Anti-monotonic Overlap-Graph Support Measures

Abstract

In graph mining, a frequency measure is anti-monotonic if the frequency of a pattern never exceeds the frequency of a subpattern. The efficiency and correctness of most graph pattern miners relies critically on this property. We study the case where the dataset is a single graph. Vanetik, Gudes and Shimony already gave sufficient and necessary conditions for anti-monotonicity of measures depending only on the edge-overlaps between the intances of the pattern in a labeled graph. We extend these results to homomorphisms, isomorphisms and homeomorphisms on both labeled and unlabeled, directed and undirected graphs, for vertex and edge overlap. We show a set of reductions between the different morphisms that preserve overlap. We also prove that the popular maximum independent set measure assigns the minimal possible meaningful frequency, introduce a new measure based on the minimum clique partition that assigns the maximum possible meaningful frequency and introduce a new measure sandwiched between the former two based on the poly-time computable Lovász θ-function. 1. We study systematically all 24 combinations of iso-, homo-, or homeomorphism, on labeled or unlabeled, directed or undirected graphs, with edge-or vertexoverlap and extend the anti-monotonicity results. 2. In our proofs, we use reductions which are also of interest in their own right, as they allow to transfer results for different types of morphisms and overlap from one setting to another. For an overview of the different reductions, see Figure 1. 3. An interesting consequence of the reductions is that any unlabeled, undirected graph is a potential vertexand edge-overlap graph in all considered settings.

BiQL: A Query Language for Analyzing Information Networks

“…Several techniques have been developed that target subsets of graph representations, such as sequences or trees [39]. Recently, there has been increasing interest in applying graph mining techniques to the network setting, that is, to a single graph [10,9,26].…”

Mining Graph Evolution Rules

“…Defining a concept of support for the single graph setting is a non-trivial task, which has received attention recently [14,8,4,5]. The most important property that a definition of support must satisfy is anti-monotonicity, that is, for all graphs G, P and P ′ , where P is a subgraph of P ′ , it must hold that σ(P, G) ≥ σ(P ′ , G).…”

“…A recent paper by Calders et al [5] introduces a new measure named minimum clique partition, which analogous to the maximal independent set is based on the notion of an overlap graph and thus requires solving an NP-complete problem. They prove that support measures based maximal independent set and minimum clique partition are the minimal and the maximal possible meaningful overlap measures, and show that [12] introduced a function which is sandwiched between these two measures; computable in polynomial time.…”

A survey of frequent subgraph mining algorithms

“…In Vanetik et al (2006), the formal definition were provided together with proofs for the sufficient and necessary conditions required for occurrence based support measures to maintain the DCP. Their work was further extended to introduce a new occurrence based support measure, which maintained the DCP, and was computable in polynomial time (Calders et al 2008). Karypis (2004c, 2005) proposed two algorithms: HSIGRAM and VSIGRAM to find all frequent subgraphs in a large sparse graph.…”