2025 Theses Doctoral

Optimal Control of Network Spreading Models in Varied Applications

Ihrig, Madison Ann

This work presents novel models for optimal control and spreading for use in a variety of network settings. These systems exploit existing analytic frameworks of network epidemiological models to generate new methods of simulating and controlling spreading dynamics.

Chapter 2 presents a novel optimal control model which considers the nodal influence of selected subgraphs, and exerts control alongside network spreading dynamics. This model utilizes a standard SI infection model as well as driver nodes to apply control over a connected graph, and verifies performance analytically and numerically over both structured and complex networks. We demonstrate that the resulting model is able to effectively identify driver nodes which may be used to minimize selected subgraphs' interactions with network spreading dynamics.

Chapter 3 introduces a traffic network congestion model adapted from traditional network SIS infection models. We first demonstrate how this epidemiological model may be adapted to simulate traffic congestion accurately in a scalable discretized manner suitable for application in a variety of real-world conditions. We demonstrate that this model is able to accurately recreate key traffic phenomena, and test accuracy utilizing data sourced from traffic sensor collections. This model is then integrated with the optimal control framework introduced in Chapter 2, which is adapted for traffic applications. We again demonstrate model efficacy at driver node identification for minimization of subgraph influence and management of congestion conditions.

Chapter 4 presents a cluster-based model reduction technique which may be applied to optimal control models of the forms from Chapters 2-3. This reduction is utilized in a multi-step framework which reduces both computational burden and time to identify optimal driver nodes within a large network. We derive reductions of all model components from Chapters 2-3 on the cluster and population scales, and apply analytic and numerical testing to demonstrate error bounds and efficacy. We prove that the derived reduced models are able to accurately identify control regions and improve efficiency under desired network conditions.

Our research demonstrates the suitability of our original models in a variety of settings, and provides frameworks for future usage. We contribute new methodologies which allow for nuanced modeling and effective control of network spreading dynamics, which may be further tailored to a number of real-world applications.