Separating internal and external dynamics of complex systems

Marcio Argollo de MenezesAlbert-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 topologyM. Argollo de Menezes and A.-L. Barabási, Phys. Rev. Lett. 92, 028701 (2004).

• Our approach: identify and study simultaneously dynamical variables fi(t) on different regions/nodes of the system

Internet

•fi(t)=number of bytes passing through router i at time t.

•347 routers

•tmax=2 days (5 minutes resolution)

World Wide Web

•fi(t) = number of visits to web site i on day t

•3000 web sites.

•Daily visitation for a 30 day period

Highways

•fi(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

•fi(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 ~ <fi(t)>

= 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 fi

<fi(t)> i

5) Repeat (2-4) T times, generating for each node i a series fi(1), fi(2), … fi(T).

i ≈ <fi(t)>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 fi

5) Repeat (2-4) T times, generating for each node fi(1), fi(2), … fi(T).

Let the number of walkers fluctuate:

W(t)= <W> + (t)

<(t)> = 0, <(t) (t’)> = (W)1/2 tt’

W: magnitude of external fluctuations

i ≈ <fi(t)> = 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): =1www, 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 fluctuationsSeparating external and internal fluctuations

• External perturbations affect nodes differently.

• Ai: node i’s share of the total traffic:

• fiext(t)=AiF(t),

where F(t)=ifi(t): total flux on the network at time t

• fiint(t) = fi(t) - fi

ext(t)

Model with sinusoidal external signal W(t)=W0+ W sin(t)

large W small W

Fluctuation ratios and the Fluctuation ratios and the exponent exponent

• From fiext(t) and fi

int(t) calculate iext and i

int for each node i.

• i = iext/i

int: ratio between external and internal fluctuations.

• P(i): quantifies the impact of external fluctuations.

=1/2 =1 =1=1/2

Scaling of fluctuations and theScaling of fluctuations and the exponentexponent

From fiext(t) and fi

int(t) measure <fiext(t)>, i

ext and <fiint(t)>, i

int

• =1/2

• Internal dynamics dominate

• iint > i

ext

• =1

• External dynamics dominate

• iint ~ i

ext

Measuring electric activity on the brain 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, 330-337, 2001).

Higher excitability

stronger internal dynamics

Smaller iext/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.

ConclusionsConclusions

• Monitor the simultaneous dynamics of numerous nodes

=Obtain more information about the system.