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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Priority model for diffusion in lattices and complex networks

A

B

Priority Model for Diffusion in Lattices and Complex Networks

Shai Carmi

Pula July 2007


My collaborators

My collaborators

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

  • Supervised by Prof. Shlomo Havlin.


My collaborators1

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

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


Model definition

A

B

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.


Model definition1

A

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B

A

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B

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B

A

B

B

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A

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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

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

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 lattices1

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

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

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 protocol1

Priority diffusion – site protocol

  • Result:

various densities


Priority diffusion particle protocol

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 protocol1

Priority diffusion – particle protocol

  • Agrees with simulations too.

various densities

large densities


Complex networks

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


Empty sites in a network

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

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

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

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

The end

Thank you for

your attention!


Priority diffusion in networks quantitative discussion

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

Priority diffusion in networks –More simulations

SF,ER

Real Internet

SF

Lattice, ER


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