| Summary: | 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).
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