###### Brennan Klein

Complex networks are the syntax of complex systems; they are *models* that allow us to study phenomena across nature and society. And because they are models, the famous “all models are wrong, but some are useful” quotation rings especially true. We need to use the *right* networks to properly study complex systems, and in order to do so, the methods we use to create and analyze networks must be fit for purpose. This motivation has guided much of my dissertation, and in it, I explore three related themes around *constructing*, *comparing*, and *reconstructing* complex networks.

In the first chapter, I describe a theoretical and computational infrastructure that allows us to ask whether a given network captures the most informative scale to model the dynamics in the system. We see that many real world networks (especially heterogeneous networks) exhibit an information holarchy whereby a coarse-grained, macroscale representation of the network has more effective information than the original microscale network. In the next chapter, I consider the challenging problem of comparing pairs of networks and quantifying their differences. These tools are broadly referred to as “graph distance” measures, and there are dozens used throughout Network Science. However, unlike in other domains of Network Science where rigorous benchmarks have been established to compare our surplus of tools, there is still no theoretically-grounded benchmark for characterizing these tools. To address this, I propose that simple, well-understood ensembles of random networks are natural benchmarks for network comparison methods. In this chapter, I characterize over 20 different graph distance measures, and I show how this simple *within-ensemble graph distance* can lead to the development of new tools for studying complex networks. The final chapter is an example of exactly that: I show how the within-ensemble graph distance can be used to characterize and evaluate different techniques for reconstructing networks from time series data. Tying together the original theme of using the “right” network, this chapter addresses one of the most fundamental challenges in Network Science: how to study networks when the network structure is not known. Whether it’s reconstructing the network of neurons from time series of their activity, or identifying whether one stock’s price fluctuations cause changes in another’s, this problem is ubiquitous when studying complex systems; not only that, there are (again) dozens of techniques for transforming time series data into a network. In this chapter, I measure the within-ensemble graph distance between pairs of networks that have been reconstructed from time series data using a given reconstruction technique. What I find is that different reconstruction techniques have characteristic distributions of distances and that certain techniques are either redundant or underspecified given other more comprehensive methods. Ultimately, the goal of this dissertation is to stress the importance of rigorous standards for the suite of tools we have in Network Science, which ultimately becomes an argument about how to make Network Science more useful as a science.

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