Average weighted trapping time of the node- and edge-weighted fractal networks. (English) Zbl 1510.05270
Summary: In this paper, we study the trapping problem in the node- and edge- weighted fractal networks with the underlying geometries, focusing on a particular case with a perfect trap located at the central node. We derive the exact analytic formulas of the average weighted trapping time (AWTT), the average of node-to-trap mean weighted first-passage time over the whole networks, in terms of the network size \(N_g\), the number of copies \(s\), the node-weight factor \(w\) and the edge-weight factor \(r\). The obtained result displays that in the large network, the AWTT grows as a power-law function of the network size \(N_g\) with the exponent, represented by \(\theta(s,r,w)=\log_s(srw^2)\) when \(srw^2\neq 1\). Especially when \(srw^2=1\), AWTT grows with increasing order \(N_g\) as \(\log N_g\). This also means that the efficiency of the trapping process depend on three main parameters: the number of copies \(s>1\), node-weight factor \(0<w\leq 1\), and edge-weight factor \(0<r\leq 1\). The smaller the value of \(srw^2\) is, the more efficient the trapping process is.
MSC:
| 05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
| 28A80 | Fractals |
| 90B10 | Deterministic network models in operations research |
Keywords:
fractal network; node- and edge- weighted network; average weighted trapping time; weight factorReferences:
| [1] | Albert, R.; Barabási, A. L., Statistical mechanics of complex networks[j], Rev Modern Phys, 74, 1, 47-54 (2002) · Zbl 1205.82086 |
| [2] | Dorogovtsev, S. N.; Mendes, J. F.F., Evolution of networks: from biological nets to the internet and WWW[M] (2003), Oxford University Press: Oxford University Press Oxford · Zbl 1109.68537 |
| [3] | Achard, S.; Salvador, R.; Whitcher, B.; Suckling, J.; Bullmore, E., A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs[j], J Neurosci, 26, 63-75 (2006) |
| [4] | Zhou, C.; Zemanová, L.; Zamora, G.; Hilgetag, C. C.; Kurths, J., Hierarchical organization unveiled by functional connectivity in complex brain networks [j], Phys Rev Lett, 97, 23, 238103-238115 (2006) |
| [5] | Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J., The backbone of the climate network[j], EPL, 87, 48007-48014 (2009) |
| [6] | Donges, J. F.; Schultz, H. C.H.; Marwan, N.; Zou, Y.; Kurths, J., Investigating the topology of interacting networks[j], Eur Phys J B, 84, 635-651 (2011) |
| [7] | Garlaschelli, D.; Loffredo, M. I., Structure and evolution of the world trade network[j], Phys A, 355, 1, 138-144 (2005) |
| [8] | Hidalgo, C. A.; Klinger, B.; Barabási, A. L.; Hausmann, R., The product space conditions the development of nations[j], Science, 317, 482-487 (2007) |
| [9] | Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J., Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes[j], Eur Phys J B, 85, 38-57 (2012) |
| [10] | Zemp, D. C.; Wiedermann, M.; Kurths, J.; Rammig, A.; Donges, J. F., Node-weighted measures for complex networks with directed and weighted edges for studying continental moisture recycling[j], EPL, 107, 58005-58010 (2014) |
| [11] | Wiedermann, M.; Donges, J. F.; Heitzig, J.; Kurths, J., Node-weighted interacting network measures improve the representation of real-world complex systems[j], EPL, 102, 28007-28012 (2013) |
| [12] | Zhang, L. L.; Wang, Y. H.; Wang, Q. Y., Synchronization for time-varying complex dynamical networks with different-dimensional nodes and non-dissipative coupling[j], Commun Nonlinear Sci Numer Simul, 24, 64-74 (2015) · Zbl 1463.93234 |
| [13] | Wu, X. J.; Lu, G. T., Projective lag synchronization of the general complex dynamical networks with distinct nodes[j], Commun Nonlinear Sci Numer Simul, 17, 4417-4429 (2012) · Zbl 1248.93096 |
| [14] | Lü, L.; Li, C. R.; Chen, L. S.; Wei, L. L., Lag projective synchronization of a class of complex network constituted nodes with chaotic behavior[j], Commun Nonlinear Sci Numer Simul, 19, 2843-2849 (2014) · Zbl 1510.93190 |
| [15] | Solís-Perales, G.; Ruiz-Velázquez, E.; Valle-Rodríguez, D., Synchronization in complex networks with distinct chaotic nodes[j], Commun Nonlinear Sci Numer Simul, 14, 2528-2535 (2009) · Zbl 1221.34148 |
| [16] | Barrat, A.; Barthélemy, M.; Pastor-Satorras, R.; Vespignani, A., The architecture of complex weighted networks[j], Proc Natl Acad Sci USA, 101, 3747-3752 (2004) |
| [17] | Cai, G. L.; Yao, Q.; Shao, H. J., Global synchronization of weighted cellular neural network with time-varying coupling delays[j], Commun Nonlinear Sci Numer Simul, 17, 3843-3847 (2012) · Zbl 1250.92013 |
| [18] | Krause, A. E.; Frank, K. A.; Mason, D. M.; Ulanowicz, R. E.; Taylor, W. W., Compartments revealed in food-web structure[j], Nature, 426, 282-285 (2003) |
| [19] | Pastor-Satorras, R.; Vespignani, A., Evolution and structure of the internet-a statistical physics approach[M] (2004), Cambridge University Press: Cambridge University Press Cambridge |
| [20] | Carletti, T.; Righi, S., Weighted fractal networks[j], Phys A, 389, 2134-2142 (2010) |
| [21] | Dai, M. F.; Ye, D. D.; Hou, J.; Li, X. Y., Scaling of average weighted receiving time on double-weighted koch networks[j], Fractals, 23, 02, 1550011-1550017 (2015) |
| [22] | Barnsley, M., Fractals everywhere[M] (1988), Academic Press: Academic Press London · Zbl 0691.58001 |
| [23] | Edgar, G. A., Measure, topology and fractal geometry[M], UTM (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0727.28003 |
| [24] | Singh, S. L.; Mishra, S. N.; Sinkala, W., A new iterative approach to fractal models[j], Commun Nonlinear Sci Numer Simulat, 17, 521-529 (2012) · Zbl 1242.28016 |
| [25] | Singh, S. L.; Mishra, S. N.; Sarika, J., Orbit of an image under iterated system[j], Commun Nonlinear Sci Numer Simulat, 16, 1469-1482 (2011) · Zbl 1221.68287 |
| [26] | Wei, D. J.; Liu, Q.; Zhang, H. X.; Hu, Y.; Deng, Y.; Mahadevan, S. K., Box-covering algorithm for fractal dimension of weighted networks[j], Sci Rep, 03, 3049-3055 (2013) |
| [27] | Huang, Y.; Gu, Y.; Wang, S., How to get the reliable fractal dimension by the box counting[j], Commun Nonlinear Sci Numer Simul, 02, 22-27 (1996) · Zbl 0919.28007 |
| [28] | Song, C.; Galios, L. K.; Havlin, S.; Makse, H. A., How to calculate the fractal dimension of a complex network: the box covering algorithm[j], J Stat Mech: Theory Exp, 03, 03006-03021 (2007) |
| [29] | Christian, M. S.; Tobias, A. K.; Josá, S. A.J.; Hans, J. H., Box-covering algorithm for fractal dimension of complex networks[j], Phys Rev E, 86, 1, 016707-016717 (2012) |
| [30] | Volchenkov, D., Random walks and flights over connected graphs and complex networks[j], Commun Nonlinear Sci Numer Simulat, 16, 21-55 (2011) · Zbl 1221.60069 |
| [31] | Zhang, Z. Z.; Zhou, S. G.; Xie, W. L.; Chen, L. C.; Lin, Y.; Guan, J. H., Standard random walks and trapping on the koch network with scale-free behavior and small-world effect[j], Phys Rev E, 79, 061113-061124 (2009) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
