Priority Model for Diffusion in Lattices and Complex Networks

1. A B Priority Model for Diffusion in Lattices and Complex Networks Shai Carmi Pula July 2007

2. My collaborators • I am a Ph.D. student at the Department of Physics, Bar-Ilan University, Israel. • Supervised by Prof. Shlomo Havlin.

3. 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.

4. 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

5. 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.

6. A 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?

7. 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.

8. 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!

9. 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

10. Priority diffusion – Lattices • Both particles diffuse normally: <R2>=Dt. • But how is time shared between A and B? ρ=10 ρ=1

11. 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:

12. Priority diffusion – site protocol • Result: various densities

13. 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.

14. Priority diffusion – particle protocol • Agrees with simulations too. various densities large densities

15. 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

16. 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.

17. 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.

18. 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

19. 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.

20. The end Thank you for your attention!

21. 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,

22. Priority diffusion in networks –More simulations SF,ER Real Internet SF Lattice, ER