Cascades on correlated and
modular networks
James P. Gleeson
Department of Mathematics and Statistics,
University of Limerick, Ireland
www.ul.ie/gleesonj
Collaborators and funding
Sergey Melnik, UL
Diarmuid Cahalane, UCC (now Cornell)
Rich Braun, University of Delaware
Donal Gallagher, DEPFA Bank
SFI Investigator Award
MACSI (SFI Maths Initiative)
IRCSET Embark studentship
Some areas of interest
Noise effects on oscillators

Applications: Microelectronic circuit design
Diffusion in microfluidic devices

Applications: Sorting and mixing devices
Complex systems



Agent-based modelling
Dynamics on complex networks
Applications: Pricing financial derivatives
Some areas of interest
Noise effects on oscillators

Applications: Microelectronic circuit design
Diffusion in microfluidic devices

Applications: Sorting and mixing devices
Complex systems



Agent-based modelling
Dynamics on complex networks
Applications: Pricing financial derivatives
Overview
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
Overview
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
What is a network?
A collection of N “nodes” or “vertices” which can be labelled i…
…connected by links or “edges”, {i,j}.
Examples:
• World wide web
• Internet
• Social networks
• Networks of neurons
• Coupled dynamical systems
Examples of network structure
The Erdós-Rényi random graph
Consider all possible links,
create any link with a given
probability p.
Degree distribution is Poisson
with mean z:
e z z k
pk 
k!

z   k pk
k 0
Examples of network structure
The Small World network
Start with a regular ring having links to k nearest neighbours.
Then visit every link and rewire it with probability p.
[Watts & Strogatz, 1998]
Examples of network structure
Scale-free networks
Many real-world networks (social, internet, WWW) are found to have scale-free degree
distributions.
“Scale-free” refers to the
power law form:
p k ~ k 
Examples
[Newman, SIAM Review 2003]
Overview
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
Dynamics on networks
• Binary-valued nodes:
• Epidemic models (SIS, SIR)
• Threshold dynamics (Ising model, Watts)
• ODEs at nodes:
• Coupled dynamical systems
• Coupled phase oscillators (Kuramoto model)
Global Cascades and Complex Networks
Initially small localized effects can propagate over the
whole network, causing a global cascade
Examples of global cascades:
• Epidemics, computer viruses
• Spread of fads and innovations
• Cascading failures in infrastructure (e.g. power grid) networks
Similarity: initial failures increase the likelihood of subsequent failures
Cascade dynamics depends strongly on:
• Network topology (degree distribution, degree-degree correlations,
community structure, clustering)
• Resilience of individual nodes (node response function)
Structures and dynamics review see:
• M.E.J. Newman, SIAM Review 45, 167 (2003).
• S.N. Dorogovtsev et al., arXiv:0705.0010 (2007)
Watts` model
D.J. Watts,
Proc. Nat. Acad. Sci. 99, 5766
(2002).
Threshold dynamics
The network:
• aij is the adjacency matrix (N ×N)
• un-weighted
• undirected
aij  {0,1}
aij  a ji
The nodes:
• are labelled i , i from 1 to N;
• have a state ;
vi (t ) {0,1}
• and a threshold ri from some distribution.
Threshold dynamics
Node i has state vi (t ) {0,1}
and threshold ri
Neighbourhood average: r i 
Updating:
1
ki
a v
ij
j
if ri  ri
1
vi  
 unchanged otherwise
The fraction of nodes in state vi=1 is r(t):
j
Watts` model
D.J. Watts,
Proc. Nat. Acad. Sci. 99, 5766
(2002).
Watts` model
z k e z
pk 
k!
P( r )   ( r  R )
R
Cascade condition:
k ( k  1)
pk F  k1   1

z
k 1

r
Thresholds CDF: F ( r ) 
 P( s) ds

Watts` model
Watts: initially activate single node (of N), determine if r
at steady state.
Us:
1
initially activate a fraction r 0 of the nodes, and
determine the steady state value of r .
Conditions for global cascades (and dependence on the
size of the seed fraction) follow…
Main result
Our result:

k
k
k 1
m 0
 
r  r0  (1  r0 ) pk    qm (1  q )k m F  mk 
m
with
qn1  r0  (1  r0 )G( qn ),
q0  r0 ,
and
k k 1  k  1 m
k 1m
G( q)   pk  
q
(1

q
)
F  mk 

k 1 z
m 0  m 

Derivation: Generalizing zero-temperature random-field Ising model
results from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna,
J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.
Results
R  0.18
N  105
z k e z
pk 
k!
P( r )   ( r  R )
r0  103
r 0  5  103
r 0  102
Results
r 0  104
z k e z
pk 
k!
P( r )   ( r  R )
r 0  102
Main result
Our result:

k
k
k 1
m 0
 
r  r0  (1  r0 ) pk    qm (1  q )k m F  mk 
m
with
qn1  r0  (1  r0 )G( qn ),
q0  r0 ,
and
k k 1  k  1 m
k 1m
G( q)   pk  
q
(1

q
)
F  mk 

k 1 z
m 0  m 

1.5
Cascade condition
1.25
qn1  r0  (1  r0 )G( qn ),
q0  r0 ,
G(q) 1
0.75
0.5
0.25
0.25
0.5
q
0.75
1
1.25
1.5
Cascade condition
1.25
qn1  r0  (1  r0 )G( qn ),
q0  r0 ,
G(q) 1
0.75
0.5
0.25
0.25
0.5
q
0.75
1
1.25
Simple cascade condition
qn 1
First-order cascade condition: using
qn1  r0  (1  r0 )G( qn ),
slope>1
q0  r0 ,
slope=1
demand
qn
(1  r0 )G(0)  1 (slope>1)
for global cascades to be possible. This yields the condition
k (k  1)
1
1
p
F

F
(0)

,


k 

k 

z
1  r0
k 1

reproducing Watts’ percolation result when r0  0 and F (0)  0.
Simple cascade condition
r 0  104
z k e z
pk 
k!
P( r )   ( r  R )
r 0  102
Extended cascade condition
qn 1
Second-order cascade condition: expand
qn1  r0  (1  r0 )G( qn ),
q0  r0 ,
above
slope=1
to second order and demand no
positive zeros of the quadratic
aq2  bq  c  0
for global cascades to be possible.
The extension is, to first order in r 0 :
G(0)  12  2G(0)G(0)  2r0 G(0)  G(0)2  G(0)  2G(0)G(0)  0.
qn
Extended cascade condition
r 0  104
R
z k e z
pk 
k!
P( r )   ( r  R )
r 0  102
Gaussian threshold distribution
r0  0
  0.05
z k e z
pk 
k!
 ( r  R)2 
P( r ) 
exp  

2
2
2

2


1
  0.2
Gaussian threshold distribution
R  0.2
  0.05
R  0.362
R  0.38
N  105
r0  0
z k e z
pk 
k!
 ( r  R)2 
P( r ) 
exp  

2
2
2

2


1
  0.2
Bifurcation analysis
qn1  r0  (1  r0 )G( qn ),
R  0.35
r0  0;   0.2
q  G ( q)  0
R  0.371
R  0.375
Results: Scale-free networks
r 0  103
r 0  10
2
P( r )   ( r  R )
z  10.5
pk  k  exp(  k  )
  100
  0.2
  0.45
 ( r  R)2 
P( r ) 
exp  

2
2
2

2


r0  0
1
z6
Results: Scale-free networks
z k e z
pk 
k!
r 0  102
P( r )   ( r  R )
pk  k  exp(  k  )
  100
Overview
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
Watts` model of global cascades
Consider undirected unweighted network of N nodes
(N is large) defined by degree distribution pk
Each node i has:
vi {0,1}
• fixed threshold ri
•
binary state
given by thresholds CDF
r
F (r) 
(probability that a node has threshold < r)
 P( s) ds

Initially activate fraction ρ0<<1 of N nodes.
mi

1
,
if
 ri
Updating: node i becomes active if the

vi  
ki
active fraction of its neighbours exceeds

unchanged otherwise
its threshold
The average fraction of active nodes
r (t ) 
1
N

N
v (t )
i 1 i
Derivation of result
Derivation: Generalizing zero-temperature random-field Ising model
results from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna,
J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.
Derivation of result
Main idea: pick a node A at random and calculate its probability of
becoming active. This will give ρ(∞).
A
Derivation of result
Main idea: pick a node A at random and calculate its probability of
becoming active. This will give ρ(∞).
Re-arrange the network in the form of a tree with A being the root.
∞
A
…
………………
n+2 …
…
…
n+1
n
…………………..
…
qn 1
qn
q:nprobability that a node on level n is active,
conditioned on its parent (on level n+1) being
inactive.
Derivation of result
Main idea: pick a node A at random and calculate its probability of
becoming active. This will give ρ(∞).
Re-arrange the network in the form of a tree with A being the root.
∞
qn 1  r 0 
A
…
………………
n+2 …
(1  r 0 )  (initially inactive)
…
…
n+1
n
…………………..
(initially active)
…
qn 1
qn
q:nprobability that a node on level n is active,
conditioned on its parent (on level n+1) being
inactive.
Derivation of result
Main idea: pick a node A at random and calculate its probability of
becoming active. This will give ρ(∞).
Re-arrange the network in the form of a tree with A being the root.
∞
qn 1  r 0 
A
…
………………
n+2 …
(1  r 0 )  (initially inactive)
…

…………………..
p
k 1
…
n+1
n
(initially active)
…
qn 1
qn
k
 (has degree k; k-1 children)
 k  1 m

qn 1  qn k 1 m
 m 
(m out of k-1
children active)
k-1 children
q:nprobability that a node on level n is active,
conditioned on its parent (on level n+1) being
inactive.
Degree distribution of nearest
neighbours:
pk 
k pk
.
z
Derivation of result
Main idea: pick a node A at random and calculate its probability of
becoming active. This will give ρ(∞).
Re-arrange the network in the form of a tree with A being the root.
∞
qn 1  r 0 
A
…
………………
n+2 …
(1  r 0 )  (initially inactive)
…

…………………..
p
k 1
k 1
…
n+1
n
(initially active)
…
qn 1
qn
k-1 children
q:nprobability that a node on level n is active,
conditioned on its parent (on level n+1) being
inactive.
k
 (has degree k; k-1 children)
 k  1 m
m
k 1 m




q
1

q
F
 

n
n


m 0  m 
k
(m out of k-1
children active)
(activated by m
active neighbours)
Derivation of result
q0  r 0
k 1 k  1

 m
k 1 m  m 
~


qn 1  r 0  (1  r 0 ) pk  
qn 1  qn 
F 

k
k 1
m 0  m 

Our result for the
average fraction of
active nodes
k m
m
r  r 0  (1  r 0 ) pk   q 1  q k m F  
k
k 1
m 0  m 

k
Valid when:
(i) Network structure is locally tree-like (vanishing clustering coefficient).
(ii) The state of each node is altered at most once.
Conclusions
• Demonstrated an analytical approach to determine the average
avalanche size in Watts’
model of threshold dynamics.
• Derived extended condition for global cascades to occur; noted
strong dependence on seed size.
• Results apply for arbitrary degree distribution, but zero clustering
important.
• Further work…
Overview
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
Extensions
• Generalized dynamics:
• SIR-type epidemics
• Percolation
• K-core sizes
• Degree-degree correlations
• Modular networks
• Asynchronous updating
• Non-zero clustering
Derivation of result
q0  r 0
k 1 k  1

 m
k 1 m  m 
~


qn 1  r 0  (1  r 0 ) pk  
qn 1  qn 
F 

k
k 1
m 0  m 

Our result for the
average fraction of
active nodes
k m
m
r  r 0  (1  r 0 ) pk   q 1  q k m F  
k
k 1
m 0  m 

k
Generalization to other dynamical models
q0  r 0
k 1 k  1

 m
k 1 m
~


qn 1  r 0  (1  r 0 ) pk  
qn 1  qn 
F m, k 

k 1
m 0  m 

Our result for the
average fraction of
active nodes
k m
r  r0  (1  r0 ) pk   q 1  q k m F m, k 
k 1
m 0  m 

Fraction of active neighbours (Watts):
Absolute number of active neighbours:
Bond percolation:
Site percolation:
k
m
F m, k   CDFthr  
k
F m, k   CDFthr m
F m, k   1  (1  p) m
 0, if m  0
F m, k   
Qk , if m  0
Generalization to other dynamical models
q0  r 0
k 1 k  1

 m
k 1 m
~


qn 1  r 0  (1  r 0 ) pk  
qn 1  qn 
F m, k 

k 1
m 0  m 

Our result for the
average fraction of
active nodes
k m
r  r0  (1  r0 ) pk   q 1  q k m F m, k 
k 1
m 0  m 

k
K-core: the largest subgraph of a network whose nodes have degree at least K
Initially activate (damage) fraction ρ0 of nodes.
A node becomes active if it has fewer than K inactive neighbours:
0, if k  m  K
F m, k   
1, if k  m  K
Final inactive fraction (1- ρ) of the total network gives the size of K-core
K-core sizes on degree-degree correlated networks
Theory vs Numerics:
7-cores in Poisson random graphs with z = 10
r = -0.5
r=0
r = 0.98
Initial damage ρ0
Case r = 0 considered in
S.N. Dorogovtsev et al., PRL 96, 040601 (2006).
Degree-degree correlated networks
Adopt approach of M. Newman for percolation problems
(PRE 67, 026126 (2003), PRL 89, 208701 (2002)).
P(k,k’) – joint PDF that an edge connects vertices with degrees k, k’
………………
…
…
Consider a k-degree node at level n+1:
qn( k1)
n+1
n
…………………..
…
– probability that a k-degree node is active
(conditioned on its parent being inactive)
qn( k ) 
( k )

P
(
k
,
k
)
q
k 
n

k
P (k , k )
– probability that a child
of an inactive k-degree
node is active
Degree-degree correlated networks
q0( k )  r 0( k )
qn( k ) 
q
(k )
n 1
r
(k )
n 1
( k )

P
(
k
,
k
)
q
k 
n
r

(k )
0
r
(k )
0
k
P (k , k )
 (1  r
 k  1 ( k )
 qn
)  
m 0  m 
k 1
(k )
0
 (1  r
 k  (k )
)    qn
m 0  m 
k
(k )
0
  1  q 
( k ) k 1 m
n
m
  1  q 
m
( k ) k m
n
r n  k pk r n(k )
(Also obtain a cascade condition in matrix form).
F m, k 
F m, k 
Degree-degree correlated networks
Theory (curves) vs Numerics (symbols):
7-cores in Poisson random graphs with z = 10
r = -0.5
(zero initial
damage)
r=0
r = 0.98
Pearson correlation r
r

k P(k , k )  
k ,k  kk P(k , k ) 

2
k ,k 

kP(k , k ) 
kP(k , k )
k ,k 
k ,k 
Initial damage ρ0
2
2
Case r = 0 considered in
S.N. Dorogovtsev et al., PRL 96, 040601 (2006).
Correlated networks (105 nodes) generated using Gaussian copula.
Predicting K-cores in CAIDA internet router network
Internet router network structure from www.caida.org
k
pk
k
k
Degree distribution
pk
Degree-degree
correlation matrix
P(k , k )
Predicting K-cores in CAIDA internet router network
Internet router network structure from www.caida.org
Us: Predicted from analysis of
degree distribution and
degree-degree correlation.
Actual size
Predicted from analysis of degree distribution only
(see S.N. Dorogovtsev et al., PRL 96, 040601 (2006)).
Modular networks; asynchronous updating
Similar idea, but instead of P(k,k’) use the mixing matrix e, which
quantifies connections between different communities.
qn(i ) 
( j)
e
q
 j ij n
e
j ij
qn(i)1  g (i ) (qn(i ) )
r n(i)1  h (i ) (qn(i ) )
Asynchronous updating gives
continuous time evolution:
   
  h q   r 
q (i )   g (i ) q (i )  q (i )
r (i )
(i )
(i )
(i )
Modular networks example
Summary
Structure of complex networks
Dynamics on complex networks
Derivation of main result
Extensions and applications
•
•
J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75,
056103 (2007).
J.P. Gleeson, Phys. Rev. E. 77, 046117 (2008).
Cascades on correlated and
modular networks
James P. Gleeson
Department of Mathematics and Statistics,
University of Limerick, Ireland
www.ul.ie/gleesonj
What is best “random” model for the Internet?
Medusa model:
Carmi et al., Proc. Nat. Acad. Sci. ‘07
Jellyfish model:
Siganos et al., J. Comm. Networks ‘06
Internet structure using router data from CAIDA
pk
k
Transmissibility (bond
Occupation probability)
DEPFA Bank collaboration: CDO pricing
m1
m2
p1
p2
m3
p3
m4
p4
S1
0 to 5%
Definitions
mi
S2
10% to 15%
Notional of credit i
pi
S3
15% to 25%
S4
25% to 35%
Default probability
of credit i, (derived
from the CDS
quote).
Sq
mN
pN
{m1, m2,…,mN}
{p1, p2,…,pN}
Correlation
Structure
S5
35% to 100%
Fair price for
protection against
losses in tranche q
Problem
?
{S1,
S2,…,Ss}
Existing models fail
to reproduce the
prices (Sq)
observed on the
market.
An external field
Stochastic Dynamics on Networks
Hysteresis: PRG
Stochastic Dynamics on Networks
Hysteresis: PRG
Stochastic Dynamics on Networks
Stochastic dynamics
Aim: Fundamental understanding of the interactions
between nonlinear dynamical systems and
random fluctuations.
External noise sources
e.g. transistor noise,
thermal noise.
Heterogeneity within system
e.g. agent-based models,
large-scale networks.
Tools:
• Numerical simulations
…guiding fundamental understanding via…
• Asymptotic methods
• Perturbation techniques
• Exact solutions
Noise in oscillators (Theme 1)
Prof. M. P. Kennedy,
Microelectronic Engineering,
UCC
New computational and
asymptotic methods for the
spectrum of an oscillator
subject to white noise
Stochastic perturbation
methods for effects of
coloured noise
Papers: • SIAM J. Appl. Math.
• IEEE TCAS
Collaboration (Feely/Kennedy):
Noise effects in digital phaselocked loops
Microfluidic mixing and sorting (Theme 3)
Experimentalists at Tyndall
National Institute, Cork
Analysis of MHD
micromixing in annular
geometries
Modelling of micro-sorting
methods
Papers: • SIAM J. Appl. Math.
• Phys. Rev E
• Phys. Fluids
Collaborations: (Lindenberg/Sancho)
Noise-induced sorting techniques for
microparticles
Cascades on correlated and
modular networks
James P. Gleeson
Department of Mathematics and Statistics,
University of Limerick, Ireland
www.ul.ie/gleesonj