  We present a stability study of the class of multivariate self-excited Hawkes
point processes, that can model natural and social systems, including
earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts
of exchanges in social communities, interactions between Internet bloggers,
bank network fragility and cascading of failures, national sovereign default
contagion, and so on. We present the general theory of multivariate generating
functions to derive the number of events over all generations of various types
that are triggered by a mother event of a given type. We obtain the stability
domains of various systems, as a function of the topological structure of the
mutual excitations across different event types. We find that mutual triggering
tends to provide a significant extension of the stability (or subcritical)
domain compared with the case where event types are decoupled, that is, when an
event of a given type can only trigger events of the same type.
