Dynamical processes on complex networks · Books and reviews on processes on complex networks •...
Transcript of 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
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
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)
Graph theory: basics
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.
Graph theory: basics
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
Modularity [Newman & Girvan PRE 2004]
= 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:
Modularity [Newman & Girvan PRE 2004]
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
Weighted assortativity
ki=5; si=21; knn,i=1.8 ; knn,iw=1.2: knn,i > knn,i
w
155
5
5i
Weighted assortativity
ki=5; si=9; knn,i=1.8 ; knn,iw=3.2: knn,i < knn,i
w
511
1
1i
Weighted assortativity
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)
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
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
Watts-Strogatz (1998)
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
Watts-Strogatz (1998)
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
Fitness model (hidden variables)
• Degree
• Degree distribution
Fitness model (hidden variables)
• 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
Barabasi-Albert (1999)
(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
!=" )(
Barabasi-Albert (1999)
Barabasi-Albert (1999)
Barabasi-Albert (1999)
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
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
2. Some empirical results
3. Models Erdos-Renyi random graph Watts-Strogatz small-world Barabasi-Albert and variants
III. Dynamical processes on complex networks
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
Percolation in complex networks
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
Percolation in complex networks
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
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
2. Some empirical results
3. Models Erdos-Renyi random graph Watts-Strogatz small-world Barabasi-Albert and variants
III. Dynamical processes on 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
Simple Models of Epidemics
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
Simple Models of Epidemics
• 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:
SIS model on SF networks
[ 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:
SIS model on SF networks
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
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
2. Some empirical results
3. Models Erdos-Renyi random graph Watts-Strogatz small-world Barabasi-Albert and variants
III. Dynamical processes on 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
Metapopulation model
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)
Metapopulation model
Theoretical studies
Predictability ?
Epidemic threshold ?
Arrival times distribution ?
Propagation pattern
Epidemics starting in Hong Kong
Epidemics starting in Hong Kong
Propagation pattern
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
Collaborators and links
http://cxnets.googlepages.com
PhD students
- P. Crepey (Inserm, Paris)
- A. Gautreau (LPT, Orsay)