Note: This is a joint talk, with Mark Novak presenting the biological context and results, and David Koslicki presenting the math behind their approach.
The need to understand how complex networks respond to perturbations of their constituent entities pervades many disciplines, including applications in communications, human health, and fisheries management. Many of these perturbations involve sustained, chronic changes (a.k.a. press perturbations) imposed on particular target nodes or on the network as a whole. Ecosystem-based fisheries management, for example, entails the need to consider how alternative harvest scenarios will alter the abundance of a particular focal species or stock, and how these press perturbations will ripple through the ecosystem to affect other, non-target species of interest.
In this talk, we will present recent joint work where we use a few classic results in matrix analysis to determine the qualitative and quantitative sensitivity of the inverse of the so-called "community matrix" to perturbations/uncertainty in the community matrix entries. These results provide insight into the variability in the response of an ecological network to press perturbations when there is uncertainty in the network's topology and interaction strengths. Two recurring examples in this talk include a trophic chain (TC) network and an intraguild predation (IGP) network. Interestingly, we observe that while the TC network is more qualitatively stable than the IGP network, the IGP is more quantitatively stable.
An open question about the distribution of certain minors of a Gaussian matrix ensemble will also be discussed.