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

The Impact of Time Delays in Network Synchronization in a Noisy Environment

• G. Korniss

David Hunt B.K. Szymanski

Supported by ARL NS‐CTA, DTRA, ONR MAPCON 2012

• Phys. Rev. Lett. 105, 068701 (2010)

http://arxiv.org/abs/1209.4240 (2012)

2

Hutchinson model (logistic growth with delay in population dynamics)

         K t N t rN t N

t

) ( 1 ) ( ) (

) ( ) ( t x K t N  

K N N  

 

), (

) ( ) (      t rx t x

t

 

population size carrying capacity intrinsic growth rate

          K t N t rN t N

t

) ( 1 ) ( ) ( 

Hutchinson (1948); Maynard Smith (1971); R.M. May (1973)

• S. Ruan, in NATO Sci. Ser. II Math. Phys. Chem. (Springer, 2006) p. 477

) (t N

∗

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

3

IP activity

(Zeus load balancer)

spontaneous brain activity (fMRI)

(Justin Vincent; http://martinos.org/~vincent/ )

flocking birds

http://www.youtube.com/watch?v=6AmSpHxnKm8 http://www.youtube.com/watch?v=VaQ66lDZ‐08

4

Synchronization/Coordination in a Noisy Environment with Time Delays

) ( ) ( ) ( t t t

t

η h Γ h      

ij i ij ij

C C    

network Laplacian:

) ( ~ ) ( ~ ) ( ~ t t h t h

k k k k t

       



 



1 1 2 2

) ( ~ 1 ) (

N k k t

h N t w



   

j j i ij i t

t h t h C t h )] ( ) ( [ ) (



     

j j i ij i t

t h t h C t h )] ( ) ( [ ) (  

) ( )] ( ) ( [ ) ( t t h t h C t h

i j j i ij i t

         



noise network/coupling strength delay

2 1 2

)] ( ) ( [ 1 ) ( t h t h N t w

N i i





  spread or width: (measure of de‐coordination):

max 1 2 1

          

 N



5

) ( ) ( ) ( t t h t h

t

       

Coordination, Noise, Time Delay

) ( 2 ) ( ) ( t t D t t         

) (   

 s

e s s g





characteristic equation:

 , 2 , 1

), , (     

 

s s

) ) ( (

st

ce t h 

infinitely many relaxation “rates”, {s}, for  > 0

synchronizability condition:





  0 ) Re(s

Frisch & Holme (1935); Hayes (1950); Hutchinson (1948); Maynard Smith (1971); R.M. May (1973)

2 /   



  



    

 

     

, ) ( 2

) )( ( ) (

] 1 [ 2 ) (

s s s g s g

t s s

e D t h

synchronizability:

     ) (

2

h

6

Coordination, Noise, Time Delay

57 . 1 2 /     c

50 100 150 200

−4 −2 2 4

h

~(t)

50 100 150 200

−12 −6 6 12

h

~(t)

50 100 150 200

t

−120 −60 60 120

h

~(t)

(a) (b) (c)

01 . , 1 , 1    dt D 

10 .   50 . 1   60 . 1  

) (t h

2 / ) (   

c

e 1  

2 1     e

   2

7

Coordination, Noise, Time Delay

01 . , 1 , 1    dt D 

10

−2

10

−1

10 10

1

10

2

10

3

t

10

−2

10

−1

10 10

1

10

2

10

3

10

4

<h

~2(t)>

τ=2.00 τ=1.60 τ=1.50 τ=1.00 τ=0.30 τ=0.10

57 . 1 2 /     c

  ) (

2 t

h

2 / ) (   

c

) ( ) ( ) ( t t h t h

t

       

monotonically decreasing function of the coupling 

0.2 0.4 0.6 0.8 1

λ

10 20 30 40 50

<h

~2(∞)>

∞ λ

(Ornstein–Uhlenbeck)

Coordination, Noise, Time Delay

8

) ( 2 ) ( ) ( t t D t t         

non‐monotonic function of the coupling 

0.2 0.4 0.6 0.8 1

λ

10 20 30 40 50

<h

~2(∞)>

∞ 1 sin

• λcos
• (

)

2 / ) (   

c

Küchler and Mensch, SSR 40, 23 (1992).

9

Implications for Networks:

 

   

 

1 1 1 1 2 2

) ( ) ( ~ 1 ) (

N k k N k k

f N D t h N t w   

k

k

  2 /    k hk       ) ( ~2

Synchronizability and Coordination:

2 /

max

   

Olfati‐Saber and Murray (2004) (deterministic consensus problems)

Hunt et al., PRL (2010)

10

Limitations of Network Synchronization

• networks with potentially large degrees can be extremely

vulnerable to intrinsic network delays while attempting to synchronize, coordinate, or balance their tasks, load, etc.

) ( , 2 1

max max max max max

k O k k N N      

: 2 /

max

   k

synchronization/stability breaks down

: 4 /

max

   k

sufficient for synchronizability/stability

Simple example: unweighted graphs

Fiedler (1973); Anderson and Morley (1985); Mohar (1991)

largest eigenvalue of the network Laplacian largest degree

11

Limitations of Network Synchronization

10 100 1000 N 0.2 0.4 0.6 0.8 1 ps ER τ = 0.08 ER τ = 0.09 ER τ = 0.10 ER τ = 0.11 BA τ = 0.08 BA τ = 0.09 BA τ = 0.10 BA τ = 0.11

Fraction of Synchronizable Networks

Comparison of ER and BA Networks

2 / 1 max max

~ ~ : BA N k     ) ln( ~ ~ : ER

max max

N k    

heterogeneous vs. homogeneous unweighted random graphs ps(,N) fraction of synchronizable networks

6   k

) 2 / (

max

   

Hunt et al., PRL (2010)

12

Scaling in the Synchronizable Regime

) ( ) ( ) (

1 1 , 2

     

 

g f N D w

N k k

  



 

k k ij ij

A C       '

) ( ) (

, 2

 

 

g w   BA network, N=1000, k6

10

−2

10

−1

10 10

1

10

2

10

3

t

10

−2

10

−1

10 10

1

10

2

10

3

<w

2(t)>

p=1.0

synchronize frantically (at rate 1)

13

Trade‐Offs

2 / 2 . 1

max

   

BA network, N=200, k6

Synchronization rate: p Hunt et al., PRL (2010)

) ( )] ( ) ( [ ) ( ) ( t t h t h C t p t h

i j j i ij i t

         



10

−2

10

−1

10 10

1

10

2

10

3

t

10

−2

10

−1

10 10

1

10

2

10

3

<w

2(t)>

p=0.8

reduce local synch. rate to 0.80

14

Trade‐Offs

2 / 2 . 1

max

   

BA network, N=200, k6

Synchronization rate: p

) ( )] ( ) ( [ ) ( ) ( t t h t h C t p t h

i j j i ij i t

         



Hunt et al., PRL (2010)

15

Trade‐Offs

• reducing the local synch.

frequency can stabilize the system (in fact, even no synchronization at all is better than “over‐ synchronization”: power‐law divergence vs exp. divergence of the fluctuations with time)

2 / 2 . 1

max

   

BA network, N=200, k6

Synchronization rate: p

10

−2

10

−1

10 10

1

10

2

10

3

t

10

−2

10

−1

10 10

1

10

2

10

3

<w

2(t)>

p=1.0 p=0.0 p=0.8

synchronize frantically (at rate 1) do not synchronize at all (rate 0) reduce local synch. rate to 0.80

) ( )] ( ) ( [ ) ( ) ( t t h t h C t p t h

i j j i ij i t

         



Hunt et al., PRL (2010)

16

Coordination with Multiple Time Delays

Complete Graph with N nodes (“normalized”):

5 10 15 20 25 τtr 0.5 1 1.5 2 2.5 τo N = 11 N = 8 N = 5 N = 2

Synchronization Boundary

tr



• 
• reentrant behavior in tr
• local delay is dominant (more harmful)

Hunt, Korniss, and Szymanski, PLA (2011).

) ( )] ( ) ( [ 1 ) ( t t h t h N t h

i i j tr

• j
• i

i t

             



 Hunt et al. (2012)

17

) ( )] ( ) ( [ ) ( t t h t h A k t h

i j j

• i

ij i i t

          



• local delay is dominant (more harmful)

Hunt et al. (2012)

o : local delays (reaction, decision, execution)  = o + tr : local delays + transmission, queuing delays

Coordination with Multiple Time Delays

18

Global vs. Local Weighted Coupling

) ( )] ( ) ( [ ) ( t t h t h A k t h

i j j i ij i t

            



) ( )] ( ) ( [ ) ( t t h t h A k t h

i j j i ij i i t

          



N A k A k

j i ij i j i ij

      

 

, ,

identical total interaction cost:

  

• Hunt et al. (2012)

19

• nodes/individuals constantly react to endogenous and exogenous information:

coordination/agreement/consensus/alignment

• they react to the information or signal received from their neighbors possibly

with some time lag  (as result of finite transmission, decision, or execution delays)

Hunt et al., PRL 105 , 068701 (2010)

The Impact of Time Delays in Information and Communication Networks

de-coordination

network connectivity or communication frequency network connectivity or communication frequency

de-coordination high connectivity / “too much communication” low connectivity / no communication low connectivity / no communication

20

Summary

• Delays can destroy synchronization/coordination in networks
• Networks with large hubs can be particularly vulnerable in this regard
• Too much communication can cause more harm than good
• On the other hand, understanding the fundamental scaling properties of the

underlying fluctuations (in particular the ones associated with the largest‐ eigenvalue mode) can guide optimization and trade‐offs to control and to reduce these large fluctuations

• Currently studying the effects of heterogeneously distributed time delays {ij}
• D. Hunt, B.K. Szymanski, G. Korniss, http://arxiv.org/abs/1209.4240 (2012).
• D. Hunt, G. Korniss, and B.K. Szymanski, Phys. Lett. A 375, 880 (2011).
• D. Hunt, G. Korniss, and B.K. Szymanski, Phys. Rev. Lett. 105, 068701 (2010).

Supported by DTRA, ARL NS‐CTA, ONR

c

  

 