20 problems

Let $f$ be an interaction function satisfying the conditions of Theorem 2, and let $n$ be finite. A function … should exist such that $\{h(\Phi_t)\}_{t\in\mathbb{N}}$ is a non-nega…

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…

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…

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 $\vec{z}_t$ 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…

Let $\mathcal{C}$ be the compact convex set of coherent probability laws of $(X,Y)$ on $[0,1]^2$, and let $\operatorname{ext}(\mathcal{C})$ denote its extreme points. A law is a…

Let $(X,Y)$ be a coherent random vector with values in $[0,1]^2$, and suppose that $X$ and $Y$ are independent. Independent-opinions bound. Then … for every…

Let the initial opinions $x_1(0),\dots,x_N(0)$ be chosen independently and uniformly from $[0,L]$, and let $q_{L,N}$ be the probability that the resulting equilibrium is stable, me…

Let $m\geq 1$, and consider opinions in $\mathbb{R}^m$ with a norm replacing the absolute value in the update formula. Let $\Omega$ be an arbitrary generalized weighted asynchronou…

Let $K$ be the set of truth seekers and let $\overline{K}$ be the set of ignorants in a generalized weighted asynchronous bounded-confidence opinion system (WASBOCOS) with structur…

Let $x(t)\in\mathbb{R}^n$ be a trajectory of an SBC or SBI system, and let a trajectory have pseudo-stable behavior after a time $\tau$ when the agents split into non-empty fixed a…

Let $n$ interacting agents have opinion vector $x(t)\in\mathbb{R}^n$ and bounds vector $r\in\mathbb{R}^n_{>0}$. In the synchronized bounded confidence (SBC) and synchronized bounde…

Let $r=\{r_1,\dots,r_n\}\in\mathbb{R}^n_{>0}$ be the agents' confidence bounds and let $x(t)\in\mathbb{R}^n$ evolve according to the heterogeneous Hegselmann–Krause system … where…

Consider the continuous-time opinion-dynamics system with $n$ agents, and choose the agents' initial opinions independently and randomly according to a bounded probability density…