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

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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...

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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

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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

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Mean cluster radius as a function of the temperature

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Critical temperature as a function of the leaders strength

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

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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

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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

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What is the proper geometry for social networks ?

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

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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:

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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

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

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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).

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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

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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

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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

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The additional decrease of the magnetization Susceptibility

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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 ?

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