Percolation and diffusion in network models
Shai Carmi, Department of Physics, Bar-Ilan University
Advisor: Prof. Shlomo Havlin
Percolation
Networks
Diffusion
What is the relation between network topology and dynamics?
Percolation theory of networks addresses the question:
Networks are everywhere: technology, society, economy, transportation,
biology, and others.
Consider a simple random walk, or diffusion.
Are networks stable?
How to model networks?
For scale-free networks, depends on how nodes are removed:
Random network model (Erdos-Renyi, ER)
Connect each pair of nodes with equal small probability p.
•Studied since the 1960s.
•Nodes have typical degree (number of
connections). Degree distribution is
Poissonian.
•Easy to solve for various properties.
•Small world.
•But cannot account for properties of real
networks.
Scale free (SF) network model
The distribution of node degrees is broad, usually a power-law.
Some nodes have extremely high degree (hubs).
•Nodes do not have typical degree; degrees
span several order of magnitudes.
•Asymmetry (or heterogeneity) in network
structure.
•Hubs can be attracted or repelled from other
hubs.
•SF topology naturally results from growth with
preferential attachment.
•Almost all natural networks are scale-free.
Failure
Attack
Nodes are randomly removed
Nodes of highest degree are removed first
•Almost all remaining
nodes are connected.
•Scale-free networks are
resilient to random
failure!
•The percolation
threshold is:
2
•Attacked network is
fragmented.
•Scale-free networks are
highly sensitive to
targeted attack on their
hubs!
•The critical fraction of
nodes that need to remain
before the network breaks
down is called the
percolation threshold and
is denoted by pc.
For attack on scale-free
networks, pc>0.
k
1
pc =
;κ =
k
κ −1
where k is the degree.
•For scale-free networks,
κ→∞, and thus pc=0,
meaning all nodes need
to be removed before
the network
disintegrates.
•Removal of nodes might leave the network connected, but eliminates many shortcuts.
•In many real systems, nodes connected only through very long paths are practically disconnected.
Communication:
Packets
accumulate
errors or are
discarded after many
transfers.
Society:
Information is
irrelevant if it
arrives
too late.
Transportation:
Commuting is pointless
if commute
time is too
long
Epidemics:
Individuals very
distant from the
origin of the epidemics
might not be infected due
to mutation or vaccination.
New percolation model is needed!
Limited Path Percolation (LPP):
Technology
Biology
Society
Others
•Internet
•Wireless networks
•Electric circuits
•Protein interaction
•Genetic regulation
•Biochemical
pathways
•Social networks
•Online networks
•Business relations
•Transportation networks
•Linguistic networks
•WWW
•Energy landscapes
A fraction p of the network nodes are removed. However, nodes are considered disconnected if
the new distance is more than a times the original distance.
~
We find new results for the percolation threshold pc and for the size of the largest connected
component Sa. SF networks are assumed to have degree distribution P (k ) ∝ k −γ .
Questions
In a lattice, equilibrium particle density is homogeneous.
Networks are heterogeneous substrates: particle density is
proportional to degree.
If the total particle density is ρ, the density at a node of degree
k is: ρk = ρ k/<k>.
→ hubs are loaded.
p=1/5
In communication networks, packets often have
inherent priorities. Low priority packets are
handled only after high priority packets are handled.
How is the diffusion affected?
Model: Particles of two types: A (high priority) and B (low priority), are randomly
traversing a network. At each time step, a B particle is allowed to move only if its current
node is clear of A particles. Selection of particles is random, by uniformly choosing either a
site (the site protocol) or a particle (the particle protocol).
What are the diffusion coefficients when priority is applied?
Analytically solvable for lattices!
Find first the fraction f0 of sites empty of
the high priority particles.
f 0( site ) =
1
ρ +1
; f 0( particle ) = e −ρ
Proceed to find the diffusion coefficients (site protocol):
DA =
ρ A (1 + ρ A + ρ B )
ρB
; DB =
(1 + ρ A )( ρ A + ρ B )
(1 + ρ A )( ρ A + ρ B )
For the particle protocol, calculate first the probability of a low priority particle B to be free
r = 1 − 2 ρ A + 13 / 4 ρ A2 + O ( ρ 3 )
from which the diffusion coefficients are easily derived.
In networks, the hubs are almost always occupied:
f 0( particle ,k ) = exp(− ρk / k )
The waiting times of the B particles increase exponentially
with node degree: τ = exp( ρ A k / k )
Real
Internet
In scale-free networks, freely moving A’s aggregate at
the hubs. B’s can’t escape from the hubs and aggregate
there as well, leading to total inhibition of their motion.
•How and why did so many networks evolve to be scale-free?
•Which are the most important nodes and links?
•Do networks have hierarchical organization?
•What is the typical distance between nodes?
•Can scale-free networks be embedded in a lattice/fractal?
•How to find communities in a network?
•Is scale-free topology optimal and in what sense?
•How does scale-free topology affect network function and dynamics
(random walk, routing, synchronization, conductance)?
•How does the network structure affect its stability?
The waiting times of low priority particles in
scale-free networks is compatible with sub-diffusion.
ψ
SF
(t ) ~
1
,
t ⋅ ln γ − 1 t
Conclusion: network topology should be taken into
account when designing routing protocols.
E. Lopez, R. Parshani, R. Cohen, S. Carmi, and S. Havlin. Phys. Rev. Lett. 99, 188701 (2007).
M. Maragakis, S. Carmi, D. ben-Avraham, S. Havlin, and P. Argyrakis. Phys. Rev. E (RC) 77, 020103 (2008).
Background picture: The Internet at the Autonomous Systems level. Nodes are colored by their proximity to the network’s core. Node size is proportional to its number of links (degree). S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, and E. Shir. PNAS, 104, 11150 (2007).