On Automorphism Groups of Networks

  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},
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. 

Figures and Tables from this paper

A Structured Table of Graphs with Symmetries and Other Special Properties
We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These
Generating symmetric graphs.
This work presents an algorithm that is able to generate graphs with any desired symmetry pattern and can be coupled with other graph generating algorithms to tune the final graph's properties of interest such as the degree distribution.
Characterization of Symmetry of Complex Networks
Using three indexes based on the natural action of the automorphism group Aut ( Γ ) of Γ on the vertex set V of a given network Γ = Γ ( V, E ) , one can get a quantitative characterization of how symmetric a network is and can compare the symmetry property of different networks.
Symmetry-based structure entropy of complex networks
Precisely quantifying the heterogeneity or disorder of network systems is important and desired in studies of behaviors and functions of network systems. Although various degree-based entropies have
Hidden Symmetries in Real and Theoretical Networks
  • Dallas Smith, B. Webb
  • Computer Science, Physics
    Physica A: Statistical Mechanics and its Applications
  • 2019
A generalization of network symmetry called latent symmetry is presented, which is an extension of the standard notion of symmetry on networks, which can be directed, weighted or both and is defined in terms of standard symmetries in a reduced version of the network.
Emergence of symmetry in complex networks.
An improved version of the Barabaśi-Albert model integrating similar linkage pattern successfully reproduces the symmetry of real networks, indicating that similar linkagepattern is the underlying ingredient that is responsible for the emergence of symmetry in complex networks.
Network Analyzing by the Aid of Orbit Polynomial
The present paper studied some properties of the orbit polynomial with respect to the stabilizer elements of each vertex and constructed graphs with a small number of orbits and characterized some classes of graphs in terms of calculating their orbits polynomials.
A General Model of Dynamics on Networks with Graph Automorphism Lumping
It is proved that symmetries of the network can be used to lump equivalent states in state-space and connects a wide range of models specified in terms of node-based dynamical rules to their exact continuous-time Markov chain formulation.
Exploiting symmetry in network analysis
The effect of symmetries on network measures and how they can be exploited to increase computational efficiency are studied and the spectral signatures of symmetry for an arbitrary network measure such as the graph Laplacian are uncovered.
Invariant Graph Partition Comparison Measures
This article starts with the proof that all partition comparison measures found in the literature fail on symmetric graphs, because they are not invariant with regard to the graph automorphisms.


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
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
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
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
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.