Correlation function for generalized P\'olya urns: Finite-size scaling analysis

0.29 MB·English

by  Shintaro Mori

Correlation function for generalized Po´lya urns:
Finite-size scaling analysis
Shintaro Mori∗
5
1 Department of Physics, Kitasato University
0
2
Kitasato 1-15-1, Sagamihara, Kanagawa 252-0373, JAPAN

v
o
N
Masato Hisakado

6
Financial Services Agency

]
h Kasumigaseki 3-2-1, Chiyoda-ku,
c
e
Tokyo 100-8967, Japan
m
- (Dated: November 9, 2015)
t
a
t
s Abstract
.
t
a
We describe a universality class for the transitions of a generalized Po´lya urn by studying the
m
- asymptotic behavior of the normalized correlation function C(t) using finite-size scaling analy-
d
n
o sis. X(1),X(2), are the successive additions of a red (blue) ball (X(t) = 1(0)) at stage t
···
c
[ and C(t) Cov(X(1),X(t + 1))/Var(X(1)). Furthermore, z(t) = t X(s)/t represents the
≡ s=1

3 P
successive proportions of red balls in an urn to which, at the t+1-th stage, a red ball is added
v
4
6 (X(t+1) = 1) with probability q(z(t)) = (tanh[J(2z(t) 1)+h]+1)/2,J 0, and a blue ball is
− ≥
7
0 added (X(t+1) = 0) with probability 1 q(z(t)). A boundary (Jc(h),h) exists in the (J,h) plane
−
0
. between a region with one stable fixed point and another region with two stable fixed points for
1
0
5 q(z). C(t) c+c′ tl−1 with c = 0(> 0) for J < Jc(J > Jc), and l is the (larger) value of the
∼ ·
1
v: slope(s) of q(z) at the stable fixed point(s). On the boundary J = Jc(h), C(t) ≃ c+c′ ·(lnt)−α′
i
X and c = 0(c > 0),α′ = 1/2(1) for h = 0(h = 0). The system shows a continuous phase transition
6
r
a for h = 0 and C(t) behaves as C(t) (lnt)−α′g((1 l)lnt) with a universal function g(x) and a
≃ −
length scale 1/(1 l) with respect to lnt. β = ν α′ holds with β = 1/2 and ν = 1.
|| ||
− ·
PACS numbers: 05.70.Fh,89.65.Gh
∗
[email protected]
1
I. INTRODUCTION
The contagion process is one of the most studied topics in statistical physics and has
attracted the attention of many researchers from various disciplines [1–8]. Po´lya urn is one
of the simplest models for this process [9–11]. In this model, an urn consists of t balls, where
the proportion of red balls is z(t) (0,1) and the rest of the balls are blue. The probability
∈
of a new red ball being added to the urn is z(t), while it is 1 z(t) for a new blue ball; the
−
proportion of red balls then becomes z(t+1). This procedure is iterative, which produces a
sequence of proportions z(t ),z(t +1),z(t +2), , where the urn contained t z(t ) red
0 0 0 0 0
··· ·
balls at t = t . The limit value lim z(t) obeys a beta distribution with shape parameters
0 t→∞
α = t z(t ) and β = t z(t ).
0 0 0 0
· ·
As the Po´lya urn process is very simple and there are many reinforcement phenomena in
natureandthesocialenvironment, manyvariantsoftheprocesshavebeenproposed, referred
to as generalized Po´lya urn processes[12]. In the nonlinear generalizations of this model, a
continuous function q : [0,1] [0,1] determines the probability q(z(t)) of a red ball being
→
added at stage t+1. This nonlinear version is referred to as a nonlinear Po´lya process[12–
14]. In contrast to the original linear model, the nonlinear model can have many isolated
stable states. The fixed point z of q(z), where q(z ) = z , is (un)stable if (z z )(q(z) z)
∗ ∗ ∗ ∗
− −
is negative (positive) for all z in the vicinity of z [13]. z is referred to as downcrossing
∗ ∗
(upcrossing) as the graph y = f(z) crosses the curve y = z in the downwards (upwards)
direction. The slope of q(z) at z is smaller (larger) than one when z is downcrossing
∗ ∗
(upcrossing). When q(z) touches the diagonal q = z in the (z,q) plane at z , a point that
t
is referred to as the touchpoint, the stability of z depends on the difference between the
t
slope of q(z) and the diagonal z in the left neighborhood of z [14]. If it is less (more) than
t
1/2, z is (un)stable. The multiplicity of the stable states provides a convenient picture
t
that explains the lock-in phenomena in the technology and product adoption processes [15].
Suppose two selectively neutral technologies enter the market at the same time. Because
economies of scale play the role of an externality that persuade a new consumer to buy the
dominant technology, determining which technology to buy depends on the proportion of
each technology possessed by previous consumers. If the dependence is described by the
non-linear function q(z), the technology adoption process is described by a non-linear Po´lya
urn. An S-shaped q(z) function with two stable fixed points suggests the random monopoly
2
formed when one technology dominating over the other depends on chance fluctuations at
the start [? ].
Information cascade provides a good experimental setup for verifying theoretical predic-
tions [16–19]. Here, participants sequentially answer questions with two possible choices
(two-choice questions). In addition to their own information or knowledge, they refer to
social information about the number of previous subjects that chose each option. In an
experiment where subjects answer general knowledge two-choice questions, it is possible to
change the number of stable states by controlling the difficulty of the question [18, 19]. A
subject that knows the answer to a question chooses the correct answer with a probability
of 1. A subject who does not know the answer tends to choose the majority choice. By
changing the difficulty of the question, an experimenter can control the ratio p of the latter,
no-knowledge, subject. We denote the probability that the no-knowledge subject choose the
correct answer as q (z), when the ratio of correct choices among previous subjects is z. The
h
probability that a subject choose the correct answer is then q(z) = (1 p) 1 + p q (z).
h
− · ·
The sequential voting process in the experiment is described by a non-linear Po´lya urn. It
was shown that q(z) has one (two) stable fixed point(s) for p < p (> p ).
c c
If there is only one stable fixed point, z(t) converges to it. In the case of multiple stable
fixed points, the stable fixed point to which z converges will be random[13]. By controlling
the model parameters, the number of stable fixed points can be changed, which induces a
non-equilibrium transition. In an exactly solvable case, where q(z) is a combination of the
constantq (1/2,1]andtheHeavisidefunctionθ(z 1/2),i.e., q(z) = (1 p) q +p θ(z 1/2)
∗ ∗
∈ − − · · −
with a correlation control parameter p [0,1], q(z) touches the diagonal at z = 1/2 for
t
∈
p = p = 1 1/2q . For p < p , there is a unique stable state at z = (1 p)q + p. For
c ∗ c + ∗
− −
p > p , there are two stable states at z = (1 p)q p. Because the slope of q(z) in the left
c ±
− ±
neighborhood of z is 0, the touchpoint at z is unstable, and z(t) converges to the stable
t t
fixed point at z for p = p [20].
+ c
The probability of convergence to a stable fixed point depends strongly on the color of
the first ball when there are multiple stable states. If the color is red and z(1) = 1 (blue
and z(1) = 0), the probability of convergence to a larger stable fixed point becomes higher
(lower). The difference in the probabilities is given by the limit value c of the normalized
correlation function C(t) between the color of the first ball and that of the t + 1-th ball.
Furthermore, c plays the role of the order parameter for the phase transition. In the afore-
3
mentioned exactly solvable model, which we refer to as the ”digital” model, C(t) at p = p
c
shows a power law dependence on t as C(t) t−α with α = 1/2. The order parameter be-
∝
haves as c (p p )β with β = 1. In addition, C(t) obeys the scaling form C(t) t−αg(t/ξ)
c
∝ − ∝
near p with a universal function g and correlation length ξ. ξ diverges as ξ p p −ν||
c c
∝ | − |
with ν = 2 [21]. The scaling relation β = ν α holds as in the absorbing states phase
|| ||
·
transition [22, 23].
In this work, we use finite-size scaling (FSS) analysis in order to study the asymptotic
behavior of the correlation function for generalized Po´lya urns. We adopt a logistic-type
model q(z) = (tanh[J(2z 1) + h] + 1)/2 with two parameters J and h. Here, J is the
−
parameter of the strength of the correlation, and h is the parameter of the asymmetry. The
motivation for adopting this model was derived from experimental findings [18, 19]. With
this choice, there is a threshold value J (h), and at J = J (h), q(z) becomes tangential to
c c
the diagonal at z . From the above discussion, the touchpoint at z is stable, which differs
t t
from the digital model. There are two stable states at z and z for h = 0. If the order
t +
6
parameter c takes a positive value at J = J (h), the phase transition becomes discontinuous.
c
For h = 0, the touchpoint at z = 1/2 is the unique stable state. c should be equal to zero
t
and the phase transition becomes continuous. We also clarify the universality class of the
continuous transition.
The remainder of the paper is organized as follows. Section II introduces the model. In
section III, we discuss the asymptotic behavior of C(t) for J = J (h) using previous results,
c
6
andpropose the asymptotic form C(t) c+c′(lnt)−α′ for J = J (h). In section IV, we study
c
≃
the FSSrelations of thesystem. Section Vis devoted to the FSSstudy of C(t) for J = J (h).
c
We show that c > 0 (c = 0) and α′ = 1 (α′ = 1/2) for h > 0 (h = 0). The system shows
a continuous phase transition for h = 0. In section VI, we study the universality class of
the continuous transition. We show that C(t) (lnt)−α′g(lnt/ξ) with a universal function
∝
g(x) and a length scale ξ. We define the critical exponents β and ν as c (J J )β and
|| c
∝ −
ξ J J −ν||, respectively. Using the scaling relation β = ν α′ with ν = 1, we obtain
c || ||
∝ | − | ·
β = 1/2. Section VII provides our summary and further comments. In Appendix A, we
derive the explicit form of g(x) for the digital model.
4
II. MODEL
We define the stochastic process X(t) 0,1 ,t 1,2, ,T , where the probability
∈ { } ∈ { ··· }
that X(t) takes value of 1 is given by a function q(z) of the proportion z(t 1) of the
−
variables X(1), ,X(t 1) that are equal to 1.
··· −
1
q(z) Pr(X(t) = 1 z(t 1) = z) = (tanh[J(2 z 1)+h]+1),
≡ | − 2 · −
t
1 1
z(t) = X(s) for t > 0, and z(0) = . (1)
t 2
Xs=1
The choice of q(z) is arbitrary, and we adopt the above form, which is familiar in the field
of physics [24, 25]. The fixed point of q(z) is a solution to q(z) = z. Using the mapping
m = 2z 1, we obtain the self-consistent equation m = tanh(J m+h) for the magnetization
− ·
m in the mean-field Ising model. Here, we consider only the case for which J 0. Because
≥
of the symmetry under (X,h) (1 X, h), we also assume that h 0.
↔ − − ≥
1
J (h)
c
0.8
0.6
h
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3
J
FIG. 1. Plot of J (h) in (J,h).
c
The number of fixed points for q(z) depends on (J,h). There is a threshold value J =
J (h) as a function of h (Fig. 1). For J < J (h), there is only one fixed point at z = z
c c +
(Fig.2(a)). With increasing J, q(z) becomes tangential to the diagonal z at z for J = J (h).
t c
For h > 0, z = z , and both z and z are stable (Fig.3(b)). For h = 0, z = z , and
t + t + t +
6
it is also stable (Fig.3(a)). In both cases, the slope of the curve at z is equal to one. For
t
J > J (h), there are three fixed points, and we denote them as z < z < z ; z is stable,
c − u + ±
and z is unstable (Fig.2(b)). We denote the value q(z ),q(z ) as q ,q and the slope of q(z)
u ± t ± t
at z ,z as l q′(z ),l = q′(z ) = 1. As z is stable and downcrossing, l < 1.
± t ± ± t t ± ±
≡
5
(a) 1 (b) 1
q(z) q(z)
z z
0.8 0.8
0.6 0.6
z) z)
q( q(
0.4 0.4
0.2 0.2
0 0
z z z z
0 + 1 0 - u + 1
z z
FIG. 2. Plot of q(z) vs. z for h > 0. The intersection between y = q(z) and y = z is the fixed
point of q(z). (a) J < J (h), with one fixed point at z = z . (b) J > J (h), with three fixed points
c + c
at z z ,z ,z .
− u +
∈{ }
(a) 1 (b) 1
h=0,J=J h>0,J=J
c c
J>J J>J
c c
0.8 z 0.8 z
0.6 0.6
z) z)
q( q(
0.4 0.4
0.2 0.2
0 0
z z z z z z z z’
0 + t - 1 0 - t u + +
z z
FIG. 3. Plot of q(z) vs. z for J = J (h) (thin solid line) and J > J (h) (thick solid line) for (a)
c c
h = 0 and (b) h> 0.
We note a crucial difference between h = 0 and h > 0. For h = 0, the touchpoint at
z = 1/2 for J = J (0) = 1 coincides with the stable fixed point at z = 1/2 for J < J (0).
t c + c
It splits into the two stable fixed points at z = z for J > J (0). (Fig.3(a)). z continuously
± c ±
moves away from z and z z (J J )1/2 for J J << 1 as in the case of the mean
t + − c c
− ∝ − | − |
field Ising model. On the other hand, for h > 0, the touchpoint z appears at a different
t
position from z for J = J (h) (Fig.3(b)). As J increases from J (h), z splits into z and
+ c c t −
z . The change from J < J (h) to J > J (h) is discontinuous. This difference suggests that
u c c
the phase transition is continuous for h = 0 and discontinuous for h > 0.
6
III. ASYMPTOTIC BEHAVIOR OF C(t)
In this section, we derive the asymptotic form of the correlation function C(t) using the
previous results for J = J (h). Based on them, we assume the functional form of C(t) for
c
6
J = J (h). C(t) is defined as the covariance between X(1) and X(t + 1) divided by the
c
variance of X(1),Var(X(1)):
C(t) Cov(X(1),X(t+1))/Var(X(1)).
≡
Through normalization, C(0) = 1. C(t) can be expressed as the difference between two
conditional probabilities.
C(t) = Pr(X(t+1) = 1 X(1) = 1) Pr(X(t+1) = 1 X(1) = 0). (2)
| − |
In general, C(t) is positive for J > 0.
A. C(t) for J = J (h)
c
6
The asymptotic behavior of C(t) depends on (J,h). If J < J (h), there is one stable fixed
c
point at z andz(t) converges toz throughthe power-law relationE(z(t) z ) tl+−1 [26].
+ + +
− ∝
Here, the expectation value E(A) of a certain quantity A is defined as the ensemble average
over the paths of the stochastic process. If J > J (h), there is another stable fixed point at
c
z . Both z are stable, and z(t) converges to one of the fixed points. The convergence of
− ±
z(t) to z also exhibits a power-law behavior E(z(t) z ) tl±−1 [27]. We assume that the
± ±
− ∝
probability that z(t) converges to one of the z depends on X(1) and we denote this as
±
p (x) Pr(z(t) z X(1) = x).
± ±
≡ → |
For J < J (h), z(t) always converges to z irrespective of the value of X(1) = x, and
c +
p (x) = 1 holds. In this case, we set p (x) = 0. Regarding the asymptotic behavior of the
+ −
convergence of z(t) z , which also depends on X(1), we assume
±
→
E(z(t) z X(1) = x) W (x)tl±−1
± ±
→ | ≃
7
We write the dependence of the coefficients W (x) on the value of X(1) explicitly. Using
±
these behaviors and notations, we estimate the asymptotic behavior of C(t) as
C(t) = Pr(X(t+1) = 1 X(1) = 1) Pr(X(t+1) = 1 X(1) = 0)
| − |
= E(q(z(t)) X(1) = 1) E(q(z(t)) X(1) = 0)
| − |
1
= ( 1)x−1 E(q(z(t)) x)Pr(z(t) z x)+E(q(z(t)) x)Pr(z(t) z x)
+ −
− { | → | | → | }
Xx=0
1
( 1)x−1 (q +l E(z z x))p (x)+(q +l E(z z x))p (x)
+ + + + − − − −
≃ − { − | − | }
Xx=0
1
= ( 1)x−1 (q +l W (x)tl+−1)p (x)+(q +l W (x)tl−−1)p (x)
+ + + + − − − −
−
Xx=0 (cid:8) (cid:9)
1
= q p (x)+q p (x)+(l W (x)p (x)tl+−1 +l W (x)p (x)tl−−1) ( 1)x−1.
+ + − − + + + − − −
−
Xx=0(cid:2) (cid:3)
(3)
Here, we expand q(z) as
q(z) = q(z +l (z z )) q +l (z z ).
± ± ± ± ± ±
· − ≃ · −
Given that p (x)+p (x) = 1 for x = 0,1, the limit value c lim C(t) is estimated to
+ − t→∞
≡
be
c = (q q )(p (1) p (0)). (4)
+ − + +
− −
For J < J (h), p (x) = 1 and c = 0. As z is stable for J > J (h), the probability for the
c + − c
convergence of z(t) to z is positive. It is natural to assume that p (1) > p (0) and c > 0
− + +
for J > J (h).
c
The asymptotic behavior of C(t) is governed by the term with the largest value among
l ,l for J > J (h). We define l as
+ − c max
{ }
l , J < J (h),
l  + c (5)
max
≡ 
Max l ,l , J > J (h).
+ − c
{ }


8
We summarize the asymptotic behavior of C(t) as
C(t) c+c′ tl−1 and l = l . (6)
max
≃ ·
Here, wewrite thecoefficient ofthetermproportionaltotl−1 asc′. IfJ > J (h), theconstant
c
term c is the leading term. If J < J (h), the power law term c′ tl−1 is the leading term.
c
·
There also exists a sub-leading term to c′ tl−1 that we do not write explicitly. One reason
·
for this is that we do not understand the asymptotic behavior. The second reason is that
our interest is focused on the value of l.
B. J = J (h)
c
On the boundary J = J (h), there are two stable points z and z for h > 0. As z is
c + t t
stable, the probability for the convergence of z(t) to z is positive. It is natural to assume
t
that p (1) > p (0) and c > 0. If h = 0, there is only one stable point at z = z = 1/2 and
+ + + t
c = 0. As l = l = 1, we anticipate that C(t) c becomes a decreasing function of lnt.
max t
| − |
One possibility is a power-law behavior of lnt such as
C(t) c+c′ (lnt)−α′. (7)
≃ ·
In the case of the digital model, C(t) t−α with α = 1/2 for p = p . We denote the power
c
∝
law exponent for lnt as α′.
We derive α′ by a simple heuristic argument. At first, we consider the case of h = 0.
There is only one stable touchpoint at z , and z(t) converges to it. Eq.(3) suggests that the
t
asymptotic behavior of C(t) is governed by E(z z x) as z is the only stable state. As
t t
− |
q(z ) = z and q′(z ) = 1 at z , q(z) can be approximated in the vicinity of z as
t t t t t
q(z) = δ(z z )3 +z.
t
− −
Here δ is a positive constant, as z is stable (Fig.3a). The time evolution of E(z z x) is
t t
− |
given as
1 δ
E(z(t+1) z x) E(z(t) z x) = E(q(z(t)) z(t) x) E((z z )3 x).
t t t
− | − − | t+1 − | ≃ −t − |
9
Here the denominator t + 1 in the middle of the equation comes from the fact that there
occurs a 1 change in E(z(t) x) for X(t+1) 0,1 . We also assume E((z(t) z )3 x)
t+1 | ∈ { } − t | ≃
E(z(t) z x)3 and the equation can be written as
t
− |
d δ
E(z(t) z x) = E(z(t) z x)3.
t t
dt − | −t − |
The solution to this shows the next asymptotic behavior
E(z z x) (lnt)−1/2,
t
− | ∝
and we obtain α′ = 1/2.
Likewise, for h > 0, there are two stable states q and q . The subleading term in C(t)
+ t
is governed by E(z z x). We can approximate q(z) in the vicinity of z to be
t t
− |
q(z) = δ(z z )2 +z.
t
−
Here δ is a positive constant (Fig.3b). If z(t) > z , z(t) moves toward the right-hand
t
direction, on average, and converges to z . We only need to consider the case z(t) < z and
+ t
z(t) converges to z . In the case, E(z(t) z x) obeys the next differential equation.
t t
− |
d δ
E(z(t) z x) = E(z(t) z x)2.
t t
dt − | t − |
The solution shows the next asymptotic behavior
E(z(t) z x) (lnt)−1,
t
− | ∝
and we obtain α′ = 1.
C. Numerical check of C(t) c+c′ tl−1
≃ ·
We perform the numerical integration of the master equation of the system and check
the asymptotic forms for C(t) numerically. We denote the joint probability function for
t X(s) andX(1) asP(t,n,x ) Pr( t X(s) = n,X(1) = x ). For t = 1, P(1,1,1) =
s=1 1 ≡ s=1 1
P P
10