Signed networks from signed topological overlap

I have previously described the differences between signed and unsigned networks and the two closely related ways of constructing a signed network in WGCNA. WGCNA now contains a third option for constructing a signed network, implementing a signed version of the standard Topological Overlap described by Katia Nowick and collaborators already in 2009 and also implemented in their R package wTO.

The concept of topological overlap was first introduced by Ravasz et al. for unweighted networks and adapted for weighted networks in the first paper describing WGCNA by Zhang and Horvath. The original expressions for TOM assumed that all adjacencies were non-negative (0 or 1 for unweighted networks, and between 0 and 1 for weighted networks), and the resulting TOM is also always non-negative.

In contrast, Nowick et al. were motivated by correlation networks in which the underlying correlation could be positive or negative, and the sign may be important. They modified the original TOM definition by what look like a few tweaks, and out came a measure that preserves the sign of the relationship between the nodes while taking account the connections through shared neighbors in the same way the standard unsigned TOM does. One can construct a signed network using a Nowick-type signed topological overlap, then simply set the connections strengths of all pairs of nodes with negative TO to zero.

Nowick-type TO in WGCNA

I recently implemented the signed Nowick TO in WGCNA where it can be selected using the argument TOMType with value “signed Nowick”. This works for all TOM calculation functions as well as for the “blockwise” network construction and module identification functions. By default, the returned values can be negative; the argument suppressNegativeTOM can be used to set the result to zero whenever it is negative. Calculating the signed Nowick TO and setting negative TOM values to zero results in a network that is similar to what one would get from the previously available “signed” and “signed hybrid” networks.

One application where the signed Nowick TO could make a more substantial difference is construction of consensus modules where the consensus is defined using a quantile. Imagine a pair of genes with a (strong) negative correlation in set 1 and similar but positive correlation in set 2, which would result in standard TOM being near zero in set 1, and (usually) strong positive TOM in set 2. A quantile (say q=0.25) consensus of the two numbers would be positive, roughly a quarter of the strong TO in set 2. In contrast, were one to use signed Nowick TO, the two TOM values could be similar in absolute value but of opposite sign (negative in set 1 and positive in set 2), resulting in negative consensus TO. All this is a bit speculative since I do not have a good practical example, but certainly worth keeping in mind.