Priority model for diffusion in lattices and complex networks
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A. B. Priority Model for Diffusion in Lattices and Complex Networks. Shai Carmi. Pula July 2007. My collaborators. I am a Ph.D. student at the Department of Physics, Bar-Ilan University, Israel. Supervised by Prof. Shlomo Havlin. My collaborators. Michalis. Panos. Dani.

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Priority Model for Diffusion in Lattices and Complex Networks

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A

B

Priority Model for Diffusion in Lattices and Complex Networks

Shai Carmi

Pula July 2007


My collaborators

  • I am a Ph.D. student at the Department of Physics, Bar-Ilan University, Israel.

  • Supervised by Prof. Shlomo Havlin.


My collaborators

Michalis

Panos

Dani

Michalis Maragakis, Ph.D. student; and Prof. Panos Argyrakis,Aristotle University of Thessaloniki, Greece.

Prof. Daniel ben-Avraham, Clarkson University, NY, USA.


Motivation

  • Many communication networks use random walk to search other computers or spread information.

  • Some data packets have higher priority than others.

  • How does priority policy affect diffusion in the network?

God bless Google Images


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Model definition

  • Two species of particles, A and B.

  • 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.

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

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


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Model definition

  • Example- lattice (1-d):

  • Who is mobile?

  • Condition for B to be mobile is being in a site empty of A. What is the probability for this?


Empty sites

  • Assume only A particles.

  • What is the probability fj for a site to have exactly j particles?

  • Define a Markov Chain on the states {0,1,2,…} which are the number of particles in a given site.

  • The {fj}j=0,1,2,.. are the equilibrium probabilities of the chain.


Empty sites – Lattices

  • Write transition probabilities for the chain (lattices):Choosing by siteChoosing by particle

  • Write equations for equilibrium probabilities:

  • Use normalization and conservation of material:

ρis the number of particles per site

Same in every dimension!


Empty sites – Lattices

  • Results:

  • So we know how many empty sites to expect for one species. What happens when A and B are moving together?

f0

f0

ρ

ρ


Priority diffusion – Lattices

  • Both particles diffuse normally: <R2>=Dt.

  • But how is time shared between A and B?

ρ=10

ρ=1


Priority diffusion – site protocol

  • Densities are ρA and ρB.

  • Fraction of sites with any A:

  • Fraction of sites with no A and no B:

  • Therefore, the fraction of time A is moving (PA) satisfies:


Priority diffusion – site protocol

  • Result:

various densities


Priority diffusion – particle protocol

  • No miracles here 

  • Define r as the ratio of free B's to total B’s.

  • Solvable for low densities

  • Happens to be always independent of ρB.

  • For large densities, r approaches (the fraction of sites with no A) from below.

  • Using r, easy to find PA and PB.


Priority diffusion – particle protocol

  • Agrees with simulations too.

various densities

large densities


Complex networks

  • What happens for particles diffusing in a network?

Internet as seen with DIMES project www.netdimes.org

S.C. et al. PNAS 104, 11150 (2007)

Using Lanet-vi program of I. Alvarez-Hamelin et al.http://xavier.informatics.indiana.edu/lanet-vi


SF & ER networks

Empty sites in a network

  • Consider one species only, in the particle protocol.

  • Follow the same Markov chain formalism as before, but with transition probabilities:For a site with degree k.

  • Fraction of empty sites is:

Consistent with total number of particles in a site proportional to its degree k.


Priority diffusion in networks – Qualitative discussion

  • A’s move freely, and tend to aggregate at the hubs.

  • Therefore, B’s at the hubs have very low probability to escape.

  • In lattices and ER networks hubs do not exist so B’s can move.

  • In scale-free networks hubs exist. B’s also tend to aggregate at these hubs and therefore become immobile.


Priority diffusion in networks – Simulations

Real Internet

various <k>

SF,ER

various γ

SF

Lattice, ER

Distribution of waiting times (for B):narrow for lattices and ER, broad for SF.

Waiting time for the B’s grows exponentially with the degree


Priority Diffusion – Summary

  • Use Markov chain formulation to calculate number of sites empty of the high priority species.

  • In lattices use this number to calculate diffusion coefficients for the normal diffusion of both species.

  • For networks, probability for a low priority particle to be in an empty site decreases exponentially with the degree.

  • In heterogeneous networks where particles stick to the hubs, low priority particles are immobile.

  • Conclusion– when priority constraints exist, network structure and protocols should be designed with care.


The end

Thank you for

your attention!


Priority diffusion in networks – Quantitative discussion

  • B can move if site is empty of A, which happens with probability

  • In an average sense, in every time step a site can become empty with probability p.

  • Leads to exponential waiting time distribution:

  • For SF networks with P(k)~k-γ,


Priority diffusion in networks –More simulations

SF,ER

Real Internet

SF

Lattice, ER


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