The trapping problem and the average shortest weighted path of the weighted pseudofractal scale-free networks

2019 ◽  
Vol 30 (01) ◽  
pp. 1950010 ◽  
Author(s):  
Meifeng Dai ◽  
Changxi Dai ◽  
Huiling Wu ◽  
Xianbin Wu ◽  
Wenjing Feng ◽  
...  
Keyword(s):  
Network Size ◽  
Exceptional Case ◽  
Weight Factor ◽  
Rigorous Solution ◽  
Network Nodes ◽  
Trapping Time ◽  
Scale Free ◽  
Wide Range ◽  
Scale Free Networks ◽  
Power Law Function

In this paper, we study the trapping time in the weighted pseudofractal scale-free networks (WPSFNs) and the average shortest weighted path in the modified weighted pseudofractal scale-free networks (MWPSFNs) with the weight factor [Formula: see text]. At first, for exceptional case with the trap fixed at a hub node for weight-dependent walk, we derive the exact analytic formulas of the trapping time through the structure of WPSFNs. The obtained rigorous solution shows that the trapping time approximately grows as a power-law function of the number of network nodes with the exponent represented by [Formula: see text]. Then, we deduce the scaling expression of the average shortest weighted path through the iterative process of the construction of MWPSFNs. The obtained rigorous solution shows that the scalings of average shortest weighted path with network size obey three laws along with the range of the weight factor. We provide a theoretical study of the trapping time for weight-dependent walk and the average shortest weighted path in a wide range of deterministic weighted networks.

RANDOM WALKS WITH A TRAP IN SCALE-FREE FRACTAL HIERARCHICAL LATTICES

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750058 ◽  
Author(s):  
CHANGMING XING ◽  
LIN YANG ◽  
LEI GUO
Keyword(s):  
Random Walks ◽  
Influence Factor ◽  
Substantial Effect ◽  
Network Size ◽  
The Other ◽  
Trapping Time ◽  
Scale Free ◽  
Scale Free Networks ◽  
Power Law Function ◽  
Hierarchical Lattices

In this paper, we study two kinds of random walks with a trap in a class of scale-free fractal hierarchical lattices. One is standard random walks belonging to unbiased random walks, while the other one named mixed random walks is biased. The structural properties of these hierarchical lattices are controlled by a parameter [Formula: see text]. We derive exact solutions of the average trapping time (ATT) for the two trapping issue, respectively. The results show that in large networks, both of the ATT grow asymptotically as a power-law function of network size with the exponent related to the parameter [Formula: see text]. It indicates that network structure has a substantial effect on the efficiency of trapping processes performed in scale-free networks. Comparing the results obtained for the two different random walks, we find that changes of the walking rule have no effect on the leading exponent of the ATT, but could modify the coefficient of the formula for the ATT. The findings are helpful for better understanding the influence factor of random walks in complex systems.

scholarly journals NONUNIVERSALITY IN SEMI-DIRECTED BARABÁSI–ALBERT NETWORKS

2012 ◽  
Vol 23 (09) ◽  
pp. 1250062 ◽  
Author(s):  
M. A. SUMOUR ◽  
M. A. RADWAN
Keyword(s):  
Type A ◽  
Network Nodes ◽  
Scale Free ◽  
Scale Free Networks ◽  
Usual Scale

In usual scale-free networks of Barabási–Albert type, a newly added node selects randomly m neighbors from the already existing network nodes, proportionally to the number of links these had before. Then the number n(k) of nodes with k links each decays as 1/kγ where γ = 3 is universal, i.e. independent of m. Now we use a limited directedness in building the network, as a result of which the exponent γ decreases from 3 to 2 for increasing m.

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).

SCALING OF THE AVERAGE RECEIVING TIME ON A FAMILY OF WEIGHTED HIERARCHICAL NETWORKS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650038 ◽  
Author(s):  
YU SUN ◽  
MEIFENG DAI ◽  
YANQIU SUN ◽  
SHUXIANG SHAO
Keyword(s):  
Bipartite Graph ◽  
Power Law ◽  
Complete Bipartite Graph ◽  
Network Size ◽  
Large Network ◽  
Weight Factor ◽  
Hierarchical Networks ◽  
Current Node ◽  
Edge Linking ◽  
Power Law Function

In this paper, based on the un-weight hierarchical networks, a family of weighted hierarchical networks are introduced, the weight factor is denoted by [Formula: see text]. The weighted hierarchical networks depend on the number of nodes in complete bipartite graph, denoted by [Formula: see text], [Formula: see text] and [Formula: see text]. Assume that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them. We deduce the analytical expression of the average receiving time (ART). The obtained remarkable results display two conditions. In the large network, when [Formula: see text], the ART grows as a power-law function of the network size [Formula: see text] with the exponent, represented by [Formula: see text], [Formula: see text]. This means that the smaller the value of [Formula: see text], the more efficient the process of receiving information. When [Formula: see text], the ART grows with increasing order [Formula: see text] as [Formula: see text] or [Formula: see text].

EFFECT OF CLIENT DEMANDS ON DESTRUCTIVENESS OF TARGETED ATTACKS IN DIRECTED WEIGHTED SCALE-FREE NETWORKS

2011 ◽  
Vol 25 (19) ◽  
pp. 1603-1617 ◽  
Author(s):  
LI-LI MA ◽  
XIN JIANG ◽  
ZHAN-LI ZHANG ◽  
ZHI-MING ZHENG
Keyword(s):  
Real World ◽  
Weight Distribution ◽  
Network Robustness ◽  
Network Nodes ◽  
Scale Free ◽  
Node Importance ◽  
Targeted Attacks ◽  
Link Weight ◽  
Scale Free Networks ◽  
Unique Mechanism

Network resilience is vital for the survival of networks, and scale-free networks are fragile when confronted with targeted attacks. We survey network robustness to targeted attacks from the viewpoint of network clients by designing a unique mechanism based on the undeniable roles of network clients in real-world networks. Especially, the mechanism here is designed on the actual phenomenon that the vital nodes in a network may be totally different for clients with different demands. Concretely, node client-demand centrality is proposed to quantify the contributions of nodes to network clients and we show that it is a proper index to assign an order to network nodes according to node importance for network clients. Great discrepancy of node importance order for clients with different demands is found in scale-free networks with four different kinds of link weight distribution, which suggests that the destructiveness of fatal attacks on networks can be greatly reduced by adjusting the demands of network clients.

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.

Synchronization of Chaotic Circuits with Stochastically-Coupled Network Topology

2021 ◽  
Vol 31 (01) ◽  
pp. 2150015
Author(s):  
Yoko Uwate ◽  
Yoshifumi Nishio ◽  
Thomas Ott
Keyword(s):  
Secure Communication ◽  
Communication Systems ◽  
Small World ◽  
Chaotic Circuit ◽  
Scale Free ◽  
Wide Range ◽  
Scale Free Networks ◽  
Dynamic Topology ◽  
Chaotic Circuits ◽  
Fully Coupled

In recent years, research on synchronization between coupled chaotic circuits has attracted interest in a wide range of fields. This is because the synchronization of coupled chaotic circuits is a multidisciplinary phenomenon that occurs in various applications, such as broadband communication systems or secure communication. In this study, we propose a coupled chaotic circuit network model with stochastic couplings. We investigate the synchronization phenomena observed for the proposed network using different network structures such as fully-coupled, random, small world and scale-free networks. We find that the same synchronization characteristics can be obtained for these networks with a dynamic topology as when the coupling strength is changed in static networks.

EXACT SOLUTIONS FOR AVERAGE TRAPPING TIME OF RANDOM WALKS ON WEIGHTED SCALE-FREE NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750013 ◽  
Author(s):  
CHANGMING XING ◽  
YIGONG ZHANG ◽  
JUN MA ◽  
LIN YANG ◽  
LEI GUO
Keyword(s):  
Random Walks ◽  
Exact Solutions ◽  
Weight Distribution ◽  
Close Relation ◽  
The Other ◽  
Trapping Time ◽  
Scale Free ◽  
Weight Parameter ◽  
Scale Free Networks ◽  
Fractal Network

In this paper, we present two deterministic weighted scale-free networks controlled by a weight parameter [Formula: see text]. One is fractal network, the other one is non-fractal network, while they have the same weight distribution when the parameter [Formula: see text] is identical. Based on their special network structure, we study random walks on network with a trap located at a fixed node. For each network, we calculate exact solutions for average trapping time (ATT). Analyzing and comparing the obtained solutions, we find that their ATT all grow asymptotically as a power-law function of network order (number of nodes) with the exponent [Formula: see text] dependent on the weight parameter, but their exponent [Formula: see text] are obviously different, one is an increasing function of [Formula: see text], while the other is opposite. Collectively, all the obtained results show that the efficiency of trapping on weighted Scale-free networks has close relation to the weight distribution, but there is no stable positive or negative correlation between the weight distribution and the trapping time on different networks. We hope these results given in this paper could help us get deeper understanding about the weight distribution on the property and dynamics of scale-free 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.

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