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Complex Networks Luis Miguel Varela COST meeting, Lisbon March 27 th 2013

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Complex Networks Luis Miguel Varela COST meeting, Lisbon March 27 th 2013 Complex Networks Luis Miguel Varela Cabo Introduction Main properties of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Introduction Complex networks analysis in socioeconomic models Statistical mechanics of complex networks Computational algorithms Applications - Market models - Regional trade and development - Other social network models of interest Suggested trends Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Statistical mechanics of complex networks Graph: an undirected (directed) graph is an object formed by two sets, V and E, a set of nodes ( V ={v 1,…,v N }) and an unordered (ordered) set of links ( E ={e 1 …e K }). Adjacency matrix: - Contains most of the relevant information about the graph - A symmetric: undirected graph (a) - A non symmetric: directed acyclic graph (b) Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Measures on networks Applications social networks financial networks Economic networks Small-worlds: Erdös-Bacon number Degree distribution - Assortativeness - Preferential attachment Clustering coefficient Betweenness Statistical mechanics of complex networks Objects kindly provided by G. Rotundo Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social networks models of interest Suggested trends

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Statistical mechanics of complex networks Small-worlds 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 l ij : number of steps from one point to the other following the shortest path. In most real networks, is a very small quantity (small-world) In a square lattice of size N: In a complex network of size N: Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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The diameter of the network is small if compared to the number of nodes ~(log(N)) Examples (social networks): sociological experiment of Stanley Milgram (1967): anybody can be contacted through at most d=6 intermediaries Hollywood actors (mean) d=3.65 Co-authors in maths (mean) d =9.5 Photo of a poster in the metro station of Paris, advertising on music events small world Slide kindly provided by G. Rotundo

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Kevin Bacon number: Kevin Bacon. Nick Nolte CAPE FEARRobert De Niro GOODFELLASJoe Pesci JFK Degree of separation of Nick Nolte: 3 Val Kilmer Tom Cruise Kevin Bacon Number of intermediaries of Hollywood actors to have worked with Kevin Bacon (social game popular in 1994) Degree of separation of Val Kilmer: 2 TOP GUNA FEW GOOD MAN Example 1 Example 2 small world-Bacon number (analogously to Erdos for mathematicians) Slide kindly provided by G. Rotundo

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Statistical mechanics of complex networks 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. Betweenness, number of shortest paths that passes through a given node for all the possible paths between two nodes. Measures the “importance” of a node in a network. Number of shortest paths including v C(v)= Total number of shortest paths Betweenness (red=0,blue=max) Example: node has the most high C(v) Objects kindly provided by G. Rotundo Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Statistical mechanics of complex networks Clustering coefficient, ratio between the number E i of edges that actually exist between these k i nodes and the total number k i (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 Clustering spectrum: Average clustering coefficient of the vertices of degree k Average clustering coefficient Clustering coefficient of real networks and random graphs R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002) Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Counting triangles: friends of friends are friends of mine, too? clustering coefficient Clustering coefficient = N. triangles N. Tringles less one links + + = A B C D A is friend of B, that is friend of A and C, but A and C are not friends Transitivity property Slide kindly provided by G. Rotundo

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

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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. Uncorrelated network: independent of 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. Consequence of the conservation of edges Number of edges k k’ = number of edges k’ k Statistical mechanics of complex networks Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Fundamental concepts for network topology description Correlation measures Average degree of the nearest neighbors of the vertices of degree. Alternative to p(k’|k) k nn (k) dependent on k: correlations Assortative: k nn (k) increasing function of k Disassortative: k nn (k) decreasing function of k Statistical mechanics of complex networks Two-vertex correlations Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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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 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 Statistical mechanics of complex networks Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Nodes are divided into groups - high internal connection - low connection to the other groups community structure Example: Network of friends at school Divided by younger age, Older age, White and black Slide kindly provided by G. Rotundo

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Bocaletti et al. Physics Reports 424, 175 – 308 (2006). Directed networks and weighted networks Weighted networks: strong and weak ties between individuals in social networks Nodes Links weights Weighted degree Weighted clustering coeff. Statistical mechanics of complex networks Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Statistical mechanics of complex networks Ising model in networks: paradigm of order-disorder transitions in agent-based models 2D spin system 1D regular lattice (Ising, 1925) 2D regular lattice (onsager, 1944) No phase transition! Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends

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Statistical mechanics of complex networks Ising model in networks: paradigm of order-disorder transitions in agent-based models 2D spin system 1D WS network (Viana-Lopes et al., 2004) Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Phase transition in 1D! (long-distance correlations dramatic increase in connectivity)

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Statistical mechanics of complex networks Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Ising model in networks: paradigm of order-disorder transitions in agent-based models 2D spin system Mean-field for 1D AB network (Bianconi, 2004) Phase transition in 1D! (long-distance correlations dramatic increase in connectivity)

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Watts-Strogatz algorithm Start with order Randomize rewiring with probability p excluding self- connections and duplicate edges Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms Duncan J. Watts, Steven H. Strogatz, Nature 393, 440 (1998)

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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 Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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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. Scale free networks (e.g. Albert-Barabasi) Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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preferential attachment New nodes are attached to the hubs (preferentially) Scale-free networks Social networks models Some objects kindly provided by G. Rotundo

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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. Analysis and visualization interactive programs. POPULATION ANALYSIS Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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Pajek: http://pajek.imfm.si/doku.php Others: Cytoscape (http://www.cytoscape.org/), UCINet. Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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Much more time-consuming than ODE-based methods Automatization need: Long unattended runs. Parallelization Most convenient option: high-level language + network algorithm libraries. 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) Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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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 POSTPROCESSING Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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SIMULATION NetworkX: http://networkx.lanl.gov/, Included in Python(x,y). Generators, algebra, input/output, representation... Optimized algorithms, programmed in low level languages. Nodes can contain any type of data. Integration with NumPy. Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms SIMULATION NetworkX: http://networkx.lanl.gov/, Included in Python(x,y). Generators, algebra, input/output, representation... Optimized algorithms, programmed in low level languages. Nodes can contain any type of data. Integration with NumPy.

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Computational algorithms

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Applications: market models GDP and other macroeconomic indicators Miskiewicz and Ausloos Gligor and Ausloos Lambiotte and Ausloos Redelico, Proto and Ausloos

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Applications: market models Market correlations and concentrations. Tax evasion. Rotundo and coauthors Westerhoff and coauthors

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Applications: market models Spreading of innovations

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Applications: regional trade and development

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends Applications: other social networks

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends - Directed diffusion of commodities and people: complex networks based description - Nonlinear production processes: emergence of time and space patterns (enzyme model of production…) J. D. Murray, Mathematical Biology, Springer, 2001.

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Complex Networks Luis Miguel Varela Cabo Introduction Statistical mechanics of complex networks Computational algorithms Applications - Market models Regional trade and development - Other social network models of interest Suggested trends -Go beyond spin ½ systems (Potts models) (richness of decision). -Apply known statistical physics models (phase transitions, percolation, non-Markovian processes, linear response theory… - Combine with dynamic processes for: a) Spreading of innovations and market models b) Financial models c) Companies/banks networks d) ETC.

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MANY THANKS FOR YOUR ATTENTION

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Complex Networks Luis Miguel Varela COST meeting, Lisbon March 27 th 2013

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