*Context: this question is a translation of a common informal phrasing of the friendship paradox ("Most people have fewer friends than most of their friends"). Note that the question is similar to, but distinct from, Average degree of neighbors in a simple graph (-> Friendship paradox).*

Given a simple graph $G=(V,E)$ define:

- $\mathrm{N}(v) = \{w\in V:\{v,w\} \in E\}$, the neighbors of a vertex.
- $\mathrm{deg}(v) = |\mathrm{N}(v)|$, the degree of a vertex.
- $\mathrm{pop}(v) = |\{w \in \mathrm{N}(v) : \text{deg}(v) > \text{deg}(w)\}|$, the popularity of a vertex (
*i.e. the number of neighbors with lower degree*).

A vertex is said to be **popular** iff $\mathrm{pop}(v) > \frac{\mathrm{deg(v)}}2$: i.e. *the vertex has higher degree than most of its neighbors*. **Are there any graphs where more than half of the vertices are popular?** Note that the counterexamples given in Average degree of neighbors in a simple graph (-> Friendship paradox) do not work here.