2. Leaders and clusters in social space Janusz Hołyst Faculty of Physics and Center of Excellence for Complex Systems Research (CSR), Warsaw University of Technology

3. Interdisciplinary trade balance social impact theory historical role of political and religious leaders Ising model phase transitions nucleation theory bistability and hysteresis spontaneous self-localization scale-free networks imported from sociology:exported to sociology: trade is NOT balanced...

5. Social impact theory (B. Latane, 1981) N - individuals holding one of two opposite opinions: yes - no,  i =  1, i =1,2,3,...N (spins) Each individual is characterised by a strength parameter s i and is located in a social space, every (i,j) is ascribed a „social distance” d ij Individuals change their opinions according to  i (t+1) =  i (t) sign [-I i (t)] where I i (t) is the social impact (local field) acting on the individual i click here for demonstration

6. Condition for the cluster radius a(S L ): impact at the cluster border I(a)=0 (metastable state) After some integration: where R- radius of the social space, h – external social impact

8. Mean cluster radius as a function of the temperature

10. Critical temperature as a function of the leaders strength

11. How are leaders created ? Concentration of power, capital... Self –localization of active Brownian particles

12. System entropy S = -  p(i) ln(p(i)) as the function of the coupling parameter , where p(i)=N(i)/N Active Brownian particles clik here for demonstration

13. Influence of geometry on social dynamics Two and three dimensional systems - similar results for several immediace functions g(x) Random geometry - random distribution of social immediacies m ij

14. What is the proper geometry for social networks ?

16. (a)Internet routers; (b) movie actors; (c- d) coauthorships

17. Barabasi-Albert model of scale free-networks (Science 286, 509 1999) Network grows up: at every time step a new node with m links to old nodes is added Preferential attachment: probability  that a new node will be connected to the node i is:

18. 2 2 k=2 2 2 2 2 3 3 3 6 7 8 Ising interactions in BA model (Aleksiejuk, Holyst, Stauffer, 2002) = 1=s i = -1=s i

19. Fig. 1a: Mean magnetization versus temperature for 2 million nodes and various m Fig. 1b: Effective T c versus N for m =5

20. Fig. 2: Correlation between the number of neighbours and the local magnetization for one network of N = 4000 at T = 2; 9 and 16. The curve is the mean field prediction tanh(kM/T).

21. What is the order parameter ? k 1 =6k 2 =2 s 1 +s 2 =0 s 1 k 1 +s 2 k 2 <0 s1s1 s2s2 no order ? order ! local field created by the spin s 1 local field created by the spin s 2

22. Fig. 3: Total magnetization versus time, summed over 100 networks of N = 30; 000 when after every 50 iterations the most-connected free spin is forced down permanently. For higher temperatures the sign change of the magnetization happens sooner. Effect of leader(s) in scale-free networks –nucleation of a new phase due to pinning of most connected spins

23. Theory of spin pinning If we pine j of the most connected spins (leaders of social group ?) to the state S=-1 then all spins of the degree k >  > N are pinned where The direct decrease of the mean magnetization (per one spin)  M=2j/N The effective internal (pinning) field

24. The additional decrease of the magnetization Susceptibility

25. Conclusions In the presence of a strong leader a discontinous phase transition to the social homogenous state can take place in a social group. The transition can be induced by the social temperature. The effect is generic and occures for several geometries, random and scale-free models (by pinning a few most connected group members). Where is the hydrogen atom for this model ?