Separating internal and external dynamics of complex systems Marcio Argollo de Menezes Albert-László Barabási

Scale-free: P(k) ≈ k -  Hierarchical: C(k) ≈ k -  Small World:

Networks support dynamical processes

Understand the dynamical processes that take place on networks. Inspiration: real data WWW Internet Metabolic networks Social network maps Network dynamics: diversity of the observed behavior, rather than any degree of universality. Beyond topology M. Argollo de Menezes and A.-L. Barabási, Phys. Rev. Lett. 92, (2004). Our approach: identify and study simultaneously dynamical variables f i (t) on different regions/nodes of the system

Internet f i (t)=number of bytes passing through router i at time t. 347 routers t max =2 days (5 minutes resolution)

World Wide Web f i (t) = number of visits to web site i on day t 3000 web sites. Daily visitation for a 30 day period

Highways f i (t)=traffic at a given point of a road i on day t. Daily traffic on 127 roads of the Colorado highway network from 1998 to 2001.

Computer chip f i (t)=state of a given logic component i at clock cycle t. 462 signal carriers 8,862 clock cycles.

1)For each node i: 2) Create a scatter plot:

Scaling of fluctuations  i ~   = 1/2  = 1

1) Start with an arbitrary network (SF/SW or ER). A simple diffusion model 3) Let each walker perform N steps. 2) Place W walkers on randomly selected nodes. 4) Record for each node i the number of visitations f i  i 5) Repeat (2-4) T times, generating for each node i a series f i (1), f i (2), … f i (T).  i ≈ 1/2  = 1/2

The origin of  =1/2 Random connections: decoupling of nodes What about  =1? After walkers perform N steps:

Internal fluctuations Randomness of the particle arrival or diffusion process External fluctuations Fluctuations of the number of agents/particles Two sources of fluctuations

1) Start with an arbitrary network (SF/SW or ER). Introducing external fluctuations 3) Each walker performs N steps. 2) Place W walkers on randomly selected nodes. 4) Record for each node i the total visitation f i 5) Repeat (2-4) T times, generating for each node f i (1), f i (2), … f i (T). Let the number of walkers fluctuate: W(t)= +  (t) = 0, = (  W) 1/2  tt’  W: magnitude of external fluctuations  i ≈  = 1 For large  W

The origin of  =1 Random connections: decoupling of nodes After walkers perform N steps:

 W large External fluctuations dominate (large  W large):  =1 www, highways  W=0 Small external fluctuations (small  W):  =1/2 Internet, chip Summary B. Huberman et al., Science 280, 95 (1998).

Separating external and internal fluctuations External perturbations affect nodes differently. A i : node i’s share of the total traffic: f i ext (t)=A i F(t), where F(t)=  i f i (t): total flux on the network at time t f i int (t) = f i (t) - f i ext (t) Model with sinusoidal external signal W(t)=W 0 +  W sin(  t) large  W small  W

Fluctuation ratios and the  exponent From f i ext (t) and f i int (t) calculate  i ext and  i int for each node i.  i =  i ext /  i int : ratio between external and internal fluctuations. P(  i ): quantifies the impact of external fluctuations.  =1/2  =1  =1/2

Scaling of fluctuations and the  exponent From f i ext (t) and f i int (t) measure,  i ext and,  i int  =1/2 Internal dynamics dominate  i int >  i ext  =1 External dynamics dominate  i int ~  i ext

Measuring electric activity on the brain EEG: local voltage differences in the brain  neural activity. Time resolved activity measured simultaneously in 64 regions in the head. (i=1..64; t=1..256) Two different systems: Alcoholic vs. non-alcoholic person Alcoholics: deficit ininhibition (hyperexcitability) in the central nervous system. (Alcohool Clin. Exp. Res. vol. 25, , 2001). Higher excitability stronger internal dynamics Smaller  i ext /  i int ratios

-empirical data: two universality classes -modeling data:  =1/2: internal dyamics  = 1: externally driven dyamics -separating internal/external components:  =1/2: Internet, chip: internal fluctuations dominate  = 1 : www, highways: external fluctuations dominate Are the exponents universal?  =1 is, the  =1/2 perhaps not. Conclusions Monitor the simultaneous dynamics of numerous nodes = Obtain more information about the system.