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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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关于时效网络的研究层出不穷,其中一些结果进一步拓展和加深了对时效结构特性参量及相关统计力学的理解。本文主要从以下两方面介绍时效网络的相关研究进展:1) 时效网络的时效模体和社团结构的研究[7, 26, 42];2) 时效网络重要节点挖掘及各类指标研究[1, 35, 43-44]。
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近几年来,大量的研究集中于时效网络的中尺度特性上,包括网络的时效模体[45]和社团结构[26]。在时效模体方面,文献[27]引入了流模体的概念来量化模体(见图 8) ,以此区别此概念在静态网络和时效网络中的差异,并发现在电子邮件网络和面对面接触网络中,个体接触的规则性和持续性导致了两种表示框架下流模体的不同。图 8中有10个三联图(序号1~10) 和16个三角形(序号11~26) ,深色连边对应于高流量和浅色连边对应于低流量。文献[28]分析了大量移动电话通话数据中与性别和年龄相关的时效模体,通过跟参考零模型的比较后发现时效同质性的存在:相似用户更倾向于出现在时效通讯模式中。文献[29]提出了一种混合马尔科夫链的方法检测随机时效网络中的模体结构,此方法相比于确定性检测方法具有更好的鲁棒性。
图 8 26种类型的连接三节点图,即具有3个顶点的子图[27]
社团结构是复杂网络的一个基本拓扑特性,影响网络中的信息传播,对网络中的动力学过程有着重要的意义。社团结构的检测在静态网络中已经有了大量的研究,其中包括谱平分检测法[46-47]、模块度检测法[48-49]和多片网络检测法[50]等。谱平分检测法主要采用拉普拉斯矩阵的最小非零特征值所对应的特征向量的元素的正负作为评判标准。文献[48]提出的模块度检测法是每步寻找使模块度增大最大的节点,进而得到具有最大模块度的社团分隔方法。模块度的方法也被推广到3个社团划分[49]。多片网络检测其实是模块度检测法的另一种推广[50]。然而,时效网络的社团结构划分研究却相对较少。文献[20]通过优化时效网络在多层表示框架下的模块化函数评估了已有社团检测算法的鲁棒性。文献[22]提出了社团活力的概念,表示社团结构在一个时间层内的生命强度,并可以用于描述社团结构的时间演化过程。文献[26]指出如果将时效网络用多片网络表征(见图 9,实线代表层内的连接,虚线代表层间的连接),就可以采用多片网络社团检测法进行社团划分,然而该论文中提出的多片网络社团检测法仅适用于网络片之间只有相同节点有连边的情况。文献[42]提出了利用时效网络随时间的改变量(例如连边或节点的增加或减少)和上一时刻的社团结构来判断下一时刻的社团划分方法。文献[51]提出一种基于一致性聚类的社团检测算法,不仅可以很好地发现静态网络中的社团结构,也能用于检测时效网络中的社团结构及其演化过程。文献[52]运用了非负张量分解的方法来提取时效网络中的社团结构进而追踪其时效活动模式,发现学校内的时效活动具有极强的相关性。
图 9 多层网络示意图[26]
尽管如此,中尺度层次上的时效网络的模体和社团结构的检测还处于探索阶段,许多问题仍困扰着研究学者,例如模体和社团的快速检测算法和结构特征分类等。
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复杂网络中的重要节点是指相比网络其他节点而言,能够在更大程度上影响网络的结构与功能的一些特殊节点。科学家研究时效网络的结构特征,提出各类指标的一个主要应用就是挖掘网络中的重要节点。挖掘网络中重要节点的研究受到越来越多的关注,不仅因为其重大的理论研究意义,更因为其广泛的实际应用价值。在得到网络的重要节点相关信息后,可以更加准确地预测和控制网络上的动力学过程[53],例如流行病传播中哪些节点最具有传播力;在流行病疫苗中如何考虑接种重点人群[54-55];在社会传播网络中怎样通过关键人物控制谣言的扩散[56];在对一个无标度网络的蓄意攻击中,少量重要节点被攻击就会导致整个网络瓦解[57];在商业市场中如何制定宣传策略、开拓市场;在工程中哪些节点需要优先控制等。同时已有许多通过分析网络中重要节点而取得成功的例子,如2012年美国大学生数学建模竞赛中,利用犯罪克星建立模型寻找出犯罪头目;google搜索引擎利用Pagerank算法给每个网页打分,将网页按重要性进行排序等。由于应用领域极广,且不同类型的网络中节点的重要性评价方法各有侧重,研究者从不同的实际问题出发设计出各种各样的方法包括社交网络分析、交通网络搜索、复杂网络分析等。
节点重要性评价的方法有很多种,不同的评价方法的侧重点各有不同,这些方法都是从不同的实际问题出发所设计得到的[58-60]。静态网络中已有大量参量被用来描述网络的拓扑特性,例如考虑连边关系的度参量和聚类参量;考虑节点间距离的最小路径参量、介数中心性参量和接近度中心性参量;考虑图谱特性的邻接矩阵主特征向量和拉普拉斯向量等。文献[61]从基于节点近邻的排序方法,基于路径的排序方法,基于特征向量的排序方法,基于节点移除和收缩的排序方法4个方面对静态网络中常用的30多种排序指标的计算思路、应用场景和优缺点进行了系统地比较、归纳与总结[61]。在此不再赘述。
尽管对于节点重要性排序的研究在静态网络上已经取得一定进展,但时效网络中,由于时间维度的引入,节点中心性参量的定义及排序需要重新审视和改进。经过近几年的广泛研究,人们已经发展出一套时效结构测度[1, 6, 45],如时效路径、可达性、连通性、平均等待时间、网络效率、最小生成树、邻近中心性、介数中心性和边持续模式等。时效网络有其独特的特征,对其统计特性的研究主要分为两大类。一类是在整个观察窗口内将网络作为一个整体进行研究。文献[1]将网络作为一个整体,提出了时效网络中平均时效距离等概念,并在此基础上定义了时效网络节点的接近中心度;另一类是将网络先进行切片,然后分别研究每个小网络内部的统计特性,然后再根据每个切片的特性归纳出整个网络的统计特性。文献[36-38]则是将网络进行切片,提出了在切片研究中网络的最短时效路径、接近中心性、介数中心性、聚类系数等各类统计特性,并提出了节点重要性预测以及网络切片方法等。
目前时效网络特征向量中心性的研究主要采用的方法是在初始时刻给每个节点分一个值为1的中性量值,然后在每次连接建立的时候,参加交互的个体按一定比例共享中心性量值,最后进行归一化处理[1, 61]。此方法存在的问题是这个比例值反映了最近一次有效连接对中心性量值影响的程度,是需要额外数据支持或者随机设定的。另外,某些节点在很长一段时间内没有和其他节点建立连接,会使这些节点的中心性量值远大于或远小于其他节点。文献[43]提出为使中心性量值随时间更加均匀分布,可以在每一步都进行中心性量值的归一化处理。文献[35]证明经常移动的节点很难定义它们的各种特点,但可以定义节点的中心度(temporal degree),节点的时效中心度和它引起的传播比例、或者移除它以外其他节点引起的传播比例成正比。节点时效中心度可以定位一个时间窗内的有效连边的个数,且随时间的推演不断变化,节点v∈V的时效中心度的表达式如下:
$${{D}_{i,j}}(v)=\frac{\sum\limits_{t=i}^{j}{{{D}_{t}}(v)}}{(|V|-1)(j-i)}$$ 式中,${{D}_{t}}(v)$为聚合网络${{G}_{t}}$在时间段$[i,j]$的度。同时文献[35]提出了时效接近中心度参量(temporal closeness):
$${{C}_{i,j}}(v)=\sum\limits_{i\le t <j}{\sum\limits_{u\in V\backslash v}{\frac{1}{{{\Delta }_{t,j}}(v,u)}}}$$ 式中,${{\Delta }_{t,j}}(v,u)$为节点v到节点u在时间段u的最短时效路径,如果节点[t, j]到节点v在时间段[t, j]之间不存在的最短时效路径,则定义${{\Delta }_{t,j}}(v,u)$为$\infty $。
此后,文献[40]对时效网络中的接近中心度做了研究,发现具有较高中心度的节点离其他节点越近,越是在信息传播中不依赖于其他节点。除此之外,文献[35]还提出了时效的介数中心度参量(temporal betweenness):
$${{B}_{i,j}}(v)=\sum\limits_{i\le t <j}{\sum\limits_{\begin{smallmatrix} s\ne v\ne d\in V \\ {{\sigma }_{i,j}}(s,d)>0 \end{smallmatrix}}{\frac{{{\sigma }_{t,j}}(s,d,v)}{{{\sigma }_{t,j}}(s,d)}}}$$ 式中,定义${{S}_{x,y}}(u,v)$为在时间段$[x,y]$中从源节点s到目标节点d的所有最短时效路径集合;${{S}_{x,y}}(s,d,v)$为${{S}_{x,y}}(s,d)$的子集,表示这些路径中包含节点v的所有路径。在定义时效的介数中心度的公式中${{\sigma }_{t,j}}(s,d)\equiv $ $|{{S}_{t,j}}(s,d)|$,${{\sigma }_{t,j}}(s,d,v)\equiv |{{S}_{t,j}}(s,d,v)|$。基于此,文献[18]验证了介数偏好性广泛存在于真实的时效网络中,影响着最短时效路径的长度,对于介数偏好性的忽视将得出错误的传播动力学结论。
需要说明的是,大部分时效网络的拓扑结构参量,例如时效的接近度参量和时效的介数参量,都是基于时间相关路径提出的[1, 44]。时效网络中的时间相关路径(time-respecting path)是需要遵从于不同连边的先后排序的。例如在时效网络中,信息可以从节点i经过节点k最终传到节点j,但并不代表节点i和节点j之间存在一条真实的路径,仅需要在时间维度上表明从节点k到节点j的有效连接是发生在从节点i到节点k的有效连接之后。时效路径的有效连接都是临时的,会在某个特定的时间点建立或断开,所以时间窗的选择对时效网络路径的研究至关重要。
此外,文献[16]提出了一个衡量单个交互事件的重要性测度,发现阵发活动模式导致了接触事件的重要性具有极强的异质性,且少量的重要性事件在维持时效网络的连通性和提高效率方面具有显著作用。文献[17]定义了一个新的时效鲁棒性测度,用于评估单个节点的移除对于时效连通性的影响,发现高连接节点与高影响力节点之间存在一定的相关性。文献[19]提出了一个基于随机游走的中心性测度,发现节点的稳态粒子密度依赖于等效网络的入度强度和游走者的逗留概率。文献[63]在研究时效网络可控性的同时,提出了控制中心度指标。文献[65]在静态网中基于节点边缘贡献值评价节点重要性的基础上[64],考虑网络的时间属性并提出事件相关节点感染方式,将改进算法应用到时效社交网络中。文献[23]提出一个刻画个体重要性的新指标“参与活动潜力”来替代对应聚合网络中的度中心指标,能够非常准确地预测出面对面接触网络中的中心节点。此外,他们也基于复旦大学校园内交互式无线用户数据比较了时效网络与聚合网络的可达性和路径长度分布等指标,展示了时效网络所特有的结构特征[24-25]。
这些已有的工作强调了连边的前后顺序对特征向量中心性的作用,基于其他时效特征的节点中心性指标及时效网络中重要节点挖掘的方法仍然有待深入研究。
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在时效网络理论分析处理方面,本文介绍n阶聚合网络近似理论和多层耦合网络分析方法,这两种方法一般用于描述实证网络及配置网络中的传播过程,有望从本质上理解不同时效结构-个体行为关联性在传播过程中的影响效应。
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在时效网络中,由于两个节点之间的联系不仅依赖于上一个时间片段内的信息,还可依赖于之前多个片段内的信息,因此其具有非马尔科夫特性[7],这就需要用新的方法来描述时效网络。一个简单而直接的方法是把时间表示成网络的另一个维度,即时间展开表示方式(time-unfolded representations)(见图 11) 。利用时间展开表示法,就可以将时效结构的非马尔科夫过程近似为一个马尔科夫过程。为此,首先考虑最简单的一种近似,即两个节点在这一时片段内的联系只依赖于时效网络在上一时片段和上上时片段内的连接信息。为了方便理论上的分析处理,将两个节点在一个时间片段内的连接定义为一个节点,如图 11b中一个时间片段内的一条连边可视为一个节点。如果网络的大小为N,那么可能的节点数就为$C_{N}^{2}$。如果两条连边在相邻两个时间片段内首尾相连,那么就认为这两条连边所对应的节点相连(例如在时间片段t内节点a和节点b相连,在时间片段t +1内节点b和节点点c相连,那么按照这里的定义,连接节点a和节点b以及节点cb和节点c的两条连边被视为两个节点,并且这两个节点相连)。由于上述过程将一条连边合成了一个节点以及将多条边合成了一条边,因此以这种方式所构成的网络被形象地称为聚合网络。以图 11为例,图 11a显示的是一阶聚合网络${{G}^{(1)}}$,即原始网络结构,连边上的数字表示相对应的两端节点之间连接的次数,但并没有提供具体的时效信息。图 11b是对${{G}^{(1)}}$做的时间展开表示(横坐标为时间轴),${{G}^{T}}$和${{\hat{G}}^{T}}$表示的是${{G}^{(1)}}$可能所具有的两种时效结构,显然${{G}^{T}}$和${{\hat{G}}^{T}}$包含了时效网络的所有结构信息。图 11c中,${{G}^{(2)}}$和${{\hat{G}}^{(2)}}$分别对应于${{G}^{T}}$和${{\hat{G}}^{T}}$这两种时效结构下的二阶聚合网络。由于这里只考虑了相邻的两个时间片段,因此此处为二阶聚合网络。这个方法可以方便地推广到n阶聚合网络的情形。显然,阶数越高越逼近原始的时效结构。通过这个方法,具有非马尔科夫性质的时效网络结构,就近似成了一个马尔科夫模型。如此就能利用传统的成熟的统计学理论对时效网络进行进一步分析研究了。
图 11 二阶聚合网络生成过程[7]
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此方法也是将网络分成许多时间片段,进而记录下每个时间片段的邻接矩阵${{\mathbf{A}}_{t}}$。然而为了完整地表示时效网络的结构演变以及其上的传播动力学,只定义了时间片段的邻接矩阵${{\mathbf{A}}_{t}}$是不够的。还需要考虑不同的时间片段之间的关系。这种关系可以是因果关系或者某种关联。由于这种关系发生于两个邻接矩阵之间,因此必须引入张量代数的方法来进行描述[150]。为此,本文将某个时间t的连接张量定义为$W_{\beta }^{\alpha }(t)$,这等价于之前所述的邻接矩阵${{\mathbf{A}}_{t}}$表示为四阶的含时连接张量。此外,定义不同时间片段s , t之间的相互关系张量为$C_{\beta }^{\alpha }(s,t)$。$C_{\beta }^{\alpha }(s,t)$是这个理论方法中的重要参量,其取值取决于具体所研究的问题,例如可以通过节点的某种因果性作用进行设定,或者是对实际数据作大时间尺度内的统计关联而得。在定义了$W_{\beta }^{\alpha }(t)$和$C_{\beta }^{\alpha }(s,t)$之后,可以将整个时效网络所包含的信息统一成四阶含时连接张量$M_{\beta \delta }^{\alpha \gamma }$,其满足如下关系:
$$\begin{align} & M_{\beta \delta }^{\alpha \gamma }=\sum\limits_{t,s=1}^{T}{C_{\beta }^{\alpha }(s,t){{e}^{\gamma }}(s){{e}_{\delta }}(t)}= \\ & \sum\limits_{t,s=1}^{T}{\sum\limits_{i,j=1}^{N}{{{w}_{ij}}(s,t)\varepsilon _{\beta \delta }^{\alpha \gamma }(ijst)}} \\ \end{align}$$ 式中,${{w}_{ij}}(s,t)$是$C_{\beta }^{\alpha }(s,t)$的表示矩阵元,${{e}^{\gamma }}(s)$定义为${{e}^{\gamma }}(s)$空间中的反协变正则向量,$\varepsilon _{\beta \delta }^{\alpha \gamma }(ijst)=$ ${{e}^{\alpha }}(i){{e}_{\beta }}(j){{e}^{\gamma }}(s){{e}_{\delta }}(t)$为${{R}^{N\times N\times T\times T}}$空间中正则基的四阶向量。在这样一个理论框架下,许多含时网络的统计量就可以用张量的形式方便的给出。
图 12 时效网络用耦合的多层网络表示[150]
Review on the Research Progress of the Structure and Dynamics of Temporal Networks
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摘要: 在时效网络中连边激活的时效特征能够显著影响相同时间尺度下网络系统的动力学行为,是当前网络研究的热点课题之一。该文从时效网络的建模方法、时效网络的结构特性及相关统计力学、时效网络中的传播动力学、时效网络与人类行为结合的统计特性及目前常用的处理时效网络的理论方法等多方面对时效网络的研究进展进行综述,并对目前的国内外研究现状进行分析,提出了时效网络面临的几个关键科学问题,展望了该领域未来的研究方向。Abstract: In temporal networks, the temporal structure of edge activations can remarkably affect dynamics of systems interacting through the network at the same time scale, which is one of hot research topics in complex networks. The research progress is reviewed in this paper, covering the temporal network modeling, temporal network structure and related statistical mechanics, temporal network propagation dynamics, the combination statistical characteristics of temporal network and human behaviors, as well as some theoretical analysis methods of dealing with temporal networks. In addition, some significant scientific problems are put forward by analyzing the current research situation at home and aboard. Finally, the future research direction and development trend of this field are prospected.
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Key words:
- key nodes /
- network modeling /
- network structure /
- propagation dynamics /
- temporal networks
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图 2 Time-Varying Graphs示意图[33]
图 3 社团大小随时间的演化图[34]
图 4 左图为聚合网络,右图为随时间的演化图[35]
图 5 A 、B 、C 、D四个节点随时间接触的序列线图聚合为右侧的静态权重网[1]
图 6 两类基本的时效网络表达模型[1]
图 7 用多层网络模型描述时效过程[39]
图 8 26种类型的连接三节点图,即具有3个顶点的子图[27]
图 9 多层网络示意图[26]
图 10 Email数据集中处于1.05×106~1.08×106之间某一节点随时间的阵发行为[134]
图 11 二阶聚合网络生成过程[7]
图 12 时效网络用耦合的多层网络表示[150]
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