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...

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 si and is located in a social space, every (i,j) is ascribed a „social distance” dij

• Individuals change their opinions according to i (t+1) = i (t) sign [-Ii(t)] where Ii(t) is the social impact (local field) acting on the individual i

click here for demonstration

Condition for the cluster radius a(SL):

impact at the cluster border I(a)=0 (metastable state)

After some integration:

LShRhRa 32)2(2

16

1 2

where

R- radius of the social space,

h – external social impact

What is the proper geometry for social networks ?

From Euclidean geometry

to Newton interactions

Geometry = interactions

2 2

k=2

2

2

2

2

3

3

36

78

Ising interactions in BA model (Aleksiejuk, Holyst, Stauffer, 2002)

= 1=si

= -1=sij

jiissJH

***

,

Fig. 1a: Mean magnetization versus temperature for 2 million nodes and various m

Fig. 1b: Effective Tc versus Nfor m =5

What is the order parameter ?

k1=6 k2=2

s1+s2=0

s1k1+s2k2<0

s1 s2

no order ?

order !

local field created by the spin s1

local field created by the spin s2

Fig. 3: Total magnetization versus time, summed over 100 networks ofN = 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

Magnetic liquids

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

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

Applications

• Dynamical sealing

S

N

High-pressure region

Low-pressure region

MF

Applications

• Cooling and vibrations damping

Applications• Magnetic drug targeting

Modeling of ferrofluids

i

iji

ijexjiij BrrH )()(

Hamiltonian

)()(

),()(

,)(

rar

drrar

drr

mgd

ex

dattr

Interactions

rdr

rdrR

attr

ex

)(

)(

Characteristic parameter

Ferrofluid-like model of social impact

jijiji

jijiji

tJtJ

tJH

1)()1(

)(

,,

,

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

• New kind of art

Applications

The same is boring... different is attractive...

σi(t) = σj(t) σi(t) ≠ σj(t)

)()()( ,, tJttJ jijiji

Social meaning of the model

System phase diagram

jijiji tJtJ 1)()1( ,,

1.

13.

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.

11 2 c

J(t) J(t + 1) J(t + 2) … J(t + 2n) = J(t)

(1 + + )n (1 + - )n = 1

Algorithm1. 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 Ji,j(t).

jijiji tJtJ 1)()1( ,,

Ji,j(t) ii

Time scales

- time scale ratio

jijiji tJtJ 1)()1( ,,

large faster changes of J(t) than opinions (spin) dynamics temporary ferro- and paramagnetism

m

t

c < , 1

Results

Second order phase transition

No dependence on temperature !!!

Why temperature does not play any role ?

< ij > ~ <m> ~ f ( exp(-Ji,j / T) )but

thus

in mean field

m is just a function of ( / ) It follows:

Temperature dependence of Ji,j

jiJ ,

Distributiuon of interactions strengths

• system is described by a full weighted graph• for high temperatures – scale free distribution with 0.85

)(, tJ ji

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 <Ji,j>/T is just a function of η/α

Thank you for your attention