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Complex Networks

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

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Complex Networks

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

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

  3. 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 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

  4. 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 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={v1,…,vN}) and an unordered (ordered) set of links (E={e1…eK}). Adjacency matrix: - Contains most of the relevant information about the graph - A symmetric: undirected graph (a) - A non symmetric: directed acyclic graph (b)

  5. 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 Statistical mechanics of complex networks Measures on networks Small-worlds: Erdös-Bacon number Applications social networks Clustering coefficient financial networks Degree distribution - Assortativeness - Preferential attachment Economic networks Betweenness Objectskindlyprovidedby G. Rotundo

  6. 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 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 lij: number of steps from one point to the other following the shortest path. In most real networks, < l > is a very small quantity (small-world) In a square lattice of size N: In a complex network of size N:

  7. small world The diameter of the network is small if compared to the number of nodes ~(log(N)) Photo of a poster in the metro station of Paris, advertising on music events 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 Slide kindly provided by G. Rotundo

  8. small world-Bacon number (analogously to Erdos for mathematicians) Kevin Bacon number: Number of intermediaries of Hollywood actors to have worked with Kevin Bacon (social game popular in 1994) Example 1 Kevin Bacon. CAPE FEAR Robert De Niro GOODFELLAS Joe Pesci Nick Nolte   JFK Degree of separation of Nick Nolte: 3 Example 2 TOP GUN A FEW GOOD MAN Val Kilmer Tom Cruise Kevin Bacon Degree of separation of Val Kilmer: 2 Slidekindlyprovidedby G. Rotundo

  9. 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 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 Example: node has the most high C(v) Betweenness (red=0,blue=max) Objectskindlyprovidedby G. Rotundo

  10. 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 Statistical mechanics of complex networks 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 Average clustering coefficient Clustering spectrum: Average clustering coefficient of the vertices of degree k Clustering coefficient of real networks and random graphs R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

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

  12. 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 Statistical mechanics of complex networks 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. Objectkindlyprovidedby G. Rotundo Average degree degree 1 degree 2 degree 3 A network is called sparse if its average degree remains finite when taking the limit N>> . In real (finite) networks, <k> <<N

  13. 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 Number of edges k k’ = number of edges k’ k Statistical mechanics of complex networks 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. Fundamental concepts for network topology description independent of k Uncorrelated network: 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

  14. 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 Statistical mechanics of complex networks Two-vertex correlations Correlation measures Fundamental concepts for network topology description Average degree of the nearest neighbors of the vertices of degree. Alternative to p(k’|k) knn(k) dependent on k: correlations Assortative: knn(k) increasing function of k Disassortative: knn(k) decreasing function of k

  15. 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 Statistical mechanics of complex networks - 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. Fundamental concepts for network topology description - 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

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

  17. 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 Statistical mechanics of complex networks Weighted networks: strong and weak ties between individuals in social networks Directed networks and weighted networks Nodes Links weights Weighted degree Weighted clustering coeff. Bocaletti et al. Physics Reports 424, 175 – 308 (2006).

  18. 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 Statistical mechanics of complex networks Ising model in networks: paradigm of order-disorder transitions in agent-based models 1D regular lattice (Ising, 1925) No phase transition! 2D regular lattice (onsager, 1944) 2D spin system

  19. 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 Statistical mechanics of complex networks Ising model in networks: paradigm of order-disorder transitions in agent-based models 1D WS network (Viana-Lopes et al., 2004) Phase transition in 1D! (long-distance correlations  dramatic increase in connectivity) 2D spin system

  20. 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 Statistical mechanics of complex networks Ising model in networks: paradigm of order-disorder transitions in agent-based models Mean-field for 1D AB network (Bianconi, 2004) Phase transition in 1D! (long-distance correlations  dramatic increase in connectivity) 2D spin system

  21. 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 Watts-Strogatz algorithm Start with order Randomize rewiring with probability p excluding self-connections and duplicate edges Duncan J. Watts, Steven H. Strogatz, Nature 393, 440 (1998)

  22. 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 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

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

  24. Scale-free networks preferential attachment New nodes are attached to the hubs (preferentially) Social networks models Someobjectskindlyprovidedby G. Rotundo

  25. 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 POPULATION ANALYSIS • 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.

  26. 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 Pajek: http://pajek.imfm.si/doku.php Others: Cytoscape (http://www.cytoscape.org/), UCINet.

  27. 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 • 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)

  28. 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 POSTPROCESSING • 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

  29. 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.

  30. 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.

  31. 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

  32. 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

  33. 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 Miskiewicz and Ausloos Gligor and Ausloos Lambiotte and Ausloos Redelico, Proto and Ausloos Applications: market models GDP and other macroeconomic indicators

  34. 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 Rotundo and coauthors Westerhoff and coauthors Applications: market models Market correlations and concentrations. Tax evasion.

  35. 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

  36. 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

  37. 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

  38. 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 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.

  39. 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 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.

  40. MANY THANKS FOR YOUR ATTENTION

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

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