On a Duality between Metrics and $\Sigma$-Proximities

In studies of discrete structures, functions are frequently used that express the proximity of objects but do not belong to the family of metrics. We consider a class of such functions that is characterized by a normalization condition and an inequality that plays the same role as the triangle inequality does for metrics. We show that the introduced functions, named Σ-proximities (" sigma-proximities "), are in a definite sense dual to metrics: there exists a natural one-to-one correspondence between metrics and Σ-proximities defined on the same finite set; in contrast to metrics, Σ-proximities measure comparative proximity; the closer the objects, the greater the Σ-proximity; diagonal entries of the Σ-proximity matrix characterize the centrality of objects. The results are extended to the case of arbitrary infinite sets of objects. A metric on a set A is a function d : A 2 → R such that for any x, y, z ∈ A, (1) d(x, y) = 0 if and only if x = y; (2) d(x, y) + d(x, z) − d(y, z) ≥ 0 (triangle inequality). It follows from this definition that for any x, y ∈ A, d(x, y) = d(y, x) (symmetry); d(x, y) ≥ 0 (nonnegativity). Functions that express proximity are not necessarily metrics. Let us consider another class of functions, whose representatives are frequently encountered and implicitly used in both applied and theoretical studies, for instance, in analyses of linear statistical models, Markov processes, electrical circuits and economic models, and also in graph theory and network theory [1-9]. Definition 1. Suppose that A is a nonempty finite set and Σ is a real number. A function σ : A 2 → R will be referred to as a Σ-proximity (read as " sigma-proximity ") on A, if for any x, y, z ∈ A, the following statements are true: (1) normalization condition: t∈A σ(x, t) = Σ; (2) triangle inequality: σ(x, y) + σ(x, z) − σ(y, z) ≤ σ(x, x), and if z = y and x = y, then the inequality is strict. The reason why this inequality is referred to as a property of metrics that has a different form will be clear from what follows. By virtue of the normalization condition, every matrix that represents a Σ-proximity has an eigenvector of all ones, Σ being the corresponding eigenvalue. When considering Σ-proximities, we will assume that the set A and the number …