The weight, which can vary depending on implementation (see section below), is intended to indicate how closely related the vertices are.
Then, starting with all the nodes in the network disconnected, begin pairing nodes from highest to lowest weight between the pairs (in the divisive case, start from the original network and remove links from lowest to highest weight).
Additionally, the communities found in the network are highly dependent on the choice of weighting function.
Hence, when compared to real-world data with a known community structure, the various weighting techniques have been met with varying degrees of success.
[1] This technique is similar to a divisive hierarchical clustering algorithm, except the weights are recalculated with each step.
[2] This method provides a computationally less-costly alternative to the Girvan-Newman algorithm while yielding similar results.