Correlation function for generalized Pólya urns: Finite-size scaling analysis

Abstract

We describe a universality class for the transitions of a generalized Pólya urn by studying the asymptotic behavior of the normalized correlation function C(t) using finite-size scaling analysis. X(1), X(2), · · · are the successive additions of a red (blue) ball (X(t) = 1 (0)) at stage t and C(t) ≡ Cov(X(1), X(t + 1))/Var(X(1)). Furthermore, z(t) = t s=1 X(s)/t represents the successive proportions of red balls in an urn to which, at the t + 1-th stage, a red ball is added (X(t + 1) = 1) with probability q(z(t)) = (tanh[J(2z(t) − 1) + h] + 1)/2, J ≥ 0, and a blue ball is added (X(t + 1) = 0) with probability 1 − q(z(t)). A boundary (J c (h), h) exists in the (J, h) plane between a region with one stable fixed point and another region with two stable fixed points for q(z). C(t) ∼ c + c ′ · t l−1 with c = 0 (> 0) for J < J c (J > J c ), and l is the (larger) value of the slope(s) of q(z) at the stable fixed point(s). On the boundary J = J c (h), C(t) ≃ c + c ′ · (ln t) −α ′ and c = 0 (c > 0), α ′ = 1/2 (1) for h = 0 (h = 0). The system shows a continuous phase transition for h = 0 and C(t) behaves as C(t) ≃ (ln t) −α ′ g((1 − l) ln t) with a universal function g(x) and a length scale 1/(1 − l) with respect to ln t. β = ν || · α ′ holds with β = 1/2 and ν || = 1.

First passage properties of a generalized Pólya urn

“…Another possible avenue to explore is the recent study of random walks with extreme value memory discussed in [38]. Finally, we note a strong parallel with ideas emerging from studies of other generalised Pólya urn processes based on a scaling analysis of the correlation function relating the first and last step [39]. …”

Phase transition of social learning collectives and the echo chamber

“…As the interaction is strong, the statistical mechanical approach is promising [13,14]. One example is the phase transition caused by the information cascade [15,16]. Information cascade is the tendency to choose the majority choice even if one thinks the minority option is correct or optimal [17,18].…”

“…It was observed that as the difficulty of the question increases, the number of stable states changes. The change in the number of the stable states induces a nonequilibrium discontinuous phase transition in the infinite population limit [16]. The performance of humans is greatly improved by social learning, and this is called the collective intelligence effect [21,22].…”

Information cascade on networks

“…On the phase boundary between the one-peak and two-peaks phase, the result for a generalized Pólya urn process suggests the following asymptotic behavior [27].…”

“…The phase transition of the voting models was detected by considering that the normalized correlation function C(t) between the first voter's choice and (t +1)-th voter's choice plays a key role [26], [27]. The function is defined as the covariance divided by the variance of the first voter's choice.…”