Network-Oriented Modeling of the Interaction of Adaptive Joint Decision Making, Bonding and Mirroring

Abstract

In this paper the interaction of joint decision making, Hebbian learning of mirroring and bonding is analysed. An adaptive network model is designed, and scenarios are explored using this computational model. The results show that Hebbian learning of mirroring connections plays an important role in decision making, and that bonding can be necessary to make a joint decision, but also that conversely joint decisions strengthen the bonding and the mirroring connections.

Appendix Mathematical Analysis

In this section a mathematical analysis of the equilibria of the network model is presented; for the proofs, see the Appendix 2². A state Y or connection weight ω has a stationary point at t if ${\mathbf{d}}Y\left( t \right)/{\mathbf{d}}t = 0$ or ${\mathbf{d}}\upomega\left( t \right)/{\mathbf{d}}t = 0$ The adaptive network model is in an equilibrium state at t if every state Y and every connection weight ω of the model has a stationary point at t. Considering the differential equation for a temporal-causal network model, and assuming a nonzero speed factor a more specific criterion can be found:

Lemma (Criterion for a stationary point in a temporal-causal network)

Let Y be a state (similarly for a connection weight ω) and $X_{1} , \ldots ,X_{k}$ the states with outgoing connections to state Y. Then

Y has a stationary point at $t \Leftrightarrow {\mathbf{c}}_{Y} (\upomega_{{X_{1} ,Y}} X_{1} \left( t \right), \ldots ,\upomega_{{X_{k} ,Y}} X_{k} \left( t \right)) = Y\left( t \right)$

Using this, first the equilibria of the homophily connections have been determined. The values found are $\upomega_{{X_{A} ,Y_{B} }} \left( t \right) = 0$ or $\upomega_{{X_{A} ,Y_{B} }} \left( t \right) = 1$ or $\left| {X_{A} (t) - Y_{B} (t)} \right| =\uptau_{{X_{A} ,Y_{B} }}$ In the simulation experiments indeed it is seen that eventually these values 0 and 1 occur for the homophily links. The third option does not show up, so presumably it is non-attracting. Similarly the equilibrium values for the Hebbian learning links were found:

$$ \upomega_{{X_{1} ,X_{2} }} \left( t \right) = X_{1} \left( t \right)X_{2} (t)/[X_{1} \left( t \right)X_{2} (t) + (1 -\upmu)] $$

The maximal value of this occurs when $X_{1} \left( t \right) = 1$ and $X_{2} \left( t \right) = 1$ and is $\upomega_{{X_{1} ,X_{2} }} \left( t \right) = 1/[2 -\upmu]$. In the simulations μ = 0.95, so it should be expected that the values of the Hebbian learning links will never exceed 1/[2 − 0.95] = 0.95238; so in particular when they are adaptive, they never will become 1 like the homophily links. This can indeed be observed in the simulations.

Next it was shown how based on the above solutions for equilibrium values, for certain cases the implications found in the scenarios can be proven mathematically:

Joint decisions ⇒ Bonding

Assume that an achieved equilibrium state a positive joint decision occurs with ps_a,A(t) ≥ 0.8 and ps_a,B(t) ≥ 0.8. Then the only solution for ω_{psa,B,srsa,AB,A}(t) is 1. This proves that eventually bonding takes place. For the other case, assume that an achieved equilibrium state a negative joint decision occurs with ps_a,A(t) ≤ 0.2 and ps_a,B(t) ≤ 0.2. Then also the only solution is ω_{psa,B,srsa,AB,A}(t) =1. This proves again that eventually bonding takes place.

Similarly, proofs have been found for

Bonding & mirroring links & stimulus impact ≥ 0.7 ⇒ Positive joint decisions

Positive decision ⇒ Hebbian learning

For more proofs and more details, see Appendix 2.