Introduction
Modeling framework
-
Global network interactions should be considered when a particular agent’s belief is updated;
-
Continuous, time-varying connection weights should be utilized for each pair of interacting agents;
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A non-uniform confidence distribution should be employed on the agents’ beliefs that are taken into consideration by a particular agent when forming her next-period belief;
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Higher connection weights should be assigned on pairs of agents whose private beliefs present lower deviation.
The two-agent case
Convergence and consensus analysis
Limiting consensus beliefs
The three-agent case
Convergence and consensus analysis
Steady states | Eigen values | ||
---|---|---|---|
\(\overline{u}\)
|
\(\overline{v}\)
|
\(\lambda _{1}\)
|
\(\lambda _{2}\)
|
0 | 0 | 0 | 0 |
0 | 1 | 1 | 6 |
1 |
\(-\)1 | 6 | 1 |
1 | 0 | 1 | 6 |
\(-1\)
| 0 | 1 | 6 |
\(-1\)
| 1 | 6 | 1 |
0 |
\(-\)1 | 1 | 6 |
Limiting consensus beliefs
Approximate solutions for x(t), y(t), z(t)
Statistical measures |
\({x_1(t)}\)
|
\({y_1(t)}\)
|
\({z_1(t)}\)
|
---|---|---|---|
RMSE | 0.0003916 | 0.0005626 | 0.0009010 |
Rsquare | 0.999 | 0.9999 | 0.999 |
Node beliefs functions |
\({x_1(t)}\)
|
\({y_1(t)}\)
|
\({z_1(t)}\)
|
---|---|---|---|
\(\alpha _0\)
| 0.2614 (0.2609, 0.2618) | 0.2614 (0.2608, 0.2620) | 0.2609 (0.2599, 0.2618) |
\(\alpha _1\)
|
\(-0.1114~(-0.1125,-0.1103)\)
|
\(-0.1614~(-0.1630,-0.1599)\)
|
\(0.2892~(0.2867,-0.2917)\)
|
\(\lambda ^{\alpha }_1\)
|
\(-2.374~(-2.268,-2.479)\)
|
\(-2.378~(-2.273,-2.483)\)
|
\(-2.433~(-2.334,-2.533)\)
|
Statistical measures |
\({x_1(t)}\)
|
\({y_1(t)}\)
|
\({z_1(t)}\)
|
---|---|---|---|
RMSE | 0.0001734 | 0.0002472 | 0.0003845 |
Rsquare | 1.0000 | 1.0000 | 1.0000 |
Node beliefs functions |
\({x_1(t)}\)
|
\({y_1(t)}\)
|
\({z_1(t)}\)
|
---|---|---|---|
\(\beta _0\)
| 0.2611 (0.2608, 0.2615) | 0.2611 (0.2606, 0.2616) | 0.2614 (0.2606, 0.2622) |
\(\beta _1\)
|
\(-0.4078~(-55.23,54.41)\)
|
\(-1.075~(-600,597.9)\)
|
\(-0.3297~(-18.28,17.62)\)
|
\(\beta _2\)
|
\(0.2967~(-54.52,55.11)\)
|
\(0.9137~(-598.1,599.9)\)
|
\(0.6183~(-17.34,18.57)\)
|
\(\lambda ^{\beta }_1\)
|
\(-1.783~(-13.15,9.585)\)
|
\(-1.751~(-24.87,21.37)\)
|
\(-1.599~(-8.81,5.611)\)
|
\(\lambda ^{\beta }_2\)
|
\(-1.627~(-14.94,11.68)\)
|
\(-1.671~(-26.73,23.38)\)
|
\(-1.908~(-7.186,3.37)\)
|
Refining approximate solutions for x(t), y(t), z(t)
Node beliefs functions |
x(t) |
y(t) |
z(t) |
---|---|---|---|
RMSE mean | 0.000368386 | 0.000368386 | 0.000368386 |
RMSE variance | 0.000000495 | 0.000000495 | 0.000000495 |
Rsquare mean | 0.9993 | 0.9992 | 0.9993 |
Rsquare variance | 0.000005193 | 0.000005590 | 0.000003516 |
Fit parameters | Range mean | Range variance |
---|---|---|
\(\alpha _0\)
| 0.037131686 | 0.000906785 |
\(\alpha _1\)
| 0.001504342 | 0.000007971 |
\(\lambda ^{\alpha }_1\)
| 0.000150795 | 0.000000080 |
Node beliefs functions |
x(t) |
y(t) |
z(t) |
---|---|---|---|
RMSE mean | 0.000239287 | 0.000240794 | 0.000240050 |
RMSE variance | 0.000000294 | 0.000000298 | 0.000000294 |
Rsquare mean | 0.9996 | 0.9996 | 0.9996 |
Rsquare variance | 0.00000145 | 0.00000156 | 0.00000148 |
Fit parameters | Range mean | Range variance |
---|---|---|
\(\beta _0\)
|
\(0.013\times 10^3\)
|
\(0.00007\times 10^9\)
|
\(\beta _1\)
|
\(4.567\times 10^3\)
|
\(9.71979\times 10^9\)
|
\(\beta _2\)
|
\(0.065\times 10^3\)
|
\(0.00079\times 10^9\)
|
\(\lambda ^{\beta }_1\)
|
\(0.032\times 10^3\)
|
\(0.00183\times 10^9\)
|
\(\lambda ^{\beta }_2\)
|
\(4.585\times 10^3\)
|
\(9.72008\times 10^9\)
|
Approximate expressions for \(p^{*}\)
RMSE |
\(2.4001\times 10^{-5}\)
|
Rsquare | 0.9991 |
Coefficients | Value (\({95\%}\) confidence interval) |
---|---|
\(P_{00}\)
|
\(-1.408\times 10^{-5}~(-2.135\times 10^{-5},-6.804\times 10^{-6})\)
|
\(P_{10}\)
|
\(-3.175\times 10^{-5}~(-6.475\times 10^{-5},1.252\times 10^{-6})\)
|
\(P_{01}\)
|
\(3.175\times 10^{-5}~(-1.252\times 10^{-5},6.475\times 10^{-6})\)
|
\(P_{20}\)
|
\(-4.451\times 10^{-5}~(-8.273\times 10^{-5},-6.303\times 10^{-6})\)
|
\(P_{11}\)
|
\(-0.0001117~(-0.0001686,-5.468\times 10^{-5})\)
|
\(P_{02}\)
|
\(-4.451\times 10^{-5}~(-8.273\times 10^{-5},-6.303\times 10^{-6})\)
|
\(P_{30}\)
|
\(-0.6981~(-0.06991,-0.06970)\)
|
\(P_{21}\)
|
\(-0.1048~(-0.1050,-0.1046)\)
|
\(P_{12}\)
| 0.1048 (0.1046, 0.1050) |
\(P_{03}\)
| 0.6981 (0.06970, 0.06991) |