I'm able to generate a tikz node with shadow (lower right) on the node, but this is different than shading not the entire node, but just the circle and edges like in the picture below.
The shadow on each edge is not really the problem, but the shadow in the nodes, that is dropped by the circle not only externally but also inside the node is something that I was not able to do in Tikz.
Minimum working tex:
\documentclass{article}
\usepackage{tikz}
\usepackage{fullpage}
\usetikzlibrary{shadows}
\begin{document}
\begin{tikzpicture}[main/.style = {draw, circle, text=black,
fill=white, circular drop shadow={opacity=0.25}}]
\node[main] (1) {$x_1$};
\end{tikzpicture}
\end{document}



