Fibration symmetries uncover the building blocks of biological networks

Abstract

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.

References in corpus (3)

• Synchronization in complex networks
• Introduction to Focus Issue: Patterns of Network Synchronization
• Computation of Balanced Equivalence Relations and their Lattice for a Coupled Cell Network

Cited by in corpus (17)

• Does the brain behave like a (complex) network? I. Dynamics
• Local Dirac Synchronization on Networks
• Circuits with broken fibration symmetries perform core logic computations in biological networks
• Matryoshka and Disjoint Cluster Synchronization of Networks
• Predicting synchronized gene coexpression patterns from fibration symmetries in gene regulatory networks in bacteria
• Network Renormalization
• Parallel Strong Connectivity Based on Faster Reachability
• Symmetry-driven network reconstruction through pseudobalanced coloring optimization
• Fast algorithm to identify cluster synchrony through fibration symmetries in large information-processing networks
• Synchrony in Gene Regulatory Networks
• Quasifibrations of Graphs to Find Symmetries in Biological Networks
• Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks
• Higher order interactions lead to "reluctant" synchrony breaking
• Approximate Network Symmetry
• Discovering sparse control strategies in C. elegans
• Beyond holography: the entropic quantum gravity foundations of image processing
• Designing topological cluster synchronization patterns with the Dirac operator