ON CONSENSUS IN THE CUCKER–SMALE TYPE MODEL ON ISOLATED TIME SCALES

. This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual inﬂuences in the group of agents do not change. A suﬃcient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.


1.
Introduction. Recently we can observe quite intensive increase of interest in research on groups of autonomous agents, in which everyone influences on others' behavior and opinions. The mechanism of cooperation in the group of agents seems to be very interesting to study: it creates qualitatively new behaviors like reaching a consensus without any central direction. Examples of this are the emergence of common languages in primitive societies, the way in which populations of animals move together (referred as flocking for birds) or reaching a consensus in a group of people like party or society. The description of this kind of interaction between agents is well suited by opinion dynamics. First steps towards the creation of opinion dynamics models were made in 1956 by John French (see [18]), who studied the interpretation of many complex phenomena about groups taking into account interpersonal relations among the individuals. Since then many researchers have been studying the behavior of a group of autonomous agents and a consensus formation [14,17,26,31]. In 1997 Krause [25] presented a nonlinear opinion dynamics model under bounded confidence, that is, each agent's opinion is influenced by those opinions, which are not too far. In the literature this model is also known as the Hegselmann-Krause model because of the profound study that was done by Hegselmann and Krause in [23]. In recent years several authors dedicated their attention to the study of Krause's model, see for example [6,7,19,20,30] and references therein. An important generalization of the Krause model was presented by Cucker and Smale in [15]. In this case the dynamics is described by a system of two equations: for i = 1, . . . , N , where x i (t) ∈ R k and v i (t) ∈ R k denote, respectively, the state and consensus parameters of agent i at time t, h is the magnitude of the step size and the weights a ij = H 1 + x i − x j 2 β , for fixed H > 0 and β ≥ 0, quantify the way that the members of the group influence each other. Cucker and Smale in [16] showed that under certain conditions the convergence of the consensus parameters to a common value is guaranteed.
In this paper we propose the following Cucker-Smale type model on an isolated time scale T: for i = 1, 2, ..., N , where σ and µ denote, respectively, the forward jump operator and the graininess function on T (for the definitions see Section 2), and a ij ∈ R + 0 := [0, ∞), for i, j ∈ {1, ..., N }. In other words, the magnitude of the step size is a function of time (not necessarily a constant). Our aim is to analyze how the convergence to a consensus depends on the graininess function, under the assumption that the way in which the members of the group influence each other does not change. As a motivating example, let us consider model (2) with N = 5 and the weights a ij given byĀ MatrixĀ describes a situation when agent 1 influences agents 2,3,4,5 and agent 2 influences agent 5 and no other influence occurs. Figures 1-2 illustrate the behavior of the system on two different time scales with the same initial conditions. The performed numerical simulations suggest that on T = 1 − 1 k : k = 1, . . . , 10 ∪ t k = 1 + 2.5 k i=0 sin(i) : k ∈ N 0 the system tends to a consensus, whereas on T = 1 − 1 k : k = 1, . . . , 10 ∪ t k = 1 + 6 k i=0 sin(i) : k ∈ N 0 there is no consensus. This example shows strikingly how important is the choice of a time scale in a consensus phenomenon.  Our discussions will be organized as follows: Section 2 sets up notation and terminology. In Section 3 we prove a sufficient condition (regarding the range of the graininess function) under which the considered model tends to a consensus (Theorem 3.2) and we apply this condition to specific cases of system (2). Moreover, it is proved that under certain conditions all consensus parameters converge to a weighted average mean, which is proved to be an invariant of the dynamics. Several examples showing the behavior of dynamic system (2) on different time scales are given in order to illustrate the results. We end with Section 4 of conclusions and future work.
2. Preliminaries on time scales calculus. The origin of calculus on time scales goes back to 1988 [2,24] when S. Hilger wrote his PhD thesis under the supervision of B. Aulbach. The idea was to unify and extend discrete and continuous analysis. With time this theory proved to be useful in various fields that require modeling of discrete and continuous data simultaneously, like the calculus of variations, control theory, economics and biology (see, e.g. [1,3,4,5,8,11,12,21,22,27,28,29]). For a wider look on the time scale calculus and its applications we refer the reader to the books [9,10].
A time scale, denoted by T, is an arbitrary nonempty closed subset of real numbers R. We assume that the time scale T has the topology inherited from R with the standard topology. Trivial examples of time scales are R, integer numbers Z and natural numbers N. Other examples include periodic numbers . We call σ the forward jump operator. Analogously we define the backward jump operator ρ : T → T by ρ(t) = sup {s ∈ T : s < t} with sup ∅ = inf T (i.e., ρ(m) = m if T has a minimum m). Using operators σ and ρ, we can classify points of any time scale T. Namely, a point t is called right-dense, right-scattered, left-dense and left-scattered if σ(t) = t, σ(t) > t, ρ(t) = t and ρ(t) < t, respectively. We say that t is isolated if ρ(t) < t < σ(t), and that t is dense if ρ(t) = t = σ(t). The (forward) graininess function µ : Hence, for a given t ∈ T, µ(t) measures the distance of t to its right neighbor.
1 k : n ∈ N}, then for every t n ∈ T, σ(t n ) = t n + 1 n+1 and µ(t n ) = 1 n+1 . In the case when sup T is finite and left-scattered, we define Main results. Assume that T is an isolated time scale such that sup T = ∞ and sup{µ(t), t ∈ T} < ∞. Let us analyze system (2) with initial conditions where E = R k and t 0 ∈ T. System (2) can be rewritten in the matrix form as where I is the N × N identity matrix, L = D − A is the Laplacian of the matrix a kj . The matrix notation Let us define what we mean by a consensus in the context of isolated time scales.
Definition 3.1. We say that system (5) initiated at t 0 tends to a consensus if the consensus parameters tend to a common value, namely lim n→∞ v i (σ n (t 0 )) =v, for every i = 1, ..., N .
Remark 2. Note that if there existst ∈ T such that v i (t) =v for every i = 1, ..., N , then for each i the dynamics originating from (x i (t),v) is given by the rigid translation x i (σ(t)) = x i (t) + µ(t)v, meaning that for each t ≥t the distance between the states of each two agents is preserved.
In order to establish conditions under which system (5) tends to a consensus we introduce some terminology and recall useful facts. Consider Q d := {(q 1 , q 2 , ..., q N ) ∈ E N : q 1 = q 2 = · · · = q N }, the diagonal of E N and Q ⊥ d := {(q 1 , q 2 , ..., q N ) ∈ E N : q 1 + q 2 + · · · + q N = 0}, its orthogonal complement with respect to the standard inner product. Therefore, every v ∈ E N can be uniquely written as The same holds for the state parameter. This implies that the projection onto Q ⊥ d of the solutions of (5) are the solutions of the restriction of the system to Q ⊥ d . Observe that convergence of all v i to a common value (see Definition 3.1) means convergence to the diagonal of E N or, if we set Q = E N /Q d , convergence to 0 in this quotient space.
In Q ≃ Q ⊥ d , fix an inner product and denote by · Q the norm induced by this inner product. Now we are in position to formulate our main result giving a sufficient condition under which system (5) converges to a consensus.
Moreover, there exists a constant C such that x(σ n (t 0 )) Q ≤ C, for all n ∈ N, and there existsx ∈ Q ⊥ d such that x ⊥ (σ n (t 0 )) →x, when n → ∞.
Generally, checking the condition I −µ(t)L Q < 1, for all t ≥ t 0 , is computationally cumbersome. However, there are two special cases that are easier to analyze. Namely, when A is a symmetric matrix or A is such that N j=1 a ij = N , for all i = 1, . . . , N , with all its rows equal.
Recall that the matrix norm induced by the Euclidean norm coincides with the spectral norm. That is, if A is a N × N real matrix, then where σ max (A) is the largest eigenvalue of the symmetric matrix A T A. In particular, if A is symmetric, then the Laplacian L is also symmetric and its eigenvalues l 1 , . . . , l N satisfy 0 = l 1 ≤ l 2 ≤ · · · ≤ l N = L .

In this case
where λ 1 (µ(t)), . . . , λ m (µ(t)) denotes the set of all distinct eigenvalues of I − µ(t)L. Corollary 1. If A is a symmetric matrix and |λ(µ(t))| < 1, for all t ≥ t 0 , then system (2) with the initial conditions (4) tends to the consensus (v,v, . . . ,v), Proof. Observe that the average mean of the consensus parameters is invariant in time. Indeed, summing from i = 1 to N both sides of equations and using symmetry of matrix A, we get It means that for all n ∈ N. Taking the limit (n → ∞) on both sides of the last equation one Example 3.4. Let us consider system (5) with N = 5, E = R and the symmetric matrix Using a software, it can be checked that the interval (0, 1.2871) is an approximate solution to I − µL Q < 1. Figures 3-4 illustrate the time evolution of consensus parameters and their states on different time scales when 0 < µ(t) < 1.2871, for all t ∈ T. In all those cases, the system tends to a consensus. Figure 5 shows the situation for µ = 1.2871, where there is no consensus. However, when we choose the time scale with µ = 1.2771, 1.2871, 1.2771, 1.2871, . . ., simulations show that the system tends to a consensus ( Figure 6). where, for convenience, a j = a ij for all i, j = 1, . . . , N . Therefore, Since 0 < µ(t) < 2, for all t ≥ t 0 , we have |1 − µ(t)| < 1, for all t ≥ t 0 . Finally, applying Theorem 3.2 we conclude that system (5) tends to a consensus. By the same method as in the proof of Corollary 1 one gets the following equality for all n ∈ N. Now to finish the proof it is enough to take the limit (n → ∞) on both sides of the last equation.

Remark 3.
Observe that numerical simulations in Example 3.4 and Example 3.5 demonstrate that the condition given in Theorem 3.2 is not necessary for system (5) to tend to a consensus ( Figure 6 and Figure 9). (v j (t) − v i (t)) . Observe that when |v j (t) − v i (t)| < 1, for all i, j = 1, . . . , N and µ(t) ∈ 0, 2 , for all t ≥ t 0 , then by Proposition 3 system (6) initiated at t 0 tends to a consensus. This improves the sufficient condition for the Krause model to reach a consensus presented in [20] for the case when all agents belong to the same confidence set (i.e., everyone influences on all the others opinions).

Conclusions.
It is well known that the continuous and the discrete time models behave sometimes differently, sometimes similarly. However, even in the discrete case when using different time measurements the dynamic behavior of a model can be changed. Here one may consider as an example the Cucker-Smale type model proposed in this paper. The theory of time scales proved to be a powerful mathematical tool that allows to analyze the dynamic behavior of a model as the time measurement is changed. Therefore it was natural to apply this theory to the Cucker-Smale model. We have proved a sufficient condition under which the Cucker-Smale type model tends to a consensus. This condition shows clearly the importance of the graininess function in a consensus phenomenon. The advantage of using this result lies in the fact that in applications the graininess function can be interpreted as a frequency of meetings of the group members. We want to emphasize that this work was intended as an attempt to motivate studies on the Cucker-Smale model by using the theory of time scales. There are many open problems: How to compute the maximum value of µ, for an arbitrary matrix A, in order to satisfy the hypothesis of Theorem 3.2? How to establish a sufficient condition for having a consensus in the case when T is an arbitrary time scale? How to prove a counterpart of Theorem 3.2 in the case when T is an arbitrary time scale and the weights a ij are functions of x i , x j ? Those are directions of our further research.