  Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets
and let $\Delta(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the
maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The
\emph{diversity} of $\mathcal{F}$ is defined as $d(\mathcal{F}) :=
|\mathcal{F}| - \Delta(\mathcal{F})$.
  Diversity can be viewed as a measure of distance from the `trivial'
maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed,
the diversity of this family is $0$. Moreover, the diversity of the largest
non-trivial intersecting family \`a la Hilton-Milner is $1$.
  It is known that the maximum possible diversity of an intersecting family
$\mathcal{F}\subseteq \binom{[n]}{r}$ is $\binom{n-3}{r-2}$ as long as $n$ is
large enough.
  We introduce a generalization called the \emph{$C$-weighted diversity} of
$\mathcal{F}$ as $d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot
\Delta(\mathcal{F})$. We determine the maximum value of $d_C(\mathcal{F})$ for
intersecting families $\mathcal{F} \subseteq \binom{[n]}{r}$ and characterize
the maximal families for $C\in \left[0,\frac{7}{3}\right)$ as well as give
general bounds for all $C$. Our results imply, for large $n$, a recent
conjecture of Frankl and Wang concerning a related diversity-like measure. Our
primary technique is a variant of Frankl's Delta-system method.
