Revealing structure in large graphs: Szemerédi’s regularity lemma and its use in pattern recognition

Abstract

Introduced in the mid-1970's as an intermediate step in proving a long-standing conjecture on arithmetic progressions, Szemerédi's regularity lemma has emerged over time as a fundamental tool in different branches of graph theory, combinatorics and theoretical computer science. Roughly, it states that every graph can be approximated by the union of a small number of random-like bipartite graphs called regular pairs. In other words, the result provides us a way to obtain a good description of a large graph using a small amount of data, and can be regarded as a manifestation of the all-pervading dichotomy between structure and randomness. In this paper we will provide an overview of the regularity lemma and its algorithmic aspects, and will discuss its relevance in the context of pattern recognition research.

Separating Structure from Noise in Large Graphs Using the Regularity Lemma

“…In particular, the higher is the value of the M AP ∈ [0, 1], the higher is the quality of the proposed graph search algorithm. Figures 7,8,9,10 show that the proposed summarization based approach improved the query quality. Figure 8: The MAP@k of the top-k graphs given as output in a database of 180 graphs.…”

“…However, despite being polynomial in the size of the underlying graph, all these algorithms have a hidden tower-type dependence on an accuracy parameter. To overcome this limitation, in the last years we have proposed some simple heuristics that, most of the times, allowed us to construct a regular partition [7,8,9].…”

You Can't See Me: Anonymizing Graphs Using the Szemerédi Regularity Lemma

“…In the first step we apply the algorithm implemented by Fiorucci et al (2019) 2 . This extends the previous algorithm of Fiorucci et al (2017) by proposing a novel heuristic procedure where the node set is first partitioned into two groups of nodes and then these are recursively split into smaller groups until a desired cardinality is met and certain conditions that measure quality of the ε-regularity of the partition are satisfied (Pelillo et al, 2017). In particular Fiorucci et al propose two different heuristics to split the groups, one called degree based, which groups together nodes with similar degrees (Fiorucci et al, 2017), and a second one called indeg guided, which splits a sparse (dense) partition into two sparse (dense) partitions.…”

k-Anonymity on Graphs Using the Szemerédi Regularity Lemma

“…Their approach is based on the Szemerédi regularity lemma [7], a well-known result of extremal graph theory. The Szemerédi regularity lemma has been successfully applied to several problems, from graph theory [18] to computer vision and pattern recognition [25], [34]. The lemma roughly states that every sufficiently large and dense graph can be approximated by the union of Fig.…”