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Complex Networks: Structures and Dynamics

Complex Networks: Structures and Dynamics. Changsong Zhou AGNLD, Institute für Physik Universität Potsdam. Summary. Part - I Characterization of Complex Networks. Part - II Dynamics on Complex Networks. Part - III Relevance to Neurosciences. Reductionism and complexity.

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Complex Networks: Structures and Dynamics

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  1. Complex Networks:Structures and Dynamics Changsong Zhou AGNLD, Institute für Physik Universität Potsdam

  2. Summary • Part - I • Characterization of Complex Networks. • Part - II • Dynamics on Complex Networks. • Part - III • Relevance to Neurosciences.

  3. Reductionism and complexity • Brain in ``DynamicsLand´´

  4. Reductionism and complexity • Connection topology Crystal Lattices

  5. Internet Reductionism and complexity • Connection topology Crystal Lattices All-to-all interactions

  6. Internet Reductionism and complexity • Connection topology Crystal Lattices All-to-all interactions ? Mean field Diffusion

  7. Technological Networks World-Wide Web Internet Power Grid

  8. Social Networks Citation Networks Friendship Net Movie Actors Sexual Contacts Collaboration Networks

  9. Transportation Networks Airport Networks Local Transportation Road Maps

  10. Neural Networks Biological Networks Protein interaction Ecological Webs Genetic Networks Metabolic Networks

  11. A food web A Unified Approach towards the Connection Topology of various Complex Systems

  12. Symmetrical Adjacency Matrix 1 3 2 4 5 6 7 Aij = 8 Networks Approach Basic Graphs

  13. 1 3 2 4 5 6 7 8 Networks Approach Non-Symmetrical Adjacency Matrix Basic Graphs DiGraphs Aij =

  14. 7.4 0.4 1.8 4.5 0.6 0.7 2.8 0.4 8 0.5 8 5 2 Networks Approach Basic Graphs DiGraphs Weighted Graphs

  15. Characterization • Vertex degree: k(v) Basic Graphs Friendship

  16. Characterization • Clustering Coeficient: C(v) Basic Graphs Friendship

  17. Characterization • Clustering Coeficient: C(v) • Number of existing connections: 2 Basic Graphs Friendship

  18. Characterization • Clustering Coeficient: C(v) • Number of existing connections: 2 • Total Number of possible connections: • ½·kv·(kv-1) = ½·(4·3) = 6 Simple Graphs Friendship

  19. Characterization • Clustering Coeficient: C(v) • Number of existing connections: 2 • Total Number of possible connections: • ½·kv·(kv-1) = ½·(4·3) = 6 • Cv = 2 / 6 = 0.333 Basic Graphs Friendship

  20. Characterization • Clustering Coeficient: C(v) • Number of existing connections: 2 • Total Number of possible connections: • ½·kv·(kv-1) = ½·(4·3) = 6 • Cv = 2 / 6 = 0.333 Basic Graphs Friendship How well are the neighbours connected!

  21. Characterization • Distance (pathlength) Basic Graphs j Friendship i

  22. Characterization • Distance (pathlength) Basic Graphs j Friendship i

  23. Characterization • Distance (pathlength) Basic Graphs j Friendship i

  24. 3 2 3 1 2 1 0 1 2 Characterization • All-to-all distance matrix: Length of the shortest paths Lij =

  25. General Features of Real Networks • Scale-free structure Power-lawdistribution of degrees

  26. General Features of Real Networks • Small world structure Small distance High clustering

  27. Random Network Models ERDOS - RÉNYI MODELL (E-R) Connecting a pair of nodes with probability p

  28. Random Network Models ERDOS - RÉNYI MODELL (E-R) • Degree distribution: Poissonian! Mean degree K=NP

  29. Random Network Models ERDOS - RÉNYI MODELL (E-R) • Degree distribution: • Giant Component: Poissonian!

  30. Random Network Models WATTS - STROGATZ MODELL (W-S): • Degree? • Clustering? • Pathlength?

  31. Random Network Models WATTS - STROGATZ MODELL (W-S): Rewiring a link with probability p

  32. Random Network Models WATTS - STROGATZ MODELL (W-S): Having shortcuts now!

  33. Watts, Strogatz. Nature 393/4, 1998 Random Network Models WATTS - STROGATZ MODELL (W-S): • SMALL - WORLD NETS = • High clustering • Short distance

  34. Comparison Regular Lattice Small-World Net Random Graph P(k) = δ(k-Z) : Z= number of neighbours Poissonian!

  35. Random Network Models EVOLVING NETWORKS, Barabási-Albert model (B-A) • Ingredients: • Growing AND • Preferential attachment

  36. Random Network Models EVOLVING NETWORKS, Barabási-Albert model (B-A) • Ingredients: • Growing AND • Preferential attachment • Results: • “Richer-Gets-Richer” • k distribution: Scale Invariant!

  37. SCALE - FREE NETWORKS Barabási, Albert. Science 286 (1999) Random Network Models EVOLVING NETWORKS, Barabási-Albert model (B-A)

  38. Properties of the models Lattice Small-World Random Scale-Free Pathlength Long > Short ≥ Short ≥ Short Clustering Large ≥ Large ≥ Small Small Large in many real scale-free networks !

  39. Significant Impacts • Network Resiliance: • Highly robust agains RANDOM failure of node.

  40. Significant Impacts • Network Resiliance: • Highly robust agains RANDOM failure of node.

  41. Significant Impacts • Network Resiliance: • Highly robust agains RANDOM failure of node. • Highly vulnerable to deliberate attack on HUBS.

  42. Significant Impacts • Network Resiliance: • Highly robust agains RANDOM failure of node. • Highly vulnerable to deliberate attack on HUBS.

  43. Significant Impacts • Network Resiliance: • Highly robust agains RANDOM failure of node. • Highly vulnerable to deliberate attack on HUBS. • Applications: • Inmunization in computer networks and populations Cohen et al PRL, (2000, 2002)

  44. Communities and Overlapping Nodes Palla et al. Nature 435, 9 (2005) Cat cortico-cortical connections Physics collaboration network

  45. 9.3 6.0 9.3 7.2 7.0 0.3 0.3 5.1 5.1 Weighted and Directed Networks Graphs Weighted Directed In/out-degree In/out-intensity Intensity Degree k

  46. Weighted Networks • Are weights correlated with degrees? • NO ⇒ Scientific Collaborations (SCN) • YES ⇒ World-Airport-Networks (WAN)

  47. Weighted Networks • Are weights correlated with degrees? • NO ⇒ SCN • YES ⇒ WAN

  48. Weighted Networks • Weighted Clustering Coeficient: (WAN) v h v j Barrat et al. (2004) PNAS vol.101, 11

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