We study generalizations of the Hegselmann-Krause (HK) model for opinion dynamics, incorporating features and parameters that are natural components of observed social systems. The first generalization is one where the strength of influence depends on the distance of the agents' opinions. Under this setup, we identify conditions under which the opinions converge in finite time, and provide a qualitative characterization of the equilibrium. We interpret the HK model opinion update rule as a quadratic cost-minimization rule. This enables a second generalization: a family of update rules which possess different equilibrium properties. Subsequently, we investigate models in which a external force can behave strategically to modulate influence user updates. We consider cases where this external force can introduce additional agents and cases where they can modify the cost structures for other agents. We describe and analyze some strategies through which such modulation may be possible in an order-optimal manner. Our simulations demonstrate that generalized dynamics differ qualitatively and quantitatively from traditional HK dynamics.
@cite_4: We study a model of opinion dynamics introduced by Krause: each agent has an opinion represented by a real number, and updates its opinion by averaging all agent opinions that differ from its own by less than one. We give a new proof of convergence into clusters of agents, with all agents in the same cluster holding the same opinion. We then introduce a particular notion of equilibrium stability and provide lower bounds on the inter-cluster distances at a stable equilibrium. To better understand the behavior of the system when the number of agents is large, we also introduce and study a variant involving a continuum of agents, obtaining partial convergence results and lower bounds on inter-cluster distances, under some mild assumptions.
@cite_3: We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions towards an average opinion, whereas low thresholds result in several opinion clusters: members of the same cluster share the same opinion but are no longer influenced by members of other clusters.
@cite_2: We study convergence of the following discrete-time non-linear dynamical system: @math agents are located in Rd and at every time step, each moves synchronously to the average location of all agents within a unit distance of it. This popularly studied system was introduced by Krause to model the dynamics of opinion formation and is often referred to as the Hegselmann-Krause model. We prove the first polynomial time bound for the convergence of this system in arbitrary dimensions. This improves on the bound of nO(n) resulting from a more general theorem of Chazelle [4]. Also, we show a quadratic lower bound and improve the upper bound for one-dimensional systems to O(n3).
@cite_12: We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions toward an average opinion, whereas low thresholds result in several opinion clusters. The model is further generalized to network interactions, threshold heterogeneity, adaptive thresholds, and binary strings of opinions. © 2002 Wiley Periodicals, Inc.
@cite_20: 
Bounded-confidence opinion dynamics @cite_12 @cite_3 have been proposed as agent-based dynamics for modeling opinion formation. Unlike previous models for social dynamics where interactions among social agents are governed by an underlying graph (fixed or random, but independent of opinions), bounded confidence models make interactions opinion-dependent. In the Hegselmann-Krause model @cite_12 every agent updates its opinion (modeled as a scalar) by averaging all other opinions within a certain distance (threshold) from its own. This dynamics has been studied extensively via numerical techniques as well as analytically. @cite_4 proved that this dynamics converges in finite time and provided upper and lower bounds for the convergence times in terms of number of agents. @cite_2 and Nedic and Touri @cite_20 studied multi-dimensional (vector) Hegselmann-Krause dynamics.