Complex Networks

1. Complex Networks Albert Diaz Guilera Universitat de Barcelona

2. Complex Networks • Presentation • Introduction • Topological properties • Complex networks in nature and society • Random graphs: the Erdos-Rényi model • Small worlds • Preferential linking • Dynamical properties • Network dynamics • Flow in complex networks

3. 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

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

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

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

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

8. Complexity • Challenge: “Accurate and complete description of complex systems” • Emergent properties out of very simple rules • unit dynamics • interactions

9. 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

10. 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?

11. 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

12. 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

13. 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)

14. Example • People that participate in social events • Incidence matrix:

15. Adjacence matrix: event by event Adjacence matrix: person by person

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

17. Bipartite graph • Board of directors

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

19. Topological properties • Degree distribution • Clustering • Shortest paths • Betweenness • Spectrum

20. 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

21. 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

22. 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

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

24. Clustering coefficient • Clustering coefficient of a node • Clustering coefficient of the network

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

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

27. 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

28. 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

29. Local vs global centrality

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

31. 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

32. 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

33. 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

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

35. 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

36. 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

37. 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

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

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

40. Complex networks in nature and society • NOT regular lattices • NOT random graphs • Huge databases and computer power “simple” mathematical analysis

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

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

43. Networks of citations of scientific papers • Nodes: papers • Links (directed): citations • =3

44. 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

45. 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)

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

47. Average path length

48. Clustering

49. Degree distribution