Transcript Weighted networks: analysis, modeling A. Barrat, LPT
Complex networks
A. Barrat, LPT, Université Paris-Sud, France I. Alvarez-Hamelin (LPT, Orsay, France) M. Barthélemy (CEA, France) L. Dall’Asta (LPT, Orsay, France) R. Pastor-Satorras (Barcelona, Spain) A. Vespignani (LPT, Orsay, France) http://www.th.u-psud.fr/
Plan of the talk
● Complex networks: examples ● Small-world networks ● Scale-free networks: evidences, modeling, tools for characterization ● Consequences of SF structure ● Perspectives: weighted complex networks
Examples of complex networks
●
Internet
●
WWW
●
Transport networks
●
Protein interaction networks
●
Food webs
●
Social networks
● ...
Social networks: Milgram’s experiment
Milgram, Psych Today 2, 60 (1967) Dodds et al., Science 301, 827 (2003)
“Six degrees of separation”
Small-world properties: also in the Internet
Distribution of chemical distances between two nodes Average fraction of nodes within a chemical distance d
Usual random graphs: Erd
ös-Renyi model (1960)
N points, links with proba p: static random graphs Poisson distribution (p=O(1/N))
short distances (log N) BUT...
Clustering coefficient n 3 1 2 Higher probability to be connected
C = # of links between 1,2,…n neighbors n(n-1)/2
Clustering
: My friends will know each other with high probability!
(typical example: social networks)
Asymptotic behavior
Lattice
L
(
N
)
N
1 /
d
C
(
N
)
const
.
Random graph
L
(
N
) log
N C
(
N
)
N
1
In-between: Small-world networks N nodes forms a regular lattice. With probability p, each edge is rewired randomly =>Shortcuts
N = 1000 •Large clustering coeff.
•Short typical path Watts & Strogatz, Nature 393, 440 (1998)
Size-dependence p >> 1/N => Small-world structure
Amaral & Barthélemy Phys Rev Lett 83, 3180 (1999) Newman & Watts, Phys Lett A 263, 341 (1999) Barrat & Weigt, Eur Phys J B 13, 547 (2000)
Is that all we need ?
NO, because
...
Random graphs, Watts-Strogatz graphs are homogeneous graphs (small fluctuations of the degree k): While.....
Airplane route network
CAIDA AS cross section map
Topological characterization
•
The Internet and the World-Wide-Web
•
Protein networks
P(k) =
probability that a node has k links
•
Metabolic networks
•
Social networks
•
Food-webs and ecological networks
P(k) ~ k -
(
• •
Are
3) Scale-free properties
Diverging fluctuations
Exp. vs. Scale-Free
Poisson distribution Power-law distribution Exponential Network Scale-free Network
Main Features of complex networks
•
Many interacting units
•
Self-organization
•
Small-world
•
Scale-free heterogeneity
•
Dynamical evolution Standard graph theory
Random graphs •
Static
•
Ad-hoc topology
Origins SF
Two important observations
(1) The number of nodes (N) is NOT fixed.
Networks continuously expand by the addition of new nodes Examples: WWW : addition of new documents Citation : publication of new papers (2) The attachment is NOT uniform.
A node is linked with higher probability to a node that already has a large number of links.
Examples : WWW : Citation : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) well cited papers are more likely to be cited again
Scale-free model (1) GROWTH :
A t every timestep we add a new node with
m
edges (connected to the nodes already present in the system).
(2) PREFERENTIAL ATTACHMENT
The probability
Π :
that a new node will be connected to node
i
depends on the connectivity
k i
of that node (
k i
)
k i j k j
P(k) ~k -3
A.-L.Barabási, R. Albert, Science
286,
509 (1999)
Connectivity distribution BA network
More models
•
Generalized BA model Non-linear preferential attachment :
(k) ~ k
(Redner et al. 2000) Initial attractiveness :
(k) ~ A+k
(Mendes et al. 2000) Rewiring (Albert et al. 2000)
•
Highly clustered (Dorogovtsev et al. 2001) (Eguiluz & Klemm 2002)
(
k i
)
i
k i j j k j
•
Fitness Model (Bianconi et al. 2001) (....)
•
Multiplicative noise (Huberman & Adamic 1999)
Tools for characterizing the various models
●
Connectivity distribution P(k) =>Homogeneous vs. Scale-free
●
Clustering
●
Assortativity
●
...
=>Compare with real-world networks
Topological correlations: clustering
i
k i i =5 c i i =0.
a ij : Adjacency matrix
Topological correlations: assortativity
k=4 i k=4 k=7 k=3
k i =4 k nn,i =(3+4+4+7)/4=4.5
Assortativity
● Assortative behaviour: growing k nn (k)
Example: social networks Large sites are connected with large sites
● Disassortative behaviour: decreasing k nn (k)
Example: internet Large sites connected with small sites, hierarchical structure
Consequences of the topological heterogeneity
●
Robustness and vulnerability
●
Propagation of epidemics
Robustness
Complex systems maintain their basic functions (cell even under errors and failures mutations; Internet router breakdowns) 1 S: fraction of giant component
S
f c
0 1
Fraction of removed nodes,
f
node failure
s Case of Scale-free Networks
f c
Random failure
f c
=1 (
3) Attack =progressive failure of the most connected nodes
f c
<1 1 Internet maps
R. Albert, H. Jeong, A.L. Barabasi, Nature
406
378 (2000)
Failures vs. attacks Failures Topological error tolerance S
1 0
f c f
3 :
f c
=1 (R. Cohen et al PRL, 2000) 1
Attacks
Other attack strategies
● Most connected nodes ● Nodes with largest betweenness ● Removal of links linked to nodes with large k ● Removal of links with largest betweenness ● Cascades ● ...
j
Betweenness
measures the “centrality” of a node i: for each pair of nodes (l,m) in the graph, there are s lm shortest paths between l and m b i s i lm shortest paths going through i is the sum of s i lm / s lm over all pairs (l,m) i b i b j is large is small
Other attack strategies
● Most connected nodes ● Nodes with largest betweenness ● Removal of links linked to nodes with large k ● Removal of links with largest betweenness ● Cascades ● ...
Problem of reinforcement ?
P. Holme et al., P.R.E
65
(2002) 056109 A. Motter et al., P.R.E
66
(2002) 065102, 065103 D. Watts, PNAS
99
(2002) 5766
Epidemic spreading on SF networks
Natural computer virus •DNS-cache computer viruses •Routing tables corruption Data carried viruses •ftp, file exchange, etc.
Computer worms •e-mail diffusing •self-replicating Epidemiology Internet topology E-mail network topology Air travel topology
Mathematical models of epidemics Coarse grained description of individuals and their state
•
Individuals exist only in few states:
•
Healthy or Susceptible * Infected * Immune * Dead
•
Particulars on the infection mechanism on each individual are neglected.
Topology of the system: the pattern of contacts along which infections spread in population is identified by a network
•
Each node represents an individual
•
Each link is a connection along which the virus can spread
SIS model: •
Each node is infected with rate
n
if connected to one or more infected nodes
•
Infected nodes are recovered (cured) with rate
d without loss of generality d =1 (sets the time scale) •
Definition of an effective spreading rate
l
=
n/d r
Absorbing phase Virus death
r
=prevalence The epidemic threshold is a general result Active phase Finite prevalence
l c l •
Non-equilibrium phase transition
•
epidemic threshold = critical point
•
prevalence
r
=order parameter The question of thresholds in epidemics is central (in particular for immunization strategies)
What about computer viruses?
● Very long average lifetime (years!) compared to ●
Absorbing
Small prevalence in the endemic case
Finite prevalence Virus death
l c l
Computer viruses ???
Long lifetime + low prevalence = computer viruses always tuned infinitesimally close to the epidemic threshold ???
SIS model on SF networks
SIS= Susceptible – Infected – Susceptible Mean-Field usual approximation: all nodes are “equivalent” (same connectivity) => existence of an epidemic threshold 1/
k =density of infected nodes of connectivity k =>epidemic threshold
l c
=
Epidemic threshold in scale-free networks
l c
=
l c
0 Order parameter behavior in an infinite system
Rationalization of computer virus data
•Wide range of spreading rate with low prevalence (no tuning) •Lack of healthy phase = standard immunization cannot drive the system below threshold!!!
Results can be generalized to generic scale-free connectivity distributions P(k)~ k -
•
If
3 we have absence of an epidemic threshold and no critical behavior .
•
If
3
4 an epidemic threshold appears, but it is approached with vanishing slope ( no criticality ).
•
If
>
4 the usual MF behavior is recovered.
SF networks are equal to random graph.
Main results for epidemics spreading on SF networks
•
Absence of an epidemic/immunization threshold
•
The network is prone to infections (endemic state always possible)
•
Small prevalence for a wide range of spreading rates
•
Progressive random immunization is totally ineffective
•
Infinite propagation velocity
Very important consequences of the SF topology!
(NB: Consequences for immunization strategies) Pastor-Satorras & Vespignani (2001, 2002), Boguna, Pastor-Satorras, Vespignani (2003), Dezso & Barabasi (2001), Havlin et al. (2002), Barthélemy, Barrat, Pastor-Satorras, Vespignani (2004)
Perspectives: Weighted networks
● Scientific collaborations ● Internet ● Emails ● Airports' network ● Finance, economic networks ● ...
=> are weighted networks !!
Weights: examples
● Scientific collaborations
:
(M. Newman, P.R.E. 2001) i, j: authors; k: paper; n
k
: number of authors : 1 if author i has contributed to paper k ● Internet, emails: traffic, number of exchanged emails ● Airports: number of passengers for the year 2002
Weights
●
Weights: heterogeneous (broad distributions)?
●
Correlations between topology and traffic ?
●
Effects of the weights on the dynamics ?
Weights: recent works and perspectives
● Empirical studies (airport network; collaboration network: PNAS 2004) ● New tools (PNAS 2004) ● strength ● weighted clustering coefficient (vs. clustering coefficient) ● weighted assortativity (vs. assortativity) ● New models (PRL 2004) ● New effects on dynamics (resilience, epidemics...) on networks (work in progress)
http://www.th.u-psud.fr/