Cluster Transitions in a Multi-Agent Clustering Model
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThB17.1
Cluster transitions in a multi-agent clustering model Filip De Smet and Dirk Aeyels
Abstract— Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behavior. It is observed in fields ranging from the exact sciences to social and life sciences; consider e.g. swarm behavior of animals or social insects, dynamics of opinion formation, or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modeling brain or heart cells. We consider the formation of clusters in a multi-agent system with each agent belonging to a multi-dimensional space. We mainly focus on cluster transitions with varying coupling strength. While for the one-dimensional case the transition generically involves at most two clusters, this is not necessarily true for higher dimensions. For a system with three agents, all-to-all coupling, and equal weights, we investigate necessary and sufficient conditions guaranteeing the occurrence of only two possible cluster structures: a single cluster containing all three agents, and a configuration with three clusters; for no value of the coupling strength is there an intermediate stage consisting of two clusters. We illustrate the results with a numeric example on opinion formation.
I. INTRODUCTION The basic principles explaining the emergence of one (or several) clusters in multi-agent systems, as documented in the literature, invariably adopt a simple model for the behavior of the agents. Swarming models mostly focus on the collective behavior and the cohesion of a single cluster maintained by the attraction between the animals (or the alignment of their velocities), in counterbalance with the repulsive interactions and/or the drift induced by random walk behavior [15], [9], [21]. Models for opinion formation often focus on reaching a consensus [12], i.e. the process of developing into a single cluster of people agreeing unanimously on a particular issue, or the coexistence of only two opposite opinions, usually involving nearest neighbour interactions [3], [19]. However, other models generating multiple clusters with different opinions as a general outcome have been proposed [8]. A particular type of clustering is observed in systems of coupled oscillators, such as aggregations of flashing fireflies and coupled Josephson junction arrays [18]. (The term clustering is then often replaced by synchronization or entrainment.) One distinguishes between phase clustering and frequency clustering. The first form (see e.g. [17], [10], [23]) is associated with networks of oscillators with identical natural frequencies. (The natural frequency of an oscillator Both authors are with the SYSTeMS Research Group, Ghent University, Technologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium
[email protected], [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
characterizes its behavior when in isolation, i.e. without interaction with other oscillators.) Each cluster consists of oscillators with (asymptotically) equal phases, with the number of different clusters depending on the interaction. For non-identical natural frequencies for which the differences are sufficiently small, it may still be possible to distinguish different phase clusters [20]. Larger variations in the natural frequencies may induce oscillators having different long term average frequencies, resulting in frequency clustering [16], [4]: each cluster is characterized by the long term average frequency of its members. For more details on both phenomena and for examples of clustering in chaotic systems we refer to [14]. Due to the complexity and richness of the dynamics of some of these models, analytical results are often hard to come by and restricted to the existence and local stability properties of some of their solutions [10], [20], [2], and exploration of the parameter space is usually done through simulations [11], [13]. In previous papers [5], [1], [6] we introduced a dynamical model for clustering, corresponding to a system of nonidentical attracting agents, which may be connected according to an arbitrary network structure. Each agent has an autonomous component — its natural velocity — and attracts other agents by a saturating interaction function. Depending on the value of the coupling strength, different cluster structures may arise. In contrast with many of the aforementioned models, an in-depth analysis is possible: we were able to analytically characterize the cluster structure by a set of necessary and sufficient conditions in the model parameters, we showed that there exists a unique cluster structure satisfying these inequalities for a given set of parameters and that, in general, the distances between agents from the same cluster approach constant values. In this paper we extend this model, considering agents in a multi-dimensional space. In section II we present the model, and in section III we review some results from [6] concerning the model with one-dimensional agents. In section IV we discuss the clustering behavior of the multi-dimensional model. The transitions between the cluster structures with increasing coupling strength may involve more than two clusters — even for generic cases — as is shown in section V, where we investigate a system with three agents and we derive conditions for three clusters to merge and form a single cluster when the coupling strength is increased above a transition value. We conclude with an illustration on opinion formation.
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ThB17.1 II. MODEL DESCRIPTION
B. Results Theorem 1: Consider the following inequalities
We consider the model described by
> > e(G< ve(G< k , Gk , Gk ) < v k+1 , Gk+1 , Gk+1 ),
N
x˙i (t) = bi + K ∑ γ j fi j ( x j (t) − xi (t) )ex j (t)−xi (t) ,
∀ k ∈ IM−1 , < > e(G< ve(Gk ∪ Gk,1 , Gk,2 , G> k ) 0, K ≥ 0, N > 1, and xi (t), bi ∈ RP , with P ≥ 1. The differentiable functions fi j are non-decreasing with fi j (0) = 0, and satisfy fi j = f ji ,
and
lim fi j (ξ ) = Fi j ,
ξ →+∞
for all i and j in {1, . . . , N}, for some symmetric matrix F ∈ (R+ )N×N . Furthermore, ex ,
x , kxk
∀ x ∈ RP \ {0};
e0 , 0.
III. PRELIMINARY RESULTS The results formulated in this section concern the model (1) for the case P = 1, where each agent belongs to a one-dimensional space: xi (t) ∈ R, ∀ i ∈ {1, . . . , N}. They immediately follow from results in [6]. A. Some notation Consider a non-empty set G0 ⊂ IN . For a vector b ∈ (RP )N we define hbiG0 as the weighted average of bi over G0 : hbiG0
∑i∈G0 γi bi , . ∑i∈G0 γi
ve(G− , G0 , G+ ) , hbiG0 +
!
∑ γi ∑
∑i∈G0 γi i∈G0
j∈G+
γ j Fi j −
∑
γ j Fi j ,
j∈G−
for all G− , G0 , G+ ⊂ IN with G0 non-empty. We consider the following definition of clustering behavior — which we term as distance clustering behavior — of a solution x of (1) with P = 1 with respect to a cluster structure G: •
•
•
(2b)
∀ k ∈ IM . Denote by (2a’), resp. (2b’), the inequalities (2a), resp. (2b), in which the strict inequalities are replaced by non-strict inequalities. Then the conditions (2), resp. (2’), are sufficient, resp. necessary, for clustering behavior with respect to G of all solutions of the system (1) with P = 1. When the conditions (2’) are satisfied, but not (2), any clustering behavior depends on whether the interaction functions fi j attain their saturation values or not, or the system may not exhibit clustering behavior at all. In this latter case, there are two agents with their mutual distance growing unbounded on the one hand, but returning infinitely many times below some finite value on the other hand. Theorem 2: For every b ∈ RN and every symmetric matrix F ∈ (R+ )N×N , there exists a partition of R+ in a finite number of intervals, such that the interior of each interval represents values of K corresponding to a unique ordered set partition G of IN , satisfying (2). Theorem 3: Let x be a solution of (1) with P = 1 and with cluster structure G = (G1 , . . . , GM ). For each k ∈ IM , if i ∈ Gk , then > lim x˙i (t) exists and equals ve(G< k , Gk , Gk ).
t→+∞
Let G = (G1 , . . . , GM ) be an ordered set partition of IN . S S > Set G< k , k0 k Gk0 . With b ∈ R (i.e. for P = 1), set
K
∀ Gk,1 , Gk,2 ( Gk , with Gk,2 = Gk \ Gk,1 ,
(2a)
The distances between agents in the same cluster remain bounded (i.e. xi (t) − x j (t) is bounded for all i, j ∈ Gk , for any k ∈ IM , for t ≥ 0). For any D > 0 there exists a time after which the distances between agents in different clusters are and remain at least D. The agents are ordered by their membership to a cluster: k < l ⇒ xi (t) < x j (t), ∀ i ∈ Gk , ∀ j ∈ Gl , ∀t ≥ T , for some T > 0.
If furthermore all functions fi j are increasing in R+ , then for each k ∈ IM , if i, j ∈ Gk , then lim (xi (t) − x j (t)) exists and is independent of x(0). t→+∞ In other words, within a cluster the distances between the agents will approach constants, which are independent of the initial condition, and the velocities of the agents will approach the asymptotic average cluster velocity. When the coupling strength is varied, transitions between cluster structures may take place. The transition values correspond to values for K for which one of the inequalities in (2) becomes an equality. Depending on whether the inequality corresponds to (2a) or (2b), two clusters will merge or one cluster will split into two clusters. Generically, at most one equality will occur for a single value of K, and thus at most two clusters are involved in the cluster transition. As we will show in this paper, when P > 1, the occurrence of a transition involving more than two clusters is possible in a generic way. C. Remarks on clustering behavior The ordering of the clusters in the definition of clustering behavior is included to enable a convenient formulation of the results. In the next section we will redefine clustering behavior by the two other characteristics only (i.e. bounded clusters, with unbounded separation between the clusters),
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ThB17.1 since no natural order exists when the dimension is larger than one. Furthermore, we will also consider a slightly different type of clustering behavior — velocity clustering behavior, as opposed to distance clustering behavior — where a cluster is defined by a common long term average velocity of its members. From the previous results it follows that both types of clustering behavior coincide for the model (1) (with P = 1) except in non-generic cases where equalities arise in (2’), i.e. at transitions between different cluster structures.
velocity of the agents: x j (t)/t − xi (t)/t
lim ex j (t)−xi (t) = lim t→+∞ x j (t)/t − xi (t)/t = ev j −vi .
t→+∞
Denoting the (velocity) cluster structure by H = {H1 , . . . , HM }, and considering (weighted) averages over the clusters Hk of the equations of motion (1), one can verify that the asymptotic velocities vi , limt→+∞ x˙i (t) satisfy vi = hbiHk +
IV. MULTI-DIMENSIONAL AGENTS A. Definition of clustering behavior As mentioned before, the definition of clustering behavior of a solution x of (1) cannot simply be extended to P > 1; we will drop the ordering of the clusters: x exhibits distance clustering behavior with respect to the (unordered) set partition G if the following conditions are satisfied. •
•
The distances between agents
in the same cluster remain bounded (i.e. xi (t) − x j (t) is bounded for all i, j ∈ Gk , for any Gk ∈ G, for t ≥ 0). For any D > 0 there exists a time after which the distances between agents in different clusters are and remain at least D.
We say that a solution x of (1) exhibits velocity clustering behavior with respect to a set partition H of IN if limt→+∞ xi t(t) exists for all i in IN , and all agents belonging to the same cluster in H have the same limit value, while agents from different clusters have different limit values. As we will show in a forthcoming paper [7], the solutions of (1) are such that for each i ∈ IN , limt→+∞ x˙i (t) exists and is independent of the initial condition, implying that every solution x of (1) exhibits velocity clustering behavior with respect to some partition H, with each cluster Hk characterized by a common asymptotic velocity for the agents it contains. If x also exhibits distance clustering behavior with respect to a partition G then it follows that G is a refinement of H, i.e. ∀ G0 ∈ G : ∃ H0 ∈ H, G0 ⊂ H0 . Only in some non-generic cases may G differ from H or not exist at all (i.e. there may be no distance clustering behavior). In the remainder of the paper we will simply refer to ‘clustering behavior’ when the distinction between both types is not relevant for the aspects we are considering.
K ∑l∈Hk γl
∑ γl ∑ γ j Fl j ev j −vl , l∈Hk
(3)
j∈IN
for all i in Hk , for all k ∈ IM . The existence of velocities vi solving these equations can be considered as the equivalent of (2a) for P > 1. The conditions (2b) state that (for P = 1) a cluster cannot split into two subsets for the given velocities of the other subsets. To formulate the equivalent of (2b) for P > 1 we therefore consider an arbitrary partition H˜ k = {H˜ k,1 , . . . , H˜ k,M0 } of Hk (with H˜ k 6= {Hk }), and we impose that the equations v0i = hbiH˜ k,m +
K ∑l∈H˜ k,m γl
∑
l∈H˜ k,m
γl
∑ γ j Fl j ev0j −v0l ,
(4)
j∈IN
for all i ∈ H˜ k,m , and for all m ∈ IM0 (notice that only i ∈ Hk is considered), do not have a solution for v0 with v0i = vi for all i ∈ / Hk (where v is defined by (3)). In other words, in the previous equation we consider only v0i with i ∈ Hk as free variables, while v0i for any i ∈ IN \ Hk is fixed and equal to vi . As opposed to the conditions (2b) which correspond to partitions in only two subsets, any partition of Hk has to be considered. This will be illustrated in section V, where we investigate conditions for which three clusters may merge and form a single cluster when increasing the coupling strength K, while no intermediate cluster structure with two clusters exists for any value of K. C. Independence of the initial condition In this section we show that the existence of a solution x∗ exhibiting distance (resp. velocity) clustering behavior implies the same distance (resp. velocity) clustering behavior for any other solution x. This follows from a similar argument as in [6], by considering the function V : R → R : t 7→
∑ γi kxi (t) − xi∗ (t)k2 .
i∈IN
For convenience we introduce the notation B. Conditions for clustering behavior
f¯i j (x) , fi j (kxk)ex ,
Conditions replacing (2) and (2’) for the characterizing clustering behavior for P > 1 are much harder to derive. In this paper we will discuss the equations (3) below and look into a special case with 3 agents, all-to-all interaction and equal weights. The interaction between agents from different clusters will approach its saturation value (in norm), and will asymptotically be aligned according to the difference in asymptotic
for all x ∈ RP , and i, j ∈ IN . Since for any a, b ∈ RP and i, j ∈ IN , (a − b)T f¯i j (a) − f¯i j (b)
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= (kak ea − kbk eb )T ( fi j (kak)ea − fi j (kbk)eb ) ≥ (kak − kbk) ( fi j (kak) − fi j (kbk)) ≥0
(5)
ThB17.1 × f¯0 (xm (t) − xl (t)) +C0 ,
(since fi j is non-decreasing), it follows that = −K
dV (t) = 2K ∑ γi γ j (xi (t) − xi∗ (t))T dt i, j∈IN
× f0 (kxm (t) − xl (t)k) +C0 ,
× f¯i j (x j (t) − xi (t)) − f¯i j x∗j (t) − xi∗ (t) = −K
which would lead to a contradiction if W (t) would grow unbounded with t, and therefore agent m belongs to the cluster Gk .
(x j (t) − xi (t) − x∗j (t) + xi∗ (t))T
∑
i, j∈IN
× f¯i j (x j (t) − xi (t)) − f¯i j x∗j (t) − xi∗ (t)
V. A TRANSITION FROM THREE CLUSTERS TO ONE CLUSTER
≤ 0, and therefore V (t) and kxi (t) − xi∗ (t)k (i ∈ IN ) remain bounded when t → +∞, and thus distance (resp. velocity) clustering behavior of x∗ implies distance (resp. velocity) clustering behavior of x with respect to the same cluster structure. D. Convexity of the clusters for all-to-all coupling In this section we consider all-to-all coupling, i.e. we assume that the functions fi j are all equal to a (nondecreasing) function denoted by f0 , with saturation value F0 > 0. We show that for a solution exhibiting distance clustering behavior, an agent with a natural velocity that is a convex combination of natural velocities of agents from the same cluster also belongs to this cluster. This property does not hold when there is no all-to-all coupling (see e.g. [6] for an example with P = 1). Assume the solution x of (1) exhibits distance clustering behavior with respect to G = {G1 , . . . , GM }, with bm = ∑ j∈Gk µ j b j , for some k ∈ IM , m ∈ IN , µ ∈ [0, 1]N with µ j = 0 if j ∈ / Gk , µm = 0, and ∑ j∈IN µ j = ∑ j∈Gk µ j = 1. We will show that this convexity property implies that m ∈ Gk . Consider the function W , defined by
2
1
W : R → R : t 7→ W (t) , xm (t) − ∑ µ j x j (t) .
2 j∈G k
Then dW (t) dt !T xm (t) −
=
∑
µ j x j (t)
K
∑ γn f¯0 (xn (t) − xm (t))
n∈IN
j∈Gk
! −
∑ µl K ∑ γn f¯0 (xn (t) − xl (t)) n∈IN
l∈Gk
=K
In this section we investigate the conditions for which three clusters (each containing one agent) may merge and form a single cluster of three agents when increasing the coupling strength K above some critical value Kc , under the assumptions of all-to-all coupling ( f12 = f13 = f23 , f0 and F12 = F13 = F23 , F0 ) and equal weights: γ1 = γ2 = γ3 = 13 . We assume that b1 , b2 and b3 are not all equal and we consider the limit K → Kc (K < Kc ) in the equations (3) for the velocity cluster structure H = {{1}, {2}, {3}}. The limit values of the left hand sides are all equal to the average velocity hbi{1,2,3} , since for K = Kc the asymptotic velocities coincide: v1 = v2 = v3 = hbi{1,2,3} . The limit values of the terms ev j −vi will however be different from e0 = 0, as their norm remains equal to one. We therefore introduce xie (i ∈ {1, 2, 3}) as a limit point (when K approaches Kc from below) of the corresponding values of q
j∈Gk l∈Gk n∈IN
vi − hbi{1,2,3}
2 , ∑i∈{1,2,3} 13 vi − hbi{1,2,3}
and we rewrite the equations (3) as Kc F0 hbi{1,2,3} = bi + ∑ exej −xie , 3 j∈I 3
∀ i ∈ {1, 2, 3},
(6)
where we assume that x1e , x2e and x3e are pairwise different. If f0 (ξ ) = F0 for some ξ > 0 (i.e. f0 reaches its saturation value) and kxej − xie k ≥ ξ (i, j ∈ {1, 2, 3}, i 6= j), then the choice xi (t) , xie + hbi{1,2,3}t, for all t in R, results in a possible solution of (1). This shows how the xie can be interpreted as equilibrium positions in RP (up to a common linear term in t), hence our notation. Setting b0i , bi − hbi{1,2,3} ,
∑ ∑ ∑ µ j µl γn (xm (t) − xn (t) + xn (t) − xl (t))T
∀ i ∈ {1, 2, 3},
we obtain
× f¯0 (xn (t) − xm (t)) − f¯0 (xn (t) − xl (t)) +K
∑ µl (γm + γl ) kxm (t) − xl (t)k l∈Gk
3b01 = ex2e −x1e + ex3e −x1e , Kc F0 3b0 − 2 = ex1e −x2e + ex3e −x2e , Kc F0 3b0 − 3 = ex1e −x3e + ex2e −x3e . Kc F0 −
T
∑ ∑ ∑ µ j µl γn (xl (t) − x j (t))
j∈Gk l∈Gk n∈IN
× f¯0 (xn (t) − xm (t)) − f¯0 (xn (t) − xl (t)) . The second term is bounded by some constant C0 because of the distance clustering behavior. In the first term we use the property (5) for n 6= m and n 6= l, resulting in
It follows that
dW (t) ≤ −K ∑ µl (γm + γl )(xm (t) − xl (t))T dt l∈G
0 2
3b1
Kc F0 = 2 + 2ζ1 ,
k
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(7a) (7b) (7c)
ThB17.1
0 2
3b2
Kc F0 = 2 + 2ζ2 ,
0 2
3b3
Kc F0 = 2 + 2ζ3 ,
0 2 0 2 0 2
b3 + b1 > b2 .
where ζ1 , eTxe −xe ex3e −x1e , 2
1
ζ2 , eTxe −xe ex3e −x2e , 1
2
ζ3 , eTxe −xe ex2e −x3e . 3 1 e e e e Setting E , ex1 −x3 ex2 −x1 ex3e −x2e ∈ RP×3 , it follows from the coplanarity of the columns of E that rank(E) < 3 and thus 1 −ζ1 −ζ3 1 −ζ2 = 0, det(E T E) = det −ζ1 −ζ3 −ζ2 1 or 1 − 2ζ1 ζ2 ζ3 − ζ12 − ζ22 − ζ32 = 0, (9) which is rewritten in terms of b01 , b02 and b03 as follows:
2 2
2 2
2 2 2 b01 b02 + 2 b02 b03 + 2 b01 b03
0 4 0 4 0 4 kb01 k2 kb02 k2 b03 2
− b1 − b2 − b3 = . (10) 2 Kc F0 3
The left hand side can be shown to be non-negative by considering the values b0k as sides of a triangle (since b01 + b02 + b03 = 0) and applying triangle inequalities. It is zero only when b01 , b02 and b03 are collinear. If this is not the case then a value for Kc follows from (10). Equation (10) guarantees that we can find coplanar vectors exej −xie satisfying (7), without relating these vectors to points xie . To guarantee the existence of points xie satisfying (7), we need to impose a condition on the orientation of the vectors exej −xie ; we require the existence of positive coefficients such that the corresponding linear combination of columns of E is zero: 3 ∃ c ∈ R+ : Ec = 0, or equivalently E T Ec = 0, 0 resulting in the conditions ζ1 ζ2 + ζ3 > 0,
(11a)
ζ2 ζ3 + ζ1 > 0,
(11b)
ζ3 ζ1 + ζ2 > 0.
(11c)
In section A of the appendix we show that under the condition (9) and for ζ1 , ζ2 , ζ3 ∈ [−1, 1], (11) is equivalent to ζ1 + ζ2 − ζ3 + 1 > 0,
(12a)
ζ2 + ζ3 − ζ1 + 1 > 0,
(12b)
ζ3 + ζ1 − ζ2 + 1 > 0,
(12c)
0 2 0 2 0 2
b1 + b2 > b3 ,
0 2 0 2 0 2
b2 + b3 > b1 ,
(13a)
or
(13b)
(13c)
In other words, the triangle with sides kb01 k, kb02 k and b03 is acute, or equivalently, the angles between the vectors b01 , b02 and b03 in RP are obtuse. Notice that the inequality signs are strict. In case of equality, two of the corresponding points x1e , x2e and x3e would coincide, and this (non-generic) case has to be considered separately; the behavior of the solutions of (1) then depends on whether or not f0 reaches its saturation value, but we will not elaborate on this. Up to now, we have shown that (10) and (13) are necessary for the existence of points xke satisfying (7). We will show that these conditions are also sufficient. It can be easily verified that if ζ1 = cos α and ζ2 = cos β for some α, β ∈ [0, π], then (9) and (11a) imply that ζ3 = − cos(α + β ), and therefore there exists a triangle with vertices x˜ke (k ∈ {1, 2, 3}), for which the cosines of the angles equal ζ1 , ζ2 and ζ3 . Defining b˜ 0k (k ∈ {1, 2, 3}) by equations similar to (7), it then follows that kb˜ 0k k = kb0k k, for all k in {1, 2, 3}, and also b˜ 01 + b˜ 02 + b˜ 03 = 0. Therefore b01 , b02 and b03 , and b˜ 01 , b˜ 02 and b˜ 03 define two congruent triangles and there exists an orthogonal transformation mapping b˜ 0k onto b0k , ∀ k ∈ {1, 2, 3}. The images xke of the vertices x˜ke under this transformation then satisfy (7), and as a consequence we have proven the following. Theorem 4: The equations (10) and (13) are necessary and sufficient for the existence of pairwise different xke ∈ RP (k ∈ {1, 2, 3}) satisfying (7). We conclude that the phenomenon where three clusters merge when increasing the coupling strength above some critical value, is not restricted
to non-generic cases, such as when kb01 k = kb02 k = b03 , but the range of triples (b1 , b2 , b3 ) ∈ (RP )3 for which it occurs has non-zero measure, as follows from the conditions (13). VI. OPINION FORMATION In this section we illustrate how the system (1) may model an opinion formation process regarding an issue with several aspects that have to be taken into account. An opinion is represented by a point in RP , and each coordinate corresponds to one of the P various aspects. (We assume that each aspect can be represented by a single variable — possibly by further decomposing aspects into several partial aspects. Furthermore we assume that each of these aspects can be represented by a real variable.) As an example, the issue might be the question of how to respond to an economic recession, with severel aspects corresponding to e.g. lowering taxes, stimulating innovative research, or a preference for protective measures instead of cooperation with other countries on a larger scale. We consider N individuals taking part in a meeting; each individual has his own opinion on the issue on the agenda, which may evolve in time due to discussion with the other members. Opinions cannot grow unbounded, and therefore xi in (1) is not an appropriate quantity to represent an opinion. Instead we will take the derivative yi = x˙i as a measure of someone’s opinion. The equations for yi can be written
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ThB17.1 3
as (assuming xi (0) = 0, ∀ i ∈ IN , without loss of generality regarding the long-term behavior)
2
K N
yi (t) = bi + K ∑ γ j f¯i j
Rt
0 0 0 0 (y j (t ) − yi (t ))dt ,
1
(14)
j=1
0
∀ i ∈ IN . The value bi ∈ RP represents the a priori opinion of agent i (corresponding to no discussion). With yi (t) representing theRopinion of agent i at time t, each component of the integral 0t (y j (t 0 ) − yi (t 0 ))dt 0 may reflect the level of disagreement accumulated over time for one aspect of the issue, and may correspond to the amount of discussions taking place between agents i and j on this aspect, proportional with time and with difference in opinion. One may be willing to make concessions for each aspect separately, but when combined, large concessions for one aspect will lead to smaller concessions for other aspects, as is reflected by the dependence of each interaction term on the norm of the aformentioned integral. In general, everyone starts with his own opinion bi while with time and through interaction, different groups are formed, each group characterized by a final opinion vi obtained through discussion. The pressure to reach a decision, or the tendency to adapt one’s opinion by paying attention to each other’s arguments, is reflected by the coupling strength K. As an important distinction with other existing models (for an overview, see [12]) we want to emphasize that the model (14) allows the coexistence of several groups (see also [6]), each characterized by its own group opinion — as opposed to models focusing on total consensus or the coexistence of only two opinions (such as in [19] or [22]). In Fig. 1(a) we consider P = 2 and we show the evolution of the opinions vi eventually reached as a function of K. The vi -values are calculated by simulations. We consider 50 agents with both components of bi chosen from a Gaussian distribution with zero mean and standard deviation one. The parameters γi are all equal to N1 , and the Fi j (i 6= j) are all taken equal to one. Notice the emergence of one big cluster containing most agents for K ≈ 2. In Fig. 1(b) the time evolution of the opinions yi for K = 2, again obtained by numerical integration of the mathematical model, is shown. (For the numerical integration in Fig. 1 the Euler method was used with a time step of 0.03.) For comparison, we also show a similar result for the case P = 1 in Fig. 2, with again N = 50. The bi -values are the first components of the bi -values from the previous simulation with P = 2. Fig. 2 was obtained by means of an algorithm based on the inequalities (2) (instead of a simulation of the differential equations). Compared to Fig. 1, the formation of the central cluster is more gradual, with each transition corresponding to merging two clusters. This difference is reminiscent of the difference between first-order and secondorder phase transitions, which will be further investigated in future research.
1 0 vi,2 −1 −1.5
−0.5
−1
vi,1
0
1.5
1
0.5
(a) 1
0.5
t 0 2
yi,2
0 −2
−2
−1.5
−1
−0.5
yi,1
0
0.5
1.5
1
2
(b) Fig. 1. Opinion formation with P = 2: Fig. (a) shows the opinions vi as a function of the coupling strength K. Fig. (b) shows the time evolution for K = 2. The parameters values are: N = 50, γi = N1 , and Fi j = 1 (i, j ∈ IN , j 6= i).
2.5 2 1.5
K 1 0.5 0 −3
−2
−1
0
vi
1
2
3
Fig. 2. Opinion formation with P = 1: The opinions vi are shown as a function of the coupling strength K. The parameters values are: N = 50, γi = N1 , and Fi j = 1 (i, j ∈ IN , j 6= i).
VII. CONCLUSION
We have considered a multi-agent clustering model with a multi-dimensional state-space for each agent. While for the one-dimensional case a transition in the cluster structure generically involves at most two clusters, we have shown that, for higher dimensions, more than two clusters may be involved in a transition. We analysed a system with three agents, and characterized the different cluster structures that may arise with varying coupling strength. As an illustration, we have simulated an example of an opinion formation process.
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ThB17.1 VIII. ACKNOWLEDGMENTS This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its authors. During this research Filip De Smet was supported by a Ph.D. fellowship of the Research Foundation - Flanders (FWO). A PPENDIX A. Proof of the equivalence of (11) and (12) under (9) for ζ ∈ [−1, 1]3 Adding the equations in (11) two by two, and considering that ζ ∈ [−1, 1]3 , it follows that ζ1 + ζ2 > 0, ζ2 + ζ3 > 0, ζ3 + ζ1 > 0, implying (12). For the other direction of the equivalence relation, we multiply the equations (12) two by two, obtaining ζ32 + 2ζ3 + 1 − ζ12 − ζ22 + 2ζ1 ζ2 > 0, ζ12 + 2ζ1 + 1 − ζ22 − ζ32 + 2ζ2 ζ3 > 0, ζ22 + 2ζ2 + 1 − ζ32 − ζ12 + 2ζ3 ζ1 > 0.
[10] D. Golomb, D. Hansel, B. Shraiman, and H. Sompolinsky. Clustering in globally coupled phase oscillators. Physical Review A, 45(6):3516– 3530, March 1992. [11] G. Gr´egoire, H. Chat´e, and Y. Tu. Moving and staying together without a leader. Physica D, 181(3-4):157–170, July 2003. [12] R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5(3), 2002. [13] Y. Maistrenko, O. Popovych, O. Burylko, and P. A. Tass. Mechanism of desynchronization in the finite-dimensional Kuramoto model. Physical Review Letters, 93(8):084102, August 2004. [14] S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette. Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems, volume 2 of World Scientific Lecture Notes in Complex Systems. World Scientific, 2004. [15] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros. Mutual interactions, potentials, and individual distance in a social aggregation. Journal of Mathematical Biology, 47:353–389, May 2003. [16] L. G. Morelli, H. A. Cerdeira, and D. H. Zanette. Frequency clustering of coupled phase oscillators on small-world networks. European Physical Journal B, 43:243–250, Jan. 2005. [17] K. Okuda. Variety and generality of clustering in globally coupled oscillators. Physica D, 63(3-4):424–436, 1993. [18] S. H. Strogatz. SYNC: The Emerging Science of Spontaneous Order. Hyperion Press, New York, 2003. [19] K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community. International Journal of Modern Physics C, 11(6):1157–1165, September 2000. [20] P. Tass. Phase and frequency shifts in a population of phase oscillators. Physical Review E, 56(2):2043–2060, August 1997. [21] T. Vicsek, A. Czir´ok, I. J. Farkas, and D. Helbing. Application of statistical mechanics to collective motion in biology. Physica A, 274(1–2):182–189, December 1999. [22] F. Wu and B. A. Huberman. Social structure and opinion formation. Computational Economics 0407002, EconWPA, July 2004. [23] D. H. Zanette and A. S. Mikhailov. Dynamical clustering in oscillator ensembles with time-dependent interactions. Europhysics Letters, 65(4):465–471, February 2004.
Using (9) this can be written as 2(ζ1 ζ2 + ζ3 )(1 + ζ3 ) > 0, 2(ζ2 ζ3 + ζ1 )(1 + ζ1 ) > 0, 2(ζ3 ζ1 + ζ2 )(1 + ζ2 ) > 0, implying (11). R EFERENCES [1] D. Aeyels and F. De Smet. A mathematical model for the dynamics of clustering. Physica D: Nonlinear Phenomena, 273(19):2517–2530, October 2008. [2] D. Aeyels and J. A. Rogge. Existence of partial entrainment and stability of phase locking behavior of coupled oscillators. Progress of Theoretical Physics, 112(6):921–942, December 2004. [3] C. M. Bordogna and E. V. Albano. Statistical methods applied to the study of opinion formation models: a brief overview and results of a numerical study of a model based on the social impact theory. Journal of Physics: Condensed Matter, 19(6):065144 (16pp), February 2007. [4] F. De Smet and D. Aeyels. Partial entrainment in the finite KuramotoSakaguchi model. Physica D, 234(2):81–89, October 2007. [5] F. De Smet and D. Aeyels. Clustering in a network of mutually attracting agents. In Proceedings of the 47th IEEE Conference on Decision and Control, pages 1827–1832, December 2008. [6] F. De Smet and D. Aeyels. Clustering in a network of non-identical and mutually interacting agents. Proceedings of the Royal Society A, 465:745–768, March 2009. [7] F. De Smet and D. Aeyels. A multi-dimensional model for clustering exhibiting a first-order phase transition. In preparation, 2009. [8] G. Deffuant, F. Amblard, G. Weisbuch, and T. Faure. How can extremism prevail? a study based on the relative agreement interaction model. Journal of Artificial Societies and Social Simulation, 5(4), 2002. [9] V. Gazi and K. M. Passino. Stability analysis of swarms. IEEE Transactions on Automatic Control, 48(4):692–697, April 2003.
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ThB17.1
Cluster transitions in a multi-agent clustering model Filip De Smet and Dirk Aeyels
Abstract— Clustering is a phenomenon that may emerge in multi-agent systems through self-organization: groups arise consisting of agents with similar dynamic behavior. It is observed in fields ranging from the exact sciences to social and life sciences; consider e.g. swarm behavior of animals or social insects, dynamics of opinion formation, or the synchronization (which corresponds to cluster formation in the phase space) of coupled oscillators modeling brain or heart cells. We consider the formation of clusters in a multi-agent system with each agent belonging to a multi-dimensional space. We mainly focus on cluster transitions with varying coupling strength. While for the one-dimensional case the transition generically involves at most two clusters, this is not necessarily true for higher dimensions. For a system with three agents, all-to-all coupling, and equal weights, we investigate necessary and sufficient conditions guaranteeing the occurrence of only two possible cluster structures: a single cluster containing all three agents, and a configuration with three clusters; for no value of the coupling strength is there an intermediate stage consisting of two clusters. We illustrate the results with a numeric example on opinion formation.
I. INTRODUCTION The basic principles explaining the emergence of one (or several) clusters in multi-agent systems, as documented in the literature, invariably adopt a simple model for the behavior of the agents. Swarming models mostly focus on the collective behavior and the cohesion of a single cluster maintained by the attraction between the animals (or the alignment of their velocities), in counterbalance with the repulsive interactions and/or the drift induced by random walk behavior [15], [9], [21]. Models for opinion formation often focus on reaching a consensus [12], i.e. the process of developing into a single cluster of people agreeing unanimously on a particular issue, or the coexistence of only two opposite opinions, usually involving nearest neighbour interactions [3], [19]. However, other models generating multiple clusters with different opinions as a general outcome have been proposed [8]. A particular type of clustering is observed in systems of coupled oscillators, such as aggregations of flashing fireflies and coupled Josephson junction arrays [18]. (The term clustering is then often replaced by synchronization or entrainment.) One distinguishes between phase clustering and frequency clustering. The first form (see e.g. [17], [10], [23]) is associated with networks of oscillators with identical natural frequencies. (The natural frequency of an oscillator Both authors are with the SYSTeMS Research Group, Ghent University, Technologiepark Zwijnaarde 914, 9052 Zwijnaarde, Belgium
[email protected], [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
characterizes its behavior when in isolation, i.e. without interaction with other oscillators.) Each cluster consists of oscillators with (asymptotically) equal phases, with the number of different clusters depending on the interaction. For non-identical natural frequencies for which the differences are sufficiently small, it may still be possible to distinguish different phase clusters [20]. Larger variations in the natural frequencies may induce oscillators having different long term average frequencies, resulting in frequency clustering [16], [4]: each cluster is characterized by the long term average frequency of its members. For more details on both phenomena and for examples of clustering in chaotic systems we refer to [14]. Due to the complexity and richness of the dynamics of some of these models, analytical results are often hard to come by and restricted to the existence and local stability properties of some of their solutions [10], [20], [2], and exploration of the parameter space is usually done through simulations [11], [13]. In previous papers [5], [1], [6] we introduced a dynamical model for clustering, corresponding to a system of nonidentical attracting agents, which may be connected according to an arbitrary network structure. Each agent has an autonomous component — its natural velocity — and attracts other agents by a saturating interaction function. Depending on the value of the coupling strength, different cluster structures may arise. In contrast with many of the aforementioned models, an in-depth analysis is possible: we were able to analytically characterize the cluster structure by a set of necessary and sufficient conditions in the model parameters, we showed that there exists a unique cluster structure satisfying these inequalities for a given set of parameters and that, in general, the distances between agents from the same cluster approach constant values. In this paper we extend this model, considering agents in a multi-dimensional space. In section II we present the model, and in section III we review some results from [6] concerning the model with one-dimensional agents. In section IV we discuss the clustering behavior of the multi-dimensional model. The transitions between the cluster structures with increasing coupling strength may involve more than two clusters — even for generic cases — as is shown in section V, where we investigate a system with three agents and we derive conditions for three clusters to merge and form a single cluster when the coupling strength is increased above a transition value. We conclude with an illustration on opinion formation.
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ThB17.1 II. MODEL DESCRIPTION
B. Results Theorem 1: Consider the following inequalities
We consider the model described by
> > e(G< ve(G< k , Gk , Gk ) < v k+1 , Gk+1 , Gk+1 ),
N
x˙i (t) = bi + K ∑ γ j fi j ( x j (t) − xi (t) )ex j (t)−xi (t) ,
∀ k ∈ IM−1 , < > e(G< ve(Gk ∪ Gk,1 , Gk,2 , G> k ) 0, K ≥ 0, N > 1, and xi (t), bi ∈ RP , with P ≥ 1. The differentiable functions fi j are non-decreasing with fi j (0) = 0, and satisfy fi j = f ji ,
and
lim fi j (ξ ) = Fi j ,
ξ →+∞
for all i and j in {1, . . . , N}, for some symmetric matrix F ∈ (R+ )N×N . Furthermore, ex ,
x , kxk
∀ x ∈ RP \ {0};
e0 , 0.
III. PRELIMINARY RESULTS The results formulated in this section concern the model (1) for the case P = 1, where each agent belongs to a one-dimensional space: xi (t) ∈ R, ∀ i ∈ {1, . . . , N}. They immediately follow from results in [6]. A. Some notation Consider a non-empty set G0 ⊂ IN . For a vector b ∈ (RP )N we define hbiG0 as the weighted average of bi over G0 : hbiG0
∑i∈G0 γi bi , . ∑i∈G0 γi
ve(G− , G0 , G+ ) , hbiG0 +
!
∑ γi ∑
∑i∈G0 γi i∈G0
j∈G+
γ j Fi j −
∑
γ j Fi j ,
j∈G−
for all G− , G0 , G+ ⊂ IN with G0 non-empty. We consider the following definition of clustering behavior — which we term as distance clustering behavior — of a solution x of (1) with P = 1 with respect to a cluster structure G: •
•
•
(2b)
∀ k ∈ IM . Denote by (2a’), resp. (2b’), the inequalities (2a), resp. (2b), in which the strict inequalities are replaced by non-strict inequalities. Then the conditions (2), resp. (2’), are sufficient, resp. necessary, for clustering behavior with respect to G of all solutions of the system (1) with P = 1. When the conditions (2’) are satisfied, but not (2), any clustering behavior depends on whether the interaction functions fi j attain their saturation values or not, or the system may not exhibit clustering behavior at all. In this latter case, there are two agents with their mutual distance growing unbounded on the one hand, but returning infinitely many times below some finite value on the other hand. Theorem 2: For every b ∈ RN and every symmetric matrix F ∈ (R+ )N×N , there exists a partition of R+ in a finite number of intervals, such that the interior of each interval represents values of K corresponding to a unique ordered set partition G of IN , satisfying (2). Theorem 3: Let x be a solution of (1) with P = 1 and with cluster structure G = (G1 , . . . , GM ). For each k ∈ IM , if i ∈ Gk , then > lim x˙i (t) exists and equals ve(G< k , Gk , Gk ).
t→+∞
Let G = (G1 , . . . , GM ) be an ordered set partition of IN . S S > Set G< k , k0 k Gk0 . With b ∈ R (i.e. for P = 1), set
K
∀ Gk,1 , Gk,2 ( Gk , with Gk,2 = Gk \ Gk,1 ,
(2a)
The distances between agents in the same cluster remain bounded (i.e. xi (t) − x j (t) is bounded for all i, j ∈ Gk , for any k ∈ IM , for t ≥ 0). For any D > 0 there exists a time after which the distances between agents in different clusters are and remain at least D. The agents are ordered by their membership to a cluster: k < l ⇒ xi (t) < x j (t), ∀ i ∈ Gk , ∀ j ∈ Gl , ∀t ≥ T , for some T > 0.
If furthermore all functions fi j are increasing in R+ , then for each k ∈ IM , if i, j ∈ Gk , then lim (xi (t) − x j (t)) exists and is independent of x(0). t→+∞ In other words, within a cluster the distances between the agents will approach constants, which are independent of the initial condition, and the velocities of the agents will approach the asymptotic average cluster velocity. When the coupling strength is varied, transitions between cluster structures may take place. The transition values correspond to values for K for which one of the inequalities in (2) becomes an equality. Depending on whether the inequality corresponds to (2a) or (2b), two clusters will merge or one cluster will split into two clusters. Generically, at most one equality will occur for a single value of K, and thus at most two clusters are involved in the cluster transition. As we will show in this paper, when P > 1, the occurrence of a transition involving more than two clusters is possible in a generic way. C. Remarks on clustering behavior The ordering of the clusters in the definition of clustering behavior is included to enable a convenient formulation of the results. In the next section we will redefine clustering behavior by the two other characteristics only (i.e. bounded clusters, with unbounded separation between the clusters),
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ThB17.1 since no natural order exists when the dimension is larger than one. Furthermore, we will also consider a slightly different type of clustering behavior — velocity clustering behavior, as opposed to distance clustering behavior — where a cluster is defined by a common long term average velocity of its members. From the previous results it follows that both types of clustering behavior coincide for the model (1) (with P = 1) except in non-generic cases where equalities arise in (2’), i.e. at transitions between different cluster structures.
velocity of the agents: x j (t)/t − xi (t)/t
lim ex j (t)−xi (t) = lim t→+∞ x j (t)/t − xi (t)/t = ev j −vi .
t→+∞
Denoting the (velocity) cluster structure by H = {H1 , . . . , HM }, and considering (weighted) averages over the clusters Hk of the equations of motion (1), one can verify that the asymptotic velocities vi , limt→+∞ x˙i (t) satisfy vi = hbiHk +
IV. MULTI-DIMENSIONAL AGENTS A. Definition of clustering behavior As mentioned before, the definition of clustering behavior of a solution x of (1) cannot simply be extended to P > 1; we will drop the ordering of the clusters: x exhibits distance clustering behavior with respect to the (unordered) set partition G if the following conditions are satisfied. •
•
The distances between agents
in the same cluster remain bounded (i.e. xi (t) − x j (t) is bounded for all i, j ∈ Gk , for any Gk ∈ G, for t ≥ 0). For any D > 0 there exists a time after which the distances between agents in different clusters are and remain at least D.
We say that a solution x of (1) exhibits velocity clustering behavior with respect to a set partition H of IN if limt→+∞ xi t(t) exists for all i in IN , and all agents belonging to the same cluster in H have the same limit value, while agents from different clusters have different limit values. As we will show in a forthcoming paper [7], the solutions of (1) are such that for each i ∈ IN , limt→+∞ x˙i (t) exists and is independent of the initial condition, implying that every solution x of (1) exhibits velocity clustering behavior with respect to some partition H, with each cluster Hk characterized by a common asymptotic velocity for the agents it contains. If x also exhibits distance clustering behavior with respect to a partition G then it follows that G is a refinement of H, i.e. ∀ G0 ∈ G : ∃ H0 ∈ H, G0 ⊂ H0 . Only in some non-generic cases may G differ from H or not exist at all (i.e. there may be no distance clustering behavior). In the remainder of the paper we will simply refer to ‘clustering behavior’ when the distinction between both types is not relevant for the aspects we are considering.
K ∑l∈Hk γl
∑ γl ∑ γ j Fl j ev j −vl , l∈Hk
(3)
j∈IN
for all i in Hk , for all k ∈ IM . The existence of velocities vi solving these equations can be considered as the equivalent of (2a) for P > 1. The conditions (2b) state that (for P = 1) a cluster cannot split into two subsets for the given velocities of the other subsets. To formulate the equivalent of (2b) for P > 1 we therefore consider an arbitrary partition H˜ k = {H˜ k,1 , . . . , H˜ k,M0 } of Hk (with H˜ k 6= {Hk }), and we impose that the equations v0i = hbiH˜ k,m +
K ∑l∈H˜ k,m γl
∑
l∈H˜ k,m
γl
∑ γ j Fl j ev0j −v0l ,
(4)
j∈IN
for all i ∈ H˜ k,m , and for all m ∈ IM0 (notice that only i ∈ Hk is considered), do not have a solution for v0 with v0i = vi for all i ∈ / Hk (where v is defined by (3)). In other words, in the previous equation we consider only v0i with i ∈ Hk as free variables, while v0i for any i ∈ IN \ Hk is fixed and equal to vi . As opposed to the conditions (2b) which correspond to partitions in only two subsets, any partition of Hk has to be considered. This will be illustrated in section V, where we investigate conditions for which three clusters may merge and form a single cluster when increasing the coupling strength K, while no intermediate cluster structure with two clusters exists for any value of K. C. Independence of the initial condition In this section we show that the existence of a solution x∗ exhibiting distance (resp. velocity) clustering behavior implies the same distance (resp. velocity) clustering behavior for any other solution x. This follows from a similar argument as in [6], by considering the function V : R → R : t 7→
∑ γi kxi (t) − xi∗ (t)k2 .
i∈IN
For convenience we introduce the notation B. Conditions for clustering behavior
f¯i j (x) , fi j (kxk)ex ,
Conditions replacing (2) and (2’) for the characterizing clustering behavior for P > 1 are much harder to derive. In this paper we will discuss the equations (3) below and look into a special case with 3 agents, all-to-all interaction and equal weights. The interaction between agents from different clusters will approach its saturation value (in norm), and will asymptotically be aligned according to the difference in asymptotic
for all x ∈ RP , and i, j ∈ IN . Since for any a, b ∈ RP and i, j ∈ IN , (a − b)T f¯i j (a) − f¯i j (b)
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= (kak ea − kbk eb )T ( fi j (kak)ea − fi j (kbk)eb ) ≥ (kak − kbk) ( fi j (kak) − fi j (kbk)) ≥0
(5)
ThB17.1 × f¯0 (xm (t) − xl (t)) +C0 ,
(since fi j is non-decreasing), it follows that = −K
dV (t) = 2K ∑ γi γ j (xi (t) − xi∗ (t))T dt i, j∈IN
× f0 (kxm (t) − xl (t)k) +C0 ,
× f¯i j (x j (t) − xi (t)) − f¯i j x∗j (t) − xi∗ (t) = −K
which would lead to a contradiction if W (t) would grow unbounded with t, and therefore agent m belongs to the cluster Gk .
(x j (t) − xi (t) − x∗j (t) + xi∗ (t))T
∑
i, j∈IN
× f¯i j (x j (t) − xi (t)) − f¯i j x∗j (t) − xi∗ (t)
V. A TRANSITION FROM THREE CLUSTERS TO ONE CLUSTER
≤ 0, and therefore V (t) and kxi (t) − xi∗ (t)k (i ∈ IN ) remain bounded when t → +∞, and thus distance (resp. velocity) clustering behavior of x∗ implies distance (resp. velocity) clustering behavior of x with respect to the same cluster structure. D. Convexity of the clusters for all-to-all coupling In this section we consider all-to-all coupling, i.e. we assume that the functions fi j are all equal to a (nondecreasing) function denoted by f0 , with saturation value F0 > 0. We show that for a solution exhibiting distance clustering behavior, an agent with a natural velocity that is a convex combination of natural velocities of agents from the same cluster also belongs to this cluster. This property does not hold when there is no all-to-all coupling (see e.g. [6] for an example with P = 1). Assume the solution x of (1) exhibits distance clustering behavior with respect to G = {G1 , . . . , GM }, with bm = ∑ j∈Gk µ j b j , for some k ∈ IM , m ∈ IN , µ ∈ [0, 1]N with µ j = 0 if j ∈ / Gk , µm = 0, and ∑ j∈IN µ j = ∑ j∈Gk µ j = 1. We will show that this convexity property implies that m ∈ Gk . Consider the function W , defined by
2
1
W : R → R : t 7→ W (t) , xm (t) − ∑ µ j x j (t) .
2 j∈G k
Then dW (t) dt !T xm (t) −
=
∑
µ j x j (t)
K
∑ γn f¯0 (xn (t) − xm (t))
n∈IN
j∈Gk
! −
∑ µl K ∑ γn f¯0 (xn (t) − xl (t)) n∈IN
l∈Gk
=K
In this section we investigate the conditions for which three clusters (each containing one agent) may merge and form a single cluster of three agents when increasing the coupling strength K above some critical value Kc , under the assumptions of all-to-all coupling ( f12 = f13 = f23 , f0 and F12 = F13 = F23 , F0 ) and equal weights: γ1 = γ2 = γ3 = 13 . We assume that b1 , b2 and b3 are not all equal and we consider the limit K → Kc (K < Kc ) in the equations (3) for the velocity cluster structure H = {{1}, {2}, {3}}. The limit values of the left hand sides are all equal to the average velocity hbi{1,2,3} , since for K = Kc the asymptotic velocities coincide: v1 = v2 = v3 = hbi{1,2,3} . The limit values of the terms ev j −vi will however be different from e0 = 0, as their norm remains equal to one. We therefore introduce xie (i ∈ {1, 2, 3}) as a limit point (when K approaches Kc from below) of the corresponding values of q
j∈Gk l∈Gk n∈IN
vi − hbi{1,2,3}
2 , ∑i∈{1,2,3} 13 vi − hbi{1,2,3}
and we rewrite the equations (3) as Kc F0 hbi{1,2,3} = bi + ∑ exej −xie , 3 j∈I 3
∀ i ∈ {1, 2, 3},
(6)
where we assume that x1e , x2e and x3e are pairwise different. If f0 (ξ ) = F0 for some ξ > 0 (i.e. f0 reaches its saturation value) and kxej − xie k ≥ ξ (i, j ∈ {1, 2, 3}, i 6= j), then the choice xi (t) , xie + hbi{1,2,3}t, for all t in R, results in a possible solution of (1). This shows how the xie can be interpreted as equilibrium positions in RP (up to a common linear term in t), hence our notation. Setting b0i , bi − hbi{1,2,3} ,
∑ ∑ ∑ µ j µl γn (xm (t) − xn (t) + xn (t) − xl (t))T
∀ i ∈ {1, 2, 3},
we obtain
× f¯0 (xn (t) − xm (t)) − f¯0 (xn (t) − xl (t)) +K
∑ µl (γm + γl ) kxm (t) − xl (t)k l∈Gk
3b01 = ex2e −x1e + ex3e −x1e , Kc F0 3b0 − 2 = ex1e −x2e + ex3e −x2e , Kc F0 3b0 − 3 = ex1e −x3e + ex2e −x3e . Kc F0 −
T
∑ ∑ ∑ µ j µl γn (xl (t) − x j (t))
j∈Gk l∈Gk n∈IN
× f¯0 (xn (t) − xm (t)) − f¯0 (xn (t) − xl (t)) . The second term is bounded by some constant C0 because of the distance clustering behavior. In the first term we use the property (5) for n 6= m and n 6= l, resulting in
It follows that
dW (t) ≤ −K ∑ µl (γm + γl )(xm (t) − xl (t))T dt l∈G
0 2
3b1
Kc F0 = 2 + 2ζ1 ,
k
4781
(7a) (7b) (7c)
ThB17.1
0 2
3b2
Kc F0 = 2 + 2ζ2 ,
0 2
3b3
Kc F0 = 2 + 2ζ3 ,
0 2 0 2 0 2
b3 + b1 > b2 .
where ζ1 , eTxe −xe ex3e −x1e , 2
1
ζ2 , eTxe −xe ex3e −x2e , 1
2
ζ3 , eTxe −xe ex2e −x3e . 3 1 e e e e Setting E , ex1 −x3 ex2 −x1 ex3e −x2e ∈ RP×3 , it follows from the coplanarity of the columns of E that rank(E) < 3 and thus 1 −ζ1 −ζ3 1 −ζ2 = 0, det(E T E) = det −ζ1 −ζ3 −ζ2 1 or 1 − 2ζ1 ζ2 ζ3 − ζ12 − ζ22 − ζ32 = 0, (9) which is rewritten in terms of b01 , b02 and b03 as follows:
2 2
2 2
2 2 2 b01 b02 + 2 b02 b03 + 2 b01 b03
0 4 0 4 0 4 kb01 k2 kb02 k2 b03 2
− b1 − b2 − b3 = . (10) 2 Kc F0 3
The left hand side can be shown to be non-negative by considering the values b0k as sides of a triangle (since b01 + b02 + b03 = 0) and applying triangle inequalities. It is zero only when b01 , b02 and b03 are collinear. If this is not the case then a value for Kc follows from (10). Equation (10) guarantees that we can find coplanar vectors exej −xie satisfying (7), without relating these vectors to points xie . To guarantee the existence of points xie satisfying (7), we need to impose a condition on the orientation of the vectors exej −xie ; we require the existence of positive coefficients such that the corresponding linear combination of columns of E is zero: 3 ∃ c ∈ R+ : Ec = 0, or equivalently E T Ec = 0, 0 resulting in the conditions ζ1 ζ2 + ζ3 > 0,
(11a)
ζ2 ζ3 + ζ1 > 0,
(11b)
ζ3 ζ1 + ζ2 > 0.
(11c)
In section A of the appendix we show that under the condition (9) and for ζ1 , ζ2 , ζ3 ∈ [−1, 1], (11) is equivalent to ζ1 + ζ2 − ζ3 + 1 > 0,
(12a)
ζ2 + ζ3 − ζ1 + 1 > 0,
(12b)
ζ3 + ζ1 − ζ2 + 1 > 0,
(12c)
0 2 0 2 0 2
b1 + b2 > b3 ,
0 2 0 2 0 2
b2 + b3 > b1 ,
(13a)
or
(13b)
(13c)
In other words, the triangle with sides kb01 k, kb02 k and b03 is acute, or equivalently, the angles between the vectors b01 , b02 and b03 in RP are obtuse. Notice that the inequality signs are strict. In case of equality, two of the corresponding points x1e , x2e and x3e would coincide, and this (non-generic) case has to be considered separately; the behavior of the solutions of (1) then depends on whether or not f0 reaches its saturation value, but we will not elaborate on this. Up to now, we have shown that (10) and (13) are necessary for the existence of points xke satisfying (7). We will show that these conditions are also sufficient. It can be easily verified that if ζ1 = cos α and ζ2 = cos β for some α, β ∈ [0, π], then (9) and (11a) imply that ζ3 = − cos(α + β ), and therefore there exists a triangle with vertices x˜ke (k ∈ {1, 2, 3}), for which the cosines of the angles equal ζ1 , ζ2 and ζ3 . Defining b˜ 0k (k ∈ {1, 2, 3}) by equations similar to (7), it then follows that kb˜ 0k k = kb0k k, for all k in {1, 2, 3}, and also b˜ 01 + b˜ 02 + b˜ 03 = 0. Therefore b01 , b02 and b03 , and b˜ 01 , b˜ 02 and b˜ 03 define two congruent triangles and there exists an orthogonal transformation mapping b˜ 0k onto b0k , ∀ k ∈ {1, 2, 3}. The images xke of the vertices x˜ke under this transformation then satisfy (7), and as a consequence we have proven the following. Theorem 4: The equations (10) and (13) are necessary and sufficient for the existence of pairwise different xke ∈ RP (k ∈ {1, 2, 3}) satisfying (7). We conclude that the phenomenon where three clusters merge when increasing the coupling strength above some critical value, is not restricted
to non-generic cases, such as when kb01 k = kb02 k = b03 , but the range of triples (b1 , b2 , b3 ) ∈ (RP )3 for which it occurs has non-zero measure, as follows from the conditions (13). VI. OPINION FORMATION In this section we illustrate how the system (1) may model an opinion formation process regarding an issue with several aspects that have to be taken into account. An opinion is represented by a point in RP , and each coordinate corresponds to one of the P various aspects. (We assume that each aspect can be represented by a single variable — possibly by further decomposing aspects into several partial aspects. Furthermore we assume that each of these aspects can be represented by a real variable.) As an example, the issue might be the question of how to respond to an economic recession, with severel aspects corresponding to e.g. lowering taxes, stimulating innovative research, or a preference for protective measures instead of cooperation with other countries on a larger scale. We consider N individuals taking part in a meeting; each individual has his own opinion on the issue on the agenda, which may evolve in time due to discussion with the other members. Opinions cannot grow unbounded, and therefore xi in (1) is not an appropriate quantity to represent an opinion. Instead we will take the derivative yi = x˙i as a measure of someone’s opinion. The equations for yi can be written
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as (assuming xi (0) = 0, ∀ i ∈ IN , without loss of generality regarding the long-term behavior)
2
K N
yi (t) = bi + K ∑ γ j f¯i j
Rt
0 0 0 0 (y j (t ) − yi (t ))dt ,
1
(14)
j=1
0
∀ i ∈ IN . The value bi ∈ RP represents the a priori opinion of agent i (corresponding to no discussion). With yi (t) representing theRopinion of agent i at time t, each component of the integral 0t (y j (t 0 ) − yi (t 0 ))dt 0 may reflect the level of disagreement accumulated over time for one aspect of the issue, and may correspond to the amount of discussions taking place between agents i and j on this aspect, proportional with time and with difference in opinion. One may be willing to make concessions for each aspect separately, but when combined, large concessions for one aspect will lead to smaller concessions for other aspects, as is reflected by the dependence of each interaction term on the norm of the aformentioned integral. In general, everyone starts with his own opinion bi while with time and through interaction, different groups are formed, each group characterized by a final opinion vi obtained through discussion. The pressure to reach a decision, or the tendency to adapt one’s opinion by paying attention to each other’s arguments, is reflected by the coupling strength K. As an important distinction with other existing models (for an overview, see [12]) we want to emphasize that the model (14) allows the coexistence of several groups (see also [6]), each characterized by its own group opinion — as opposed to models focusing on total consensus or the coexistence of only two opinions (such as in [19] or [22]). In Fig. 1(a) we consider P = 2 and we show the evolution of the opinions vi eventually reached as a function of K. The vi -values are calculated by simulations. We consider 50 agents with both components of bi chosen from a Gaussian distribution with zero mean and standard deviation one. The parameters γi are all equal to N1 , and the Fi j (i 6= j) are all taken equal to one. Notice the emergence of one big cluster containing most agents for K ≈ 2. In Fig. 1(b) the time evolution of the opinions yi for K = 2, again obtained by numerical integration of the mathematical model, is shown. (For the numerical integration in Fig. 1 the Euler method was used with a time step of 0.03.) For comparison, we also show a similar result for the case P = 1 in Fig. 2, with again N = 50. The bi -values are the first components of the bi -values from the previous simulation with P = 2. Fig. 2 was obtained by means of an algorithm based on the inequalities (2) (instead of a simulation of the differential equations). Compared to Fig. 1, the formation of the central cluster is more gradual, with each transition corresponding to merging two clusters. This difference is reminiscent of the difference between first-order and secondorder phase transitions, which will be further investigated in future research.
1 0 vi,2 −1 −1.5
−0.5
−1
vi,1
0
1.5
1
0.5
(a) 1
0.5
t 0 2
yi,2
0 −2
−2
−1.5
−1
−0.5
yi,1
0
0.5
1.5
1
2
(b) Fig. 1. Opinion formation with P = 2: Fig. (a) shows the opinions vi as a function of the coupling strength K. Fig. (b) shows the time evolution for K = 2. The parameters values are: N = 50, γi = N1 , and Fi j = 1 (i, j ∈ IN , j 6= i).
2.5 2 1.5
K 1 0.5 0 −3
−2
−1
0
vi
1
2
3
Fig. 2. Opinion formation with P = 1: The opinions vi are shown as a function of the coupling strength K. The parameters values are: N = 50, γi = N1 , and Fi j = 1 (i, j ∈ IN , j 6= i).
VII. CONCLUSION
We have considered a multi-agent clustering model with a multi-dimensional state-space for each agent. While for the one-dimensional case a transition in the cluster structure generically involves at most two clusters, we have shown that, for higher dimensions, more than two clusters may be involved in a transition. We analysed a system with three agents, and characterized the different cluster structures that may arise with varying coupling strength. As an illustration, we have simulated an example of an opinion formation process.
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ThB17.1 VIII. ACKNOWLEDGMENTS This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with its authors. During this research Filip De Smet was supported by a Ph.D. fellowship of the Research Foundation - Flanders (FWO). A PPENDIX A. Proof of the equivalence of (11) and (12) under (9) for ζ ∈ [−1, 1]3 Adding the equations in (11) two by two, and considering that ζ ∈ [−1, 1]3 , it follows that ζ1 + ζ2 > 0, ζ2 + ζ3 > 0, ζ3 + ζ1 > 0, implying (12). For the other direction of the equivalence relation, we multiply the equations (12) two by two, obtaining ζ32 + 2ζ3 + 1 − ζ12 − ζ22 + 2ζ1 ζ2 > 0, ζ12 + 2ζ1 + 1 − ζ22 − ζ32 + 2ζ2 ζ3 > 0, ζ22 + 2ζ2 + 1 − ζ32 − ζ12 + 2ζ3 ζ1 > 0.
[10] D. Golomb, D. Hansel, B. Shraiman, and H. Sompolinsky. Clustering in globally coupled phase oscillators. Physical Review A, 45(6):3516– 3530, March 1992. [11] G. Gr´egoire, H. Chat´e, and Y. Tu. Moving and staying together without a leader. Physica D, 181(3-4):157–170, July 2003. [12] R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5(3), 2002. [13] Y. Maistrenko, O. Popovych, O. Burylko, and P. A. Tass. Mechanism of desynchronization in the finite-dimensional Kuramoto model. Physical Review Letters, 93(8):084102, August 2004. [14] S. C. Manrubia, A. S. Mikhailov, and D. H. Zanette. Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems, volume 2 of World Scientific Lecture Notes in Complex Systems. World Scientific, 2004. [15] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros. Mutual interactions, potentials, and individual distance in a social aggregation. Journal of Mathematical Biology, 47:353–389, May 2003. [16] L. G. Morelli, H. A. Cerdeira, and D. H. Zanette. Frequency clustering of coupled phase oscillators on small-world networks. European Physical Journal B, 43:243–250, Jan. 2005. [17] K. Okuda. Variety and generality of clustering in globally coupled oscillators. Physica D, 63(3-4):424–436, 1993. [18] S. H. Strogatz. SYNC: The Emerging Science of Spontaneous Order. Hyperion Press, New York, 2003. [19] K. Sznajd-Weron and J. Sznajd. Opinion evolution in closed community. International Journal of Modern Physics C, 11(6):1157–1165, September 2000. [20] P. Tass. Phase and frequency shifts in a population of phase oscillators. Physical Review E, 56(2):2043–2060, August 1997. [21] T. Vicsek, A. Czir´ok, I. J. Farkas, and D. Helbing. Application of statistical mechanics to collective motion in biology. Physica A, 274(1–2):182–189, December 1999. [22] F. Wu and B. A. Huberman. Social structure and opinion formation. Computational Economics 0407002, EconWPA, July 2004. [23] D. H. Zanette and A. S. Mikhailov. Dynamical clustering in oscillator ensembles with time-dependent interactions. Europhysics Letters, 65(4):465–471, February 2004.
Using (9) this can be written as 2(ζ1 ζ2 + ζ3 )(1 + ζ3 ) > 0, 2(ζ2 ζ3 + ζ1 )(1 + ζ1 ) > 0, 2(ζ3 ζ1 + ζ2 )(1 + ζ2 ) > 0, implying (11). R EFERENCES [1] D. Aeyels and F. De Smet. A mathematical model for the dynamics of clustering. Physica D: Nonlinear Phenomena, 273(19):2517–2530, October 2008. [2] D. Aeyels and J. A. Rogge. Existence of partial entrainment and stability of phase locking behavior of coupled oscillators. Progress of Theoretical Physics, 112(6):921–942, December 2004. [3] C. M. Bordogna and E. V. Albano. Statistical methods applied to the study of opinion formation models: a brief overview and results of a numerical study of a model based on the social impact theory. Journal of Physics: Condensed Matter, 19(6):065144 (16pp), February 2007. [4] F. De Smet and D. Aeyels. Partial entrainment in the finite KuramotoSakaguchi model. Physica D, 234(2):81–89, October 2007. [5] F. De Smet and D. Aeyels. Clustering in a network of mutually attracting agents. In Proceedings of the 47th IEEE Conference on Decision and Control, pages 1827–1832, December 2008. [6] F. De Smet and D. Aeyels. Clustering in a network of non-identical and mutually interacting agents. Proceedings of the Royal Society A, 465:745–768, March 2009. [7] F. De Smet and D. Aeyels. A multi-dimensional model for clustering exhibiting a first-order phase transition. In preparation, 2009. [8] G. Deffuant, F. Amblard, G. Weisbuch, and T. Faure. How can extremism prevail? a study based on the relative agreement interaction model. Journal of Artificial Societies and Social Simulation, 5(4), 2002. [9] V. Gazi and K. M. Passino. Stability analysis of swarms. IEEE Transactions on Automatic Control, 48(4):692–697, April 2003.
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