The global switching off of arrows can be achieved as in esdd's comment or bay saying very loop/.append style={-}. Local switching off works essentially the same, I added a style no arrow for that.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=3cm,
thick,main node/.style={circle,draw,font=\sffamily\Large\bfseries},
no arrow/.style={-,every loop/.append style={-}}
]
\node[main node] (1) {1};
\node[main node] (2) [below left of=1] {2};
\node[main node] (3) [below right of=2] {3};
\node[main node] (4) [below right of=1] {4};
\path[every node/.style={font=\sffamily\small}]
(1) edge node [left] {0.6} (4)
edge [bend right] node[left] {0.3} (2)
edge [no arrow,loop above] node {0.1} (1)
(2) edge node [right] {0.4} (1)
edge node {0.3} (4)
edge [loop left] node {0.4} (2)
edge [bend right] node[left] {0.1} (3)
(3) edge node [right] {0.8} (2)
edge [bend right] node[right] {0.2} (4)
(4) edge node [left] {0.2} (3)
edge [loop right] node {0.6} (4)
edge [bend right] node[right] {0.2} (1);
\end{tikzpicture}
\end{document}

->andshorten>=1ptfrom the option list of thetikzpictureenvironment. For the loops you have to addevery loop/.style={}. – esdd Apr 24 '19 at 06:27every loop/.append style={-}in the options of thetikzpicture. Notice that I did not (vote to) close your question, but I can see why people could want to close it. It would be much better if you added a simple example such that it is clear what you want (unless the above solved it already). Once this is done I will be happy to help you reopening the question. – Apr 25 '19 at 02:14