# On Automorphism Groups of Networks

@article{MacArthur2007OnAG, title={On Automorphism Groups of Networks}, author={Ben D. MacArthur and Rub'en J. S'anchez-Garc'ia and James W. Anderson}, journal={Discrete Applied Mathematics}, year={2007} }

We consider the size and structure of the automorphism groups of a variety of empirical `real-world' networks and find that, in contrast to classical random graph models, many real-world networks are richly symmetric. We relate automorphism group structure to network topology and discuss generic forms of symmetry and their origin in real-world networks.

## 56 Citations

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## References

SHOWING 1-10 OF 72 REFERENCES

Graphs whose full automorphism group is a symmetric group

- Mathematics
- 1988

We address the problem of describing all graphs Γ such that Aut Γ is a symmetric group, subject to certain restrictions on the sizes of the orbits of Aut Γ on vertices. As a corollary of our general…

Topics in Graph Automorphisms and Reconstruction

- Mathematics
- 2003

1. Graphs and groups: preliminaries 2. Various types of graph symmetry 3. Cayley graphs 4. Orbital graphs and strongly regular graphs 5. Graphical regular representations and pseudosimilarity 6.…

The Symmetry Ratio of a Network

- Mathematics, Computer ScienceCATS
- 2005

A number of results are proved placing bounds on the symmetry ratio for several families of networks, including distance-transitive networks, prisms, twistedPrisms, antiprisms, tori, Cayley graphs, and random graphs.

The Structure and Function of Complex Networks

- Physics, Computer ScienceSIAM Rev.
- 2003

Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

Spectral Graph Theory

- Computer Science
- 1996

Eigenvalues and the Laplacian of a graph Isoperimetric problems Diameters and eigenvalues Paths, flows, and routing Eigenvalues and quasi-randomness Expanders and explicit constructions Eigenvalues…

Eigenvalue spectra of complex networks

- Mathematics
- 2005

We examine the eigenvalue spectrum, ρ(μ), of the adjacency matrix of a random scale-free network with an average of p edges per vertex using the replica method. We show how in the dense limit, when p…

Groups Acting on Graphs

- Mathematics
- 1989

Preface Conventions 1. Groups and graphs 2. Cutting graphs and building trees 3. The almost stability theorem 4. Applications of the almost stability theorem 5. Poincare duality 6. Two-dimensional…

Detecting degree symmetries in networks.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

It is found that most studied examples of degree symmetry are weakly positively degree symmetric, and the exceptions are an airport network (having a negative degree-symmetry coefficient) and one-mode projections of social affiliation networks that are rather strongly degree asymmetric.

Algebraic Graph Theory

- Mathematics, Computer ScienceGraduate texts in mathematics
- 2001

The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.

Nonlinear dynamics of networks: the groupoid formalism

- Mathematics
- 2006

A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that preserve
the network topology, has existed for some time.…