# Will a large stable system be complex?

#### Will a large stable system be complex?

- Event time: 11:30am until 12:30pm
- Event date: 29th April 2020
- Speaker: Dr Pierpaolo Vivo (Dept of Mathematics, King's College London)
- Location: Online - see email.

### Event details

We consider a first-order dynamical system where the NxN pairwise interaction matrix is random and sparse, i.e. it is the (weighted) adjacency matrix of an underlying random graph. We focus on the transient behaviour in the vicinity of a stationary point. The squared norm of the population vector — averaged over the random graph ensemble and uniform initial conditions — may exhibit a large degree of universality at short times. For oriented graphs, we can universally characterise the transient behaviour solely in terms of the average connectivity c, the variance σ^{2} of the bond disorder, and simple spectral properties (e.g. presence and location of outliers). Our predictions are numerically tested on a wide variety of graph topologies and bond disorder with excellent agreement, as well as on simulated first-order dynamics on small empirical networks. On the technical side, the results follow from the calculation of a Dunford-Taylor integral over the two-point resolvent of random graphs, which can be efficiently computed for large N using the celebrated ‘cavity method’ — borrowed from the physics of disordered systems.

Joint work with Izaak Neri (King's College London) and Wojciech Tarnowski (Krakow). [https://arxiv.org/abs/1906.10634]

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