dynamical processes on complex networks · books and reviews on processes on complex networks •...

Dynamical processes oncomplex networks

Marc BarthélemyCEA, IPhT, France

Lectures IPhT 2010

Outline I. Introduction: Complex networks

1. Example of real world networks and processes

II. Characterization and models of complex networks

1. Tool: Graph theory and characterization of large networks Diameter, degree distribution, correlations (clustering, assortativity), modularity

2. Some empirical results

3. Models Fixed N: Erdos-Renyi random graph, Watts-Strogatz, Molloy-Reed algorithm,Hidden variables Growing networks: Barabasi-Albert and variants (copy model, weighted, fitness)

III. Dynamical processes on complex networks

1. Overview: from physics to search engines and social systems

2. Percolation

3. Epidemic spread: contact network and metapopulation models

• Statistical mechanics of complex networksReka Albert, Albert-Laszlo BarabasiReviews of Modern Physics 74, 47 (2002)cond-mat/0106096

• The structure and function of complex networksM. E. J. Newman, SIAM Review 45, 167-256 (2003)cond-mat/0303516

• Evolution of networksS.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)cond-mat/0106144

References: reviews on complex networks

References: books on complex networks

• Evolution and structure of the Internet: A statistical PhysicsApproachR. Pastor-Satorras, A. VespignaniCambridge University Press, 2004.

• Evolution of networks: from biological nets to the Internet andWWWS.N. Dorogovtsev, J.F.F. MendesOxford Univ. Press, 2003.

• Scale-free networksG. CaldarelliOxford Univ. Press, 2007.

Books and reviews on processes oncomplex networks

• Complex networks: structure and dynamicsS. Boccaletti et al.Physics Reports 424, 175-308 (2006).

• Dynamical processes on complex networksA. Barrat, M. Barthélemy, A. VespignaniCambridge Univ. Press, 2008.

What is a network ?

Network=set of nodes joined by links G=(V,E)

- very abstract representation- very general

- convenient to describe many different systems:biology, infrastructures, social systems, …

Networks and Physics

• Complex

Most networks of interest are:

• Very large

Statistical tools needed ! (see next chapter)

Studies on complex networks (1998- )

• 1. Empirical studies Typology- find the general features

• 2. Modeling Basic mechanisms/reproducing stylized facts

• 3. Dynamical processesImpact of the topology on the properties of dynamicalprocesses: epidemic spread, robustness, …

Empirical studies: Unprecedentedamount of data…..

Transportation infrastructures (eg. BTS)

Census data (socio-economical data)

Social networks (eg. online communities)

Biological networks (-omics)

Empirical studies: sampling issues

Social networks: various samplings/networks Transportation network: reliable data Biological networks: incomplete samplings Internet: various (incomplete) mapping processes WWW: regular crawls …

possibility of introducing biases in themeasured network characteristics

Networks characteristics

Networks: of very different origins

Do they have anything in common?Possibility to find common properties?

- The abstract character of the graph representationand graph theory allow to give some answers…- Important ingredients for the modeling

Modeling complex networks

Microscopical processes

Properties at the macroscopic level

Statistical physics

• many interacting elements• dynamical evolution• self-organisation

Non-trivial structureEmergent properties, cooperative

phenomena

Networks inInformation technology

Networks in Information technology:processesImportance of Internet and the web

Congestion Virus propagation Cooperative/social phenomena (online

communities, etc.) Random walks, search (pagerank algo,…)

- Nodes=routers- Links= physical connections

different granularities

Internet

Many mapping projects (topology and performance): CAIDA, NLANR, RIPE, …(http://www.caida.org)

Internet mapping

• continuously evolving and growing• intrinsic heterogeneity• self-organizing

Largely unknown topology/properties

Nodes: Computers, routersLinks: physical lines

Large-scale visualization

Internet backbone

Internet-map

World Wide Web

Over 1 billion documents

ROBOT: collects all URL’sfound in a document andfollows them recursively

Nodes: WWW documentsLinks: URL links

Virtual network to find and shareinformations

Social networks

Social networks: processes

Many social networks are the support of somedynamical processes

Disease spread Rumor propagation Opinion/consensus formation Cooperative phenomena …

Scientific collaboration network

Nodes: scientistsLinks: co-authored papers

Weights: depending on• number of co-authored papers• number of authors of each paper• number of citations…

Citation network Nodes: papers

Links: citations

Science citation indexS. Redner

3212

33

Hopfield J.J., PNAS1982

Actor’s network Nodes: actorsLinks: cast jointly

N = 212,250 actors〈k〉 = 28.78

John Carradine

The Sentinel

(1977)

The story of

Mankind (1957)

http://www.cs.virginia.edu/oracle/star_links.html

Ava Gardner Groucho Marx

distance(Ava, Groucho)=2

Distance ?

Character network Nodes: characters Links: co-appearance in a scene

Les Miserables-V. HugoNewman & Girvan, PRE (2004)-> Community detection problem

The web of Human sexualThe web of Human sexualcontactscontacts

Liljeros et al., Nature (2001)

Transportation networks

Transportation networks

Transporting energy, goods or individuals

- formation and evolution

- congestion, optimization, robustness

- disease spread

NB: spatial networks (planar): interesting modeling problems

Nodes: airportsLinks: direct flight

Transporting individuals: global scale(air travel)

Transporting individuals: intra city

TRANSIMS project

Nodes: locations(homes, shops,offices, …) Links: flow ofindividuals

Chowell et al Phys. Rev. E (2003) Nature (2004)

Transporting individuals: intra city

Transporting goods

State of Indiana (Bureau of Transportation statistics)

Nodes: powerplants, transformers,etc,…) Links: cables

Transporting electricity

New York state power grid

Transporting electricity US power grid

Transporting gas European pipelines

Transporting water

Nodes: intersections, auxinssourcesLinks: veins

Example of aplanar network

Networks in biology

Networks in biology

(sub-)cellular level: Extracting usefulinformation from the huge amount ofavailable data (genome, etc). Link structure-function ?

Species level: Stability of ecosystems,biodiversity

Neural Network Nodes: neuronsLinks: axons

Metabolic Network

Nodes: proteinsLinks: interactions

Protein Interactions

Nodes: metabolitesLinks:chemical reactions

Genetic (regulatory) network

Nodes: genesLinks: interaction

Genes are colored according to theircellular roles as described in the YeastProtein Database

Understanding these networks: one of themost important challenge in complexnetwork studies

Food webs Nodes: species Links: feeds on

N. Martinez



Lectures IPhT 2010









2. Percolation


II. 1 Graph theory basics and statistical tools

Graph theory: basics

Graph G=(V,E)

V=set of nodes/vertices i=1,…,N E=set of links/edges (i,j)

Undirected edge:

Directed edge:

i j

i j

Bidirectional communication/interaction

Maximum number of edges Undirected: N(N-1)/2 Directed: N(N-1)


Complete graph :

(all-to-all interaction/communication)

Planar graph: can be embedded in the plane, i.e., it canbe drawn on the plane in such a way that its edges mayintersect only at their endpoints.


planar Non planar

Adjacency matrix

N nodes i=1,…,N

1 if (i,j) E0 if (i,j) ∉ E

011131011211011111003210 0

3

1

2

Symmetricfor undirected networks

Adjacency matrix

N nodes i=1,…,N

1 if (i,j) E0 if (i,j) ∉ E

011030010200001101003210

0

3

1

2

Non symmetricfor directed networks

Sparse graphs

Density of a graph D=|E|/(N(N-1)/2)

Number of edges

Maximal number of edgesD=

Sparse graph: D <<1 Sparse adjacency matrix

Representation: lists of neighbours of each node

l(i, V(i))

V(i)=neighbourhood of i

Paths

G=(V,E)Path of length n = ordered collection of n+1 vertices i0,i1,…,in V n edges (i0,i1), (i1,i2)…,(in-1,in) E

i2i0 i1

i5i4

i3

Cycle/loop = closed path (i0=in)

Paths and connectedness

G=(V,E) is connected if and only if there exists apath connecting any two nodes in G

is connected

• is not connected• is formed by two components

G=(V,E)In general: different components with different sizes

Giant component= component whosesize scales with the number of vertices N

Existence of agiant component

Macroscopic fraction of thegraph is connected

Paths and connectedness

Paths and connectedness: directed graphs

Tendrils Tube Tendril

Giant SCC: Strongly Connected Component Giant OUT

ComponentGiant IN Component

Disconnectedcomponents

Paths can be directed (eg. WWW)

Paths and connectedness: directed graphs

“Bow tie” diagram for the WWW (Broder et al,2000)

Shortest paths

i

j

Shortest path between i and j: minimum numberof traversed edges

distance l(i,j)=minimum number ofedges traversed on a pathbetween i and j

Diameter of the graph:

Average shortest path:

Shortest paths

Complete graph:

Regular lattice:

“Small-world” network

Degree of a node

k>>1: Hubs

How many friends do you have ?

Statistical characterization

• List of degrees k1,k2,…,kN Not very useful!

• Histogram: Nk= number of nodes with degree k

• Distribution: P(k)=Nk/N=probability that a randomly chosen node has degree k

• Cumulative distribution: probability that a randomly chosen node has degree at least k (reduced noise)

Statistical characterization

• Average degree:

• Second moment (variance):

Sparse graph:

Heterogeneous graph:

Degree distribution P(k) : probability that a node has k links

Degree distribution

Poisson/Gaussian distribution Power-law distribution

Centrality measuresHow to quantify the importance of a node?

Degree=number of neighbours=∑j aij

iki=5

Closeness centrality

gi= 1 / ∑j l(i,j)

(directed graphs: kin, kout)

Betweenness Centrality

σst = # of shortest paths from s to tσst(ij)= # of shortest paths from s to t via (ij)

ij

kij: large centrality

jk: small centrality

P(k): not enough to characterize a network

Large degree nodes tend toconnect to large degree nodesEx: social networks

Large degree nodes tend toconnect to small degree nodesEx: technological networks

Multipoint degree correlations

Multipoint degree correlations

Measure of correlations:P(k’,k’’,…k(n)|k): conditional probability that a node of degreek is connected to nodes of degree k’, k’’,…

Simplest case:P(k’|k): conditional probability that a node of degree k isconnected to a node of degree k’

Case of random uncorrelated networks

• P(k’|k) independent of k

• proba that an edge points to a node of degree k’:

proportional to kʼ

number of edges from nodes of degree k’number of edges from nodes of any degree

2-points correlations: Assortativity

Are your friends similar to you ?

• P(k’|k): difficult to handle and to represent

Exemple:

i

k=3k=7

k=4k=4

ki=4knn,i=(3+4+4+7)/4=4.5

Assortativity

Correlation spectrum:Putting together nodes whichhave the same degree:

class of degree k

Assortativity

Also given by:

Assortativity

Assortative behaviour: growing knn(k)Example: social networksLarge sites are connected with large sites

Disassortative behaviour: decreasing knn(k)Example: internetLarge sites connected with small sites, hierarchical structure

# of links between neighbors

k(k-1)/2

Do your friends know each other ?

3-points: Clustering coefficient

• P(k’,k’’|k): cumbersome, difficult to estimate from data

Correlations: Clustering spectrum

• Average clustering coefficient

= average over nodes withvery different characteristics

• Clustering spectrum:

putting together nodes whichhave the same degree class of degree k

(link with hierarchical structures)

Real networks are fragmented into group or modules

Modularity vs.Fonctionality ?

More on correlations: communities and modularity

Extract relevant anduseful information inlarge complex network

More links “inside”than “outside”

More on correlations: communities and modularity

Mesoscopic objects:communities(or modules)

Applications

Biological networks: function ?

Social networks: social groups

Geography/regional science: urbancommunities

Neurophysiology (?)

…

How can we compare differentpartitions?

?

Modularity

Partition P1

Partition P2

- Modularity: quality function Q- P1 is better than P2 if Q(P1)>Q(P2)

Modularity [Newman & Girvan PRE 2004]

Number of modules in P Number of links in module s Number of links in the network Total degree of nodes in module s

= # links in module s

Average number oflinks in module s for arandom network


= Total degree of nodes in module s

probability that a half-link randomlyselected belongs in module s:

probability that a link begins in sand ends in s’:

average number of links internal tos:


Motifs

• Find n-node subgraphs in real graph.

• Find all n-node subgraphs in a set of randomized graphs with

the same distribution of incoming and outgoing arrows.

• Assign Z-score for each subgraph.

• Subgraphs with high Z-scores are denoted as Network

Motifs.

rand

randreal NNZ

!

"=

Milo et al, Science (2002)

Summary: characterization of large networks

Statistical indicators:-degree distribution

- diameter

- clustering

- assortativity

- modularity

Internet, Web, Emails: importance of traffic

Ecosystems: prey-predator interaction

Airport network: number of passengers

Scientific collaboration: number of papers

Metabolic networks: fluxes hererogeneous

Are:- Weighted networks

- With broad distributions of weights

Beyond topology: weights

Weighted networks

General description: weights

i jwij

aij: 0 or 1wij: continuous variable

usually wii=0symmetric: wij=wji

Weighted networks

Weights: on the links

Strength of a node:

Naturally generalizes the degree to weighted networks

Quantifies for example the total traffic at a node

Weighted clustering coefficient

si=16ci

w=0.625 > ci

ki=4ci=0.5

si=8ci

w=0.25 < ci

wij=1

wij=5

i i

Random(ized) weights: C = Cw

C < Cw : more weights on cliquesC > Cw : less weights on cliques

i j

k(wjk)

wij

wik

Average clustering coefficientC=∑i C(i)/NCw=∑i Cw(i)/N

Clustering spectra

Weighted clustering coefficient

Weighted assortativity

ki=5; knn,i=1.8

1

55

5

5i

ki=5; knn,i=1.8

5

11

1

1i


ki=5; si=21; knn,i=1.8 ; knn,iw=1.2: knn,i > knn,i

w

155

5

5i


ki=5; si=9; knn,i=1.8 ; knn,iw=3.2: knn,i < knn,i

w

511

1

1i


Participation ratio “disparity”

1/ki if all weights equal

close to 1 if few weights dominate

Participation ratio “disparity”

In general:

Heterogeneous

Homogeneous (all equal)



Lectures IPhT 2010









2. Percolation


II. 3 Models of complexnetworks

Some models of complex networks

Static networks: N nodes from the beginning- The archetype: Erdos-Renyi random graph (60’s)- A small-world: Watts-Strogatz model (1998)- Fitness models (hidden variables)

Dynamic: the network is growing- Scale-free network: Barabasi-Albert (1999)- Copy model- Weighted network model

Optimal networks

Simplest model of random graphs:Erdös-Renyi (1959)

N nodes, connected with probability p

Paul Erdős and Alfréd Rényi(1959)"On Random Graphs I" Publ.Math. Debrecen 6, 290–297.

Erdös-Renyi (1959): recap

Some properties:

- Average number of edges

- Average degree

Finite average degree

Erdös-Renyi model: recap

Proba to have a node of degree k=• connected to k vertices, • not connected to the other N-k-1

Large N, fixed : Poisson distribution

Exponential decay at large k

• N points, links with proba p:

Erdös-Renyi model: clustering and averageshortest path

• Neglecting loops, N(l) nodes at distance l:

For

: many small subgraphs

: giant component + small subgraphs

Erdös-Renyi model: components

Erdös-Renyi model: summary

- Small clustering

- Small world

- Poisson degree distribution

Generalized random graphs

Desired degree distribution: P(k)

Extract a sequence ki of degrees taken fromP(k)

Assign them to the nodes i=1,…,N

Connect randomly the nodes together,according to their given degree

Generalized random graphs

Average clustering coefficient

Average shortest path

Small-world and randomness

Watts-Strogatz (1998)

Lattice Random graphReal-Worldnetworks*

Watts & Strogatz, Nature 393, 440 (1998)

* Power grid, actors, C. Elegans


N nodes forms a regular lattice.With probability p,each edge is rewired randomly =>Shortcuts

• Large clustering coeff.• Short typical path

MB and Amaral, PRL 1999

Barrat and Weight EPJB 2000

N = 1000


Fitness model (hidden variables)

Erdos-Renyi: p independent from the nodes

• For each node, a fitness

Soderberg 2002Caldarelli et al 2002

• Connect (i,j) with probability

• Erdos-Renyi: f=const


• Degree

• Degree distribution


• If power law -> scale free network

• If and

Generates a SF network !

Barabasi-Albert (1999) Everything’s fine ?

Small-world network

Large clustering

Poisson-like degree distribution

Except that for

Internet, Web

Biological networks

…

Power-law distribution:

Diverging fluctuations !

Internet growth

Moreover - dynamics !

Barabasi-Albert (1999)

(1) The number of nodes (N) is NOT fixed.Networks continuously expand by theaddition of new nodes Examples:

WWW : addition of new documentsCitation : publication of new papers

(2) The attachment is NOT uniform.A node is linked with higher probability to a node thatalready has a large number of links: ʻʼRich get richerʼʼ

Examples :WWW : new documents link to wellknown sites (google, CNN, etc)Citation : well cited papers are morelikely to be cited again


(1) GROWTH : At every time step we add a new node with medges (connected to the nodes already present in thesystem).(2) PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node idepends on the connectivity ki of that node

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

jj

ii

k

kk

!=" )(

P(k) ~k-3

Barabási & Albert, Science 286, 509 (1999)

jj

ii

k

kk

!=" )(



Clustering coefficient

Average shortest path

Copy model

a. Selection of a vertex

b. Introduction of a new vertex

c. The new vertex copies m linksof the selected one

d. Each new link is kept with proba 1-α, rewiredat random with proba α

1−α

α

Growing network:

Copy model

Probability for a vertex to receive a new link at time t (N=t):

• Due to random rewiring: α/t

• Because it is neighbour of the selected vertex: kin/(mt)

effective preferential attachment, withouta priori knowledge of degrees!

Copy model

Degree distribution:

model for WWW and evolution of genetic networks

=> Heavy-tails

Preferential attachment: generalization

Rank known but not the absolute value

Who is richer ?

Fortunato et al, PRL (2006)

Preferential attachment: generalization

Rank known but not the absolute value

Scale free network even in the absence of the value of the nodesʼ attributes

Fortunato et al, PRL (2006)

Weighted networks

• Topology and weights uncorrelated

• (2) Model with correlations ?

Weighted growing network

• Growth: at each time step a new node is added with m links tobe connected with previous nodes

• Preferential attachment: the probability that a new link isconnected to a given node is proportional to the nodeʼs strength

Barrat, Barthelemy, Vespignani, PRL 2004

Redistribution of weights

New node: n, attached to iNew weight wni=w0=1Weights between i and its other neighbours:

The new traffic n-i increases the traffic i-j

Onlyparameter

n i

j

Mean-field evolution equations

Mean-field evolution equations

Correlations topology/weights:

Numerical results: P(w), P(s)

(N=105)

Another mechanism:Heuristically Optimized Trade-offs (HOT)

Papadimitriou et al. (2002)

New vertex i connects to vertex j by minimizing the function Y(i,j) = α d(i,j) + V(j)d= euclidean distanceV(j)= measure of centrality

Optimization of conflicting objectives



Lectures IPhT 2010




1. Tool: Graph theory and characterization of large networks


3. Models Erdos-Renyi random graph Watts-Strogatz small-world Barabasi-Albert and variants


1. Percolation

2. Epidemic spread

3. Other processes: from physics to social systems

III. 1 Percolation, randomfailures, targetedattacks

Consequences of the topologicalheterogeneity

Robustness and vulnerability

RobustnessComplex systems maintain their basic functions

even under errors and failures(cell: mutations; Internet: router breakdowns)

node failure

fc

0 1Fraction of removed nodes, f

1

SS: fraction of giantcomponent

Case of Case of Scale-free NetworksScale-free Networks

Random failure fc =1 (2 < γ ≤ 3)

Attack =progressive failure of the mostconnected nodes fc <1

R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

Random failures: percolation

p=probability of anode to be presentin a percolationproblem

Question: existence or not of a giant/percolating cluster, i.e. of a connected cluster of nodes of size O(N)

f=fraction ofnodes removedbecause of failure

p=1-f

Percolation in complex networks

q=probability that a randomly chosen link does notlead to a giant percolating cluster

Proba that none of theoutgoing k-1 links leadsto a giant cluster

Proba that the link leadsto a node of degree k

Average overdegrees

q=probability that a randomly chosen link does not lead to agiant percolating cluster

Other solution iff F’(1)>1


q=probability that a randomly chosen link does not lead to agiant percolating cluster

“Molloy-Reed” criterion for the existenceof a giant cluster in a random uncorrelated network


Random failures

Initial network: P0(k), <k>0, <k2>0

After removal of fraction f of nodes: Pf(k), <k>f, <k2>f

Node of degree k0 becomes of degree k with proba

Initial network: P0(k), <k>0, <k2>0

After removal of fraction f of nodes: Pf(k), <k>f, <k2>f

Molloy-Reed criterion: existence of a giant cluster iff

Robustness!!!

Random failures

Attacks: various strategies

Most connected nodes Nodes with largest betweenness Removal of links linked to nodes with large k Removal of links with largest betweenness Cascades ...

Failure cascade

(a) and (b): random(c): deterministic and dissipative

Moreno et al, Europhys. Lett. (2003)

Dynamical processes on complexnetworks


Lectures IPhT 2010

Outline

I. Introduction: Complex networks







1. Overview: from physics to social systems

2. Percolation

3. Epidemic spread

III. 3 Epidemic spread

a. Contact networkb. Metapopulation model

General references

Pre(Re-)prints archivePre(Re-)prints archive:

- http://arxiv.org (condmat, physics, cs, q-bio)

Books (Books (epidemiologyepidemiology):):

• J.D. Murray, Mathematical Biology, Springer, New-York, 1993

• R.M. Anderson and R.M. May, Infectious diseases of humans:

Dynamics and control. Oxford University Press, 1992

Epidemiology vs. Etiology: Two levels

Microscopic level (bacteria, viruses): compartments

Understanding and killing off new viruses

Quest for new vaccines and medicines

Macroscopic level (communities, species)

Integrating biology, movements and interactions

Vaccination campaigns and immunization strategies

Epidemiology One population:

nodes=individuals

links=possibility of disease transmission

Metapopulation model: many populations coupled together

Nodes= cities

Links=transportation (air travel, etc.)

Modeling in Epidemiology

R.M. May Stability and Complexity in Model Ecosystems 1973

Epidemics spread on a ʻcontact networkʼ: Differentscales, different networks:

• Individual: Social networks (STDs on sexual contact network)NB: Sometimes no network !

• Intra-urban: Location network (office, home, shops, …)

• Inter-urban: Railways, highways

• Global: Airlines (SARS, etc)

21st century: Networks and epidemics

Networks and epidemiologyHigh School datingwww.umich.edu/~mejn

Nodes: kids (boys & girls)

Links: date each other

Networks and Epidemiology

Urban scale

Global scale

The web of Human sexual contacts (Liljeros et al. , Nature, 2001)

Broad degree distributions

E-mail networkEbel et al., PRE (2002)

Internet

Broad degree distributions

Topology of the system: the pattern of contactsalong which infections spread in population isidentified by a network

• Each node represents an individual

• Each link is a connection along which the viruscan spread

Simple Models of Epidemics

Stochastic model:

• SIS model:

• SIR model:

• SI model:

λ: proba. per unit time of transmitting the infection µ: proba. per unit time of recovering


Numerical simulation:For all nodes:

(i) Check all neighbors and if infected ->(ii). If no infected => Next

neighbor

(ii) Node infected with proba λdt

(iii) proceed to next node

Infection proba is then:[1 – (1 - λ dt)(ki)] = λ ki dt


• Random graph:• Equation for density of infected i(t)=I/N:

SIS on Random Graphs

• µ-1 is the average lifetime of the infected state

• [1-i(t)]=prob. of not being infected

• <k>i(t) is the prob. of at least one infected neighbor

• λ= prob(S->I)

• s(t)+i(t) = 1 ∂ts = -∂ti

With:

• In the stationary state ∂t i=0, we obtain:

Or:

SIS on Random Graphs

• Equation for s(t), i(t) and r(t):

SIR on Random Graphs

∂t s(t) = -λ <k> i [1-i] (S I)

∂t i(t) = -µ i+λ <k> i [1-i] (S I and I R)

∂t r(t) = +µ i (I R)

• µ-1 is the average lifetime of the infected state• s(t)+i(t)+r(t) = 1

• Stationary state: ∂t x=0

s=i=0, r=1

• Outbreak condition: ∂ti(t=0)>0

λ>λc=µ/<k>

µ >> 1 then λc>> 1 (eg. hemo. fever)

SIR on Random Graphs

• Epidemic threshold =critical point• Prevalence i =orderparameter

The epidemic threshold is a general result (SIS, SIR)

The question of thresholds in epidemics is central

Epidemic Threshold λc

i

λλ c

Active phaseAbsorbingphase

Finite prevalenceVirus death

Density of infected

Epidemic spreading on heterogeneousnetworks

Number of contacts (degree) can vary a lothuge fluctuations

Mean-field approx by degree classes: necessary tointroduce the density of infected having degree k, ik

• Scale-free network:• Mean-field equation for the relative density of infected of degree k ik(t):

where Θ is the proba. that any given link points to an infected node

SIS model on SF networks

• Estimate of Θ: proba of finding - (i) a neighbor of degree kʼ - (ii) and infected

Proba. that degree(neighbor)=kʼProba. that the neighbor is infected

SIS model on uncorrelated SF networks

We finally get:

Stationary state ∂tik=0:


[ Pastor Satorras &Vespignani, PRL 86, 320 (2001)]

F’(Θ=0)<1

F’(Θ=0)>1

Solution of Θ=F(Θ) non zero if Fʼ(Θ=0)>1:


Effect of topology: Epidemic threshold

Effect of (strong) heterogeneities of the contact network

(Pastor-Satorras & Vespignani, PRL ‘01)

g = density of immune nodes

λ (1-g) λ → λ (1− g )

Epidemic dies if λ (1-g) < λc

Regular orrandomnetworks

Immunization threshold:gc=(λ-λc)/λ

Consequence: Random immunization

Immunization threshold gc =1

• Random immunization is totally ineffective

• g: fraction of immunized nodes

• Different immunization specifically devised forhighly heterogeneous systems

• Epidemic dies if• Scale-free network λc = 0

Consequence: Random immunization

Targeted immunization strategies

Progressive immunizationof crucial nodes

Epidemic thresholdis reintroduced

[ Pastor Satorras &Vespignani, PRE 65, 036104 (2002)]

[ Dezso & Barabasi cond-mat/0107420; Havlin et al. Preprint (2002)]

Acquaintances immunization strategies: Findinghubs

Cohen & al., PRL (2002)

Average connectivity of neighbor:

Vaccinate acquaintances !

Proba of finding a hub:

Dynamics of Outbreaks

Stationary state understood

What is the dynamics of the infection ?

Stages of an epidemic outbreak

Infected individuals => prevalence/incidence

Dynamics: Time scale of outbreak

At short times:

- SI model S I

-

• Exponential networks:

with:

Dynamics: Time scale of outbreak

• Scale-free networks:

At first order in i(t)

Dynamics: Time scale of outbreak (SF)

• Scale-free networks:

with:

[ MB, A. Barrat, R. Pastor-Satorras & A. Vespignani, PRL (2004)]

Dynamics: Time scale of outbreak (SF)

Dynamics: Cascade

Which nodes are infected ?

What is the infection scenario ?

1. Absence of an epidemic threshold 2. Random immunization is totally ineffective (target !)

3. Outbreak time scale controlled by fluctuations

4. Cascade process:

Seeds Hubs Intermediate Small k

Epidemics: Summary

Dramatic consequences of the heterogeneous topology !

• Spread of an information among connected individuals- protocols for data dissemination- marketing campaign (viral marketing)

• Difference with epidemic spread:- stops if surrounded by too many non-ignorant

• Categories:- ignorant i(t) (susceptible)- spreaders s(t) (infected)- stifflers r(t) (recovered)

Rumor spreading

• Process:- ignorant --> spreader (λ)- spreader+spreader --> stiffler (!)- spreader+stiffler --> stiffler

• Equations:

∂t i(t) = -λ <k> i(t) s(t) (i-> s)

∂t s(t) = +λ <k> i(t) s(t) - α <k> s(t)[s(t)+r(t)] (i->s and s-> r)

∂t r(t) = + α i <k> s(t)[s(t)+r(t)] (s-> r)

Rumor spreading

• Random graph-Epidemic threshold:

Always spreading !

Rumor spreading

Dynamical processes on complexnetworks


Lectures IPhT 2010

Outline

I. Introduction: Complex networks







1. Overview: from physics to social systems

2. Percolation

3. Epidemic spread

II. 2 Empirical results Main features of real-world networks

Plan

Motivations

Global scale: metapopulation model

Theoretical results

Predictability

Epidemic threshold, arrival times

Applications

SARS

Pandemic flu: Control strategy and Antivirals

Epidemiology: past and current Human movements and disease spread

Complex movement patterns: different means, different scales (SARS):Importance of networks

Black deathSpatial diffusionModel:

Nov. 2002

Mar. 2003

Motivations

Modeling: SIR with spatial diffusion i(x,t):

where: λ = transmission coefficient (fleas->rats->humans) µ=1/average infectious period S0=population density, D=diffusion coefficient

Airline network and epidemic spread

Complete IATA database:

- 3100 airports worldwide

- 220 countries

- ≈ 20,000 connections

- wij #passengers on connection i-j

- >99% total traffic

Modeling in Epidemiology

Metapopulation model

Baroyan et al, 1969:≈40 russian cities

Rvachev & Longini, 1985: 50 airports worldwide

Grais et al, 1988: 150 airports in the US

Hufnagel et al, 2004: 500 top airports worldwide

l j popj

popl

Colizza, Barrat, Barthelemy & Vespignani, PNAS (2006)

Meta-population networks

City a

City j

City i

Each node: internal structureLinks: transport/traffic


Rvachev Longini (1985)

Flahault & Valleron (1985); Hufnagel et al, PNAS 2004, Colizza, Barrat, Barthelemy, VespignaniPNAS 2006, BMB, 2006. Theory: Colizza & Vespignani, Gautreau & al, …

Inner city term Travel term

Transport operator:

Reaction-diffusion modelsFKPP equation

Stochastic model

compartmental model + air transportation data

Susceptible

Infected

Recovered

!

pPAR,FCO ="PAR,FCO

NPAR

#t

Travel probabilityfrom PAR to FCO:

# passengersfrom PAR to FCO:Stochastic variable,multinomial distr.

Stochastic model: travel term

ξPAR,FCO

ξPAR,FCO

Transport operator:

other source of noise:

two-legs travel:

ingoing outgoing

Stochastic model: travel term

ΩPAR ({X}) = ∑l (ξl,PAR ({Xl}) - ξPAR,l ({XPAR}))

Ωj({X})= Ωj(1)({X})+ Ωj

(2)({X})

wjlnoise = wjl [α+η(1-α)] α=70%

Discrete stochastic Model

})({)()( 1, StN

SItSttS jj

j

jj

jj !++"#=#"+ $%

})({)()( 2,1, ItItN

SItIttI jjjj

j

jj

jj !++"#"#+="#+ $$µ%

})({)()( 2, RtItRttR jjjjj !+"#+="#+ $µ

2,1,,

jj!! Independent Gaussian noises

S I R

=> 3100 x 3 differential coupled stochastic equationsColizza, Barrat, Barthélemy, Vespignani, PNAS 103, 2015 (2006); Bull. Math. Bio. (2006)


Theoretical studies

Predictability ?

Epidemic threshold ?

Arrival times distribution ?

Propagation pattern

Epidemics starting in Hong Kong

Epidemics starting in Hong Kong

Propagation pattern

One outbreak realization:

Another outbreak realization ? Effect of noise ?

? ? ? ? ? ?

Predictability

Overlap measure

Similarity between 2 outbreak realizations:

Overlap function

time t time t

1)( =! t

time t time t

1)( <! t

Predictability

no degree fluctuationsno weight fluctuations

+ degreeheterogeneity

+ weightheterogeneity

Colizza, Barrat, Barthelemy & Vespignani, PNAS (2006)

Effect of heterogeneity:

degree heterogeneity: decreases predictability

Weight heterogeneity: increases predictability !

Good news: Existence of preferred channels !

Epidemic forecast, risk analysis of containment strategies

j

lwjl

Predictability

Theoretical result: Threshold

moments of the degree distribution

travel probability

Colizza & Vespignani, PRL (2007)

Reproductive number for a population:

Effective for reproductive number for a network ofpopulations: Travel restrictions not efficient !!!

Travel limitations….

Colizza, Barrat, Barthélemy, Valleron, Vespignani. PLoS Medicine (2007)

Theoretical result: average arrival time

Population of city kTraffic between k and lTransmissibilitySet of paths between s and t

Ansatz for the arrival time at site t (starting from s)

1. SARS: test of the model

2. Control strategy testing: antivirals

Applications

Application: SARS

popj

popi

refined compartmentalization

parameter estimation: clinical data + local fit

geotemporal initial conditions: available empirical data

modeling intervention measures: standard effective modeling

SARS: predictions

SARS: predictions (2)

Colizza, Barrat, Barthelemy & Vespignani, bmc med (2007)

Colizza, Barrat, Barthelemy & Vespignani, bmc med (2007)

SARS: predictions (3) - “False positives”

[1-11]1Saudi Arabia[1-32]1Mauritius[1-13]1Israel[1-8]1Brunei[1-20]1Nepal[1-8]1Cambodia[1-20]1Bahrain[1-14]2Bangladesh

[1-11]2United Arab Emir.[1-10]2Netherlands[9-114]30Japan

90% CIMedianCountry

More from SARS - Epidemic pathways

For every infected country:

where is the epidemic coming from ?

- Redo the simulation for many disorder realizations(same initial conditions)

- Monitor the occurrence of the paths(source-infected country)

Existence of pathways:

confirms the possibility of epidemic forecasting !

Useful information for control strategies

SARS- what did we learn ?

Metapopulation model, no tunable parameter:

good agreement with WHO data !

Application of the metapopulation model: effect ofantivirals

Threat: Avian Flu

Question: use of antivirals

Best strategy for the countries ?

Model:

Etiology of the disease (compartments)

Metapopulation+Transportation mode (air travel)

2. Antivirals

Flu type disease: Compartments

Feb 2007 May 2007 Jul 2007

Dec 2007 Feb 2008 Apr 2008 0

rmax

Pandemic flu with R0=1.6 starting from Hanoi (Vietnam) in October (2006)Baseline scenario

Pandemic forecast…

Colizza, Barrat, Barthélemy, Valleron, Vespignani. PLoS med (2007)

Effect of antivirals

Comparison of strategies (travel restrictions not efficient)

“Uncooperative”: each country stockpiles AV

“Cooperative”: each country gives 10% (20%) of its own stock

Baseline: reference point (no antivirals)

Effect of antivirals: Strategy comparison

Best strategy: Cooperative !

Colizza, Barrat, Barthelemy, Valleron, Vespignani, PLoS Med (2007)

Conclusions and perspectives

Global scale (metapopulation model)

Pandemic forecasting

Theoretical problems (reaction-diffusion on networks)

Smaller scales-country, city (?)

Global level: “simplicity” due to the dominance of air travel

Urban area ? Model ? What can we say about the spread of adisease ? Always more data available…

Outlook

Prediction/Predictability vs diseaseparameters, initial conditions, errors etc.

Integration of data sources: wealth, census,traveling habits, short range transportation.

Individual/population heterogeneity

Social behavior/response to crisis

Collaborators

- A. Barrat (LPT, Orsay)

- V. Colizza (ISI, Turin)

- A.-J. Valleron (Inserm, Paris)

- A. Vespignani (IU, Bloomington)

Collaboratory

[email protected]

Collaborators and links

http://cxnets.googlepages.com

PhD students

- P. Crepey (Inserm, Paris)

- A. Gautreau (LPT, Orsay)

dynamical processes on complex networks · books and reviews on processes on complex networks •...

Documents