13 problems

Let $\mathcal{G}(n)$ be a symmetric connected graph. Denote by $\mathcal{A}(\mathcal{G}(n))$ the set of strictly diagonally dominant weight matrices for $\mathcal{G}(n)$, and by…

Consider a voter model on a possibly disconnected random graph, and define the consensus time as the first time at which every connected component reaches local consensus. Let the…

Let a social network evolve through a minor adjustment of its network structure, and let the consensus time denote the time required for the system to reach consensus. Consensus-sp…

Consensus-exponent conjecture. The same exponents should hold for all smaller $\beta$ in the region

Majority-consensus conjecture. A majority consensus can be attained in this model, but the number of required communication rounds should be higher than three. The paper motivates…

Let the underlying random graph be used for communication among the agents, with the initial states chosen randomly. Benjamini et al.'s majority-consensus conjecture. Majority cons…

Consider the systems referred to as the delayed consensus models with proportional and reactive anticipation, denoted by (in the source) systems … . The question concerns whether a…

Multidimensional consensus conjecture. Asymptotic global consensus is always found; equivalently,

The paper considers adaptive networks with a linear average consensus protocol, including a slowly varying edge whose weight may range over the real numbers. The resulting dynamics…

Let $\mathcal{J}_j=\{p_1,\ldots,p_{s_j}\}$ be a topic set, and let $\tilde{\mathcal{J}}_j$ denote the topics upon which one or more topics in $\mathcal{J}_j$ depend. For each topic…

Almost global synchronization conjecture. The consensus manifold $\mathcal{C}$ is almost globally asymptotically stable for all such $p$; equivalently, $p\leq n-2$ is both necessar…

Liggett's conjecture. Fixation to consensus holds on $\mathbbm{Z}^d$ for every $\theta>\frac{1}{2}$. This conjecture concerns whether a sufficiently positive initial bias forces as…

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a connected graph on $N$ vertices. Let $\mathbf{1}$ be the column vector of ones, and let $D(\mathcal{G})$ denote the number of synch…