Topologies of Complex Networks: Functions and Structures

During the last decade, significant efforts have been made toward improving our understanding of the topological structures underlying complex networks and illuminating some of the intriguing large-scale properties exhibited by these systems. The dominant theme of these efforts has been on studying the graph-theoretic properties of the corresponding connectivity structures and on developing universal theories and models that transcend system-specific details and describe the different systems well in a statistical sense. However, in this thesis we argue that these efforts have had limited success and are in need of substantial correction. Using a highly engineered system, the Internet, as a case study we demonstrate that networks are designed for a purpose, and ignoring that aspect or obscuring it with the use of some generic but random mechanism can result in models that misrepresent what matters for system functions. By accounting in a minimal manner for both the functional requirements and structural features inherent in the design of an engineered system, we propose an alternative, optimization-based modeling approach that highlights the necessary trade-offs between system performance and the technological and economic constraints that are crucial when designing the system. We show that our proposed approach yields network models that not only match the large-scale graph-theoretic properties of measured router-level topologies well but are also fully consistent with engineering intuition and networking reality, especially as far as their performance aspects and robustness properties are concerned. In fact, we show that our design-inspired network models can be easily distinguished from previously considered probabilistic network models and efficiently achieve the level of performance for which they were designed in the first place. While this thesis focuses on the Internet, it has much broader implications for complex networks and graph theory generally. To better differentiate between different graphs that are identical in certain graph statistics, we introduce a structural metric, the $s$-metric, and demonstrate that it provides insights into the diversity of graphs constrained by certain common properties and sheds new light on many classic graph concepts such as the various notions of self-similarity, likelihood, and assortativity. Our $s$-metric clarifies much of the confusion surrounding the sensational qualitative claims in the current graph theory literature for complex networks and offers a rigorous and quantitative alternative. Moreover, to examine the space of graphs that satisfy certain common properties, we propose a new approach that is based on establishing a link between two graphs if and only if one can be obtained from the other via a local transformation. Exploring the resulting connected space of graphs by dividing it into countable subspaces provides a much clearer picture on the whole space. We also show that this space of graphs has a rich and interesting structure and that some properties of the latter can be related to features of the individual graphs in this space (e.g., degree variability of a node $g$ in the space of graphs and the $s$-metric for $g$).