Priority Model for Diffusion in Lattices and Complex Networks

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Priority Model for Diffusion in Lattices and
Complex Networks

• Shai Carmi

Boston University October 2007
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About myself

• I am a Ph.D. student at the Department of
  Physics, Bar-Ilan University, Israel visiting
  scholar at the Center for Polymer Studies at BU.
• Supervised by Prof. Shlomo Havlin( Prof. H.
  Eugene Stanley).

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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.
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Motivation

• Many communication networks use random walk to
  search for 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 As 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?

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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 fjj0,1,2,.. are the stationary
  probabilities of the chain.

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Empty sites Lattices

• Write transition probabilities for the chain
  (lattices)Choosing by siteChoosing by
  particle
• Write equations for stationary probabilities
• Use normalization and conservation of material

? is the number of particles per site
Same in every dimension!
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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
?
?
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Priority diffusion Lattices

• Both particles diffuse normally ltR2gtDt.
• But how is time shared between A and B?

?10
?1
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Priority diffusion site protocol

• Densities are ?A and ?B.
• 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

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Priority diffusion site protocol

• Result

various densities
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Priority diffusion particle protocol

• No miracles here ?
• Define r as the fraction of free B's to total
  Bs.
• Solvable for low densities

• Happens to always be independent of ?B.
• For large densities, r approaches (the fraction
  of sites with no A) from below.
• Using r, easy to find PA and PB.

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Priority diffusion particle protocol

• Agrees with simulations too.

small densities
various densities
large densities
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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
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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.
• Probability of a site to be empty

Consistent with total number of particles in a
site proportional to its degree k.
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Priority diffusion in networks Qualitative
discussion

• As move freely, and tend to aggregate at the
  hubs.
• Therefore, Bs at the hubs have very low
  probability to escape.
• In lattices and ER networks hubs do not exist so
  Bs can move.
• In scale-free networks hubs exist. Bs also tend
  to aggregate at these hubs and therefore become
  immobile.

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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 distribution of Bs waiting
  times
• For SF networks with P(k)k-?,

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Priority diffusion in networks Simulations
various ltkgt
various ?
Real Internet
SF,ER
SF
Lattice, ER
Waiting time for the Bs grows exponentially with
the degree
Distribution of waiting times (for B)narrow for
lattices and ER, broad for SF.
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Priority diffusion in networks Simulations
various degrees
Concentration of B particles vs. node degree.
Theory (solid lines)
Exponential distribution of waiting times (of B)
for different degrees.
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How to make the Bs more mobile?

• In the following, we consider some strategies and
  model variations to avoid the trapping of the Bs
  at the hubs.
• Consider the particle protocol when a non-free B
  is chosen, one of its cohabitant As is moving
  instead.
• As are effectively kicked-out of sites with
  large concentration of Bs.
• For lattices, we find analytically .
• Probability for B to be free is higher than
  previous protocol, but converges to for
  large densities.

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New protocol in networks

• As cannot aggregate at the hubs anymore, and
  thus the Bs waiting time grows slowly with the
  degree.

other protocol
Bs waiting time
New protocol
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Soft priorities

• What happens when Bs have some probability x to
  jump even if they coexist with an A?
• For the particle protocol in lattices, obtain
  analytically
• Behavior on x?0 and x?1 is well explained.
• For networks .
• Thus, for k?8, , and the Bs waiting
  time does not diverge even in the hubs.
• Divergence only for x?0.

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Soft priorities - simulations
peaks at 1/x
small x
x0, power-law
large x
xgt0, exponential
Average waiting time for Bs vs. the degree, for
various values of x.
Distribution of waiting time for Bs, for various
values of x.
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Other strategies

• Instead of random walk, visit the ith neighbor
  with probability proportional to .
• If alt0, particles avoid the hubs and dont
  aggregate.
• Fraction of empty sites is expected to be
  , so for a-1 it is independent of k and
  we recover the lattice case.
• Impose natural constraint on the number of
  particles a node can contain particles will not
  aggregate.

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Priority Diffusion Summary

• In lattices use the number of sites empty of the
  high priority species to calculate diffusion
  coefficients for the normal diffusion of both
  species.
• For networks, probability for the low priority
  particles to be in an empty site decreases
  exponentially with the degree, and leads to their
  trapping in heterogeneous networks.
• Several strategies can be applied to avoid the
  aggregation of the particles at the hubs and
  allow improved mobility.
• Conclusion when priority constraints exist,
  network structure and protocols should be
  designed with care.

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The end
Thank you for your attention!