Calculating Point Density using Python
Yes, you do have edges, and they are the distances between the nodes. In your case, you have a complete graph with weighted edges.
Simply derive the distance from each node to each other node -- which gives you O(N^2) in time complexity --, and use both nodes and edges as input to one of these approaches you found.
Happens though your problem seems rather an analysis problem other than anything else; you should try to run some clustering algorithm on your data, like K-means, that clusters nodes based on a distance function, in which you can simply use the euclidean distance.
The result of this algorithm is exactly what you'll need, as you'll have clusters of close elements, you'll know what and how many elements are assigned to each group, and you'll be able to, according to these values, generate the coefficient you want to assign to each node.
The only concern worth pointing out here is that you'll have to determine how many clusters -- k-means, k-clusters -- you want to create.
I have a list of X and Y coordinates from geodata of a specific part of the world. I want to assign each coordinate, a weight, based upon where it lies in the graph.
For Example: If a point lies in a place where there are a lot of other nodes around it, it lies in a high density area, and therefore has a higher weight. 
The most immediate method I can think of is drawing circles of unit radius around each point and then calculating if the other points lie within in and then using a function, assign a weight to that point. But this seems primitive.
I've looked at pySAL and NetworkX but it looks like they work with graphs. I don't have any edges in the graph, just nodes.