Ratio of Symmetries Between any two n-Node Graphs

Given any two graphs on the same vertex set, $G_1 = (V, E_1)$ and $G_2 = (V, E_2)$, along with the difference between the two graphs $\Delta = (E_1 \setminus E_2) \cup (E_2 \setminus E_1)$, we prove that the ratio of the sizes of the two graphs' automorphism groups is equivalent to the ratio of the sizes of $\Delta$'s automorphism orbits in $G_1$ and $G_2$ respectively. This result provides a link between graphs' symmetries that might otherwise seem to be unrelated

Inherent Limits on Topology-Based Link Prediction

Link prediction systems (e.g. recommender systems) typically use graph topology as one of their main sources of information. We calculate hard upper bounds on how well the structure alone enables link prediction for a wide variety of real-world graphs. We find that in the sparsest of these graphs the upper bounds are non-trivially low, thereby demonstrating that if a link prediction system is going to perform well in a sparse context, it will almost certainly need more information than the network topology alone

Towards Interpretable Graph Modeling with Vertex Replacement Grammars

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.Comment: 10 pages, 9 figures, accepted at IEEE BigData 201

Modeling Graphs with Vertex Replacement Grammars

One of the principal goals of graph modeling is to capture the building blocks of network data in order to study various physical and natural phenomena. Recent work at the intersection of formal language theory and graph theory has explored the use of graph grammars for graph modeling. However, existing graph grammar formalisms, like Hyperedge Replacement Grammars, can only operate on small tree-like graphs. The present work relaxes this restriction by revising a different graph grammar formalism called Vertex Replacement Grammars (VRGs). We show that a variant of the VRG called Clustering-based Node Replacement Grammar (CNRG) can be efficiently extracted from many hierarchical clusterings of a graph. We show that CNRGs encode a succinct model of the graph, yet faithfully preserves the structure of the original graph. In experiments on large real-world datasets, we show that graphs generated from the CNRG model exhibit a diverse range of properties that are similar to those found in the original networks.Comment: Accepted as a regular paper at IEEE ICDM 2019. 15 pages, 9 figure