complex networks - gecomplexity-cost.eu

COMPLEX NETWORKS

Luis Miguel VarelaGrupo de Nanomateriales y Materia Blanda

Dpto. Física de la Materia CondensadaUniversidad de Santiago de Compostela

[email protected]

COST Action IS1104: The EU in the new complex geography of economic systems: models, tools and policy evaluation

WG4 meeting. Urbino. September, 2012

COMPLEX NETWORKS



[email protected]

“I want to know God’s mind. The rest are details.”A. Einstein

Simplex sigillum veri.Physics Faculty Auditorium. University of Göttingen



WHAT IS COMPLEXITY?

COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

Emergence approach vs. reductionist approach

Simple systems: systems that can be decomposed in a sum of parts: the whole equals the sum of the parts.

V

N

Complex systems: can not be decomposed in a sum of parts, the whole is more than the mere sum of its parts ⇒ emergence (something in the whole is not in its parts).

Classical mechanics: reductionist approach


WHAT IS COMPLEXITY?

Novum Organum1620

Dialogos… andConsiderationes…

1632-38

1700 1800 1900

Nova Methodus1684

Principia 1687

Statisticaltheoryof radiation

2nd Law~1850

Maxwell1873

Stat. Mech.Boltzmann

1st Law

Carnot1824

Lagrange1788

D. BernouilliHydrodynamica

1738

Celsius1742

CavendishFranklin~1850

AmpereGauss

Coulomb1780’s

Cavendish experiment1797-1798

1600

FresnelFraunhoffer

Sir Isaac Newton(1642-1727)

12312

21 rr

mmGF

dt

pdF

rr

rr

−=

= Newtoniansynthesis

Neoclassicaleconomics

WHAT IS COMPLEXITY?


- Static universe: In classical physics structures are given previously to any consideration and they do not evolve in time.

-They are supposed to be separableof the rest of the universe and the laws that govern the whole are the same that govern the parts.

- Superposition principle: net effects are the deterministic result of the addition of causes (forces).

Classical physics: reductionist approach

∫=V

t df ϕϕϕ )(

)()( rdrV

rr ψψ ∫=

http://www.acturban.org/biennial/ElectronicCatalogue/Delft/BlueBanana.jpg

http://mrbarlow.files.wordpress.com/2008/03/traffic_jam.jpg?w=171&h=257

OK, BUT WE HAVE TO EXPLAIN…


OK, BUT WE HAVE TO EXPLAIN…


WHAT IS COMPLEXITY?


- No unanimously admitted definition of complexity: up to 45 definitions (Seth Lloyd, 1997).

Complex system: a system with many degrees of freedom strongly coupled in a nonlinear way through complex feedback loops. Complexity implies structure with variation.

- Relationships are non-linear- Relationships contain feedback loops

- Complex systems are open- Complex systems have a memory (hysteresis)

- Complex systems may be nested- Boundaries are difficult to determine

- Dynamic network of multiplicity-May produce emergent phenomena (structures)

-Large complex networks: typically self-organized and chaotic

IS ECONOMY A COMPLEX SYSTEM?


FORMAL DESCRIPTION OF THE STRUCTURE AND DYNAMICS OF A COMPLEX SYSTEM


Mathematical description: highly interdisciplinary field (that’s what makes this WG4 the fascinating, chaotically self-organized centerpiece of the Action…)

-Statistical theory of stable processes. Fractals.- Theory of nonlinear dynamic systems. Chaos theory.- Statistical mechanics (phase transitions, self-organized criticality…)- Thermodynamics of irreversible processes (nonlinear regime)-Network theory-Computer simulations



STRUCTURE: Underlying every complex system there is always some kind of networkformed by the agents (nodes) and interactions between them (edges).

www-personal.umich.edu/~mejn/networks/



DYNAMICS: Complex network as a substrate for nonlinear dynamic processes. Interplay between the structure and dynamics of complex networks.

- Evolution of the network itself (evolving networks, rewiring…)

- Spreading processes (epidemics, rumors…)

- Dynamic processes (dynamics of the state of nodes)

- Synchronization of networks and collective dynamics.

COMPLEX NETWORKS

HISTORICAL OUTLOOK

Leonhard Euler (Seven Bridges of Königsberg,1736)

Origins of complex network theory: graph theory.

n-Cayley treeErdös-Rényi random

graph theory (On random graphs, 1959)

COMPLEX NETWORKS

HISTORICAL OUTLOOK

Small-world networks: interpolation between regular networks and random graphs

Exponentially distributed homogeneous network (Watts and Strogatz, 1998)

Duncan J. Watts, Steven H. Strogatz, Nature 393, 440 (1998)Exponential networks (SW, random)

Origins of network theory: graph theory.

COMPLEX NETWORKS

HISTORICAL OUTLOOK

Financial market model, Carrete, Varela et al.

N=10000

Alun L. Lloyd, Robert M. May, Science 292, 1316 (2001).N=100, formalism de AB

Scale free network

Scale-free heterogeneous networks (Barabási and Albert, 1999).

Small-world networks: interpolation between regular networks and random graphs

Origins of network theory: graph theory.

AB Scale free network W=100, M=90. Sexually transmitted

diseases

COMPLEX NETWORKS

DEFINITION OF NETWORK

Graph: an undirected (directed) graph is an object formed by two sets, N and L, a set of nodes (N={n1,…,nN}) and an unordered (ordered) set of links (L={l 1…lK}).

Many systems fit in this scheme (social networks, biological networks, computer networks…)

COMPLEX NETWORKS

CLASSIFICATION OF NETWORKS

A) Classification by network topology- Regular networks- Small world networks

- Random networks (binomial degree distribution)- “Small-world” (exponential degree distribution , homogeneous)- Scale free (power law degree distribution, heterogeneous)

B) Classification by network directedness- Directed and undirected networks

C) Classification by heterogeneity in the capacity and the intensity of the connections in the network- Weighted and unweighted networks- Diluted and fully connected networks.

D) Classification by the occurrence or not of time evolution of the network structure- Static and evolving networks

COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description

- Small world property (average path length), in most networks there is a relatively short path between any two nodes, defined as the number of edges along the shortest path connecting them. The connectedness can also be measured by means of the diameter of the graph,d, defined as the maximum distance between any pair of its nodes.

Networks do not have a “distance” : no proper metricspace. Chemical distance between two verticeslij : number of steps from one pointto the other following the shortest path.

COMPLEX NETWORKS

NETWORKS TOPOLOGY


Average shortest path length

∑∑ =−

=< lji

ij llplNN

l )()1(

2

In most real networks, < l > is a very small quantity (small-world)

In a square lattice of size N: Nl ≈

In a complex network of size N: Nl log≈

COMPLEX NETWORKS

NETWORKS TOPOLOGY


∑=i

icN

c1

Clustering coefficient, ratio between the number Ei of edges that actually exist between these ki nodes and the total number ki (ki-1)/2 gives the value of the clustering coefficient of node i. The clustering coefficient provides a measure of the local connectivity structure of the network

)1(2

2

−=

=

ii

i

i

ii kk

EkE

c c Low c Large

Clustering spectrum: Average clustering coefficient of the vertices of degree k

ii

kk ckNp

kci∑= δ

)(1

)(

Average clustering coefficient

A network is called sparse if its average degree remains finite when taking the limit N . In real (finite) networks, <k> <<N

COMPLEX NETWORKS

NETWORKS TOPOLOGY


Average degree

- Degree distribution: p(k) probability that a node has a definite amount of edges. In directed networks the in-degree and out-degree are defined.

∑ ∑==i k

i kkpkN

k )(1

−

≈−

−

−

AB

WS

random)1(

)(α

γ

k

e

ppk

N

kp k

kNk

CentralityTo go from one vertex to other in the network, following theshortest path, a series ofother vertices and edges are visited. The ones visited more frequently will be more central in the network. We can define quantitatively this concept of centrality by means of the betweenness of a vertex bi or an edgebij . Number of shortest paths that pass through the vertexi (edge(i,j)), for all the possible pairs of vertices in the network.

COMPLEX NETWORKS

NETWORKS TOPOLOGY


COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology descriptionTwo-vertex correlationsReal networks are usually correlated: degrees of the nodes at the ends of a given vertexare not in general independent. P(k’ | k)= probability that a k-node points to a k’-node.

Uncorrelated network: k

kpkkkp

)'(')|'( = independent ofk

Correlated network: p(k’|k) depends on bothk’ andk

Degree of detailed balanced condition: P(k) and P(k’ | k) are not independent, but are related by a degree detailed balance condition.

Consequence of the conservation of edges

)'|(')'()|'()(

)'|(')'()|'()(

kkpkkpkkkpkp

kkpkkNpkkkpkNp

==

Number of edgesk k’=number of edgesk’ k

COMPLEX NETWORKS

NETWORKS TOPOLOGY


Correlation measures

∑='

)|'(')(k

nn kkpkkk

Average degree of the nearest neighbors of thevertices of degree. Alternative top(k’|k)

knn(k) dependent onk: correlations

Assortative: knn(k) increasing function ofkDisassortative: knn(k) decreasing function ofk

COMPLEX NETWORKS

NETWORKS TOPOLOGY


R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

COMPLEX NETWORKS

NETWORKS TOPOLOGY


- Motifs: A motif M is a pattern of interconnections occurring either in a undirected or in a directed graph G at a number significantly higher than in randomized versions of the graph, i.e. in graphs with the same number of nodes, links and degree distribution as the original one, but where the links are distributed at random.

- Community (or cluster, or cohesive subgroup) is a subgraph G(N,L), whose nodes are tightly connected, i.e. cohesive.

S. Boccaletti et al. Physics Reports 424 (2006) 175 –308

Regular networks (order)

Constant number of connections per node

1. Binomial degree distribution: in a randomgraph with connectionp the degree of the node i follows a binomial distribution

2. Average path length(scales logarithmically)

k

Nl rand

ln

ln≈ Average path length of random networksR. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

Small world

COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM NETWORKS

Random network (disorder; e.g. Erdös-Renyi)

1)1(1

)( −−−

−== kNk

ii ppk

Nkkp

Random networks (disorder; e.g. Erdös-Renyi)

3. Clustering coefficient

N

kCrand ≈

Clustering coefficient of real networks and random graphsR. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

COMPLEX NETWORKS


Random graph models lead to a strong infraestimation of the clustering degree of real networks (independent of N)

COMPLEX NETWORKS

NETWORKS TOPOLOGY: SMALL WORLD NETWORKS

Watts-Strogatz algorithm

Start with order Randomize rewiring with probability pexcluding self-connections and duplicate edges

“Small-world” networks (e.g. Watts-Strogatz)

1. Small average path length (small-world property)

2. Relatively high clustering coefficient

Watts-Strogatz model

3. High homogeneity degree: all the nodes have approximately the same number of edges.

4. Equivalent to mean field.

5. Exponential degree distribution (exponentially distributed model).

COMPLEX NETWORKS


ii kk ≈

COMPLEX NETWORKS


Albert-Barabási algorithm

1. Network growth: start with a small number of nodes and at each time step add anew node that links to m already existing nodes

2. Preferential attachment (evolving network): the probability that a new node links to node i depends on the degree of the already existing node:

Albert-Barabási

Dorogovtsev- Mendes-Samukhin

1. Potential degree distribution (extreme events, superspreader, hierarchies):

P(k) ~ k-g

2. Average path length shorter than in exponentially distributed networks.

3. Degree of correlation of the degree of the different nodes

4. Clusterization degree ~ 5 times greater than that ofrandom networks.

Scale free networks (e.g. Albert-Barabasi)

75.0N

kCrand ≈

COMPLEX NETWORKS


Bocalettiet al.Physics Reports 424, 175 – 308 (2006).

COMPLEX NETWORKS


Directed networks and weighted networksWeighted networks:strong and weak ties between individuals in social networks

NodesLinks

weights Weighted degree

Weighted clustering coeff.

COMPLEX NETWORKS


Spatial networks: networks whose nodes occupy a definite position in the three dimensional Euclidean space (e.g. neural networks, transportation networks, Internet, electric power grid, highways, streets, pipelines, etc.).


-Small world structure

-Scale free models (Xulvi-Brunet and Sokolov)

COMPLEX NETWORKS


Spatial networks: networks whose nodes occupy a definite position in the three dimensional Euclidean space (e.g. neural networks, transportation networks, Internet, electric power grid, highways, streets, pipelines, etc.).


COMPLEX NETWORKS

DYNAMIC PROCESSES IN COMPLEX NETWORKS


- Evolution of the network itself (evolving networks, rewiring…)

- Spreading processes (epidemics, rumors…)

- Synchronization and collective dynamics of networks.

COMPLEX NETWORKS



Evolution of the network itself (evolving networks, assortative mixing…)

A) Preferential attachment (evolving network): the probability that a new node links to node i depends on the degree of the already existing node:

Albert-Barabási Dorogovtsev- Mendes-Samukhin

B) Assortative mixing: nodes with many connections tend to be connected to other highly connected nodes.

COMPLEX NETWORKS



Spreading processes

)(tiψ

));,(;,()( tLNGft ijii ≠= ψψψ&

COMPLEX NETWORKS



Spreading processes: Sprott model of chaotic dynamics in a complex network (J.C. Sprott, Chaos 18, 023135 (2008)).

Sigmoidal nonlinearity

Sigmoidal nonlinearities(saturation)

COMPLEX NETWORKS



Spreading processes: epidemic dynamics

)()(

)()()()(

)()()(

tIdt

tdR

tStIktIdt

tdI

tStIkdt

tdS

µ

λµ

λ

=

+−=

−=

S(t)

I(t)

R(t)

)()( tStIkλ−

)(tIµ−

Watts-Strogatz network: Homogeneous mixing hypothesis

Homogeneous populations

• Flu epidemic dynamics: SEIR model• Network topology: Watts-Strogatz (<k>=5)• Epidemiologic parameters

Latence period = 1.9 days; Infection rate

Infectious period = 4.1 days; Recovery rate

Effective infectious period Deff= 2.6 days

Pecentage of exposed that do not suffer the illness = 35%

Birth and death rates µ= 2.10-5.

Reproductive basic number, R0=1.6-2.4. effDkR λ≈0

COMPLEX NETWORKS



Spreading processes: avian flu pandemics in Galicia (Spain) (A: González-Vázquez,, M. Otero-Barrós, J Carrete, E..Pis, L.M. Varela, C. Ricoy, Eur. J. Health Econ., submitted)

( )

( )

( ) ( )IEdt

dR

IIEdt

dI

EESIkdt

dE

SSIkdt

dS

γµχµ

λγµα

αχµλ

µλµ

+++=

++−=

−+−=

−−=

COMPLEX NETWORKS



R0(%

Infected peoples)

tmax(day

s)

New cases

/day in the

maximum

(Gross Attack

Rate %)Case

Fatality Rate)

A (a) 1.85

60,90% 61 64.500 (2.15%)

30,45 % 18.270

B (b) 1.60

52,87% 78 41.700 (1,39%)

26,44% 15.864

C (c) 1.70

56,62% 71 50.700 (1,69%)

28,26% 16.956

D(d) 2.00

64,31% 55 78.600 (2,62%)

32,16% 19.296

E 2.20

67,76% 48 96.600 (3,22%)

33.88% 20.328

F 2.40

70,31% 43 114.600 (3,82%)

35,16% 21.096

Spreading processes: avian flu pandemics in Galicia (Spain) (L.M. Varela, C. Ricoy)

COMPLEX NETWORKS



Spreading processes: avian flu pandemics in Galicia (Spain) (L.M. Varela, C. Ricoy)

COMPLEX NETWORKS



COMPLEX NETWORKS



Spreading processes: spreading of rumors

Ignorantsi(t)Spreaders s(t)Stiflersr(t)

COMPLEX NETWORKS



Spreading processes: market models

J. Pombo, L.M. Varela, C. Ricoy, Eur. J. Health Econ. In press

COMPLEX NETWORKS


Spreading processes: financial market models

N=10.000Days:500 Time step: 360/2000Nunber of links of a new node = 5Number of initial shares per agent = 50Initial amount of money = 500 m.u.Fraction of opening price for price discretization = 0.001Iterations between consecutive portfolio changes = 10Probability of imitation to a higher degree neighbor= 0.4Probability of random portfolio change = 0.1Number of high-degree nodes receiving informational shock = 100

Continuous auction-driven market. (Based on a market model A. Consiglio, V.

Lacagnina, A. Russino, Quantitative Finance 5, 71 (2005))

COMPLEX NETWORKS

DYNAMIC PROCESSES IN COMPLEX NETWORKSSpreading processes: financial market models

Before the shock, and after the equilibration, the log-returns distribution is close to a gaussian, although somewhat leptocurtic. After the shock, however, it presents "heavy tails". A Shapiro-Wilk test at the closing of each day shows its evolution. The time it takes the system to reach a new equilibrium is strongly dependent on the imitation probability and several progressively smaller "rebounds" are clearly seen. The shock is not clearly visible in the volume and price plots.

COMPLEX NETWORKS

DYNAMIC PROCESSES IN COMPLEX NETWORKSSpreading processes: financial market models

J. Voit, The Statistical Mechanics of FinancialMarkets. Springer, Berlín, 2003)

COMPLEX NETWORKS

COMPUTER SIMULATION OF PROCESSES IN COMPLEX NETWORKS

Program flux 3 steps

COMPLEX NETWORKS


• Real or hypothetical.• Depends on the amount of data:

• Intrinsic characteristics (e. g. classes). • Full description (e. g. contact tracing).• Building algorithm (e. g. Barabási-Albert).• Sampling of the degree distribution (e. g. polls).• Tools:• Standard statistical methods and software.• Analisys and visualization interactive programs.

POPULATION ANALYSIS

COMPLEX NETWORKS


Pajek: http://pajek.imfm.si/doku.php

Otros: Cytoscape(http://www.cytoscape.org/), UCINet.

COMPLEX NETWORKS


• Much more time-consuming than ODE-based methods• Automatization need:

• Long unattended runs.• Parallelization

• Most convenient option: high-level language + network algorithm libraries.

SIMULATION

Python (http://www.python.org)

NumPy: array treatment (MATLAB-like).Scipy: scientific functions on NumPy.

RPy : integrates R in Python with NumPy.Parallelism, access to databases, text processing and binary files, user graphic interfaces, 2D/3D

plots, geographical information systems...Windows distribution: Python(x,y ( http://www.pythonxy.com)

•General methods: Calculus sheets [catastrophic precission: G. Almiron et al., Journal of Statistical Software 34 (2010)].•Analysis environments: MATLAB, Mathematica, Octave, etc.•Specific: R+statnet, Python+NetworkX

COMPLEX NETWORKS


POSTPROCESSING

COMPLEX NETWORKS


SIMULATION

NetworkX: http://networkx.lanl.gov/ ,Included in Python(x,y).Generatiors, algebra, input/output, representation...Optimized algorithms, programmed in low level languages.Nodes can contain any type of data.Integration with NumPy.

COMPLEX NETWORKS


SIMULATION

NetworkX: http://networkx.lanl.gov/ ,Included in Python(x,y).Generatiors, algebra, input/output, representation...Optimized algorithms, programmed in lowlevel languages.Nodes can contain any type of data.Integration with NumPy.

REFERENCES

1. P. Samuelson, P., W. D. Nordhaus, Economía, 16ª edición (McGraw-Hill, Madrid, 1999). 2. http://es.wikipedia.org3. W. Brian Arthur, Science, 284, 107 (1999). 7. H. Takayasu, Fractals in the Physical Science, (Manchester University Press, Manchester, 1990).5. R. N. Mantegna, H. E. Stanley, An Introduction to Econophysics(Cambridge University Press, Cambridge, 2000).6. The Complex Networks of Economic Interactions: Essays in Agent-Based Economics and Econophysics. Akira Namatame, Taisei Kaizouji, Yuuji Aruka (Eds) (Springer, 2006).7. Statistical Mechanics of Complex Networks. Romualdo Pastor-Satorras, Miguel Rubi, Albert Diaz-Guilera (Eds.) (Springer, 2003).8. The Economy As An Evolving Complex System(Santa Fe Institute Studies in the Sciences of Complexity Proceedings) Philip W. Anderson, Kenneth Arrow, David Pines et al. (Westview, 1988)9. J. Voit, The Statistical Mechanics of Financial Markets. (Springer-Verlag, Berlín, 2003).10. Per Bak, C. Tang, K.Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987 ).11. Duncan J. Watts, Steven H. Strogatz, Nature 393, 440 (1998) 12. Alun L. Lloyd, Robert M. May, Science 292, 1316 (2001).13. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)14. S. Boccaletti et al. Physics Reports 424 (2006) 175 –308.15. Y. Moreno, R. Pastor-Satorras, A. Vespignani Eur. Phys. J. B 26, 521 (2002).16. R. Pastor-Satorras, A. Vespignani, Phys. Rev. E, 65, 036104 (2002)17. T. Saramäki, K. Kaski, J. Theor. Biol. 234, 413 (2005). 18 J.C. Sprott, Chaos 18, 023135 (2008)19. A. Consiglio, V. Lacagnina, A. Russino, Quantitative Finance 5, 71 (2005)

COMPLEX NETWORKS



[email protected]



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