I was wondering how spring_layout takes edge weight into account. From wikipedia,
'An alternative model considers a spring-like force for every pair of nodes (i,j) where the ideal length \delta_{ij} of each spring is proportional to the graph-theoretic distance between nodes i and j, without using a separate repulsive force. Minimizing the difference (usually the squared difference) between Euclidean and ideal distances between nodes is then equivalent to a metric multidimensional scaling problem.'
How is edge weight factored in, specifically?
This isn't a great answer, but it gives the basics. Someone else may come by who actually knows the Fruchterman-Reingold algorithm and can describe it. I'm giving an explanation based on what I can find in the code.
From the documentation,
weight : string or None optional (default=’weight’)
The edge attribute that holds the numerical value used for the edge weight. If None, then all edge weights are 1.
But that doesn't tell you what it does with the weight, which is your question.
You can find the source code. If you send in weighted edges, it will create an adjacency matrix A with those weights and pass A to _fruchterman_reingold.
Looking at the code there, the meat of it is in this line
displacement=np.transpose(np.transpose(delta)*\
(k*k/distance**2-A*distance/k))\
.sum(axis=1)
The A*distance is calculating how strong of a spring force is acting on the node. A larger value in the corresponding A entry means that there is a relatively stronger attractive force between those two nodes (or if they are very close together, a weaker repulsive force). Then the algorithm moves the nodes according to the direction and strength of the forces. It then repeats (50 times by default). Interestingly, if you look at the source code you'll notice a t and dt. It appears that at each iteration, the force is multiplied by a smaller and smaller factor, so the steps get smaller.
Here is a link to the paper describing the algorithm, which unfortunately is behind a paywall. Here is a link to the paper on the author's webpage
