I met this problem when dealing with a social network problem. It is a subproblem of that. The problem is like that:We look at a graph $G(V,E)$, where $V$ is the set of vertices and $E$ set of edges. We define $x \in V$ as an 'important' vertex if at least $\frac 1 3 deg(x)$ ($deg(x)$ refers to the degree of $x$)of the vertices in $x$'s neighborhood have degree no more than degree of $x$. We then define 'important ' edge as the edge when either one of the two vertices of the edge or both are 'important' vertices. We define the set of 'important' edge as $E_1$. We want to show $|E_1| \geq \frac 1 2 |E|$. I cannot really think out any method. Could anyone provide some suggestions? Many thanks.     
Find proportion of 'important' edge among all edges in a graph
algorithms;graphs;graph theory