distancere

Distancere is a generalized distance concept used in some theoretical and applied contexts to blend intrinsic feature similarity with relational structure among items. It is not a single standard metric, but a family of measures that accommodate both feature space and connection information.

Formal definition: Let X be a set of objects with feature vectors f(x) in R^d and let G = (X, E) be a relation graph over X. Distancere d_R: X×X → R+ is defined by d_R(x,y) = α d_f(x,y) + (1−α) d_r(x,y), where 0 ≤ α ≤ 1. Here d_f is a feature-based distance (such as Euclidean distance on f) and d_r is a relational distance derived from G (for example a shortest-path distance or a diffusion-based distance). If α = 1, distancere reduces to d_f; if α = 0, it reduces to d_r.

Computation: To compute d_R for a dataset, one can precompute the feature distance matrix and a relational distance matrix and then combine them. When d_r is graph-based, shortest-path algorithms or diffusion processes on G can be used to obtain the relational component. The overall complexity depends on the chosen d_r and the size of X, with trade-offs between accuracy and scalability.

Applications: Distancere is used in clustering, nearest-neighbor search, and similarity-based retrieval where both feature similarity and relational context matter. It appears in semi-supervised learning, recommendation systems, network analysis, and bioinformatics, particularly when relationships among items convey information beyond their individual features.

Properties and limitations: Distancere does not necessarily satisfy all properties of a metric for all α and choices of d_r; triangle inequality may fail if either component lacks it. Selecting α and the relational measure is problem-specific, and the quality of the underlying relation graph strongly influences results.

See also: distance metric, graph distance, diffusion distance, kernel methods, relational learning.