Applications of Laplacian spectrum for the weighted scale-free network with a weight factor

2018 ◽  
Vol 32 (32) ◽  
pp. 1850353 ◽  
Author(s):  
Meifeng Dai ◽  
Tingting Ju ◽  
Jingyi Liu ◽  
Yu Sun ◽  
Xiangmei Song ◽  
...  
Keyword(s):  
Laplacian Matrix ◽  
Weight Matrix ◽  
Weight Factor ◽  
Scale Free Network ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Scale Free ◽  
Free Network ◽  
Eigenvalue And Eigenvector ◽  
Binary Scale

Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network.

scholarly journals Bipartite subgraphs and the signless Laplacian matrix

2011 ◽  
Vol 5 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Steve Kirkland ◽  
Debdas Paul
Keyword(s):  
Laplacian Matrix ◽  
Rayleigh Quotient ◽  
Scale Free Network ◽  
Power Law Distribution ◽  
Laplacian Eigenvalue ◽  
Scale Free ◽  
Protein Protein Interaction ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Free Network

For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalized Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subgraphs. Our results are applied to some graphs with degree sequences approximately following a power law distribution with exponent value 2:1 (scale-free networks), and to a scale-free network arising from protein-protein interaction.

scholarly journals Eigenvalue-based entropy in directed complex networks

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0251993
Author(s):  
Yan Sun ◽  
Haixing Zhao ◽  
Jing Liang ◽  
Xiujuan Ma
Keyword(s):  
Complex Networks ◽  
Complex Network ◽  
Network Models ◽  
Small World ◽  
Laplacian Matrix ◽  
Directed Network ◽  
Small World Network ◽  
Scale Free Network ◽  
Scale Free ◽  
Free Network

Entropy is an important index for describing the structure, function, and evolution of network. The existing research on entropy is primarily applied to undirected networks. Compared with an undirected network, a directed network involves a special asymmetric transfer. The research on the entropy of directed networks is very significant to effectively quantify the structural information of the whole network. Typical complex network models include nearest-neighbour coupling network, small-world network, scale-free network, and random network. These network models are abstracted as undirected graphs without considering the direction of node connection. For complex networks, modeling through the direction of network nodes is extremely challenging. In this paper, based on these typical models of complex network, a directed network model considering node connection in-direction is proposed, and the eigenvalue entropies of three matrices in the directed network is defined and studied, where the three matrices are adjacency matrix, in-degree Laplacian matrix and in-degree signless Laplacian matrix. The eigenvalue-based entropies of three matrices are calculated in directed nearest-neighbor coupling, directed small world, directed scale-free and directed random networks. Through the simulation experiment on the real directed network, the result shows that the eigenvalue entropy of the real directed network is between the eigenvalue entropy of directed scale-free network and directed small-world network.

ENHANCING THE SYNCHRONIZABILITY OF SCALE-FREE NETWORKS BY ADDING EDGES

2010 ◽  
Vol 21 (01) ◽  
pp. 67-77 ◽  
Author(s):  
SHENG-JUN WANG ◽  
ZHI-XI WU ◽  
HAI-RONG DONG ◽  
GUANRONG CHEN
Keyword(s):  
Nearest Neighbors ◽  
Laplacian Matrix ◽  
New Method ◽  
Scale Free Network ◽  
Largest Eigenvalue ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
The Largest Eigenvalue

To efficiently enhance the synchronizability of a scale-free network by adding some edges, we numerically study the effect of edge-adding on the spectrum of the network Laplacian matrix. Based on the relation between the largest eigenvalue of the Laplacian matrix and the largest degree of the scale-free network, we show that adding a new edge to the node of largest degree will generally weaken the synchronizability of a scale-free network. We consequently propose a method to effectively enhance the network synchronizability by attaching the new edge to a node whose nearest-neighbors have small degrees. The effect of the new method is analyzed and demonstrated with comparisons.

Research on new evolution model of scale-free network

2009 ◽  
Vol 29 (5) ◽  
pp. 1230-1232
Author(s):  
Hao RAO ◽  
Chun YANG ◽  
Shao-hua TAO
Keyword(s):  
Evolution Model ◽  
Scale Free Network ◽  
Scale Free ◽  
Free Network

scholarly journals Analysis on Invulnerability of Wireless Sensor Network towards Cascading Failures Based on Coupled Map Lattice

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Xiuwen Fu ◽  
Yongsheng Yang ◽  
Haiqing Yao
Keyword(s):  
Random Network ◽  
Small World ◽  
Coupled Map Lattice ◽  
Wireless Sensor ◽  
Cascading Failures ◽  
Scale Free Network ◽  
Scale Free ◽  
Network Topologies ◽  
Free Network ◽  
Entire Network

Previous research of wireless sensor networks (WSNs) invulnerability mainly focuses on the static topology, while ignoring the cascading process of the network caused by the dynamic changes of load. Therefore, given the realistic features of WSNs, in this paper we research the invulnerability of WSNs with respect to cascading failures based on the coupled map lattice (CML). The invulnerability and the cascading process of four types of network topologies (i.e., random network, small-world network, homogenous scale-free network, and heterogeneous scale-free network) under various attack schemes (i.e., random attack, max-degree attack, and max-status attack) are investigated, respectively. The simulation results demonstrate that the rise of interference R and coupling coefficient ε will increase the risks of cascading failures. Cascading threshold values Rc and εc exist, where cascading failures will spread to the entire network when R>Rc or ε>εc. When facing a random attack or max-status attack, the network with higher heterogeneity tends to have a stronger invulnerability towards cascading failures. Conversely, when facing a max-degree attack, the network with higher uniformity tends to have a better performance. Besides that, we have also proved that the spreading speed of cascading failures is inversely proportional to the average path length of the network and the increase of average degree k can improve the network invulnerability.

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