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Journal of Geo-information Science >

2017 , Vol. 19 >Issue 10: 1279 - 1286

DOI: https://doi.org/10.3724/SP.J.1047.2017.01279

Dynamic Analysis of Interactive Transmission of Warning Information and Traffic Congestion

  • ZHOU Yan , 1, 2 ,
  • LI Yanxi , 1, * ,
  • JIANG Ronggui 1 ,
  • GENG Erhui 1
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  • 1. School of Resources and Environment, University of Electronic Science and Technology of China, Chengdu 611731, China
  • 2. Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
*Corresponding author: LI Yanxi, E-mail:

Received date: 2017-04-30

  Request revised date: 2017-05-12

  Online published: 2017-10-20

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Abstract

The problem of urban traffic congestion has become a serious problem in the development of many cities in the world. To solve this problem, pan-spatial information system provides a new way of solving urban traffic congestion by multi-granularity abstracting, multi-scale modeling and multi-level comprehensive analysis of dynamic and complex traffic jam processes. In reality, the process of traffic congestion is usually accompanied by the dissemination of traffic warning information. Accordingly, when the competition occurs, which is generated by traffic congestion and the spreading of warning information in different network layers, the interplay between traffic congestion and warning information plays an important role. Thus, in order to study the interplay between information spreading and traffic congestion spreading, we constructed a multiplex network with road intersections or sites to analyze the interplay between information spreading and traffic congestion spreading. Firstly, we considered the effect of the surrounding nodes and proposed an improved SIS model. Then, based on the improved SIS model, we used the method of state transition probability to study the competing spreading processes of multiplex network. Finally, using the Monte Carlo method, we analyzed and simulated the traffic congestion threshold in both homogeneous network and heterogeneous network. This study indicates that the process of traffic congestion depends on dynamics of warning information spreading through transport network.

Key words： traffic congestion; multiplex network; epidemic spreading; information spreading

Cite this article

ZHOU Yan , LI Yanxi , JIANG Ronggui , GENG Erhui . Dynamic Analysis of Interactive Transmission of Warning Information and Traffic Congestion[J]. Journal of Geo-information Science, 2017 , 19(10) : 1279 -1286 . DOI: 10.3724/SP.J.1047.2017.01279

1 引言

城市交通拥堵问题已经成为当今世界许多城市发展过程中面临的一个严峻问题。交通拥堵不仅影响着城市交通系统的运行效率,还给城市发展和人类生活带来了诸多不便。城市交通系统是一个复杂巨系统,传统的分析理论与方法已无法满足城市交通系统的研究,全空间信息系统把现实世界抽象为由多粒度时空对象组成的数据世界,通过对现实世界中复杂、动态的交通拥堵过程进行多粒度抽象、多尺度建模和综合分析,为解决城市交通拥堵提供了新的途径[1-2]。
在交通拥堵传播领域,有学者通过研究人们对车辆路径的选择行为,发现不同网络拓扑结构和交通量产生率会影响交通拥堵的传播[3]。有研究分别针对匀质和异质交通需求,探讨了匀质及异质能力分配条件下的复杂交通网络的拥堵和效率情况[4]。根据实际交通传播的情况改进了中观交通流模型,并利用该模型分析了复杂网络上的交通传播动力学特征及传播规律[5]。部分研究网络基于拓扑结构与网络传输过程之间的关系,通过改变网络结构或改进路由算法的方法研究交通拥堵问题[6-8]。已有的研究主要集中在对网络拓扑结构、网络流量负荷与交通拥堵之间的关系进行分析,并没有考虑交通拥堵的传播过程。最新的研究通过建立考虑时间延迟影响的交通拥堵传播的SIS病毒传播模型,进而研究交通拥堵传播过程中的影响因素[9]。虽然是针对交通拥堵的传播过程,但是只单独研究了拥堵传播过程,没有考虑信息传播对拥堵传播的影响。然而,在现实生活中,交通拥堵传播过程往往会伴随着“道路拥堵”预警信息传播。这种“道路拥堵”预警信息可以通过人与人之间或者公共媒介等工具在人群中传播开来,进而出现交通拥堵传播与预警信息在网络中的交互传播。因此考虑交通拥堵-预警信息的全空间多层网络的交互传播是一个值得研究的重要问题。
综上所述,为了深入分析城市交通网络拥堵动态演化过程,本文根据交通流传播的特点,引入全空间信息系统多层次网络交互分析技术思想,建立了交通拥堵传播的改进SIS病毒传播模型,利用状态转移概率的方法,研究多层网络中交通拥堵传播和预警信息传播相互交互动力学。基于无标度网络和小世界网络对模型进行仿真,进一步比较分析了相关影响因素的作用规律,可为交通拥堵的管理或控制策略的制定提供一定的参考。

2 基于状态概率转移的交通拥堵-预警信息交互模型

2.1 基于交通拥堵传播的改进SIS病毒传播模型分析

经典的SIS病毒传播模型一般用来描述痊愈后的个体不能拥有免疫能力的疾病,比如流行性感冒等。因此,在SIS病毒传播模型中,节点被划分为两类：易感节点和感染节点。在病毒传播过程中,当感染节点遇到易感节点时,会以感染率β把传染病传给易感节点。同时,由于感染后的个体自身的安全机制的作用,将以恢复率δ转变为易感节点。而且,易感节点一旦被感染,将会充当感染节点的角色继续传播病毒。
与SIS病毒传播过程相比,交通拥堵传播具有相似的传播过程,但由于交通系统自身的复杂特性,其传播过程又具有典型的多尺度交互影响特点（节点-路口-路段-网络）。在交通拥堵传播过程中,当拥堵节点以恢复率δ转变为畅通节点时,其恢复过程与下游道路的路况有关,交叉口下游的畅通道路越多,拥堵消散过程耗时越短,而感染节点的恢复过程与周围的节点的状态无关。
因此,在改进的SIS病毒传播模型中,节点被划分为2类：畅通节点（对应易感节点）和拥堵节点（对应感染节点）。当拥堵传播率β超过一定阈值时,拥堵节点会以传播率β使周围的畅通节点变为拥堵节点;同时,在拥堵消散过程中,拥堵节点会因为周围节点的交通状况,以消散率δ转变为畅通节点。

2.2 交通拥堵-预警信息交互模型分析

由于交通拥堵传播过程往往会伴随着“道路拥堵”预警信息传播,这使得具有多尺度交互影响特点的交通拥堵传播过程变得更加复杂,因此本文借鉴全空间信息系统多层次网络交互分析思想,引入复杂网络技术理论,对交通拥堵与信息传播过程进行多粒度抽象（节点-路口-路段-网络）,同时构建出交通拥堵-预警信息多层网络模型。该多层网络包含两层复杂网络,一层是演化交通拥堵传播过程的城市交通网络,一层是描述“道路拥堵”预警信息传播的通信网络。在城市交通网络中,道路交叉口或站点对应节点,连接它们的道路对应边,其中采用改进的SIS病毒传播模型演化交通拥堵在网络中的传播。
在信息通信网络中,节点集与城市交通网络的节点集相同,边则代表节点之间是否能进行正常通信。对信息通信网络,采用正常-预警-正常（normal-warning-normal,NWN）信息传播模型描述“道路拥堵”预警信息在网络中的传播。在NWN信息传播模型中,网络节点具有2种形式：具有并能转发预警信息的节点（即节点的状态为W）和没有预警信息或通信受限的节点（即节点的状态为N）。在预警信息传播过程中,当驾驶人员获得“道路拥堵”预警信息时,往往会考虑重新选择路线,从而减少了进入拥堵路段的车流量,加快拥堵消散,但是,没有接受到预警信息的个体则不会采取措施来减少拥堵风险。假设预警信息的主要来源有2个方面：① 来源于信息通信层中产生并发出了预警信息的节点,即没有预警信息的节点与发出预警信息的节点进行通信,以概率λ接受到预警信息并转化为具有预警信息的节点;② 来源于交通网络层中已拥堵的节点,即交通网络层中已拥堵的节点在通信网络层中自发地转为具有预警信息的节点。同时,随着道路交通由拥堵状态转为畅通状态,预警信息也逐渐停止传播,从而导致具有预警信息的节点以概率μ变为无预警信息的节点。
在上述交通拥堵-预警信息交互传播模型中,假设未被预警的交通畅通路口发生拥堵的概率为βN,拥堵消散的概率为δN;而被预警的交通畅通路口发生拥堵的概率为βW,拥堵消散的概率为δW。已知被预警的交通畅通路口发生拥堵的概率一般会小于未被预警的交通畅通路口,同时,被预警的交通拥堵路口比未被预警的交通拥堵路口的拥堵消散过程耗时更少,因此,设被预警的交通畅通路口发生拥堵的概率相对于未被预警的交通畅通路口以因子γ（0≤γ≤1）倍相应地发生交通拥堵,即βW=γβN,而未被预警的交通拥堵路口相对于被预警的交通拥堵路口以因子θ（0≤θ≤1）倍相应地进行拥堵消散过程,即δN=θδW。特例,γ=0表示被预警的交通畅通路口不会发生拥堵,同理,θ=0表示未被预警的交通拥堵路口在短时间内不会改变拥堵状态。
依据上述交通拥堵-预警信息交互传播机制,多层网络中的节点可分为3类：被预警的交通畅通路口、被预警的交通拥堵路口和未被预警的交通畅通路口,其对应的3种状态分别为：预警-易感（Warning-Susceptible,WS）状态、预警-感染（Warning-Infected,WI）状态和正常-易感（Normal-Susceptible,NS）状态。由于在动力学交互过程中,假设未被预警的交通拥堵路口立即转化为被预警的交通拥堵路口,因此在描述交通拥堵-预警信息交互模型中,正常-感染（Normal-Infected,NI）状态可被忽略。综上所述,交通拥堵-预警信息交互传播模型（Susceptible-Infected-Susceptible and Normal-Warning-Normal,SIS-NWN）中不同类型节点的状态转化示意图,如图1所示,其中涉及到的主要符号及具体描述见表1。
Fig. 1 Transition probability diagram for the nodes𠈙 states in the two-layer SIS-NWN networks

图1 多层网络中交通拥堵-预警信息模型的节点状态转化示意图

Tab. 1 The main notations and descriptions

表1 主要的符号及描述

符号 描述
λ 节点由正常状态（N）转化为预警状态（W）的概率
μ 节点由预警状态（W）转化为正常状态（N）的概率
βN 处于正常状态（N）的节点发生交通拥堵的概率
βW 处于预警状态（W）的节点发生交通拥堵的概率
δN 处于正常状态（N）的节点发生拥堵后,拥堵消散的概率
δW 处于预警状态（W）的节点发生拥堵后,拥堵消散的概率
假设多层网络中每层复杂网络的节点总数均为M,用A=[aij]∈RM×M和B=[bij]∈RM×M分别表示信息通信网络和城市交通网络的邻接矩阵,当节点i和节点j相邻时,值为1,反之为0。初始时刻在网络中随机选择一个节点i作为具有拥堵信息的交通拥堵节点。设 p i WI t 、 p i WS t 和 p i NS t 分别为t时刻节点i处于WI、WS和NS状态的概率,由于该交通拥堵-预警信息交互传播模型满足连续时间的马尔科夫过程,因此其满足归一化条件（式（1））：
p i WI t + p i WS t + p i NS t = 1 （1）
根据Chakrabarti等[10]的方法,结合图1,可以利用节点i在t时刻处于各个状态的概率来描述各个节点的动力学过程,从而建立交通拥堵传播动力学方程,例如从图1中可知,t+1时刻处于NS状态的节点的概率与t时刻NS状态的节点没有转化为其它状态的概率、WI状态的节点转化为NS状态的概率以及WS状态的节点转化为NS状态的概率有关,因此t+1时刻节点i处于NS状态的概率 p i NS t + 1 计算公式如式（2）,其中 p i NS t 1 - r i t + 1 - q i N t 为t时刻状态为NS的节点保持状态不变的概率; p i WI t f i N t 为t时刻状态为WI的节点发生拥堵消散转化到NS状态的概率; p i WS t μ 为t时刻状态为WS的节点停止传播预警信息,从而转化到NS状态的概率。同理可以分析计算得到式（3）和（4）。
p i NS t + 1 = p i NS t 1 - r i t + 1 - q i N t + p i WI t f i N t + p i WS t μ （2）
p i WS t + 1 = p i WS t 1 - q i W t + 1 - μ + p i WI t f i W t + p i NS t r i t （3）
p i WI t + 1 = p i WI t 1 - f i W t + 1 - f i N t + p i WS t q i W t + p i NS t q i N t （4）
其中,ri(t)为通信网络层中节点i受到具有预警信息的邻居点影响接收到预警信息的概率,计算见式（5）; q i W t 为在具有预警信息的拥堵节点影响下节点由畅通转为拥堵的概率,计算见式（6）,反之为 q i N t ,计算见式（7）;同理, f i W t 为在具有预警信息条件下的发生拥堵消散的概率,计算见式（8）,反之为 f i N t ,具体计算见式（9）。
r i t = 1 - ∏ j = 1 M 1 - λ a ji p j W t ] （5）
q i W t = 1 - ∏ j = 1 M 1 - β W b ji p j WI t ] （6）
q i N t = 1 - ∏ j = 1 M 1 - β N b ji p j WI t ] （7）
f i W t = 1 - ∏ j = 1 M 1 - δ W b ji p j WS t ] （8）
f i N t = 1 - ∏ j = 1 M 1 - δ N b ji p j NS t ] （9）
其中, p j W t = p j WI t + p j WS t （10）

3 阈值分析

根据上面建立的交通拥堵-预警信息交互传播动力学方程可以看出,影响交通拥堵传播的参数较多,因此需要通过阈值分析找到其中具有重要作用的影响参数。
当t→∞时,3种状态NS、WS及WI节点的总数分 别 N NS = ∑ i = 1 M p i NS ∞ , N WS = ∑ i = 1 M p i WS ∞ , N WI = ∑ i = 1 M p i WI ∞ 。 采用稳态分析方法,可以利用这组动力学方程求得传播临界值。
式（2）-（4）是一个非线性动力学系统,设系统初始节点的值为0,可以通过雅可比矩阵式（11）来分析式（7）在初始点的局部稳定性：
DF | 0,0 = β N B - δ N B 0 M λA 1 - μ I M + λA （11）
式中：B=[bij]M×M,A=[aij]M×M,IM为M阶单位矩阵。当满足 max β N B - δ N B ， 1 - μ I M + λA < 1 时,初始节点局部稳定,从而可以得到式（12）：
β N < 1 Λ max B + δ N ， λ < μ Λ max A （12）
式中： Λ max B ， Λ max A 分别是B矩阵和A矩阵的最大特征值。由式（12）可以看出,参数βN和δN为影响力较大的参数,其取值的大小会极大地影响交通拥堵传播,同时,矩阵B和矩阵A的最大特征值也会影响交通拥堵传播。因此,交通拥堵的传播过程不仅与交通拥堵传播阈值和交通拥堵消散阈值有关,而且与交通网络中的预警信息传播动力学有关。

4 数值仿真

为了验证上述状态概率方程分析的传播临界值的正确性以及进一步分析影响参数在交通拥堵传播过程中的作用,本文以BA无标度网络和WS小世界网络作为复杂网络城市交通网络的拓扑结构模型[9,11-12],并假设双层网络中的两层网络均为同一个BA无标度网络或WS小世界网络,设网络规模均为M=1000。基于Monte Carlo仿真,在交通拥堵传播和预警信息传播的初始时刻,从交通网络层中随机选择一个节点作为交通拥堵节点,同时令该随机选择的节点在信息通信层中作为最先具有“道路拥堵”预警信息的节点。多层网络中拥堵节点的稳态密度为 p I = ∑ p i I / M = ∑ p i WI / M ,具有预警信息节点的稳态密度为 p W = ∑ p i W / M = ∑ ( p i WI + p i WS ) / M 。
由βW=γβW,当γ=0时表示具有预警信息的节点不会发生拥堵,即βW=0;当γ=1时表示具有预警信息的节点没有采取措施降低拥堵产生的概率,即βW=βN。图2表示在无标度网络和小世界网络下,三种不同交通拥堵传播模型（即γ=0、γ=0.5、γ=1）的拥堵稳态密度与拥堵传播率β=βN的关系,设预警信息传播率为λ=0.2,预警信息消失率为μ=0.4,具有预警信息的交通拥堵消散率为δW=0.2,倍数θ=0.2。图2显示,基于多层网络的拥堵-预警交互传播模型增强了交通拥堵传播临界值,显著降低了交通拥堵爆发规模,并且具有预警信息的节点均不发生拥堵的拥堵-预警交互模型（γ=0）,明显比具有预警信息的节点以βW=γβN概率发生拥堵的交互模型在降低交通拥堵爆发的风险上,更加有效地抑制了拥堵在路网中的传播,降低了交通拥堵最终爆发规模。
Fig. 2 The size of infected nodes pI is shown as a function of infectivity β of three kinds of traffic congestion models in Watts-Strogatz model and Barabasi-Albert model, respectively

图2 在无标度网络和小世界网络中3种交通拥堵模型的pI随β的变化图

Fig. 3 The size of infected nodes pI is shown as a function of infectivity δ of three kinds of traffic congestion models in Watts-Strogatz model and Barabasi-Albert model, respectively

图3 在无标度网络和小世界网络中3种交通拥堵模型的pI随δ的变化图

Fig. 4 Monte Carlo simulations of the two-layer SIS-NWN networks in Watts-Strogatz model and Barabasi-Albert model.The size of infected nodes pI is shown as a function of infectivity δW

图4 在无标度网络和小世界网络中不同δW值下的交通拥堵-预警信息交互模型

由δN=θδW,当θ=0时表示未被预警的节点在短时间内不会发生拥堵消散,即δN=0;当θ=1时表示交通拥堵消散的过程均受到预警信息传播的影响,即δN=δW。图3表示在无标度网络和小世界网络下,3种不同交通拥堵传播模型（即θ=0、θ=0.5、θ=1）的拥堵稳态密度与拥堵消散率δ=δW的关系,设预警信息传播率为λ=0.2,预警信息消失率为μ=0.4,具有预 警信息的交通拥堵消散率为βN=0.6,倍数γ=0.4。根据图3的仿真结果可知,具有预警信息的交通拥堵消散过程能显著降低交通拥堵的爆发规模,特别是当交通拥堵消散过程均受到预警信息传播的影响时（θ=1）,预警信息的传播对拥堵的爆发规模影响更大。
在实际路网中,交通拥堵的爆发规模不仅与拥堵路口处的交通拥堵传播临界值有关,还与该路口的拥堵消散能力有关,并且接受到预警信息的路口会提高抗拥堵能力。图4表示,在具有预警信息的条件下,不同拥堵消散概率δW=0.2、0.4、0.6、0.8对拥堵爆发规模的影响。从图中可以看出,在无标度网络和小世界网络下,预警信息对拥堵路口的消散能力影响程度越大,拥堵传播阈值越大,拥堵爆发规模越小。
Fig. 5 The relationship between pI, pW and β under differentvalues in Watts-Strogatz model and Barabasi-Albert model

图5 在无标度网络和小世界网络中不同λ值下的pI、pW与β之间的关系

图5表示不同的预警信息传播率λ=0.2、0.4、0.6、0.8下,拥堵传播规模pI、信息传播规模pW与交通拥堵传播率β之间的关系。设具有预警信息的节点降低拥堵发生的程度为γ=0.4,预警信息消失率为μ=0.4,具有预警信息的交通拥堵消散率为δW=0.2,倍数θ=0.2。图5显示,在无标度网络和小世界网络中,随着预警信息传播率λ的增大,拥堵传播规模 和信息传播规模同时降低。这意味着预警信息在网络中的蔓延减缓了拥堵的传播,减小了拥堵爆发规模。

5 结论

在实际生活中,当道路发生交通拥堵时,拥堵的传播与消散过程不仅与拥堵处周围的道路路况有关,还会受到“道路拥堵”预警信息传播的影响。同时,由于实际的交通系统通常是由多个相互作用且相互依赖的复杂网络组成,因此本文根据交通流传播的特点,引入全空间信息系统多层次网络交互分析技术思想,对复杂、动态的交通拥堵过程进行节点-路口-路段-网络的多粒度抽象和多尺度建模,建立了交通拥堵传播的改进SIS病毒传播模型,并利用状态转移概率的方法,构建多层网络中交通拥堵传播和预警信息传播的交互动力学模型。在该模型中,通过考虑拥堵处周围路况对拥堵消散过程的影响而改进的SIS病毒传播模型,用以描述城市交通网络层中的交通拥堵传播过程,并且采用NWN信息传播模型描述信息通信层中的预警信息传播过程。针对本文所提出的交通拥堵-预警信息模型,通过数学推导和数值仿真相结合的方法,发现并证明了交通拥堵的传播过程不仅与交通拥堵传播阈值和交通拥堵消散阈值有关,而且与交通网络中的预警信息传播动力学有关。同时,仿真实验结果还表明,“道路拥堵”预警信息在网络中的传播能在一定程度上减缓交通拥堵的传播,从而降低拥堵爆发的规模。多层网络中交通拥堵传播和预警信息交互传播的研究,不仅为研究城市的交通拥堵提供了一种新的视角,同时,也为全空间信息系统多层次网络交互传播分析技术研究提供了一定的应用参考。

The authors have declared that no competing interests exist.

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