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Complex Networks Albert Diaz Guilera Universitat de Barcelona

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Complex Networks 0Presentation 1Introduction –Topological properties –Complex networks in nature and society 2Random graphs: the Erdos-Rényi model 3Small worlds 4Preferential linking 5Dynamical properties –Network dynamics –Flow in complex networks

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Presentation 2 hours per session approx homework –short exercises: analytical calculations –computer simulations –graphic representations What to do with the homework? –BSCW: collaborative network tool

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BSCW Upload and download documents (files, graphics, computer code,...) Pointing to web addresses Adding notes as comments Discussions Information about access bscw.ppt

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1. INTRODUCTION Complex systems Representations –Graphs –Matrices Topological properties of networks Complex networks in nature and society Tools

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Physicist out their land Multidisciplinary research Reductionism = simplicity Scaling properties Universality

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Multidisciplinary research Intricate web of researchers coming from very different fields Different formation and points of view Different languages in a common framework Complexity

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Challenge: Accurate and complete description of complex systems Emergent properties out of very simple rules –unit dynamics –interactions

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Why is network anatomy important Structure always affects function The topology of social networks affects the spread of information Internet + access to the information - electronic viruses

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Current interest on networks Internet: access to huge databases Powerful computers that can process this information Real world structure: –regular lattice? –random? –all to all?

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Network complexity Structural complexity: topology Network evolution: change over time Connection diversity: links can have directions, weights, or signs Dynamical complexity: nodes can be complex nonlinear dynamical systems Node diversity: different kinds of nodes

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Scaling and universality Magnetism Ising model: spin-spin interaction in a regular lattice Experimental models: they can be collapsed into a single curve Universality classes: different values of exponents

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Representations From a socioeconomic point of view: representation of relational data How data is collected, stored, and prepared for analysis Collecting: reading the raw data (data mining)

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Example People that participate in social events Incidence matrix:

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Adjacence matrix: event by event Adjacence matrix: person by person

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Graphs (graphic packages: list of vertices and edges) Persons Events

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Bipartite graph Board of directors

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Directed relationships Sometimes relational data has a direction The adjacency matrix is not symmetric Examples: –links to web pages –information –cash flow

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Topological properties Degree distribution Clustering Shortest paths Betweenness Spectrum

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Degree Number of links that a node has It corresponds to the local centrality in social network analysis It measures how important is a node with respect to its nearest neighbors

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Degree distribution Gives an idea of the spread in the number of links the nodes have P(k) is the probability that a randomly selected node has k links

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What should we expect? In regular lattices all nodes are identical In random networks the majority of nodes have approximately the same degree Real-world networks: this distribution has a power tail scale-free networks

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Clustering Cycles in social network analysis language Circles of friends in which every member knows each other

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Clustering coefficient Clustering coefficient of a node Clustering coefficient of the network

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What happens in real networks? The clustering coefficient is much larger than it is in an equivalent random network

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Directedness The flow of resources depends on direction Degree –In-degree –Out-degree Careful definition of magnitudes like clustering

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Ego-centric vs. socio-centric Focus is on links surrounding particular agents (degree and clustering) Focus on the pattern of connections in the networks as a whole (paths and distances) Local centrality vs. global centrality

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Distance between two nodes Number of links that make up the path between two points Geodesic = shortest path Global centrality: points that are close to many other points in the network. (Fig. 5.1 SNA) Global centrality defined as the sum of minimum distances to any other point in the networks

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Local vs global centrality A,CBG,MJ,K,LAll other Local55211 global

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Global centrality of the whole network? Mean shortest path = average over all pairs of nodes in the network

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Betweenness Measures the intermediary role in the network It is a set of matrices, one for ach node Comments on Fig. 5.1 Ratio of shortest paths bewteen i and j that go through k There can be more than one geodesic between i and j

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Pair dependency Pair dependency of point i on point k Sum of betweenness of k for all points that involve i Row-element on column-element

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Betweenness of a point Half the sum (count twice) of the values of the columns Ratio of geodesics that go through a point Distribution (histogram) of betweenness The node with the maximum betweenness plays a central role

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Spectrum of the adjancency matrix Set of eigenvalues of the adjacency matrix Spectral density (density of eigenvalues)

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Relation with graph topology k-th moment N*M = number of loops of the graph that return to their starting node after k steps k=3 related to clustering

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A symmetric and real => eigenvalues are real and the largest is not degenerate Largest eigenvalue: shows the density of links Second largest: related to the conductance of the graph as a set of resistances Quantitatively compare different types of networks

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Tools Input of raw data Storing: format with reduced disk space in a computer Analyzing: translation from different formats Computer tools have an appropriate language (matrices, graphs,...) Import and export data

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UCINET General purpose Compute basic concepts Exercises: –How to compute the quantities we have defined so far –Other measures (cores, cliques,...)

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PAJEK Drawing package with some computations Exercises: –Draw the networks we have used –Check what can be computed –Displaying procedures

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Complex networks in nature and society NOT regular lattices NOT random graphs Huge databases and computer power simple mathematical analysis

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Networks of collaboration Through collaboration acts Examples: –movie actor –board of directors –scientific collaboration networks (MEDLINE, Mathematical, neuroscience, e-archives,..) => Erdös number

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Communication networks Hyperlinks (directed) Hosts, servers, routers through physical cables (directed) Flow of information within a company: employees process information Phone call networks ( =2)

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Networks of citations of scientific papers Nodes: papers Links (directed): citations =3

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Social networks Friendship networks (exponential) Human sexual contacts (power-law) Linguistics: words are connected if –Next or one word apart in sentences –Synonymous according to the Merrian-Webster Dictionary

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Biological networks Neural networks: neurons – synapses Metabolic reactions: molecular compounds – metabolic reactions Protein networks: protein-protein interaction Protein folding: two configurations are connected if they can be obtained from each other by an elementary move Food-webs: predator-prey (directed)

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Engineering networks Power-grid networks: generators, transformers, and substations; through high-voltage transmission lines Electronic circuits: electronic components (resistor, diodes, capacitors, logical gates) - wires

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Average path length

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Clustering

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Degree distribution

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