65 problems

Consider the opinion-dynamics system given by equations … , in which the personal conviction values $u_i$ evolve according to … with $a_i\geq 0$ for $i=1,\ldots,N$, and where…

Deffuant phase-transition conjecture. There is a phase transition from discordance to consensus at the confidence threshold $\theta=1$.

Let the H-K particle model be a bounded-confidence opinion dynamics model with confidence radius $R$, and consider its final clusters. 2R-conjecture. The final clusters typically f…

In the coevolutionary game, let each player's action-opinion strategy be $z_i=(x_i,y_i)\in\\{0,1\\}\times[0,1]$, and let the discriminant determining the best-response action be ……

Let $\bm{+1}$ and $\bm{-1}$ denote the all-positive and all-negative configurations, respectively. For a path $\rho$, let $\Phi_\rho$ be its maximal energy along the path and let…

Let $S(n)$ denote the maximum number of connected components (clusters) obtainable in a bounded confidence (BC) process with $n$ agents. Half-density limit conjecture. … Determinin…

Let $(c_1,\dots,c_l)$ be a sequence of cluster sizes of a bounded confidence (BC) process. A subsequence is connected when its corresponding clusters are connected in the final con…

Lyapunov-function conjecture. The function

Metastable consensus-time conjecture. If $p\geq p_c$, then there exists some $\bar c(p)>0$ such that

Let an initial opinion distribution be given in the Hegselmann–Krause opinion dynamics, and consider the associated sequence of $\epsilon$-switches, where an $\epsilon$-switch is a…

For each $\lambda>0$, let $\Xi_{\lambda}^{(0)}$ be a simple stationary renewal process on $\mathbb{R}$ satisfying the functional strong law of large numbers, and let…

Let $R>0$ be the interaction radius, and consider the Hegselmann–Krause dynamics in the restricted setting with $N$ points on a compact interval. A fixed point consists of clusters…

Let $f(t,A,w)$ be the population distribution, let $f_\ell(z)$ be the leaders' opinion distribution, and define the population's average opinion and the leaders' average opinion by…

Throughout the paper, the initial opinions are i.i.d. random variables, with expectation $\mu:=\mathbb{E}[f_0(v)]$. Let $f_t(v)$ denote the opinion at vertex $v$ at time $t$, and s…

Control-action uplift conjecture. After a critical mass is reached, one may uplift the control action.

Joint-control algorithm conjecture. The algorithm proposed in the preliminary work should solve the effectiveness guarantees problem in this simplified joint-control setting.

Let $u\in[0,1]$, let $\vec{v}(t,u)=(f_+(t,u),f_-(t,u))^\top$ be the vector of opinion densities, and let $\pi^g_+,\pi^g_-,\pi^r_+,\pi^r_-\in[0,1]$ and $q\in\mathbb N$ be the model…

Let $\boldsymbol{h}^{G_n(t)}$ denote the opinion-density process on the dynamic random graph $G_n$, let $\beta>0$ be the model parameter, and let $\boldsymbol{h}^{G_n(nt)}$ be the…

Divergent-edges conjecture. The set of divergent edges has positive density, and the opinion distance along every edge diverges to infinity:

Let $\vec{z}_t$ denote the disagreement component of the opinion vector at time $t$, and let $\lambda$ be the parameter appearing in the model. Here $\mathbb{E}$ denotes expectatio…

Expander-graph conjecture. There is a choice of corrupted set such that the conclusion to Theorem holds.

Approximation conjecture. There exist $\mathbf{x}'\in C_\delta(\mathbf{x})$, a sequence $\{\gamma^{(\ell)}\}_\ell$, and a sequence $\{\mathbf{x}^{(\ell)}\}_\ell$ such that

Let a graph be evaluated under an opinion-forming model with initially selected experts, binary labels, and an adversary that chooses the experts' labels. In the iterative propagat…

Consider the multilayer epidemic–opinion model in which $\tau$ is the opinion conversion rate, and let an individual's earlier opinion(s) denote the opinion or opinions held before…

Existence conjecture. The result asserting existence of Carathéodory solutions for the topological bounded confidence model should hold for $\kappa>1$ as well.