Physics of Flow in Random Media

1
Limited Path Percolation in Complex
Networks Eduardo López University of Oxford
Outline

• Motivation. Percolation and its effects.
• Presentation of new limited path length
  percolation model
• Scaling theory of new model and results
• Targeted percolation, theory and results.
• Conclusions

References
Limited path length percolation in complex
networks, López, Parshani, Cohen, Carmi and
Havlin, Phys. Rev. Lett. 99, 188701 (2007).
Collaborators
Shlomo Havlin
Roni Parshani
Reuven Cohen
Shai Carmi
2
Motivation Percolation and its limits
How far can I send a message?
Nathan goes on trip
Messages are harder to transmit on longer paths
Errors
Loss
3
Motivation Percolation and its limits

• Infectious diseases

EpiSimS model (Portland, OR)
Social contact network
Flu decays over time/season. Pathogen mutations
4
What is percolation theory?
Theory to determine connectivity in systems
p i, j distance S(p) connected nodes
i
1 lij N
lij
lt1 lij P8N
lij
lijgtlij due to removal
j
lij
pc lijgt lij Ndf/d
ltpc most disconnec. log N
p occupied fraction of links
P8 probability of random node to be in largest
cluster
pc connectivity threshold
Transition
df fractal dim., d dim.
connected
?
disconnected
5
New percolation model applied to complex networks

• Not always valid percolation accepts long
  short paths

• Definition of connection i and j are connected
  if lij(p) ? alij

• Notation
• Sa(p) Largest cluster size at occupation p,
  length condition a

Results New limited path percolation transition

• Below and above range, behavior is
• similar to regular percolation

6
Results of Limited Path Percolation on EpiSimS

• EpiSimS sample
• Individuals with
• social contacts
• lasting t gt to

• INSET
• ltlgt vs. p

• MAIN
• Sa vs. p (a1.2, 8)

7
Theory of model networks Erdos-Rényi

• Developed in the 1960s by Erdos and Rényi.
  (Publications of the Mathematical Institute of
  the Hungarian Academy of Sciences, 1960).

• N nodes and each pair connected with probability
  f.

• Define k as the degree (number of links of a
  node), and k
• is average number of links per node over the
  network.

Construction

• Distribution of degree is Poisson-like
  (exponential)

8
Outline of scaling theory for Limited Path
Percolation Example Erdos-Rényi

• Before percolation, typical path length l log
  N/log ltkgt

• Tree approx. ? Sa (k -1)l (pltkgt) a log
  N/log ltkgt Nd

• Scaling exponent 0 ? d ? a(1log p/log ltkgt) ? 1

• d ?1 because Sa cannot exceed N

9
Comparison of phase diagram of regular Limited
Path Percolation (Erdos-Rényi)
Regular percolation
Limited path percolation
Regular percolation
Communicating
Non-communicating
Limited path percolation predicts a larger
communication threshold.
10
Results for SaNd (Erdos-Rényi)
Regular Percolation
Limited path percolation
11
Complex Networks
Poisson distribution
Erdos-Rényi Network
12
Scaling theory for limited path percolation on
scale-free networks

• For lgt3

• For 2ltllt3

Tree approximation invalid. Networks are
ultra-small
Therefore
13
Phase Diagram of Limited Path Percolation on
scale-free networks
Communicating
Non-communicating
14
Results for SaNd (Scale-free)
15
Targeted attacks on scale-free networks

• Scale-free networks have sensitive nodes (hubs)
  with large k.

• Examples Airline hubs, central communication
  nodes,
• disease super-spreaders.

Model for targeted percolation

• p fraction of lowest degree nodes present.

• In targeted percolation (no length
• restriction) pc is large
• pc1 (l?2)
• pc close to 1 (lgt2)
• Network falls apart with few node removals.

hub
Question What happens for limited path
percolation?
16
Scaling theory for limited path
targeted percolation on scale-free networks

• For lgt3

• For 2ltllt3

Tree approximation valid again after percolation
Any finite a fails to produce transition to
linear phase
17
Phase Diagram of Limited Path Percolation
Scale-free targeted removal
Communicating
1
SF 2ltllt3
SF lgt3
Linear Phase (d1)
Transition line pc
Random

Concentration p
Logarithmic phase
Concentration p
Fractal Phase (dlt1)
pc(ko-1)-1
Logarithmic Phase (d0)
0
8
8
1
1
Length factor a
Length factor a
Non-communicating
18
Results for SaNd Scale-free targeted removal
19
Random removal
Erdos-Rényi
Scale-free (lgt3)
Scale-free (2?l?3)
Quantity
0
Transition
d
1
Sa
Nd
Nd
Nd
Transition
No Transition
Targeted removal
1
-
-
d
Sa
Nd
-
(log N)d
20
Conclusions

• We define a new percolation model which takes
  into
• account the length restriction of useful paths.

• This model is important in real-world
  applications such as
• epidemics, data transfer, and transportation.

• We find a new percolation transition at
• which implies when lengths are constrained, more
  connections
• are necessary to percolate. Transition preserves
  path length scaling.

• We encounter two typical phases i) power-law
  with Sa Nd,
• and ii) a linear phase Sa N.