In applied mathematics, network theory and machine learning, there is increasing interest in the characterization of the dynamics of topological signals or cochains, i.e. dynamical signals defined both on the nodes and on the links of a network, however little attention has been given so far to the study of topological wave equations coupling the dynamics of topological signals of different dimension.

Here we formulate the topological Dirac equation describing the evolution of a topological wave function on networks and on simplicial complexes and we investigate its properties. On networks, the topological wave function describes the dynamics of topological signals or cochains defined on nodes on links. On simplicial complexes the wave function is also defined on higher-dimensional simplices. Therefore the topological wave function satisfies a relaxed condition of locality as it acquires the same value along simplices of dimension larger than zero. The topological Dirac equation defines eigenstates whose dispersion relation is determined by the spectral properties of the Dirac operator defined on networks and generalized network structures including simplicial complexes and multiplex networks. On networks the dispersion relation is relativistic and the degeneracy of zero eigenvector of the Dirac operator is given by the sum of the zeroth and the first betti number of the network. On simplicial complexes the Dirac equation leads to multiple energy bands. On multiplex networks, i.e. network in which nodes are related by different types of links, the topological Dirac equation can be generalized to distinguish between different multilinks indicating the different ways two nodes can be connected, leading to a natural definition of rotations of the topological spinor.

The topological Dirac equation is here initially formulated on spatial networks or simplicial complexes for describing the evolution of the topological wave function in continuous time. This framework is then extended to treat the topological Dirac equation on 1+d lattices describing a discrete space-time with one temporal dimension and d spatial dimensions with dimension d smaller or equal to three. From this analysis it emerges that in this framework time-like and space-like links are structurally indistinguishable and their different nature is only determined by the choice of the boundary operator acting on them. Therefore, in this context, the time-like and space-like nature of the links is inherently related to dynamics of cochains more than on their structural properties. This presentation includes also the discussion of numerical results obtained by implementing the topological Dirac equation on simplicial complex models and on real simple and multiplex network data.

[1] Bianconi, Ginestra. "The topological Dirac equation of networks and simplicial complexes." JPhys Complexity 2021 (in press) https://iopscience.iop.org/article/10.1088