Growing Graphs from Hyperedge Replacement Graph Grammars

Abstract

Discovering the underlying structures present in large real world graphs is a fundamental scientific problem. In this paper we show that a graph's clique tree can be used to extract a hyperedge replacement grammar. If we store an ordering from the extraction process, the extracted graph grammar is guaranteed to generate an isomorphic copy of the original graph. Or, a stochastic application of the graph grammar rules can be used to quickly create random graphs. In experiments on large real world networks, we show that random graphs, generated from extracted graph grammars, exhibit a wide range of properties that are very similar to the original graphs. In addition to graph properties like degree or eigenvector centrality, what a graph "looks like" ultimately depends on small details in local graph substructures that are difficult to define at a global level. We show that our generative graph model is able to preserve these local substructures when generating new graphs and performs well on new and difficult tests of model robustness.

Edge Replacement Grammars : A Formal Language Approach for Generating Graphs

“…First, we introduce clique trees and then define hyperedge replacement grammars. The content in this section is based on [1].…”

“…Finding the minimal-width clique tree is NP-complete [3]. [1] uses a Maximum Cardinality Search (MCS) heuristic introduced by [21] to compute a clique tree with a reasonably-low, but not necessarily minimal, width. Then, this clique tree induces an HRG in a natural way as shown below.…”

“…The approach differs based on the type of node of the clique tree that is being processed. We refer the reader to [1] for a more detailed discussion and visualization of the HRG learning process.…”

“…The likelihood of a graph belonging to a particular family is the probability of the derivation under the appropriate grammar. The idea of using graph grammars for graph generation was also explored in [1] where authors propose a hyperedge replacement grammar (HRG) based generative model. We compare our approach with their approach.…”

Towards Interpretable Graph Modeling with Vertex Replacement Grammars

“…Renewed interest in graph grammars provides a promising route towards the goal of building a non-parametric, interpretable graph model. Previous work has investigated the relationship between graph mining and formal language theory by extracting Vertex Replacement Grammars (VRGs) [10] and (Hyper)edge Replacement Grammars (HRGs) [3], [11]. Unfortunately, the composition of grammar rules in HRGs, and some VRGs are known to produce clunky patterns that are difficult to interpret.…”