Spreading dynamics on
small-world networks with a
power law degree distribution
Alexei Vazquez
The Simons Center for Systems Biology
Institute for Advanced Study
Epidemic outbreak
External source
Population
Population structure
Contact graph
N individuals
pk connectivity distribution
D average distance
Sexual contacts
Sexually transmitted diseases Sweden
1 year
-1
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pk~k-  , 2<<5
•Liljeros et al. Nature (2001)
•Jones & Handcook, Nature (2003)
•Schneeberger et al, Sex Transm Dis (2004)
lifetime
-1
Sexual contacts
STD
Colorado Springs HIV network
Potterat et al, Sex. Transm Infect 2002
k -2
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N=250
D8
city
Physical contact or proximity
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nation/world
Eubank et al, Nature 2004
1 day
Portland
k- 1.8
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USA
Barrat et al, PNAS 2004
N=3,880
D4.37
Physical contact or proximity
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Branching process model
Spanning tree
Generation
0
root
1
2
3
4
kpk/<k>
k-1
k
pk
Branching process model
Timming
d
d+1
generation
Generation time
T
Distribution
G()=Pr(T)
t
t+T1
t+T2
t+T3
time
Branching process model
1. The process start with a node (d=0) that
generates k sons with probability distribution pk.
2. Each son at generation 0<d<D generates k-1 new
sons with probability kpk/<k>.
3. Nodes at generation D does not generate any
son.
4. The generation times are independent random
variables with distribution function G().
Note: Galton-Watson, Newman
Bellman-Harris, Crum-Mode-Jagers
Recursive calculation
t=0
d
t=0
d
T1
T2
d+1
Results
Constant transmission rate :
G()=1-e-
Incidence I(t):
expected rate of new infections at time t
 e
I(t) ~ 
D1  t
N
(

t)
e

( R˜ 1)t
t  t 0
t  t 0
k(k 1)
˜
R
k
t0 
D 1 1
R˜ 
Reproductive number
Time scale


Vazquez, Phys. Rev. Lett. 2006
pk~k -, kmax~N

 ln N
t 0 ~ ln ln N

N (3 )
>3,

<3,

3
3
t<<t0 (t0 when N )
I(t) ~ e
( R˜ 1)t
t>>t0 (t00 when N )
I(t) ~ N(  t) D1 e t
Vazquez, Phys. Rev. Lett. 2006
Numerical simulations
• Network: random graph with a given degree
distribution.
• pk~k -
• Constant transmission rate 
• N=1000, 10000, 100000
• 100 graph realizations, 10000 outbreaks
Numerical simulations
log-log
linear-log
I(t)/N
I(t)/N
e(K-1)t
tD-1e-t
t
 1,000
 10,000
 100,000
t
Cumulative number
Case study: AIDS epidemics
t2
t3
t3
t3
 New York - HOM
 New York - HET
 San Francisco - HOM
 South Africa
 Kenya
t2
 Georgia
 Latvia
 Lithuania
exponential
t (years)
Szendroi & Czanyi, Proc. R Soc. Lond. B 2004
Generalizations
Degree correlations
Multitype
Degree correlations
k




q( k | k) 



k’
k pk 
uncorrelated
k
k pk 
correlated
k
  0 disassortative



K k   q( k | k)( k 1) ~ k    0 uncorrelated
  0
k 
assortative


Degree correlations
Kk
Kk
k
k
Degree correlations
k




q( k | k) 



k’
k pk 
uncorrelated
k
k pk 
correlated
k
  0 disassortative



K k   q( k | k)( k 1) ~ k    0 uncorrelated
  0
k 
assortative

N( t)D-1e-t

 e(R*-1)t

Vazquez, Phys. Rev. E 74, 056101 (2006)
Multi-type
i=1,…,M types
Ni
p(i)k
eij
D
number of type i agents
type i degree distribution
mixing matrix
average distance
Reproductive number matrix
R˜ ij 
k i ki 1
ki
eij
: largest eigenvalue
Multi-type
 Type 1
 Type 2
 Type 3
 Type 4
Type-network
eij
eii
Strongly connected type-networks
 e(  1)t
I(t) ~ 
D1  t
N (t) e
t  t 0
t  t 0
t0 
D 1 1
 

Vazquez, Phys. Rev. E (In press); http://arxiv.org/q-bio.PE/0605001
Generalizations
Non-exponential
generating time distributions
Intermediate states
 1  

(

)
e
Ý
g( )  G( ) 
( )
 e( R˜ 1)t
I(t) ~ 
D1  t
N(t) e
t  t0
t  t0
Vazquez, DIMACS Series in Discrete Mathematics… 70, 163 (2006)
Long time behavior: Email worms
Receive infected email
Sent infected emails
time
generating time (residual waiting time)
Generating time probability density
g(t) 
1


 dP( )
t
In collaboration with R. Balazs, L. Andras and A.-L. Barabasi
Email activity patterns
Left:
T
University server
3,188 users
129,135 emails sent
<  >~1 day
E~25 days
T
Right:
  
exp 
  E 

  
exp 
  E 
Comercial email server
~1,7 millions users
~39 millions emails sent
<  >~4 days
E~9 months
Incidence: model
 1
 t 
 exp  Poisson model
 
 

g(t)  
A(t)exp t 
email data



  E 


   E

 t 
FPoisson(t)exp  Poisson model

 

I(t)  
 F (t)exp t 
email data


email

  E 

Prevalence: http://www.virusbtn.com
I(t)
I(t)
I(t)
Prevalence data
Decay time ~ 1 year
Poisson model
<  >~1 day - University
<  >~4 days - Comercial
Email data
E~25 days - University
E~9 months - Comercial
Conclusions
• Truncated branching processes are a suitable
framework to model spreading processess on real
networks.
• There are two spreading regimes.
– Exponential growth.
– Polynomial growth followed by an exponential
decay.
• The time scale separating them is determined by D/R.
• The small-world property and the connectivity
fluctuations favor the polynomial regime.
• Intermediate states favor the exponential regime.