COMPLEX NETWORKS COST Action IS1104: the EU in the New Complex Geography of Economic Systems: Models, Tools and Policy Evaluation WG4 Meeting

COMPLEX NETWORKS COST Action IS1104: The EU in the new complex geography of economic systems: models, tools and policy evaluation WG4 meeting. Urbino. September, 2012 Luis Miguel Varela Grupo de Nanomateriales y Materia Blanda Dpto. Física de la Materia Condensada Universidad de Santiago de Compostela [email protected] “I want to know God’s mind. The rest are details.” A. Einstein Simplex sigillum veri. Physics Faculty Auditorium. University of Göttingen COMPLEX NETWORKS

COST Action IS1104: The EU in the new complex geography of economic systems: models, tools and policy evaluation WG4 meeting. Urbino. September, 2012

Luis Miguel Varela Grupo de Nanomateriales y Materia Blanda Dpto. Física de la Materia Condensada Universidad de Santiago de Compostela [email protected] COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

WHAT IS COMPLEXITY?

Emergence approach vs. reductionist approach V

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

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). COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

WHAT IS COMPLEXITY? Classical mechanics: reductionist approach r r dp F = dt Newtonian r r synthesis = − m1m2 F G 3 r12 r12 Neoclassical economics Stat. Mech. Boltzmann Sir Isaac Newton Dialogos… and D. Bernouilli Carnot 2nd Law (1642-1727) Considerationes… Principia Hydrodynamica Coulomb Ampere Statistical theory 1824 ~1850 1632-38 1687 1738 1780’s Gauss of radiation

1900 1600 1700 1800

Celsius Cavendish Fresnel 1st Law Novum Organum Nova Methodus Lagrange Maxwell 1742 Franklin Fraunhoffer 1620 1684 1788 1873 ~1850

Cavendish experiment 1797-1798 COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM WHAT IS COMPLEXITY? Classical physics: reductionist approach

- Static universe : In classical physics structures are given previously to any consideration and they do not evolve in time. -They are supposed to be separable of the rest of the ϕ = f (ϕ)dϕ universe and the laws that govern the whole are the t ∫ V same that govern the parts. - Superposition principle : net effects are the deterministic result of the addition of causes (forces).

r r ψ (r) = ∫ dψ (r ) V COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

OK, BUT WE HAVE TO EXPLAIN…

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

http://www.acturban.org/biennial/ElectronicCatalogue/Delft/BlueBanana.jpg COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

OK, BUT WE HAVE TO EXPLAIN… COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

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 COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM IS ECONOMY A COMPLEX SYSTEM? COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING 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 COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

FORMAL DESCRIPTION OF THE STRUCTURE AND DYNAMICS OF A COMPLEX SYSTEM STRUCTURE : Underlying every complex system there is always some kind of network formed by the agents (nodes) and interactions between them (edges).

www -personal.umich.edu/~mejn/networks/ COMPLEXITY ECONOMICS: ECONOMY AS AN EVOLVING COMPLEX SYSTEM

FORMAL DESCRIPTION OF THE STRUCTURE AND DYNAMICS OF A COMPLEX SYSTEM

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

Origins of complex network theory: graph theory.

Leonhard Euler n-Cayley tree (Seven Bridges of Erdös-Rényi random Königsberg ,1736) graph theory ( On random graphs , 1959) COMPLEX NETWORKS

HISTORICAL OUTLOOK Origins of network theory: graph theory. 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) COMPLEX NETWORKS

HISTORICAL OUTLOOK Origins of network theory: graph theory. Small-world networks: interpolation between regular networks and random graphs Scale-free heterogeneous networks (Barabási and Albert, 1999).

AB Scale free network W=100, M=90. Sexually transmitted Alun L. Lloyd, Robert M. May, diseases Science 292, 1316 (2001). Financial market model, N=100, formalism de AB Carrete, Varela et al. Scale free network N=10000 COMPLEX NETWORKS

DEFINITION OF NETWORK

Graph : an undirected (directed) graph is an object formed by two sets, N and L, a set of N L nodes ( ={n 1,…,nN}) and an unordered (ordered) set of links ( ={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 metric space. Chemical distance between two vertices lij : number of steps from one point to the other following the shortest path. COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description

Average shortest path length

2 l = l = lp (l) − ∑ ij ∑ N(N 1) i< j l

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

In a square lattice of size N: l ≈ N

In a complex network of size N: l ≈ log N COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description

Clustering coefficient , ratio between the number Ei of edges that actually exist between these ki nodes and the total number ki (k i-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 E 2E c = i = i Low c Large c i   − ki ki (ki 1)    2  Average clustering coefficient = 1 c ∑ci N i Clustering spectrum : Average clustering coefficient of the vertices of degree k = 1 δ c(k) ∑ kk ci i Np (k) i COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description - 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.

  N k N −k   p 1( − p) random  k   p(k) ≈  e−γk WS  −α  k AB  = 1 = Average degree k ∑ki ∑kp (k) N i k A network is called sparse if its average degree remains finite when taking the limit N . In real (finite) networks, <

NETWORKS TOPOLOGY

Fundamental concepts for network topology description

Centrality To go from one vertex to other in the network, following the shortest path, a series of other 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 edge bij . Number of shortest paths that pass through the vertex i (edge (i,j )), for all the possible pairs of vertices in the network. COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description Two-vertex correlations Real networks are usually correlated: degrees of the nodes at the ends of a given vertex are not in general independent. P(k’ | k)= probability that a k-node points to a k’-node. k' p(k )' Uncorrelated network: p(k |' k) = independent of k k Correlated network: p(k’|k) depends on both k’ and k

Degree of detailed balanced condition : P(k) and P(k’ | k) are not independent, but are related by a degree detailed balance condition. Number of edges k k’=number of edges k’ k Np (k)kp (k |' k) = Np (k )' k' p(k | k )' p(k)kp (k |' k) = p(k )' k' p(k | k )' Consequence of the conservation of edges COMPLEX NETWORKS

NETWORKS TOPOLOGY

Fundamental concepts for network topology description

Correlation measures

Average degree of the nearest neighbors of the vertices of degree. Alternative to p(k’|k) = knn (k) ∑k' p(k |' k) k'

knn (k) dependent on k: correlations

Assortative : knn (k) increasing function of k Disassortative : knn (k) decreasing function of k COMPLEX NETWORKS

NETWORKS TOPOLOGY Fundamental concepts for network topology description

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

NETWORKS TOPOLOGY Fundamental concepts for network topology description

- Motifs: A motif M is a pattern of interconnections occurring either in a undirected or in a directed graph G at a number signi ﬁcantly 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 COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM NETWORKS

Regular networks (order)

Constant number of connections per node Random network (disorder; e.g. Erdös-Renyi) 1. Binomial degree distribution : in a random graph with connection p the degree of the node i follows a binomial distribution  N −1 = =   k − N −k−1 pi (ki k)  p (1 p)  k  2. Average path length (scales logarithmically) ln N l ≈ Average path length of random networks rand ln k R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

Small world COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM NETWORKS

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

3. Clustering coefficient

k C ≈ rand N

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

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

NETWORKS TOPOLOGY: SMALL WORLD NETWORKS

Watts-Strogatz algorithm

Randomize rewiring with probability p Start with order excluding self-connections and duplicate edges COMPLEX NETWORKS

NETWORKS TOPOLOGY: SMALL WORLD NETWORKS

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

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

2. Relatively high clustering coefficient 3. High homogeneity degree: all the nodes have approximately the same number of edges. ≈ ki ki 4. Equivalent to mean field.

5. Exponential degree distribution (exponentially distributed model). Watts-Strogatz model COMPLEX NETWORKS

NETWORKS TOPOLOGY: SMALL WORLD NETWORKS

Albert-Barabási algorithm

1. Network growth : start with a small number of nodes and at each time step add a new 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 COMPLEX NETWORKS

NETWORKS TOPOLOGY: SMALL WORLD NETWORKS

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

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 of random networks.

k C ≈ rand N 75.0 COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM NETWORKS

Directed networks and weighted networks Weighted networks: strong and weak ties between individuals in social networks Nodes Links weights Weighted degree

Weighted clustering coeff.

Bocaletti et al. Physics Reports 424, 175 – 308 (2006). COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM 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)

Bocaletti et al. Physics Reports 424, 175 – 308 (2006). COMPLEX NETWORKS

NETWORKS TOPOLOGY: REGULAR AND RANDOM NETWORKS

Bocaletti et al. Physics Reports 424, 175 – 308 (2006). COMPLEX NETWORKS