Simulating Mutual Support Networks of Human and Artificial Agents

Abstract

In this paper, a multi-agent simulation model is presented to investigate the dynamics of ‘mutual-support networks’: online social networks consisting of both humans and artificial agents. Via such networks, human users who are coping with stress can share their problems, via text messages, with human peers as well as ‘artificial friends’. Even though not everybody feels comfortable sharing personal problems with artificial agents, a bot is always available to help human users, and does not face any negative consequence of providing help to stressed peers. Using the simulation model, the dynamics of social networks consisting of an arbitrary combination of humans and agents have been explored under various circumstances. This exploration resulted in several insights that are useful for shaping our vision on artificial friends: (1) humans can provide less emotional support than artificial agents because they have limited emotional resources, (2) the type of support that is provided has a large impact on the human’s stress level, and (3) the more open users are to receiving automated support, the more effective the support is in reducing their stress level. The model was internally validated by means of a mathematical verification.

Appendix

In order to verify our implementation, we performed a mathematical analysis of the equilibria of the model. Due to space limitations, we will not provide an exhaustive verification, but we limit the analysis to the following configuration:

• a given human $H_1$ faces a negative event $e^{-}$ at time t (a problem at school);

• this leads to a stress value $st > 0$;

• consequently, the user sends a message p to bot $B_1$;

• as a response, the user receives a supportive message s, following the cognitive change strategy.

The purpose of this section is to illustrate that the model behaves as expected for this particular configuration.

A given state S has a stationary point at a time step t if $\frac{\partial S(t)}{\partial t} = 0$. Given that, the model is considered in equilibrium when each one of its states has a stationary point at a time step t. Therefore, for continuous stimuli over time (constant values for $e^{-}$ and $e^{+}$), it is possible to verify that all the model’s states reach a stationary point at a sufficiently large t within a given time window. One can check this behaviour via Fig. 6, which contains the results for two simulation cases considering a network with one human and one bot: the human faces negative situations at school equal to 0.3 and 0.7 in both cases, respectively. These values where chosen taking into account the fact that when a human agent’s stress level exceeds the threshold 0.5, (s)he decides to ask for emotional support. As can be confirmed by checking the figure, the maximum stress level of a human agent is equal to $e^{-}$ (value of the negative event). In other words, $\lim _{t\rightarrow \infty } st_{H_n} = e^{-}$. Therefore, it is expected that the human decides to send a request for support only with a stimulus $e^{-} > 0.5$.

After performing two simulations using the settings as described above, we substituted the states’ final values in the equations used to define the model. This was done using an online linear solver tool².

For a stimulus $e^{-} < 0.5$, we have the following formulae for the states of the model:

$$\begin{aligned} X_1 = e^{-}_{H_1} = e \end{aligned}$$

(8)

$$\begin{aligned} X_2 = st(t)_{H_1} = \frac{\omega _1 \cdot e - \omega _{10} \cdot rd(s, t)_{B_1, H_1}}{\omega _1} \end{aligned}$$

(9)

$$\begin{aligned} X_3 = sd(t, p)_{H_1, B_1} = 0 \end{aligned}$$

(10)

$$\begin{aligned} X_4 = ms(p, t)_{H_1, B_1} = \omega _3 \cdot sd(p, t)_{H_1, B_1} \end{aligned}$$

(11)

$$\begin{aligned} X_5 = rd(p, t)_{H_1, B_1} = \omega _5 \cdot ms(p, t)_{H_1, B_1} \end{aligned}$$

(12)

$$\begin{aligned} X_6 = sd(s, t)_{B_1, H_1} = \omega _6 \cdot rd(p, t)_{H_1, B_1} \end{aligned}$$

(13)

$$\begin{aligned} X_7 = ms(s, t)_{B_1, H_1} = \omega _7 \cdot sd(s, t)_{B_1, H_1} \end{aligned}$$

(14)

$$\begin{aligned} X_8 = rd(s, t)_{B_1, H_1} = \omega _9 \cdot ms(s, t)_{B_1, H_1} \end{aligned}$$

(15)

The set that solves these eight equations is:

$$\begin{aligned} (X_1 = e, X_2 = e, X_3 = 0, X_4 = 0, X_5 = 0, X_6 = 0, X_7 = 0, X_8 = 0) \end{aligned}$$

(16)

In contrast, for a stimulus $e^{-} > 0.5$, we have the following formulae for the states of the model:

$$\begin{aligned} X_1 = e^{-}_{H_1} = e \end{aligned}$$

(17)

$$\begin{aligned} X_2 = st(t)_{H_1} = \frac{\omega _1 \cdot e - \omega _{10} \cdot rd(s, t)_{B_1, H_1}}{\omega _1} \end{aligned}$$

(18)

$$\begin{aligned} X_3 = sd(p, t)_{H_1, B_1} = 0 \end{aligned}$$

(19)

$$\begin{aligned} X_4 = ms(p, t)_{H_1, B_1} = 1 \end{aligned}$$

(20)

$$\begin{aligned} X_5 = rd(p, t)_{H_1, B_1} = \omega _5 \cdot ms(p, t)_{H_1, B_1} \end{aligned}$$

(21)

$$\begin{aligned} X_6 = sd(s, t)_{B_1, H_1} = \omega _6 \cdot rd(p, t)_{H_1, B_1} \end{aligned}$$

(22)

$$\begin{aligned} X_7 = ms(s, t)_{B_1, H_1} = 1 \end{aligned}$$

(23)

$$\begin{aligned} X_8 = rd(s, t)_{B_1, H_1} = \omega _9 \cdot ms(s, t)_{B_1, H_1} \end{aligned}$$

(24)

The set that solves these eight equations is:

$$\begin{aligned} (X_1 = e, X_2 = \frac{(e \cdot \omega _1)+(\omega _{10} \cdot \omega _9)}{\omega _1}, X_3 = 0, X_4 = 1, X_5 = \omega _5, X_6 = \omega _5 \cdot \omega _6, X_7 = 1, X_8 = \omega _9) \end{aligned}$$

(25)

The final results for the stress state (st) are, respectively, 0.298 and 0.407, for the first and the second case. As can be confirmed via Fig. 6, these results are satisfied by our implementation for the following set of parameters (besides the stimuli values stated above):

$$\begin{aligned} (\omega _1 = 1, \omega _2 = 1, \omega _3 = 1, \omega _3 = 1, \omega _3 = 1, \omega _3 = 1, \omega _3 = 1, \omega _3 = 1, \omega _3 = 1, \omega _{10} = -0.292) \end{aligned}$$

(26)

Fig. 6.

Simulations of the verified cases. Note that some of the lines are overlapping.

Finally, another way of verification is to check to what extent the requirements of our model’s description are satisfied. Some of the requirements are illustrated by Fig. 6: (1) a bot only sends support when a human has requested it; (2) there is an inhibiting effect of emotional support on stress level; (3) the maximum value for stress is equal to the negative event value used as stimulus; (4) a human only shares a problem seeking for emotional support when his/her stress level exceeds 0.5. All of these findings provide us confidence to state that the model is internally consistent.