39 problems

Let a network structure and a lumping partition be such that the relevant combinatorial calculations can be expressed in terms of the hypergeometric distribution. The lumped transi…

Let $\mathcal X$ be the state space of the network SIRS epidemic model, let $y$ denote the vector of infected-population variables, and suppose that an endemic equilibrium…

Let a graph have $N$ vertices, and construct a configuration-model graph by assigning each vertex a prescribed number of stubs and pairing the stubs uniformly at random, allowing s…

The stochastic model has online transmission rate $\tau$, offline recovery parameter $\gamma_p$, offline parameter $\eta$, and offline-to-online feedback strength $\theta$. Conside…

Asymptotic error-magnification conjecture. As $t\to\infty$, $\kappa_{ji}^t$ approaches a limit, denoted by $\kappa_{ji}$:

Upstream observability conjecture. For almost every choice of entries of $A$, if there is a directed path from node $i$ to node $j$, then the time series of node $i$ can be reconst…

Shallow-structure conjecture. If the error survives in a preferential-attachment CKP and $p$ is sufficiently far away from $0$, then the CKP becomes shallow.

Preferential-attachment structure conjecture. Based on simulations, CKPs evolving according to preferential attachment lose their preferential attachment structure when $p$ is suff…

Let $G$ be a graph, let $K_N$ be the complete graph on $N$ vertices, let $u$ be the all-infected initial infection-probability vector, and let $V_\infty$ be the endemic steady-stat…

Let $G$ be a graph, let $K_N$ be the complete graph on $N$ vertices, let $u$ denote the all-infected initial infection-probability vector, and let $y(t;G,\tau,u)$ be the prevalence…

Let $G$ be a graph, let $K_N$ be the complete graph on $N$ vertices, let $u$ denote the all-infected initial infection-probability vector, and let $V(0)$ be an initial infection-pr…

Superscroll occurrence conjecture. Although the world around us is only three-dimensional, superscrolls can occur in media where oscillatory or excitable elements are connected on…

Let $V$ be an infinite network satisfying the square-root decay assumption: there exists $C_0>0$ such that for every $k\leq |V|^2$ (and every $k\in\mathbb N$ when $V$ is infinite),…

The brain is organized into areas that can process different types of input separately while also integrating them into a unified representation. Tononi's conjecture. The brain nee…

Rigid Synchrony Conjecture. Every rigid global synchrony pattern $\bowtie_{\mathbb{X}}$ is balanced.

Rigid Phase Conjecture. If

Rigidity Properties conjectures. Every network has the corresponding property: RIP, RSP, RPP, and FOP.

Consider a network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $.75(n-1)$ other oscillators. Let $_c$ d…

Sliding-surface conjecture. The quantity $\frac{1}{L\beta d_{r+1}}$ acts as a sliding surface for the closed-loop dynamics. This concerns the switching behavior of the closed-loop…

Let $\mathcal{D} = \{d_1, d_2, \ldots, d_p\}$ be the set of heterogeneous node degrees, with $d_1 < d_2 < \ldots < d_p$, and let $\mathbf{x}^*_{cl}$ denote the nonzero endemic equi…

0.75n critical-threshold conjecture. The critical threshold could be $0.75n$.

The finite-dimensional network dynamics with $n$ nodes and its limiting Vlasov–Fokker–Planck equation (VFPE) are considered in the graphop framework. A Dobrushin-type approximation…

SIRS negative-correlation conjecture. As discussed for the SIS and SIR models, one should have

Consider the decentralized smart grid control model … where $P_i$ is the produced or consumed power at node $i$, $K=(K_{ij})$ is the weighted adjacency matrix, and…

In the Network SIRI model, let Case 6b denote the model case defined in the paper, and let $\mathcal{M}_0$ be the center subset appearing in the stated uniqueness theorem. Extensio…