arXiv:1912.11949v1 [math.DS] 27 Dec 2019 rus n hnsm fte a eg ofr igrgopf group bigger a form to merge can them of birds some time, over then change and can groups, a leaders consider that idea, observes the easily fix one To flocking. asymptotic in topologies 7]). ex [2, articles [38], survey force recent bonding (see etc descrip [36], topo hydrodynamic networks network and communication kinetic [10], metric avoidance 28], [26, collision 3 limit 34], 20, mean-field 21, [19, 9, failures [8, random tions 23], e.g [17, aspects, approximation several from discrete community math applied in studied n hydrvdsffiin rmwrslaigt h lbla global the to a introdu an leading [14] the as frameworks on [44] sufficient Smale the model derived and in Vicsek’s they of Cucker applications and spirit in 2007, the found in In and model Netwon-like 45] [31]. 44, networks 43, sensor 37, observe 7, often literat are [3, in phenomena systems interchangeably Flocking usi used velocity herding. sometimes same and swarming is the with it motion and ordered mation, an in organized are ino oe NF rn uddb h oe oenet(MSI government Korea suppor the by is funded Jung grant (NRF) J. Korea of of work tion The Korea(NRF-2017R1A2B2001864). nti ae,aogohr,w ou nteeeto h random the of effect the on focus we others, among paper, this In h agn“ jargon The e od n phrases. and words Key 2010 Date Acknowledgment. MREC FSOHSI LCIGFRTEDISCRETE THE FOR FLOCKING STOCHASTIC OF EMERGENCE eebr3,2019. 30, December : opeegraph complete ahmtc ujc Classification. Subject Mathematics UKRSAEMDLWT ADMYSWITCHING RANDOMLY WITH MODEL CUCKER-SMALE oklaigt h tcatcflcigwt rbblt one probability with flocking stochastic the to this For leading topologies. work network switching randomly with model Abstract. eut[5 ntecniuu - model. C-S a continuous flocking the our of on corollary [15] a result As framew proce given one. geometric a probability in the with conditions flocking and required process the Poisson satisfy the processes topologi use network we A of times, times. ing switching set random admissible for function an density probability of terms in formulated I-AGDN,SUGYA A IWO UG N OENKIM DOHEON AND JUNG, JINWOOK HA, SEUNG-YEAL DONG, JIU-GANG flocking esuyeegn yaiso h iceeCce-ml (in Cucker-Smale discrete the of dynamics emergent study We fe hi ok h ukrSaeC)mdlhsbe exte been has model Cucker-Smale(CS) the work, their After . h oko .Y ai upre yteNtoa eerhFou Research National the by supported is Ha S.-Y. of work The eoe olciepeoeo nwihsl-rple p self-propelled which in phenomenon collective a denotes ” ukrSaemdl admysicigtplg,directe topology, switching randomly model, Cucker-Smale 1. TOPOLOGIES 91,3D5 68M10. 34D05, 39A12, Introduction 1 ):NRF-2016K2A9A2A13003815. : P) r ota ehv stochastic a have we that so ork xmlsfrtelwo switch- of law the for examples s e yteNtoa eerhFounda- Research National the by ted ois[1 2 8 2 3 9 42], 39, 33, 32, 18, 12, [11, logies epoieasffiin frame- sufficient a provide we , ayi,w mrv h earlier the improve we nalysis, ,4] iedlydcommunica- time-delayed 41], 0, u ucetfaeokis framework sufficient Our . okcnsltit eea local several into split can flock ’ in 5 ,2,2,4] nonsym- 40], 29, 22, 6, [5, tions ru fmgaigbrs Then, birds. migrating of group ,sohsi oss[,4 3 27], 13, 4, [1, noises stochastic ., sraie ydgah and digraphs by realized es r ihohrjrossc as such jargons other with ure sadso htteetwo these that show and ss r nenlvrals[4 25], [24, variables internal tra ayial aaebemodel, manageable nalytically nmn ilgclcomplex biological many in d gol niomna infor- environmental only ng dlclaypoi flocking asymptotic local nd niern oansc as such domain engineering e ipesecond-order simple a ced o iet ie Thus, time. to time rom ysicignetwork switching ly hr,DCS) short, rps flocking. graphs, d dto of ndation articles nsively 2 DONG, HA, JUNG, AND KIM the underlying network topologies keep changing over time. Hence, it is natural to consider flocking models with randomly switching network topologies for realistic flocking modeling. In this direction, authors recently studied the emergent dynamics of the continuous CS model with randomly switching network topologies and provided a sufficient framework for asymptotic global flocking in [15] (see related works [35, 46] for linear consensus models with randomly switching topologies). The goal of this work is to revisit the discrete Cucker-Smale model with randomly switch- ing network topologies, and provide a sufficient framework for global flocking and improve the earlier result [15] on the continuous model. To address our problem, we first discuss our problem setting. Let := , , be the admissible set of network topologies SG {G1 · · · GNG } (digraphs) satisfying a certain connectivity condition to be specified later, and let xi and vi be the position and velocity of the i-th C-S particle under the influence of randomly switching network topologies. Then, their dynamics is governed by the CS model:
x˙ i = vi, t> 0, 1 i N, N ≤ ≤ (1.1) 1 σ v˙i = χijφ( xj xi ) (vj vi) , N k − k − Xj=1 where denotes the standard ℓ2-norm in Rd and the communication weight φ : [0, ) k · k ∞ → R+ is bounded, Lipschitz continuous and monotonically decreasing: φ(x) φ(y) 0 < φ(r) φ(0) =: κ, [φ]Lip := sup | − | < , (1.2) ≤ x6=y x y ∞ | − | (φ(r) φ(s))(r s) 0, r,s 0. − − ≤ ≥ σ The adjacent matrix (χij) denotes the time-dependent network topology corresponding to the randomly switching law σ : [0, ) Ω 1, ,N . The law σ is associated with ∞ × → { · · · G} a sequence tℓ ℓ≥0 of random variables, called random switching instants, such that for each fixed ω{ }Ω, σ(ω) : [0, ) 1, ,N is piecewise constant and right-continuous ∈ ∞ → { · · · G} whose discontinuities are tℓ(ω) ℓ≥0. In addition, the random law σ assigns (1.1) with a network topology at each instant.{ } Specifically, once we fix ω Ω and an instant t> 0, then σ(t,ω)= σ(t (ω)) = k for some 1 k N and ℓ 0, and we∈ equip system (1.1) with the ℓ ≤ ≤ G ≥ network topology (χσ(t,ω)) given by the 0-1 adjacency matrix of the k-th digraph . ij Gk ∈ SG Next, we consider the discrete analog of CS whose emergent dynamics we will study. The discrete counterpart of CS will be given by the forward first-order Euler method. Let h := ∆t be the time-step, and for notational simplicity, we use the following handy notation: x [t] := x (th), v [t] := v (th) and σ[t] := σ(th), t 0 Z . i i i i ∈ { }∪ + Then, the discrete C-S model reads as follows:
xi[t +1] = xi[t]+ hvi[t], t = 0, 1, , 1 i N, N · · · ≤ ≤ (1.3) h σ[t] vi[t +1] = vi[t]+ χij φ( xj[t] xi[t] )(vj [t] vi[t]). N k − k − Xj=1 In this paper, we look for a sufficient framework leading to the asymptotic global flocking: sup max xi[t] xj[t] < , lim max vi[t] vj[t] = 0. 0≤t<∞ 1≤i,j≤N k − k ∞ t→∞ 1≤i,j≤N k − k DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 3
Before we present our main result, we introduce configuration diameters: X := (x , ,x ),V := (v , , v ), 1 · · · N 1 · · · N (X) := max xi xj , (V ) := max vi vj . D 1≤i,j≤N k − k D 1≤i,j≤N k − k Below, we briefly discuss our main result for (1.3). As aforementioned, we need to impose certain conditions on the admissible set for network topologies and the probability dis- SG tribution of the random switching times tℓ . As discussed in [15], the random dwelling times between immediate switchings need{ to} be made not too short and not too long. If this happens and choice of network topologies in the admissible set is made randomly, SG then the union of network topologies in G will govern the asymptotic dynamics in a suf- ficiently long finite time interval, due toS the law of large numbers. Hence, we need to assume that the union graph contains a spanning tree so that any pair of two particles will communicate eventually in a finite-time interval. These heuristic arguments can be made rigorously in Section 3.1. Then under these assumptions on the network topologies and random switching times, if the time-step h, choice probability pk for the network topology and communication weight φ satisfy a set of assumptions: Gk 1 log 1−hκ 1 0 P ω : x∞ > 0s.t sup (X[t,ω]) x∞, and lim (V [t,ω]) = 0 = 1. ∃ 0≤t<∞ D ≤ t→∞ D The rest of this paper is organized as follows. In Section 2, we briefly review basic termi- nologies and lemmas to be used throughout this paper, such as directed graphs, scrambling matrices, and their basic properties. We also provide a monotonicity lemma for velocity di- ameter. In Section 3, we provide a framework and our main results of this paper. In Section 4, we provide technical proof for our main result. In Section 5, we consider two concrete stochastic processes for dwelling times (Poisson process and geometric process) and then show that they satisfy the proposed framework which results in the asymptotic flocking. We also discuss some improvements for the continuous model by removing the compact support assumption on the probability density function of switching times. Finally, Section 6 is devoted to a summary of our main results and some future issues to be discussed later. Notation Throughout the paper, (Ω, F , P) denotes a generic probability space. For ma- trices A = (aij)N×N and B = (bij)N×N , we denote A B if aij bij for all i and j. Moreover, we define the Frobenius norm of A as follows. ≥ ≥ A := tr(A∗A), k kF and I = diag(1, , 1) is the d d identityp matrix. d · · · × 2. Preliminaries In this section, we briefly review basic materials on the directed graphs and matrix algebra, and then study the dissipative structure of system (1.3). 4 DONG, HA, JUNG, AND KIM 2.1. Directed graphs and matrix algebra. For the modeling of network topology, we use directed graphs (digraphs). A digraph = ( , ) consists of a vertex set = 1, ,N and an edge set consisting of orderedG V connectedE pairs of vertices.V We{ say· · ·j is} a E⊂V×V neighbor of i if (j, i) and i := j : (j, i) denotes the neighbor set of i. If(i, i) , then is said to have∈E a self-loopN at{ i. For a∈ given E} digraph = ( , ), we consider its∈E 0-1 G G V E adjacency matrix χ = (χij), where 1 if (j, i) , χ := ij 0 if (j, i) ∈E/ . ∈E A path in from i to j is a sequence of distinct vertices i = i, i , , i = j such that G 0 1 · · · n (im−1, im) for every 1 m n. If there is a path from i to j, then we say j is reachable from i. Moreover,∈E a digraph≤ ≤is said to have a spanning tree (or rooted) if has a vertex i from which any other verticesG are reachable. G Next, we review some concepts and results in matrix algebra which will be crucially used in later sections. First, we introduce several concepts of nonnegative matrices in the following definition. Definition 2.1. Let A = (a ) be a nonnegative N N matrix. ij × (1) A is a stochastic matrix, if its row-sum is equal to unity: N a = 1, 1 i N. ij ≤ ≤ Xj=1 (2) A is a scrambling matrix, if for each pair of indices i and j, there exists an index k such that aik > 0 and ajk > 0. (3) A is a weighted adjacency matrix of a digraph if the following holds: G a > 0 (j, i) . ij ⇐⇒ ∈E In this case, we write = (A). (4) The ergodicity coefficientG ofG A is defined as follows. N µ(A) := min min aik, ajk . i,j { } Xk=1 Remark 2.1. The following assertions easily follow from the above definition: (1) A is scrambling if and only if µ(A) > 0. (2) For nonnegative matrices A and B, A B = µ(A) µ(B). ≥ ⇒ ≥ In the sequel, we present two lemmas without proofs. These lemmas will be crucially used in next sections. Lemma 2.1. (Lemma 2.2, [16]) Suppose that a nonnegative N N matrix A = (a ) is × ij stochastic, and let B = (bj), Z = (zj) and W = (wj ) be N d matrices such that i i i × W = AZ + B. Then, we have max wi wk (1 µ(A)) max zl zm + √2 B F , i,k k − k≤ − l,m k − k k k DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 5 where z := (z1, , zd), b := (b1, , bd), w := (w1, , wd), i = 1, , N. i i · · · i i i · · · i i i · · · i · · · Lemma 2.2. (Theorem 5.1, [46]) If A are nonnegative N N matrices with positive diago- i × nal elements such that (Ai) has a spanning tree for all 1 i N 1, then A1A2 ...AN−1 is scrambling. G ≤ ≤ − 2.2. Matrix formulation of the DCS model. In this subsection, we provide a priori estimate for the velocity diameter along the sample path. For convenience, we suppress ω-dependence in the sequel. First, we consider the R.H.S. of (1.3)2: N h RHS := v [t]+ χσ[t]φ( x [t] x [t] )(v [t] v [t]) 2 i N ij k j − i k j − i Xj=1 N h h (2.1) = 1 χσ[t]φ( x [t] x [t] )+ χσ[t]φ( x [t] x [t] ) v [t] − N ij k j − i k N ii k j − i k i Xj=1 h + χσ[t]φ( x [t] x [t] )v [t]. N ij k j − i k j Xj6=i To rewrite (2.1) as a matrix form, we introduce N N matrix L [t] (k = 1, ,N ): × k · · · G L [t] := D [t] A [t], k k − k where the matrices Ak[t] and Dk(t) are defined as follows. A [t]= ak [t] , D (t) = diag dk[t], , dk [t] , k ij k 1 · · · N (2.2) N ak [t] := χk φ( x [t] x [t] ) and dk[t]= χk φ( x [t] x [t] ). ij ij k i − j k i ij k i − j k Xj=1 Finally, we combine (2.1) and (2.2) to rewrite system (1.3) as a matrix form: X[t +1] = X[t]+ hV [t], t = 0, 1, , (2.3) h · · · V [t +1] = Id L [t] V [t], − N σ[t] and define s −1 1 h (2.4) Φ[s ,s ,ω] := Id L . 1 0 − N σ[t,ω] t=s Y0 Next, we study the monotonicity of (V [t]). D Lemma 2.3. Suppose that the time-step and coupling strength satisfy 0 Proof. First, we use φ(r) φ(0) = κ in (1.2) and hκ < 1 ≤ to see that for each i = 1, ,N and k = 1, ,N , · · · · · · G N h h dk[t]= χk φ( x [t] x [t] ) hκ < 1. N i N ij k i − j k ≤ Xj=1 h This implies that Id N Lσ[t][t] is always nonnegative and its low sum is one, i.e., stochastic. Since the ergodicity coefficient− of any nonnegative matrices is nonnegative, we apply Lemma 2.1 to get h (V [t + 1]) 1 µ Id L [t] (V [t]) (V [t]). D ≤ − − N σ[t] D ≤ D The other estimate for the spatial diameter readily follows from (2.3)1. 3. A framework and main result In this section, we briefly present a framework for stochastic flocking in terms of admis- sible network topologies and switching times, and state our main result. 3.1. Problem setup. Let (Ω, F , P) be a probability space. Then, we set the sequence of (possibly) switching times t to be a sequence of N-valued independent random { ℓ}ℓ≥0 variables on (Ω, F , P). Then, we assume that the switching law σ : N Ω 1, ,NG and the sequence t t of dwelling times satisfy the following× conditions:→ { · · · } { ℓ+1 − ℓ}ℓ≥0 For each ℓ 0, t t is given by • ≥ ℓ+1 − ℓ (3.1) t t =1+ T , ℓ+1 − ℓ ℓ where Tℓ ℓ≥0 is a given sequence of independent integer-valued nonnegative random variables.{ } For each ℓ 0 and ω Ω, σ (ω) := σ(t,ω) is constant on the interval t • ≥ ∈ t ∈ [tℓ(ω),tℓ+1(ω)). σ is a sequence of i.i.d. random variables such that for any ℓ 0, • { tℓ }ℓ≥0 ≥ P(σ = k)= p , for each k = 1, ,N , tℓ k · · · G where p , ,p > 0 are given positive constants satisfying p + + p = 1. 1 · · · NG 1 · · · NG We briefly comment on the random switching time process t as follows. Suppose that { ℓ} we have process at t = tℓ. Then, the next switching time tℓ+1 is determined by the relation (3.1) via the integer-valued and nonnegative random process Tℓ: tℓ+1 = tℓ +1+ Tℓ, In Section 5, we will consider two explicit processes (Poisson and geometric processes) and show that they satisfy our framework that will be presented soon. For each k = 1, ,N , let = ( , ) be the k-th admissible digraph. Then, for each · · · G Gk V Ek t 0 and ω Ω, the time-dependent network topology (χσ ) = (χσ[t,ω]) is determined by ≥ ∈ ij ij DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 7 σ[t,ω] 1 if (j, i) , χ := ∈Eσ[t,ω] ij 0 if (j, i) / . ∈Eσ[t,ω] For technical reasons and without loss of generality, we assume that each admissible digraph k has a self-loop at each vertex, and we define the union graph of σ[t,ω] for s0 t For each positive integer n and positive real number c > 0, we define an increasing sequence a (n, c) N of integers by the following recursive relation: { ℓ }ℓ∈ (3.2) a (n, c) = 0, a (n, c)= a (n, c)+ n + c log(ℓ + 1) , (ℓ N). 0 ℓ+1 ℓ ⌊ ⌋ ∈ In what follows, we sometimes suppress n and c dependence for aℓ(n, c): aℓ := aℓ(n, c). We impose the following assumptions ( ) on the set G of admissible digraphs and the probability density function of the switchingA times: S ( 1): The union digraph from has a spanning tree: • A SG = , has a spanning tree. Gk V Ek 1≤k[≤NG 1≤k[≤NG ( 2): The dwelling times are bounded in probability: for some M N, • A ∈ ai(N−1)−1 P ω : T (ω) M(n + c log i(N 1) ), for some i N p˜(n), ℓ ≥ ⌊ − ⌋ ∈ ≤ ℓ=a(iX−1)(N−1) wherep ˜(n) 0 as n . → →∞ Now, we are ready to state our main theorem on the emergence of flocking. Theorem 3.1. Suppose that parameters N, h, κ, pk’s and communication weight φ satisfy the following conditions: 1 (M + N 1) log 1−hκ 1 0 4. Emergence of stochastic flocking In this section, we provide a sufficient framework for the emergence of flocking in system (1.3), and provide a proof for Theorem 3.1. First, we present a priori assumptions ( ˜1) - ( ˜3): for a fixed ω Ω, A A ∈ ( ˜1): there exist n N, n> 0 and c> 0 such that • A ∈ 1 cN log < 1, 1 hκ − and the subsequence t∗ N t N defined by t∗ := t in (3.2) satisfies { ℓ }ℓ∈ ⊂ { ℓ}ℓ∈ ℓ aℓ(n,c) ([t∗,t∗ ))(ω) has a spanning tree for all ℓ 0. G ℓ ℓ+1 ≥ ( ˜2): there exist n N, M N, and c > 0 such that the random sequence • AT (ω) satisfy the∈ following∈ boundedness condition: { ℓ }ℓ≥0 ai(N−1)−1 (4.1) T (ω) < M(n + c log i(N 1) ), for each i N. ℓ ⌊ − ⌋ ∈ ℓ=a(iX−1)(N−1) ( ˜3): the position diameter is uniformly bounded in time: • A sup (X[t,ω]) x∞ < . 0≤t<∞ D ≤ ∞ For the proof of Theorem 3.1, we split its proof into three steps as follows: Step A: A priori assumptions ( ˜1), ( ˜2) and ( ˜3) imply stochastic flocking. • A A A Step B: Conditions on initial configuration implies the a priori assumption ( ˜3). • A Step C: Framework ( 1)-( 2) imply condition on initial configuration and a priori • A A assumptions ( ˜1)-( ˜2). A A In the following three subsections, we present each step one by one. 4.1. Step A (A priori assumptions imply stochastic flocking). In this subsection, we show that the a priori assumptions ( ˜1), ( ˜2) and ( ˜3) imply stochastic flocking. A A A Lemma 4.1. Suppose that a priori assumptions ( ˜1) and ( ˜3) hold, and system parameters satisfy A A 0 (1) Φ[t∗ ,t∗ ] is stochastic for each r N. r(N−1) (r−1)(N−1) ∈ ∗ ∗ (2) The ergodicity coefficient of Φ[tr(N−1),t(r−1)(N−1)] satisfies ∞ N−1 t∗ −t∗ hφ(x ) µ(Φ[t∗ ,t∗ ]) (1 hκ) r(N−1) (r−1)(N−1) , r(N−1) (r−1)(N−1) ≥ − N(1 hκ) − for each r N. ∈ Proof. For each r N and 1 i N 1, we consider Φ[t∗ ,t∗ ]. ∈ ≤ ≤ − (r−1)(N−1)+i (r−1)(N−1)+i−1 Here, we assume that t [t∗ ,t∗ ]= t , ,t . { ℓ}ℓ≥0 ∩ (r−1)(N−1)+i−1 (r−1)(N−1)+i { ℓ1 · · · ℓq+1 } Then, we get q ∗ ∗ Φ[t(r−1)(N−1)+i,t(r−1)(N−1)+i−1]= Φ[tℓj+1 ,tℓj ], jY=1 and for each j, we can choose k 1, ,N such that j ∈ { · · · G} σ[t,ω]= k for all t N [t ,t ). j ∈ ∩ ℓj ℓj+1 Thus, we have tℓ −1 tℓ −1 j+1 h j+1 h h Φ[t ,t ]= Id L [t] = Id D [t]+ A [t] ℓj+1 ℓj − N kj − N kj N kj t=t t=t Yℓj Yℓj tℓj −1 +1 h (1 hκ)Id + A [t] , ≥ − N kj t=t Yℓj and consequently, q q tℓj+1 −1 h Φ[t∗ ,t∗ ]= Φ[t ,t ] (1 hκ)Id + A [t] (r−1)(N−1)+i (r−1)(N−1)+i−1 ℓj+1 ℓj ≥ − N kj j=1 j=1 t=t Y Y Yℓj q tℓj+1 −1 t∗ −t∗ −1 h (1 hκ) (r−1)(N−1)+i (r−1)(N−1)+i−1 A [t] ≥ − N kj j=1 t=t X Xℓj ∞ t∗ −t∗ hφ(x ) (1 hκ) (r−1)(N−1)+i (r−1)(N−1)+i−1 F , ≥ − N(1 hκ) i − where F denotes the 0-1 adjacency matrix of the graph [t∗ ,t∗ ) i G (r−1)(N−1)+i−1 (r−1)(N−1)+i and the second inequality follows from the fact: m m m (aId + B )= amId + am−1 B + am−1 B , for all a> 0, B 0. i i ···≥ i i ≥ Yi=1 Xi=1 Xi=1 Hence, we have N−1 ∗ ∗ ∗ ∗ Φ[tr(N−1),t(r−1)(N−1)]= Φ[t(r−1)(N−1)+i,t(r−1)(N−1)+i−1] Yi=1 10 DONG, HA, JUNG, AND KIM ∞ N−1 N−1 t∗ −t∗ hφ(x ) (1 hκ) r(N−1) (r−1)(N−1) F . ≥ − N(1 hκ) i − Yi=1 N−1 Since each Fi is the 0-1 adjacency matrix of a graph with a spanning tree, i=1 Fi is scrambling by Lemma 2.2. Moreover, we have Q N−1 µ Fi 1. ! ≥ Yi=1 Thus, we can deduce the result (2). For the stochasticity, note that Id h L [t] is stochastic − N k for any k = 1, ,NG. This gives the stochasticity of Φ[tℓj+1 ,tℓj ] and consequently, we obtain the desired· · · result. Next, we provide the decay estimates for the velocity diameter based on a priori assump- tions. In the sequel, we set X0 := X[0],V 0 := V [0]. Proposition 4.1. (Velocity alignment) Suppose that ( ˜1)-( ˜3) hold, and system parame- ters satisfy A A 0 ai(N−1)−1 ai(N−1)−1 t∗ t∗ = (t t )= a a + T i(N−1) − (i−1)(N−1) ℓ+1 − ℓ i(N−1) − (i−1)(N−1) ℓ ℓ=a(iX−1)(N−1) ℓ=a(iX−1)(N−1) (M + N 1)(n + c log i(N 1) ). ≤ − ⌊ − ⌋ This follows from the definition of the sequence aℓ and the a priori assumption. Thus, we have r hφ(x∞) N−1 (V [t]) 1 (1 hκ)(M+N−1)(n+⌊c log i(N−1)⌋) (V 0) D ≤ " − − N(1 hκ) # D Yi=1 − DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 11 r hφ(x∞) N−1 exp (1 hκ)(M+N−1)(n+c log i(N−1)) (V 0) ≤ "− − N(1 hκ) # D Yi=1 − r hφ(x∞) N−1 = exp (1 hκ)(M+N−1)(n+c log(N−1)) ic(M+N−1) log(1−hκ) (V 0) "− − N(1 hκ) # D − Xi=1 hφ(x∞) N−1 r+1 exp (1 hκ)(M+N−1)(n+c log(N−1)) xc(M+N−1) log(1−hκ)dx (V 0) ≤ − − N(1 hκ) D " − Z1 # hφ(x∞) N−1 (r + 1)1+c(M+N−1) log(1−hκ) 1 = exp (1 hκ)(M+N−1)(n+c log(N−1)) − (V 0). − − N(1 hκ) 1+ c(M + N 1) log(1 hκ) D " − − − # This leads to the desired estimate. 4.2. Step B (Pathwisely, ( ˜1) and ( ˜2) imply flocking). In this subsection, we show that the spatial diameter is uniformlyA bounded.A For this, we present a simple lemma without a proof. Lemma 4.2. [15] For any x> 0 and δ > 0, we have the following inequality: δ δ e−x x−δ. ≤ e Now, we show that the a priori assumption ( ˜3) on the position diameter can be replaced by the assumption on initial data. A Lemma 4.3. (Pathwise uniform bound for spatial diameter) Suppose that ω Ω satisfies ∈ a priori assumptions ( ˜1)-( ˜2), and system parameters and initial data satisfy A A (i) 0 hφ(x∞) N−1 (r + 1)1+c(M+N−1) log(1−hκ) 1 exp (1 hκ)(M+N−1)(n+c log(N−1)) − × − − N(1 hκ) 1+ c(M + N 1) log(1 hκ) − − − !# ∞ (X0)+ h (V 0) (M + N 1)(n + c log((r + 1)(N 1))) ≤ D D " − − Xr=0 hφ(x∞) N−1 (r + 1)1+c(M+N−1) log(1−hκ) 1 exp (1 hκ)(M+N−1)(n+c log(N−1)) − × − − N(1 hκ) 1+ c(M + N 1) log(1 hκ) − − − !# (X0)+ h (V 0)(M + N 1)(n + c log(N 1)) ≤ D D − − ∞ δ δ + h (V 0) (M + N 1)(n + c log((r + 1)(N 1))) D − − e r=1 " X −δ hφ(x∞) N−1 (r + 1)1+c(M+N−1) log(1−hκ) 1 (1 hκ)(M+N−1)(n+c log(N−1)) − × − N(1 hκ) 1+ c(M + N 1) log(1 hκ) − − − ! # Finally, we combine Proposition 4.1 with ( ˜1)-( ˜2) and a condition on communication weight to show the emergence of flocking for anyA initialA configuration. Proposition 4.2. Suppose that ω Ω satisfies a priori assumptions ( ˜1)-( ˜2), and system parameters and communication weight∈ satisfy A A 1 0 r∗−1 ∞ 1 + log(r + 1) = C + 1+c(M+N−1) log(1−hκ) δ r=1 r=r∗! (r + 1) 1 X X − r∗−1 ∞ 1 + log(r + 1) 1 + log(r + 1) C δ + (1+c(M+N−1) log(1−hκ))δ ≤ 1+c(M+N−1) log(1−hκ) ∗ r " r=1 (r + 1) 1 r=r # X − X < , ∞ where C > 0 is a positive constant and r∗ > 0 is a natural number satisfying 1 (r + 1)1+c(M+N−1) log(1−hκ) 1 r1+c(M+N−1) log(1−hκ), a (0, 1), ∀r r∗. − ≥ 2 ∈ ≥ Note that the existence of r∗ is guaranteed since (r + 1)1+c(M+N−1) log(1−hκ) 1 lim − = 1. r→∞ r1+c(M+N−1) log(1−hκ) Furthermore, since the series in (ii) of Lemma 4.3 converges, we may deduce that (X0)+ h (V 0)(M + N 1)(n + c log(N 1)) D D − − ∞ δ δ + h (V 0) (M + N 1)(n + c log((r + 1)(N 1))) D − − e r=1 " X −δ hφ(x∞) N−1 (r + 1)1+c(M+N−1) log(1−hκ) 1 (1 hκ)(M+N−1)(n+c log(N−1)) − × − N(1 hκ) 1+ c(M + N 1) log(1 hκ) − − − ! # C 1 + (φ(x∞))−(N−1)δ , ≤ where C is a constant independent of x∞. Note that the R.H.S. in (4.2) gives φ(r)−(N−1)δ = O(r(N−1)δε) as r . →∞ This implies φ(r)−(N−1)δ (4.3) lim = 0. r→∞ r Thus, it follows from (4.3) that we can deduce the existence of x∞ satisfying the relation (ii) Lemma 4.3 for δ given in (4.2). Since the condition (ii) is now satisfied, we obtain our desired results from Lemma 4.2 and Proposition 4.1. 4.3. Step C (Framework ( 1) ( 2) implies stochastic flocking). In this subsection, A − A we show that a priori assumptions ( ˜1) ( ˜2) can be fulfilled under our framework and prove that flocking emerges with probabilityA − A one. First, we show that the first a priori assumption ( ˜1) can be satisfied almost surely with the appropriate choices of parameters. Below, we defineA a subsequence t∗ t by t∗ := t . { ℓ }ℓ≥0 ⊂ { ℓ}ℓ≥0 ℓ aℓ(n,c) 14 DONG, HA, JUNG, AND KIM Lemma 4.4. Suppose that a positive integer n and a real number c > 0 are sufficiently large enough such that NG n 1 1 (1 pk) and c> , k = 1, ,NG, − ≤ 2 −log(1 pk) · · · Xk=1 − and let (X,V ) be a solution process to (1.1). Then we have the following assertions. (1) For each k = 1, ,N , · · · G ∞ (1 p )⌊c log(ℓ+1)⌋ < . − k ∞ Xℓ=0 (2) P(ω : ([t∗,t∗ ))(ω) has a spanning tree for any ℓ 0) G ℓ ℓ+1 ≥ NG ∞ n ⌊c log(ℓ+1)⌋ exp (2 log 2) (1 pk) (1 pk) =: p1(n). ≥ − − − ! Xk=1 Xℓ=0 Proof. The proof is almost the same as that of Proposition 4.3 in [15]. Thus, we omit the proof. Proof of Theorem 3.1: We choose ε so that 1 1 (M + N 1) log 1−hκ 0 ε< − , or equivalently ≤ N 1 − (N 1) min log 1 1−pk − − 1≤k≤NG 1 1 ε(N 1) < − − , min log 1 1 1−pk (M + N 1) log 1≤k≤NG − 1−hκ and we set 1 1 1 ε(N 1) c := + − − . 2 min log 1 1 1−pk (M + N 1) log 1≤k≤NG − 1−hκ First, note that 1 1 c> , c(M + N 1) log < 1, min log 1 − 1 hκ 1−pk 1≤k≤NG − 1 1 c(M + N 1) log 1−hκ and 0 ε< − − . ≤ N 1 − Now, we choose any n N such that ∈ N G 1 (1 p )n . − k ≤ 2 Xk=1 Define the events A, B and as F A : ([t∗,t∗ ))(ω) has a spanning tree for any ℓ 0, G ℓ ℓ+1 ≥ DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 15 ai(N−1)−1 B : T (ω) < M(n + c log i(N 1) ), for each i N, ℓ ⌊ − ⌋ ∈ ℓ=a(iX−1)(N−1) : x∞ > 0s.t sup (X[t,ω]) x∞ and lim (V [t,ω]) = 0. F ∃ 0≤t<∞ D ≤ t→∞ D Then, it follows from Proposition 4.1 that P( A B) = 1. F | ∩ Hence, we use Proposition 4.2, Lemma 4.4 to see P( ) P( (A B)) = P(A B)P( A B) F ≥ F ∩ ∩ ∩ F | ∩ = 1 P(Ac Bc) 1 (P(Ac)+ P(Bc)) = P(A) P(Bc) − ∪ ≥ − − p (n) p (n) 1 as n . ≥ 1 − 2 → →∞ This implies our desired result. 5. The dwelling time process and the continuous CS model In this section, we consider two explicit processes for the dwelling time process Tℓ which is equivalent to specifying the switching time process for the change of network topologies, and show that they satisfy one of the key conditions in our framework: ai(N−1)−1 lim P ω : Tℓ(ω) M(n + c log i(N 1) ), for some i N = 0. n→∞ ≥ ⌊ − ⌋ ∈ ℓ=a(iX−1)(N−1) Moreover, we discuss an improved stochastic flocking estimate compared to authors’ recent work [15]. 5.1. Dwelling time process. In this subsection, we present two explicit random sequence (T ) satisfying ( 2). ℓ ℓ≥0 A (Poisson’s process): Suppose that the dwelling time process T is given by a sequence • ℓ of Poisson’s random variable with a rate λℓ: (λ )k P(N = k)= ℓ e−λℓ , k = 0, 1, 2, . λℓ k! · · · Proposition 5.1. Suppose that the process Tℓ = Nλℓ is a sequence of independent Poisson random variables with the parameters λℓ satisfying λmax := sup λℓ < , ℓ≥0 ∞ and let (X,V ) be a solution process to (1.1). Then, for any c > 0, we can choose M N so that we have the following estimate: ∈ ai(N−1)−1 P ω : N (ω) M(n + c log i(N 1) ), for some i N p (n), λℓ ≥ ⌊ − ⌋ ∈ ≤ 2 ℓ=a(iX−1)(N−1) wherep (n) is a sequence satisfying { 2 } lim p2(n) = 0. n→∞ 16 DONG, HA, JUNG, AND KIM Proof. First, we use P ( ∞ B ) ∞ P(B ) to get ∪i=1 i ≤ i=1 i ai(N−1)−1 P P ω : N (ω) M(n + c log i(N 1) ), for some i N λℓ ≥ ⌊ − ⌋ ∈ ℓ=a(iX−1)(N−1) ∞ ai(N−1)−1 P ω : N (ω) M(n + c log i(N 1) ) ≤ λℓ ≥ ⌊ − ⌋ Xi=1 ℓ=a(iX−1)(N−1) ∞ = P ω : N˜ (ω) M(n + c log i(N 1) ) , λi ≥ ⌊ − ⌋ Xi=1 ai(N−1)−1 where λ˜i = λℓ and we used the independence to get ℓ=a(i−1)(N−1) P ai(N−1)−1 Nλ N˜ . ℓ ∼ λi ℓ=a(iX−1)(N−1) For simplicity, we set H(i, n) := n + c log i(N 1) , ⌊ − ⌋ and we have, for i 1, ≥ ai(N−1)−1 (5.1) λ˜ = λ a a λ (N 1)λ H(i, n). i ℓ ≤ i(N−1) − (i−1)(N−1) max ≤ − max ℓ=a − − (iX1)(N 1) Then, it follows from the distribution of Poisson random variables and (5.1) that ∞ r −λ˜i MH(i,n) (λ˜i) e (λ˜i) P ω : N˜ (ω) MH(i, n) = λi ≥ r! ≤ (MH(i, n))! r=MHX(i,n) ((N 1)λ H(i, n))MH(i,n) ((N 1)eλ /M)MH(i,n) − max − max , ≤ (MH(i, n))! ≤ 2πMH(i, n) where the first inequality follows from the Taylor approximap tion for eλi˜ : MH(i,n)−1 r MH(i,n) MH(i,n) MH(i,n) λ˜ ˜ (λ˜ ) (λ˜ ) d (λ˜ ) e i eλi i = i ex i , − r! (MH(i, n))! dxMH(i,n) ˜ ≤ (MH(i, n))! r=0 x=λ∈(0,λi) X n+ 1 −n and the last one is from the Stirling’s approximation: n ! √2πn 2 e for all n N. ≥ ∈ For M > (N 1)eλ , one obtains − max ∞ P ω : N˜ (ω) M(n + c log i(N 1) ) λi ≥ ⌊ − ⌋ i=1 X ∞ ((N 1)eλ /M)MH(i,n) − max ≤ 2πMH(i, n) Xi=1 p DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 17 ∞ 1 ((N 1)eλ /M)M(n+c log i(N−1)−1) ≤ √ − max 2πMn i=1 X ∞ ((N 1)eλ /M)M(n+c log(N−1)−1) = − max iMc log((N−1)eλmax/M) √2πMn Xi=1 =: p (n) 0, as n . 2 → →∞ The last series converges if and only if Mc log((N 1)eλmax/M) < 1. This holds for sufficiently large M > 0, since − − lim Mc log((N 1)eλmax/M)= . M→+∞ − −∞ (Geometric process): Next, we consider a geometric process for T . Suppose that • ℓ Tℓ = Ypℓ is a sequence of independent random variables following geometric distributions: (5.2) P(Y = k) = (1 p )kp , k = 0, 1, 2, , pℓ − ℓ ℓ · · · with the parameters satisfying 1 pmin := inf pℓ > . ℓ≥0 2 We first begin with a technical lemma. Lemma 5.1. For positive integers A and B, one has B ∂ jℓ (1 xℓ) xℓ > 0, ∂xk − j ≥0, PB j ≤A ℓ=1 ! ℓ Xℓ=1 ℓ Y for 0 B B A−Pℓ=2 jℓ B (1 x )jℓ x = (1 x )jℓ x − ℓ ℓ − ℓ ℓ B ℓ=1 ! B j1=0 ℓ=1 ! jℓ≥0, PXℓ=1 jℓ≤A Y jℓ≥0, PXℓ=2 jℓ≤A X Y B j A−PB j +1 = (1 x ) ℓ x 1 (1 x ) ℓ=2 ℓ . − ℓ ℓ − − 1 PB ℓ=2 ! jℓ≥0, Xℓ=2 jℓ≤A Y Hence, we have B ∂ jℓ (1 xℓ) xℓ ∂xk − ℓ=1 ! jℓ≥0,XPℓ jℓ 0. − ℓ ℓ − ℓ − 1 PB ℓ=2 ! ℓ=2 jℓ≥0, Xℓ=2 jℓ≤A Y X 18 DONG, HA, JUNG, AND KIM Proposition 5.2. Suppose that (Ypℓ ) is a geometric process whose distributions are given by (5.2), and let (X,V ) be a solution process to (1.1). Then for any c> 0, we can choose M N such that ∈ ai(N−1)−1 P ω : Y (ω) M(n + c log i(N 1) ), for some i N p (n), pℓ ≥ ⌊ − ⌋ ∈ ≤ 2 ℓ=a(iX−1)(N−1) wherep (n) 0 as n . 2 → →∞ Proof. Note that ai(N−1)−1 P ω : Y (ω) M(n + c log i(N 1) ), for some i N pℓ ≥ ⌊ − ⌋ ∈ ℓ=a(iX−1)(N−1) ∞ ai(N−1)−1 P ω : Y (ω) M(n + c log i(N 1) ) . ≤ pℓ ≥ ⌊ − ⌋ Xi=1 ℓ=a(iX−1)(N−1) For simplicity, we set J(i, n) := a a ,H(i, n) := n + c log i(N 1) . i(N−1) − (i−1)(N−1) ⌊ − ⌋ Then, for i 1, ≥ ai(N−1)−1 P ω : Y (ω) ai(N−1)−1 P ω : Y (ω) MH(i, n) pℓ ≥ ℓ=a(iX−1)(N−1) MH(i,n)−1 1 r + J(i, n) 1 = pJ(i,n) − (1 p )r min J(i,n) − J(i, n) 1 − min p r=0 min X − DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 19 J(i, n)+ MH(i, n) 1 (1 p )MH(i,n) − − min ≤ J(i, n) 1 MH(i,n) − pmin J(i, n)+ MH(i, n) (1 p )MH(i,n) − min ≤ J(i, n) MH(i,n) pmin M(N 1)H(i, n) (1 p )MH(i,n) − − min ≤ (N 1)(i, n) MH(i,n) − pmin N− MH(i,n) N− 1 1 M MH(i,n) e N 1 1+ M M (1 p ) 1+ − − min , ≤ √2π M N 1 ! pMH(i,n) − min where we used J(i, n) (N 1)H(i, n) and the first inequality follows from the Taylor ≤ − approximation for (1 x)−J(i,n): − MH(i,n)−1 1 r + J(i, n) 1 − (1 p )r J(i,n) − r − min p r=0 min X (1 p )MH(i,n) dMH(i,n) 1 = − min (MH(i, n))! dxMH(i,n) (1 x)J(i,n) x=p∈(0,1−pmin) − J(i, n)+ MH(i, n) 1 (1 p )MH( i,n) − − min , ≤ MH(i, n) J(i,n)+MH(i,n) pmin and the last one is from the following inequality obtained by Stirling’s approximation: n+m+ 1 −n−m n+m+ 1 n + m (n + m)! e(n + m) 2 e e(n + m) 2 = = n+ 1 m+ 1 n+ 1 m+ 1 m n!m! ≤ √2πn 2 e−n √2πm 2 e−m 2πn 2 m 2 · e(n + m)n+m 1 1 e(n + m)n+m = + n m n m 2πn m rn m ≤ √2πn m m m n e m 1+ n n n = 1+ , n,m 1. √2π n m ≥ Now, we choose M N large enough so that ∈ N−1 N− 1 M N 1 1+ M M 1 p C(M) := 1+ − − min < 1 and Mc log C(M) < 1, M N 1 pmin − − which is possible since lim C(M)= 1−pmin < 1. Then, one obtains M→∞ pmin ∞ ai(N−1)−1 P ω : Y (ω) MH(i, n) pℓ ≥ Xi=1 ℓ=a(iX−1)(N−1) ∞ ∞ e e (C(M))MH(i,n) (C(M))M(n+c log i(N−1)−1) ≤ √2π ≤ √2π Xi=1 Xi=1 20 DONG, HA, JUNG, AND KIM ∞ e M(n+c log(N−1)−1) Mc log C(M) (C(M)) i := p2(n) 0, as n . ≤ √2π → →∞ Xi=1 5.2. The continuous CS model. In this subsection, we use our results for the discrete Cucker-Smale model to extend the previous result [15] for the continuous Cucker-Smale model. In the previous work, the random variables tℓ+1 tℓ are assumed to be i.i.d with a probability density function f with a compact support.− More precisely, the probability density function f is assumed to be supported on a finite interval [a, b] with 0 In system (1.1), we adopt the switching law σ : [0, ) Ω 1, ,NG and the sequence t t satisfy the following conditions: ∞ × → { · · · } { ℓ+1 − ℓ}l≥0 For each ℓ 0, t t is given by • ≥ ℓ+1 − ℓ t t = a + T , ℓ+1 − ℓ ℓ where a> 0 is a constant and Tℓ’s are nonnegative, independent random variables. For each ℓ 0 and ω Ω, σ (ω) is constant on the interval t [t (ω),t (ω)). • ≥ ∈ t ∈ ℓ ℓ+1 σ is a sequence of i.i.d. random variables such that for any ℓ 0, • { tℓ }ℓ≥0 ≥ P(σ = k)= p , for each k = 1, ,N , tℓ k · · · G where p , ,p > 0 are given positive constants satisfying p + + p = 1. 1 · · · NG 1 · · · NG For each k = 1, ,N , let = ( , ) be the k-th admissible digraph. Then, for each · · · G Gk V Ek t 0 and ω Ω, the time-dependent network topology (χσ ) = (χσt(ω)) is determined by ≥ ∈ ij ij 1 if (j, i) , χσt(ω) := σt(ω) ij 0 if (j, i) ∈E/ . ∈Eσt(ω) For technical reasons and without loss of generality, we assume that each admissible digraph k has a self-loop at each vertex, and we define the union graph of σ[t,ω] for s0 t = , has a spanning tree. Gk V Ek 1≤k[≤NG 1≤k[≤NG ( 2): For some M > 0, the random variables satisfy • B ai(N−1)−1 P ω : T (ω) M(n + c log i(N 1) ), for some i N p˜(n), ℓ ≥ ⌊ − ⌋ ∈ ≤ ℓ=a(iX−1)(N−1) DISCRETE CUCKER-SMALE MODEL WITH RANDOMLY SWITCHING TOPOLOGIES 21 wherep ˜(n) 0 as n . → →∞ Then, the similar analysis together with the proof in [15] used in previous sections yields the following improved result. Theorem 5.1. Suppose that parameters N, κ, choice probability pk’s and communication weight φ satisfy (a(N 1) + M)κ 1 − < 1, = (rε) as r , min log 1 φ(r) O →∞ 1−pk 1≤k≤NG where ε is a positive constant satisfying 1 (a(N 1) + M)κ 0 ε< − , ≤ N 1 − (N 1) min log 1 1−pk − − 1≤k≤NG and let (X,V ) be a solution process to (1.1). Then, one has the stochastic flocking: P ω : x∞ > 0 s.t sup (X(t,ω) x∞, and lim (V (t,ω)) = 0 = 1. ∃ 0≤t<∞ D ≤ t→∞ D Remark 5.1. In the previous work [15], we assumed that the sequence of dwelling times tℓ+1 tℓ are i.i.d with a probability density function f supported on a finite interval [{a, b] with− }0 6. Conclusion In this work, we provided a sufficient framework for the stochastic flocking for the discrete Cucker-Smale model with randomly switching topologies. Our sufficient framework involves conditions on the admissible network topologies and switching instants. First, we assume that the union of network topologies in an admissible set contains at least one spanning tree so that all the particles will be eventually connected in the space-time domain. Second, we assume that the dwelling times are short enough in a probabilistic sense so that all the particles communicate sufficiently often in a whole topological-temporal domain. Under these two assumptions, we showed that the discrete Cucker-Smale model exhibits a sto- chastic flocking for generic initial data asymptotically. For definiteness, we also show that the Poisson process and the geometric process for the dwelling time process both satisfy the second assumption in our framework. Although our current work corresponds to the dis- crete analog of the continuous one, we can improve the earlier result [15] for the continuous C-S model by removing the compact support assumption on the probability distribution function for switching time process. There are many other issues that we did not investigate in this work. For example, our analysis focuses only on global flocking. However, as noticed in literature, even for all-to-all couplings, the Cucker-Smale ensemble may not exhibit a global stochastic flocking depending on the decaying nature of the communication weight function, namely, formation of local flocking. 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Xue: Cucker-Smale flocking with randomly failed interactions. J. Franklin Inst. 352 (2015) 1099-1118. [42] J. Shen: Cucker-Smale flocking under hierarchical leadership. SIAM J. Appl. Math. 68 (2007) 694-719. [43] J. Toner and Y. Tu: Flocks, herds, and schools: A quantitative theory of flocking Phys. Rev. E 58 (1998) 4828-4858. [44] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen and O. Schochet: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (1995) 1226-1229. [45] T. Vicsek and A. Zefeiris: Collective motion. Phys. Rep. 517 (2012) 71-140. [46] C.W. Wu: Synchronization and convergence of linear dynamics in random directed networks. IEEE Trans. Automat. Control 51 (2006), 1207–1210. 24 DONG, HA, JUNG, AND KIM (Jiu-Gang Dong) Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China E-mail address: [email protected] (Seung-Yeal Ha) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of) E-mail address: [email protected] (Jinwook Jung) Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea (Republic of) E-mail address: [email protected] (Doheon Kim) School of Mathematics, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of) E-mail address: [email protected]