Synchronization Transitions in Complex Networks Y. Moreno 1,2,3 1 Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza, Zaragoza 50018, Spain 2 Department of Theoretical Physics University of Zaragoza, Zaragoza 50009, Spain. 3 ISI Foundation, Turin, Italy Thematic School on Complex Networks Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 1 / 40
Outline A. Arenas, A. Diaz-Guilera,J.Kurths, Y. Moreno, and C. Zhou, Synchronization in Complex Networks, Physics Reports, 469, 93-153 (2008). 1 Kuramoto model 2 Explosive Synchronization: Set up and Results 3 Master Stability Approach Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 2 / 40
Synchronization One of the most paradigmatic example of collective behavior Emergence of coherent rythms in the dynamics of coupled agents We will model these coupled agents as phase-oscillators... But first lets discuss a very useful approach. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 3 / 40
The Kuramoto model We have a collection of N agents, being the dynamics of each of them described by a phase θ i [0, 2π): θ i = ω i + K N sin(θ j θ i ), (1) N j=1 Synchronization is measured via the Kuramoto order parameter: r = 1 N N e iθ j (2) j r 0 dynamical incoherence r 1 full synchronization. There exist a phase transition when increasing λ = K/N. At some critical coupling λ c = 2/[πg(0)] the incoherent solution becomes unstable and r starts to increase with λ. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 4 / 40
The Synchronization transition Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 5 / 40
The Kuramoto model in Complex Networks We have a network of N agents, coupled as dictated by the adjacency matrix A. The Kuramoto dynamics of each agent is: N θ i = ω i + λ A ij sin(θ j θ i ), (3) j=1 Synchronization can be measured via the Kuramoto order parameter: r = 1 N N e iθ j, (4) and by the degree of synchronization between pairs of connected nodes: D ij = lim A 1 τ+t ij T e i[θ i (t) θ j (t)] dt. (5) T D ij 0 local incoherence τ j D ij = 1 local synchronization. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 6 / 40
Synchronization transitions in Networks Two remarkable examples of network architecture: Scale-free networks: P(k) k γ Erdös-Rényi graphs: P(k) e k k k /k! Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 7 / 40
Synchronization transitions in Networks Synchronization diagram for SF Networks: evolution of the Kuramoto order parameter as a function of the coupling r(λ). The onset of synchronization occurs at a non-zero value of λ, i.e., λ c > 0 even when N 1.0 0.8 R 0.6 0.4 0.2 N=10 3 N=5x10 3 N=10 4 N=2x10 4 N=3x10 4 N=4x10 4 N=5x10 4 0.0 0.0 0.2 0.4 0.6 0.8 1.0! Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 8 / 40
r Synchronization transitions in Networks Finite Size Scaling Analysis The idea is that: if λ < λ c (subcritical regime), then r falls of as N 1/2. if λ > λ c (supercritical regime), then r const, though with O(N 1/2 ) fluctuations. at λ = λ c r falls as a power law. Our best estimate gives λ c = 0.05 ± 0.01 Besides, when both N, t we get r (λ λ c ) β, with β = 1 2. 10 0 10!1 10!2 10!3!=0.02!=0.04!=0.06!=0.08 10!4 10 3 10 4 10 5 N Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 9 / 40
Synchronization transitions in Networks Stability of the Fully Synchronized state τ : Average time it takes for a node to be again in the synchronized state after being perturbed. 10 1 <!> 10 0 N=10 3, k max =91 τ k ν 10!1 10 1 10 2 k The more connected a node is, the more stable it is. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 10 / 40
Synchronization transitions in Networks Stability of the Fully Synchronized state As we are perturbing a single node and this perturbation, ξ i, is small, we can consider that it only affects the first neighbors of the perturbed node. The stability analysis can be locally reduced to the problem of how such a perturbation relaxes in a star-like topology (perturbed node attached to k >> 1 nodes). Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 11 / 40
Synchronization transitions in Networks Stability of the Fully Synchronized state For a start graph we have: η i = 1 for i = 1,..., N 2 η N 1 = η hub = N = k hub 1 The fastest relaxation rate corresponds to the hub and goes like: 1 k hub for k hub 1 Finally, the superposition of many perturbations, each one corresponding to a k hub + 1 star, leads to different contributions of 1 k hub with k hub = k min,..., k max. Thus, τ k 1. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 12 / 40
Synchronization transitions in Networks Synchronization diagrams in SF and ER networks: evolution of the Kuramoto order parameter as a function of the coupling r(λ) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 13 / 40
Synchronization transitions in Networks Looking at the microscopic patterns with D ij : How nodes and links are incorporated into the Giant Synchronized Component? Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 14 / 40
Synchronization transitions in Networks Different paths towards synchronization: The importance of hubs in the entrainment of oscillators Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 15 / 40
Explosive Percolation In percolation processes the addition of a new rule for link creation turns the percolation transition explosive: D. Achlioptas, R.M. D Souza, and J. Spencer, Science 323, 1453 (2009). F. Radicchi, and S. Fortunato, Phys. Rev. Lett. 103, 168701 (2009). Y.S. Cho et al., Phys. Rev. Lett. 103, 135702 (2009). J. Nagler, A. Levina, and M. Timme, to appear in Nature Phyics (2011). Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 16 / 40
Explosive Synchronization Can we change the order of the synchronization transition from microscopic dynamics? Yes! Hubs promote synchronization: by integrating a mean-field dynamics they are able to collect many nodes in the synchronized component. We will destroy this topological ability dynamically: by assigning hubs a natural frequency far from the average one Ω. To this aim we propose this setting: This condition automatically implies: ω i = k i with i = 1,..., N (6) Ω = ω i = k i (7) g(ω) = P(k) (8) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 17 / 40
Explosive Synchronization Going smoothly from ER graphs to SF networks: We confirm that for SF networks (i.e. only when hubs are present) the synchronization transition is explosive displaying a strong histeresis effect. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 18 / 40
Explosive Synchronization We compute for each value of λ the effective frequency of each node: ω eff i = 1 T t+t t θ i (τ) dτ, with T 1. (9) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 19 / 40
Explosive Synchronization What happens with a general model of SF networks (configurational ensembles with P(k) k γ ): The rule ω i = k i changes the order of the transition in SF networks Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 20 / 40
Explosive Synchronization In SF networks P(k) k γ so (when ω i = k i and thus P(k) = g(ω)) we have g(ω) ω γ. Is this broad frequency distribution behind the first-order transition? No. We do need the correlation structure-dynamics to obtain the explosive synchronization! Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 21 / 40
The star graph The star graph is a first proxy of a SF network capturing the neiborhood of a hub: One central hub connecting with K leaves. We set the natural frequency of the leaves identical with value ω The hub naturally pace at frequency ω h. The average frequency is Ω = (Kω + ω h )/(K + 1) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 22 / 40
The star graph The equations of motion for the hub and the leaves read: θ h = ω h + λ K sin(θ j θ h ), (10) j=1 θ j = ω + λ sin(θ h θ j ), with j = 1,..., K. (11) Hub motion: We settle in a rotating frame with frequency Ω so that the hub motion is described as: φ h = (ω h Ω) + λ(k + 1)r sin(φ h ), (12) Imposing that the phase of the hub is locked, φ h = 0, we obtain: sin φ h = (ω h Ω) λ(k + 1)r. (13) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 23 / 40
The star graph Leaves motion: Again, in a rotating frame with frequency Ω so that the motion of leaf is described as: φ j = (ω Ω) + λ sin(φ h φ j ), with j = 1,..., K. (14) Imposing that the phase of the hub is locked, φ h = 0, we obtain: [1 cos φ j = (Ω ω) sin φ h ± sin 2 ] φ h [λ 2 (Ω ω) 2 ]. (15) λ This equation implies that locking is lost at λ c = (Ω ω). The order parameter is r = cosφ. Thus, at λ c, r takes the value: r c = 1 K (Ω ω) (ω h Ω) λ c(k+1)r c + 1 (ω h Ω) 2 K + 1 λ c λ 2 c(k + 1) 2 rc 2. (16) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 24 / 40
The star graph In our case (ω i = k i ): ω = 1, ω h = K, Ω = 2K/(K + 1) thus: λ c = K 1 K + 1 and r c = K K + 1. (17) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 25 / 40
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Master Stability Function formalism Derivation follows the paper Dynamical Systems on Networks: a tutorial a highly recommendable review by MAP & JPG, arxiv preprint 1403.7663 Let s suppose that each node i is associated with a single variable x i. We use x to denote the vector of variables. Consider the continuous dynamical system ẋ i = f i (x i ) + N A ij g ij (x i, x j ), i {1,..., N}, (18) j=1 A ij = adjacency matrix g ij (x i, x j ) effects coupling to neighbors. the equilibrium points Eq. (18) satisfy x i = 0 for all nodes i. local stability of these points via linear stability analysis: let x i = xi (where ɛ i 1) and take a Taylor expansion. + ɛ i Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 30 / 40
For each i, we obtain Master Stability Function formalism N ẋ i = ɛ i = f i (x i + ɛ i ) + A ij g ij (x i + ɛ i, x j + ɛ j ), j=1 N = f i (x i ) + A ij g ij (x i, x j ) + ɛ i f N i xi =x + ɛ i A ij j=1 i j=1 g ij x i xi =x i,x j =x j N + ɛ j A ij j=1 g ij x j xi =x i,x j =x j +... = α 1 + α 2 + α 3 + α 4 + α 5 +..., (19) where f i := df i dx i and the α l terms are defined in order. Because x is an equilibrium, it follows that α 1 + α 2 = 0. The terms α 3 and α 4 are linear in ɛ i, and α 5 is linear in ɛ j. We now neglect all higher-order terms. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 31 / 40
Master Stability Function formalism To simplify notation, we define a i := ɛ i f i b ij := g ij x i c ij := g ij x j xi =x i, xi =x i,x j =x j xi =x i,x j =x j,. (20) That is, ɛ = Mɛ +..., (21) where M = [M ij ] and [ M ij = δ ij a i + ] b ik A ik + c ij A ij. (22) k Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 32 / 40
Master Stability Function formalism Assume that M has N unique eigenvectors (not always the case, although it will usually be if away from a bifurcation point) and is thus diagonalizable, we obtain ɛ = N α r (t)v r, (23) r=1 where v r (with corresponding eigenvalue µ r ) is the rth (right) eigenvector of the matrix M. It follows that ɛ = N r=1 N α r v r = M α r (t)v r = r=1 r=1 N µ r α r (t)v r. (24) Separately equating the linearly independent terms in equation (24) then yields α r = µ r α r, which in turn implies that α r (t) = α r (0) exp(µ r t). Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 33 / 40
Master Stability Function formalism As usual for dynamical systems, we obtain local asymptotic stability if Re(µ r ) < 0 for all r, instability if any Re(µ r ) > 0, and a marginal stability (for which one needs to examine nonlinear terms) if any Re(µ r ) = 0 for some r and none of the eigenvalues have a positive real part. Example, let s consider a (significantly) simplified situation in which every node has the same equilibrium location: i.e., xi = x for all nodes i. (This arises, for example, in the SI model of a biological contagion.) We will also assume that f i f for all nodes and g ij g for all node pairs. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 34 / 40
Master Stability Function formalism With these simplifications, it follows that f (x ) + N A ij g(x, x ) = f (x ) + k i g(x, x ) = 0, (25) j=1 where we recall that k i is the degree of node i. Equation (25) implies that either all nodes have the same degree (i.e., that our graph z-regular ) or that g(x, x ) = 0. We do not wish to restrict the network structure severely, so we will suppose that the latter condition holds. It follows that f (x ) = 0, so the equilibria of the coupled equation (18) in this simplified situation are necessarily the same as the equilibria of the intrinsic dynamics that are satisfied by individual (uncoupled) nodes. This yields a simplified version of the notation from equation (20): a i a := f x=x, b ij b := g, x i xi =x j =x c ij c := g. x j xi =x j =x (26) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 35 / 40
Master Stability Function formalism We thus obtain ɛ i = (a + bk i )ɛ i + c N A ij ɛ j, i {1,..., N}. (27) j=1 If we assume that g(x i, x j ) = g(x j ), which is yet another major simplifying assumption, we obtain Consequently, b = 0 and ẋ i = f (x i ) + where I is the N N identity matrix. N A ij g(x j ). (28) j=1 ɛ = (ai + ca)ɛ, (29) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 36 / 40
Master Stability Function formalism An equilibrium of (29) is (locally) asymptotically stable if and only if all of the eigenvalues of P := ai + ca = P T are negative. (The matrix P is symmetric, so all of its eigenvalues are guaranteed to be real.) Letting w r denote an eigenvector of A with corresponding eigenvalue λ r. It follows that (ai + ca)w r = (a + cλ r )w r (30) for all r (where there are at most N eigenvectors and there will be exactly N of them if we are able to diagonalize A), so w r is also an eigenvector of the matrix P, with corresponding eigenvalue (a + cλ r ). For (local) asymptotic stability, we thus need a + cλ r < 0 to hold for all λ r. This, in turn, implies that we need a < 0 because the adjacency matrix A is guaranteed to have both positive and negative eigenvalues. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 37 / 40
Master Stability Function formalism We thus need: (i) λ r < a/c for c > 0 and (ii) λ r > a/c for c < 0. If (i) is satisfied for the most positive eigenvalue λ 1 of A, then it (obviously) must be satisfied for all eigenvalues of A. If (ii) is satisfied for the most negative eigenvalue λ N of A, then it (obviously) must be satisfied for all eigenvalues of A. It follows that 1 λ N < c a < 1 λ 1, (31) which is more insightful when we insert the definitions of a and c. This yields g xi 1 x j =x < j =x λ N f x=x < 1 λ 1. (32) Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 38 / 40
Master Stability Function formalism The left and right terms in equation (32), which is called a master stability condition, depend only on the structure of the network, and the central term depends only on the nature (i.e., functional forms of the individual dynamics and of the coupling terms) of the dynamics. It also illustrates that the eigenvalues of adjacency matrices have important ramifications for dynamical behavior when studying dynamical systems on networks. Indeed, investigations of the spectra (i.e., set of eigenvales) of adjacency matrices can yield crucial insights about dynamical systems on networks. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 39 / 40
Master Stability Function formalism N ẋ = F (x i ) + σ L ij H(x j ), j=1 i = 1,..., N where L ij is the Laplacian matrix: L ij = 1 if i and j are connected, L ii = k i, and 0 otherwise. It can be shown that the linear stability of the synchronized state is determined by the eigenvalues of L, effectively decoupling the dynamics and the topology. In particular, if λ 1 = 0 λ 2... λ N, are the eigenvalues of L, then it follows that: The larger the ratio λ N /λ 2 is, the more stable the synchronized state is and vice versa. Y. Moreno (BIFI) Synchronization in Complex Networks Les Houches, 7-18/04/14 40 / 40