mathematical modelling and networksghergu/summerschool/gleeson.pdf · 2012-06-07 · mathematical...

Mathematical Modelling and Networks

James Gleeson MACSI,

Dept of Mathematics and Statistics,

University of Limerick

www.ul.ie/gleesonj

james.gleeson@ul.ie

What is MACSI?

• See www.macsi.ul.ie for details of research, vacancies, summer

schools, internships, etc.

• 1-year taught MSc in Mathematical Modelling, with summer research

project.

What is MACSI?

Mathematical Modelling in MACSI

E. S. Benilov, C. P. Cummins and W. T.

Lee, “Why do bubbles in Guinness sink?”

ArXiv:1205.5233 (2012)

What is a network?

A collection of N “nodes” or “vertices”, connected by links or “edges”

Examples:

• World wide web

• Internet

• Social networks

• Networks of neurons

• Coupled dynamical systems

• Bank networks

see, for example, M. E. J. Newman, Networks: an Introduction, OUP 2010

M. E. J. Newman, SIAM Review, 45, 167 (2003)

6

Six Degrees of Kevin Bacon

THE ORACLE

OF BACON

OracleOfBacon.org

Stephanie

Berry

Andy

Garcia

7

Six Degrees of Kevin Bacon

The Untouchables (1987)

The Air I Breathe (2007)

Sandra

Bullock Infamous (2006)

Loverboy (2005)

Finding Forrester (2000)

The Invasion (2007)

Examples of network structure

Examples of network structure

The Erdős–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

0

)1(k

kpkNpz

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 of network structure

[Newman, SIAM Review 2003]

Examples of degree distributions

[Boss et al, 2007]

Examples of degree distributions: directed networks

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)

see, for example, A. Barrat et al., Dynamical Processes on Complex Networks,

CUP 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)

see, for example, A. Barrat et al., Dynamical Processes on Complex Networks,

CUP 2008

Watts` model

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

The fraction of active nodes is:

Threshold dynamics

Updating: 1 if

unchanged otherwise

i i

i

rv

Neighbourhood average: 1

i 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 if is of order 1 at steady state. Us: initially activate a fraction of the nodes, and determine the steady state value of

0

.

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

• J.P. Gleeson and D.J. Cahalane, Phys. Rev. E. 75, 056103 (2007)

Main result

Our result:

with

0 0

1 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 k

k 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

3

0 10 3

0 5 10 2

0 10 ( ) ( )P r r R

!

k z

k

z ep

k

510N

0.18R

Results

( ) ( )P r r R

!

k z

k

z ep

k

0.18R

1.01

610N

random

seeds

targeting high-

degree seeds

Results

4

0 10

2

0 10

( ) ( )P r r R

!

k z

k

z ep

k

Main result

Our result:

with

0 0

1 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 k

k m

kkG q p q q F

mz

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 k

k

k kp F F

z

0 0 (0) 0.F

slope>1

(slope>1)

Simple cascade condition

4

0 10

2

0 10

( ) ( )P r r R

!

k z

k

z ep

k

Extended cascade condition

4

0 10

2

0 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

2

0 10

( ) ( )P r r R

!

k z

k

z ep

k

exp( )kp k k

100

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

A

Main idea: pick a node A at random and calculate its probability of

becoming active. This will give ρ(∞).

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.

Derivation of result

…………………..

∞

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.

nq

1nq

nq

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.

1 0

0(1 )

nq

(initially active)

(initially inactive)

Derivation of result

…………………..

∞

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.

nq

1nq

nq

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.

1 0

0(1 )

nq

1

k

k

p

(initially active)

(initially inactive)

(has degree k; k-1 children)

Derivation of result

…………………..

∞

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.

nq

1nq

nq

(m out of k-1

children active)

mk

n

m

n qqm

k

11

1

k-1 children

Degree distribution of nearest

neighbours:

.kk

k pp

z

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.

1 0

0(1 )

nq

1

k

k

p

(initially active)

(initially inactive)

(has degree k; k-1 children)

(m out of k-1

children active)

(activated by m

active neighbours)

Derivation of result

…………………..

∞

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.

nq

1nq

nq

1

0

k

m

k

mFqq

m

k mk

n

m

n

11

1

k-1 children

k

mFqq

m

kp

k

m

mkm

k

k

01

00 1)1(

k

mFqq

m

kpq

k

m

mk

n

m

n

k

kn

1

0

1

1

001 11~)1(

00 q

Derivation of result

This is a pair approximation theory, 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

Extensions of analytical approach

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

• Directed networks

• Degree-degree correlations

• Modular networks

• Asynchronous updating

• Models of networks with non-zero clustering

Extensions

N

10

N

20

Conclusions: Part I

Developed a tree-based theory to calculate cascades on large

networks without use of Monte-Carlo simulations

• cascade condition gives analytical insight

Described for the Watts threshold model, but also applied to other

types of cascade dynamics

Described for configuration model networks, but also applied to

other random graph ensembles

References: see www.ul.ie/gleesonj

Experiments: an open problem for modellers?

D. Centola,

Science 329,

1194 (2010)

Experiments

D. Centola,

Science 329,

1194 (2010)

Experiments

D. Centola,

Science 329,

1194 (2010)

Nature, Feb 2008

Examples of network analysis

Nature, Feb 2008

Science, July 2009 Science, July 2009

Examples of network analysis

Banking networks and systemic risk

“Can network structure be altered to improve network robustness?

Answering that question is a mighty task for the current generation of

policymakers.”

A. G. Haldane, Executive Director, Financial Stability, Bank of England, in a

speech entitled “Rethinking the Financial Network”, April 2009.

Bank

i

IB loans by

bank i

(assets of

bank i)

IB borrowings

of bank i

(liabilities of

bank i) Creditors

of bank i

Debtors

of bank i

Systemic risk and network models

• E. Nier et al. “Network models and financial stability,” J. Econ. Dyn. Control (2007)

[Bank of England Working Paper No. 346]

• P. Gai and S. Kapadia, “Contagion in financial networks,” Proc. R. Soc. A (2010)

[Bank of England Working Paper No. 383]

• RM May and N Arinaminpathy, “Systemic risk: the dynamics of model banking

systems,” J. R. Soc. Interface (2009)

liabilities of bank i

IB borrowings

of bank i IB loans by

bank i

assets of bank i

deposits

“net worth”

external assets

“shock” “net worth”

shocks

transmitted

to creditors

of bank i

ia

ia

Network topology, loans, shocks

j k

ij

GK model ij

1

ij

1

Total IB assets of each bank sum

to 1. Each asset loan is of size ij

1

Zero recovery. Default occurs when number m of defaulted debtors

satisfies , so a directed-network version of Watts’ model. ij

m

ik

Nier et al

model

wTotal IB assets of each bank depends

on j. Each loan is of fixed size w

Non-zero recovery. Shock transmitted from default of bank i is

w

k

as

i

ii ,minin

w

ij

[c.f. Eisenberg and Noe

Management Sci., 2001]

w

pjk : probability a random node (bank) has j debtors

(asset loans, in-degree) and k creditors (liability

loans, out-degree)

Nier et al. Results of Monte-Carlo simulations

• E. Nier et al. “Network models and financial stability,” J. Econ. Dyn. Control

(2007) [Bank of England Working Paper No. 346]

2.0,25 pN

Erdős-Rényi

random graph,

mean degree z=5.

Our results: Nier et al model

zk

zj

jk ek

ze

j

zp

!!

25N

Small network; Erdős–Rényi random graph.

Randomly chosen initial seed.

7.1 kCp jkjk

200N

50maxseed kk

Results: Nier et al model

Larger network; skew degree distribution.

Target largest bank as initial seed.

7.1 kCp jkjk

200N

Results: Nier et al model

Larger network; skew degree distribution.

Dependence of cascade size on degree of initial seed.

maxseed 30 kk

7.1 kCp jkjk

200N

50maxseed kk

min

in

crita

s

Results: Nier et al model

Larger network; skew degree distribution.

Dependence of cascade size on degree of initial seed.

7.1 kCp jkjk

200N

min

in

crita

s

Results: Nier et al model

Larger network; skew degree distribution.

Dependence of cascade size on degree of initial seed.

Summary and references

• E. Nier et al. “Network models and financial stability,” J. Econ. Dyn. Control (2007)

[Bank of England Working Paper No. 346]

• P. Gai and S. Kapadia, “Contagion in financial networks,” Proc. Roy. Soc. A (2010)

[Bank of England Working Paper No. 383]

• R. M. May and N Arinaminpathy, “Systemic risk: the dynamics of model banking

systems,” J. R. Soc. Interface (2009)

• Our work: see www.ul.ie/gleesonj

Certain classes of cascade dynamics can be solved (semi-)

analytically on random network models

We have shown how two models for systemic risk in banking

networks (Nier et al and Gai & Kapadia) may be analysed without

use of Monte-Carlo simulations

Our methods may be extended to other (less stylized) models,

and used to consider amelioration strategies for default contagion

Overall summary

Structure of complex networks

Threshold dynamics

Experiments

Banking networks and systemic risk

Further reading:

M. E. J. Newman, Networks: an Introduction, OUP 2010

M. E. J. Newman, SIAM Review, 45, 167 (2003)

Interested?

See www.macsi.ul.ie for details of research,

vacancies, summer schools, internships, etc.

1-year taught MSc in Mathematical Modelling, with

summer research project.

For more on networks research:

www.ul.ie/gleesonj

Further reading:

M. E. J. Newman, Networks: an Introduction, OUP 2010

M. E. J. Newman, SIAM Review, 45, 167 (2003)

Adam Hackett, UL

Diarmuid Cahalane, Cornell

Sergey Melnik, UL

Davide Cellai, UL

Jonathan Ward, Reading

Mason Porter, Oxford

Peter Mucha, U. North Carolina

Rick Durrett, Duke

Science Foundation Ireland

MACSI: Mathematics Applications

Consortium for Science &

Industry

IRCSET Inspire

Collaborators and funding

Seeking PhD students and (soon)

postdoctoral researchers: see

www.ul.ie/gleesonj

Mathematical Modelling and Networks

James Gleeson MACSI,

Dept of Mathematics and Statistics,

University of Limerick

www.ul.ie/gleesonj

james.gleeson@ul.ie