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. 

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References

SHOWING 1-10 OF 72 REFERENCES
Graphs whose full automorphism group is a symmetric group
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
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
TLDR
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
  • M. Newman
  • Physics, Computer Science
    SIAM Rev.
  • 2003
TLDR
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
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
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
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.
  • P. Holme
  • Mathematics, Medicine
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
TLDR
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
TLDR
The Laplacian of a Graph and Cuts and Flows are compared to the Rank Polynomial.
Nonlinear dynamics of networks: the groupoid formalism
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.
...
1
2
3
4
5
...