On the Steady State of Continuous Time Stochastic Opinion Dynamics with Power Law Confidence

This paper introduces a class of non-linear and continuous-time opinion dynamics model with additive noise and state dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed form expressions for this class of dynamics are obtained for the two agent case. However the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph

Opinion influence and evolution in social networks: a Markovian agents model

In this paper, the effect on collective opinions of filtering algorithms managed by social network platforms is modeled and investigated. A stochastic multi-agent model for opinion dynamics is proposed, that accounts for a centralized tuning of the strength of interaction between individuals. The evolution of each individual opinion is described by a Markov chain, whose transition rates are affected by the opinions of the neighbors through influence parameters. The properties of this model are studied in a general setting as well as in interesting special cases. A general result is that the overall model of the social network behaves like a high-dimensional Markov chain, which is viable to Monte Carlo simulation. Under the assumption of identical agents and unbiased influence, it is shown that the influence intensity affects the variance, but not the expectation, of the number of individuals sharing a certain opinion. Moreover, a detailed analysis is carried out for the so-called Peer Assembly, which describes the evolution of binary opinions in a completely connected graph of identical agents. It is shown that the Peer Assembly can be lumped into a birth-death chain that can be given a complete analytical characterization. Both analytical results and simulation experiments are used to highlight the emergence of particular collective behaviours, e.g. consensus and herding, depending on the centralized tuning of the influence parameters.Comment: Revised version (May 2018

Learning and Forecasting Opinion Dynamics in Social Networks

Social media and social networking sites have become a global pinboard for exposition and discussion of news, topics, and ideas, where social media users often update their opinions about a particular topic by learning from the opinions shared by their friends. In this context, can we learn a data-driven model of opinion dynamics that is able to accurately forecast opinions from users? In this paper, we introduce SLANT, a probabilistic modeling framework of opinion dynamics, which represents users opinions over time by means of marked jump diffusion stochastic differential equations, and allows for efficient model simulation and parameter estimation from historical fine grained event data. We then leverage our framework to derive a set of efficient predictive formulas for opinion forecasting and identify conditions under which opinions converge to a steady state. Experiments on data gathered from Twitter show that our model provides a good fit to the data and our formulas achieve more accurate forecasting than alternatives

Continuous transition from the extensive to the non-extensive statistics in an agent-based herding model

Systems with long-range interactions often exhibit power-law distributions and can by described by the non-extensive statistical mechanics framework proposed by Tsallis. In this contribution we consider a simple model reproducing continuous transition from the extensive to the non-extensive statistics. The considered model is composed of agents interacting among themselves on a certain network topology. To generate the underlying network we propose a new network formation algorithm, in which the mean degree scales sub-linearly with a number of nodes in the network (the scaling depends on a single parameter). By changing this parameter we are able to continuously transition from short-range to long-range interactions in the agent-based model.Comment: 12 pages, 6 figure

Model of human collective decision-making in complex environments

A continuous-time Markov process is proposed to analyze how a group of humans solves a complex task, consisting in the search of the optimal set of decisions on a fitness landscape. Individuals change their opinions driven by two different forces: (i) the self-interest, which pushes them to increase their own fitness values, and (ii) the social interactions, which push individuals to reduce the diversity of their opinions in order to reach consensus. Results show that the performance of the group is strongly affected by the strength of social interactions and by the level of knowledge of the individuals. Increasing the strength of social interactions improves the performance of the team. However, too strong social interactions slow down the search of the optimal solution and worsen the performance of the group. In particular, we find that the threshold value of the social interaction strength, which leads to the emergence of a superior intelligence of the group, is just the critical threshold at which the consensus among the members sets in. We also prove that a moderate level of knowledge is already enough to guarantee high performance of the group in making decisions.Comment: 12 pages, 8 figues in European Physical Journal B, 201

Noisy continuous--opinion dynamics

We study the Deffuant et al. model for continuous--opinion dynamics under the influence of noise. In the original version of this model, individuals meet in random pairwise encounters after which they compromise or not depending of a confidence parameter. Free will is introduced in the form of noisy perturbations: individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside the whole opinion space. We derive the master equation of this process. One of the main effects of noise is to induce an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion groups are formed, although with some opinion spread inside them. Using a linear stability analysis we can derive approximate conditions for the transition between opinion groups and the disordered state. The master equation analysis is compared with direct Monte-Carlo simulations. We find that the master equation and the Monte-Carlo simulations do not always agree due to finite-size induced fluctuations that we analyze in some detail

Distributed Evaluation and Convergence of Self-Appraisals in Social Networks

We consider in this paper a networked system of opinion dynamics in continuous time, where the agents are able to evaluate their self-appraisals in a distributed way. In the model we formulate, the underlying network topology is described by a rooted digraph. For each ordered pair of agents $(i,j)$, we assign a function of self-appraisal to agent $i$, which measures the level of importance of agent $i$ to agent $j$. Thus, by communicating only with her neighbors, each agent is able to calculate the difference between her level of importance to others and others' level of importance to her. The dynamical system of self-appraisals is then designed to drive these differences to zero. We show that for almost all initial conditions, the trajectory generated by this dynamical system asymptotically converges to an equilibrium point which is exponentially stable

Markovian SIR model for opinion propagation

In this work, we propose a new model for the dynamics of single opinion propagation at a size-limited location with a low population turnover. This means that a maximum number of individuals can be supported by the location and that the allowed individuals have a long sojourn time before leaving the location. The individuals can have either no opinion (S), a strong opinion that they want to spread (I), or an opinion that they keep for themselves (R); the letters stem from the popular Susceptible-Infectious-Recovered (SIR) epidemic model. Furthermore, we consider a variable opinion transmission rate. Hence, the opinion spreading is modelled as a Markovian non-standard SIR epidemic model with stochastic arrivals, departures, infections and recoveries. We evaluate the system performance by two complementary approaches: we apply a numerical but approximate solution approach which relies on Maclaurin series expansions and we investigate the fluid limit of the system at hand. Finally, we illustrate our approach by some numerical examples