Multi-Diensional Hegselann-Krause Dynaics A. Nedić Industrial and Enterprise Systes Engineering Dept. University of Illinois Urbana, IL 680 angelia@illinois.edu B. Touri Coordinated Science Laboratory University of Illinois Urbana, IL 680 touri@illinois.edu Abstract We consider ulti-diensional Hegselann- Krause odel for opinion dynaics in discrete-tie for a set of hoogeneous agents. Using dynaic syste point of view, we investigate stability properties of the dynaics and show its finite tie convergence. The novelty of this work lies in the use of dynaic syste approach and the developent of Lyapunov-type tools for the analysis of the Hegselann-Krause odel. Furtherore, soe new insights and results are provided. The results are valid for any nor that is used to define the neighbor sets. I. INTRODUCTION Recently, odeling of social networks has attracted a significant attention and several odels for opinion dynaics have been proposed and studied [], []. These odels have also found applications in engineered network systes of autonoous agents such as, for exaple, in robotic networks [3]. One of the coonly used odels for opinion dynaics is the Hegselann- Krause proposed by R. Hegselann and U. Krause in [], where agent opinions are scalar valued. The stability and convergence of the Hegselann-Krause dynaics and its extensions have been investigated in [4], [5], while this dynaics for continuous tie and tie-varying networks have been proposed in [6] [8]. The convergence rate of the Hegselann-Krause odel has been investigated in [9] (see also [0]). All of the aforeentioned work considers scalar-valued opinions, where the analysis rests on the fact that the Hegselann- Krause dynaics preserves the initial ordering of the opinion values. In this paper, we consider the Hegselann-Krause odel extended to the case where the agent opinions are finite-diensional vectors. In the vector case, we cannot use the line of analysis applicable to the scalar case, as its basic preise that we can order the opinions is violated. To deal with this, we approach the Hegselann-Krause odel by adopting a dynaic syste point of view, and by using novel tools and results This research is supported by NSF Award CMMI-074538. established in [9] for weighted-averaging dynaics. We develop a new terination criterion for the dynaics. Then, we construct a Lyapunov function based on an adjoint dynaics, and we use this function to establish the finite-tie convergence of the Hegselann-Krause dynaics. While developing these results, we were unaware that the stability and finite tie convergence for ulti-diensional odel has already been proven by Lorenz in [], Theore 3.3., where the analysis relies on non-negative atrix theory. While our finite-tie convergence result is not new, our approach and analysis of the Hegselann-Krause dynaics is novel. Besides this, we provide soe insights into the dynaics which lead to a new terination criterion (Proposition ). We also assert the existence of adjoint dynaics, which plays critical role in Lyapunov-based arguent for the finite-tie convergence (Propositions 3 and 4). Finally, the approach proposed in this paper is better suited for the developent of tight bounds on the terination tie than an approach based on non-negative atrix theory (the atrix product analysis typically yields a rather pessiistic bounds when the atrices are tie-varying). The paper is organized as follows: In Section II, we discuss discrete-tie Hegselann-Krause dynaics. In Section III, we investigate soe properties of this dynaics. In particular, we define and study several types of agents and also, discuss the adjoint dynaics for Hegselann-Krause dynaics which plays an iportant role in our developent. Using the developed tools, in Section IV we establish a finite-tie terination tie for the Hegselann-Krause dynaics. We derive this result by bounding the decrease rate of a quadratic Lyapunov function along the trajectories of the dynaics. In Section V we conclude with a discussion on the established results and soe suggestions for further studies. II. HEGSELMANN-KRAUSE OPINION DYNAMICS We consider the discrete tie Hegselann-Krause opinion dynaics odel of [] extended to the case
where the agent opinions are vectors. We assue that there are agents that interact in tie, and we use [] = {,..., } to denote the agent index set. The interactions occur at tie instances denoted by nonnegative integers t = 0,,,.... At each tie t, agent i has an opinion represented by a row-vector x i (t) R n. The collection of profiles {x i (t), i []} is referred to as the opinion profile at tie t. Initially, each agent i has an opinion vector x i (0) R n. At each tie t, the agents interact with their neighbors and update opinions. The neighbors of each agent are deterined based on the difference of the opinions and a prescribed axiu difference ɛ > 0, tered a confidence, which liits the agent interactions in tie. More precisely, the set of neighbors of agent i at tie t, denoted by N i (t), is given as follows: N i (t) = {j [] x i (t) x j (t) ɛ}, () where is any nor in R n. Note that i N i (t) for all i [] and all t 0. The opinion profile evolves in tie according to the Hegselann-Krause dynaics: x i (t+) = N i (t) j N i(t) x j (t) for all t 0, () where N i (t) denotes the cardinality of the set N i (t). Note that the opinion dynaics is copletely deterined by the confidence value ɛ and the initial profile {x i (0) i []}. Before we proceed with our study of the odel, let us discuss the notation that we use. The vectors are assued to be colun vectors, unless clearly stated otherwise. We use x to denote the transpose of a vector x and, siilarly, A for the transpose of a atrix A. We write e to denote a colun vector with all entries equal to. A vector v R is a stochastic (or probability) vector if v i 0 and v e =. Given a vector v R, diag(v) denotes the diagonal atrix with v i, i [], on its diagonal. For a atrix A, we write A ij or [A] ij to denote its ij-th entry. A atrix A is stochastic if Ae = e and A ij 0 for all i, j. We write S to denote the cardinality of a finite set S. A. Isolated Agent Groups At a given tie t, let agent sets S, S [] consist of connected agents. We say that groups S and S are isolated fro each other if the convex hull of {x i (t), i S } and the convex hull of {x j (t), j S } are at the distance greater than ɛ. Two isolated groups are illustrated in Figure. We next show that isolated groups never interact. Proposition : The agent groups that are initially isolated will reain isolated at all ties. Fig.. An illustration of two isolated agent groups in a plane. Proof: Let S and S be two isolated agent groups at tie t = 0. At tie t =, each opinion x i () for i S is a convex cobination of the opinions {x i (0), i S }, and therefore the convex hull of {x i (), i S } is contained in the convex hull of {x i (0), i S }. By proceeding in this anner, we can conclude that at each tie t, the convex hull of opinions {x i (t), i S } is contained in the convex hull of {x i (0), i S }. The sae is true for opinions {x j (t), i S }. Hence, the distance between the convex hulls of {x i (t), i S } and {x j (t), j S } is non-decreasing in tie. Therefore, at any tie t this distance is at least as large the distance between the initial convex hulls of {x i (0), i S } and {x i (0), i S }, which is larger than ɛ. Thus, the agent groups S and S will be isolated at all ties. In light of Proposition, it suffices to analyze the stability of an isolated group identified at the initial tie. Thus, for the rest of the paper, without loss of generality, we assue that the initial profile {x i (0), i []} defines a single isolated group. III. PROPERTIES OF DYNAMICS Evidently, the dynaics will be stable if the neighbor sets stabilize for all agents, i.e., there is soe T 0 such that N i (t) = N i (t + ) for all i and all t T. To capture this, we classify agents into those with sall opinion spread and those with large opinion spread. In the next section, we forally define these agent types and investigate their properties. A. Agent Types At any tie, we distinguish two types of agents, M(t) and G(t) = [] \ M(t), defined as follows: M(t) = {i [] G(t) = {i [] ax x i(t) x j (t) > ɛ }, (3) j N i(t) 3 ax x i(t) x j (t) ɛ j N i(t) 3 }. To interpret these sets, we view the distance ax j Ni(t) x i (t) x j (t) as a easure of the localopinion spread for agent i. Then, the set M(t) consists of
the agents that have a large local-opinion spread, while the set G(t) consists of the agents that have a sall local-opinion spread. These two sets of agents will play an iportant role in the subsequent developent. Soe iportant relations for the agents in the set G(t) are provided in the forthcoing lea. Lea : There holds for all t 0: (a) N i (t) N j (t) for all i G(t) and j N i (t); (b) N j (t) = N i (t) for all i G(t) and j N i (t) G(t); (c) For every i, l G(t), either N i (t) = N l (t) or N i (t) N l (t) =. Proof: Let s N i (t). Then, by the triangle inequality for the nor, we have x s (t) x j (t) x s (t) x i (t) + x i (t) x j (t) ɛ 3 + ɛ 3 < ɛ, where we use s N i (t), j N i (t) and i G(t). Thus, N i (t) N j (t) showing that part (a) holds. For part (b), note that i N j (t) and j N i (t) thus, fro part (a) it follows that N i (t) N j (t) and also N j (t) N i (t) and hence, N i (t) = N j (t). To show part (c), we distinguish two cases: the case when l and i are neighbors, and the case when l and i are not neighbors. If l and i are neighbors, then we have N i (t) = N l (t) by part (b) since l, i G(t). Suppose now that l and i are not neighbors. To arrive at a contradiction assue that N i (t) N l (t) and let s N i (t) N l (t). Then, since l, i G(t) it follows x l (t) x s (t) ɛ 3, x i(t) x s (t) ɛ 3. By the triangle inequality for the nor, we have x l (t) x i (t) ɛ, iplying that l and i are neighbors, which is a contradiction. Hence, N i (t) N l (t). In the following lea, we further investigate the properties of the agents with a sall local-opinion spread. Lea : Let i G(t) be such that N i (t) G(t). Then, at tie t+, one of the following two cases occurs (i) N i (t + ) = N i (t); (ii) N i (t + ) N i (t). Furtherore, in case (ii), if there is an agent l G(t) such that l N i (t + ) \ N i (t), then l N j (t + ) for all j N i (t) and x j (t + ) x l (t + ) > ɛ for all j N i (t). 3 Proof: By Lea (b), it follows that N i (t) = N j (t) for every j N i (t). Let us use S = N i (t) for ease of notation. Thus, at tie t +, the agents in S are all neighbors of each other, and they will reach a local-group agreeent, denoted by x S (t + ) i.e., x j (t + ) = x S (t + ) for all j S. (4) Therefore, S N i (t + ). At the tie t +, there are two possibilities: (i) the agent group ay reain the sae, i.e., N i (t + ) = S, which eans that their group-agreeent x S (t + ) reains isolated fro the other agents opinions; or (ii) the group acquires a new neighbor. In the latter case, there ust be an agent l who was outside of the group S at tie t, but becoes a eber of the group at tie t+, i.e., l N i (t+)\s. Let us now consider case (ii) in detail. For this case to happen, the group-agreeent x S (t + ) ust fall within ɛ-distance fro the opinion x l (t + ) of an agent l with l S. Thus, at tie t +, there ust be an agent l such that l S and x l (t + ) x S (t + ) ɛ, which in view of (4) iplies x j (t + ) x l (t + ) ɛ for all j S. In other words l N j (t + ) for all j S. Since l was not a neighbor of any j S at tie t, we ust have x j (t) x l (t) > ɛ for all j S. Assue now agent l is such that l G(t). Then, by the triangle inequality for the nor, we have for all j S, x j (t) x l (t) x j (t) x S (t + ) + x S (t + ) x l (t + ) + x l (t + ) x l (t). (5) Using the convexity of the nor, we further have for all j S, x j (t) x S (t + ) x j (t) x s (t) S s S ax s S x j(t) x s (t) ɛ 3, (6) where the last inequality follows fro j G(t) for all j S. Siilarly, since l G(t), we have x l (t) x l (t + ) ɛ 3. (7) By substituting estiates (6) and (7) in relation (5), we conclude that for all j S, x j (t) x l (t) ɛ 3 + x S(t + ) x l (t + ), 3
which by x j (t) x l (t) > ɛ iplies ɛ ɛ 3 < x S(t + ) x l (t + ). Thus, since x S (t + ) represents group S agreeent we have x j (t + ) x l (t + ) > ɛ 3 B. Terination Criterion for all j S. (8) At first, we introduce the terination tie for the dynaics, which is the instance at which all the agent reach their corresponding steady states. Forally, the terination tie T of the dynaics () is the sallest t 0 such that x i (k + ) = x i (k) for all k t and all i [], i.e., T = inf t 0 {t x i(k+) = x i (k) for all i [] and k t}. At the oent, we are not eliinating the possibility that the tie T is infinite i.e., we ay have T = +. Using Lea we establish a finite-tie terination criterion for the ulti-diensional dynaics. Specifically, we show that the dynaics reaches the stable state when the agents neighbor sets N i (t), i [] becoe stable and all agents have sall local-opinion spread, as seen in the following proposition. Proposition : Suppose that the tie ˆt is such that G(ˆt) = [] and G(ˆt + ) = []. Then, there holds N i (ˆt + ) = N i (ˆt) for all i [], and the terination tie T of the Hegselann-Krause dynaics satisfies T ˆt +. Proof: Since G(ˆt) = [], by Lea, it follows that either N i (ˆt + ) = N i (ˆt) for all i [], or N i (ˆt + ) N i (ˆt) for soe i []. We show that the latter case cannot occur. Specifically, every l N i (ˆt + ) \ N i (ˆt) ust belong to the set G(ˆt) since M(ˆt) =. Therefore, by Lea it follows that x j (ˆt + ) x l (ˆt + ) > ɛ 3 for all j N i (ˆt). In particular, this iplies i, l M(ˆt + ) which is a contradiction since M(ˆt + ) = (due to G(ˆt + ) = []). Therefore, we ust have N i (ˆt + ) = N i (ˆt) for all i, which eans that each agent group has reached a local group-agreeent and this agreeent persists at tie ˆt + and onward. Observe that Proposition characterizes a situation where we can conclude the terination tie is finite. However, the proposition does not guarantee that such a situation is going to occur. In what follows, we develop additional insights and results that will enable us to ensure that the situation described in Proposition will indeed happen. C. Adjoint Dynaics Here, we show that the Hegsellan-Krause dynaics has an adjoint dynaics with soe special properties. To do so, we first copactly represent the dynaics by defining the agent-interaction atrix B(t) with the entries given as follows: { B ij (t) = N i(t) if j N i (t), 0 otherwise. The dynaics in () can now be written as: X(t + ) = B(t)X(t) for all t 0, (9) where X(t) is the n atrix with the rows given by the opinion vectors x (t),..., x (t). We say that the Hegselann-Krause dynaics has an adjoint dynaics if there exists a sequence of probability vectors {π(t)} R such that π (t) = π (t + )B(t) for all t 0. We say that an adjoint dynaics is uniforly bounded away fro zero if there exists a scalar p (0, ) such that π i (t) p for all i [] and t 0. To show the existence of the adjoint dynaics, we use a variant of Theore 4.8 in [9], as stated below. Theore : Let {A(t)} be a sequence of stochastic atrices such that the following two properties hold: (i) There exists a scalar γ (0, ] such that A ii (t) γ for all i [] and t 0; (ii) There is a scalar α (0, ] such that for every nonepty S [] and its copleent S = [] \ S, there holds A ij (t) α A ji (t) for all t 0. i S,j S j S,i S Then, the dynaics z(t + ) = A(t)z(t), t 0, has an adjoint dynaics {π(t)} which consists of probability vectors that are uniforly bounded away fro zero. We have the following result as a consequence of Theore. Proposition 3: The Hegselann-Krause opinion dynaics has an adjoint dynaics which is uniforly bounded away fro zero. Proof: We show that the conditions of Theore are satisfied. Condition (i) is obviously satisfied with 4
γ =. For condition (ii), we note that for every nonzero entry of B(t), we have B ij (t) = N i (t) N j (t) = B ji(t). Noting that the preceding relation also holds when B ij (t) = 0, we have for all i, j [] and all t 0, B ij (t) B ji(t). (0) Then, for any non-epty S [], by suing the relations in (0) over i S and j S, we see that B ij (t) B ji (t) for all t 0. i S,j S j S,i S Thus, the sequence {B(t)} satisfies condition (ii) of Theore, and the result follows. By Proposition 3, there is a probability sequence {π(t)} for which we have π (t) = π (t + )B(t) for all t 0. () The adjoint dynaics is iportant in our construction of a Lyapunov coparison function, which we use to prove the finite convergence tie of the Hegselann-Krause dynaics, as seen in the next section. IV. TERMINATION TIME We are now ready to show that the terination tie T of the Hegselann-Krause dynaics is finite. We establish this by constructing a Lyapunov coparison function and showing that the function is decreasing along the trajectories of the opinion dynaics. At this point, we consider the dynaics defined by the Euclidean nor in (). To construct a Lyapunov coparison function we use the adjoint dynaics in (). The coparison function for the dynaics is a function V (t), which is defined for all t 0, as follows V (t) = π i (t) x i (t) π (t)x(t), () i= where the row X i,: (t) is given by the opinion vector x i (t) of agent i and π(t) is the adjoint dynaics of (). For this function, we have following essential relation, which we extensively use in the sequel. Lea 3: For any t 0, we have V (t + ) = V (t) π l (t + ) N l (t) x i (t) x j (t). Proof: For θ [n], let us define V θ (t) = i= π i(t)(x θ i (t) π (t)x :,θ (t)), where x θ i (t) is the θth entry of x i (t) and X :,θ (t) is the θth colun of X(t). In fact, V θ (t) is nothing but the restriction of the Lyapunov function V (t) to the θth coponent of the dynaics. Note, that X :,θ (t + ) = B(t)X :,θ (t) for all θ [n]. Since B(t) is a stochastic atrix, by Theore in [], we have V θ (t + ) = V θ (t) H ij (t)(x θ i (t) x θ j(t)) = V θ (t) i= j>i i= j= H ij (t)(x θ i (t) x θ j(t)), where H(t) = B (t)diag(π(t + ))B(t). Therefore, i= j= = (3) H ij (t)(x θ i (t) x θ j(t)) (4) i= j= π l (t + )B li (t)b lj (t)(x θ i (t) x θ j(t)) = π l (t + ) B li (t)b lj (t)(x θ i (t) x θ j(t)) = i= j= π l (t + ) N l (t) (x θ i (t) x θ j(t)), where the last equality follows fro B li (t) = N l (t) if i N l (t) and B li (t) = 0 otherwise. Thus, by cobining (3) and (4), we obtain V θ (t + ) = V θ (t) π l (t + ) N l (t) (x θ i (t) x θ j(t)). Finally, suing up the above relation for θ [n] and using the properties of the -nor, we have V (t + ) = n θ= = V (t) n V θ (t + ) = θ= n V θ (t) θ= π l (t + ) N l (t) π l (t + ) N l (t) (x θ i (t) x θ j(t)) x i (t) x j (t). Lea 3 provides a key relation for our subsequent developent. In particular, it provides the basis for 5
our proof of the stability of the Hegselann-Krause dynaic, as seen in the following proposition. Proposition 4: The Hegselann-Krause opinion dynaics (9), defined using any nor in (), reaches its steady state in a finite tie. Proof: By suing the relation of Lea 3 for t = 0,,..., τ for soe τ, and by rearranging the ters, we obtain V (τ) + D(τ ) = V (0). where for s 0, s π l (t + ) D(s) = N l (t) x i (t) x j (t). t=0 Letting τ, since V (τ) 0 for all τ, we have π l (t + ) N l (t) x i (t) x j (t) <. t=0 Thus, we ust have π l (t + ) li t N l (t) x i (t) x j (t) = 0. By Proposition 3 the adjoint dynaics {π(t)} is uniforly bounded away fro zero, i.e., π l (t) p for soe p > 0, and for all l [] and t 0. Furtherore, N l (t) for all l and t. Therefore, for every l [], li x i (t) x j (t) = 0. t We further have x i (t) x j (t) i N l (t) x i (t) x l (t) ax i N l (t) x i(t) x l (t), where the first inequality follows fro the fact l N l (t) for all l [] and t 0. Consequently, for all l [], li ax x i(t) x l (t) = 0. i N l (t) t Hence, for every l [], there is a tie t l such that ax i N l (t) x i(t) x l (t) ɛ 3 for all t t l. In view of the definition of the set M(t) in (3), the preceding relation iplies that M(t) = for all t ax l [] t l. By Proposition, it follows that the terination tie T satisfies T ax l [] t l +. Since all nors in R are equivalent, it follows that the Hegselann-Krause dynaics terinates in a finite tie for any nor. V. DISCUSSION We provided an alternative establishent of a finite terination tie for Hegselann-Krause dynaics in ulti-diensional case. Our ethod is based on the analysis of a quadratic Lyapunov coparison function for the dynaics, which to the best of the authors knowledge is a novel ethod to analyze this dynaics. A generic upper bound for the terination tie has been established in [9] (Theore 4.7), which ay be iproved using the properties of the Hegselann- Krause dynaics developed in this paper. Another interesting direction to be explored is the stability of the Hegselann-Krause dynaics for infinitely any agents, which sees possible using the theory established in []. REFERENCES [] R. Hegselann and U. Krause, Opinion dynaics and bounded confidence odels, analysis, and siulation, Journal of Artificial Societies and Social Siulation, vol. 5, 00. [] J. Lorenz, Repeated averaging and bounded confidenceodeling, analysis and siulation of continuous opinion dynaics, Ph.D. dissertation, Universität Breen, 007. [3] F. Bullo, J. Cortés, and S. Martínez, Distributed Control of Robotic Networks, ser. Applied Matheatics Series. Princeton University Press, 009. [4] J. Lorenz, A stabilization theore for continuous opinion dynaics, Physica A: Statistical Mechanics and its Applications, vol. 355, p. 73, 005. [5] L. Moreau, Stability of ulti-agent systes with tiedependent counication links, IEEE Transactions on Autoatic Control, vol. 50, no., pp. 69 8, 005. [6] R. Hegselann, Opinion dynaics insights by radically siplifying odels, Laws and Models in Science, pp. 9, 004. [7] V. Blondel, J. Hendrickx, and J. Tsitsiklis, On Krause s ultiagent consensus odel with state-dependent connectivity, IEEE Transactions on Autoatic Control, vol. 54, no., pp. 586 597, nov. 009. [8], Continuous-tie average-preserving opinion dynaics with opinion-dependent counications, SIAM J. Control Opti., vol. 48, pp. 54 540, 00. [9] B. Touri, Product of rando stochastic atrices and distributed averaging, 0, phd Thesis, University of Illinois at Urbana- Chapaign, to appear in Springer Theses series. [0] B. Touri and A. Nedić, Hegselann-krause dynaics: An upper bound on terination tie, 0, preprint subitted to IEEE Conference on Decision and Control. [], On existence of a quadratic coparison function for rando weighted averaging dynaics and its iplications, in Decision and Control and European Control Conference (CDC- ECC), 0 50th IEEE Conference on, dec. 0, pp. 3806 38. [] B. Touri, T. Başar, and A. Nedić, On averaging dynaics in general state spaces, 0, preprint subitted to IEEE Conference on Decision and Control. 6