Symmetry in complex networks

MacArthur, Ben D.; Sánchez-García, Rubén J.; Anderson, James A.

doi:10.1016/j.dam.2008.04.008

Cited by 155 publications

(165 citation statements)

References 32 publications

(25 reference statements)

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“…It has the intuitive property that completely asymmetric graphs have a measure of 0 and completely symmetric graphs a measure of 1. In contrast, the original definition by MacArthur et al [19] yields different values for different asymmetric graphs, which makes them incomparable by the measure.…”

mentioning

confidence: 63%

“…MacArthur et al [19] introduced a relative measure for the degree of symmetry, which they call "network redundancy":…”

Section: Data Analysis Proceduresmentioning

confidence: 99%

“…Often, for example, [19], non-simple graphs are "simplified" by removing multiple edges, loops and edge weights, and replacing edge 2-tuples (directed) by edge 2-sets (undirected). We refrained from doing so as this simplification is likely to "add" symmetry that is actually not present in the original version of the graph (see the example in Figure 1).…”

Section: Data Analysis Selectionmentioning

confidence: 99%

“…Therefore, the conclusion "symmetries in real-world graphs are unlikely" is false without further evidence. Several empirical investigations on the actual symmetry of complex/real-world networks already exist [10,17,18], with an emphasis on [19], but all these publications have in common that only a very small selection of graphs (about 25 maximal) was taken into account. A theoretical overview of symmetry in complex networks is given by Garrido [20].…”

Section: Introductionmentioning

confidence: 99%

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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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mentioning

confidence: 63%

“…MacArthur et al [19] introduced a relative measure for the degree of symmetry, which they call "network redundancy":…”

Section: Data Analysis Proceduresmentioning

confidence: 99%

Section: Data Analysis Selectionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Ball

Geyer-Schulz

2018

Symmetry

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“…We have already realised this for synchronous states (in networks built from identical units), and we advocate for a push to develop this methodology further to treat more exotic network states, including those expected in the presence of noise [44]. Moreover, it seems very fruitful to classify emergent dynamics based upon the group of structural symmetries of the network [45], as has recently been done in [34] to determine cluster states.…”

Section: Discussionmentioning

confidence: 99%

Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

Coombes

Thul

2016

Eur. J. Appl. Math

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The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

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