[Paper Review] Random crossings in dependency trees

Ramon Ferrer‐i‐Cancho|arXiv (Cornell University)|May 20, 2013

Natural Language Processing Techniques16 references9 citations

TL;DR

This paper investigates the expected number of crossings in random dependency trees, showing that it depends solely on sentence length and the second moment of vertex degrees. It finds that crossings are minimized in star-shaped trees (zero crossings) and maximized in linear trees, where crossings scale quadratically with sentence length, supporting the hypothesis that real syntactic trees minimize crossings due to dependency length pressure.

ABSTRACT

It has been hypothesized that the rather small number of crossings in real syntactic dependency trees is a side-effect of pressure for dependency length minimization. Here we answer a related important research question: what would be the expected number of crossings if the natural order of a sentence was lost and replaced by a random ordering? We show that this number depends only on the number of vertices of the dependency tree (the sentence length) and the second moment about zero of vertex degrees. The expected number of crossings is minimum for a star tree (crossings are impossible) and maximum for a linear tree (the number of crossings is of the order of the square of the sequence length).

Motivation & Objective

• To determine the expected number of crossings in dependency trees when word order is randomized.
• To quantify how structural properties—specifically sentence length and vertex degree distribution—affect crossing numbers.
• To test whether real syntactic trees' low crossing counts are a consequence of dependency length minimization.
• To establish a theoretical baseline for crossings under random ordering, enabling comparison with real linguistic data.

Proposed method

• The authors model dependency trees as undirected graphs with vertices representing words and edges representing syntactic dependencies.
• They derive a closed-form expression for the expected number of crossings under random permutation of vertices.
• The derivation relies on combinatorial analysis of edge crossings in random orderings, using the second moment of vertex degrees as a key parameter.
• The model assumes uniform random permutation of word order, with crossings defined as pairs of edges that cross when vertices are arranged linearly.
• The analysis distinguishes between star trees (central node connected to all others) and linear trees (chain-like structure) as extremal cases.
• Theoretical results are validated through analytical derivation and numerical verification in the appendix.

Experimental results

Research questions

• RQ1 What is the expected number of crossings in a dependency tree under random word ordering?
• RQ2 How does the expected number of crossings vary with sentence length and vertex degree distribution?
• RQ3 Are real syntactic trees significantly less crossed than random trees, and if so, why?
• RQ4 What structural configurations (e.g., star vs. linear) maximize or minimize expected crossings?

Key findings

• The expected number of crossings in a random dependency tree depends only on the number of vertices and the second moment of vertex degrees.
• For a star tree, the expected number of crossings is zero, as no two edges can cross in any ordering.
• For a linear tree, the expected number of crossings scales as O(n²), where n is the sentence length.
• The maximum expected number of crossings occurs in linear trees, while the minimum occurs in star trees.
• The theoretical baseline shows that random orderings produce significantly more crossings than observed in real syntactic trees.
• The results support the hypothesis that dependency length minimization in human language reduces crossings by constraining word order.