Asymmetric Proximity Matrices
- Represent relationships where distance or similarity can differ depending on direction between two objects.
- Commonly appear in directed networks (e.g., social graphs) and transportation models where A→B may differ from B→A.
- Used in clustering and classification to capture non-reciprocal relationships.
Definition
Section titled “Definition”An asymmetric proximity matrix is a type of proximity matrix that allows different distances between objects in each direction. Unlike a symmetric proximity matrix — where, for example, if object A is two units away from object B, then object B is also two units away from object A — an asymmetric proximity matrix permits the distance from A to B to differ from the distance from B to A.
Explanation
Section titled “Explanation”Asymmetric proximity matrices represent relationships or distances between objects or entities when those relationships are not necessarily reciprocal. They are suitable when connections or costs are direction-dependent, enabling a more accurate representation of relationships in contexts where A→B and B→A may have different values. In data analysis and machine learning, these matrices are employed in algorithms that rely on pairwise distances or similarities, such as clustering and classification, to reflect directional or non-symmetric relationships among items.
Examples
Section titled “Examples”Network graph
Section titled “Network graph”Objects are nodes and distances are edges. Distances between nodes can differ by direction depending on connections. For example, in a social network, the distance between two people may be different depending on whether the connection is a mutual friend or a one-way friendship.
Transportation network
Section titled “Transportation network”Objects are cities or locations and distances are the time or cost to travel between them. The distance from city A to city B may be different from the distance from city B to city A, depending on the availability of transportation options and the route taken.
Use cases
Section titled “Use cases”- Clustering algorithms: use directional distances to group objects into clusters based on asymmetric similarity.
- Classification algorithms: use asymmetric distances to predict the class or category of new objects with direction-dependent relationships.
Related terms
Section titled “Related terms”- Proximity matrix
- Symmetric proximity matrix
- Network graph
- Transportation network
- Clustering
- Classification