Mathematical Modelling and Networksghergu/SummerSchool/Gleeson.pdf · 2012-06-07 · Mathematical...
Transcript of 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
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