥ i := q ( x i , u i , b ⊥ i ) (21) where ˆ S ( x i ) := ˆ S ( α i x i ) − ˆ S ( β i x i ) and ˆ S , ˆ S are definedas in Theorem V.1. The following assumption ensures that ˆ S is a monotonically increasing sigmoid. Assumption 1. α i > > β i for all i = 1 , . . . , N a . Assumption 1 is satisfied for self-reinforcing, mutually ex-clusive options and we let it hold for the rest of the paper. In allsimulations we let ˆ S ( x ) = tanh( x ) and ˆ S ( x ) = tanh(2 x ) .The following proposition classifies the steady-state opinionbehaviors of (21); see Fig. 6 for an illustration. Proposition VI.1 (Opinion Dynamics of Uncoupled Agents) . For opinion dynamics (21) for each agent i , the following hold:A. For b ⊥ i = 0 and ≤ u i ≤ u ∗ i with u ∗ i = d i α i − β i , theneutral equilibrium x i = 0 is globally asymptotically stable.It is also locally exponentially stable for < u i < u ∗ i , andglobally exponentially stable for u i = 0 ;B. For b ⊥ i = 0 , the bifurcation problem q ( x i , u i ,
0) = 0 hasa pitchfork singularity at ( x ∗ i , u ∗ i ) = (cid:16) , d i α i − β i (cid:17) . For u i >u ∗ i the neutral equilibrium x i = 0 loses stability and twoadditional branches of locally exponentially stable equilibria x i = ± x si emerge, implicitly defined by − d i x si + u i ˆ S ( x si ) ;C. For b ⊥ i (cid:54) = 0 , q ( x i , u i , b ⊥ i ) is a 1-parameter unfoldingof the symmetric pitchfork. For all u i ≥ , a single locallyexponentially stable equilibrium x si exists in the half-plane sign( x i ) = sign( b ⊥ i ) . Moreover, x si varies smoothly with u i and ∂x si ∂u i > . For sufficiently large values of u i awayfrom the singularity u ∗ i , a second locally exponentially stableequilibrium exists in the half-plane sign( x i ) = − sign( b ⊥ i ) .Proof. Uncoupled agent dynamics (21) are a one-dimensionalinstance of the nonlinear consensus model studied in [22].Conclusions A − C follow from [22, Theorem 1].We next establish how the shape of the bifurcation diagramwith no input (Fig. 6, left) from Proposition VI.1 A,B changeswith input b ⊥ i (Fig. 6, right) as described in Proposition VI.1 C . Lemma VI.2 (Input Response Without Feedback) . Let x si bea hyperbolic equilibrium of (21) with input b ⊥ i and u i ≥ .Then ∂x si ∂b ⊥ i > if x si is stable, and ∂x si ∂b ⊥ i < if it is unstable.Proof. Differentiate q ( x si , u i , b ⊥ i ) = 0 by b ⊥ i at constant u i : ∂x si ∂b ⊥ i = 1 d i − u i ˆ S (cid:48) ( x si ) = − λ (22)where λ ∈ R is the eigenvalue of the linearization of (21)evaluated at equilibrium x s . At any stable equilibrium x s of(21), λ < and thus ∂x si ∂b ⊥ i > . The unstable case follows.We can couple the agents with the attention feedbackdynamics (19). For simplicity, let the feedback weights be ¯ A ik = ¯ A ik = ¯ A ik = ¯ A ik := ¯ a ik ∈ { , } (23)with ¯ a ik = 1 whenever u i is influenced by the state of agent k . The attention feedback law (19) then simplifies to τ u ˙ u i = − u i + S u (cid:32) N a (cid:88) k =1 (¯ a ik x k ) (cid:33) . (24) Fig. 6: Bifurcation diagrams of (21) with α i = 2 , β i = − , d i =1 with no input (left) and a positive input (right). Black lines plotsteady-state solutions (nullclines) and gray arrows are streamlinesshowing direction of the flow. Fig. 7: Sensitivity of opinion formation to input magnitude. Trajec-tories of (21),(25) with g = 10 , y m = 1 , u f = 2 , α i = 2 , β i = − , d i = 1 , τ u = 1 for b ⊥ i = 0 . (left) and b ⊥ i = 1 . (right). Initialstate ( u i (0) , x i (0)) = (0 , is a blue circle, and final state a yellowdiamond. Nullclines of (21) are black solid and (25) are red dashed.Gray arrows show streamlines of the flow. Color scale is time. When agent i is decoupled from all other agents, i.e., ¯ a ii = 1 and ¯ a ik = 0 for all k (cid:54) = i , (24) reduces to τ u ˙ u i = − u i + S u ( x i ) := h ( x i , u i ) . (25)As shown in Fig. 7, the equilibria of (21),(25) correspondto intersections of the x i -nullcline q ( x i , u i , b ⊥ i ) = 0 (blacksolid) and u i -nullcline h ( x i , u i ) = 0 (red dashed). PropositionVI.1 defines the shape of the x i -nullcline; see Fig. 6 (right);Lemma VI.2 describes how the shape changes with input b ⊥ i .Let agent i be strongly opinionated when it is opinionated andits attention is close to its saturation value, i.e., u i (cid:39) u sat . Tunable sensitivity of opinion formation to input b ⊥ i canthen be understood by comparing the plots of Fig. 7, where thetrajectory for agent i is plotted on the left for b ⊥ i = 0 . and onthe right for b ⊥ i = 1 . . For the given parameters and b ⊥ i = 0 . ,the nullclines intersect at three points. For unopinionatedinitial conditions, the opinion state is attracted to the pointcorresponding to an unopinionated equilibrium: agent i rejectsthe input b ⊥ i = 0 . and does not form a strong opinion. Forthe same parameters and b ⊥ i = 1 . , the nullclines intersectat only one point, corresponding to a strongly opinionatedequilibrium. Thus, for the same initial conditions, agent i accepts the input b ⊥ i = 1 . and forms a strong opinion. Theimplicit sensitivity threshold that distinguishes rejected fromaccepted inputs can be tuned by design parameter y m . Tunable robustness of opinion formation to changes in input b ⊥ i can be understood by comparing the sequence of plots inFig. 8(a) to the sequence of plots in Fig. 8(b). In (a) and (b)the plot on the left shows agent i forming a strong opinionin the direction of the input b ⊥ i = 1 . In (a) and (b) the plot (a)(b) Fig. 8: Robustness of opinion formation to changes in input. Tra-jectories of (21),(25) with g = 10 , y m = 1 , α i = 2 , β i = − , d i =1 , τ u = 1 . (Left) Input is b ⊥ i = 1 , initial state ( u i (0) , x i (0)) = (0 , is a blue circle, and final state is a cyan diamond. (Right) Inputchanges to b ⊥ i = − , initial state is final state on left and final stateis yellow square. (a) u f = 1 , and agent changes opinion in directionof new input. (b) u f = 2 . , and agent retains opinion in originaldirection. Nullclines, streamlines, and time are drawn as in Fig. 7. on the right shows what happens to agent i ’s opinion whenthe input switches to b ⊥ i = − , i.e., an input that is in theopposite direction of the original input. In (a), when u f = 1 ,agent i accepts the change of input and forms a strong opinionin the direction of the new input. In (b), when u f = 2 . ,agent i rejects the change of input and retains a strong opinionin the direction of the original input. The implicit robustnessthreshold that distinguishes rejected from accepted changes ininput can be tuned by design parameter u f .To develop systematic tools for tuning design parameters u f and y m , we provide some geometric intuition about thebehavior of the opinion and attention dynamics (21),(25). Lemma VI.3 (Trapping Region) . There exists ε i > suchthat the set u i ≥ , x i ≥ − ε i is forward-invariant for thecoupled dynamics (21),(25) whenever b ⊥ i > . Similarly, theset u i ≥ , x i ≤ ε i is forward-invariant whenever b ⊥ i < .Proof. This follows from the observation that ˙ u i ≥ when-ever u i = 0 and ˙ x i = b ⊥ i whenever x i = 0 . Lemma VI.4 (Flow Monotonicity) . The flow generated byvector field (21),(25), restricted to the forward-invariant quad-rants from Proposition VI.3, is monotone in the sense of [64].Proof.
Consider b ⊥ i > and restrict the system to the positivequadrant u i ≥ , x i ≥ . In this quadrant ∂q∂u i ≥ and ∂h∂x i ≥ , and the proposition follows by [64, Chapter 3 Proposition1.1, Remark 1.1]. The result for b ⊥ i < and the negativequadrant u i ≥ , x i ≤ follows analogously.Lemmas VI.3,VI.4 show how agent i ’s input picks out thetrapping region. In the forward-invariant quadrant determined by b ⊥ i , the equilibrium opinions x si are solutions of − d i x si + S u (cid:0) ( x si ) (cid:1) ˆ S ( x si ) + b ⊥ i , (26)and they all belong to the monotone, continuous branch of the x i -nullcline (Fig. 6, right). This branch is made of hyperbolicequilibria of (21), and thus Lemma VI.2 applies. It followsthat if slope g in S u is sufficiently large, then, generically, the x i and u i -nullclines intersect three times (when | b ⊥ i | is small,Fig. 7 left) or one time (when | b ⊥ i | is large, Fig. 7 right) in theforward-invariant quadrant. The transition between three andone equilibria happens through a fold bifurcation.Fig. 9 plots x si for four different parameter combinations.For sufficiently large g , (26) exhibits two fold singularities inthe half-plane x si ≥ as b ⊥ i is varied. The right fold occurs forpositive values of b ⊥ i and corresponds to the annihilation of theunopinionated equilibrium with a saddle point. The left foldoccurs for negative or positive values of b ⊥ i and corresponds tothe annihilation of the strongly opinionated equilibrium with asaddle point. Equilibria and fold singularities in the half-plane x si ≤ are obtained by reflecting Fig. 9 along the x si = 0 axis. We formalize this geometric discussion in Assumption 2,which we let hold for the rest of the paper. Assumption 2.
Parameter g in S u of (20) is sufficiently largeso (26) has two fold points in each half-plane x i > and x i < of the ( b ⊥ i , x si ) plane. Let ( b LF , ± x LF ) be the coordinatesof the left folds and ( b RF , ± x RF ) the coordinates of the rightfolds, where x RF < √ y m < x LF , b LF < b RF , < b RF . Fold points of the opinion plus attention dynamics areimportant because they determine the appearance and disap-pearance of unopinionated and strongly opinionated equilibria,thus tuning sensitivity and robustness thresholds. The fact thatthe fold point locations depend on the model parameters asillustrated in Fig. 9 is proved in the following theorem.
Theorem VI.5 (Tunable Fold Point Location) . Consider thesystem (21),(25). Suppose u sat > u ∗ i = d i α i − β i . Then, atthe four fold points of (26) , ∂b p ∂y m > and ∂b p ∂u f < , p ∈ { RF, LF } . Further, the distance between the two folds inthe same half-plane increases with u f : ∂∂u f ( b LF − b RF ) > .Proof. Let the right hand side of (26) be f ( x si , b ⊥ i ) . Theexpressions f ( x p , b p ) = 0 and ∂f ∂x p := f ( x p , b p ) = 0 for p ∈ { RF, LF } together define the two fold points. The foldequations define x p and b p as implicit functions of y m . Usingthe implicit function theorem, we find ∂b p ∂y m = − ∂f ∂y m at a fold: ∂b p ∂y m = u f g (cid:18) F (cid:48) (cid:0) g (cid:0) ( x p ) − y m (cid:1)(cid:1) − F (cid:48) ( − gy m ) (cid:19) ˆ S ( x p ) > .Analogously, the fold equations define x p and b p as implicitfunctions of u f . Using the implicit function theorem andthe monotonicity of F , we find ∂b∂u f = − ∂f ∂u f at a fold: ∂b p ∂u f = − (cid:18) F (cid:0) g (cid:0) ( x p ) − y m (cid:1)(cid:1) − F ( − gy m ) (cid:19) ˆ S ( x p ) < . That ∂b LF ∂u f − ∂b RF ∂u f > follows from the above expression andmonotonicity of F and ˆ S since x RF < x LF .The following theorem shows how uncoupled agents withattention feedback dynamics can become strongly opinionated -3 205 y m = 1 y m = 3 -3 205 u f = 1u f = 2.5Upper Branch LF RFRFLF
Lower Branch
LF LF RF (a) (b)
Fig. 9: Solutions of (26). g = 10 , d i = 1 , α i = 2 , β i = − . (a) u f = 2 . , y m = 1 (solid), y m = 3 (dashed). (b) y m = 2 . , u f = 1 (solid), u f = 2 . (dashed). in response to a sufficiently strong input b ⊥ i . It also shows thatthe sensitivity threshold, which distinguishes between smalland large inputs, can be tuned by tuning the right fold locationaccording to Theorem VI.5. Theorem VI.6 (Input Response for Uncoupled Agents withAttention Dynamics) . Let u sat > u ∗ i = d i α i − β i . There exists (cid:15) > such that if (cid:107) ( x (0) , u (0)) (cid:107) < (cid:15) , the following hold foropinion and attention dynamics (21),(25):A. For all b ⊥ i (cid:54) = 0 , ( x i ( t ) , u i ( t )) converges asymptotically to anonzero equilibrium ( x si , u si ) with u si > and x si of the samesign as b ⊥ i . The value of x si is the value of the continuoussolution branch in the pitchfork unfolding of (21) at u i = u si ;B. There exist constants < b < b RF and C ≥ such thatfor all | b ⊥ i | ≤ b , there exists an equilibrium ( x si , u si ) with (cid:107) ( x si , u si ) (cid:107) ≤ C | b ⊥ i | , and (cid:107) ( x i ( t ) − x si , u i ( t ) − u si ) (cid:107) ≤ ce − t/τ for some c, τ > .C. For any ν > , there exists a threshold b th ≥ b RF > suchthat if b ⊥ i > b th , then there exists a unique, locally exponen-tially stable equilibrium ( x si , u si ) in the quadrant x i , u i ≥ ,with | u si − u sat | < ν . Moreover, lim t →∞ (cid:107) ( x i ( t ) − x si , u i ( t ) − u si ) (cid:107) = 0 .Proof. A. By Lemma VI.4, the flow is monotone in theforward invariant set determined by Lemma VI.3. All tra-jectories are bounded and by [64, Chapter 3 Theorem 2.2]convergence to an equilibrium in the positive quadrant follows.By Proposition VI.1- C , the equilibrium values x si on thecontinuous branch of solutions are of the same sign as b ⊥ i . B. When b ⊥ i = 0 , the Jacobian of (21),(25) at equilibrium ( x si , u si ) = (0 , has eigenvalues − d i , − . Thus, (0 , isexponentially stable. Both ∂q∂b ⊥ i and ∂h∂b ⊥ i are bounded uni-formly in time. The conclusion follows by [49, Corollary 5.1],definition of small-signal finite-gain stability [49, Definition5.2], and monotonicity of the flow. C. After the right fold, i.e., for b ⊥ i > b RF , the onlyequilibrium in the quadrant x i , u i ≥ is the strongly opinion-ated equilibrium. By Lemma VI.2, the x i -coordinate of thisequilibrium can be made arbitrarily large by increasing b ⊥ i and thus u si = S u (( x si ) ) can be made arbitrarily close to itssaturation value u sat . Remark VI.2 (Sensitivity Threshold of Opinion Formation) . The coordinate of the right fold b RF in (26) (Fig. 9) cor-responds to the minimum threshold b th from Theorem VI.6.Thus, b RF indicates how sensitive the agent is to input: agent i strongly amplifies inputs b ⊥ i > b RF and does not amplifyinputs b ⊥ i < b RF ; see Fig. 7. Remark VI.3 (Robustness Threshold of Opinion Formation) . The coordinate of the left fold b LF in (26) (Fig. 9) indicateshow robust the opinion formation is to changes in input.Suppose agent i becomes strongly opinionated with input b ⊥ i = b > , and then the input switches to b ⊥ i = b < b .Due to the hysteresis shown in Fig. 9, as long as b > b LF ,agent i will remain strongly opinionated in the direction of b ,even if b favors the alternative option; see Fig. 8. Theorems VI.5 and VI.6 show how design parameters u f and y m have complementary roles in tuning system response: y m tunes the sensitivity threshold and u f tunes the robustnessthreshold . Increasing y m shifts b RF to the right, increasing thesensitivity threshold above which inputs will affect the opinionformation process. Increasing u f has the primary effect ofshifting b LF to the left, rendering the opinion formation morerobust to input fluctuations. See Figs. 8 and 9 for illustration. C. Tunable Cascade Dynamics under Attention Coupling
We next examine a network of partly homogeneous agentscoupled by their attention dynamics, i.e., system (21),(24).
Assumption 3 (Partly Homogeneous Agents) . Let parametersin a group of N a agents with opinion dynamics (21) satisfy α i = α > , β i = β < , d i = d > for all i = 1 , . . . , N a . Remark VI.4 (Mode Interaction and Control) . The completelydecoupled system γ = δ = 0 is a special instance of themode interaction regime γ = δ of the opinion dynamics, asdiscussed in Remark IV.8. In this regime it is possible for stableagreement and disagreement solutions to coexist. Becauseeach agent’s opinion dynamics satisfy Proposition VI.1, if b ⊥ i = 0 for all agents, then there are exactly N a possibleopinion formation bifurcation branches emerging at u = u c .Two of them are of agreement type and N a − of themare of disagreement type. If b ⊥ i (cid:54) = 0 , then multistability isbroken close to u = u c in favor of each agent’s input but itis recovered for larger u . Availability of such versatile andinterpretable configurations of opinions can prove useful forapplications involving dynamic task allocation, for example. Let the inter-agent coupling in the attention dynamics (24)be defined by the attention graph network adjacency matrix ¯ A with elements ¯ a ik ∈ { , } as in (23). We define the regionof influence of agent i in the network as the set of all nodes k to which there exists a directed path from node i in ¯ A .In the following theorem we show that it is enough for asingle agent with sufficient influence in the network to receivea strong input in order for the entire network of agents tobecome strongly opinionated through an opinion cascade. Theorem VI.7 (Coupled Attention Dynamics and OpinionCascades) . Consider N a agents with opinion and attentiondynamics governed by (21),(24), attention dynamics couplingdefined by ¯ A as in (23) . Let Assumption 3 hold. Assume b ⊥ i (cid:54) = 0 for all i = 1 , . . . , N a . Then, there exists (cid:15) > such that if (cid:107) ( x (0) , u (0)) (cid:107) < (cid:15) , the following hold: A. The system asymptotically converges to a nonzero equi-librium ( x s , u s ) with u si > and x si of the same sign as input b ⊥ i for all i = 1 , . . . , N a . x si takes the value of the continuoussolution branch in the pitchfork unfolding of (21) at u i = u si ;B. There exist constants b > and C ≥ such that for all (cid:107) b ⊥ (cid:107) ≤ b , (cid:107) ( x s , u s ) (cid:107) ≤ C (cid:107) b ⊥ (cid:107) . There exists (cid:15) > suchthat for all (cid:107) ( x (0) , u (0)) (cid:107) < (cid:15) , (cid:107) ( x ( t ) − x s , u ( t ) − u s ) (cid:107) ≤ ce − t/τ for some c, τ > .C. Let agent i ’s region of influence be the entire networkand ¯ a i i = 1 . For any ν > , there exist thresholds b th > and u th > such that, if | b ⊥ i | > b th and u f > u th , there existsan equilibrium ( x s , u s ) in the orthant { u k ≥ , sign( x k ) =sign( b ⊥ k ) , k = 1 , . . . , N a } , satisfying | u sk − u sat | < ν , for all k ∈ { , . . . , N a ) , and lim t →∞ (cid:107) ( x ( t ) − x s , u ( t ) − u s ) (cid:107) = 0 .Proof. Proofs of
A, B are directly analogous to proofs of The-orem VI.6-A,B, and we omit details due to space constraints. C. Modulo coordinate change x i (cid:55)→ ± x i , i = 1 , . . . , N a , wecan assume b ⊥ i > for all i = 1 , . . . , N a . Then there exists ε > such that U − = { u k ≥ , x k ≥ − ε, k = 1 , . . . , N a } is forward invariant for (21),(24). Also, any equilibrium in U − is contained in U + = { u k ≥ , x k ≥ + ε, k =1 , . . . , N a } ⊂ U − . The flow of (21),(24) is monotone in U − and every trajectory converges to an equilibrium ( x s , u s ) .It remains to show that | u sk − u sat | < ν for all k =1 , . . . , N a , if b ⊥ i and u f are large enough. Let i , . . . , i N a be the vertices along the influence path starting from i ,i.e., ¯ a i i = ¯ a i k +1 i k = 1 , k = 2 , . . . , N a − . Attentiondynamics of i are ˙ u i = − u i + S u (cid:16) x i + (cid:80) k (cid:54) = i a i k x k (cid:17) .If a i k = 0 for all k (cid:54) = i , by Theorem VI.6 x si can be madearbitrary large and u si arbitrarily close to u sat by increasing b ⊥ i . Since S u (cid:16) x i + (cid:80) k (cid:54) = i a i k x k (cid:17) ≥ S u (cid:0) x i (cid:1) , the sameholds true in the general case. Attention dynamics of thesecond node i are ˙ u i = − u i + S u (cid:16) x i + (cid:80) k (cid:54) = i a i k x k (cid:17) .At steady-state, u si = S u (cid:16) ( x si ) + (cid:80) k (cid:54) = i a i k ( x sk ) (cid:17) . Since x si is arbitrarily large for sufficiently large b ⊥ i , u si is arbi-trarily close to u sat for sufficiently large b ⊥ i . Recall u sat = u f e gy m / (1 + e gy m ) . Thus u sat can be made arbitrarily largeby increasing u f . By differentiation of steady-state equation − x si + u si ˆ S ( x si ) + b ⊥ i , it follows that ∂x si ∂u si ≥ c > for all ( x si , u si ) ∈ U + . Thus, by increasing u f , x si can alsobe made arbitrarily large. The result follows by repeating thesame steps for remaining nodes i , . . . , i N a in the path.Fig. 10 illustrates a cascade as predicted by Theorem VI.7for 5 agents with only agent 1 getting a large input. Becauseagent 1 is at the root of the path graph that defines the attentiondynamic coupling, its response to its input sets off a cascadesuch that all the agents form a strong opinion in the directionof their small inputs. Without the influence of agent 1, theother agents would not have formed strong opinions.Coexistence and multistability of consensus and dissensusequilibria, as well as other interpretable equilibria, make thedecoupled opinion dynamics maximally flexible and sensitiveto input. Theorem VI.7 shows that, with the addition ofattention feedback dynamics, agents reliably favor the optioninformed by their input, and (in the possible presence of t Fig. 10: Simulation of (21),(24) shows a cascade for N a = 5 agents from initial condition ( x i (0) , u i (0)) = (0 , , for all i , andagent 1 the only agent getting a large input. Attention dynamics selfweights are ¯ a ii = 1 and nonzero inter-agent weights are given bythe pictured directed path graph: ¯ a = ¯ a = ¯ a = ¯ a = 1 . α = 1 , β = − , u f = 2 , g = 10 , y m = 1 . , τ u = 1 , b = (1 . , − . , − . , . , . . uncertainty) agents without input reliably flip a coin and forman opinion in favor of one of the options. Due to the bistabilityof equilibrium branches, a change in input can easily facili-tate a transition from one opinion configuration to another,including transitions between consensus and dissensus. Designparameters u f , y m can be used to tune the sensitivity androbustness thresholds of an opinion cascade in the same wayoutlined for uncoupled agents in Section VI-B. D. Tunable Transitions between Consensus and Dissensus
We illustrate (without formal analysis due to space restric-tions) how introducing feedback dynamics to social influenceweights in the opinion dynamics can be used to facilitateconsensus and dissensus and transitions between them.We consider a network with N o = 2 options and N a agents.Suppose agents comprise two clusters of size N and N , withpositive and negative inputs, respectively, i.e., in the notationof (17), b ⊥ i = ˆ b ⊥ for the agents in the first cluster and b ⊥ i = ˆ b ⊥ for the agents in the second cluster. Let I p , p = 1 , , be theindex set of each cluster and ˆ x p = N p (cid:80) i ∈I p x i , p ∈ { , } ,be the average opinion state of the first and second cluster,respectively. Define intra-agent coupling as A jjii = α/N p > and A jlii = β/N p < , l (cid:54) = j , p = 1 , , d i = d for all i , andagent attention dynamics by (24) with ¯ a ik = 1 for all i, k .We define feedback dynamics for inter-agent coupling asfollows. For k (cid:54) = i , let A jjik ( t ) = γ i ( t ) /N p if i, k ∈ I p , p =1 , , and A jjik ( t ) = 0 if i ∈ I p , k ∈ I s , s (cid:54) = p . For k (cid:54) = i and l (cid:54) = j , let A jlik ( t ) = 0 if i, k ∈ I p , p = 1 , , and A jlik ( t ) = δ i ( t ) /N s if i ∈ I p , k ∈ I s , s (cid:54) = p . Dynamics of γ i and δ i are τ γ ˙ γ i = − γ i + σS γ (ˆ x ˆ x ) (27a) τ δ ˙ δ i = − δ i − σS δ (ˆ x ˆ x ) (27b)where σ ∈ { , − } , τ γ , τ δ > are time scales, S γ ( y ) = γ f tanh( g γ y ) , S δ ( y ) = δ f tanh( g δ y ) , and γ f , δ f , g γ , g δ > .Let the timescale separation assumption, τ γ , τ δ (cid:29) τ u (cid:29) hold. It follows by contractiveness [65] of the attentionand inter-agent coupling dynamics, similarly to the proofof Theorem III.5, that resulting opinion dynamics possessan attractive five-dimensional invariant manifold parametrizedby (ˆ x , ˆ x , ˆ u, ˆ γ, ˆ δ ) , where ˆ u , ˆ γ , ˆ δ are the network average -4040 200 400 600-3030 5-202 (a) (b) t t x i x i Fig. 11: (a) Transient opinion trajectories settling to the clusteredattractive manifold from random initial conditions in a simulation;(b) Full simulation. Top: opinion trajectories; Bottom: parametertrajectories. Seven agents form two clusters of sizes N = 3 (dashed-line opinion trajectories), N = 4 (solid-line opinion trajectories).Parameters are d = 1 , α = 1 , β = − , ˆ b ⊥ = 0 . , ˆ b ⊥ = − . τ u = 10 , τ γ = τ δ = 100 , γ f = 2 , δ f = 1 , u f = 2 , g = g γ = g δ = 10 , y m = 1 . . Initial conditions x i (0) are drawnfrom N (0 , , u i (0) from N (0 , . , γ i (0) from N ( − , . , and δ i (0) from N (1 , . . Also, parameters d i , α i , β i , b ⊥ i have additiveperturbations drawn from N (0 , . independently for each agent i .For t < , σ = 1 and for t ≥ , σ = − . attention and inter-agent coupling parameters, respectively.Fig. 11 illustrates the convergence of the network state to thismanifold from arbitrary random initial conditions.The sign of design parameter σ in (27) determines whetherthe system tends towards consensus or dissensus, and switch-ing the sign can reliably trigger a transition between consenusand dissensus. For generic initial conditions that cause agentsto become opinionated, γ i will approach σγ f and δ i willapproach − σδ f for all i = 1 , . . . , N a . When σ = 1 thisfinal state corresponds to a clustered consensus state and when σ = − it corresponds to a clustered dissensus state. Fig. 11illustrates the opinion formation of 7 agents that form twoclusters, one with 3 agents and the other with 4 agents. Onecluster has input favoring option 1 and the second favoringoption 2. Initially, γ − δ < on average and the clusters evolveto a dissensus state which is informed by the agents’ inputs.However, because σ = 1 , the two clusters eventually evolvetowards a consensus state once γ − δ > despite the inputsfavoring disagreement. At time t = 300 , σ switches sign to σ = − and the two clusters evolve back towards a clustereddissensus state once γ − δ < . In Fig. 11 we emphasizerobustness of the clustered trajectories by perturbing modelparameters and inputs for each agent, as well as throughheterogeneous initial conditions.VII. F INAL R EMARKS
We have proposed and analyzed a general model ofcontinuous-time opinion dynamics for an arbitrary number ofagents that interact over a network as they form real-valuedopinions about an arbitrary number of options. The modelgeneralizes linear and nonlinear models of opinion dynamicsin the literature. Significantly, it exhibits the range of opinionformation behaviors predicted by model-independent theory,but not captured by existing linear and nonlinear models.We have shown that opinion formation can be modeled asa bifurcation, a fundamentally nonlinear phenomenon, withopinions forming rapidly and reliably even in the absence of inputs. This reveals, among other things, how groupscan break deadlock among equally valuable options. It alsoreveals that formation of both consensus and dissensus statesdoes not require tailored heterogeneity such as a structurallybalanced inter-agent network; indeed, consensus and dissensusare generic for symmetric and homogeneous groups.We have shown that multistability of opinion formation out-comes is generic when options are (near) equally valuable, andwe have investigated important implications for flexibility androbustness of transitions between opinion states and for opin-ion cascades. Flexibility is facilitated by the ultra-sensitivity ofthe opinion formation process near the bifurcation. Robustnessof opinion formation to small uncertainty and variation in allsystem parameters and all inputs results from hyperbolicity ofstable opinion states.We have shown how the range of possible behaviors can bedistinguished by a small number of parameters, and how thislends analytical tractability to investigations of mechanisms ofcollective behavior as well as to opportunity for systematizingcontrol of networked multi-agent dynamics. We have leveragedthis opportunity by proposing feedback dynamics for theparameter we refer to as the agent’s attention (or susceptibility)to social influence. We have shown how design parameters canbe used to tune implicit thresholds that govern sensitivity androbustness of opinion formation, opinion cascades, and opinionstate transitions to (possibly changing) inputs.In ongoing work we are investigating the role of nontriv-ial heterogeneity, including how irregular network structurepredicts distribution of opinions across a network (see [54]for first steps). We are also investigating how increasing thenumber of options affects the temporal dynamics and sensi-tivities of opinion formation. We are using the model to studyopinion formation in the case of multiple issues using multi-layer networks. In design, we are applying the general modelto problems in multi-robot coordination, task switching, anddecision making for spatial navigation. For natural systems,we are using the model to explore mechanisms that explaincognitive control in the brain [66] and social behavior, fromforaging in animal groups to political polarization.R EFERENCES[1] M. H. DeGroot, “Reaching a consensus,”
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