A Network Theoretical Approach to Real-World Problems: Application of the K-Core Algorithm to Various Systems

The study of complex networks is, at its core, an exploration of the mechanisms that control the world in which we live at every scale, from particles no bigger than a grain of sand and amino acids that comprise proteins, to social networks, ecosystems, and even countries. Indeed, we find that, regardless of the physical size of the network\u27s components, we may apply principles of complex network theory, thermodynamics, and statistical mechanics to not only better understand these specific networks, but to formulate theories which may be applied to problems on a more general level. This thesis explores several networks at vastly different scales, ranging from the microscopic (amino acids and frictional packed particles) to the macroscopic (human subjects asked to view a set of videos) to the massive (real ecosystems and the financial ecosystem (Haldane 2011, May 2008) of stocks in the S&P500 stock index). The networks are discussed in chronological order of analysis. We begin with a review of k-core theory, including its applications to certain dynamical systems, as this is an important concept to understand for the next two sections. A discussion of the network structure (specifically, a k-shell decomposition) of both ecological and financial dynamic networks, and the implications of this structure for determining a network\u27s tipping point of collapse, follows. Third, this same k-shell structure is examined for networks of frictional particles approaching a jamming transition, where it is seen that the jamming transition is a k-core transition given by random network theory. Lastly comes a thermodynamical examination of human eye-tracking networks built from data of subjects asked to watch the commercials of the 2014 Super Bowl Game; we determine, using a Maximum Entropy approach, that the collective behavior of this small sample can be used to predict population-wide preferences. The behavior of all of these networks are explained using aspects of network theoretical and statistical mechanics frameworks and can be extended beyond the specific networks analyzed herein

K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases

We consider the $k$-core decomposition of network models and Internet graphs at the autonomous system (AS) level. The $k$-core analysis allows to characterize networks beyond the degree distribution and uncover structural properties and hierarchies due to the specific architecture of the system. We compare the $k$-core structure obtained for AS graphs with those of several network models and discuss the differences and similarities with the real Internet architecture. The presence of biases and the incompleteness of the real maps are discussed and their effect on the $k$-core analysis is assessed with numerical experiments simulating biased exploration on a wide range of network models. We find that the $k$-core analysis provides an interesting characterization of the fluctuations and incompleteness of maps as well as information helping to discriminate the original underlying structure