Priority Model for Diffusion in Lattices and Complex Networks

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# Priority Model for Diffusion in Lattices and Complex Networks - PowerPoint PPT Presentation

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

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?

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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