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以通信网、电力网、交通网为代表的很多复杂网络以传输负载为基本功能. 在这些网络中, 网络的吞吐量是衡量网络传输性能的重要指标, 如何提升网络的吞吐量是研究热点之一. 不少研究人员提出了不同的路由算法, 通过调节传输路径来提高网络吞吐量. 但之前的研究很少考虑网络中节点的空间位置. 本文针对空间网络提出了一种高效的路由策略, 通过节点位置得到路径长度; 采用该算法, 负载从源节点沿着最短长度的路径传输到目标节点. 为了检验算法的有效性, 采用网络从自由流状态转变成拥塞状态的相变点Rc来衡量网络的吞吐量. 在匀质和异质空间网络上的仿真表明, 与传统的最少跳数路由策略相比, 本文提出的基于最短路径长度的路由算法能有效提高空间网络的吞吐量.In many complex networks, such as communication networks, power grids, and transportation networks, the main task is load transmission from sources to destinations. Therefore, the transmission throughput is a very important indicator to measure the network performance, and improving the throughput becomes one of the hotspots in the research of these complex networks. Many researchers have proposed different routing algorithms to improve the network throughput. However, few of them considered the spatial location of nodes in the network. Indeed, many real-world networks can be modeled by spatial networks, where the spatial location of nodes plays a vital role in determining the structure and dynamic behaviors of such networks. Specifically, when the locations of nodes are considered, each link has a length. And the shortest path may have different meaning. Traditionally, the shortest path indicates the path which passes the least number of links from source to destination, or the least number of hops. However, when the length of link is taken into account, the least number of links does not mean the least summation of link lengths along the path. The latter can be called the shortest path length. To this end, we proposes an efficient routing strategy for spatial networks based on the shortest path length in this work. In order to test the effectiveness of the algorithm, the network throughput
${R}_{\rm c}$ is used, at which the network changes from a free flow state to a congestion state, to measure the performance of the network. Simulations of homogeneous and heterogeneous spatial networks show that compared with the traditional least number of hops routing strategy, the routing algorithm based on the shortest path length proposed in this paper can effectively improve the throughput of the network. The routing algorithm proposed in this paper can be applied to many real-world spatial networks for improving their performances.-
Keywords:
- routing strategy /
- spatial network /
- congestion
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图 2 N = 1500,
$ \left\langle{k}\right\rangle $ = 8时的LAEE网络的度分布Fig. 2. Degree distribution of the LAEE network. N = 1500,
$ \left\langle{k}\right\rangle $ = 8.
图 4 节点i与其连通范围内的节点以p概率进行连接
Fig. 4. Node i connects nodes within its connection range with probability p.
图 5 N = 1500,
$ \left\langle{k}\right\rangle $ = 8时改进的随机几何图的度分布Fig. 5. Degree distribution of the improved random geometric graph. N = 1500,
$ \left\langle{k}\right\rangle $ = 8.
图 6 平均度
$ \left\langle{k}\right\rangle=4 $ 时, 不同节点规模下两种路由方式平均路径长度的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 6. With the average degree
$ \left\langle{k}\right\rangle=4 $ , the average path length under two routing strategies with different node sizes: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 7 平均度
$ \left\langle{k}\right\rangle=4 $ 时, 不同节点规模下两种路由方式平均跳数的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 7. With the average degree
$ \left\langle{k}\right\rangle=4 $ , the average number of hops under two routing strategies with different node sizes: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 8 平均度
$ \left\langle{k}\right\rangle=4 $ 时, 不同节点规模下两种路由方式的网络最大吞吐量$ {R}_{\rm{c}} $ 的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 8. With the average degree
$\left\langle{k}\right\rangle=4$ ,$ {R}_{\rm{c}} $ under two routing strategies with different node sizes: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 9 节点数
$ N=1000 $ 时, 不同平均度下两种路由方式平均路径长度的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 9. With N = 1000, the average path length under two routing strategies with different average degrees: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 10 节点数
$ N=1000 $ 时, 不同平均度下两种路由方式平均跳数的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 10. With N = 1000, the average number of hops under two routing strategies with different average degrees: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 11 节点数
$ N=1000 $ 时, 不同平均度下两种路由方式的网络最大吞吐量$ {R}_{\rm{c}} $ 的比较 (a) LAEE演化模型; (b)改进的随机几何图模型. 图中每个点是10次仿真的平均值Fig. 11. With
$ N=1000 $ ,$ {R}_{c} $ under two routing strategies with different average degrees: (a) LAEE evolution model; (b) improved random geometric graph model. Each point is the average of 10 runs.
图 12 LAEE演化模型中, 节点数N = 1000,
$ \left\langle{k}\right\rangle $ = 4时的介数分布 (a)最少跳数路由; (b)最短路径长度路由Fig. 12. Betweenness distribution of the LAEE network with N = 1000,
$ \left\langle{k}\right\rangle $ = 4: (a) Least number of hops routing strategy; (b) shortest path length routing strategy.
图 13 改进的随机几何图模型中, 节点数N = 1000,
$ \left\langle{k}\right\rangle $ = 4时的介数分布 (a)最少跳数路由; (b)最短路径长度路由Fig. 13. Betweenness distribution of the improved random geometric graph with N = 1000,
$ \left\langle{k}\right\rangle $ = 4: (a) Least number of hops routing strategy; (b) shortest path length routing strategy. -
[1] Erdös P, Rényi A 1959 Publ. Math. 6 290
[2] Barabási A L, Albert R 1999 Science 286 509
Google Scholar
[3] Watts D J, Strogatz S H 1998 Nature 393 440
Google Scholar
[4] 陈关荣 2008 力学进展 38 653
Google Scholar
Chen G R 2008 Adv. Mech. 38 653
Google Scholar
[5] 马金龙, 张俊峰, 张冬雯, 张红斌 2021 物理学报 70 078902
Google Scholar
Ma J L, Zhang J F, Zhang D W, Zhang H B 2021 Acta Phys. Sin. 70 078902
Google Scholar
[6] 于晓 2020 软件工程 23 1
Google Scholar
Yu X 2020 Software Engineering 23 1
Google Scholar
[7] 任卓明 2020 物理学报 69 048901
Google Scholar
Ren Z M 2020 Acta Phys. Sin. 69 048901
Google Scholar
[8] Hearnshaw E J S, Wilson M M J 2013 Int. J. Operat. Product. Manage. 33 442
Google Scholar
[9] Chen S Y, Huang W, Cattani C, Altieri G 2012 Math. Probl. Eng. 2012 732698
Google Scholar
[10] Tan F, Wu J J, Xia Y X, Tse C K 2014 Phys. Rev. E 89 062813
Google Scholar
[11] 陈华良, 刘忠信, 陈增强, 袁著祉 2009 物理学报 58 6068
Google Scholar
Chen H L, Liu Z X, Chen Z Q, Yuan Z Z 2009 Acta Phys. Sin. 58 6068
Google Scholar
[12] 刘伟彦, 刘斌 2014 物理学报 63 248901
Google Scholar
Liu W Y, Liu B 2014 Acta Phys. Sin. 63 248901
Google Scholar
[13] 赵磊, 周金和 2019 计算机科学 46 137
Google Scholar
Zhao L, Zhou J H 2019 Computer Science 46 137
Google Scholar
[14] 刘金良 2013 物理学报 62 040503
Google Scholar
Liu J L 2013 Acta Phys. Sin. 62 040503
Google Scholar
[15] 赵宇红 2019 博士学位论文 (北京: 北京科技大学)
Zhao Y H 2019 Ph. D. Dissertation (Beijing: University of Science and Technology Beijing) (in Chinese)
[16] 李世宝, 娄琳琳, 陈瑞祥, 洪利 2014 物理学报 63 028901
Google Scholar
Li S B, Lou L L, Chen R X, Hong L 2014 Acta Phys. Sin. 63 028901
Google Scholar
[17] 邵斐, 蒋国平 2011 物理学报 60 078902
Google Scholar
Shao F, Jiang G P 2011 Acta Phys. Sin. 60 078902
Google Scholar
[18] Yan G, Zhou T, Hu B, Fu Z Q, Wang B H 2005 Phys. Rev. E 73 046108
Google Scholar
[19] Huang W, Chow T W S 2009 Chaos 19 043124
Google Scholar
[20] Wang W X, Wang B H, Yin C Y, Xie Y B, Zhou T 2006 Phys. Rev. E 73 026111
Google Scholar
[21] Xia Y X, Wang C, Shen H L, Song H N 2020 Physica A 559 125071
Google Scholar
[22] Zhang M, Chen S, Sun L, Du W, Cao X 2021 Engineering 7 465
Google Scholar
[23] Du W B, Zhou X L, Jusup M, Wang Z 2016 Sci. Rep. 6 19059
Google Scholar
[24] Jiang L, Xu Q, Pan H, Dai Y, Tong J 2020 Secur. Commun. Netw. 2020 6513920
Google Scholar
[25] Bai G H, Li Y J, Fang Y N, Zhang Y A, Tao J Y 2020 Reliab. Eng. Syst. Saf. 193 106602
Google Scholar
[26] Zhao L, Lai Y C, Park K, Ye N 2005 Phys. Rev. E 71 026125
Google Scholar
[27] Dall J, Christensen M 2002 Phys. Rev. E 66 016121
Google Scholar
[28] Jiang L R, Jin X Y, Xia Y X, Ouyang B, Wu D P, Chen X 2015 Int. J. Distrib. Sens. Netw. 2014 764698
Google Scholar
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