On diagonal degrees and star networks

Given an open cover $\mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $\mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)\leq sn(X)\leq L(X)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $|X|\leq sn(X)^{\Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_\delta$-diagonal has cardinality at most $\mathfrak{c}$, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the property $Ue(X)\leq\min\{aL(X),e(X)\}$ and use the Erd\H{o}s-Rado theorem to show that $|X|\leq 2^{Ue(X)\overline{\Delta}(X)}$ for any Urysohn space $X$.