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复杂网络的研究已经持续了很多年,在网络研究的最初阶段,数据的获得相对困难,对网络的研究多数是抽象为静态网络来进行的。基于静态网络,研究者也开始考虑另一个维度——时间。时间是物质运动、变化的持续性、顺序性的表现,具有不可逆性。近些年随着数据获取越来越便利,获取带有时间属性标签的网络数据也变得容易,这使得人们对复杂网络的研究从拓扑结构固定的静态网络向带有时间标签的网络过渡,对复杂网络“实体”的关注也逐渐转移到“关系的建立”及“事物的发展”等时间不可逆的过程上。不同于静态拓扑结构的网络,加入时间维度的网络中的连边随着时间会间断性地出现和消失,这样的网络被称为时效网络(temporal networks)[1-6]。
2012年,文献[1]强调,现实世界里各种被复杂网络表征的物理、技术、社会和经济系统都是随时间动态变化的。由于之前的静态网络研究忽略了网络的时间属性,因此在研究时高估了节点间的有效连接,却低估了网络的最短路径。同时很多网络事件的发生具有非连续性、多次性等特点,静态网络不能很好地刻画网络事件的这些特点,造成研究结果的真实性存在偏差,进而影响传播预测、社团划分的准确性。引入时间维度后,最直接的变化在于网络连接拓扑结构决定的节点之间的相互作用被改变,从而导致以传播动力学为代表的复杂网络动力学过程的基础需要被重新审视。具体说来,由于时效网络引入了时间标签,网络中节点之间的(有效)连边与不考虑时效属性的静态网络相比,增加了不同连边的先后排序、连边的持续时间、个体的接触频率等新特征,导致信息在节点间的传递由静态拓扑决定延伸到由时效-拓扑共同决定的层面,因此,时效网络能够反映出静态网络所不具备的性质,如因果性、阵发性等[7]。
时效网络数据获得的渠道很多。在过去的几十年中,人们花费大量时间,利用现代信息技术,诸如因特网、万维网和移动通讯网进行交流沟通、工作和娱乐。这个虚拟世界中,存在着大量的人类(实时)交互行为的电子记录,包括电子邮件、手机通话和短消息等,为时效复杂网络的研究提供了丰富的数据资源[8-10]。此外,通过蓝牙、无线射频、无线感应和Wi-Fi等信息技术获得的人类线下交互时效网络也为成功分析人类线下交互行为规律提供了丰富的数据[11-14]。随着高分辨率数据的出现,从各种复杂系统中获得的交互行为的时间戳或时间序列,为研究时效网络的动态演化过程对网络性质的影响提供了条件,为分析人类交互行为特征和提出人类行为动力学的恰当模型提供了巨大的机遇,例如,已有研究成功表征了人类交互行为中的非泊松动力学特征[15]。因此,研究时效网络自身变化的规律及对传播动力学过程的作用机制是亟待重视的科学问题。深入分析时效网络的时效拓扑特性与传播动力学的关系成为理解掌握现实世界中各种各样传播过程的理论基础,是设计合理有效干预控制手段的必要前提,这对整个社会安全、有效的运转有着重要的现实意义。
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[1] | HOLME P, SARAMAKI J. Temporal networks[J]. Physics Reports, 2012, 519(3): 97-125. doi: 10.1016/j.physrep.2012.03.001 | |
[2] | VESPIGNANI A. Modelling dynamical processes in complex socio-technical systems[J]. Nature Physics, 2012, 8(): 32-39. | |
[3] | PERUANI F, TABOURIER L. Directedness of information flow in mobile phone communication networks[J]. PLoS ONE, 2011, 6(12): e28860-. doi: 10.1371/journal.pone.0028860 | |
[4] | BARRAT A, CATTUTO C, COLIZZA V. Empirical temporal networks of face-to-face human interactions[J]. The European Physical Journal Special Topics, 2013, 222(6): 1295-1309. doi: 10.1140/epjst/e2013-01927-7 | |
[5] | KOREN Y. Collaborative filtering with temporal dynamics[J]. Communications of the ACM, 2010, 53(4): 89-97. doi: 10.1145/1721654 | |
[6] | BLONDER B, WEY T W, DOMHAUS A. Temporal dynamics and network analysis[J]. Methods in Ecology and Evolution, 2012, 3(6): 958-972. doi: 10.1111/j.2041-210X.2012.00236.x | |
[7] | SCHOLTES I, WIDER N, PFITZNER R. Slow-down vs. speed-up of diffusion in non-Markovian temporal networks[J]. Nature Communications, 2014, 5(): 5024-. doi: 10.1038/ncomms6024 | |
[8] | ZHANG Y Q, LI X, LIANG D, et al. Characterizing bursts of aggregate pairs with individual poissonian activity and preferential mobility[J]. IEEE Communications Letters, 19(7):1225-1228. | |
[9] | BARABASI A L. The origin of bursts and heavy tails in human dynamics[J]. Nature, 2005, 435(): 207-211. doi: 10.1038/nature03459 | |
[10] | WU Y, ZHOU C, XIAO J. Evidence for a bimodal distribution in human communication[J]. Proceedings of the National Academy of Sciences of the United States of America, 2010, 107(44): 18803-18808. doi: 10.1073/pnas.1013140107 | |
[11] | EAGLE N, PENTLAND A, Reality mining:Sensing complex social systems[J]. Personal and Ubiquitous Computing, 2006, 10(4):255-268. | |
[12] | CATTUTO C, BROECK W V D, BARRAT A. Dynamics of person-to-person interactions from distributed RFID sensor networks[J]. PLoS ONE, 2010, 5(7): e11596-. doi: 10.1371/journal.pone.0011596 | |
[13] | STEHLE J, VOIRIN N, BARRAT A. High-resolution measurements of face-to-face contact patterns in a primary school[J]. PLoS ONE, 2011, 6(8): e23176-. doi: 10.1371/journal.pone.0023176 | |
[14] | CHAINTREAU A, HUI P, CROWCROFT J. Impact of human mobility on opportunistic forwarding algorithms[J]. IEEE Transactions on Mobile Computing, 2007, 6(6): 606-620. doi: 10.1109/TMC.2007.1060 | |
[15] | STARNINI M, BARONCHELLI A, PASTOR-SATORRAS R. Modeling human dynamics of face-to-face interaction networks[J]. Physical Review Letters, 2013, 110(16): 168701-. doi: 10.1103/PhysRevLett.110.168701 | |
[16] | TAKAGUCHI T, SATO N, YANO K. Importance of individual events in temporal networks[J]. New Journal of Physics, 2012, 14(): 093003-. doi: 10.1088/1367-2630/14/9/093003 | |
[17] | TRAJANOVSKI S, SCELLATO S, LEONTIADIS I. Error and attack vulnerability of temporal networks[J]. Physical Review E, 2012, 85(6): 066105-. doi: 10.1103/PhysRevE.85.066105 | |
[18] | PFITZNER R, SCHOLTES I, GARAS A. Betweenness preference:Quantifying correlations in the topological dynamics of temporal networks[J]. Physical Review Letters, 2013, 110(19): 198701-. doi: 10.1103/PhysRevLett.110.198701 | |
[19] | ROCHA L E C, MASUDA N. Random walk centrality for temporal networks[J]. New Journal of Physics, 2014, 16(): 063023-. doi: 10.1088/1367-2630/16/6/063023 | |
[20] | BASSETT D S, PORTER M A, WYMBS N F. Robust detection of dynamic community structure in networks[J]. CHAOS, 2013, 23(): 013142-. doi: 10.1063/1.4790830 | |
[21] | GAUVIN L, PANISSON A, CATTUTO C. Detecting the community structure and activity patterns of temporal networks:a non-negative tensor factorization approach[J]. PLoS ONE, 2014, 9(1): e86028-. doi: 10.1371/journal.pone.0086028 | |
[22] | FU C, LI M, ZOU D. Community vitality in dynamic temporal networks[J]. International Journal of Distributed Sensor Networks, 2013, (): 281565-. | |
[23] | ZHANG Y Q, LI X. Characterizing large-scale population's indoor spatio-temporal interactive behaviors[C]//Proceedings of the ACM SIGKDD International Workshop on Urban Computing.[S.l.]:ACM, 2012:25-32. | |
[24] | ZHANG Y, WANG L, ZHANG Y Q. Towards a temporal network analysis of interactive WiFi users[J]. Europhysics Letters, 2012, 98(6): 68002-. doi: 10.1209/0295-5075/98/68002 | |
[25] | ZHANG Y Q, LI X. Temporal dynamics and impact of event interactions in cyber-social populations[J]. CHAOS, 2013, 23(): 013131-. doi: 10.1063/1.4793540 | |
[26] | MUCHA P J, RICHARDSON T, MACON K. Community structure in time-dependent, multiscale, and multiplex networks[J]. Science, 2010, 328(5980): 876-878. doi: 10.1126/science.1184819 | |
[27] | ROCHA L E C, BLONDEL V D. Flow motifs reveal limitations of the static framework to represent human interactions[J]. Physical Review E, 2013, 87(4): 042814-. doi: 10.1103/PhysRevE.87.042814 | |
[28] | KOVANEN L, KASKI K, KERTESZ J. Temporal motifs reveal homophily, gender-specific patterns, and group talk in call sequences[J]. Proceedings of the National Academy of Sciences of the United States of America, 2013, 110(45): 18070-18075. doi: 10.1073/pnas.1307941110 | |
[29] | LIU K, CHEUNG W K, LIU J. Detecting stochastic temporal network motifs for human communication patterns analysis[C]//Proceedings of 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM).[S.l.]:IEEE, 2013:533. | |
[30] | ZHANG Y, LI X, XU J. Human interactive patterns in temporal networks[J]. IEEE Transactions on Systems, Man, Cybernetics:Systems, 2015, 45(2): 214-222. doi: 10.1109/TSMC.2014.2360505 | |
[31] | CUI J, ZHANG Y Q, LI X. On the clustering coefficients of temporal networks and epidemic dynamics[C]//Proceedings of 2013 IEEE International Symposium on Circuits and Systems (ISCAS).[S.l.]:IEEE, 2013:2299. | |
[32] | LI X, ZHANG Y Q, VASILAKOS A V. Discovering and predicting temporal patterns of WiFi-interactive social populations, in opportunistic mobile social networks[M].[S.l.]:CRC, 2014. | |
[33] | CASTEIGTS A, FLOCCHINI P, QUATTROCIOCCHI W. Time-varying graphs and dynamic networks[J]. Ad-hoc, Mobile, and Wireless Networks, 2011, 6811(): 346-359. doi: 10.1007/978-3-642-22450-8 | |
[34] | ROSVALL M, BERGSTROM C T. Mapping change in large networks[J]. PLoS ONE, 2010, 5(1): e8694-. doi: 10.1371/journal.pone.0008694 | |
[35] | KIM H, ANDERSON R. Temporal node centrality in complex networks[J]. Physical Review E, 2012, 85(2): 26107-. doi: 10.1103/PhysRevE.85.026107 | |
[36] | TANG J, LEONTIADIS L, SCELLATO S, et al. Temporal network metrics and their application to real world networks[EB/OL]. (2013-05-09). http//:arxiv.org/abs/1305.6974. | |
[37] | TANG J, MUSOLESI M, MASCOLO C, et al. Temporal distance metrics for social network analysis[C]//Proceedings of the 2nd ACM Workshop on Online Social Networks.[S.l.]:ACM, 2009:31-36. | |
[38] | TANG J, MUSOLESI M, MASCOLO C, et al. Analyzing information flows and key mediators through temporal centrality metrics[C]//Proceedings of the 3rd ACM Workshop on Social Network Systems.[S.l.]:ACM, 2010:1-6. | |
[39] | VALDANO E, FERRERI L, POLETTO C. Analytical computation of the epidemic threshold on temporal networks[J]. Physical Review X, 2015, 5(2): 021005-. doi: 10.1103/PhysRevX.5.021005 | |
[40] | PAN R K, SARAMAKI J. Path lengths, correlations, and centrality in temporal networks[J]. Physical Review E, 2011, 84(1): 16105-. doi: 10.1103/PhysRevE.84.016105 | |
[41] | STARNINI M, MACHENS A. CATTUTO C. Immunization strategies for epidemic processes in time-varying contact networks[J]. Journal of Theoretical Biology, 2013, 337(): 89-100. doi: 10.1016/j.jtbi.2013.07.004 | |
[42] | NGUYEN N P, DINH T N, XUAN Y, et al. Adaptive algorithms for detecting community structure in dynamic social networks[C]//INFOCOM, Proceeding IEEE.[S.l.]:IEEE, 2011:2282-2290. | |
[43] | GRINDROD P, PARSONS M C, HIGHAM D J. Communicability across evolving networks[J]. Physical Review E, 2011, 83(4): 046120-. doi: 10.1103/PhysRevE.83.046120 | |
[44] | SCHOLTERS I, WIDER N, GARAS A. High-order aggregate networks in the analysis of temporal networks:Path structures and centralities[J]. The European Physical Journal B, 2016, 89(): 61-. doi: 10.1140/epjb/e2016-60663-0 | |
[45] | HOLME P, SARAMAKI J. Temporal networks[M].[S.l.]:Springer, 2013. | |
[46] | POTHEN A, SIMON H D, LIOU K P. Partitioning sparse matrices with eigenvectors of graphs[J]. SIAM Journal on Matrix Analysis and Applications, 1990, 11(3): 430-452. doi: 10.1137/0611030 | |
[47] | 卢鹏丽. 图谱理论与复杂网络相关算法[M]. 北京:国防工业出版社, 2013. | LU Peng-li. Spectral graph theory and some related algorithms in complex network[M]. Beijing:National Defense Industry Press, 2013. |
[48] | NEWMAN M E J. Modularity and community structure in networks[J]. Proceedings of the National Academy of Sciences of the United States of America, 2006, 103(23): 8577-8582. doi: 10.1073/pnas.0601602103 | |
[49] | RICHARDSON T, MUCHA P J, PORTER M A. Spectral tripartitioning of networks[J]. Physical Review E, 2009, 80(3): 036111-. doi: 10.1103/PhysRevE.80.036111 | |
[50] | ZHANG Y, LI X. When susceptible-infectious-susceptible contagion meets time-varying networks with identical infectivity[J]. Europhysics Letters, 2014, 108(2): 28006-. doi: 10.1209/0295-5075/108/28006 | |
[51] | LANCICHINETTI A, FORTUNATO S. Consensus clustering in complex networks[J]. Scientific Reports, 2012, 2(): 336-. | |
[52] | GAUVIN L, PANISSON A, CATTUTO C. Detecting the community structure and activity patterns oftemporal networks:a non-negative tensor factorization approach[J]. PLoS ONE, 2014, 9(): e86028-. doi: 10.1371/journal.pone.0086028 | |
[53] | CHEN D, LU L, SHANG M S. Identifying influential nodes in complex networks[J]. Physica A, 2012, 391(4): 1777-1787. doi: 10.1016/j.physa.2011.09.017 | |
[54] | LI C, WANG H, MIEGHEM P V. Bounds for the spectral radius of a graph when nodes are removed[J]. Linear Algebra and Its Applications, 2012, 437(): 319-323. doi: 10.1016/j.laa.2012.02.023 | |
[55] | LI C, WANG H, MIEGHEM P V. Degree and principal eigenvectors in complex networks[J]. Networking, 2012, 7289(): 149-160. | |
[56] | LI C, LI Q, MIEGHEM P V. Correlation between centrality metrics and their application to the opinion model[J]. European Physical Journal B, 2015, 88(3): 65-. doi: 10.1140/epjb/e2015-50671-y | |
[57] | GALLAWAY D S, NEWMAN M E J, STROGATZ S H. Network robustness and fragility:Percolation on random graphs[J]. Physical Review Letters, 2000, 85(25): 5468-. doi: 10.1103/PhysRevLett.85.5468 | |
[58] | 赫南, 李德毅, 淦文燕. 复杂网络中重要性节点发掘综述[J]. 计算机科学, 2007, 34(12): 1-5. | HE Nan, LI De-yi, GAN wen-yan. Mining vital nodes in complex networks[J]. Computer Science, 2007, 34(12): 1-5. |
[59] | 孙睿, 罗万伯. 网络舆论中节点重要性评估方法综述[J]. 计算机应用研究, 2012, 29(10): 3606-3608. | SUN Rui, LUO Wang-bo. Review on evaluation of node importance in public opinion[J]. Application Research of Computers, 2012, 29(10): 3606-3608. |
[60] | 刘建国, 任卓明, 郭强. 复杂网络中节点重要性排序的研究进展[J]. 物理学报, 2013, 62(17): 179801-. | LIU Jian-guo, REN zhuo-ming, GUO Qiang. Review on evaluation of node importance in complex networks[J]. Acta Physica Sinica, 2013, 62(17): 179801-. |
[61] | 任晓龙, 吕琳媛. 网络重要节点排序方法综述[J]. 科学通报, 2014, 59(13): 1175-1197. doi: 10.1360/972013-1280 | REN Xiao-long, LÜ Lin-yuan. Review on vital nodes mining in complex networks[J]. Chinese Science Bulletin, 2014, 59(13): 1175-1197. doi: 10.1360/972013-1280 |
[62] | TAYLOR D, MYERS S A, CLAUSET A, et al. Eigenvector-based centrality measures for temporal networks[EB/OL]. (2016-02-21). http://arxiv.org/abs/1507.01266. | |
[63] | PAN Y J, LI X. Structural controllability and controlling centrality of temporal networks[J]. PLoS ONE, 2014, 9(4): e94998-. doi: 10.1371/journal.pone.0094998 | |
[64] | NARAYANAM R, NARAHARI Y. A shapley value-based approach to discover influential nodes in social networks[J]. IEEE Transactions on Automation Science and Engineering, 2010, 8(1): 130-147. | |
[65] | 邓冬梅, 朱建, 陈端兵. 时效阵发性对信息传播的影响[J]. 计算机科学, 2013, 40(s2): 26-28. | DENG Dong-mei, ZHU Jian, CHEN Duan-bing. Influence of bursty on information diffusion[J]. Computer Science, 2013, 40(s2): 26-28. |
[66] | ANDERSON R M, MAY R M. Infectious diseases of humans:Dynamics and control[M]. Oxford:Oxford University Press, 1991. | |
[67] | CASTELLANO C, PASTOR-SATORRAS R. Thresholds for epidemic spreading in networks[J]. Physical Review Letters, 2010, 105(21): 218701-. doi: 10.1103/PhysRevLett.105.218701 | |
[68] | MIEGHEM P V, OMIC J S, KOOIJ R E. Virus spread in networks[J]. IEEE/ACM Transaction on Networking, 2009, 17(1): 1-14. doi: 10.1109/TNET.2008.925623 | |
[69] | LI C, BOVENKAMP R V D, MIEGHEM P V. Susceptible-infected-susceptible model:a comparison of N-intertwined and heterogeneous mean-field approximations[J]. Physical Review E, 2012, 86(2): 026116-. doi: 10.1103/PhysRevE.86.026116 | |
[70] | PARK S M, KIM B J. Dynamic behaviors in directed networks[J]. Physical Review E, 2006, 74(2): 026114-. doi: 10.1103/PhysRevE.74.026114 | |
[71] | LI C, WANG H. MIEGHEM P V. Epidemic threshold in directed networks[J]. Physical Review E, 2013, 88(6): 062802-. doi: 10.1103/PhysRevE.88.062802 | |
[72] | ZHANG Z Z, LIN Y, GUO X Y. Eigenvalues for transition matrix of a small-world scale-free network:Explicit expressions and application[J]. Physical Review E, 2015, 91(6): 062808-. doi: 10.1103/PhysRevE.91.062808 | |
[73] | FU X C, SMALL M, WALKER D M. Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization[J]. Physical Review E, 2008, 77(3): 036113-. doi: 10.1103/PhysRevE.77.036113 | |
[74] | WANG W, TANG M, ZHANG H F. Epidemic spreading on complex networks with general degree and weight distributions[J]. Physical Review E, 2014, 90(4): 042803-. doi: 10.1103/PhysRevE.90.042803 | |
[75] | MIEGHEM P V, WANG H, GE X. Influence of assortativity and degree-preserving rewiring on the spectra of networks[J]. European Physical Journal B, 2010, 76(4): 643-652. doi: 10.1140/epjb/e2010-00219-x | |
[76] | MIEGHEM P V, STEVANOVIC D, KUIPERS F A. Decreasing the spectral radius of a graph by link removals[J]. Physical Review E, 2011, 84(1): 016101-. doi: 10.1103/PhysRevE.84.016101 | |
[77] | WU X, LIU Z H. How community structure influences epidemic spread in social networks[J]. Physica A, 2008, 387(2): 623-630. | |
[78] | WANG H, LI Q, AGOSTINO G D. Effect of the interconnected network structure on the epidemic threshold[J]. Physical Review E, 2013, 88(2): 022801-. doi: 10.1103/PhysRevE.88.022801 | |
[79] | 周涛, 韩筱璞, 闫小勇. 人类行为时空特性的统计力学[J]. 电子科技大学学报, 2013, 42(4): 481-540. | ZHOU Tao, HAN Xiao-pu, YAN Xiao-yong. Statistical mechanics on temporal and spatial activities of human[J]. Journal of University of Electronic Science and Technology of China, 2013, 42(4): 481-540. |
[80] | MASUDA N, HOLME P. Predicting and controlling infectious disease epidemics using temporal networks[J]. F1000 Prime Reports, 2013, 5(): 6-. | |
[81] | RUAN Z Y, WANG C Q, HUI P M. Integrated Travel network model for studying epidemics:Interplay between journeys and epidemic[J]. Scientific Report, 2015, 5(): 11401-. doi: 10.1038/srep11401 | |
[82] | GROSS T, LIMA C J D, BLASIUS B. Epidemic dynamics on an adaptive network[J]. Physical Review Letters, 2006, 96(20): 208701-. doi: 10.1103/PhysRevLett.96.208701 | |
[83] | GUO D, TRAJANOVSKI S, BOVENKAMP R V D. Epidemic threshold and topological structure of susceptible-infectious-susceptible epidemics in adaptive networks[J]. Physical Review E, 2013, 88(4): 042802-. doi: 10.1103/PhysRevE.88.042802 | |
[84] | YANG H, TANG M, ZHANG H F. Efficient community-based control strategies in adaptive networks[J]. New Journal of Physics, 2012, 14(12): 123017-. doi: 10.1088/1367-2630/14/12/123017 | |
[85] | GHOSHAL G, CHI L P, BARABASI A L. Uncovering the role of elementary processes in network evolution[J]. Scientific Reports, 2013, 3(): 2920-. | |
[86] | LEE S, ROCHA L E C, LILJEROS F. Exploiting temporal network structures of human interaction to effectively immunize populations[J]. PLoS ONE, 2012, 7(): e36439-. doi: 10.1371/journal.pone.0036439 | |
[87] | HABIBA, YU Y, BERGER-WOLF T Y, et al. Finding spreading blockers in dynamic networks[C]//Advances in Social Network Mining and Analysis. Berlin, Heidelberg:Springer, 2010, 5498:55-76. | |
[88] | ZHOU Y Z, XIA Y J. Epidemic spreading on weighted adaptive networks[J]. Physica A, 2014, 399(): 16-23. doi: 10.1016/j.physa.2013.12.036 | |
[89] | ZHOU J, XIAO G X, CHEN G R. Link-based formalism for time evolution of adaptive networks[J]. Physical Review E, 2013, 88(3): 032808-. doi: 10.1103/PhysRevE.88.032808 | |
[90] | ZHOU J, YAN G, LAI C H. Efficient routing on multilayered communication networks[J]. Europhysics Letters, 102(2):28002. | |
[91] | ZHOU Y Z, LIU Z H, ZHOU J. Periodic wave of epidemic spreading in community networks[J]. Chinese Physics Letters, 2007, 24(2): 581-. doi: 10.1088/0256-307X/24/2/078 | |
[92] | ZHOU J, CHUNG N N, CHEW L Y. Epidemic spreading induced by diversity of agents' mobility[J]. Physical Review E, 2012, 86(2): 026115-. doi: 10.1103/PhysRevE.86.026115 | |
[93] | ZHOU J, ZHOU Y Z, LIU Z H. Amplification of signal response at an arbitrary node of a complex network[J]. Physical Review E, 2011, 83(4): 046107-. doi: 10.1103/PhysRevE.83.046107 | |
[94] | ZHOU Y Z, WANG X F. Epidemic spreading on dualistic social networks[C]//Proceedings of the 34th Chinese Control Conference. Hangzhou, China:[s.n.], 2015:1252-1255. | |
[95] | ZHOU Y Z, XU X. Analysis of telecom fraud users behavior based on human dynamics[C]//Proceedings of the 34th Chinese Control Conference. Hangzhou, China:[s.n.], 2015:1351-1355. | |
[96] | ZHOU Y Z, ZHOU J. Epidemic spreading in partial dynamic networks[C]//Proceedings of the 31st Chinese Control Conference. Hefei, China:[s.n.], 2012:1190-1194. | |
[97] | ZHOU Y Z, ZHOU J, WANG X F. Epidemic spreading on complex networks with weighted adaptive strategy[C]//Proceedings of the World Congress on Intelligent Control and Automation. Beijing, China:[s.n.], 2012:3491-3496. | |
[98] | ZHOU J, XIAO G X, CHEONG S A. Epidemic reemergence in adaptive complex networks[J]. Physical Review E, 2012, 85(3): 036107-. doi: 10.1103/PhysRevE.85.036107 | |
[99] | ZHOU J, LIU Z H. Epidemic spreading in communities with mobile agents[J]. Physica A, 2009, 388(): 1228-. doi: 10.1016/j.physa.2008.12.014 | |
[100] | ZHOU J, LIU Z H. Epidemic spreading in complex networks[J]. Frontiers of Physics in China, 2008, 3(): 331-. doi: 10.1007/s11467-008-0027-x | |
[101] | ZHOU J, LIU Z H, LI B W. Influence of network structure on rumor propagation[J]. Physics Letters A, 2007, 368(6): 458-463. doi: 10.1016/j.physleta.2007.01.094 | |
[102] | CELLAI D, LOPEZ E, ZHOU J. Percolation in multiplex networks with overlap[J]. Physical Review E, 2013, 88(5): 052811-. doi: 10.1103/PhysRevE.88.052811 | |
[103] | CHUNG N N, CHEW L Y, ZHOU J. Impact of edge-removal on the centrality betweenness of the best spreaders[J]. Europhysics Letters, 2012, 98(5): 58004-. doi: 10.1209/0295-5075/98/58004 | |
[104] | PARKASH B A, TONG H, VALLER N, et al. Virus propagation on time-varying networks:Theory and immunization algorithms[C]//Machine Learning and Knowledge Discovery in Databases. Berlin, Germany:Springer, 2010:99-114. | |
[105] | SANATKAR M R, WHITE W N, NATARAJAN B. Epidemic threshold of an SIS model in dynamic switching networks[J]. IEEE Transactions on Systems, Man, and Cybernetics:Systems, 2016, 46(3): 345-355. doi: 10.1109/TSMC.2015.2448061 | |
[106] | VALLER N C, PARKASH B A, TONG H, et al. Epidemic spread in mobile Ad Hoc networks:Determining the tipping point[C]//Proceedings of 10th International IFIP TC 6 Networking Conference. Valencia, Spain:Springer, 2011:266-280. | |
[107] | ISHIKAWA S, TATEYA I, HAYASAKA T. Epidemics scenarios in the "romantic network"[J]. PLoS ONE, 2012, 7(11): e49009-. doi: 10.1371/journal.pone.0049009 | |
[108] | PERRA N, GONÇALVES B, PASTOR-SATORRAS R. Activity driven modeling of time varying networks[J]. Scientific Reports, 2012, 2(): 469-. | |
[109] | STARNINI M, PASTOR-SATORRAS R. Topological properties of a time-integrated activity driven network[J]. Physical Review E, 2013, 87(6): 062807-. doi: 10.1103/PhysRevE.87.062807 | |
[110] | STARNINI M, PASTOR-SATORRAS R. Temporal percolation in activity driven networks[J]. Physical Review E, 2014, 89(3): 032807-. doi: 10.1103/PhysRevE.89.032807 | |
[111] | LIU S Y, BARONCHELLI A, PERRA N. Contagion dynamics in time-varying metapopulation networks[J]. Physical Review E, 2013, 87(3): 032805-. doi: 10.1103/PhysRevE.87.032805 | |
[112] | KOTNIS B, KURI J. Stochastic analysis of epidemics on adaptive time varying networks[J]. Physical Review E, 2013, 87(6): 062810-. doi: 10.1103/PhysRevE.87.062810 | |
[113] | PERRA N, BARONCHELLI A, MOCANU D. Random walks and search in time-varying networks[J]. Physical Review Letters, 2012, 109(23): 238701-. doi: 10.1103/PhysRevLett.109.238701 | |
[114] | GAUVIN L, PANISSON A, CATTUTO C. Activity clocks:Spreading dynamics on temporal networks of human contact[J]. Scientific Reports, 2013, 3(): 3099-. | |
[115] | HOLME P, MASUDA N. The basic reproduction number as a predictor for epidemic outbreaks in temporal networks[J]. PLoS ONE, 2015, 10(3): e0120567-. doi: 10.1371/journal.pone.0120567 | |
[116] | MYERS S A, ZHU C, LESKOVEC J. Information diffusion and external influence in networks[C]//Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining.[S.l.]:ACM, 2012:33-41. | |
[117] | LAMBIOTTE R, TABOURIER L, DELVENNE J C. Burstiness and spreading on temporal networks[J]. The European Physical Journal B, 2013, 86(): 320-. doi: 10.1140/epjb/e2013-40456-9 | |
[118] | VAZQUEZ A, RACZ B, LUKACS A. Impact of non-Poissonian activity patterns on spreading processes[J]. Physical Review Letters, 2007, 98(15): 158702-. doi: 10.1103/PhysRevLett.98.158702 | |
[119] | MIN B, GOH K I, VAZQUEZ A. Spreading dynamics following bursty human activity patterns[J]. Physical Review E, 2011, 83(3): 036102-. | |
[120] | MIN B, GOH K I, KIM I M. Suppression of epidemic outbreaks with heavy-tailed contact dynamics[J]. Europhysics Letters, 2013, 103(5): 50002-. doi: 10.1209/0295-5075/103/50002 | |
[121] | KARSAI M, KIVELA M, PAN R K. Small but slow world:How network topology and burstiness slow down spreading[J]. Physical Review E, 2011, 83(2): 025102-. doi: 10.1103/PhysRevE.83.025102 | |
[122] | ROCHA L E C, LILJEROS F, HOLME P. Simulated epidemics in an empirical spatiotemporal network of 50,185 sexual contacts[J]. PLoS Computational Biology, 2011, 7(3): e1001109-. doi: 10.1371/journal.pcbi.1001109 | |
[123] | ROCHA L E C, BLONDEL V D. Bursts of vertex activation and epidemics in evolving networks[J]. PLoS Computational Biology, 2013, 9(3): e1002974-. doi: 10.1371/journal.pcbi.1002974 | |
[124] | TAKAGUCHI T, MASUDA N, HOLME P. Bursty communication patterns facilitate spreading in a threshold-based epidemic dynamics[J]. PLoS ONE, 2013, 8(7): e68629-. doi: 10.1371/journal.pone.0068629 | |
[125] | IRIBARREN J L, MORO E. Impact of human activity patterns on the dynamics of information diffusion[J]. Physical Review Letters, 2009, 103(3): 038702-. doi: 10.1103/PhysRevLett.103.038702 | |
[126] | IRIBARREN J L, MORO E. Branching dynamics of viral information spreading[J]. Physical Review E, 2011, 84(4): 046116-. doi: 10.1103/PhysRevE.84.046116 | |
[127] | CUI A X, WANG W, TANG M. Efficient allocation of heterogeneous response times in information spreading process[J]. Chaos, 2014, 24(): 033113-. doi: 10.1063/1.4890612 | |
[128] | LIU S, PERRA N, KARSAI M, et al. Controlling contagion processes in time-varying networks[EB/OL]. (2013-09-26). http://arxiv.org/abs/1309.7031. | |
[129] | COHEN R, HAVLIN S. Complex networks:Structure, robustness and function[M]. Cambridgeshire:Cambridge University Press, 2010. | |
[130] | MIRITELLO G. Temporal patterns of communication in social networks[M]. Berlin:Springer, 2013. | |
[131] | MIRITELLO G, MORO E, LARA R. Dynamical strength of social ties in information spreading[J]. Physical Review E, 2011, 83(2): 045102-. | |
[132] | YANG Z, CUI A X, ZHOU T. Impact of heterogeneous human activities on epidemic spreading[J]. Physica A, 2011, 390(23-24): 4543-4548. doi: 10.1016/j.physa.2011.06.068 | |
[133] | ZHAO Z D, LIU Y, TANG M. Epidemic variability in hierarchical geographical networks with human activity patterns[J]. Chaos, 2012, 22(): 023150-. doi: 10.1063/1.4730750 | |
[134] | KARSAI M, KIVELÄ M, PAN R K. Small but slow world:How network topology and burstiness slow down spreading[J]. Physical Review E, 2011, 83(2): 25102-. doi: 10.1103/PhysRevE.83.025102 | |
[135] | 郭进利, 汪丽娜. 幂律指数在1与3之间的一类无标度网络[J]. 物理学报, 2007, 56(10): 5635-5639. | GUO Jin-li, WANG Li-na. A class of scale free networks with power law exponent between 1 and 3[J]. Acta Physica Sinica, 2007, 56(10): 5635-5639. |
[136] | KARRER B, NEWMAN M E J. Message passing approach for general epidemic models[J]. Physical Review E, 2010, 82(1): 016101-. doi: 10.1103/PhysRevE.82.016101 | |
[137] | MIEGHEM P V, BOVENKAMP R V D. Non-markovian infection spread dramatically alters the Susceptible-Infected-Susceptible epidemic threshold in networks[J]. Physical Review Letters, 2013, 110(10): 108701-. doi: 10.1103/PhysRevLett.110.108701 | |
[138] | CATOR E, BOVENKAMP R V D, MIEGHEM P V. Susceptible-infected-susceptible epidemics on networks with general infection and cure times[J]. Physical Review E, 2013, 87(6): 062816-. doi: 10.1103/PhysRevE.87.062816 | |
[139] | BOGUÑÁ M, LAFUERZA L F, TORAL R. Simulating non-markovian stochastic processes[J]. Physical Review E, 2014, 90(4): 042108-. | |
[140] | LAMBIOTTE R, SALNIKOV V, ROSVALL M. Effect of memory on the dynamics of random walks on networks[J]. Journal of Complex Networks, 2014, 3(2): 177-188. | |
[141] | SALATHÉ M, VU D Q, KHANDELWAL S. The dynamics of health behavior sentiments on a large online social network[J]. EPJ Data Science, 2013, 2(1): 1-12. doi: 10.1140/epjds13 | |
[142] | KARSAI M, KASKI K, KERTESZ J. Correlated dynamics in egocentric communication networks[J]. PLoS ONE, 2012, 7(7): e40612-. doi: 10.1371/journal.pone.0040612 | |
[143] | SONG C, WANG D, BARABASI A L. Connections between human dynamics and network science[EB/OL]. (2013-04-08). http://arxiv.org/abs/1209.1411. | |
[144] | SUN L, AXHAUSEN K W, LEE D H. Understanding metropolitan patterns of daily encounters[J]. Proceedings of the National Academy of Sciences of the United States of America, 2013, 110(34): 13774-13779. doi: 10.1073/pnas.1306440110 | |
[145] | MIRITELLO G, LARA R, CEBRIAN M. Limited communication capacity unveils strategies for human interaction[J]. Scientific Reports, 2013, 3(): 1950-. | |
[146] | MIRITELLO G, LARA R, MORO E. Time allocation in social networks:Correlation between social structure and human communication dynamics[J]. Chapter Temporal Networks Part of the series Understanding Complex Systems, 2013, (): 175-190. | |
[147] | BAGROW J P, BROCKMANN D. Natural emergence of clusters and bursts in network evolution[J]. Physical Review X, 2013, 3(2): 021016-. doi: 10.1103/PhysRevX.3.021016 | |
[148] | KARSAI M, PERRA N, VESPIGNANI A. Time varying networks and the weakness of strong ties[J]. Scientific Reports, 2014, 4(): 4001-. | |
[149] | ODOR G. Slow, bursty dynamics as a consequence of quenched network topologies[J]. Physical Review E, 2014, 89(4): 042102-. | |
[150] | DOMENICO M D, SOLE-RIBALTA A, COZZO E. Mathematical formulation of multilayer networks[J]. Physical Review X, 2013, 3(4): 041022-. doi: 10.1103/PhysRevX.3.041022 |