Fibration symmetries uncover the building blocks of biological networks

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.