Physical networks

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In this thesis, I focus on physical networks, and aim to develop network theories that are tailored to describe and explain their properties. The structural characteristics of a network are uniquely determined by its adjacency matrix, which encodes the complete information about the interactions between a system's components. However, in physical networks, like the brain or the vascular system, the network's three dimensional layout also affects the system's structure and function. Therefore, when analyzing physical networks, it is important to take the physical embedding of the network into account. We lack, however, the tools to systematically analyze and interpret the physical embedding topology of networks combined with their structural topology. Therefore, I first focus on developing a theory that describes the topological properties of physical network embeddings. In this theory, a new invariant is introduced, the graph linking number, that captures the knots that are formed in physical network embeddings. I also find connections between topology and energy of physical network embeddings. The theory that I developed is applicable to any 3D network embeddings, and I show that by applying it to several real-world physical networks, including the mouse brain network. In the second project, I analyze one specific real-world physical networks, the Drosophila larval brain system. I first find the best method to build a network from the given data of neuron embedding in the Drosophila larval brain, which is applicable to other similar physical systems. Next I apply the developed theories on physical networks to the Drosophila brain network, and analyze various properties of the network to gain insights into the system. Ultimately, the goal of this thesis is to better describe and understand physical network, and to provide tool sets that are applicable to all physical networks, which will benefit future research of various physical systems.--Author's abstract