Statistical physics of social dynamics

Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Figure 1

Snapshots of equilibrium configurations of the Ising model (from left to right) below, at, and above $T_c$.

Figure 2

(Color online) Evolution of a two-dimensional voter model starting from a droplet (top) or a fully disordered configuration (bottom). From 132.

Figure 3

Log-log plot of the fraction $n_A$ of active bonds between nodes with different opinions. Empty symbols are for the one-dimensional case $(p=0)$. Filled symbols are for rewiring probability $p=0.05$. Data are for $N=200$ (circles), $N=400$ (squares), $N=800$ (diamonds), $N=1600$ (triangles up), and $N=3200$ (triangles left). From 73.

Figure 4

MR model. The majority opinion inside a discussion group (here of size 3) is taken by all agents.

Figure 5

Sznajd model. In the most common version of the model (Sznajd A), a pair of neighboring agents with the same opinion convince all their neighbors (top), while they have no influence if they disagree (bottom).

Figure 6

(Color online) Deffuant model. Opinion profile of a population of 500 agents during its time evolution, for $ϵ=0.25$. The population is fully mixed, i.e., everyone may interact with everybody else. The dynamics leads to a polarization of the population in two factions.

Figure 7

(Color online) Distribution of electoral performance for candidates in proportional elections held in Italy, Poland, and Finland. The overlap shows that the curve is a universal feature of the voting process. From 160.

Figure 8

Relation between the maximal speed of change and the duration of the change for birth rates and the number of mobile phones in several European countries. The linear regression of the data points in double logarithmic scale is consistent with the universal behavior predicted by the random field Ising model at zero temperature. From 345.

Figure 9

Axelrod model. Behavior of the order parameter $⟨S_{\max }⟩∕L^2$ vs $q$ for three different system sizes and $F=10$. In the inset the same quantity is reported for $F=2$. From 72.

Figure 10

Plot of the density of active links $n_a\left(t\right)$ for $F=10$, $L=150$, and (top to bottom) $q=1$, 100, 200, 230, 240, 250, 270, 300, 320, 400, 500, 10000. The inset reports the dependence of the freezing time $t_{\mathrm{co}}$ on $L$ for $F=10$ and $q=100<q_c$. The bold line has slope 2. From 72.

Figure 11

Order parameter $⟨S_{\max }⟩∕N$ as a function of the effective noise rate $r^{\prime }=r(1-1∕q)$ for different values of $q$. Simulations have been run in systems of size $N={50}^2$ with $F=10$. From 259.

Figure 12

(Color online) Naming game. Examples of the dynamics of the inventories in a failed (top) and a successful (bottom) game. The speaker selects the word highlighted. If the hearer does not possess that word, he includes it in his inventory (top). Otherwise both agents erase their inventories only keeping the winning word (bottom).

Figure 13

(Color online) Naming game. (a) Total number of words present in the system $N_w\left(t\right)$; (b) number of different words, $N_d\left(t\right)$; (c) success rate $S\left(t\right)$, i.e., probability of observing a successful interaction at time $t$. The inset shows the linear behavior of $S\left(t\right)$ at small times. The system reaches the final absorbing state, described by $N_w\left(t\right)=N$, $N_d\left(t\right)=1$, and $S\left(t\right)=1$, in which a global agreement has been reached. From 29.

Figure 14

$AB$ model. Time evolution of the average density $⟨\rho ⟩$ of bilingual individuals in small-world networks for different values of the rewiring parameter $p$. From left to right: $p=1.0$, 0.1, 0.05, 0.01, 0.0. The inset shows the dependence of the characteristic lifetime $\tau$ on the rewiring parameter $p$. The dashed line corresponds to a power-law fit $\tau \sim p^{-0.76}$. From 74.

Figure 15

(Color online) Distribution of language sizes. The $x$ axis represents the number of individuals speaking a language. In the upper diagram, on the $y$ axis the number of languages spoken by $x$ individuals is reported. In the lower diagram, the number of languages is divided by the size of the corresponding bins, obtaining the probability distribution, well fitted by a log-normal function (continuous line). The total number of languages sampled is 6142. Data from http://www.ethnologue.com/.

Figure 16

(Color online) Dynamics of language extinction according to the model of Abrams and Strogatz. The four panels show the comparison of the model with real data on the proportion of speakers over time for (a) Scottish Gaelic in Sutherland, Scotland, (b) Quechua in Huanuco, Peru, (c) Welsh in Monmouthshire, Wales, and (d) Welsh in all of Wales. In (d) both historical data (squares) and the results of a recent census (triangles) are plotted. From 1.

Figure 17

Velocity fields of particles in the model introduced by 515. A phase transition from (a) a disordered to (d) an ordered state of collective motion is observed by decreasing the amount of noise in the system. From 515.

Figure 18

(Color online) Panic behavior. (Top) Escape from a room with a single exit. The exit is clogged by the people, who can leave the room only from time to time in bunches of individuals. (Bottom) Escape from a smoky room. The time to empty the room is minimal if people maintain their self-control and look at what the others are doing. Adapted from 220.

Figure 19

Applause dynamics. Time evolution of the sound intensity of applauses in theaters and concert halls. The rhythmic applause occurs in the time window between 12 and $25\mathrm{s}$. (a) Global noise intensity, measured by a microphone (averaged over a narrow time window of $0.2\mathrm{s}$). (b) Local noise intensity. (c) Average noise intensity over a moving time window of $3\mathrm{s}$. (d) Order parameter $q$. (e) Clapping period. Adapted from 369.

Figure 20

Distribution of the response time until an email message is answered. (Inset) The same distribution is measured in ticks, i.e., units of messages sent in the system. Binning is logarithmic. The solid lines follow $\Delta t^{-1}$ and are meant as guides for the eye. From 139.

Figure 21

(Color online) Frequency-rank plots for tags co-occurring with a selected tag. Experimental data (black symbols) are shown for $\mathit{del.icio.us}$ (circles for tags co-occurring with the popular tag blog, squares for ajax, and triangles for xml). For the sake of clarity, the curves for ajax and xml are shifted down by one and two decades, respectively. All curves exhibit a power-law decay for high ranks (a dashed line corresponding to the power law $R^{-5∕4}$ is provided as a guide for the eye) and a shallower behavior for low ranks. Gray symbols are theoretical data obtained by computer simulations of the stochastic process described by 78, 77. (Inset) The same graph for the much younger system $\mathit{Connotea}$. From 77.

Figure 22

(Color online) Coevolution of opinions and networks. Histograms of cluster sizes in the consensus state for values of $\varphi$ (a) above, (b) at, and (c) below the critical point. In (b) the distribution appears to follow a power law for part of its range with exponent $-3.5\pm 0.3$, as indicated by the solid line. From 234.