Draw curve over nodes in a Tikz network

Question

How to plot a curve path along nodes in a tikz plot? Just like the image below, the path must be thick and crossing nodes

Here is a MWE; the desired curve should be like the ones in this answer, but drawn over the nodes. The naive \draw plot [smooth] (A) -- (B) -- (C) would draw only a polygonal line connecting them. I have coordinates for all the nodes, but inserting coordinates would be tedious.

\documentclass{article}
\usepackage{siunitx}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage[capposition=top]{floatrow}

\definecolor{nodefill}{RGB}{250,247,196}
\definecolor{linkfill}{RGB}{177,177,151}

\begin{document}

\begin{figure}
    \centering
    \begin{tikzpicture}[yscale=0.5, every node/.style={draw=gray,thick,circle,inner sep=-1pt, minimum size=1cm, fill=nodefill}]
    \node[] (A)     at (0.0,-0.2)       {} ;
    \node[] (B)     at (2.0,-1.5)       {} ;
    \node[] (C) at (-0.5,-4.0)      {} ;
    \node[] (D)     at (1.0,-4.3)       {} ;
    \node[] (E)     at (-0.2,-8.0)      {} ;
    \node[] (F)     at (2.4,-7.3)       {} ;

    \draw [draw=linkfill, line width=1.00mm] (A) -- (B) ;
    \draw [draw=linkfill, line width=1.00mm] (A) -- (D) ;
    \draw [draw=linkfill, line width=1.50mm] (B) -- (D) ;
    \draw [draw=linkfill, line width=1.25mm] (B) -- (F) ;
    \draw [draw=linkfill, line width=0.35mm] (C) -- (D) ;
    \draw [draw=linkfill, line width=1.25mm] (D) -- (E) ;
    \draw [draw=linkfill, line width=0.75mm] (E) -- (F) ;
    \draw [draw=linkfill, line width=1.25mm] (D) -- (F) ;

    \end{tikzpicture}
\end{figure}    


\end{document}

Answer 1

You can use two ways:

\draw [blue, line width=2pt, double, rounded corners=9pt, ->] (A.center) -- (B.center) -- (C.center) -- (D.center) -- (E.center);
\draw [red, line width=2pt, double, ->] plot[smooth, tension=2] coordinates {(A.center) (D.center) (B.center) (E.center) (F.center)};

The trick is that (A) is a node, while (A.center) is a coordinate.

As suggested by a friendly cat, you can make it more beautiful and simplify the plot syntax; you add

\usetikzlibrary{arrows.meta}

in the preamble and then

\draw [blue, line cap=round, line width=2pt, double, rounded corners=9pt, -Implies] 
    (A.center) -- (B.center) -- (C.center) -- (D.center) -- (E.center);
\draw [red, line cap=round, line width=2pt, double, -{Implies}] 
    plot[smooth,  tension=2] coordinates {(A.center) (D.center) (B.center) 
    (E.center) (F.center)};
\draw [green!50!black, line cap=round, line width=2pt, double, -{Implies}] 
    plot[smooth, tension=2, samples at={A,...,E,B,F},] (\x.center);