Talk by Ginestra Bianconi

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.

Here we investigate the properties of the topological Dirac equation describing the evolution of a topological wave function on networks and on simplicial complexes. 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 (or chiral) operator defined on networks and generalized network structures including simplicial complexes and multiplex networks. 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 a spatial network or simplicial complex for describing the evolution of the topological wave function in continuous time. This framework is also extended to treat the topological Dirac equation on $1+d$ spaces describing a discrete space-time with one temporal dimension and d spatial dimensions with dimension d smaller or equal to three. 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.