1. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 International Workshop International Workshop on Challenges and Visions in the Social Sciences in the Social Sciences ETH Zürich ETH Zürich August 18-23, 2008 August 18-23, 2008 New Complex Networks for Social Relations Hans J. Herrmann Computational Physics, IfB ETH, Zürich, Switzerland Marta C. González José Soares Andrade Jr. Pedro Lind Herman Singer

2. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008   The statistical spreading of information or contacts in societies has many practical implications. It involves non-equilibrium phenomena where fluctuations and correlations play an important role.   Usually these processes are studied by sociologists using surveys including many parameters and obtaining qualitative results. Recently techniques from statistical physics and in particular complex networks and non- linear dynamics have been used to make simplified models and obtain quantitative laws. Motivation

3. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Data from schools visualization using „pajec“ Survey interviewing 90118 student from 84 schools in US with T. Vicsek and J. Kertész (Add Health Program)

4. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Schools3-cliques 4-cliques

5. Schools affinities between black and white students

6. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Food web of the North Atlantic Ocean Complex Networks

7. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Scale-free networks scientific collaborations WWW: Internetactors HEPneuroscience Model: Barabasi-Albert  = 3

8. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Apollonian network scale-free (ultra) small world Euclidean space-filling matching

9. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Apollonian packing

10. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Other applications Force networks in polydisperse packings Highly fractured porous media Networks of roads Systems of electrical supply lines

11. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Degree distribution

12. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Small-world properties

13. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Percolation threshold epidemics epidemics random failure

14. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Electrical conductance from center to all sites of generation n: from flux flux fuses = malicious attack

15. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Critical temperature of Ising model opinion

16. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Generalized Apollonians m=0 m=1 m=2 m=3 m=4 m= ∞ n=∞

17. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Osculatory packing

18. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Apollonian packing in 3D Apollonian packing in 3D Tetrahedron

19. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Conclusions on Apollonian networks Stable against random failure. Opinions stabilize at any interaction strength. Supply current distribution follows power-law while point-to-point distribution is log-normal. Apollonian networks are between small world and ultrasmall world Force networks of polydisperse packings have few contacts between grains and their rigidity increases with density like a power-law. Porous medium follows Archie‘s law..

20. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip

21. Gossip on network

22. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip propagation Spreading time  is the minimum time it takes for a gossip to reach all accessible persons. n f is the total number of persons that eventually get the gossip. We also define the „spread factor“ f: where k is the degree of the victim.

23. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip in schools

24. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip on random graph

25. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Spreading time Barabasi-Albert m=7 m=5 m=3

26. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Spreading time schools Apollonian B-A, m=7

27. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Spreading factor schools Barabasi-Albert network m=7

28. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Spreading factor k 0 is optimal degree for Barabasi-Albert m=3 m=5 m=7 Barabasi-Albert Watts-Strogatz

29. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Distribution of  Apollonian Schools B-Am=7

30. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip with probability q schools

31. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Gossip about famous people symbols = schools lines = Barabasi-Albert

32. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Degree distribution K nn (k) is the average degree of the K nn (k) is the average degree of the nearest neighbors of a site of degree k. average over all schools all schools one school

33. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks females males F. Liljeros et al, Nature 411, 907 (2001) homosexual (Colorado Springs) heterosexual (Sweden) B-A model agents

34. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Agent model Power-law distribution of Infection Time collision time interaction potential soft-disks in 2d Marta MartaGonzález

35. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Evolution of network

36. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Evolution of network with life-times

37. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Shortest path and clustering coefficient González, Lind and Herrmann, Physical Review Letters 96, 088702 (2006) Comparison to schools

38. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Degree distribution K nn (k) is the K nn (k) is the average degree of the nearest neighbors of a site of degree k. average over all schools all schools one school Comparison to schools

39. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 3-clique communities one school average over all schools all schools symbols:schools full lines: agent model

40. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 SIS model of epidemic Susceptible Infected Δt inf = (MD time-steps)  t Susceptible infection rate λ directed percolation

41. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 SIS model for epidemics Steady State Susceptible Infected F IM = fraction of infected agents λ = 5.45 λ =1.2

42. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 scaling law for λ > λ c : order parameter Data collapse with

43. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Non-equilibrium phase transitions System goes into the absorbing state

44. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Influence of the density

45. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Critical exponent β

46. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Other critical properties =1.07, 1.08, 1.09 =0.93, 0.94, 0.95 P(t) = probability that infection stays n(t) = average number of infected agents R 2 (t) = mean square spreading distance δ η z

47. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Critical exponents P. Grassberger, J. Phys. A., (1989) 3673-3679 J. Marro and R. Dickman, “Non-equilibrium Phase Transitions…”, (1999) Cambridge University Press.

48. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Mean Field Moving Agents  Moving Agents  Contact Process 2D c 11.0(8)0.9(4)1.6488(1)  10.9(2)0.7(7)0.583(4)  10.5(9)0.5(3)0.4505(10)  00.1(5)0.2(5)0.2295(10) z11.3(0)1.2(7)1.1325(10) Critical exponents 4  +2  =dz holds.

49. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Networks properties density = 0.1 T = 1,  density = 0.23 T = 1,  Rules for generating the network Each time one agent infects other a bond between the two is created. The infection lasts (  t inf time steps), when one of the agents is recovered the bond disappears and the agent becomes susceptible to infection again. Labeling of Cluster sizes Susceptible RGB scale for cluster size

50. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Cluster size distribution N =16,  Cluster numbers Second moment of the cluster size distribution Probability that an agent belongs to the biggest cluster In Collab. with: A.D. Araújo, UFC, Brazil Susceptible Clusters size same same scaling scaling as in as inpercolation

51. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Cluster numbers

52. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Clustering coefficient and degree distribution    

53. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Power-law distribution of infection time  Susceptible Clusters size 

54. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Conclusions on epidemics Novel effects are observed studying the SIS model of infection on a system of moving agents. A continuous range of critical exponents is found as a function of the density number of agents. Geometrical properties like the degree distribution and the clustering coefficient of the network of infected agents depend on the rate of infection and not on the density of agents. The distribution of cluster sizes is not a power law at the transition to spreading. Introducing a power law distribution of infection times the epidemic threshold becomes zero, but there is still a critical rate of infection that depends on the exponents of the distribution and the mean interaction time among the agents.

55. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks females males F. Liljeros et al, Nature 411, 907 (2001) homosexual (Colorado Springs) heterosexual (Sweden) B-A model agents

56. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Comparison with real data J.J. Potterat et al, Sex.Transm.Infect. 78, 59 (2002) Barabasi-Albert scale-free network Velocity of agent proportional to k  with  = 1.0 – 1.2.

57. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Bipartite growing network

58. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Bipartite growing network

59. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Comparison with real data heterogeneous: homogeneous:

60. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks Sexual contacts only with uninitiated agents. uninitiated agents. Sexual contacts with any agent. any agent. Degree distribution

61. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks females males F. Liljeros et al, Nature 411, 907 (2001) homosexual (Colorado Springs) heterosexual (Sweden) B-A model agents

62. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks F. Liljeros et al, Nature 411, 907 (2001) heterosexual 58% females 42% males t = 0.12 t = 0.77

63. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Sexual contact networks

64. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Another friendship model Goal –friendship network model (spatially independent) for a fixed setting (e.g. enrolling at a university) –reproduce experimentally measurable quantities –natural emergence of community structures Herman Singer, ETH

65. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Properties of the agent agent properties –affinity –list of past contacts, length n i –maximal acquaintance parameter λ i –absolute importance –relative importance behavior –agent can choose between meeting new contacts and meeting again already established contacts –similar to aging but important difference: agent can accept new contacts throughout the simulations at the expense of dropping old ones → every agent optimizes its interest → local, self organized community structure emerges

66. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Friendship in formulas friendship is written as the weighted sum of the contributions –number of times n ij –interest/affinity a i, a j –with γ >0.8 friendship functions: affinity f 1 –with κ ≥ 2, here 2,3,5,10 frequency f 2 –exponential –with λ f =0.2 ↔ saturation on ~25 contacts

67. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Algorithm t=0: initialize N agents, no connections with parameters a i, n i =0, λ i t→t+1: –choose an agent randomly: i –decide on contact mode with probability select an agent from pool with probability Π j –add to contact list otherwise select an agent in contact list with propability p ij prevent social isolation and take into account random encounters: –introduce threshold p≥θ p

68. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Degree distribution

69. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Degree correlation P(k|k'): probability at a node with k to find a neighbor with k' →

70. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Clustering coefficient

71. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Friendship distribution Experimental results –417 high school students –"who are your best friends?" –probability that a person is mentioned n times

72. International Workshop on Challenges and Visions in the Social Sciences, ETH Zürich, Aug. 18-23, 2008 Social relation is difficult to quantify. Many simplified models are possible. Parameters can be tuned to agree with some data. One can find general conclusions. But does anybody believe these generalities? Does anybody care about this research? Do we need socio-physics? What does one learn from modeling? Conclusion