Threshold learning dynamics in social networks

Abstract

Social learning is defined as the ability of a population to aggregate information, a process which must crucially depend on the mechanisms of social interaction. Consumers choosing which product to buy, or voters deciding which option to take with respect to an important issue, typically confront external signals to the information gathered from their contacts. Economic models typically predict that correct social learning occurs in large populations unless some individuals display unbounded influence. We challenge this conclusion by showing that an intuitive threshold process of individual adjustment does not always lead to such social learning. We find, specifically, that three generic regimes exist separated by sharp discontinuous transitions. And only in one of them, where the threshold is within a suitable intermediate range, the population learns the correct information. In the other two, where the threshold is either too high or too low, the system either freezes or enters into persistent flux, respectively. These regimes are generally observed in different social networks (both complex or regular), but limited interaction is found to promote correct learning by enlarging the parameter region where it occurs.

Figure 1. Phase diagram of the threshold model on a fully connected network.

The colors represent the fraction of agents choosing action (from red, to blue, ). System size given by agents; averaged over realizations.

Figure 2. Typical realizations of the time evolution of the fraction of agents choosing action 1, , in a fully connected network of system size with , and (A) ; (B) ; (C) .

Figure 3. The average survival time in fully connected networks for different system sizes for and .

The continuous line corresponds to an exponential fit of the form , being a constant; averaged over realizations.

Figure 4. Phase diagram of the threshold model on a two-dimensional lattice with ().

The colors represent the fraction of agents choosing action 1 (from red, , to blue, ). System size ; average over realizations.

Figure 5. Time evolution of the threshold model on a two-dimensional lattice with for different values of and .

Panels (A–C): and time steps (A) , (B) and (C) . Panels (D–F): and time steps (D) , (E) and (F) . Panels (G–I): and time steps (G) , (H) and (I) . Black color represents an agent using action , while white color represents action . The system size is .

Figure 6. Phase diagram of the threshold model in a (A) ER network and in a (B) scale-free network with average degree .

The colors represent the fraction of agents choosing action 1 (from red, , to blue ). System size , average over realizations.

Figure 7. Influence of local connectivity in social learning (A).

The initial probability density that a node using action has a fraction of neighbor nodes with action , computed on a two-dimensional lattice for , , , and a completely connected network (from the broadest to the narrowest probability density distribution). [Inset: (black, continuous) and (red, dotted) for .] Time evolution of the probability densities (black) and (red) in a two-dimensional lattice with for (B) , (C) 5 and (D) 10. For all panels, the dashed line indicates the threshold ; parameter values: system size is , , and .