Cascades on correlated and modular networks James P. Gleeson Department of Mathematics and Statistics, University of Limerick, Ireland

Cascades on correlated and modular networks

James P. GleesonDepartment 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







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:

!k

zep

kz

k

0k

k

z k p

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]


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:

kpk ~


Examples

[Newman, SIAM Review 2003]

Overview







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

Structures and dynamics review see: • M.E.J. Newman, SIAM Review 45, 167 (2003).• S.N. Dorogovtsev et al., arXiv:0705.0010 (2007)

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)

Initially small localized effects can propagate over the whole network, causing a global cascade

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

The nodes:

• are labelled i , i from 1 to N;

• have a state ;

• and a threshold ri from some distribution.

{0,1}ija

jiij aa

}1,0{)( tvi

The fraction of nodes in state vi=1 is (t):

Threshold dynamics

Updating: 1 if

unchanged otherwisei i

i

rv

Neighbourhood average:

1i ij j

ji

a vk

}1,0{)( tviNode i has state

irand threshold

Watts` model

D.J. Watts, Proc. Nat. Acad. Sci. 99, 5766 (2002).

Watts` model

R

Cascade condition: 1

1

( 1)1k k

k

k kp F

z

Thresholds CDF: ( ) ( )r

F r P s ds

( ) ( )P r r R !

k z

k

z ep

k

Watts` model

Watts: initially activate single node (of N), determine ifat steady state.

Us: initially activate a fraction of the nodes, anddetermine the steady state value of

1

0.

Conditions for global cascades (and dependence on thesize of the seed fraction) follow…

Main result

Our result:

with

0 01 0

(1 ) (1 )k

m k m mk k

k m

kp q q F

m

1 0 0(1 ) ( ),n nq G q 0 0 ,q

and

1

1

1 0

1( ) (1 )

km k m m

k kk m

kkG q p q q F

mz

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

30 10

30 5 10

20 10

( ) ( )P r r R !

k z

k

z ep

k

510N

0.18R

Results

40 10

20 10

( ) ( )P r r R

!

k z

k

z ep

k

Main result

Our result:

with

0 01 0

(1 ) (1 )k

m k m mk k

k m

kp q q F

m

1 0 0(1 ) ( ),n nq G q 0 0 ,q

and

1

1

1 0

1( ) (1 )

km k m m

k kk m

kkG q p q q F

mz

Cascade condition

q

0.25 0.5 0.75 1 1.25 1.5

0.25

0.5

0.75

1

1.25

1.5

1 0 0(1 ) ( ),n nq G q 0 0 ,q

G(q)

Cascade condition

q

0.25 0.5 0.75 1 1.25 1.5

0.25

0.5

0.75

1

1.25

1.5

1 0 0(1 ) ( ),n nq G q 0 0 ,q

G(q)

nq

1nq

slope=1

Simple cascade condition

First-order cascade condition: using

demand

1 0 0(1 ) ( ),n nq G q 0 0 ,q

for global cascades to be possible. This yields the condition

reproducing Watts’ percolation result when and

0(1 ) (0) 1G

1

1 0

( 1) 1(0) ,

1k kk

k kp F F

z

0 0 (0) 0.F

slope>1

(slope>1)

Simple cascade condition

40 10

20 10

( ) ( )P r r R

!

k z

k

z ep

k

nq

1nq

slope=1

Extended cascade condition

Second-order cascade condition: expand

to second order and demand no positive zeros of the quadratic

1 0 0(1 ) ( ),n nq G q 0 0 ,q

for global cascades to be possible.

The extension is, to first order in :

2 20(0) 1 2 (0) (0) 2 (0) (0) (0) 2 (0) (0) 0.G G G G G G G G

0

above

02 cbqaq

Extended cascade condition

40 10

20 10

( ) ( )P r r R

!

k z

k

z ep

k

R

Gaussian threshold distribution

0.05

0.2

2

22

1 ( )( ) exp

22

r RP r

!

k z

k

z ep

k

0 0

Gaussian threshold distribution

0.05

0.2

2

22

1 ( )( ) exp

22

r RP r

!

k z

k

z ep

k

0.362

0.2

0.38

R

R

R

510N

0 0

Bifurcation analysis

0.35R

0.371R

0.375R

1 0 0(1 ) ( ),n nq G q

( ) 0q G q

0 0; 0.2

Results: Scale-free networks

( ) ( )P r r R

exp( )kp k k 100

0

0

2

310

10

2

22

1 ( )( ) exp

22

r RP r

0 5

0.2

.4

10.5z

0 0 6z

Results: Scale-free networks

20 10

( ) ( )P r r R

!

k z

k

z ep

k

exp( )kp k k

100

Overview







Consider undirected unweighted network of N nodes (N is large) defined by degree distribution pk

Watts` model of global cascades

Updating: node i becomes active if the active fraction of its neighbours exceeds its threshold

Each node i has:

• binary state

• fixed threshold given by thresholds CDF

Initially activate fraction ρ0<<1 of N nodes.

The average fraction of active nodes

( ) ( )r

F r P s ds

(probability that a node has threshold < r)

otherwise unchanged

if ,1 ii

i

i

rk

mv

N

i iN tvt1

1 )()(

}1,0{iv

ir

Derivation: Generalizing zero-temperature random-field Ising modelresults from Bethe lattices (D. Dhar, P. Shukla, J.P. Sethna, J. Phys. A 30, 5259 (1997)) to arbitrary-degree random networks.

Derivation of result

A

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.


…………………..

∞

n+2

n+1

n

…………………

…

……

A

…

: probability that a node on level n is active, conditioned on its parent (on level n+1) being inactive.

nq1nq

nq



1 0

0(1 )nq

(initially active)

(initially inactive)


…………………..

∞

n+2

n+1

n

…………………

…

……

A

…


nq1nq

nq



1 0

0(1 )nq

1k

k

p

(initially active)


(has degree k; k-1 children)


…………………..

∞

n+2

n+1

n

…………………

…

……

A

…


nq1nq

nq

(m out of k-1 children active)

mkn

mn qq

m

k

111

k-1 children

Degree distribution of nearest neighbours:

.kk

k pp

z



1 0

0(1 )nq

1k

k

p

(initially active)


(has degree k; k-1 children)

(m out of k-1 children active)

(activated by m active neighbours)


…………………..

∞

n+2

n+1

n

…………………

…

……

A

…


nq1nq

nq

1

0

k

m

k

mFqq

m

k mkn

mn

111

k-1 children

k

mFqq

m

kp

k

m

mkm

kk

0100 1)1(

k

mFqq

m

kpq

k

m

mkn

mn

kkn

1

0

1

1001 1

1~)1(

00 q


Valid when:

(i) Network structure is locally tree-like (vanishing clustering coefficient).

(ii) The state of each node is altered at most once.

Our result for the average fraction of active nodes

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







Extensions

• Generalized dynamics:• SIR-type epidemics• Percolation • K-core sizes

• Degree-degree correlations • Modular networks

• Asynchronous updating

• Non-zero clustering

k

mFqq

m

kp

k

m

mkm

kk

0100 1)1(

k

mFqq

m

kpq

k

m

mkn

mn

kkn

1

0

1

1001 1

1~)1(

00 q



Generalization to other dynamical models

k

mCDFkmF thr,Fraction of active neighbours (Watts):

mCDFkmF thr,Absolute number of active neighbours:

mpkmF )1(1, Bond percolation:

0m if,

0m if ,0,

kQkmFSite percolation:

kmFqqm

kp

k

m

mkm

kk ,1)1(

0100

kmFqqm

kpq

k

m

mkn

mn

kkn ,1

1~)1(1

0

1

1001

00 q


Kmk

KmkkmF

if,1

if ,0,

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:

Final inactive fraction (1- ρ) of the total network gives the size of K-core

Generalization to other dynamical models

kmFqqm

kp

k

m

mkm

kk ,1)1(

0100

kmFqqm

kpq

k

m

mkn

mn

kkn ,1

1~)1(1

0

1

1001

00 q


K-core sizes on degree-degree correlated networks

Initial damage ρ0

r = 0

r = -0.5

r = 0.98

Theory vs Numerics:7-cores in Poisson random graphs with z = 10

Case r = 0 considered in S.N. Dorogovtsev et al., PRL 96, 040601 (2006).

Adopt approach of M. Newman for percolation problems (PRE 67, 026126 (2003), PRL 89, 208701 (2002)).

Degree-degree correlated networks

P(k,k’) – joint PDF that an edge connects vertices with degrees k, k’

)(1

knq – probability that a k-degree node is active

(conditioned on its parent being inactive)

– probability that a child of an inactive k-degree node is active

k

knkk

n kkP

qkkPq

),(

),( )()(

n+1

…………………..

………………

…

…… Consider a k-degree node at level n+1:

n

kmFqqm

kq

k

m

mkkn

mkn

kkkn ,1

1)1(

1

0

1)()()(0

)(0

)(1

)(0

)(0

kkq

k

knkk

n kkP

qkkPq

),(

),( )()(

kmFqqm

kk

m

mkkn

mkn

kkkn ,1)1(

0

)()()(0

)(0

)(1

)(knk kn p


(Also obtain a cascade condition in matrix form).

Pearson correlation r


Initial damage ρ0

r = 0

r = -0.5

r = 0.98

Correlated networks (105 nodes) generated using Gaussian copula.

Theory (curves) vs Numerics (symbols):7-cores in Poisson random graphs with z = 10

kk kk

kkkk

kkkPkkPk

kkkPkkPkkr

,

2

,

2

2

,,

),(),(

),(),( Case r = 0 considered in S.N. Dorogovtsev et al., PRL 96, 040601 (2006).

(zero initial damage)

Predicting K-cores in CAIDA internet router network

Internet router network structure from www.caida.org

Degree distributionDegree-degree

correlation matrix

k

k

k

( , )P k kkp

kp

http://www.caida.org/

Predicting K-cores in CAIDA internet router network

Predicted from analysis of degree distribution only (see S.N. Dorogovtsev et al., PRL 96, 040601 (2006)).

Actual size

Us: Predicted from analysis of degree distribution and degree-degree correlation.

Internet router network structure from www.caida.org

http://www.caida.org/

Similar idea, but instead of P(k,k’) use the mixing matrix e, which quantifies connections between different communities.

Modular networks; asynchronous updating

j ij

jnj iji

n e

qeq

)(

)(

)( )()()(1

in

iin qh

Asynchronous updating gives continuous time evolution:

)()()()( iiii qh

)( )()()(1

in

iin qgq

)()()()( iiii qqgq

Modular networks example

Summary










www.ul.ie/gleesonj

What is best “random” model for the Internet?

Jellyfish model:Siganos et al., J. Comm. Networks ‘06

Medusa model:Carmi et al., Proc. Nat. Acad. Sci. ‘07

Internet structure using router data from CAIDA

kp

k

Transmissibility (bond Occupation probability)

DEPFA Bank collaboration: CDO pricing

m1 p1

mN pN

m2 p2

m3 p3

m4 p4

Definitions

mi

Notional of credit i

pi

Default probability of credit i, (derived from the CDS quote).

Sq

Fair price for protection against losses in tranche q

ProblemExisting models fail to reproduce the prices (Sq) observed on the market.

{m1, m2,…,mN}{p1, p2,…,pN} {S1, S2,

…,Ss}Correlation Structure

?

0 to 5%

10% to 15%

15% to 25%

25% to 35%

S1

S2

S3

S4

S5 35% to 100%

An external fieldStochastic Dynamics on Networks

Hysteresis: PRGStochastic Dynamics on Networks

Hysteresis: PRGStochastic 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

Collaboration (Feely/Kennedy):Noise effects in digital phase-

locked loops

Papers: • SIAM J. Appl. Math.• IEEE TCAS

Microfluidic mixing and sorting (Theme 3)

Experimentalists at Tyndall National Institute, Cork

Analysis of MHD micromixing in annular geometries

Modelling of micro-sortingmethods

Collaborations: (Lindenberg/Sancho)Noise-induced sorting techniques for

microparticles

Papers: • SIAM J. Appl. Math.• Phys. Rev E• Phys. Fluids




www.ul.ie/gleesonj

Documents

Cascades on correlated and modular networks James P. Gleeson Department of Mathematics and Statistics, University of Limerick, Ireland