17 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…

Let $X_0$ be an initial state, let $v(X_0)$ denote the quantity appearing in Theorem 1 of the paper, and let $\boldsymbol{0}_{1,d}$ be the zero row vector in…

Let $X_0$ be an initial state for the consensus algorithm, and let asymptotic consensus of directions mean that the normalized state vectors asymptotically converge to the same dir…

Consider an arbitrary unbiased cyclic game with an odd number of strategies, meaning that its cyclic-game parameter satisfies $a=0$, and let the game evolve under the imitation dyn…

Common-left-eigenvector convergence conjecture. If the matrices $C(t_k)$, $k\geq 0$, have a common left eigenvector corresponding to the eigenvalue one, then the state of system co…

Consider the noise-intervened opinion dynamical system described above, with effective noise strength $\delta_2$ and parameter $\epsilon$. Effective-noise-strength conjecture. The…

Impossibility claim. Given these communication constraints, there exists no distributed algorithm that can solve TP without further conditions on the sequence of communication grap…

Let $m$ denote the measurement noise in the plant, and consider the decentralized consensus algorithm in equation … . ISS-like robustness conjecture. When $m$ is not zero, the algo…

Let $x_0:[0,1]\to\mathbb{R}$ be a non-decreasing linear function, and let $x_t$ denote its successive continuous-agent updates. Write $L_c$ for the critical value in the discrete r…

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

Degree-to-node scaling conjecture. Based on simulations, the mean square error of the BGA is proportional to the ratio between the degree and the number of nodes,

Let the initial opinion vector and bounds vector be generated uniformly at random, and compare the synchronized bounded influence (SBI) and synchronized bounded confidence (SBC) sy…

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…

Let $V$ be the finite set of agents, let $X_v(t)$ be the estimator of agent $v$ at time $t$ in the model where agents retain their previous values, and let $X(\infty)$ denote the l…