网络维数:一种度量复杂网络的新方法
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摘要
如何对复杂网络进行刻画与度量,一直是人们关注的热点。在研究自相似复杂网络分形维数的基础上,提出了一种度量复杂网络的新方法——网络维数,即复杂网络边权重和的对数值与节点权重和的对数值的比值,可以将边权重及点权重推广到实数域和复数域;同时给出了不同类型权重对应的网络维数的计算方法;最后以几个代表性的经典复杂网络模型为例,讨论了所提出的网络维数的若干性质。
How to measure complex networks has always received much attention.This paper proposed a new method based on the analysis of fractal dimension of self-similarity complex networks,named network dimension,to measure complex networks.Network dimension is expressed as the division of logarithm of the sum of edges' weights and logarithm of the sum of nodes' weights of complex networks.The weights of both edge and node are extended to real and complex number fields.The calculation methods of network dimensions of weighted networks with different types of weights were presented.Finally,several representative classical complex network models were taken as examples to discuss some properties of the proposed network dimension.
How to measure complex networks has always received much attention.This paper proposed a new method based on the analysis of fractal dimension of self-similarity complex networks,named network dimension,to measure complex networks.Network dimension is expressed as the division of logarithm of the sum of edges' weights and logarithm of the sum of nodes' weights of complex networks.The weights of both edge and node are extended to real and complex number fields.The calculation methods of network dimensions of weighted networks with different types of weights were presented.Finally,several representative classical complex network models were taken as examples to discuss some properties of the proposed network dimension.
引文
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[2]WATTS D J,STROGATZ S H.Collective dynamics of‘smallworld’networks[J].Nature,1998,393(6684):440-442.
[3]NEWMAN M E J,WATTS D J.Renormalization group analysis of the small-world network model[J].Physics Letter A,1999,263(4-6):341-346.
[4]BARABASI A L,ALBERT R.Emergence of scaling in random networks[J].Science,1999,286(5439):509-512.
[5]LIU S J,LI T R,HONG X J,et al.Complex network construction based on matrix operation[J].Scientia Sinica Informationis,2016,46(5):610-626.(in Chinese)刘胜久,李天瑞,洪西进,等.基于矩阵运算的复杂网络构建方法研究[J].中国科学:信息科学,2016,46(5):610-626.
[6]LIU S J,LI T R,HONG X J,et al.Hypernetwork model and its properties[J].Journal of Frontiers of Computer Science and Technology,2017,11(2):194-211.(in Chinese)刘胜久,李天瑞,洪西进,等.超网络模型构建及特性分析[J].计算机科学与探索,2017,11(2):194-211.
[7]ZHU D Z,WU J,TAN Y J,et al.Degree-Rank function:A new statistic characteristic of complex network[J].Complex System and Complex Science,2006,3(4):28-34.(in Chinese)朱大智,吴俊,谭跃进,等.度秩函数---一个新的复杂网络统计特征[J].复杂系统与复杂性科学,2006,3(4):28-34.
[8]XU Z B,WANG J Y,ZHANG D S,et al.Fractal Dimension Description of Complexity of fault network in coal mines[J].Journal of China Coal Socity,1996,21(4):358-363.(in Chinese)徐志斌,王继尧,张大顺,等.煤矿断层网络复杂程度的分维描述[J].煤炭学报,1996,21(4):358-363.
[9]ZHENG X,CHEN J P,SHAO J L,et al.Analysis on topological properties of Beijing urban public transit based on complex network theory[J].Acta Physica Sinica,2012,61(19):95-105.(in Chinese)郑啸,陈建平,邵佳丽,等.基于复杂网络理论的北京公交网络拓扑性质分析[J].物理学报,2012,61(19):95-105.
[10]张先迪,李正良.图论及其应用[M].北京:高等教育出版社,2005.
[11]NEWMAN M E J.The structure and function of complex networks[J].SIAM Review,2003,45(2):167-256.
[12]MANDELBROT B.How long is the coast of britain?Statistical self-similarity and fractional dimension[J].Science,1967,156(3775):636-638.
[13]BALKA R,BUCZOLICH Z,ELEKES M.A new fractal dimension:The topological Hausdorff dimension[J].Advances in Mathematics,2015,274(1):881-927.
[14]SREENIVASAN K R,MENEVEAU C.The fractal facets of turbulence[J].Journal of Fluid Mechanics,1986,173(173):357-386.
[15]HARTE D.Multifractals:Theory and Applications[M].Chapman&Hall/CRC,2001.
[16]SONG C M,HAVLIN S,MAKSE H A.Self-similarity of complex networks[J].Nature,2005,433(7024):392-395.
[17]SONG C M,GALLOS L K,HAVLIN S,et al.How to calculate the fractal dimension of a complex network:the box covering algorithm[J].Journal of Statistical Mechanics Theory&Experiment,2007,2007(3):297-316.
[18]KIM J S,GOH K I,SALVI G,et al.Fractality in complex networks:critical and supercritical skeletons[J].Physical Review E,2007,75(2):016110.
[19]ZHOU W X,JIANG Z Q,SORNETTE D.Exploring self-similarity of complex cellular networks:The edge-covering method with simulated annealing and log-periodic sampling[J].Physica a Statistical Mechanics&Its Applications,2006,375(2):741-752.
[20]GAO L,HU Y,DI Z.Accuracy of the ball-covering approach for fractal dimensions of complex networks and a rank-driven algorithm[J].Physical Review E Statistical Nonlinear&Soft Matter Physics,2008,78(4Pt 2):046109.
[21]LIU J L,YU Z G,ANH V.Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions[J].Physical Review E Statistical Nonlinear&Soft Matter Physics,2014,89(3):032814.
[22]WEI D J,LIU Q,ZHANG H X,et al.Box-covering algorithm for fractal dimension of weighted networks[J].Scientific Report,2013,3(6157):3049.
[23]LIU S J,LI T R,HONG X J,et al.Supernetwork based on matrix operation and its properties[J].CAAI Transactions on Intelligent Systems,2018,13(3):359-365.(in Chinese)刘胜久,李天瑞,洪西进,等.基于矩阵运算的超网络构建方法研究及特性分析[J].智能系统学报,2018,13(3):359-365.