Variants of Hegselmann-Krause Model

Abstract

The Hegselmann-Krause system (HK system for short) is one of the most popular models for the dynamics of opinion formation in multi agent systems. Agents are modeled as points in opinion space, and at every time step, each agent moves to the mass center of all the agents within unit distance. The rate of convergence of HK systems has been the subject of several recent works and the current best bounds are O(n3) in one dimension and O(n4) in higher dimension where n being the number of agents. In this work, we investigate the convergence behavior of a few natural variations of the HK system and their e act on the dynamics. In the rest variation, we only allow pairs of agents who are friends in an underlying social network to communicate with each other and we can construct conjurations. In the second variation, only one of the agents updates its position at each time step and selection of such an agent may be at random or based on some preened order; as before, these updates of agents also take social information into consideration. In the third variant, agents may not move exactly to the mass center but somewhere close to it. In the fourth variant, we allow all agents to interact with one another, but instead of assigning equal weights to all neighbors as in the HK model, we assign Gaussian weights which are inversely proportional to the distance between agents. In the fifth variant, we consider the Synchronized Bounded In hence model where the agents have in hence bounds instead of con dance bounds, which changes the way agents interact with each other. In our nil variant, we consider the dynamics of HK systems with strategic agents where we have an additional set of agents called as strategic agents whose opinions are chosen freely at each time step. One of the goals using these strategic agents is to lower the convergence time. The dynamics of all the variants are qualitatively very different from that of the classical HK system. Nevertheless, we prove convergence or show some other interesting results for all of these models. To be more specific, for the rest and third variant we show that these systems make only polynomial number of non-trivial steps, regardless of the social network in the rest vary-ant and noise patterns in the third variant. For the second variant, however, we again show polynomial number of non-trivial steps but in expectation regardless of the social network and interestingly different dynamics. For the fourth variant, we prove an upper bound for the convergence time of Gaussian weighted HK model. For the fifth variant, we consider a special case of this SBI model and prove convergence for this case. For the final variant, we improve the existing results for the optimal convergence time for dumb-bell and equidistant configurations.