A Pairwise Averaging Procedure with Application to Consensus Formation in the Deffuant Model

Abstract

Place a water glass at each integer point, the one at the origin being full and all others empty, and consider averaging procedures where we repeatedly pick a pair of adjacent glasses and pool their contents, leaving the two glasses with equal amounts but with the total amount unchanged. Some simple results are derived for what kinds of configurations of water levels are obtainable via such procedures. These are applied in the analysis of the so-called Deffuant model for social interaction, where individuals have opinions represented by numbers between 0 and 1, and whenever two individuals interact they take a step towards equalizing their opinion, unless their opinions differ beyond a fixed amount θ in which case they make no adjustment. In particular, we reprove and sharpen the recent result of Lanchier which identifies the critical value θ c for consensus formation in the Deffuant model on Z to be 1 2 .

An agent based model for opinion dynamics with random confidence threshold

“…Helbing and Yu [24] argue that randomness in opinion dynamicsand more broadly the collective phenomena emerging from social systems-plays a key functional role. Albeit extremely difficult due to the nonlinear interaction in BC models, rigorous analyses are conducted by Lanchier [25] and Häggström [26] recently using probabilistic frameworks. They successfully identified the critical value of confidence threshold for consensus formation in the DW model on the real line Z to be 1=2, given the initial opinions are distributed uniformly on ½0; 1.…”

“…In the Fukushima-triggered panic buying case, for example, an agent may have a low or high confidence level on the opinion that iodized salt could be an antidote for radiation. The results in [25][26][27] do not apply when it comes to describing the dynamics of groups of heterogeneous agents.…”

Lie algebraic discussion for affinity based information diffusion in social networks

“…The substrate social network is regarded as an evolving network with average degree k(t) at time t. The affinity threshold θ(t) ∈ [0, 1] is interpreted as the information fitness to trigger the activation-an individual i can change her state from susceptible to informed or known only if a i ≥ θ(t). This is of course reminiscent of the confidence threshold studied in the Deffuant model [18,19], but the meaning here is totally different.…”

“…Solving the set of ordinary differential equations derived from (18) for each basis operator in L yields…”

“…While a wealth of models have been developed to understand various social dynamics, most of them are based on either numerical simulations or empirical data, with the noteworthy exception of the Deffuant opinion interaction model analytically explored by several mathematicians [17][18][19][20]. However, analytical treatment is of interest in social science, mainly because real data are not always accessible and numerical simulations tend to be difficult for large systems.…”