| Summary: | Traditionally, the study of complex networks has employed mathematical toolsfrom graph theory and statistical physics. More recently, topological data analysis has
introduced the application of algebraic topology to network science, opening up new
possibilities. This dissertation takes a similar direction by drawing on topological and
geometrical aspects of graph theory to develop data mining algorithms that provide
new scientific insights into complex networks.
I consider the geometric structure of networks at many levels. First, I study
the intrinsic metric structure of a graph using the theory of length spectrum and its
relation to the eigenvalues of the non-backtracking matrix of the graph. I use this
relationship to develop a principled algorithm for measuring graph distance. Second,
I study the geometric structure of graph embeddings. Graph embedding techniques
place a graph in a surrounding metric space. I use the geometry of this space to
develop two embedding algorithms that enable the performance of such tasks such
as link prediction and anomaly detection in an efficient and interpretable manner.
Third, because considering distances between graphs depends on the assumption that
graphs with similar structure exhibit similar behavior, I study how certain dynamical
properties of graphs change under small perturbations of their structure. In particular,
I focus on how small perturbations to the structure of the graph affect the epidemic
threshold of certain dynamics. I introduce two algorithms that effectively manipulate
the epidemic threshold by exploiting the relationship between the graph structure
and its spectral properties.
I conclude with a reflection on the current trend in network science to rethink
the study of complex systems in terms of polyadic relations. My collaborators and I
find that a necessary step in this direction is to focus on the properties of real-world
systems that determine which relationships affect the existence of other relationships.
I discuss this property, which I call dependency, in the context of graphs, simplicial
complexes, and hypergraphs.--Author's abstract
|
|---|
