  We propose a theoretical framework that explains how the mass of simple and
higher-order networks emerges from their topology and geometry. We use the
discrete topological Dirac operator to define an action for a massless
self-interacting topological Dirac field inspired by the Nambu-Jona Lasinio
model. The mass of the network is strictly speaking the mass of this
topological Dirac field defined on the network; it results from the chiral
symmetry breaking of the model and satisfies a self-consistent gap equation.
Interestingly, it is shown that the mass of a network depends on its spectral
properties, topology, and geometry. Due to the breaking of the
matter-antimatter symmetry observed for the harmonic modes of the discrete
topological Dirac operator, two possible definitions of the network mass can be
given. For both possible definitions, the mass of the network comes from a gap
equation with the difference among the two definitions encoded in the value of
the bare mass. Indeed, the bare mass can be determined either by the Betti
number $\beta_0$ or by the Betti number $\beta_1$ of the network. We provide
numerical results on the mass of different networks, including random graphs,
scale-free, and real weighted collaboration networks. We also discuss the
generalization of these results to higher-order networks, defining the mass of
simplicial complexes. The observed dependence of the mass of the considered
topological Dirac field with the topology and geometry of the network could
lead to interesting physics in the scenario in which the considered Dirac field
is coupled with a dynamical evolution of the underlying network structure.
