The evolving cobweb of relations among partially rational investors

To overcome the limitations of neoclassical economics, researchers have leveraged tools of statistical physics to build novel theories. The idea was to elucidate the macroscopic features of financial markets from the interaction of its microscopic constituents, the investors. In this framework, the model of the financial agents has been kept separate from that of their interaction. Here, instead, we explore the possibility of letting the interaction topology emerge from the model of the agents’ behavior. Then, we investigate how the emerging cobweb of relationship affects the overall market dynamics. To this aim, we leverage tools from complex systems analysis and nonlinear dynamics, and model the network of mutual influence as the output of a dynamical system describing the edge evolution. In this work, the driver of the link evolution is the relative reputation between possibly coupled agents. The reputation is built differently depending on the extent of rationality of the investors. The continuous edge activation or deactivation induces the emergence of leaders and of peculiar network structures, typical of real influence networks. The subsequent impact on the market dynamics is investigated through extensive numerical simulations in selected scenarios populated by partially rational investors.


S1 Supporting Information -Supplementary scenarios 2016
Here, we present an additional set of numerical simulations providing evidence of the robustness of our model to the relaxation of some assumptions. In particular, we made several modifications to the reputation function to verify that the main results illustrated in the manuscript still hold.
we have conducted additional numerical analyses modifying the reputation function with the aim of detecting possible failures of the designed influence process and/or discrepancies with our conclusions.

Alternative reputation functions Noise in the wealth evaluation
We start our robustness analysis by evaluating the impact of adding measurement noise, affecting the agents' evaluation of the wealth of their peers, on the observed evolution of the state variables and of the network. To this aim, for each value of ν ∈ {0, 0.75, 1}, we have performed a set of 10 simulations in which the reputation r i (k), instead of following Eq. (3), updates according to where ρ is the measurement noise is selected from a uniform distribution such that the wealth measurement error is between the 2.5% and 5% of its true value. As we can  To avoid overly complex modeling, in our work we have assumed that the agents' charisma intensity c j is independent from the agent assigning it. To provide evidence that this assumption does not impact on our results, we present a set of 10 simulations where each agent j performs its individual evaluation of the charisma of agent i i.e., in general, c j i = c l i i = j. Namely, Eq. (3) of the main text has been modified as where c j i , i.e. the charisma intensity of i as perceived by j. Notice that the self-perceptions (j = i) of the charisma intensities, c i i , i = 1, . . . , n, are randomly selected from an exponential distribution. For all i = 1, . . . , n, j = i where ρ ij is selected from a uniform distribution in [−0.05c i i , 0.05c i i ]. 1 As we can observe by comparing Fig. S2-3 with Fig. 13, the impact on the risk attitude is negligible, while Fig. S2-4 shows that the in-degree distribution is still uniform-like.
1 Notice that in the rational scenario, where ν = 0, the reputation function is unchanged if compared with that of the main manuscript.

Derivative-like term in the reputation function
In the main text, the rational component of the agents' reputation is proportional to their current wealth. One may argue that also the rate of wealth change can contribute to building its reputation. Indeed, this would model the case in which the reputation of an agent climbing the market is larger than that of a declining agent when their current wealth is similar. To evaluate the impact of a derivative-like term in the reputation, we have simulated our artificial market replacing Eq. (3) of the main text with where γ determines the relevance of the rate of wealth change of the agents' wealth in determining their reputation. Again, as we may observe from Fig. S2-5, these additional numerical analyses are consistent with the presented results.

Alternative distributions for the charisma intensity
The numerical analyses presented in the main text refer to scenarios in which the agents' charisma is randomly selected from an exponential distribution with parameter λ = 1. Indeed, it can well be the case that the distribution be different. To evaluate the robustness of our model to variations of the charisma distribution, we have performed two additional sets of simulations, one in which the charisma is selected from a uniform distribution in the interval [−2, 4] and the other one in which it is selected form a normal distribution with mean and variance both equal to 1. 2 Both distributions were tested in the semirational as well as in the irrational scenario. From Figs. S2-6, and S2-7, we can conclude that there are no significant differences among the three distributions. In-degree distribution in semirational (in green) and irrational (in red) market with agent charisma generated from a Gaussian distribution. The rational case is not reported as it is independent from the charisma. In-degree distribution in semirational (in green) and irrational (in red) market with agent charisma generated from a uniform distribution. The rational case is not reported as it is independent from the charisma.