  A fibration of graphs is an homomorphism that is a local isomorphism of
in-neighbourhoods, much in the same way a covering projection is a local
isomorphism of neighbourhoods. Recently, it has been shown that graph
fibrations are useful tools to uncover symmetries and synchronization patterns
in biological networks ranging from gene, protein,and metabolic networks to the
brain. However, the inherent incompleteness and disordered nature of biological
data precludes the application of the definition of fibration as it is; as a
consequence, also the currently known algorithms to identify fibrations fail in
these domains. In this paper, we introduce and develop systematically the
theory of quasifibrations which attempts to capture more realistic patterns of
almost-synchronization of units in biological networks. We provide an
algorithmic solution to the problem of finding quasifibrations in networks
where the existence of missing links and variability across samples preclude
the identification of perfect symmetries in the connectivity structure. We test
the algorithm against other strategies to repair missing links in incomplete
networks using real connectome data and synthetic networks. Quasifibrations can
be applied to reconstruct any incomplete network structure characterized by
underlying symmetries and almost synchronized clusters.
