Clustering

The main question for the classical HK model is whether the society converges to consensus, splits into polarization with two opinions or fragments into many more opinions. Due to the averaging dynamics and the lack of noise in the classical model, the agents will group into very sharp clusters of practically the same opinion value, within the precision of the used data type (here single precision IEEE 754 floats and a tolerance of 10-4 to generously account for numerical errors), which are therefore quite easy to classify. So a good observable of the level of consensus is the mean size of the largest cluster.

In contrast to those sharp clusters, the introduction of cost will result in broader, quasi-continuous distributions of opinions. The fundamental reason for this qualitative change lies in the presence of the frozen agents combined with the heterogeneity in the confidences. Similar to an effect already observed¹⁵, agents of different confidence εi interact with different sets of frozen agents, which results in slightly different final states. This mechanism is sketched in Fig. 1a. In the limit of n→∞, this should actually result in a continuous spectrum of final opinions. The higher the fraction of frozen agents, the broader the peaks can become. As an example, for the broader peaks, a histogram giving the distribution of final opinions is shown in Fig. 1b.

(a) Sketch to visualize the broadening of the clusters. Due to the frozen agents (light blue), agents with different confidences (horizontal bars below the opinion axis) interact with different sets of agents leading to different final positions. Note that each of the non-blue lines represents possibly multiple agents with similar confidence εi. (b) Histogram (with 100 bins) of the opinions of agents in the final state for a realization of n=4096 agents with a cost of η=4. The dashed horizontal line is drawn at 1/3 of the maximum height. We classify two clusters from the four intersections of the dashed line with the peaks. All agents with a final opinion in the range marked by vertical violet lines are assigned to one cluster.

Unfortunately, established methods to classify the clusters of a HK model, e.g. binning of the opinion space, do not work reliably with the relatively broad distributions. Therefore, this requires a new robust criterion, to define clusters of agents carrying the “same” opinion. The observable we use is loosely inspired by the full width at half maximum measure applied in, e.g. spectroscopy. From each final realization we create a histogram to estimate the probability density function p(x) of an agent to have opinion x in the final state, as shown in Fig. 1b. Then we classify the clusters using a threshold of p∗=13maxp(x). The clusters are defined using the maximal intervals [xl,xu], such that

For each interval, the set of agents {i|xl≤xi≤xu} constitutes a cluster. In other words, each peak larger than the threshold is identified as a cluster. The choice of the factor 1/3 to calculate p∗ is arbitrary but empirically chosen to be as small as possible while always being larger than the noise floor. Note that this method does not assign each agent to a cluster but it is suited as a robust estimate for the size of the largest cluster.