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Symmetry, Volume 8, Issue 2 (February 2016) – 2 articles

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583 KiB  
Article
New Upper Bound and Lower Bound for Degree-Based Network Entropy
by Guoxiang Lu, Bingqing Li and Lijia Wang
Symmetry 2016, 8(2), 8; https://doi.org/10.3390/sym8020008 - 19 Feb 2016
Cited by 3 | Viewed by 4178
Abstract
The degree-based network entropy which is inspired by Shannon’s entropy concept becomes the information-theoretic quantity for measuring the structural information of graphs and complex networks. In this paper, we study some properties of the degree-based network entropy. Firstly we develop a refinement of [...] Read more.
The degree-based network entropy which is inspired by Shannon’s entropy concept becomes the information-theoretic quantity for measuring the structural information of graphs and complex networks. In this paper, we study some properties of the degree-based network entropy. Firstly we develop a refinement of Jensen’s inequality. Next we present the new and more accurate upper bound and lower bound for the degree-based network entropy only using the order, the size, the maximum degree and minimum degree of a network. The bounds have desirable performance to restrict the entropy in different kinds of graphs. Finally, we show an application to structural complexity analysis of a computer network modeled by a connected graph. Full article
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Article
On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map
by Da Wang and ShuTang Liu
Symmetry 2016, 8(2), 7; https://doi.org/10.3390/sym8020007 - 26 Jan 2016
Cited by 9 | Viewed by 4263
Abstract
A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos [...] Read more.
A complex map can give rise to two kinds of fractal sets: the Julia sets and the parameters sets (or the connectivity loci) which represent different connectivity properties of the corresponding Julia sets. In the significative results of (Int. J. Bifurc. Chaos, 2009, 19:2123–2129) and (Nonlinear. Dyn. 2013, 73:1155–1163), the authors presented the two kinds of fractal sets of a class of alternated complex map and left some visually observations to be proved about the boundedness and symmetry properties of these fractal sets. In this paper, we improve the previous results by giving the strictly mathematical proofs of the two properties. Some simulations that verify the theoretical proofs are also included. Full article
(This article belongs to the Special Issue Symmetry and Fractals)
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