# Dynamics of Complex Flow Networks

by Debsankha Manik

Date of Examination:2018-02-02

Date of issue:2019-01-28

Advisor:Prof. Dr. Marc Timme

Referee:Prof. Dr. Marc Timme

Referee:Prof. Dr. Reiner Kree

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## Abstract

### English

Flow networks consist of individual units called nodes connected by edges transporting flows of some quantity – such as electricity, water or cars. Each of us encounters more than one flow network every day. They form the backbone of much of our technical infrastructure, such as road networks and the electrical power grid. They also enable many biological transport processes, such as venation networks in plant leaves and trachea networks in animal lungs. To perform well, such networks need to be stable - i.e. the flows must return to some steady values following a reasonably small perturbation. They should also be resilient - i.e. damaging small parts of the network should not render the whole network dysfunctional. At the same time they should also be economical. Since adding or strengthening edges costs money, nutrients or some other resource, economy in most flow networks means having as few or as weak edges as possible. The goal of this thesis is to understand how, and to which extent, the topological properties of such networks influence their flows. First, we analyze the stability and bifurcations in oscillator models describing electric power grids and demonstrate that these networks exhibit instabilities without overloads. This phenomenon may well emerge also in other sufficiently complex supply or transport networks, including biological transport processes. Second, we study multistability in phase locked states in the same networks. We first establish the existence of geometrically frustrated states in such systems - where although a steady state flow pattern exists, no fixed point exists in the dynamical variables of phases due to geometrical constraints. We then describe the stable fixed points of the system with phase differences along each edge not exceeding pi=2 in terms of cycle flows - constant flows along each simple cycle - as opposed to phase angles or flows. The cycle flow formalism allows us to compute tight upper and lower bounds to the number of fixed points in ring networks. We show that long elementary cycles, strong edge weights, and spatially homogeneous distribution of natural frequencies (for the Kuramoto model) or power injections (for the oscillator model for power grids) cause such networks to have more fixed points. We generalize some of these bounds to arbitrary planar topologies and derive scaling relations in the limit of large capacity and large cycle lengths, which we show to be quite accurate by numerical computation. Also, we present an algorithm to compute all phase locked states - both stable and unstable - for planar networks. Last, we study the phenomenon of Braess paradox in a class of flow networks. Enhancing the capacity of an edge of a flow network intuitively improves the robustness of the system's function. However, it may also deteriorate it by overloading other edges. This latter, counterintuitive phenomenon known as Braess' paradox emerges across a wide range of natural and engineered transport, traffic and abstract flow networks. Yet, how to predict which edges trigger Braess' paradox remains unknown to date. Here, we exploit a differential perspective on how network wide flow patterns change upon enhancing any edge to predict such Braessian edges. First, we exactly map the prediction problem to a dual problem of determining the flows in the same network topology with a single- source single-sink input. Second, we propose a simple approximate criterion - flow alignment - to efficiently predict Braessian edges, thereby providing an intuitive topological understanding of the phenomenon. Based on this intuition, we show how to intentionally reduce the capacity of Braessian edges to mitigate network overload due to other infrastructure damages, with beneficial consequences for network functionality.**Keywords:**Network science; oscillator networks