Metric Representations Of Networks
Electrical and Electronics
The goal of this thesis is to analyze networks by first projecting them onto structured metric-like spaces -- governed by a generalized triangle inequality -- and then leveraging this structure to facilitate the analysis. Networks encode relationships between pairs of nodes, however, the relationship between two nodes can be independent of the other ones and need not be defined for every pair. This is not true for metric spaces, where the triangle inequality imposes conditions that must be satisfied by triads of distances and these must be defined for every pair of nodes. In general terms, this additional structure facilitates the analysis and algorithm design in metric spaces. In deriving metric projections for networks, an axiomatic approach is pursued where we encode as axioms intuitively desirable properties and then seek for admissible projections satisfying these axioms. Although small variations are introduced throughout the thesis, the axioms of projection -- a network that already has the desired metric structure must remain unchanged -- and transformation -- when reducing dissimilarities in a network the projected distances cannot increase -- shape all of the axiomatic constructions considered. Notwithstanding their apparent weakness, the aforementioned axioms serve as a solid foundation for the theory of metric representations of networks. We begin by focusing on hierarchical clustering of asymmetric networks, which can be framed as a network projection problem onto ultrametric spaces. We show that the set of admissible methods is infinite but bounded in a well-defined sense and state additional desirable properties to further winnow the admissibility landscape. Algorithms for the clustering methods developed are also derived and implemented. We then shift focus to projections onto generalized q-metric spaces, a parametric family containing among others the (regular) metric and ultrametric spaces. A uniqueness result is shown for the projection of symmetric networks whereas for asymmetric networks we prove that all admissible projections are contained between two extreme methods. Furthermore, projections are illustrated via their implementation for efficient search and data visualization. Lastly, our analysis is extended to encompass projections of dioid spaces, natural algebraic generalizations of weighted networks.