Introduction to Ising Model and Opinion Dynamics for non-physicists (hopefully) Sang Hoon Lee, Complex Systems and Statistical Physics Lab., Dept. of Physics, KAIST, June 23, 2007. What is the Ising model? • Ising model: a crude attempt to simulate the structure of a physical ferromagnetic substance • Main virtue(?): a 2D Ising model yields to an exact treatment in statistical mechanics • The only nontrivial example of a phase transition that can be worked out with mathematical rigor Basic terminologies in statistical physics, but (possibly) not familiar to non-physicists~ cf) ferromagnetic substance (강자성체): Fe(철), Co(코발트), Ni(니켈), etc What’s the relationship with opinion dynamics? • The system is an array of N fixed points called lattice sites • Associated with each lattice site is a spin variable si (i = 1, …, N) which is a number that is either +1(↑; spin-up) or -1(↓; spin-down) • A given set of numbers {si} specifies a configuration of the whole system 3D spin configuration spin configuration in a non-regular lattice (or even a complex network!) What is spin? • Spin is the angular momentum intrinsic to particles such as atoms, protons, or electrons. Such particles and the spin of quantum mechanical systems (“particle spin”) possesses several non-classical features and for such systems spin angular momentum cannot be associated “net” magnetization = +2 (x↑) with rotation but instead refers only to the presence of angular momentum … • For the purpose of qualitative understanding of Ising model, just consider the spin variable as the basic unit of magnetization~ N S S N N or S +1 -1 N S N S N S S N N S “Thermodynamic” quantities … Energy E {si } J si s j H ij … s i i interaction with the neighbor-interaction (ferromagnetic coupling) external magnetic field External +J if si & sj have different signs field H -J if s & s have the same sign Entropy i j … “Energy-minimized” configuration Entropy … < … S ( E ) k B log E # of possible configurations for a given energy value E … Helmholtz free energy F E TS T: temperature (“control parameter”) cf) if T = 0, only the energy term is considered! Minimize F = “competition” between the energy and entropy with the control parameter T (energy minimization vs entropy maximization) Main question in statistical mechanics: What’s the configuration that minimize F for given temperature values like? What’s the average of {si}, the variance of {si}, etc? magnetization m susceptibility χ Phase transition (相轉移) We can expect that the magnetization is reduced as the temperature rises … Mathematical singularity: not differentiable, etc. (“singularity” class or type of phase transition) m disorder phase T TC (critical or Curie temperature) H=0 “spontaneous” magnetization: magnetized without any external magnetic field from the TV series “CSI” … If each spin variable represents the “individual opinion” … • Magnetization (average spin) → average opinion (majority vote) How to interpret T in the opinion dynamics? m disorder phase T ~ consensus Things to consider: “real” phase transition occurs only in the thermodynamic limit (N → ∞) & many other things, such as “how to simplify people’s opinions to the magnetic Ising model?” Discontinuous (1st order) & continuous (2nd order) transition m m T Continuous transition T Discontinuous transition Solution of self-consistency equation Topology matters! Ising model energy E J ss i i, j Regular lattice in Euclidean space – dimension, coordination number Existence of shortcuts, randomness – Watts-Strogatz small-world network Heterogeneity of connections – scale-free network j Ising model on complex networks Review paper about the critical phenomena in complex network: S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks”, e-print arXiv:0705.0010 for physicists … scale-free network : probability distribution of degree follows the power law. P( k ) k in the log-log scale, of course … p (k ) p (k ) p (k ) 2 3 3 5 The hub (opinion) dominates !!! 5 k “always magnetized”: no phase-transition k k similar to mean-field theory: mean-field theory: same different technical details as all-to-all interaction The “voter model” resembles the Ising model, but focuses on different aspects of the system the Ising model the voter model the Ising-Glauber dynamics equilibrium state pathway to equilibrium thermodynamic variables (consensus) state Monte Carlo simulation time to reach consensus (Metropolis algorithm) “Humans may dislike to be simulated like Ising spins, and clearly the brain is more complicated than one binary variables.” – D. Stauffer The voter model is the simplest and most completed solved examples of cooperative behavior The Rule of the Model Each site of a graph is endowed with two states – spin up ↑(s = +1) and down ↓(s = -1) like the Ising model For each evolution time step, i) pick a random site ii) the selected site adopts the state of a randomly-chosen neighbor These steps are repeated until a finite system necessarily reaches consensus The voter model dynamics - illustration Random Field Ising Model on Scale-Free Networks adjacency matrix: 1 if i and j connected, 0 if not external field given by random variable f (m) f (m) p (h) p (h) discontinuous (1st order) transition continuous transition S. H. Lee, H. Jeong, and J. D. Noh, Phys. Rev. E 74, 031118 (2006). Interpretation as opinion dynamics model? In the original manuscript of our paper … An anonymous referee said, Investigating the Ising model • Analytic approach – 1D Ising model: no phase transition (transfer matrix) – 2D Ising model: Onsager solution (advanced graduate level) – 3D Ising model: no exact solution (yet!), approximation based on series expansion, etc. – Mean-field theory: (relatively) easy calculation with the self-consistency equation – exact for all-to-all coupling & valid for D > 4 (upper-critical dimension) • Numerical simulation – Monte Carlo simulation • Boltzmann factor and Metropolis algorithm • Voter model dynamics ~ T=0 Ising model simulation Boltzmann factor (before introducing the basic algorithm for Monte Carlo simulation) exp( Ei ~ probability that an equilibrium system has the energy Ei in the canonical ensemble with the temperature T ) k BT probabilit y pi exp( E i / k B T ) exp( E j exp( E i / k B T ) / k BT ) Z (partition function) j Basic idea: let the system configuration “evolve” by considering the Boltzmann factor and record the average magnetization, variance, etc. Energy E {si } J si s j H ij s i i Basic Monte Carlo Simulation: Metropolis algorithm • Prepare the system & initialize • System configuration evolution “loop” for each T … – Randomly pick a site and calculate the energy change if the spin at the site is flipped – If the energy is lowered, the spin flip is accepted – If the energy gets larger, the spin flip is accepted with the probability exp( E / k B T ) – Record the average, variance (2nd moment), 4th moment, etc. (after the initial transient period; the equilibration time) • After the simulation, thermodynamic quantities are calculated with the recorded data Monte Carlo Simulation with what …? • Nothing is impossible! Anything that can calculate the exponential function (even this is not necessary for T = 0) & generate random numbers • Usually, programming languages such as C, C++, python, Fortran, Java, etc. Surprisingly, even with Microsoft Excel !!! 2D Ising model Monte Carlo simulation: C code by me “spin flipping” part Boltzmann factor spin configuration initialization record various quantities call the spin flipping function Result: time series of magnetization T ~ TC T < TC spontaneous magnetization “critical slowing down” T > TC disorder phase Result: thermodynamic quantities depending on the temperature Exact T C 1 / ln( 1 2 ) 2 . 269 ... (from the Onsager solution) Java Applets • Prof. Jae Dong Noh’s website • http://statphy1.uos.ac.kr/noh/java.html Summary • Ising model: “immortal” subject of statistical mechanics – application to opinion dynamics (with caution!) • Monte Carlo simulation based on Metropolis algorithm • Now, you’ll take a look at how we can do the simulation with Microsoft Excel~ :) Any question or comment?

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# A Very Brief Introduction to Wiki