EIGENTIME IDENTITY FOR WEIGHT-DEPENDENT WALK ON A CLASS OF WEIGHTED FRACTAL SCALE-FREE HIERARCHICAL-LATTICE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950138
Author(s):  
BO WU ◽  
ZHIZHUO ZHANG ◽  
YINGYING CHEN ◽  
TINGTING JU ◽  
MEIFENG DAI ◽  
...  
Keyword(s):  
Iteration Process ◽  
Laplacian Matrix ◽  
Network Size ◽  
Hierarchical Lattice ◽  
Analytical Expression ◽  
Scale Free ◽  
Scale Free Networks ◽  
Reciprocal Sum ◽  
Successive Generations ◽  
Relationship Of

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates [Formula: see text] connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when [Formula: see text] is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks.

Coherence analysis of a class of weighted tree-like polymer networks

2018 ◽  
Vol 32 (05) ◽  
pp. 1850064 ◽  
Author(s):  
Jiaojiao He ◽  
Meifeng Dai ◽  
Yue Zong ◽  
Jiahui Zou ◽  
Yu Sun ◽  
...  
Keyword(s):  
Polymer Networks ◽  
Laplacian Matrix ◽  
Network Size ◽  
Second Order ◽  
Weight Factor ◽  
Stochastic Disturbances ◽  
Weighted Tree ◽  
Consensus Dynamics ◽  
Successive Generations ◽  
Relationship Of

Complex networks have elicited considerable attention from scientific communities. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. Firstly, we introduce a class of weighted tree-like polymer networks with the weight factor. Then, we deduce the recursive relationship of the eigenvalues of Laplacian matrix at two successive generations. Finally, we calculate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor and the scalings of second-order coherence with network size obey five laws along with the range of the weight factor.

scholarly journals Diagonal Degree Correlations vs. Epidemic Threshold in Scale-Free Networks

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. L. Bertotti ◽  
G. Modanese
Keyword(s):  
Correlation Matrix ◽  
Large Scale ◽  
Network Size ◽  
Epidemic Threshold ◽  
Correlation Matrices ◽  
Scale Free ◽  
Degree Correlation ◽  
Scale Free Networks ◽  
Degree Correlations

We prove that the presence of a diagonal assortative degree correlation, even if small, has the effect of dramatically lowering the epidemic threshold of large scale-free networks. The correlation matrix considered is P h | k = 1 − r P h k U + r δ h k , where P U is uncorrelated and r (the Newman assortativity coefficient) can be very small. The effect is uniform in the scale exponent γ if the network size is measured by the largest degree n . We also prove that it is possible to construct, via the Porto–Weber method, correlation matrices which have the same k n n as the P h | k above, but very different elements and spectra, and thus lead to different epidemic diffusion and threshold. Moreover, we study a subset of the admissible transformations of the form P h | k ⟶ P h | k + Φ h , k with Φ h , k depending on a parameter which leaves k n n invariant. Such transformations affect in general the epidemic threshold. We find, however, that this does not happen when they act between networks with constant k n n , i.e., networks in which the average neighbor degree is independent from the degree itself (a wider class than that of strictly uncorrelated networks).

Complex networks

2018 ◽  
Author(s):  
Graziano Vernizzi ◽  
Henri Orland
Keyword(s):  
Complex Networks ◽  
Small World ◽  
Laplacian Matrix ◽  
Square Lattice ◽  
Small World Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
Scale Free Graphs ◽  
Regular Networks

This article deals with complex networks, and in particular small world and scale free networks. Various networks exhibit the small world phenomenon, including social networks and gene expression networks. The local ordering property of small world networks is typically associated with regular networks such as a 2D square lattice. The small world phenomenon can be observed in most scale free networks, but few small world networks are scale free. The article first provides a brief background on small world networks and two models of scale free graphs before describing the replica method and how it can be applied to calculate the spectral densities of the adjacency matrix and Laplacian matrix of a scale free network. It then shows how the effective medium approximation can be used to treat networks with finite mean degree and concludes with a discussion of the local properties of random matrices associated with complex networks.

Applications of Laplacian spectrum for the vertex–vertex graph

2019 ◽  
Vol 33 (17) ◽  
pp. 1950184 ◽  
Author(s):  
Tingting Ju ◽  
Meifeng Dai ◽  
Changxi Dai ◽  
Yu Sun ◽  
Xiangmei Song ◽  
...  
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
First Order ◽  
Laplacian Energy ◽  
Wide Range ◽  
Vertex Graph ◽  
Network Properties ◽  
Successive Generations ◽  
Relationship Of

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like.

scholarly journals SURVIVING OPINIONS IN SZNAJD MODELS ON COMPLEX NETWORKS

2005 ◽  
Vol 16 (11) ◽  
pp. 1785-1792 ◽  
Author(s):  
F. A. RODRIGUES ◽  
L. DA F. COSTA
Keyword(s):  
Complex Networks ◽  
Scaling Law ◽  
Network Size ◽  
Random Networks ◽  
Scale Free ◽  
Scale Free Networks ◽  
Network Topologies

The Sznajd model has been largely applied to simulate many sociophysical phenomena. In this paper, we applied the Sznajd model with more than two opinions on three different network topologies and observed the evolution of surviving opinions after many interactions among the nodes. As result, we obtained a scaling law which depends of the network size and the number of possible opinions. We also observed that this scaling law is not the same for all network topologies, being quite similar between scale-free networks and Sznajd networks but different for random networks.

scholarly journals Characterizing the intrinsic correlations of scale-free networks

2016 ◽  
Vol 27 (03) ◽  
pp. 1650024 ◽  
Author(s):  
J. B. de Brito ◽  
C. I. N. Sampaio Filho ◽  
A. A. Moreira ◽  
J. S. Andrade
Keyword(s):  
Extreme Values ◽  
Network Models ◽  
Network Size ◽  
Scale Free Network ◽  
Scale Free ◽  
Scale Free Networks ◽  
Free Network ◽  
Degree Correlations ◽  
Multiple Edges ◽  
Random Scale

When studying topological or dynamical properties of random scale-free networks, it is tacitly assumed that degree–degree correlations are not present. However, simple constraints, such as the absence of multiple edges and self-loops, can give rise to intrinsic correlations in these structures. In the same way that Fermionic correlations in thermodynamic systems are relevant only in the limit of low temperature, the intrinsic correlations in scale-free networks are relevant only when the extreme values for the degrees grow faster than the square root of the network size. In this situation, these correlations can significantly affect the dependence of the average degree of the nearest neighbors of a given vertex on this vertices degree. Here, we introduce an analytical approach that is capable to predict the functional form of this property. Moreover, our results indicate that random scale-free network models are not self-averaging, that is, the second moment of their degree distribution may vary orders of magnitude among different realizations. Finally, we argue that the intrinsic correlations investigated here may have profound impact on the critical properties of random scale-free networks.

AN EFFICIENT BANDWIDTH ALLOCATION STRATEGY FOR SCALE-FREE NETWORKS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250065 ◽  
Author(s):  
ZHONG-YUAN JIANG ◽  
MAN-GUI LIANG ◽  
SHUAI ZHANG ◽  
SHU-JUAN WANG ◽  
DONG-CHAO GUO
Keyword(s):  
Bandwidth Allocation ◽  
Service Providers ◽  
Network Size ◽  
Urban Transport ◽  
Allocation Strategy ◽  
Transport Networks ◽  
Scale Free ◽  
Traffic Capacity ◽  
Optimal Value ◽  
Scale Free Networks

Traffic capacity is critical for various networks and strongly depends on the distribution of link's bandwidth resources. In this paper, we propose a betweenness-based bandwidth allocation strategy in which the bandwidth of each link lij is allocated proportionally to the product (1 + Bi)α(1 + Bj)α, where α is a tunable parameter, and Bi and Bj are the betweenness of node i and node j, respectively. The optimal value of α is achieved by extensive simulations and slightly increases with the network size. Our new bandwidth allocation strategy achieves the highest traffic capacity when compared with the average bandwidth allocation strategy and the previously proposed degree-based bandwidth allocation strategy. Our work will be beneficial for network service providers to improve the traffic capacity by efficiently allocating or reallocating the overall finite link's bandwidth resources of networks such as the Internet, urban transport networks and airway networks.

scholarly journals The contact process on scale-free networks evolving by vertex updating

2017 ◽  
Vol 4 (5) ◽  
pp. 170081 ◽  
Author(s):  
Emmanuel Jacob ◽  
Peter Mörters
Keyword(s):  
Phase Transition ◽  
Power Law ◽  
Mathematical Proof ◽  
Contact Process ◽  
Network Size ◽  
Mean Field Approximation ◽  
Infection Rates ◽  
Scale Free ◽  
Power Law Exponent ◽  
Scale Free Networks

We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most polynomially in the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast with that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time at which the infection spends in metastable states.

SYNCHRONIZABILITY OF COMPLEX NETWORKS WITH COMMUNITY STRUCTURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250029 ◽  
Author(s):  
MAHDI JALILI
Keyword(s):  
Community Structure ◽  
Real World ◽  
Order Parameter ◽  
Small World ◽  
Laplacian Matrix ◽  
Small World Networks ◽  
Scale Free ◽  
Community Connections ◽  
Scale Free Networks ◽  
Kuramoto Oscillators

Many real-world networks show community structure characterized by dense intra-community connections and sparse inter-community links. In this paper we investigated the synchronization properties of such networks. In this work we constructed such networks in a way that they consist of a number of communities with scale-free or small-world structure. Furthermore, with a probability, the intra-community connections are rewired to inter-community links. Two synchronizability measures were considered as the eigenratio of the Laplacian matrix and the phase order parameter obtained for coupled nonidentical Kuramoto oscillators. We found a power-law relation between the eigenratio and the inter-community rewiring probability in which as the rewiring probability increased, the eigenratio decreased, and hence, the synchronizability enhanced. The phase order parameter also increased by increasing the rewiring probability. Also, small-world networks with community structure showed better synchronization properties as compared to scale-free networks with community structure.

EXTENDED FIBER BUNDLE MODEL FOR TRAFFIC JAMS ON SCALE-FREE NETWORKS

2008 ◽  
Vol 19 (11) ◽  
pp. 1727-1735 ◽  
Author(s):  
JIAN-FENG ZHENG ◽  
ZI-YOU GAO ◽  
XIAO-MEI ZHAO ◽  
BAI-BAI FU
Keyword(s):  
Failure Rate ◽  
Fiber Bundle ◽  
Load Distribution ◽  
Network Size ◽  
Traffic Load ◽  
Scaling Exponents ◽  
Scale Free ◽  
Traffic Jams ◽  
Fiber Bundle Model ◽  
Scale Free Networks

In this paper, we extend a fiber bundle model to study the propagation of traffic jams on scale-free networks. For the special distributions of traffic handling capacities of the links and traffic load on the nodes, the critical behavior of the jamming transition on scale-free networks is studied analytically. It is found that the links connecting to the nodes with larger degrees are more prone to suffering from traffic jams. This feature is associated with a propagation that follows a precise hierarchical dynamics. Finally, the average failure rate of the networks, which is defined as the fraction of total broken links of the network, is investigated analytically and by simulations in scale-free networks. We mainly find that, when β > γ (β and γ are the scaling exponents of the load distribution and degree distribution, respectively), there is a scaling between the average failure rate of the scale-free networks 1 - G and the network size N, 1 - G ~ N-1, independent of γ.

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