# Diffusion-driven instability of topological signals coupled by the Dirac operator.

@article{Giambagli2022DiffusiondrivenIO, title={Diffusion-driven instability of topological signals coupled by the Dirac operator.}, author={Lorenzo Giambagli and Lucille Calmon and Riccardo Muolo and Timot{\'e}o Carletti and Ginestra Bianconi}, journal={Physical review. E}, year={2022}, volume={106 6-1}, pages={ 064314 } }

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now, reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces, and higher-dimensional cells…

## 6 Citations

### Higher-order signal processing with the Dirac operator

- Computer Science2022 56th Asilomar Conference on Signals, Systems, and Computers
- 2022

The Dirac operator is introduced as a novel kind of shift operator for signal processing on complexes that couples signals defined on cells of neighboring dimensions in a principled fashion and it is demonstrated how this enables us to leverage node signals for the processing of edge flows.

### Dirac signal processing of higher-order topological signals

- Computer ScienceArXiv
- 2023

Dirac signal processing is proposed, an adaptive, unsupervised signal processing algorithm that learns to jointly learn to jointly topological signals supported on nodes, links and triangles of simplicial complexes in a consistent way.

### Local Dirac Synchronization on networks

- PhysicsChaos: An Interdisciplinary Journal of Nonlinear Science
- 2023

We propose Local Dirac Synchronization that uses the Dirac operator to capture the dynamics of coupled nodes and link signals on an arbitrary network. In Local Dirac Synchronization, the harmonic…

### Persistent Dirac for molecular representation

- Chemistry
- 2023

Molecular representations are of fundamental importance for the modeling and analysis of molecular systems. Representation models and in general approaches based on topological data analysis (TDA)…

### Turing patterns in systems with high-order interactions

- Computer ScienceChaos, Solitons & Fractals
- 2023

### Dirac gauge theory for topological spinors in 3+1 dimensional networks

- Mathematics, Physics
- 2022

. Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation. Here we propose a Dirac…

## References

SHOWING 1-10 OF 68 REFERENCES

### Higher-order simplicial synchronization of coupled topological signals

- Computer Science
- 2020

This work investigates a framework of coupled topological signals where oscillators are defined both on the nodes and the links of a network, showing that this leads to new topologically induced explosive transitions.

### Dirac synchronization is rhythmic and explosive

- PhysicsCommunications Physics
- 2022

Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing…

### The higher-order spectrum of simplicial complexes: a renormalization group approach

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are…

### Stability of synchronization in simplicial complexes

- Computer ScienceNature Communications
- 2021

It is shown that complete synchronization exists as an invariant solution, and the necessary condition for it to be observed as a stable state is given, and in some relevant instances, such a necessary condition takes the form of a Master Stability Function.

### Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes.

- MathematicsChaos
- 2022

Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional "simplices") and how…

### Higher-Order Networks

- Computer Science
- 2021

This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics.

### Turing instabilities in reaction-diffusion systems with cross diffusion

- Physics
- 2013

The Turing instability paradigm is revisited in the context of a multispecies diffusion scheme derived from a self-consistent microscopic formulation. The analysis is developed with reference to the…

### Network Geometry and Complexity

- Computer ScienceJournal of Statistical Physics
- 2018

This work investigates the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor and relates the spectral dimension of the higher-order network to the dimension and nature of its building blocks.

### Pattern formation for reactive species undergoing anisotropic diffusion

- Physics
- 2015

AbstractTuring instabilities for a two species reaction-diffusion system is studied under
anisotropic diffusion. More specifically, the diffusion constants which characterize the
ability of the…