Structure-Aware Bounds for the Distance to Instability in Hierarchical Non-Normal Networks

The distance to instability provides a natural robustness measure for stable continuous-time dynamical systems, but its direct computation can become computationally expensive for large non-normal matrices. In this paper, we develop a structure-aware framework for obtaining certified lower bounds for the distance to instability by exploiting block decompositions of the system matrix. Building on an existing block comparison result, we formulate a recursive two-block procedure in which local stability margins can be supplied by different certified estimates. A new theoretical ingredient is a Frobenius-resolvent lower estimate which, for the class of substochastic network blocks considered here, leads to an explicit DTI bound depending only on the block dimension and the model parameters, thereby avoiding singular-value and resolvent computations for such blocks. The resulting framework admits adaptive and hybrid implementations: individual blocks may be treated by direct computation, explicit estimates, or further partitioning according to the quality and computational cost of the available local information. Particular attention is given to hierarchical and nearly triangular interaction patterns, for which directional coupling can substantially simplify the admissibility condition. Numerical experiments show that finer partitions do not necessarily produce sharper bounds and that carefully selected coarse, possibly highly unbalanced partitions can provide a favorable balance between accuracy and computational cost. A large hierarchical socio-economic network example illustrates the ability of the explicit local estimate to replace direct DTI computation for a high-dimensional subsystem while retaining a substantial portion of the corresponding block bound.