Towards Interpretable Graph Modeling with Vertex Replacement Grammars

Abstract

An enormous amount of real-world data exists in the form of graphs. Oftentimes, interesting patterns that describe the complex dynamics of these graphs are captured in the form of frequently reoccurring substructures. Recent work at the intersection of formal language theory and graph theory has explored the use of graph grammars for graph modeling and pattern mining. However, existing formulations do not extract meaningful and easily interpretable patterns from the data. The present work addresses this limitation by extracting a special type of vertex replacement grammar, which we call a KT grammar, according to the Minimum Description Length (MDL) heuristic. In experiments on synthetic and real-world datasets, we show that KT-grammars can be efficiently extracted from a graph and that these grammars encode meaningful patterns that represent the dynamics of the real-world system.

Attributed Graph Modeling with Vertex Replacement Grammars

“…A recent addition to the class of graph generators are hyperedge [9,45] and node replacement [7,8] grammars. Graph grammars contain graphical rewriting rules that match and replace graph fragments, similar to how a context-free string grammar rewrites characters in a string.…”

“…1(A). Attributed Vertex Replacement Grammar (AVRG) builds on recent advances in graph grammars [7][8][9], which were designed 1 The source code is available at https://github.com/satyakisikdar/Attributed-VRG only to model and generate homogeneous graphs, to handle attributed graphs, increasing its applicability to a broad class of modern problems.…”

The Infinity Mirror Test for Graph Models

“…Since SBMs' introduction, they have been extended to handle edge-weighted [14], bipartite [15], temporal [16], and hierarchical networks [17]. Likewise, Exponential Random Graph Models (ERGMs) [18], Kronecker graph models [19,20,21], and graph grammar models [22,23,24] are able to generate graphs that are more-or-less faithful to the source graph.…”

“…These include Chung Lu model [2], clustering-based node replacement graph grammars (CNRG) [23], block two-level Erdős Réyni (BTER) [41], degree-corrected stochastic block models (SBM) [42], hyperedge replacement graph grammars (HRG) [43,22], Kronecker graphs [19,20], bottom-up graph grammar extractor (BUGGE) [24], generative adversarial network (NetGAN) [30], graph linear autoencoder (LinearAE) [26], graph convolutional neural networks (GCNAE) [27], and graph recurrent neural networks (GraphRNN) [25]. Random graphs generated using the Erdős-Rényi model with an edge probability equal to the density of the input graph are also included as a baseline.…”