Abstract
Recently, symmetry in complex network structures has attracted some research interest. One of the fascinating problems is to give measures of the extent to which the network is symmetric. In this paper, 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 ) , we propose three indexes for the characterization of the global symmetry of complex networks. Using these indexes, one can get a quantitative characterization of how symmetric a network is and can compare the symmetry property of different networks. Moreover, we compare these indexes to some existing ones in the literature and apply these indexes to real-world networks, concluding that real-world networks are far from vertex symmetric ones.
Highlights
In the past few decades, an array of discoveries has added to our understanding of complex networks [1]
In this paper, based on the orbit decompositions of the vertex set V of a network Γ(V, E) under the natural action of the automorphism group Aut(Γ), we have proposed three indexes for the characterization of the symmetry of network structures
To address a less symmetric network structure, we introduced the concept of orbit-homogeneous networks
Summary
In the past few decades, an array of discoveries has added to our understanding of complex networks [1]. By considering the size and structure of the automorphism groups of a variety of empirical real-world networks, the authors in [11] found that, in contrast to classical random graph models, many real-world networks are richly symmetric They commented on how symmetry can affect network redundancy and robustness in that same paper. Based on the natural action of the automorphism group Aut(Γ) on the vertex set V of a given network Γ(V, E), we propose three symmetry indexes SIΓ,i for the characterization of the symmetry of networks, 1 ≤ i ≤ 3. We introduce the concept of orbit-homogeneous networks, namely all the orbits in the vertex set V under the automorphism group action have the same size We take such networks as less symmetric than the vertex symmetric ones, and our second symmetry index SIΓ, characterizes how much a network differs from an orbit-homogeneous one. The following section contains our conclusions, and we summarize the notations appearing in this paper in the last section
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
