  Most of the analyses concerning signed networks have focused on the balance
theory, hence identifying frustration with undirected, triadic motifs having an
odd number of negative edges; much less attention has been paid to their
directed counterparts. To fill this gap, we focus on signed, directed
connections, with the aim of exploring the notion of frustration in such a
context. When dealing with signed, directed edges, frustration is a
multi-faceted concept, admitting different definitions at different scales: if
we limit ourselves to consider cycles of length two, frustration is related to
reciprocity, i.e. the tendency of edges to admit the presence of partners
pointing in the opposite direction. As the reciprocity of signed networks is
still poorly understood, we adopt a principled approach for its study, defining
quantities and introducing models to consistently capture empirical patterns of
the kind. In order to quantify the tendency of empirical networks to form
either mutualistic or antagonistic cycles of length two, we extend the
Exponential Random Graphs framework to binary, directed, signed networks with
global and local constraints and, then, compare the empirical abundance of the
aforementioned patterns with the one expected under each model. We find that
the (directed extension of the) balance theory is not capable of providing a
consistent explanation of the patterns characterising the directed, signed
networks considered in this work. Although part of the ambiguities can be
solved by adopting a coarser definition of balance, our results call for a
different theory, accounting for the directionality of edges in a coherent
manner. In any case, the evidence that the empirical, signed networks can be
highly reciprocated leads us to recommend to explicitly account for the role
played by bidirectional dyads in determining frustration at higher levels (e.g.
the triadic one).
