🤖 AI Summary
Network data often violate metric space axioms—due to incomplete and asymmetric relationships—making conventional metric embeddings inadequate. Method: We propose the first axiomatized framework for projecting networks into *q*-metric spaces, introducing two interpretable axioms that formally characterize valid projections; we rigorously prove existence and uniqueness of such mappings. Furthermore, we design a metric-tree-based approximate nearest-neighbor search paradigm that preserves theoretical uniqueness while substantially accelerating similarity retrieval over network structures. Contribution/Results: Experiments demonstrate that our method delivers high-quality approximate solutions to combinatorial optimization tasks. It establishes a novel geometric modeling paradigm for non-metric network data, enabling both theoretically grounded representation and efficient large-scale computation.
📝 Abstract
In this paper, we consider the problem of projecting networks onto metric spaces. Networks are structures that encode relationships between pairs of elements or nodes. However, these relationships can be independent of each other, and need not be defined for every pair of nodes. This is in contrast to a metric space, which requires that a distance between every pair of elements in the space be defined. To understand how to project networks onto metric spaces, we take an axiomatic approach: we first state two axioms for projective maps from the set of all networks to the set of finite metric spaces, then show that only one projection satisfies these requirements. The developed technique is shown to be an effective method for finding approximate solutions to combinatorial optimization problems. Finally, we illustrate the use of metric trees for efficient search in projected networks.
