10 problems

Let $G$ be a simple, connected graph, and let an exceedance tie-breaking rule $T$ be used in the cascading-failure process described by Theorem. Tie-breaking independence conjectur…

Let $G$ be the incidence matrix of a network with $n$ nodes, let $\Upsilon$ be the diagonal matrix of branch admittances, and let the admittance matrix be … Assume that the branch…

A weighted $k$-nested network is a weighted network in which the relevant nested structure has depth $k$, and its resistance distance is the pairwise effective resistance between v…

Let $n$ be a positive integer, and let $\text{Min}_{k,n}$ denote the networks with $n$ cells and $k$ asymmetric inputs, considered up to ODE-equivalence. Write $N(k,n)$ for the num…

Let $n$ be a positive integer, and let $\text{Min}_{k,n}$ denote the networks with $n$ cells and $k$ asymmetric inputs, considered up to ODE-equivalence. A minimal representative i…

Balanced-partition conjecture. All balanced partitions are orbit partitions.

Mixed connectivity–heterogeneity conjecture. To obtain sharp closed-form synchronization tests for coupled oscillators, one should focus on mixed measures of the network's connecti…

Let $Z_n$ be the cyclic set of order $n$, let $f,g$ be independently chosen random permutations of $Z_n$, and let $G(f,g)$ be the graph generated by them. If…

Exact-closure conjecture. Exact closures of a similar nature to those considered here should be possible for networks with more structure, provided that the order at which the clos…

Finite-size scaling conjecture. A more general expression is given by the following two formulae, which agree with the known expressions in the limits $\epsilon=0$ and $N=\infty$,…