1. Complex Networks: 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. 1. INTRODUCTION Reductionism and complexity Brain in ``DynamicsLand´´

4. 1. INTRODUCTION Connection topology Reductionism and complexity Crystal Lattices

5. 1. INTRODUCTION Internet All-to-all interactionsCrystal Lattices Reductionism and complexity Connection topology

6. 1. INTRODUCTION Internet All-to-all interactionsCrystal Lattices Reductionism and complexity Connection topology Diffusion Mean field ?

7. 1. INTRODUCTION Technological Networks World-Wide Web Power Grid Internet

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

9. 1. INTRODUCTION Transportation Networks Airport Networks Road Maps Local Transportation

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

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

12. 2. NETWORKS... Networks Approach Basic Graphs Symmetrical Adjacency Matrix 1 2 4 56 7 8 3 A ij = 12345678 1 01100000 2 10111000 3 11000100 4 01001010 5 01010110 6 00101001 7 00011001 8 00000110

13. 2. NETWORKS... Basic GraphsDiGraphs 12345678 1 01100000 2 00101000 3 10000000 4 01001000 5 00000100 6 00101000 7 00011001 8 00000110 Non-Symmetrical Adjacency Matrix 1 2 4 56 7 8 3 A ij = Networks Approach

14. 2. NETWORKS... 2 8 5 0.58 Basic GraphsDiGraphsWeighted Graphs 0.4 0.7 4.5 1.8 2.8 0.6 0.4 7.4 Networks Approach

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

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

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

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

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

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

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

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

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

24. 2. NETWORKS... Distance Basic Graphs Characterization Friendship

25. 2. NETWORKS... Distance Basic Graphs Characterization

26. 2. NETWORKS... Distance Simple Graphs Characterization

27. 2. NETWORKS... Distance Simple Graphs 0 Characterization

28. 2. NETWORKS... Distance Simple Graphs 0 1 1 1 Characterization

29. 2. NETWORKS... Distance Simple Graphs 0 1 1 1 2 2 2 Characterization

30. 2. NETWORKS... Characterization Distance Simple Graphs 0 1 1 1 2 2 2 3 3 Distance: length of the shortest paths

31. 2. NETWORKS... All-to-all distance matrix: Length of the shortest paths 0 1 1 1 2 2 2 3 3 123456789 1 211222334 2 121112234 3 112221323 4 212212123 5 212121123 6 221212220 7 323112212 8 332221121 9 443332212 L ij = Characterization

32. 2. NETWORKS... Other measures Simple Graphs Characterization Neighbours` degree Average degree of the neighbouring nodes Betweenness (load) The number of shortest paths passing a node or an edge

33. 2. NETWORKS... General Features of Real Networks Scale-free structure Power-law distribution of degrees

34. 2. NETWORKS... General Features of Real Networks Small world structure Small distance High clustering

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

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

37. 2. NETWORKS... Degree distribution: Giant Component: Poissonian! ERDOS - RÉNYI MODELL (E-R) Random Network Models

38. 2. NETWORKS... WATTS - STROGATZ MODELL (W-S): Degree? Clustering? Pathlength? Random Network Models

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

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

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

42. 2. NETWORKS... Regular LatticeSmall-World NetRandom Graph P(k) = δ(k-Z) : Z= number of neighbours Poissonian! Comparison

43. 2. NETWORKS... Comparison Average Pathlength Average Clustering

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

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

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

47. 2. NETWORKS... Properties of the models Small-WorldLatticeScale-FreeRandom Pathlength Clustering LongShort Large Small > Large in many real scale-free networks !

48. 2. NETWORKS... Small-WorldLatticeScale-FreeRandom Pathlength Clustering LongShort Large Small > Properties of the models

49. 2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. Significant Impacts

50. 2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. Significant Impacts

51. 2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. –Highly vulnerable to deliberate attack on HUBS. Significant Impacts

52. 2. NETWORKS... Network Resiliance: –Highly robust agains RANDOM failure of node. –Highly vulnerable to deliberate attack on HUBS. Significant Impacts

53. 2. NETWORKS... 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) Significant Impacts

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

55. 3.... AND MORE REALISTIC CHARACTERIZATION Weighted and Directed Networks GraphsWeightedDirected 5.1 7.0 9.3 0.3 5.1 7.2 9.3 0.3 6.0 degree, k( ) = 4 out-degree, out-k( ) = 3 in- degree, in-k( ) = 2 intensity. S( ) = 21.7out-intensity, out-S( ) =24.9 in-intensity, in-S( ) = 12.3 Degree k Intensity In/out-degree In/out-intensity

56. 3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Are weights correlated with degrees? –NO Scientific Collaborations (SCN) –YES World-Airport-Networks (WAN)

57. 3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Are weights correlated with degrees? –NO SCN –YES WAN

58. 3.... AND MORE REALISTIC CHARACTERIZATION Weighted Networks Weighted Clustering Coeficient: (WAN) Barrat et al. (2004) PNAS vol.101, 11 v v j h