Retrieving Hierarchies

TL;DR

This paper investigates the problem of reconstructing hierarchical tree structures from sampled nodes, analyzing how sampling errors affect reconstruction accuracy using a simple tree model and coincidence similarity.

Contribution

It introduces a simple one-parameter tree model and applies coincidence similarity to quantify reconstruction errors under various sampling error probabilities.

Findings

Reconstruction errors depend only moderately on error extent and tree types.

Relative accuracy decreases significantly with increasing error probability.

Larger accuracy variations occur at smaller error probabilities.

Abstract

Several real-world and abstract structures and systems are characterized by marked hierarchy to the point of being expressed as trees. Because the study of these entities often involves sampling (or discovering) the tree nodes in a specific order that may not correspond to that originally shaping the tree, reconstruction errors can be obtained. The present work addresses this important problem based on two main resources: (i) the adoption of a simple model of trees, involving a single parameter; and (ii) the use of the coincidence similarity as the means to quantify the errors by comparing the original and reconstructed structures considering diverse sampling error probability and extent. Several interesting results are described and discussed, including the fact that the average and standard deviation values of the reconstruction errors depend only moderately on the extent of the errors as well as on the types of trees. At the same time, it is identified that the relative reconstruction accuracy substantially decreases markedly with the error probability, with larger reconstructions accuracy relative variations being observed for the smallest values of that probability.