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McAndrew, Thomas Charles

Mathematics and Statistics

2017

PhD

Since their discovery in the 1950's by Erdos and Renyi, network theory (the study of objects and their associations) has blossomed into a full-fledged branch of mathematics. Due to the network's flexibility, diverse scientific problems can be reformulated as networks and studied using a common set of tools. I define a network G = (V,E) composed of two parts: (i) the set of objects V, called nodes, and (ii) set of relationships (associations) E, called links, that connect objects in V. We can extend the classic network of nodes and links by describing the intensity of these associations with weights. More formally, weighted networks augment the classic network with a function f(e) from links to the real line, uncovering powerful ways to model real-world applications. This thesis studies new ways to construct robust micro powergrids, mine people's perceptions of causality on a social network, and proposes a new way to analyze crowdsourcing all in the context of the weighted network model. The current state of Earth's ecosystem and intensifying climate calls on scientists to find new ways to harvest clean affordable energy. A microgrid, or neighborhood-scale powergrid built using renewable energy sources attached to personal homes, suggest one way to ameliorate this energy crisis. We can study the stability (robustness) of such a small-scale system with weighted networks. A novel use of weighted networks and percolation theory guides the safe and efficient construction of power lines (links, E) connecting a small set of houses (nodes, V) to one another and weights each power line by the distance between houses. This new look at the robustness of microgrid structures calls into question the efficacy of the traditional utility. The next study uses the twitter social network to compare and contrast causal language from everyday conversation. Collecting a set of 1 million tweets, we find a set of words (unigrams), parts of speech, named entities, and sentiment signal the use of informal causal language. Breaking a problem difficult for a computer to solve into many parts and distributing these tasks to a group of humans to solve is called Crowdsourcing. My final project asks volunteers to `reply' to questions asked of them and `supply' novel questions for others to answer. I model this `reply and supply' framework as a dynamic weighted network, proposing new theories about this network's behavior and how to steer it toward worthy goals. This thesis demonstrates novel uses of, enhances the current scientific literature on, and presents novel methodology for, weighted networks.