  A major ambition of systems science is to uncover the building blocks of any
biological network to decipher how cellular function emerges from their
interactions. Here, we introduce a graph representation of the information flow
in these networks as a set of input trees, one for each node, which contains
all pathways along which information can be transmitted in the network. In this
representation, we find remarkable symmetries in the input trees that
deconstruct the network into functional building blocks called fibers. Nodes in
a fiber have isomorphic input trees and thus process equivalent dynamics and
synchronize their activity. Each fiber can then be collapsed into a single
representative base node through an information-preserving transformation
called 'symmetry fibration', introduced by Grothendieck in the context of
algebraic geometry. We exemplify the symmetry fibrations in gene regulatory
networks and then show that they universally apply across species and domains
from biology to social and infrastructure networks. The building blocks are
classified into topological classes of input trees characterized by integer
branching ratios and fractal golden ratios of Fibonacci sequences representing
cycles of information. Thus, symmetry fibrations describe how complex networks
are built from the bottom up to process information through the synchronization
of their constitutive building blocks.
