Recent developments in the study of

transport in random networksShai Carmi

Bar-Ilan UniversityHavlin group

Minerva meetingEilat, March 2009

Networks

Why do we care about networks?

Networks appear everywhere: Communication (Internet, p2p,…) Transportation (roads, airlines,…). Social sciences (social networks, business relations,…) Life sciences (gene regulation, food webs,…)

Transport in networks Networks are commonly used as a platform for transport of:

* Information (communication and social networks)* Passengers and commodities (transportation networks)* Current (electric circuits)* Diseases (social networks)

Quantities of interest:

* Time to reach target * Maximal capacity * Number of links crossed * Congestion and avalanches* Load at each node * Diffusion coefficients

Network models

1. Most naïve model: a regular lattice.* Only good for purely spatial, local, interactions.

2. Erdos-Renyi (ER) network model: fully random.* Fixed number of nodes N, each link exists with probability p.* Narrow degree distribution: where k is node degree.

3. Scale-free (SF) networks: emergence of hubs.* Broad degree distribution: * Nodes with extremely high degree exist (hubs).* Other ingredients possible, e.g., growth, correlations.* Found to describe most real-world systems.

!/)( kkekPkk

kkP )(

Transport models: outline

1. Random walk with priorities.

2. Random walk with trapping.

3. Maximum flow.

Transport models

1. Random walk with priorities.

2. Random walk with trapping.

3. Maximum flow.

Motivation

Some communication networks use random walkto search for other computers or spread information.

Some data packets have higher priority than others.

How does priority policy affect the diffusion in the network?

Two species of particles, A and B with densities ρA and ρB. A is high priority, B is low priority.

Symmetric random walk (nearest neighbors).

Protocols

B can move only after all the A’s in its site have already moved. If motion is impossible, choose again.

Model definition

AB

Site protocol: A site is randomly chosen and sends a particle.

Particle protocol: A particle is randomly chosen and jumps out.

Solution in lattices

Write a Markov chain for the number of particles in a site. Solve for the stationary probabilities. Derive analytically the fraction of empty sites in both protocols.

Diffusion is normal: <R2>=Dt. Apply the site protocol selection rule and find D for each species.

In the particle protocol define r as the fraction of free B's to total B’s.

independent of ρB and approaches for large densities. Derive diffusion coefficients similarly.

eff particlesite )(0

)(0 );1/(1

))(1(;))(1(

)1(

BAA

BB

BAA

BAAA DD

)(4/1321 32 Or AA Ae

Solution in networks In the particle protocol: or, the probability of a site to

be empty decreases exponentially with its degree. In scale-free networks, A’s move freely, and tend to aggregate at the hubs. Therefore, B’s at the hubs have very low probability to escape. Since the B’s themselves are attracted to the hubs, they eventually

become trapped and their motion is arrested. Time for a B to leave a site of degree k . Waiting time distribution → sub-diffusion.

)/exp()(0 kkf k

)/exp()( kkkt A11 ]ln[~)( ttt

Lattice, ER

SFSF,ER

Real Internet Average waiting time for B particles.

Distribution of waiting times for B particles.

Transport models

1. Random walk with priorities.

2. Random walk with trapping.

3. Maximum flow.

Motivation

Consider again a random walk process in a network.

In a communication or a social network, a message can disappear; for example, due to failure.

How long will the message survive before being trapped?

Particles initially evenly distributed over the network. Symmetric random walk (nearest neighbors). m of the nodes are absorbing. Whenever the particle reaches the trap it is absorbed. What is the survival probability ρ(t)?

Model definition

A simple theory

Denote the total number of links entering the traps by km. The total number of links is N<k>. Thus, the probability per unit time of a particle to enter the

trap is approximately proportional to km /N<k>. , and the problem is reduced to

evaluating km for different topologies. In ER networks:

- Approximation is good when . - Explicit dependence on both m and N.

For dense enough SF networks: -kmin is the minimum degree (one trap).

)]/(exp[)( kNtAkt m

)]]/(exp[1(exp[)( kNAtkmt

)]/(exp[)( min kNtAkt

kNm /

Results

The average time before trapping T usually scales as N. In SF networks when one of the hubs is a trap

-Only for infinite γ SF and ER networks are equivalent. SF networks become less vulnerable as links are added. For ER networks A=1-1/<k>.

Conclusion:A simple mean-field approach is usually useful to solve trapping problem in networks, and leads to interesting observations.

)1/()2( NT

Theory- linesSimulation- symbols

ER

Transport models

1. Random walk with priorities.

2. Random walk with trapping.

3. Maximum flow.

Motivation

Users in communication networks (e.g., peer-to-peer) wish to exchange files by sending them through the network links.

How many users can exchange files without interfering with each other?

What is the maximum capacity of the network for a given number of users?

Assume the network contains n sources and n sinks. Consider three types of transport:

* Maximum flow (= #of parallel paths)

* Electric current* Multi-commodity flow

Non directed,non weighted(unit capacities/resistances).

Model definition

nSinks

T2

T1

nSources

S2

S1

Rest of network

Regular flowMulti-commodity flow

Theory for small n

For a single source/sink pair with degrees k1 and k2, F≈min(k1,k2).

For small n, replace k1 by the total number of links leaving the sources, and similarly for k2.

The distribution of flows is:* For ER networks:* For SF networks:

Flow per user increases with n up tothe optimal number of users above which the approximation is invalid.

)!2

)(

)(

),((

!

)(2)(

F

ekn

F

knF

F

eknF

knFknF

n

)12()( FFn

kN /

Small n theory

Simulations

Theory for large n A different approach is needed:

Find the total flow by conditioning on the number of paths of a given length.

F = F1 + F2 + F3 + …

I ≈ F1/1 + F2/2 + F3/3 + … For direct linkage, <F1>=n2p. Implicit sum formulas for <F2>, <F3>.

SinksSources

F2

F3

F1

Theory- linesSimulation- symbols

ER networks

Multi-commodity flow Flow from a source is directed towards specific sink. Thus the

contribution of the different source/sink pairs can be separated. Result for ER network:

where kn is the effective degree of the

network when n pairs communicate. The network will saturate at the

percolation threshold, when and thus .

In SF networks the absence of percolation threshold leads to increased capacity.

)!2)(

),((

!2)(;)(

0

1

0'' j

ke

j

kj

j

kejkkF

jkN

j

jkn

nnMC

kkkN

Nkkk

nnnn 01 ;

log

log)(

1* nk

)log/(* NNOn

Small n approximation

Summary

Transport in networks is important and interesting. We introduced models inspired by problems in real networks. We used probability theory and computer simulations to obtain

analytical an numerical solutions. Deep relations between network structure and dynamics were

uncovered. For example:

* Halting of low priority particles in highly connected nodes.* Effect of failure in hubs on particle survival probability.* Optimal number of users in flow network.* Influence of inter-node distance on electrical current.* Interplay between percolation theory and maximal network flow.

Collaborators My advisor: Prof. Shlomo Havlin, Bar Ilan Univ., Israel. Other collaborators:

Prof. Daniel ben-Avraham (Clarkson Univ., NY, USA)Prof. Panos Argyrakis (Aristotle Univ., Thessaloniki, Greece)Prof. H. Eugene Stanley (Boston Univ., MA, USA).

Shlomo Dani Panos Gene

Thank you for your attention!

See also: M. Maragakis, S. Carmi, D. ben-Avraham, S. Havlin, and P. Argyrakis.

"Priority diffusion model in lattices and complex networks". Phys. Rev. E (RC) 77, 020103 (2008).

S. Carmi, Z. Wu, S. Havlin, and H. E. Stanley. "Transport in networks with multiple sources and sinks". Europhys. Lett. 84, 28005 (2008) .

A. Kittas, S. Carmi, S. Havlin, and P. Argyrakis. "Trapping in complex networks“. Europhys. Lett. 84, 40008 (2008).