Qualitative modeling of catastrophe in group opinion

Abstract

Dynamic change and polarization of group opinion happen frequently in online opinion formation of modern society. Computational modeling of catastrophe in opinion can provide useful information for studying and understanding the dynamics of group opinion. This paper presents a qualitative modeling framework that integrates cusp catastrophe model and qualitative simulation to model catastrophe in group opinions. A qualitative model of opinion dynamics is proposed, and a multi-step fitting procedure is developed to fit model parameters from text data that are fuzzy and incomplete. A graphical metering approach is also designed to help in exploring the trajectory and three phases of catastrophe in opinion. The developed framework is applied to an example of online group opinion. Experiment results demonstrate the effectiveness and utility of the proposed framework for modeling of catastrophe in group opinion.

Appendices

Appendix 1

• syms x y z;

• syms a b;

• sym A

• A = sym(zeros(1,50));

• double xz;

• double yz;

• xz = zeros(4200);

• yz = zeros(4200);

• %———–Equations at Time t from Data1 ————%

$$\begin{aligned} A(1)= & {} '((0.2859+0.3455*{x}-0.7*{y})^{\wedge }2\\&+(-0.067+0.1625*{x}-0.9*{y})^{\wedge }2)-z'\\ A(2)= & {} '((0.2859+0.3455*{x}-0.4*{y})^{\wedge }2\\&+(-0.067~+~0.1625*{x}-0.75*{y})^{\wedge }2)-z'\\ A(3)= & {} '((0.0215+0.07*{x}+0.15*{y})^{\wedge }2\\&+(-6.25{E}-05~+~0.00875*{x}+0.45*{y})^{\wedge }2)-z'\\ A(4)= & {} '((0.0215~+~0.07*{x}+0.4*{y})^{\wedge }2\\&+(-6.25{E}-05+0.00875*{x}+0.875*{y})^{\wedge }2)-z'\\ A(5)= & {} '((0.0215+0.07*{x}+0*{y})^{\wedge }2\\&+(-6.25{E}-05+0.00875*{x}+1*{y})^{\wedge }2)-z'\\ A(6)= & {} '((0.0215+0.1425*{x}+0.15*{y})^{\wedge }2\\&+(-6.25{E}-05+0.01125*{x}+0.45*{y})^{\wedge }2)-z'\\ A(7)= & {} '((0.0215+0.1425*{x}+0.4*{y})^{\wedge }2\\&+(-6.25{E}-05+0.01125*{x}+0.875*{y})^{\wedge }2)-z'\\ A(8)= & {} '((0.0215+0.1425*{x}+0*{y})^{\wedge }2\\&+(-6.25{E}-05+0.01125*{x}+1*{y})^{\wedge }2)-z'\\ A(9)= & {} '((0.235+0.32*{x}+0*{y})^{\wedge }2\\&+(0.067+-0.1625*{x}+1*{y})^{\wedge }2)-z'\\ A(10)= & {} '((0.235+0.32*{x}+0.15*{y})^{\wedge }2\\&+(0.067+-0.1625*{x}+0.45*{y})^{\wedge }2)-z'\\ A(11)= & {} '((0.235+0.32*{x}+0.4*{y})^{\wedge }2\\&+(0.067+-0.1625*{x}+0.875*{y})^{\wedge }2)-z'\\ A(12)= & {} '((0.235+0.32*{x}+0*{y})^{\wedge }2\\&+(0.067+-0.1625*{x}+1*{y})^{\wedge }2)-z'\\ A(13)= & {} '((0.0215+0.07*{x}+0*{y})^{\wedge }2\\&+(-6.25{E}-05+0.00875*{x}+1*{y})^{\wedge }2)-z'\\ A(14)= & {} '((0.0215+0.07*{x}+0.4*{y})^{\wedge }2\\&+(-6.25{E}-05+0.00875*{x}+0.875*{y})^{\wedge }2)-z'\\ A(15)= & {} '((0.0215+0.1425*{x}+0*{y})^{\wedge }2\\&+(-6.25{E}-05+0.01125*{x}+1*{y})^{\wedge }2)-z'\\ A(16)= & {} '((0.0215+0.1425*{x}+0.4*{y})^{\wedge }2\\&+(-6.25{E}-05+0.01125*{x}+0.875*{y})^{\wedge }2)-z'\\ A(17)= & {} '((0.235+0.32*{x}+0*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+1*{y})^{\wedge }2)-z'\\ A(18)= & {} '((0.235+0.32*{x}+0.4*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+0.875*{y})^{\wedge }2)-z'\\ A(19)= & {} '((0.235+0.32*{x}+0.2*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+0.75*{y})^{\wedge }2)-z'\\ A(20)= & {} '((0.235+0.32*{x}+0*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+1*{y})^{\wedge }2)-z' \end{aligned}$$

$$\begin{aligned} A(21)= & {} '((0.235+0.32*{x}+0.4*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+0.875*{y})^{\wedge }2)-z'\\ A(22)= & {} '((0.235+0.32*{x}+0.2*{y})^{\wedge }2\\&+(1.067+-0.1625*{x}+0.75*{y})^{\wedge }2)-z'\\ A(23)= & {} '((0.0215+0.07*{x}+0.15*{y})^{\wedge }2\\&+(-6.25{E}-05+0.00875*{x}+0.45*{y})^{\wedge }2)-z'\\ A(24)= & {} '((0.0215+0.07*{x}+0.4*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0.00875*{x}+0.875*{y})^{\wedge }2)-z'\\ A(25)= & {} '((0.0215+0.07*{x}+0*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0.00875*{x}+1*{y})^{\wedge }2)-z'\\ A(26)= & {} '((0.0215+0*{x}+0.15*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0*{x}+0.45*{y})^{\wedge }2)-z'\\ A(27)= & {} '((0.0215+0*{x}+0.4*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0*{x}+0.875*{y})^{\wedge }2)-z'\\ A(28)= & {} '((0.0215+0*{x}+0*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0*{x}+1*{y})^{\wedge }2)-z'\\ A(29)= & {} '((0.235+0*{x}+0.15*{y})^{\wedge }2\\&+(0.067+0*{x}+0.45*{y})^{\wedge }2)-z'\\ A(30)= & {} '((0.235+0*{x}+0.4*{y})^{\wedge }2\\&+(0.067+0*{x}+0.875*{y})^{\wedge }2)-z'\\ A(31)= & {} '((0.235+0*{x}+0*{y})^{\wedge }2\\&+(0.067+0*{x}+1*{y})^{\wedge }2)-z'\\ A(32)= & {} '((0.235+0*{x}+0.15*{y})^{\wedge }2\\&+(0.067+0*{x}+0.45*{y})^{\wedge }2)-z'\\ A(33)= & {} '((0.235+0*{x}+0.4*{y})^{\wedge }2\\&+(0.067+0*{x}+0.875*{y})^{\wedge }2)-z'\\ A(34)= & {} '((0.235+0*{x}+0*{y})^{\wedge }2\\&+(0.067+0*{x}+1*{y})^{\wedge }2)-z'\\ A(35)= & {} '((0.0215+0.07*{x}-0*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0.00875*{x}+-1*{y})^{\wedge }2)-z'\\ A(36)= & {} '((0.0215+0.07*{x}-0.4*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0.00875*{x}+-0.875*{y})^{\wedge }2)-z'\\ A(37)= & {} '((0.0215+0*{x}+-0*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0*{x}+-1*{y})^{\wedge }2)-z'\\ A(38)= & {} '((0.0215+0*{x}+-0.4*{y})^{\wedge }2\\&+(-6.25\hbox {E}-05+0*{x}+-0.875*{y})^{\wedge }2)-z'\\ \end{aligned}$$

%———-Equations at Time t+1 from Data1 ————%

$$\begin{aligned} A(39)= & {} '((0.31725+0.405*{x}+0.2*{y})^{\wedge }2\\&+(-0.104625+0.2125*{x}+-0.9*{y})^{\wedge }2)-z'\\ A(40)= & {} '((0.31725+0.405*{x}+0.45*{y})^{\wedge }2\\&+(-0.104625+0.2125*{x}+-0.45*{y})^{\wedge }2)-z'\\ A(41)= & {} '((0.31725+0.405*{x}+0*{y})^{\wedge }2\\&+(-0.104625+0.2125*{x}+-1*{y})^{\wedge }2)-z'\\ A(42)= & {} '((0.31725+0.495*{x}+0.2*{y})^{\wedge }2\\&+(-0.104625+0.415*{x}+-0.9*{y})^{\wedge }2)-z'\\ A(43)= & {} '((0.31725+0.495*{x}+0.45*{y})^{\wedge }2\\&+(-0.104625+0.415*{x}+-0.45*{y})^{\wedge }2)-z'\\ A(44)= & {} '((0.31725+0.495*{x}+0*{y})^{\wedge }2\\&+(-0.104625+0.415*{x}+-1*{y})^{\wedge }2)-z'\\ A(45)= & {} '((0.6608125+0.5075*{x}+0.2*{y})^{\wedge }2\\&+(-0.756+0.415*{x}+-0.9*{y})^{\wedge }2)-z'\\ A(46)= & {} '((0.6608125+0.5075*{x}+0.45*{y})^{\wedge }2\\&+(-0.756+0.415*{x}+-0.45*{y})^{\wedge }2)-z'\\ A(47)= & {} '((0.6608125+0.5075*{x}+0*{y})^{\wedge }2\\&+(-0.756+0.415*{x}+-1*{y})^{\wedge }2)-z'\\ A(48)= & {} '((0.6608125+0.46*{x}+0.2*{y})^{\wedge }2\\&+(-0.756+0.82*{x}+-0.9*{y})^{\wedge }2)-z'\\ A(49)= & {} '((0.6608125+0.46*{x}+0.45*{y})^{\wedge }2\\&+(-0.756+0.82*{x}+-0.45*{y})^{\wedge }2)-z'\\ A(50)= & {} '((0.6608125+0.46*{x}+0*{y})^{\wedge }2\\&+(-0.756+0.82*{x}+-1*{y})^{\wedge }2)-z' \end{aligned}$$

%—————Calculating——————%

Appendix 2

Appendix 3

Step I:

All {QS’(f, t), QS’(u, t), QS’(v, t)} are listed as follows:

{(1,dec), (low,std), (low, dec)},{ (1,dec), (low,std), (very low, dec)},{(1,dec), (low,dec), (low, dec)},{ (1,dec), (low,dec), (very low, dec)},{(1,dec), (very low,std), (low, dec)},{ (1,dec), (very low,std), (very low, dec)}.

Take {(1,dec), (low,std), (low, dec)} as example, and the successors can be reasoned by transition rules in Table 4 as follows:

(1,dec)$\rightarrow$(-1,std),(-1,dec), (1,dec)

(low,std)$\rightarrow$(low,inc), (low,std), (low,dec)

(low, dec)$\rightarrow$(very low, dec), (very low, std), (low,dec)

Then, all the possible successors of {(1,dec), (low,std), (low, dec)}, i.e., {QS’($f,t+1$), QS’($u,t+1$), QS’($v,t+1$)}, are yielded as follows:

{(-1,std), (low,inc), (very low, dec)},{(-1,std), (low,inc), (very low, std)},{(-1,std), (low,inc), (low, dec)},{(-1,std), (low,std), (very low, dec)},{(-1,std), (low,std), (very low, std)},{(-1,std), (low,std), (low, dec)},{(-1,std), (low,dec), (very low, dec)},{(-1,std), (low,dec), (very low, std)},{(-1,std), (low,dec), (low, dec)},{(-1,dec), (low,inc), (very low, dec)},{(-1, dec), (low,inc), (very low, std)},{(-1, dec), (low,inc), (low, dec)},{(-1, dec), (low,std), (very low, dec)},{(-1, dec), (low,std), (very low, std)},{(-1, dec), (low,std), (low, dec)},{(-1, dec), (low,dec), (very low, dec)},{(-1, dec), (low,dec), (very low, std)},{(-1, dec), (low,dec), (low, dec)},{(1,dec), (low,inc), (very low, dec)},{(1, dec), (low,inc), (very low, std)},{(1, dec), (low,inc), (low, dec)},{(1, dec), (low,std), (very low, dec)},{(1, dec), (low,std), (very low, std)},{(1, dec), (low,std), (low, dec)},{(1, dec), (low,dec), (very low, dec)},{(1, dec), (low,dec), (very low, std)},{(1, dec), (low,dec), (low, dec)}.

Step II:

Obtain all logical successors by QSIM Logical Filter from all possible {QS’($f,t+1$), QS’($u,t+1$), QS’($v,t+1$)} as follows:

{(-1,std), (low,std), (very low, std)},{(-1,std), (low,dec), (very low, dec)},{(-1,std), (low,dec), (low, dec)},{(-1, dec), (low,std), (very low, dec)},{(-1, dec), (low,std), (low, dec)},{(-1, dec), (low,dec), (very low, dec)},{(-1, dec), (low,dec), (low, dec)},{(1, dec), (low,std), (very low, dec)},{(1, dec), (low,std), (low, dec)},{(1, dec), (low,dec), (very low, dec)},{(1, dec), (low,dec), (low, dec)}.

Step III:

Obtain all rational successors by Psychological Filter from all logical {QS’($f,t+1$), QS’($u,t+1$), QS’($v,t+1$)} as follows (take the sensitive individual as example):

{(-1,std), (low,std), (very low, std)},{(-1,std), (low,dec), (low, dec)},{(-1, dec), (low,std), (low, dec)},{(-1, dec), (low,dec), (low, dec)},{(1, dec), (low,std), (low, dec)},{(1, dec), (low,dec), (low, dec)}.

Appendix 4

Take {(-1,std), (low,std), (very low,std)} as example($x=0.2$, $y=0.25$). (-1,std)=[$b, c, b-a, d-c$], (low,std)=[$b, c, b-a, d-c$], (very low,std)=[-1, a, 0, 0] (See Table 1 and Table 2). Calculate $FQS(f,t)^{3}+\alpha \cdot \hbox {FQS}(u, t)\cdot$ FQS($f, t)+\beta \cdot \hbox {FQS}(v, t)$ using Table 4.

$\hbox {FQS}(f,t)^{3}+\alpha \cdot \hbox { FQS}(u, t)\cdot$ $\hbox {FQS}(f, t)+\beta \cdot \hbox {FQS}(v, t)=[-0.45,-0.35, 0.35, 0.25]^{\wedge }3+\alpha \cdot$ $[-0.45,-0.35$, $0.35, 0.25]$ $\cdot [-0.45,-0.35$, 0.35, 0.25] $+\beta \cdot$ $[-1, -0.8R$, $0, 0]=$ $[-0.0911$, $-0.0428$, $0.4207,0.0546] +\alpha$ $\cdot [0.1225, 0.2025$, $0.2375, 0.4375]+\beta \cdot$ $[-1, -0.8, 0, 0]=$ $[-0.0911+0.1225\alpha -$ $\beta , -0.0428+0.2025$ $\alpha -0.8\beta$, $0.4207+0.2375\alpha$- $\beta , 0.0546+0.4375$ $\alpha -0.8\beta$].

Thus, ${D}(1)=(0.2859+0.3455\alpha -0.7\beta ) ^{\wedge }2+(-0.0669 +0.1625\alpha -0.9\beta ) ^{\wedge }2=0$.