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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

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?

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.

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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.

Thank you for

### 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,

SF,ER

Real Internet

SF

Lattice, ER