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Continuous-time quantum walks on the threshold network model

Published online by Cambridge University Press:  08 November 2010

YUSUKE IDE  and
NORIO KONNO
YUSUKE IDE
Affiliation:
Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Kanagawa, Yokohama 221-8686, Japan. Email: ide@kanagawa-u.ac.jp
NORIO KONNO
Affiliation:
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan. Email: konno@ynu.ac.jp
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Abstract

It is well known that many real world networks have a power-law degree distribution (the scale-free property). However, there are no rigorous results for continuous-time quantum walks on such realistic graphs. In this paper, we analyse the space–time behaviour of continuous-time quantum walks and random walks on the threshold network model, which is a reasonable candidate model having the scale-free property. We show that the quantum walker exhibits localisation at the starting point, although the random walker tends to spread uniformly.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 6: Quantum Algorithms , December 2010 , pp. 1079 - 1090
Copyright
Copyright © Cambridge University Press 2010

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References

Agliari, E., Blumen, A. and Mülken, O. (2008) Dynamics of continuous-time quantum walks in restricted geometries. J. Phys. A 41 445301.CrossRefGoogle Scholar
Ahmadi, A., Belk, R., Tamon, C. and Wendler, C. (2003) On mixing in continuous-time quantum walks on some circulant graphs. Quantum Inf. Comput. 3 611–618.Google Scholar
Albert, R. and Barabási, A.-L. (2002) Statistical mechanics of complex networks. Rev. Mod. Phys. 74 47–97.CrossRefGoogle Scholar
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. and Hwang, D.-U. (2006) Complex networks: structure and dynamics. Phys. Rep. 424 175–308.CrossRefGoogle Scholar
Boguñá, M. and Pastor-Satorras, R. (2003) Class of correlated random networks with hidden variables. Phys. Rev. E 68 036112.CrossRefGoogle ScholarPubMed
Bose, A. and Sen, A. (2007) On asymptotic properties of the rank of a special random adjacency matrix. Elect. Comm. in Probab. 12 200–205.Google Scholar
Caldarelli, G., Capocci, A., De Los Rios, P. and Muñoz, M. A. (2002) Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett. 89 258702.CrossRefGoogle ScholarPubMed
Diaconis, P., Holmes, S. and Janson, S. (2009) Threshold graph limits and random threshold graphs. Internet Mathematics 5 (3)267–318.CrossRefGoogle Scholar
Fujihara, A., Ide, Y., Konno, N., Masuda, N., Miwa, H. and Uchida, M. (2009) Limit theorems for the average distance and the degree distribution of the threshold network model. Interdisciplinary Information Sciences 15 (3)361–366.CrossRefGoogle Scholar
Fujihara, A., Uchida, M. and Miwa, H. (2010) Universal power laws in threshold network model: theoretical analysis based on extreme value theory. Physica A 389 1124–1130.CrossRefGoogle Scholar
Hagberg, A., Schult, D. A. and Swart, P. J. (2006) Designing threshold networks with given structural and dynamical properties. Phys. Rev. E 74 056116.CrossRefGoogle ScholarPubMed
Ide, Y., Konno, N. and Masuda, N. (2007) Limit theorems for some statistics of a generalized threshold network model. Theory of Biomathematics and its Applications III. RIMS Kokyuroku 1551 81–86.Google Scholar
Ide, Y., Konno, N. and Masuda, N. (2010a) Statistical properties of a generalized threshold network model. Methodol. Comput. Appl. Probab. 12 (3)361–377.CrossRefGoogle Scholar
Ide, Y., Konno, N. and Obata, N. (2010b) Spectral properties of the threshold network model. Internet Mathematics 6 (2)173–187.CrossRefGoogle Scholar
Inui, N., Kasahara, K., Konishi, Y. and Konno, N. (2005) Evolution of continuous-time quantum random walks on cycles. Fluctuation and Noise Letters 5 L73–L83.CrossRefGoogle Scholar
Jafarizadeh, M. A. and Salimi, S. (2007) Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix. Ann. Phys. 322 1005–1033.CrossRefGoogle Scholar
Konno, N. (2005) Limit theorem for continuous-time quantum walk on the line. Phys. Rev. E 72 026113.CrossRefGoogle ScholarPubMed
Konno, N. (2006a) Continuous-time quantum walks on trees in quantum probability theory. Inf. Dim. Anal. Quantum Probab. Rel. Topics 9 287–297.CrossRefGoogle Scholar
Konno, N. (2006b) Continuous-time quantum walks on ultrametric spaces. Int. J. Quantum Inf. 4 1023–1035.CrossRefGoogle Scholar
Konno, N. (2008) Quantum walks. In: Franz, U. and Schurmann, M. (eds.) Quantum Potential Theory. Springer-Verlag Lecture Notes in Mathematics 1954 309–452.CrossRefGoogle Scholar
Konno, N., Masuda, N., Roy, R. and Sarkar, A. (2005) Rigorous results on the threshold network model. J. Phys. A: Math. Gen. 38 6277–6291.CrossRefGoogle Scholar
Mahadev, N. V. R. and Peled, U. N. (1995) Threshold Graphs and Related Topics, Elsevier.Google Scholar
Masuda, N., Miwa, H. and Konno, N. (2004) Analysis of scale-free networks based on a threshold graph with intrinsic vertex weights. Phys. Rev. E 70 036124.CrossRefGoogle ScholarPubMed
Masuda, N., Miwa, H. and Konno, N. (2005) Geographical threshold graphs with small-world and scale-free properties. Phys. Rev. E 71 036108.CrossRefGoogle ScholarPubMed
Masuda, N. and Konno, N. (2006) VIP-club phenomenon: Emergence of elites and masterminds in social networks. Social Networks 28 297–309.CrossRefGoogle Scholar
Merris, R. (1994) Degree maximal graphs are Laplacian integral. Linear Algebr. Appl. 199 381–389.CrossRefGoogle Scholar
Merris, R. (1998) Laplacian graph eigenvectors. Linear Algebr. Appl. 278 221–236.CrossRefGoogle Scholar
Mülken, O., Bierbaum, V. and Blumen, A. (2006) Coherent exciton transport in dendrimers and continuous-time quantum walks. J. Chem. Phys. 124 124905.CrossRefGoogle ScholarPubMed
Mülken, O. and Blumen, A. (2006) Continuous time quantum walks in phase space. Phys. Rev. A 73 012105.CrossRefGoogle Scholar
Mülken, O., Pernice, V. and Blumen, A. (2007) Quantum transport on small-world networks: A continuous-time quantum walk approach. Phys. Rev. E 76 051125.CrossRefGoogle ScholarPubMed
Newman, M. E. J. (2003) The structure and function of complex networks. SIAM Rev. 45 167–256.CrossRefGoogle Scholar
Salimi, S. (2008a) Study of continuous-time quantum walks on quotient graphs via quantum probability theory. Int. J. Quantum Inf. 6 945–957.CrossRefGoogle Scholar
Salimi, S. (2008b) Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory. Int. J. Theor. Phys. 47 3298–3309.CrossRefGoogle Scholar
Salimi, S. (2009) Continuous-time quantum walks on star graphs. Annals of Physics 324 1185–1193.CrossRefGoogle Scholar
Salimi, S. (2010) Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quantum Inf. Process. 9 75–91.CrossRefGoogle Scholar
Salimi, S. and Jafarizadeh, M. (2009) Continuous-time classical and quantum random walk on direct product of Cayley graphs. Commun. Theor. Phys. 51 1003–1009.CrossRefGoogle Scholar
Servedio, V. D. P., Caldarelli, G. and Buttá, P. (2004) Vertex intrinsic fitness: How to produce arbitrary scale-free networks. Phys. Rev. E 70 056126.CrossRefGoogle ScholarPubMed
Söderberg, B. (2002) General formalism for inhomogeneous random graphs. Phys. Rev. E 66 066121.CrossRefGoogle ScholarPubMed
Venegas-Andraca, S. E. (2008) Quantum Walks for Computer Scientists, Morgan and Claypool.CrossRefGoogle Scholar
Xu, X. P. (2009) Exact analytical results for quantum walks on star graph. J. Phys. A: Math. Theor. 42 115205.CrossRefGoogle Scholar
Xu, X. P. and Liu, F. (2008) Continuous-time quantum walks on Erdős-Rényi networks. Phys. Lett. A 372 6727.CrossRefGoogle Scholar