Percolation of Networks Composed of Connectivity and
Dependency Relations
Amir Bashan, Roni Parshani and Shlomo Havlin
Department of Physics, Bar-Ilan University, Ramat-Gan, Israel
The model:
Small size of
dependency clusters
Large size of
dependency clusters
Second order
transition
First order
transition
Cascade of failures:
Dependency
phase
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
0.6
0.8
1
p
25
s=1
s=2
s=3
s=4
s=6
20
15
10
5
0.2
0.4
p
The critical size of dependency groups
1
For a given network with average degree, k,
the critical size of dependency clusters, <s>c
that leads to a change from second order to
first order transition is given by:
0.8
0.6
First
order
transition
c
Percolation
phase
Identifying the first order transition
point, pc , by the sharp peak in the
number of iterations of the cascading
process.
1
0.8
p
Initial
attack
For network with average degree, k, and
fixed size of dependency clusters, s, the
size of the giant cluster,
, is given by:
Size of giant cluster
The edges represent connectivity relations while the
(blue, red and green) groups surrounded by curves
represent dependency relations between all the nodes
of the same group (color).
Change from second to first order transition
Number of
iterations
NOI
Connectivity network with dependency clusters.
0.4
Second
order
transition
0.2
Percolation
phase
1
2
3
4
5
<s>
Effect of distribution width on pc
0.9
<s>=10
0.8
c
For normally distributed dependency groups
with an average size <s> and width σ the
system becomes more stable for a wider
distribution.
p
Dependency
phase
<s>=6
0.7
0.6
<s>=4
0.5
Percolation
phase
0
0.2
0.4
σ/<s>
0.6
0.8
References:
S. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Catastrophic cascade of failures in
coupled networks, Nature, 464, 7291 (2010).
Complete
fragmentation
of the network.
R. Parshani, S. V. Buldyrev, S. Havlin, The critical effect of dependency groups on the function
of networks, PNAS, in press (2010), [arXiv:1010.4498]
A. Bashan, R. Parshani, S. Havlin, Percolation of networks composed of connectivity and
dependency relations, preprint (2010).