Analytic treatment of the network synchronization problem with time delays
aa r X i v : . [ n li n . C G ] N ov Analytic treatment of the network synchronization problem with time delays
Shahar Hod
The Ruppin Academic Center, Emeq Hefer 40250, IsraelandThe Hadassah Institute, Jerusalem 91010, Israel (Dated: October 27, 2018)Motivated by novel results in the theory of network synchronization, we analyze the effectsof nonzero time delays in stochastic synchronization problems with linear couplings in an arbitrarynetwork. We determine analytically the fundamental limit of synchronization efficiency in a noisyenvironment with uniform time delays. We show that the optimal efficiency of the network isachieved for λτ = π / √ π +4 ≈ . λ is the coupling strength (relaxation coefficient) and τ is the characteristic time delay in the communication between pairs of nodes. Our analysis revealsthe underlying mechanism responsible for the trade-off phenomena observed in recent numericalsimulations of network synchronization problems. Synchronization processes in populations of locally in-teracting elements are in the focus of intense researchin physical, biological, chemical, technological and so-cial systems [1]. Of particular interest are situations inwhich members (usually referred to as ‘agents’ or ‘nodes’in a network) try to coordinate their state in a decen-tralized manner [1, 2]. In many real-life situations themotivation for such coordination is to improve the globalperformance of the network [2]. There has been a flurryof research focusing on the efficiency and optimizationof synchronization problems in various complex networktopologies (see [1–13] and references therein).Stochastic synchronization problems in real biological,social, and computing networks are usually characterizedby finite time delays in the communication between pairsof nodes. Recently, Hunt et al. [2] have studied theimpact of such time delays on synchronizability and onthe breakdown of synchronization in dynamical network-connected systems. They considered a stochastic modelin which each node in a network adjusts its state to matchthat of its neighbors, but with a uniform time lag in react-ing to the neighborly feedback. Hunt et al. have revealedthat there are trade-offs in the synchronization problem:when there are large lag times in communication betweennodes, reduced local coordination effort may actually im-prove the global coordination of the network [2].It is worth emphasizing that the remarkable findingof [2], that there are possible scenarios for trade-offsbetween large time lags and the coupling strength, isbased on numerical simulations of the stochastic evolu-tion equations which govern the dynamics of the network[see Eq. (6) below]. The main goal of the present Letteris to provide an analytical treatment for the network syn-chronization problem. In particular, we shall determineanalytically the fundamental limit of synchronization ef-ficiency in a noisy environment with uniform time delays.We shall first describe the synchronization model stud-ied in Ref. [2]. Consider a stochastic model where N agents in a network locally adjust their state in an at-tempt to match that of their neighbors. Such coordina- tion may improve the global performance of the network[1, 2, 5, 6, 8]. As in many real-life situations, the com-munication between pairs of nodes is not instantaneous.Rather, it is characterized by some finite time lag [2].The dynamics of the system is governed by the coupledstochastic equations of motion with linear local relax-ation and a uniform time delay, ∂h i ( t ) ∂t = − N X j =1 C ij [ h i ( t − τ ) − h j ( t − τ )] + η i ( t ) , (1)where h i ( t ) is the generalized local state variable on node i , C ij = C ji ≥ i and j , and τ is the charac-teristic time delay between two connected nodes. Here η i ( t ) is a delta-correlated noise with zero mean and vari-ance h η i ( t ) η j ( t ′ ) i = 2 Dδ ij δ ( t − t ′ ), where D is the noiseintensity [2].Stochastic synchronization problems are characterizedby competition between a relaxation mechanism and arandom noise. The physically interesting observable insuch systems is the width of the synchronization land-scape. This is given by [2–4, 8] h w ( t ) i ≡ D N N X i =1 [ h i ( t ) − ¯ h ( t )] E , (2)where ¯ h ( t ) = 1 /N P Ni =1 h i ( t ) is the global average of thelocal state variables and h···i denotes an ensemble averageover the noise. A network is considered synchronizableif its late-time asymptotic behavior is characterized by a finite width [that is, if h w ( ∞ ) i < ∞ ]. The smaller thewidth, the better the synchronization [2].The coupled equations of motion (1) can be rewrittenas [2] ∂h i ( t ) ∂t = − N X j =1 Γ ij h j ( t − τ ) + η i ( t ) , (3)where Γ ij = δ ij P l C il − C ij is the symmetric networkLaplacian. Further, by diagonalizing the network Lapla-cian, one can decompose the problem into N independent modes ∂ ˜ h k ( t ) ∂t = − λ k ˜ h k ( t − τ ) + ˜ η k ( t ) , (4)where { λ k } ( k = 0 , , , ..., N −
1) are the eigenvalues ofthe network Laplacian and h ˜ η k ( t )˜ η l ( t ′ ) i = 2 Dδ kl δ ( t − t ′ ).For a connected (single-component) network, the Lapla-cian has a single zero mode (indexed by k = 0) with λ = 0, while λ k > k ≥ h w ( t ) i = 1 N N − X k =1 h ˜ h k ( t ) i . (5)Note that the eigenmodes of the system are governedby a stochastic equation of motion [Eq. (4)] of identical form for all k ≥
1. We shall therefore omit the index k for brevity, and study the stochastic differential equation ∂ ˜ h ( t ) ∂t = − λ ˜ h ( t − τ ) + ˜ η ( t ) (6)with h η ( t ) η ( t ′ ) i = 2 Dδ ( t − t ′ ).Using a Laplace transformation with initial conditions˜ h ( t ≤
0) = 0, one finds [2]˜ h ( t ) = Z t dt ′ ˜ η ( t ′ ) X α e s α ( t − t ′ ) τ s α , (7)where { s α } ( α = 1 , , ... ) are the solutions of the charac-teristic equation s + λe − τs = 0 (8)in the complex plane. The characteristic equation (8)has an infinite number of complex solutions for τ > ℜ ( s α ) < α provided λτ < π/ h ˜ h ( t ) i = X α X β − Dτ [1 − e ( z α + z β ) t/τ ](1 + z α )(1 + z β )( z α + z β ) (9)for the noise-averaged fluctuations, where z ≡ τ s . In-spection of Eq. (9) reveals that the condition for h ˜ h ( ∞ ) i to remain finite is ℜ ( z α ) < α . As discussedabove, this requires λτ < π/ ℜ ( z α ) < α ] one finds h ˜ h ( ∞ ) i = X α X β − Dτ (1 + z α )(1 + z β )( z α + z β ) (10)for the steady-state ( t → ∞ ) behavior. Writing the characteristic equation (8) in the form z + λτ e − z = 0 , (11)one realizes that z α = z α ( λτ ) [2]. Thus, one immediatelydeduces from Eq. (10) the scaling form h ˜ h ( ∞ ) i = Dτ × f (Λ) , (12)where Λ ≡ λτ . The scaling function f (Λ) was con-structed numerically in [2]. In particular, the numericalstudy of f (Λ) in [2] yielded the remarkable finding that f (Λ) is a non -monotonic function; it exhibits a singleminimum, at approximately Λ ∗ ≈ .
73 with f (Λ ∗ ) ≈ . analytical treat-ment for the problem of network synchronization in anoisy environment with time delays. To that end, weshall first analyze the asymptotic behavior of h ˜ h ( ∞ ) i near the two boundaries of the synchronizable regime:Λ → → π/
2. As we shall show below, in theselimits the sum in (10) is dominated by solutions of thecharacteristic equation (11) with ℜ ( z ) → → f (Λ) has to scale as f (Λ → ≃
1Λ + O (1) (13)in order to reproduce the exact limiting case of zero delay, h ˜ h ( ∞ ) i ≃ D/λ [2].In the Λ → π/ z ± = ± i π − ± i + π π ) ∆ + O (∆ ) (14)to the characteristic equation (11), where ∆ ≡ π/ − Λ ≪
1. Note that z + + z − = − π π ) ∆ → → z + + z − is responsible for the divergent behavior of f (Λ → π/ f (Λ) in the Λ → π (∆ →
0) limit: f (Λ → π ≃ π ∆ + O (1) . (16)The simplest analytic function which satisfies bothasymptotic behaviors (13) and (16) is f (Λ) = 1Λ + 4 π ( π − Λ) + c , (17)where c is a constant. Note that this function has a singleminimum at Λ ∗ = π / √ π + 2) . (18)We note that the numerically computed value Λ ∗ ≈ . ∼
1% difference) to the analyt-ical expression (18).In order to fix the value of the constant c in (17),one may calculate the sub-leading (constant) term in Eq.(13). In the Λ → z = − Λ − Λ + O (Λ ) (19)to the characteristic equation (11). Inspection of the de-nominator of Eq. (10) reveals that the small value of z is responsible for the divergent behavior of f (Λ → f (Λ) in the Λ → f (Λ → ≃
1Λ + 1 . (20)Equating Eqs. (17) and (20) for Λ →
0, one finds c =1 − /π , which implies f (Λ) = 1Λ + 4 π ( π − Λ) + 1 − π (21)for the scaling function in (12).Substituting Λ ∗ from (18) into (21), one obtains theminimal value f min = f (Λ ∗ ) = 1 + 2 π − + 8 π − / . (22)Again, we note that the numerically computed value f min ≈ . ∼
1% difference) tothe analytical expression (22).In figure 1 we depict the scaling function f (Λ) = h ˜ h ( ∞ ) i /Dτ as given by Eq. (21). This figure shouldbe compared with the numerical results presented in Fig.2 of [2]. We find an almost perfect agreement betweenthe analytical function (21) and the numerical results ofRef. [2].From Eqs. (18) and (22) one learns that for a sin-gle stochastic variable governed by Eq. (6) with anonzero delay, there is an optimal value of the relax-ation coefficient λ ∗ = π / / √ π + 2) τ , at which pointthe steady-state fluctuations attain their minimum value h ˜ h ( ∞ ) i min = Dτ (1 + 2 π − + 8 π − / ) [17], see also [2].Returning to the context of network synchronization,one can calculate from Eqs. (5), (12) and (21) the steady-state width of the network-coupled system: h w ( ∞ ) i = DτN N − X k =1 h λ k τ + 4 π ( π − λ k τ ) +1 − π i . (23)Thus, for large N the fundamental limit of synchroniza-tion efficiency is given by [see Eq. (22)]: h w ( ∞ ) i min = Dτ (1 + 2 π − + 8 π − / ) . (24)This is the minimum attainable width of the synchro-nization landscape in a noisy environment with uniformtime delays. λτ S ca li ng f un c ti on f( λ τ ) FIG. 1: The scaling function f (Λ) ≡ h ˜ h ( ∞ ) i /Dτ = Λ − + π ( π − Λ) − + 1 − π in the synchronizable regime 0 < Λ <π/
2. Compare this figure with the numerically constructedfunction presented in Fig. 2 of [2].
So far we have studied the characteristics of the syn-chronization network in the steady state ( t → ∞ ) regime.Another interesting characteristic of the synchronizationproblem is the relaxation time of the network, the timeit takes for the system to relax to its finite steady-statewidth (in the synchronizable regime, 0 < λτ < π/ λτ → π/ ℜ ( α ) < smallest absolute value of thereal part.] Inspection of Eq. (9) reveals that the charac-teristic relaxation time, T ( λ, τ ) [18], is given by T ≡ τ {|ℜ ( z α ) |} . (25)Taking cognizance of Eq. (14), one finds T ∆ = τ π ) π ∆ , (26)for the diverging relaxation time of the coupled networkin the vicinity of the phase transition (the ∆ → { z + , z − } from(14) into (9), one obtains the late-time behavior of thenetwork near the phase transition: h ˜ h ( t ) i ≃ h ˜ h ( ∞ ) i − Dτπ ∆ e − t/T ∆ n π ) ] × h [( π −
1] sin( πt/τ ) + π cos( πt/τ ) io . (27)We thus find that the approach of the network to asteady-state behavior is characterized by damped tempo-ral oscillations of period 2 τ and a characteristic lifetime T ∆ . It is worth noting that these characteristic oscilla-tions are clearly visible in the numerical results of Hunt etal. [2] (see Fig. 1 of [2]. Observe, in particular, the tem-poral oscillations in the plots for λτ = 1 . λτ = 1 . λτ = π/ analytically the fundamental limit of synchro-nization efficiency (the minimum attainable value of thewidth of the synchronization landscape): h w ( ∞ ) i min = Dτ (1 + 2 π − + 8 π − / ), where τ is the characteristic timedelay in the communication between pairs of nodes. Wehave shown that the optimal efficiency of the network isachieved for λτ = π / √ π +4 , where λ is the relaxation coef-ficient (coupling strength). These analytical results arein perfect agreement with the recent numerical results ofRef. [2]. Further, we have analyzed the relaxation time ofthe network and showed that it diverges in the thresholdlimit λτ → π/ τ and the coupling strength λ ) observed in recentnumerical simulations [2] of stochastic synchronizationproblems with time delays. ACKNOWLEDGMENTS
This research is supported by the Meltzer ScienceFoundation. I thank Oded Hod, Yael Oren and ArbelM. Ongo for helpful discussions. [1] A. Arenas et al. , Phys. Rep. , 93 (2008). [2] D. Hunt, G. Korniss and B. K. Szymanski, Phys. Rev.Lett. , 068701 (2010).[3] G. Korniss, Phys. Rev. E , 051121 (2007).[4] G. Korniss et al. , Science , 677 (2003).[5] M. Barahona and L. M. Pecora, Phys. Rev. Lett. ,054101 (2002).[6] T. Nishikawa et al., Phys. Rev. Lett. , 014101 (2003).[7] F. M. Atay, Phys. Rev. Lett. , 094101 (2003).[8] C. E. La Rocca, L. A. Braunstein, and P. A. Macri, Phys.Rev. E , 026111 (2009).[9] C. Zhou, A. E. Motter, and J. Kurths, Phys. Rev. Lett. , 034101 (2006).[10] T. Nishikawa and A. E. Motter, Phys. Rev. E , 065106(R) (2006).[11] R. Olfati-Saber and R. M. Murray, IEEE Trans. Autom.Control 49, 1520 (2004).[12] S. Hod and E. Nakar, Phys. Rev. Lett. , 238702 (2002).[13] S. Hod, Phys. Rev. Lett. , 128701 (2003).[14] R. Frisch and H. Holme, Econometrica , 225 (1935)[15] N. D. Hayes, J. Lond. Math. Soc. s1-25 , 226 (1950).[16] Synchronizability of the entire network requires a finitesteady-state width, h w ( ∞ ) i = N P N − k =1 h ˜ h k ( ∞ ) i < ∞ .Thus, the synchronizability condition is given by λ k τ <π/ all k ≥ h ˜ h ( ∞ ) i = D/λ ; i.e., there the steady-state fluctuationis a monotonically decreasing function of the relaxationcoefficient λ [2].[18] It is clear from Eqs. (9) and (11) that T ( λ, τ ) = τ × g ( λτ ),where g is a scaling function.[19] For λτ ≈ π/
2, the characteristic oscillations are best seenin the interval τ < ∼ t < ∼ T ∆ . From Fig. 1 of [2] one mayconfirm that the period of the damped oscillations is in-deed 2 ττ