Complex networks A. Barrat, LPT, Universit

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Complex networksA. Barrat, LPT, Université
Paris-Sud, France
I. Alvarez-Hamelin (LPT, Orsay, France) M.
Barthélemy (CEA, France) L. DallAsta (LPT,
Orsay, France) R. Pastor-Satorras (Barcelona,
Spain) A. Vespignani (LPT, Orsay, France)
http//www.th.u-psud.fr/
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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

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Examples of complex networks

• Internet
• WWW
• Transport networks
• Protein interaction networks
• Food webs
• Social networks
• ...

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Social networksMilgrams experiment
Milgram, Psych Today 2, 60 (1967) Dodds et al.,
Science 301, 827 (2003)
Six degrees of separation
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Small-world propertiesalso in the Internet
Distribution of chemical distances between two
nodes
Average fraction of nodes within a chemical
distance d
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Usual random graphs Erdös-Renyi model (1960)
N points, links with proba p static random graphs
Poisson distribution
(pO(1/N))
BUT...
short distances (log N)
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Clustering coefficient
Clustering My friends will know each other with
high probability! (typical example social
networks)
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Asymptotic behavior
Lattice
Random graph
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In-between Small-world networks
N nodes forms a regular lattice. With probability
p, each edge is rewired
randomly gtShortcuts
N 1000

• Large clustering coeff.
• Short typical path

Watts Strogatz, Nature 393, 440 (1998)
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Size-dependence
p gtgt 1/N gt 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)
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• Is that all we need ?

NO, because...
Random graphs, Watts-Strogatz graphs
are homogeneous graphs (small fluctuations of the
degree k)
While.....
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Airplane route network
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CAIDA AS cross section map
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Topological characterization
P(k) probability that a node has k links
(??? ? ? 3)
Diverging fluctuations
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Exp. vs. Scale-Free
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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

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Origins SF
Two important observations
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BA model
Scale-free model
(1) GROWTH At every timestep we add a new node
with m edges (connected to the nodes already
present in the system). (2) PREFERENTIAL
ATTACHMENT The
probability ? that a new node will be connected
to node i depends on the connectivity ki of that
node
A.-L.Barabási, R. Albert, Science 286, 509 (1999)
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Connectivity distribution
BA network
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More models

• Generalized BA model
• (Redner et al. 2000)
• (Mendes et al. 2000)
• (Albert et al. 2000)

Non-linear preferential attachment ?(k) k?
Initial attractiveness ?(k) Ak?

• Highly clustered
• (Dorogovtsev et al. 2001)
• (Eguiluz Klemm 2002)
• Fitness Model
• (Bianconi et al. 2001)
• Multiplicative noise
• (Huberman Adamic 1999)

Rewiring
(....)
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Tools for characterizing the various models

• Connectivity distribution P(k)
• gtHomogeneous vs. Scale-free
• Clustering
• Assortativity
• ...

gtCompare with real-world networks
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Topological correlations clustering
ki5 ci0.
ki5 ci0.1
i

• aij Adjacency matrix

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Topological correlations assortativity
ki4 knn,i(3447)/44.5
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Assortativity

• Assortative behaviour growing knn(k)
• Example social networks
• Large sites are connected with large sites
• Disassortative behaviour decreasing knn(k)
• Example internet
• Large sites connected with small sites,
  hierarchical structure

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Consequences of the topological heterogeneity

• Robustness and vulnerability
• Propagation of epidemics

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Robustness
Robustness
Complex systems maintain their basic functions
even under errors and
failures
(cell ? mutations Internet ?
router breakdowns)
S fraction of giant component
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Case of Scale-free Networks
Random failure fc 1
(2 lt g ? 3)
s
Attack progressive failure of the most
connected nodes fc lt1
fc
1
Internet maps
R. Albert, H. Jeong, A.L. Barabasi, Nature 406
378 (2000)
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Robust-SF
Failures vs. attacks
1
S
0
1
f
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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
• ...

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Betweenness

• measures the centrality of a node i
• for each pair of nodes (l,m) in the graph, there
  are
• slm shortest paths between l and m
• silm shortest paths going through i
• bi is the sum of silm / slm over all pairs (l,m)

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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
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Epidemic spreading on SF networks

• Natural computer virus
• DNS-cache computer viruses
• Routing tables corruption
• Data carried viruses
• ftp, file exchange, etc.

Internet topology

• Computer worms
• e-mail diffusing
• self-replicating

E-mail network topology
Epidemiology
Air travel topology
Ebel et al. (2002)
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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

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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 ln/d

• Non-equilibrium phase transition
• epidemic threshold critical point
• prevalence r order parameter

rprevalence
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What about computer viruses?

• Very long average lifetime (years!) compared to
  the time scale of the antivirus
• Small prevalence in the endemic case

Long lifetime low prevalence computer viruses
always tuned infinitesimally close to the
epidemic threshold
???
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SIS model on SF networks

• SIS Susceptible Infected Susceptible
• Mean-Field usual approximation all nodes are
  equivalent (same connectivity) gt existence of
  an epidemic threshold 1/ltkgt for the order
  parameter r (density of infected nodes)
• Scale-free structure gt necessary to take into
  account the strong heterogeneity of
  connectivities gt rkdensity of infected nodes of
  connectivity k

gtepidemic threshold
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Epidemic threshold in scale-free networks
ltk2gt ?
?
l c
?
0
Order parameter behavior in an infinite system
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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!!!

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Results can be generalized to generic scale-free
connectivity distributions P(k) k-g

• If 2 lt g ? 3 we have absence of an epidemic
  threshold
• and no critical behavior.
• If 3 lt g ? 4 an epidemic threshold appears,
  but
• it is approached with vanishing slope (no
  criticality).
• If g gt 4 the usual MF behavior is recovered.
• SF networks are equal to random graph.

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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)
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Perspectives Weighted networks

• Scientific collaborations
• Internet
• Emails
• Airports' network
• Finance, economic networks
• ...

gt are weighted networks !!
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Weights examples

• Scientific collaborations

(M. Newman, P.R.E. 2001)
i, j authors k paper nk 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

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Weights

• Weights heterogeneous (broad distributions)?
• Correlations between topology and traffic ?
• Effects of the weights on the dynamics ?

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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)

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• Alain.Barrat_at_th.u-psud.fr
• http//www.th.u-psud.fr/