Any function that is based on the concept of distances / path lengths (i.e. mean distance, efficiency, etc.) computes the length of a path by adding up edge weights along the path, as is standard in network analysis.
The documentation of
strength() makes the answer very clear. Quoting:
Summing up the edge weights of the adjacent edges for each vertex.
transitivity(): once again, the documentation explains how weights are used. Only Barrat’s version uses weights. The formula is given in the documentation, as well as a reference to Barrat’s paper.
knn(): again, the documentation contains the formula, as a reference to the same paper where Barrat’s transitivity is defined.
alpha_centrality(): this metric is defined based on the adjacency matrix. The weights are used in the adjacency matrix in the usual manner (i.e. included directly).
diversity(): formula, as well as reference to the original paper, are given.
Frankly, I’m not sure how the documentation could be improved further, though specific suggestions are always welcome.
Generally, igraph functions will not perform ad-hoc transformations on the weight values that you pass in, such as inverting them to transform large values to small ones. The weight values will be used as they are by each analysis function. For example, if you compute betweenness, which is based on the concept shortest paths, weights will be used for path length calculations. Thus larger weights naturally indicate weaker connections.
Note though that the igraph documentation is not really the best resource to learn about network analysis concepts … It’s a good idea to read up on each metric you are planning to use, as well as its interpretation. Some metrics can be defined in subtly different ways, and indeed you will find differences between the various network analysis packages. The igraph documentation aims to make it clear (and unambiguous) which specific definition is used, but it does not aim to explain the usage/interpretation of each metric in detail.