The graph isomorphism disease

Read, Ronald C.; Corneil, Derek G.

doi:10.1002/jgt.3190010410

Cited by 383 publications

(198 citation statements)

References 23 publications

(13 reference statements)

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“…The study of graph symmetries has been a field of research for many years [5]. Two graphs are said to be isomorphic if there exists a bijection of the set of nodes so that one graph equals the other under the transformation of the bijection.…”

Section: Introductionmentioning

confidence: 99%

How Symmetric Are Real-World Graphs? A Large-Scale Study

Ball

Geyer-Schulz

2018

Symmetry

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Abstract:The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the "network redundancy" measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

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Section: Introductionmentioning

confidence: 99%

How Symmetric Are Real-World Graphs? A Large-Scale Study

Ball

Geyer-Schulz

2018

Symmetry

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“…However, in cases in which many such checks are required among the same set of subgraphs, a better way of performing this task is to assign to each graph a unique code (i.e., a sequence of bits, a string, or a sequence of numbers) that is invariant on the ordering of the vertices and edges in the graph. Such a code is referred to as the canonical label of a graph G = (V, E) [40,11], and we will denote it by cl(G). By using canonical labels, we can check whether or not two graphs are identical by checking to see whether they have identical canonical labels.…”

Section: Canonical Labelingmentioning

confidence: 99%

Finding Frequent Patterns in a Large Sparse Graph

Kuramochi

Karypis

2004

Proceedings of the 2004 SIAM International Conference on Data Mining

134

180

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This paper presents two algorithms based on the horizontal and vertical pattern discovery paradigms that find the connected subgraphs that have a sufficient number of edge-disjoint embeddings in a single large undirected labeled sparse graph. These algorithms use three different methods to determine the number of the edge-disjoint embeddings of a subgraph that are based on approximate and exact maximum independent set computations and use it to prune infrequent subgraphs. Experimental evaluation on real datasets from various domains show that both algorithms achieve good performance, scale well to sparse input graphs with more than 100,000 vertices, and significantly outperform a previously developed algorithm.

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“…Here, the size of the matched subgraph was examined as the degree of the graphs was increased. This style of testing was included because regular graphs are the most challenging type for isomorphism algorithms [33]. This makes for a difficult matching problem for the subgraph case.…”

Section: Effect Of Reduced Dynamic Range Of Node Colormentioning

confidence: 99%

An algorithm using length-r paths to approximate subgraph isomorphism

DePiero

Krout

2003

Pattern Recognition Letters

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The 'LeRP' algorithm approximates subgraph isomorphism for attributed graphs based on counts of Length-R Paths. The algorithm provides a good approximation to the maximal isomorphic subgraph. The basic approach of the LeRP algorithm differs fundamentally from other methods. When comparing structural similarity LeRP uses a neighborhood of nodes that varies in size dynamically. This approach provides sufficient evidence of similarity to permit LeRP to form a node-to-node mapping and can be computed with polynomial effort in the worst-case. Results from over 32,000 simulated cases are reported. We demonstrate that LeRP does not need a high dynamic range of node and edge coloring to perform well. For example, LeRP can input 50-node and 100-node graphs that contain a common 50-node subgraph, and then compute a matching subgraph having 49.74 +/-0.46 nodes (mean +/-one standard deviation). This takes from 0.4 to 0.5 seconds. In this example, 100 trials were evaluated and graphs had discrete coloring for nodes and edges with a dynamic range of four. Test conditions are varied and include strongly regular graphs as well as Model A.

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The graph isomorphism disease

Cited by 383 publications

References 23 publications

How Symmetric Are Real-World Graphs? A Large-Scale Study

How Symmetric Are Real-World Graphs? A Large-Scale Study

Finding Frequent Patterns in a Large Sparse Graph

An algorithm using length-r paths to approximate subgraph isomorphism

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