2. Ferromagnetic fluid as a model of social impact Piotr Fronczak, Agatka Fronczak and Janusz A. Hołyst Faculty of Physics and Center of Excellence for Complex Systems Research, Warsaw University of Technology … Individuals emerges only in society. Society emerges only due to individuals...

3. 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

4. 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

6. What is the proper geometry for social networks ?

8. From Euclidean geometry to Newton interactions

9. Geometry = interactions

10. 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

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

12. 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

13. 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

14. Magnetic liquids Solutions of single-domain magnetic particles (~10 nm) in liquids (water, oils)

15. Main features In the presence of nonhomogenous magneti field B(r) magnetic moments are ordering along the field direction and nano-particles moving to higher field regions H = 0 H  0

16. Applications Dynamical sealing S N High-pressure region Low-pressure region MF

17. Applications Cooling and vibrations damping

18. Applications Magnetic drug targeting

19. Modeling of ferrofluids Hamiltonian Interactions Characteristic parameter

20. Ferrofluid-like model of social impact system consists of N individuals (members of a social group) each of them can share one of two opposite opinions on a certain subject, denoted as  i = ±1, i = 1, 2,...N  - grogariousness  - individuality

21. New kind of art Applications

22. The same is boring... different is attractive... σ i (t) = σ j (t) σ i (t) ≠σ j (t) Social meaning of the model

23. System phase diagram 1. 3. J(t)  J(t + 1)  J(t + 2) = J(t) J(t + 1) > J(t) J(t+1) = J(t)(1 +  +  ) J(t+2) = J(t+1)(1 +  -  ) (1 +  +  ) (1 +  -  ) = 1 2. J(t)  J(t + 1)  J(t + 2) …  J(t + 2n) = J(t) (1 +  +  ) n (1 +  -  ) n = 1

24. Algorithm 1.Dynamics of opinions in opinions in social group - Monte Carlo algorithm for spin variables (Metropolis). Temperature T – stochasticity of individuals opinions 2.After updating all N-spins we modify matrix J i,j (t). J i,j (t) iiii

25. Time scales  - time scale ratio large   faster changes of J(t) than opinions (spin) dynamics  temporary ferro- and paramagnetism m t  c < ,   1

26. Results Second order phase transition No dependence on temperature !!!

27. Why temperature does not play any role ? ~ ~ f ( exp(-J i,j / T) ) but thus in mean field m is just a function of (  /  ) It follows:

28. Temperature dependence of J i,j

29. Distributiuon of interactions strengths system is described by a full weighted graph for high temperatures – scale free distribution with  0.85

30. Conclusions Ferromagnetic fluids offer interesting analogy for modeling of social dynamcis We observed a self-organized ordered state with a second order phase transition and power law distributions of interactions strengths Mean value of /T is just a function of η/α

31. Thank you for your attention