If you're okay with straight edges in the graphs (i.e. no need to route edges around nodes or anything like that), you can probably use gplot along with patch and text to get what you want. The gplot function is annoyingly limited in line customization and doesn't return individual handles for the lines it plots, but a series of calls to it can approximate what you want. I prefer to stay away from rectangle objects since they don't play well with axis limits, and use patch instead to generate my node objects.
Here's an example that more or less replicates your image:
% The data
x = [1 2 2 2 ]; % Node x coordinates
y = [3 3 2 1]; % Node y coordinates
lbl = cellstr(num2str((1:nx)')); % Node labels
r = 0.4; % radius of nodes
nx = length(x);
adj = zeros(nx); % Adjacency matrix for edges,
adj(1,2) = 1; % values specify line width
adj(1,3) = 2;
adj(1,4) = 2;
% The plot
figure;
axes;
hold on;
linewidth = unique(adj(adj>0));
for il = 1:length(linewidth)
h = findall(gca, 'type', 'line');
gplot(adj == linewidth(il), [x' y']);
hnew = setdiff(findall(gca, 'type', 'line'), h);
set(hnew, 'linewidth', linewidth(il));
end
theta = linspace(0, 2*pi, 20)';
xc = bsxfun(@plus, r .* cos(theta), x);
yc = bsxfun(@plus, r .* sin(theta), y);
patch(xc, yc, 'w');
text(x,y,lbl);
axis equal;
