I'm currently building social networks for small colonies of animals which I've observed, with the aim of comparing changes in social network structure in response to changes in certain environmental variables. Individuals in these colonies undergo dyadic dominance interactions in which one individual attempts to assert dominance over another. The result of a dominance interaction can be either win/loss (i.e. one individual successfully dominates another) or it can be a tie (neither individual successfully asserts dominance).
I want the nodes of my social networks to represents individuals, and the edges to represent these dominance interactions. I've been using the igraph package for R which works well in many ways, but I have an annoying issue: igraph allows the user to model win/loss interactions using directional edges, but it doesn't allow for ties to be modeled at the same time. That is, one cannot combine directional and non-directional edges in the same network.
Is there a standard way to deal with ties in this situation? I've considered modeling a tie as 'half a win' to each participating individual, but this seems wrong since a tie is neither a win nor a loss. The other possibility I've considered is just ignoring ties, but this is also unsatisfactory since there's reason to think that dominance interactions that conclude in ties are still significant (e.g. they still convey information between two individuals about relative status and dominance rank).