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Priority Model for Diffusion in Lattices and Complex Networks

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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

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.

A

A

B

A

A

A

B

B

A

A

B

A

B

B

B

A

B

B

B

B

B

B

A

A

A

A

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 – 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.

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

- 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.

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,

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