There are several libraries, at least one of which relies on Matplotlib, that will do what you want. I recommend Networkx (www.networkx.lanl.gov) to build your graph structure, and which you can then use to call the relevant Matplotlib methods to plot. Networkx and Matplotlib work very well together.
import networkx as NX
import matplotlib.pyplot as PLT
Gh = NX.Graph()
Gh.add_edge("You", "Bike", weight=1.0)
Gh.add_edge("Bike", "Apple", weight=0.9)
Gh.add_edge("Me", "Bike", weight=1.1)
all_nodes = Gh.nodes()
# to scale node size with degree:
scaled_node_size = lambda(node) : NX.degree(Gh, node) * 700
position = NX.spring_layout(Gh) # just choose a layout scheme
NX.draw_networkx_nodes(Gh, position, node_size=map(scaled_node_size, all_nodes))
NX.draw_network_edges(Gh, position, Gh.edges(), width=1.0, alpha=1.0, edge_color="red")
# now for the Matplotlib part:
PLT.axis("off")
PLT.show()
As you can see, you could scale the edges by applying a factor to vary the 'weight' parameter to any of the 'edge' methods, just the same way as i did it for node scaling.
I would also recommend pygraphviz (obviously using graphviz as its backend). It is very similar to Netwworkx (same lead developer).