Extending Comparison Methods for Unsigned Networks to Signed Networks

We can allow the edges of networks to have both negative and positive weights. For example, signed networks can describe the interactions of microbes. To evaluate the performance of estimators for signed networks, we need quantitative comparison methods for signed networks. Finding such comparison methods is done most easily by extending a comparison method for unsigned networks. Almost all methods reported in the literature for quantitatively comparing networks implicitly assume that edge weights are non-negative. Naive attempts to modify these methods to be applicable to signed networks can lead to nonsensical conclusions. Herein I identify requirements that should be satisfied by reasonable methods for comparing signed networks, most importantly the "double penalization principle". I extend several comparison methods for unsigned networks while satisfying these requirements. Finally, I give examples where these extensions behave reasonably but naive extensions do not.