The Impact of Time Delays in Network Synchronization in a Noisy - PowerPoint PPT Presentation

MAPCON 2012 The Impact of Time Delays in Network Synchronization in a Noisy Environment G. Korniss David Hunt B.K. Szymanski Phys. Rev. Lett. 105, 068701 (2010) http://arxiv.org/abs/1209.4240 (2012) Supported by ARL NS ‐ CTA, DTRA, ONR

Hutchinson model (logistic growth with delay in population dynamics)        N N ( ( t t ) )        0 N N ( ( t t ) ) rN rN ( ( t t ) ) 1 1     t t     K K population size     intrinsic growth rate carrying capacity ( N 0 ), N K N ( t ) S. Ruan, in NATO Sci. Ser. II Math. Phys. Chem. (Springer, 2006) p. 477   N ( t ) K x ( t )      x ( t ) rx ( t ) t ∗ Hutchinson (1948); Maynard Smith (1971); R.M. May (1973 ) 2

Synchronization/Coordination in Coupled Systems • individual units or agents (represented by static or mobile nodes) attempt to adjust their local state variables (e.g., pace, load, alignment, coordination) in a decentralized fashion. Craig Reynolds (1987); Vicsek et al . (1995); Cavagna et al . (2010). • nodes interact or communicate only with their local neighbors in the network, possibly to improve global performance or coordination. • nodes react (perform corrective actions) to the information or signal received from their neighbors possibly with some time lag (as result of finite transmission, queuing, processing, or execution delays) Applications: autonomous coordination, unmanned aerial vehicles, • microsatellite clusters, sensor and communication networks, load balancing, flocking, distributed decision making in social networks http://www.youtube.com/watch?v=6AmSpHxnKm8 http://www.youtube.com/watch?v=VaQ66lDZ ‐ 08 spontaneous brain activity (fMRI) IP activity 3 flocking birds (Justin Vincent; http://martinos.org/~vincent/ ) (Zeus load balancer)

Synchronization/Coordination in a Noisy Environment with Time Delays                          h ( ( t ) ) C [ [ h ( ( t ) ) h ( ( t )] )] ( t ) h h ( t t ) C C [ h h ( t t ) h ( t )] h t t t t i i i ij ij ij i i i j j j i j j j network/coupling strength noise delay N 1    2 2 spread or width : w ( t ) [ h ( t ) h ( t )] i N (measure of de ‐ coordination):  1 i network Laplacian:       h Γ h η    C  ( t ) ( t ) ( t ) C t ij ij i ij ~ ~ ~          h ( t ) h ( t ) ( t ) N 1 1 ~   2 2 t k k k k w ( t ) h k t ( ) N  k 1             0  4 0 1 2 N 1 max

Coordination, Noise, Time Delay                  ( t ) ( t ) 2 D ( t t ) h ( t ) h ( t ) ( t ) t  st characteristic equation: ( h ( t ) ce )       s      s s ( , ),  g ( s ) s e 0 1 , 2 ,   infinitely many relaxation “rates”, { s  }, for  > 0 synchronizability condition:    ( s s ) t   2 D [ 1 e ]   0      2 Re( s ) h ( t )     g ( s ) g ( s )( s s )   ,          2 h ( ) synchronizability:    / 2 5 Frisch & Holme (1935); Hayes (1950); Hutchinson (1948); Maynard Smith (1971); R.M. May (1973)

   ( ) / 2 Coordination, Noise, Time Delay c       1 , D 1 , dt 0 . 01 0 . 10 4 h ( t ) (a)  c    2 / 2 1 . 57 ~ (t) 0 h −2 1   −4 0 50 100 150 200 e   12 1 . 50 (b) 6  1    ~ (t) 0 h 2 e −6 −12 0 50 100 150 200   120 1 . 60 (c) 60    ~ (t) 0 h 2 −60 −120 0 50 100 150 200 t 6

   ( ) / 2 Coordination, Noise, Time Delay c     1 , D 1 , dt 0 . 01 2 t  c   4   10 / 2 h ( ) τ =2.00  τ =1.60 1 . 57 3 10 τ =1.50 τ =1.00 2 10 τ =0.30 τ =0.10 ~ 2 ( t) > 1 10 < h 0 10 −1 10 −2 10 −2 −1 0 1 2 3 10 10 10 10 10 10 t 7

Coordination, Noise, Time Delay                  ( t ) ( t ) 2 D ( t t ) h ( t ) h ( t ) ( t ) t ( ) 50 50 � � �∞� � � � � �∞� � � 1 � sin ���� λ λcos ���� 40 40 � ������� (Ornstein–Uhlenbeck) 30 30 Küchler and Mensch, SSR 40 , 23 (1992). ~2 ( ∞ )> ~2 ( ∞ )> < h < h 20 20 10 10    ( ) / 2 c 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 λ λ monotonically decreasing non ‐ monotonic function of function of the coupling  the coupling  8

Implications for Networks:    N 1 N 1 1 ~  D  Hunt et al ., PRL (2010)     2 2 w ( t ) h ( t ) f ( ) k k N N   k 1 k 1 ~ 2 Synchronizability       h k ( ) k and Coordination:      / 2 k k     / 2 max Olfati ‐ Saber and Murray (2004) (deterministic consensus problems) 9

Limitations of Network Synchronization Simple example: unweighted graphs largest degree largest eigenvalue of the network Laplacian N      k 2 k , O ( k )  max max max max max N 1 Fiedler (1973); Anderson and Morley (1985); Mohar (1991)    k / 4 : sufficient for synchronizability/stability max    k / 2 : synchronization/stability breaks down max • networks with potentially large degrees can be extremely vulnerable to intrinsic network delays while attempting to synchronize, coordinate, or balance their tasks, load, etc. 10

Limitations of Network Synchronization heterogeneous vs. homogeneous unweighted random graphs Fraction of Synchronizable Networks     p s (  , N ) fraction of synchronizable networks ( / 2 ) Comparison of ER and BA Networks max 1      0.8 ER : ~ k ~ ln( N ) max max  k   0.6 6 p s ER τ = 0.08 ER τ = 0.09 ER τ = 0.10 0.4 ER τ = 0.11 BA τ = 0.08 BA τ = 0.09 BA τ = 0.10      BA τ = 0.11 1 / 2 0.2 BA : ~ k ~ N max max 0 10 100 1000 11 Hunt et al ., PRL (2010) N

Scaling in the Synchronizable Regime       C A ' ij ij k k    2 w ( ) N 1 D          2     w ( ) f ( ) g ( ) , g ( ) k    , N  k 1 BA network, N =1000,  k  6 12

Trade ‐ Offs            h ( t ) p ( t ) C [ h ( t ) h ( t )] ( t ) t i ij i j i j     BA network, N =200,  k  6 1 . 2 / 2 Synchronization rate: p max 3 10 synchronize frantically (at rate 1) p=1.0 2 10 1 10 2 ( t )> < w 0 10 −1 10 −2 10 −2 −1 0 1 2 3 10 10 10 10 10 10 t Hunt et al ., PRL (2010) 13

Trade ‐ Offs            h ( t ) p ( t ) C [ h ( t ) h ( t )] ( t ) t i ij i j i j     BA network, N =200,  k  6 1 . 2 / 2 Synchronization rate: p max 3 10 p=0.8 2 10 1 10 2 ( t )> < w 0 10 reduce local synch. rate to 0.80 −1 10 −2 10 −2 −1 0 1 2 3 10 10 10 10 10 10 t Hunt et al ., PRL (2010) 14

Trade ‐ Offs            h ( t ) p ( t ) C [ h ( t ) h ( t )] ( t ) t i ij i j i j     BA network, N =200,  k  6 1 . 2 / 2 Synchronization rate: p max 3 10 synchronize frantically (at rate 1) p=1.0 2 p=0.0 10 p=0.8 do not synchronize at all (rate 0) 1 10 2 ( t )>  reducing the local synch. < w frequency can stabilize the 0 10 system reduce local synch. rate to 0.80 (in fact, even no synchronization −1 10 at all is better than “over ‐ synchronization”: power ‐ law divergence vs exp. divergence of −2 10 the fluctuations with time) −2 −1 0 1 2 3 10 10 10 10 10 10 t Hunt et al ., PRL (2010) 15

Coordination with Multiple Time Delays Complete Graph with N nodes (“normalized”):               h ( t ) [ h ( t ) h ( t )] ( t )  t i i o j o tr i 1 N Synchronization Boundary  j i 2.5 N = 11 N = 8 N = 5 2 N = 2 1.5  τ o o 1 reentrant behavior in  tr   local delay is dominant (more harmful) Hunt, Korniss, and Szymanski, PLA (2011). 0.5 0 5 10 15 20 25 Hunt et al . (2012)  16 τ tr tr

Coordination with Multiple Time Delays             h ( t ) A [ h ( t ) h ( t )] ( t ) t i ij i o j i k j i  o : local delays (reaction, decision, execution)  =  o +  tr : local delays + transmission, queuing delays  local delay is dominant (more harmful) Hunt et al . (2012) 17