Modeling Day-to-day Flow Dynamics on Degradable Transport Network

PLOS ONE, Dec 2016

Stochastic link capacity degradations are common phenomena in transport network which can cause travel time variations and further can affect travelers’ daily route choice behaviors. This paper formulates a deterministic dynamic model, to capture the day-to-day (DTD) flow evolution process in the presence of degraded link capacity degradations. The aggregated network flow dynamics are driven by travelers’ study of uncertain travel time and their choice of risky routes. This paper applies the exponential-smoothing filter to describe travelers’ study of travel time variations, and meanwhile formulates risk attitude parameter updating equation to reflect travelers’ endogenous risk attitude evolution schema. In addition, this paper conducts theoretical analyses to investigate several significant mathematical characteristics implied in the proposed DTD model, including fixed point existence, uniqueness, stability and irreversibility. Numerical experiments are used to demonstrate the effectiveness of the DTD model and verify some important dynamic system properties.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0168241&type=printable

Modeling Day-to-day Flow Dynamics on Degradable Transport Network

December Modeling Day-to-day Flow Dynamics on Degradable Transport Network Bo Gao 0 1 2 Ronghui Zhang 2 Xiaoming Lou 2 0 Energy Conversation and Emission Reduction Management Center of Zhejiang Provincial Communication Department , Hangzhou, Zhejiang , P.R. China , 4 School of Transportation, Southeast University , Nanjing, Jiangsu , P.R. China 1 Zhejiang Institute of Communications , Hangzhou, Zhejiang , P.R. China , 2 Research Center of Intelligent Transportation System, School of Engineering, Sun Yat-sen University , Guangzhou, Guangdong , P.R. China 2 Editor: Wen-Bo Du, Beihang University , CHINA Stochastic link capacity degradations are common phenomena in transport network which can cause travel time variations and further can affect travelers' daily route choice behaviors. This paper formulates a deterministic dynamic model, to capture the day-to-day (DTD) flow evolution process in the presence of degraded link capacity degradations. The aggregated network flow dynamics are driven by travelers' study of uncertain travel time and their choice of risky routes. This paper applies the exponential-smoothing filter to describe travelers' study of travel time variations, and meanwhile formulates risk attitude parameter updating equation to reflect travelers' endogenous risk attitude evolution schema. In addition, this paper conducts theoretical analyses to investigate several significant mathematical characteristics implied in the proposed DTD model, including fixed point existence, uniqueness, stability and irreversibility. Numerical experiments are used to demonstrate the effectiveness of the DTD model and verify some important dynamic system properties. - Data Availability Statement: Data availability statement: Experimental data appeared in the submitted manuscript can be achieved directly by running the program source code of the numerical experiment. The experiment program is written by Visual C and executed on a T2250 CPU (2.50Ghz). The Experimental data and the program source code are saved on the supporting information files "S1 Information.xlsx" and "S2 Information.txt". Funding: The authors received no specific funding for this work. Competing Interests: The authors have declared that no competing interests exist. 1. Introduction Day-to-day (DTD) traffic assignment model seems to be the most widely used approach in existing literatures to describe traveler's individual route switching behavior, and the corresponding network traffic dynamic evolution at an aggregate level. Since the early work of Horowitz [ 1 ], the field has grown to potentially encompass a rather wide range of approaches, including deterministic processes and stochastic processes ([2±6]), and in the deterministic framework, these proposed processes can be divided into more detailed categories according to different equilibrium (or convergent) points, such as user equilibrium ([7±22]), stochastic user equilibrium ([ 1 ],[22±28]), partial user equilibrium [ 29 ] and bounded rational user equilibrium ([ 30,31 ]). Readers may refer to Cantarella [22] and Watling and [32] for both synthesis and development of the dynamic evolution process of traffic flows. In existing DTD models, this adjustment process is usually demonstrated by two related traveler behavior mechanisms, including the experience learning mechanism and the route choice mechanism. In degradable transport network, travelers always suffer from within-day travel time uncertainties because of the intra-day link capacity degradations. In addition, travelers also experience day-to-day travel time variations caused by the inter-day fluctuation of traffic flow. The within-day and the day-to-day travel time variations together result in the route uncertainty or unreliability. In the context of route choice, the effect of route reliability is largely determined by traveler's risk attitude. Therefore, for a realistic representation of the DTD traffic dynamics, it is essential to contain the integration of travel time uncertainty in the experience learning mechanism, and meanwhile account for travelers' risk-taking behaviors in the route choice mechanism. The DTD models previously introduced have addressed the integration of past experiences or other information sources to estimate the perceived mean travel time. However, they do not address the updating of travel time uncertainty, nor consider travelers' risk-taking behaviors in the route choice processes. Jha et al. [ 33 ], Chen and Mahmassani [ 34 ] applied Bayesian learning model to complete the integration of travel time and its associated uncertainty. According to the learning rule governed by Bayesian theorem, these two studies only address the updating of the inherent within-day travel time uncertainty, but do not address the updating of day-to-day travel time uncertainty which is caused by the inter-day fluctuation of traffic flow. In addition, they have not considered travelers' risk-taking behaviors in the context of route choice. Risk attitudes can be captured by several theories such as prospect theory (PT) [ 35 ] or its cumulative representation (CPT) [ 36 ] and expected utility theory (EUT). In EUT, travelers are usually supposed to have exogenous risk attitudes which are reflected in the shape (concavity or convexity) of the utility function. This may be unreasonable because travelers' past travel experiences are likely to influence their risk attitudes. In contrast with EUT, PT provides an implicit way to handle with the risk attitude evolution issue by updating the locations of reference points ([37±39]). Recently, some scholars (e.g. [40±42]) applied PT or CPT to examine the role of risk attitude in travelers' DTD dynamic behaviors. Their models are potential to provide well-supported descriptive paradigm for decision making under uncertainty, but at the same time, these models adopt a quite large number of behavioral parameters which may lead to the difficulty of model calibration and validation. The main objective of this paper is to describe the aggregate network flow DTD dynamics by considering both travelers' study of uncertain travel time and their choice of risky routes. This work is mainly inspired by the objective reality that uncertainties often exist in traffic systems because of the inter-day traffic flow variations and the intra-day road capacity degradations. With the presented model, this paper also makes some efforts to examine the effects of travel time uncertainties and travelers' risk attitudes on traffic flow evolution and other dynamic system properties, particularly convergence, stability and irreversibility. In the presented DTD model, the notion of variation range is adopted to indicate travel time uncertainty information. Mathematically, this notion is expressed as the difference between the longest and the shortest travel time values. In contrast with the traditional travel time variance (or its distribution), variation range seems to be a more reasonable indicator reflecting travel time uncertainty because in the real world travelers appear more sensitive to the extreme travel time value (e.g. the longest or shortest one) than to the specific travel time distribution. In addition, in the proposed model, an endogenous risk attitude evolution schema is given to reflect that travelers constantly adjust their risk attitudes through learning their past travel experiences. This study advances previous work by making the following specific contributions. First, a simple but effective indicator about travel time, namely variation range, is used to indicate its uncertainty information. Second, the within-day and the day-to-day travel time variation ranges, which are respectively caused by the intra-day road capacity degradations and the inter-day traffic flow fluctuation, are both considered to reflect traveler's sense of route 2 / 19 unreliability or risk. Third, an endogenous risk attitude evolution schema is adopted to reflect the change of traveler's risk attitude in the context of day-to-day traffic evolution. Finally, the above ideas are integrated into a DTD model to examine their effects on the whole day-to-day behavior of traffic flows. In the next section, a new DTD model, which contains both travelers' study of uncertain travel time and their choices towards risky travel route, is proposed to describe the realistic dynamic traffic flow evolution process. In Section 3, some theoretical analyses are conducted to investigate the mathematical properties implied in the proposed DTD model. Section 4 applies the proposed model to a test network to demonstrate the effectiveness of the model and verify some important dynamic system properties such as convergence, stability and irreversibility. Section 5 concludes the paper. 2. Description of the DTD Model 2.1 Degradable Traffic Network and Relevant Notions Definition A traffic network is a directed graph (N,L) where N represents the node set and L corresponds to the link or road set. The notions that will be used in this paper are listed in Table 1. 2.2 Travelers' Perceptions of Route Travel Time and Its Variation Ranges Following the previous works (e.g. [1,4,22,23,26±28]), travelers' perceptions of mean route travel time are built up through an exponential-smoothing style of learning process, which involves a weighted combination of the perceived and actual mean time on the previous days. This learning process can be represented by the following recursion equation: P S~t‡1 ˆ a S_ tr ‡ …1 r a† S~tr ˆ a l2LLlr cl…xlt; u_lt† ‡ …1 where α(0 α 1) denotes a constant parameter (independent of t), which reflects travelers' preference between actual and expected route travel time. Actually, the exponential-smoothing filter and the preference parameter α together reflect travelers' learning mechanisms and memory characteristics about the past experiences, they may apply not only to the perceptions of mean route travel time, but also to the perceptions of travel time uncertainties. Therefore, this study uses the same exponential-smoothing filter and preference parameter to establish the updating equation of route travel time variation ranges in the following content. For the day-to-day research framework as considered in our paper, the within-day flow dynamics of the traffic system is usually neglected in existing literatures. The day-to-day dynamic model only considers the flow evolution process along the large-scale time scale `day'. The within-day realization process, on the other hand, mainly address the real-time dynamic traffic flow as the realization of the travellers' route choices on a particular day, which, in turn, results in updated information feedback to the day-to-day process. Until now, most works investigate these two dynamical processes independently, and some attempts (e.g. [43±47]) are still on the road to combine these two problems into a unified doubly dynamic traffic assignment model. In this paper, we adopt the within-day flow static assumption to eliminate the effects of within-day flow dynamics on travel time variations, and focus only on the day-to-day research framework. The within-day static assumption allows a mathematical formalization that is easier to manage in theoretical terms. From this point of view, the only cause leading to withinday link travel time variations …dclt† is the intra-day link capacity fluctuations. Mathematically, 3 / 19 dclt can be expressed as the difference between cl…xlt; ult† and cl…xlt; ult†: dclt ˆ cl…xlt; ult† cl…xlt; ult†; 8l 2 L: The actual variation range of within-day route travel time, namely dStr, can then be obtained immediately according to the link-route topological relation: X X dStr ˆ St r Str ˆ lLlr cl…xlt; ult† P ˆ lLlr …cl…xlt; ult† cl…xlt; ult†† ˆ lLlr cl…xlt; ult† P lLlr dclt; 8r 2 Rod: …2† …3† Λlr Str Str St r S__t r S~~t r dStr dS~~tr DStr DS~~tr rtod htr Ptrod 4 / 19 Then the perceived variation range …dS~tr‡1† can be achieved by applying the exponentialsmoothing updating process: dS~tr‡1 ˆ a dStr ‡ …1 a† dS~tr ˆ a P lLlr dclt ‡ …1 a† dS~tr; 8r 2 Rod: Besides of the within-day variation dStr, travelers will also perceive day-to-day travel time fluctuation …DStr† from their past experiences. Mathematically, DStr can be represented as the difference of the mean route travel time on two consecutive days: P P DStr ˆ jS_ tr S_ tr 1j ˆ j lLlr cl…xlt; u_lt† lLlr cl…xlt 1; u_lt 1†j; 8r 2 Rod: In Eq 5, the absolute value sign | | is applied to guarantee the non-negativity of DStr. Note that the effect of intra-day link capacity fluctuation is not reflected in DStr because it has been considered by the within-day variation dStr. Then the perceived day-to-day variation range can be updated still through the exponential-smoothing type recursion equation: P P DS~tr‡1 ˆ a j lLlr cl…xlt; u_lt† lLlr cl…xlt 1; u_lt 1†j ‡ …1 a† DS~tr; 8r 2 Rod: dS~tr‡1 and DS~tr‡1 together present the integrated description of travel time uncertainties. In this paper, the integrated travel time variation range is defined as the sum of dS~tr‡1 and DS~tr‡1, its updating equation can be easily achieved by combining Eqs 4 and 6 together: dS~tr‡1 ‡ DS~tr‡1 ˆ a …dStr ‡ DStr† ‡ …1 P P ˆ a … lLlr dclt ‡ j lLlr cl…xlt; u_lt† a† …dS~tr ‡ DS~tr† P ‡ DS~tr†; 8r 2Rod: lLlr cl…xlt 1; u_ lt 1†j† ‡ …1 a† …dS~tr …4† …5† …6† …7† 2.3 Travelers' Perception of Systematic Disutility Associated to Every Route To model travelers' route choice and adjustment, the key is to calculate the systematic disutility of every alternative route. Traditionally, the systematic disutility is usually defined as an affine transformation of the mean route travel time without consideration of travel time variations. In this section, the traditional disutility representation is modified to reflect the effect of travel time uncertainty. This modification is derived under some mild assumptions as stated below. Assumption I: All the possible values of the perceived route travel time are distributed continuously in an interval whose length is defined by the integrated variation range …dS~tr‡1‡ DS~tr‡1†; the perceived mean route travel time …S~tr‡1† is located at the middle point of this interval, and the other route travel time are distributed symmetrically on the left and right sides of S~tr‡1. Assumption II: If the integrated variation range is zero, then the systematic disutility …htr‡1† is equal to the perceived mean route travel time …S~tr‡1†. For ease of description, we define η(x) and μ(x), respectively, to represent the disutility function and the probability density function (PDF) of the perceived route travel time x. Assumption reveals' the following conditions: x 2 ‰S~t‡1 r 0:5…dS~tr‡1 ‡ DS~tr‡1†; S~tr‡1 ‡ 0:5…dS~tr‡1 ‡ DS~tr‡1†Š; m…x† ˆ m…2S~tr‡1 x† and @m…x†=@x ˆ @m…2S~tr‡1 x†=@x (the symmetrical distribution). In addition, according to the nature of PDF, if dS~tr‡1 ‡ DS~tr‡1 ˆ 0, then htr‡1 ˆ S~tr‡1, which actually presents an anchor point of the applied disutility function, namely Z…S~tr‡1† ˆ S~tr‡1. On 5 / 19 the other hand, if dS~tr‡1 ‡ DS~tr‡1 > 0, the systematic disutility htr‡1 can be represented as follows: htr‡1 ˆ ˆ ˆ ˆ Z S~t‡1‡0:5…dS~tr‡1‡DS~tr‡1† r S~t‡1 0:5…dS~tr‡1‡DS~tr‡1† r Z S~t‡1 r S~t‡1 0:5…dS~tr‡1‡DS~tr‡1† r Z S~t‡1 r S~t‡1 0:5…dS~tr‡1‡DS~tr‡1† r Z S~t‡1 r S~t‡1 0:5…dS~tr‡1‡DS~tr‡1† r Z…x† m…x†dx Z…x† m…x†dx ‡ Z…x† m…x†dx Z S~t‡1‡0:5…dS~tr‡1‡DS~tr‡1† r S~t‡1 r Z S~t‡1 0:5…dS~tr‡1‡DS~tr‡1† r S~t‡1 r …Z…x† ‡ Z…2S~tr‡1 x†† m…x†dx; 8r 2 Rod: Z…x† m…x†dx Z…2S~tr‡1 x† m…2S~tr‡1 x†dx …8† Although the value of htr‡1 cannot be achieved from Eq 8 since the specific formulations of η (x) and μ(x) are both not given in this study, however, some fundamental properties implied in htr‡1 can still be analyzed by considering the conditions given by Assumptions I and II as well as travelers' risk attitudes. These properties are essential to establish simplified expression and then to calculate approximate value of the systematic disutility htr‡1. Consider the first situation in which travelers are assumed to be risk averse. According to EUT, risk aversion is associated with a convex disutility function. The convexity implies the following inequality: Z…x† ‡ Z…2S~tr‡1 x† > 2S~tr‡1; …x 6ˆ S~tr‡1†: With this inequality, the systematic disutility htr‡1 given by Eq 8 can be compared with the perceived mean route travel time …S~tr‡1† as follows: R S~t‡1 htr‡1 ˆ S~trr‡1 0:5…dS~tr‡1‡DS~tr‡1†…Z…x† ‡ Z…2S~tr‡1 x†† m…x†dx > S~tr‡1; 8r 2 Rod: …9† In addition, when x 2 …S~tr‡1 0:5…dS~tr‡1 ‡ DS~tr‡1†; S~tr‡1†, the convexity of the disutility function also implies the increase velocity of Z…2S~tr‡1 x† towards its right side is larger than the decrease velocity of η(x) towards the left side.This means when the integrated variation range …dS~tr‡1 ‡ DS~tr‡1† increases, the systematic disutility htr‡1 perceived by travelers will increase simultaneously. Through the above analysis, we can find the first property implied in the systematic disutility which is associated with risk aversion. This property is stated as below. Property I: Under the conditions of Assumptions I and II, if travelers are assumed to be risk averse, then their perceived systematic disutility has a larger value than the perceived mean route travel time, this disutility is positively associated with their perceived travel time variation ranges. Besides of risk aversion, risk proneness and risk neutrality can also be reflected in EUT, which correspond respectively to concave and linear disutility functions. In both situations, the previous analysis can still be carried out to study the other properties of the systematic disutility. These properties are summarized as below. Property II: Under the conditions of Assumptions I and II, if travelers are risk prone, then their perceived systematic disutility has a smaller value than the perceived mean route travel time, which is negatively associated with the perceived travel time variation ranges. 6 / 19 Property III: Under the conditions of Assumptions I and II, if travelers are risk neutral, then their perceived systematic disutility is equal to the perceived mean route travel time, regardless of their perceived travel time variation ranges. After acquiring Properties I~III, it is time now to establish a simplified expression of htr‡1 reflecting travelers' perception of route systematic disutility. In this study, htr‡1 is formulated as the weighted sum of the perceived mean route travel time and its associated variation ranges: htr‡1 ˆ S~tr‡1 ‡ rto‡d1 …dS~tr‡1 ‡ DS~tr‡1†; 8r 2 Rod; …10† Where rto‡d1 is defined as the risk attitude parameter, its value range is set as [−ρmax,ρmax]. According to Properties I~III, a positive value of rto‡d1 indicates risk aversion, while risk proneness is associated with a negative rt‡1, and when travelers are risk neutral, rto‡d1 is equal to zero. od In reality, different travelers always have different risk attitudes, this suggests the risk attitude parameter should be defined at the individual level. However, defining specific risk attitude for every traveler is a tedious work, which may greatly hinder the execution efficiency of the proposed DTD model. The focus of this paper is not to study traveler's individual behavior, but to investigate the network traffic dynamic evolution at an aggregate level. Therefore, the unified parameter rto‡d1 in Eq 10 can be treated as an aggregated form of the realistic travelers' risk attitudes. 2.4 The Evolution of Travelers' Risk Attitudes Most of the existing studies concerning risk-taking behaviors focus only on static decision scenario in which travelers' risk attitudes are exogenous and changeless, the suitability extension to consider endogenous risk attitude evolution in the dynamical or time-varying environment is quite lacking. Barkan and Busemeyer [ 37 ] examined decision makers' risk attitude change in a sequential two-gamble scenario, and found risk prone after an anticipated loss while risk aversion after an anticipated gain, which could be explained by the reference point changes in PT or CPT. In the DTD traffic dynamics, travelers can get their travel time saving (namely travel gain), if the actual travel time cost they experienced on the current day is shorter than the perceived travel time (reference point) they estimated on the previous day. As a result, these travelers will show risk attitude change trend towards risk aversion. Conversely, travelers learn loss if the experienced travel time is longer than the perceived one, and in this situation, travelers will behave attitude change towards risk proneness. According to this rule, an updating equation about the risk attitude parameter …rtod† is proposed to reflect the endogenous risk attitude evolution schema: where ztod is defined to represent travel time saving or losing perceived by travelers on day t, its value is calculated as …Pr2Rod S_ tr frt Pr2Rod S~tr frt 1†=qod. The constant parameter σ(σ 0) in Eq 11 is defined to represent travelers' sensitivity to their travel time saving or losing. The updating process of rto‡d1 defined by Eq 11 is depicted in Fig 1. In Fig 1, all possible values of rto‡d1 are located in the range of −ρmax to ρmax. When ztod < 0 (in the domain of gain), traveler's risk attitude act out evolution trend towards risk aversion, which results in the increase of the risk attitude parameter, namely rto‡d1 > rtod. On the contrary, when ztod > 0 (in the domain of loss), travelers behave attitude change towards risk proneness, which leads to the risk attitude parameter decrease …rto‡d1 < rtod†. 7 / 19 Fig 1. Updatings of the risk attitude parameter ρto‡d1 with different values of parameter σ. A larger σvalue corresponds to a larger change rate of rtod. 2.5 The Evolution of Link Traffic Flow In reality, because of traffic congestion and perception deviation, some random residuals are introduced into the systematic disutility htr‡1 to influence travelers' route choices. If the random residuals are assumed to be independent over time scales, OD pairs and routes, and furthermore, if they are identically distributed as Gumbel random variables with zero mean, then travelers' route choice probability can be given by a Logit model: Prto‡d1 ˆ P expf y htr‡1g k2Rod expf y htk‡1g ; 8r 2 Rod: …12† The positive dispersion parameter θ in Eq 12 reflects the degree of familiarity with conditions by travelers, a higher θ-value means a smaller perception variation. With travelers' route choice probability, the evolution of link traffic flow can then be formulated by considering inertial travelers: 3. Theoretical Analysis of the Proposed DTD Model This section provides some theoretical analysis to investigate several important properties implied in the proposed dynamic system. To facilitate the analysis, the DTD model presented 8 / 19 in Section 2 is reformulated as below. S~t‡1 ˆ a r P dS~tr‡1 ˆ a P DS~tr‡1 ˆ a j lLlr cl…xlt; u_ lt† lLlr …cl…xlt; ult† P P l2LLlr cl…xlt; u_ lt† ‡ …1 3.1 Fixed Point (FP) of the DTD Evolution Process and Equilibrium State FP of the DTD dynamic process (14) is obtained from conditions S~tr‡1 ˆ S~tr ˆ S~r , dS~tr‡1 ˆ dS~tr ˆ dS~r , DS~tr‡1 ˆ DS~tr ˆ DS~r , rto‡d1 ˆ rtod ˆ rod and xlt‡1 ˆ xlt ˆ xl , thus: P S~r ˆ lLlr cl…xl ; u_lt†; 8r 2 Rod; …14† …15† …16† …17† …18† …19† Obviously, FP described by Eq 19 is equivalent to the well-known SUE state, which is extensively reviewed in the literatures. It is worth noting that, fixed-point attractor of the proposed dynamic process not only depend upon the route systematic disutility hr but also upon the route choice behavior and the various kind of parameters (including α, β, σ and θ) adopted in this model. The implicit relation between model parameters α, β and fixed-point attractor of the proposed dynamic process, in fact, is determined by the risk attitude evolution schema …rto‡d1† defined by Eq 11. This schema indicates that the endogenous risk attitude is influenced by travelers' perceived travel time saving ztod, which is further influenced by the model parameters α and β since the value of ztod is calculated as …Pr2Rod S_ tr frt Pr2Rod S~tr frt 1†=qod. Furthermore, the evolution of rtod defined by Eq 11 is an irreversible process (see Section 3.4) which means the latest risk attitude parameter rt‡1 will change under any small fluctuation od of network traffic flow state. Under this situation, the stable risk attitude rod in Eq 18 is not fixed, it is determined by the past system evolution process and therefore is influenced by parameters α and β. The risk attitude evolution schema (Eq 11) also implies the stable risk attitude rod is influenced by parameter σ. In addition, the Logit model defines the effect of parameter θ on the ultimate route choice results. As a consequence, fixed-point attractor of the proposed dynamic process is influenced together by the model parameters α, β, σ and θ. 3.2 FP Existence Sufficient conditions for fixed-point existence can be easily derived through Brouwer's fixed point theorem, requiring continuity of all involved functions. Note that fixed-point condition presented by Eq 19 actually defines a map of xl to itself. Then fixed-point existence just needs the self-map about xl to be continuous. The Logit model (12) ensures traveler's route choice probability is continuous with respect to route systematic disutility. In addition, if the separable link travel time function clt…xlt; ult† adopted in the system is assumed to be continuous, then the route systematic disutility is also continuous with respect to the link flow. This contributes to the continuous self-map of xl and therefore the FP existence. 3.3 FP Uniqueness FP uniqueness is an attractive character of traffic network model, it has been extensively studied in existing literatures ([ 22,27,48 ]). For the dynamic evolution process as considered in this paper, this issue can be analyzed by investigating the monotonicity of the self-map about xl . In DTD system (14), the route choice probability is always non-increasing with respect to the adopted route systematic disutility hr . However, the value of hr usually cannot be assured to be monotone strictly increasing with respect to the link traffic flow xl . As a result, the FP uniqueness cannot be guaranteed. In the context of DTD dynamics, the un-uniqueness of FP or equilibrium state will give rise to irreversibility issue as discussed below. 3.4 FP Stability and Irreversibility Stability is an important property of a dynamical model for its applicability in practice ([1,4,7,10,22,23,25,31,48±52]). A FP is (asymptotically) stable if from any (sufficiently close) starting state the system state tends to the fixed-point as t tends to infinity. If the value of the sensitivity parameter σ in Eq 11 is positive, then the risk attitude parameter rtod is an unstable attribute against the stability of the fixed-point in dynamic system (14). In other words, any small fluctuation of network traffic flow state will cause the change of rtod. Obviously, the evolution of rtod defined by Eq 11 is an irreversible process which means the latest risk attitude parameter rt‡1 will never return to the original value rtod by removing the fluctuation. Under od this situation, the change of route systematic disutility is also irreversible and thus the fixedpoint of the dynamic system is not asymptotically stable. On the other hand, consider the situation in which the sensitivity parameter σ is equal to zero-value. Substituting σ = 0 into Eq 11 to get rto‡d1 ˆ rtod ˆ ˆ ro0d, this actually assumes exogenous or constant risk attitudes for travelers. In this situation, the FP stability is also significantly affected by the parameters (including α, β, σ and θ) adopted in DTD model (14). To investigate FP stability of a deterministic, discrete-time dynamic model, a common method is to conduct a spectral analysis about the Jacobian matrix of the transition process contained in the underlying DTD evolution system. We here give a weaker stability condition by assigning sufficiently small values for parameters α, β and θ(σ = 0). A closely related concept to FP instability is its irreversibility. When a traffic network is disturbed by some fluctuations, its flow pattern may deviate from the original FP and evolve to another new equilibrium state. A FP or equilibrium state is said to be irreversible if it cannot be restored by revoking the fluctuation. It should be pointed out that the existence of multiple equilibria is necessary for modelling irreversibility. In the proposed DTD model (14), the irreversibility phenomenon or multiple equilibria is mainly caused by the irreversible evolution process of the risk attitude parameter rtod defined by Eq 11. In the existing literatures, non-separable link (or path) cost function or bounded rationality behavior model is applied to describe 10 / 19 the irreversibility issue ([ 30,49 ]). This paper provides a new method to model this problem by considering travelers' risk-taking behaviors. 4. Numerical Examples on a Test Network The proposed DTD model (14) is applied to a three-by-three grid network with nine nodes and twelve links, whose topology is illustrated in the left side of Fig 2. In the numerical example, the link travel time function is of the BPR type: " # cl xlt; ult ˆ clfree 1 ‡ 0:15 uxltlt 4 ; l ˆ 1; 2; 12: The values of the free flow travel time clfree and the mean link capacity u_lt are given in the right side of Fig 2. In this example, the stochastic road capacity is assumed to keep a same fluctuation range for every day, ult 2 ‰0:8 u_lt; 1:2 u_ltŠ, thus ult ˆ 0:8 u_lt, ult ˆ 1:2 u_lt. One OD pair from node 1 to node 9 is considered in this network. Clearly, there are 6 routes connecting this OD pair. The daily traffic demand of this OD pair is assumed to be 500. In all test scenarios, ρmax is set as 0.8. 4.1 The Effects of Parameters α, β, σ and θ on the Evolution System With different combinations of parameters α, β, σ and θ, the DTD system (14) will exhibit different evolution processes about link (or route) flow patterns. Firstly, the DTD model is executed repeatedly according to different values of α, β and θ, and the other parameters are kept unchanged as: r019 ˆ 0:2 (risk averse), σ = 0.9. For graph simplicity, only the flow on link 2 is adopted to illustrate the evolution of the dynamic system, as shown in Fig 3. Fig 3(A) indicates a larger α-value will contribute to a faster process for the dynamic system to reach a steady state. It also demonstrates that under the condition of endogenous risk attitudes, the preference parameter α will significantly affect the fixed-point attractor of the evolution process, this is because α can influence the past travel experiences and further affect travelers' perception of risks. In Fig 3(B), the inertial parameter β shows similar effect as the parameter α. In addition, the comparison of Fig 3(A) and 3(B) shows that, α has greater impact Fig 2. Illustration of the experiment network. The left part shows network topological structure, and the right part shows link parameters. 11 / 19 Fig 3. Evolution of the system with endogenous risk attitudes (σ = 0.9). Fig 3(a)~3(d) shows the influences of parameters α, β and θ on both steady state and evolution process of the dynamic system. than β on the FP of the system. Fig 3(C) shows the dispersion parameter θ can also greatly influence the ultimate steady state, and a smaller θ-value is corresponding to a faster evolution process. In Fig 3(D), six groups of parameters α, β and θ are adopted to examine their combined effects on the evolution system. The result reveals the larger values of α, β and smaller value of θ together help to accelerate the convergence of the system, a smaller α or β or θ will lead to a more smooth evolution process, while a combination of larger α, β and θ may cause greater fluctuations. Next, consider the situation that travelers have constant risk attitudes, this can be realized by set σ = 0. In this case, the parameters α and β can only influence travelers' past experiences through affecting the evolution process of the dynamic system, but cannot affect travelers' perception of risks since the risk attitude parameter are assumed to be constant. As a result, these two parameters will not affect the FP of the dynamic system. For the dispersion parameter θ, it contributes directly to the randomness of travelers' route choice behaviors, thus it has an inherent influence on the FP no matter the risk attitude is constant or not. These analysis results can be verified by conducting a similar numerical experiment. To avoid redundancy, these numerical results are omitted here. 12 / 19 4.2 The Effects of Travelers' Risk-taking Behaviors on Fluctuation and Evolution Convergence Compared with the traditional DTD model, traveler's risk-taking behavior is an additional component considered by our dynamic system to influence the whole DTD network traffic evolution process. In this subsection, a numerical experiment is designed to investigate the effects of risky route choices on fluctuation and evolution convergence of the dynamic system. P The fluctuation function Yt ˆ ljxlt xlt 1j is defined to represent the aggregated link flow variation between two successive days. Obviously, when network traffic flow reach a FP or equilibrium state, Θt = 0. Suppose all links on the network suffer 70% mean capacity reductions on day 60 and recover to normal on day 81. Firstly, consider the situation that travelers' risk attitudes are changeless (σ = 0), the DTD model (14) is executed respectively according to different initial risk attitude parameters. The other parameters are given as: α = β = θ = 0.3. The evolutions of Θt for this situation are shown in Fig 4(A) and 4(B). Whether the link capacities decrease (occur in day 60) or increase (occur in day 81), travelers' risk aversion route choice behaviors are found to cause some additional flow fluctuations. Fig 4(A) indicates that a more sharp risk aversion attitude (namely a larger r019) leads to some greater fluctuations of the dynamic system. The observations achieved from risk aversion situation are significantly different from that appearing in risk proneness case, as shown in Fig 4 (B). It can be found that in a reasonable bound, travelers' risk proneness route choices can contribute to a smoother DTD evolution process. Beyond this bound, however, some excessive risk proneness route choices made by travelers (e.g. r019 ˆ 0:65) will result in greater network flow fluctuation and slower convergence process. By relaxing the assumption σ = 0, this experiment can also be used to investigate travelers' risky behaviors with endogenous risk attitudes. Fig 4(C) presents the evolutions of Θt according to a same initial risk attitude (r019 ˆ 0) and four different sensitivity parameters σ. The associated risk attitude updating processes are shown in Fig 4(D). In Fig 4(C), the effect of σ reflected in the capacity reduction period is found different from that appeared in capacity restoration. When link capacity reductions occur, a larger σ can lead to small flow fluctuation on the early days but slower convergences for later days. And when link capacities recover to normal, some greater fluctuations but faster convergences are associated to larger σ. This observation can be explained through studying the updating of rt19 presented in Fig 4(D). Obviously, link capacity reductions intensify flow congestions and further cause travel losing perceived by travelers, this results in the evolution of risk attitude towards proneness, namely the decreasing of rt19. After capacity reductions, a large σ permits rt19 to reach a preferable value (e.g. -0.4) quickly, which benefits the smooth-evolution of the dynamic system on the early days. On the subsequent days, however, a large σ may also push rt19 to reach a too small value which reflecting excessive risk proneness behaviors, this certainly give rise to greater fluctuations and slower convergences. For the situation of capacity restoration, a similar analysis can be applied to explain the result appearing in Fig 4(C). 4.3 The Effects of Travelers' Risk-taking Behaviors on FP Stability and Irreversibility As introduced in the previous Subsection 3.4, the fixed-point stability or reversibility can be regarded as an indicator evaluating road network resilience against fluctuations. In this numerical experiment, some reductions of mean link capacity are still introduced into the DTD model to reflect the external disturbances of the traffic network. A positive-value and a 13 / 19 Fig 4. The effects of traveler risk-taking behaviors on system evolution processes. Fig 4(a) and 4(b) compare the effect difference between risk aversion and risk proneness attitudes. Fig 4(c) and 4(d) show the influences of parameter σ on both fluctuation function Θt and endogenous risk attitude rt19. zero-value are respectively assigned to the sensitivity parameter σ to reflect two types of travelers' risk attitude evolution schema. The initial risk attitude parameter r019 is set as 0.3 (risk averse). On two time periods respectively from the day 40 to 50 and the day 100 to 110, all the links are assumed to suffer 50% mean capacity reductions, and outside of these two periods, these link capacities all restore to their original values. Throughout the second experiment, the following model parameters are used and kept fixed: α = β = θ = 0.6. Under this situation, the DTD model (14) is executed respectively according to different risk attitude evolution schemas. The link flow evolution trajectories and the corresponding FPs are shown in Fig 5. Fig 5(A) indicates that when travelers' risk attitudes are endogenous, their route choice behaviors will cause the FP instability. That is, any fluctuation of link capacity will give rise to the deviation from the original FP, and drive the dynamic system to reach a new equilibrium state but not the original one even though the changed link capacities are revoked, this in fact 14 / 19 Fig 5. Comparison of effects between two different risk attitude evolution schemas. Fig 5(a) corresponds to the case of endogenous risk attitudes, and Fig 5(b) corresponds to the case of exogenous risk attitudes. corresponds to the irreversibility. In Fig 5(B), however, capacity reduction and restoration are found only to cause some fluctuations on the evolution process but not to change the ultimate equilibrium state, which means, under the situation of exogenous or constant risk attitudes, FP of the DTD model is stable. Note that this stability is only satisfied in the attraction domain of the FP. This is because the route systematic disutility function usually cannot be assured to be monotone strictly increasing with respect to the link traffic flow. And as a result, the FP uniqueness cannot be guaranteed. Therefore the FP only meets, strictly speaking, the asymptotically stability condition in this situation. Due to the short of empirical data, we conducted numerical experiments only on a simple grid network in this section. A real transport network is usually not so regular and its topological structure is more complicated. It is meaningful to test the proposed model on a large-scale 15 / 19 real transport network. Meanwhile, a real transport network usually contains multiple travel OD pairs and a larger number of links. This means the route-based flow assignment approach, which is defined by the Logit model in this paper, may become invalid since the number of feasible routes will increase exponentially. Therefore, it is also necessary to design a more effective method for executing the proposed DTD model under the situation of real transport network. We leave these researches to our future work. Experimental data appeared in the above-mentioned figures can be achieved directly by running the program source code of the numerical experiment. In this paper, the experiment program is written by Visual C and executed on a T2250 CPU (2.50Ghz). The Experimental data are saved on ªS1 Information. Experimental data for Figs 3, 4 and 5.º The program source code are saved on ªS2 Information. Program source code for Figs 3, 4 and 5.º 5. Conclusion and Future Work This paper aims to model DTD flow dynamics on degradable transport network by considering both travelers' study of uncertain travel time and travelers' choice of risky routes. The notion of variation range is adopted to represent travelers' perceptions of travel time uncertainty. In addition, an endogenous risk attitude evolution schema is adopted to reflect the change of traveler's risk attitude in the context of DTD traffic dynamics. The uncertain route travel time and the risk attitude parameter are both integrated into a unified systematic disutility function to reflect travelers' perception of route attractiveness. These route disability values are substituted into a Logit model to describe travelers' stochastic route choice behaviors. This paper also makes some effects to investigate several mathematical properties implied in the proposed DTD model. Numerical results obtained from a test network verify that some moderate risk proneness route choices made by travelers are beneficial to a smoother DTD evolution process, while risk aversion behaviors as well as excessive risk proneness route choices will both give rise to greater fluctuation and slower convergence of the dynamic system. In addition, when travelers' risk attitudes are endogenous, their DTD dynamic route adjustment behaviors will indeed lead to FP instability and irreversibility. Although we focus on transport network in this study, our research may also benefit other relevant fields such as traffic dynamics on complex networks (e.g. [53±56]). For the proposed DTD model, quite a number of parameters are adopted to influence its dynamic evolution trajectory. Calibration of these parameters is worth of further research effort. In this paper, a simple update Eq 11 is formulated to reflect the endogenous risk attitude evolution schema, but it may not conform to the actual case. Therefore, it is meaningful to design more realistic formulations reflecting travelers' risk attitude changes in the future work. Given that the dynamic model may have multiple equilibria, it is also interesting to analytically derive the sufficient condition that assures the asymptotically stability of each fixed point. Supporting Information S1 Information. Experimental data for Figs 3, 4 and 5. (XLSX) S2 Information. The program source code for Figs 3, 4 and 5. (TXT) Acknowledgments The comments provided by two anonymous referees are much appreciated. 16 / 19 Author Contributions Conceptualization: BG RZ. Data curation: BG RZ. Formal analysis: XL RZ. Investigation: RZ BG. Methodology: RZ XL. Resources: XL. Software: XL BG. Supervision: BG RZ XL. Validation: RZ BG XL. Visualization: RZ BG XL. Writing ± original draft: RZ XL. Writing ± review & editing: XL BG. 17 / 19 18 / 19 1. Horowitz JL . The stability of stochastic equilibrium in a two link transportation network . Transportation Research Part B: Methodological . 1984 ; 18 : 13 ± 28 . 2. Cascetta E. A stochastic process approach to the analysis of temporal dynamics in transportation networks . Transportation Research Part B: Methodological . 1989 ; 23 :1± 17 . 3. Hazelton ML . Day-to-day variation in Markovian traffic assignment models . Transportation Research Part B: Methodological . 2002 ; 36 : 637 ± 648 . Watling DP , Hazelton ML . The dynamics and equilibria of day-to-day assignment models . Networks and Spatial Economics , 2003 ; 3 : 349 ± 370 . 5. Parry K , Watling DP , Hazelton ML . A new class of doubly stochastic day-to-day dynamic traffic assignment models . EURO Journal on Transportation and Logistics . 2016 ; 5 :5± 23 . Watling DP , Cantarella GE . Model representation and decision-making in an ever-changing world: the role of stochastic process models of transportation systems . Networks and Spatial Economics . 2015 ; 15 : 843 ± 882 . 7. Smith M. The stability of a dynamic model of traffic assignment: an application of a method of Lyapunov . Transportation Science. 1984 ; 18 : 259 ± 304 . 8. Friesz TL , Bernstein D , Mehta MJ , Tobin RL , Ganjalizadeh S . Day-to-day dynamic network disequilibria and idealized traveler information systems . Operations Research . 1994 ; 42 : 1120 ± 1136 . 9. Smith M , Wisten MB . A continuous day-to-day traffic assignment model and the existence of continuous dynamic user equilibrium . Annals of Operations Research . 1995 ; 60 : 59 ± 79 . 10. Zhang D , Nagurney A . On the local and global stability of a travel route choice adjustment process . Transportation Research Part B: Methodological . 1996 ; 30 : 245 ± 262 . 11. Zhang D , Nagurney A , Wu J . On the equivalence between stationary link flow patterns and traffic network equilibria . Transportation Research Part B: Methodological . 2001 ; 35 : 731 ± 748 . 12. Cho HJ , Hwang MC . A stimulus-response model of day-to-day network dynamics . IEEE Transactions on Intelligent Transportation Systems . 2005 ; 6 : 17 ± 25 . 13. Cho HJ , Hwang MC . Day-to-day vehicular flow dynamics in intelligent transportation networks . Mathematical and Computer Modelling . 2005 ; 41 : 501 ± 522 . 14. Yang F , Zhang D . Day-to-day stationary link flow pattern . Transportation Research Part B: Methodological . 2009 ; 43 : 119 ± 126 . 15. He XZ , Guo XL , Liu HX . A link-based day-to-day traffic assignment model . Transportation Research Part B: Methodological . 2010 ; 44 : 597 ± 608 . 16. Smith M , Mounce R . A splitting rate model of traffic re-routing and traffic control . Transportation Research Part B: Methodological . 2011 ; 45 : 1389 ± 1409 . 17. He XZ , Liu HX . Modeling the day-to-day traffic evolution process after an unexpected network disruption . Transportation Research Part B: Methodological . 2012 ; 46 : 50 ± 71 . 18. Guo RY , Yang H , Huang HJ . A discrete rational adjustment process of link flows in traffic networks . Transportation Research Part C: Emerging Technologies . 2013 ; 34 : 121 ± 137 . 19. Smith M , Hazelton ML , Lo HK , Cantarella GE , Watling DP . The long term behavior of day-to-day traffic assignment models . Transportmetrica A: Transport Science . 2014 ; 10 : 647 ± 660 . 20. Guo RY , Yang H , Huang HJ , Tan Z . Link-based day-to-day network traffic dynamics and equilibria . Transportation Research Part B: Methodological . 2015 ; 71 : 248 ± 260 . 21. He X , Peeta S. A marginal utility day-to-day traffic evolution model based on one-step strategic thinking . Transportation Research Part B: Methodological . 2016 ; 84 : 237 ± 255 . 22. Cantarella GE , Cascetta E . Dynamic processes and equilibrium in transportation networks: towards a unifying theory . Transportation Science . 1995 ; 29 : 305 ± 329 . Transportation Research Part B: Methodological . 1999 ; 33 : 281 ± 312 . 24. Huang HJ , Liu TL , Yang H. Modeling the evolutions of day-to-day route choice and year-to-year ATIS adoption with stochastic user equilibrium . Journal of Advanced Transportation . 2006 ; 42 : 111 ± 127 . 25. Bie J , Lo HK . Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation . Transportation Research Part B: Methodological . 2010 ; 44 : 90 ± 107 . 26. Cantarella GE . Day-to-day dynamic models for intelligent transportation systems design and appraisal . Transportation Research Part C: Emerging Technologies. 2013 ; 29 : 117 ± 130 . 27. Cantarella GE , Watling DP . Modelling road traffic assignment as a day-to-day dynamic, deterministic process: a unified approach to discrete- and continuous-time models . EURO Journal on Transportation and Logistics . 2016 ; 5 : 69 ± 98 . 28. Lou XM , Cheng L , Chu ZM . Modeling travelers' en-route path switching in a day-to-day dynamical system . Transportmetrica B: Transport Dynamics . 2016 Mar: 1 ± 25 . 29. Jin WL . A dynamical system model of the traffic assignment problem . Transportation Research Part B: Methodological . 2007 ; 41 : 32 ± 48 . 30. Guo XL , Liu HX . Bounded rationality and irreversible network change . Transportation Research Part B: Methodological . 2011 ; 45 : 1606 ± 1618 . 31. Di X , Liu HX , Ban X , Yu JW . Submission to the DTA 2012 special issue: On the stability of a boundedly rational day-to-day dynamic . Networks and Spatial Economics . 2015 ; 15 : 537 ± 557 . Watling DP , Cantarella GE . Modelling sources of variation in transportation systems: theoretical foundations of day-to-day dynamic models . Transportmetrica B: Transport Dynamics . 2013 ; 1 :3± 32 . 33. Jha M , Madanat S , Peeta S. Perception updating and day-to-day travel choice dynamics in traffic networks with information provision . Transportation Research Part C: Emerging Technologies. 1998 ; 6 : 189 ± 212 . 34. Chen R , Mahmassani H . Learning and travel time perception in traffic networks . Transportation Research Record: Journal of the Transportation Research Board . 2004 ; No. 1894 : 209 ± 221 . 35. Kahneman D , Tversky A . Prospect theory: an analysis of decisions under risk . Econometrica . 1979 ; 47 : 263 ± 291 . 36. Tversky A , Kahneman D. Advances in prospect theory: cumulative representation of uncertainty . Journal of Risk and Uncertainty . 1992 ; 5 : 297 ± 323 . 37. Barkan R , Busemeyer JR . Changing plans: dynamic inconsistency and the effect of experience on the reference point . Psychonomic Bulletin &Review . 1999 ; 6 : 547 ± 554 . 38. Xu H , Lou Y , Yin Y , Zhou J. A prospect-based user equilibrium model with endogenous reference points and its application in congestion pricing . Transportation Research Part B: Methodological . 2011 ; 45 : 311 ± 328 . Transportation Research Part C: Emerging Technologies . 2013 ; 35 : 156 ± 179 . 40. Chen R , Mahmassani H . Learning and risk attitudes in route choice dynamics . The Expanding Sphere of Travel Behavior Research: Selected Papers from the 11th International Conference on Travel Behavior Research . 2009 . 41. Xu HL , Lam WHK , Zhou J . Modeling road users' behavioral change over time in stochastic road networks with guidance information . Transportmetrica B: Transport Dynamics . 2014 ; 2 : 20 ± 39 . 42. Lou XM , Cheng L. Travelers' risk-taking behaviors in day-to-day dynamic evolution model . Transportation Research Record: Journal of the Transportation Research Board . 2016 ; No. 2565 : 27 ± 36 . 43. Emmerink RHM , Axhausen KW , Nijkamp P , Rietveld P. Effects of information in road transport networks with recurrent congestion . Transportation . 1995 ; 22 : 21 ± 53 . 44. Hu TY , Mahmassani HS . Day-to-day evolution of network flows under real-time information and reactive signal control . Transportation Research Part C: Emerging Technologies . 1997 ; 5 : 51 ± 69 . 45. Engelson L. On dynamics of traffic queues in a road network with route choice based on real time traffic information . Transportation Research Part C: Emerging Technologies. 2003 ; 11 : 161 ± 183 . 46. Peeta S , Yu JW . Adaptability of a hybrid route choice model to incorporating driver behavior dynamics under information provision . IEEE Transactions on Systems , Man, and Cybernetics-Part A : Systems and Humans. 2004 ; 34 : 243 ± 256 . 47. Friesz TL , Kim T , Kwon C , Rigdon MA . Approximate network loading and dual-time-scale dynamic user equilibrium . Transportation Research Part B: Methodological . 2011 ; 45 : 176 ± 207 . 48. Iryo T. Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology . Transportmetrica B: Transport Dynamics . 2013 ; 1 : 52 ± 67 . 49. Bie J. The dynamical system approach to traffic assignment: the attainability of equilibrium and its application to traffic system management . Ph.D. Thesis , The Hong Kong University of Science and Technology, Hong Kong , 2008 . 50. Meneguzzer C . Dynamic process models of combined traffic assignment and control with different signal updating strategies . Journal of Advanced Transportation . 2012 ; 46 : 351 ± 365 . 51. Cantarella GE , Velonà P , Watling DP . Day-to-day dynamics & equilibrium stability in a two-mode transport system with responsive bus operator strategies . Networks and Spatial Economics . 2015 ; 15 : 485 ± 506 . 52. Kumar A , Peeta S. A day-to-day dynamical model for the evolution of path flows under disequilibrium of traffic networks with fixed demand . Transportation Research Part B: Methodological . 2015 ; 80 : 235 ± 256 . 53. Guimerà R , DÂõaz-Guilera A , Vega-Redondo F , Cabrales A , Arenas A . Optimal network topologies for local search with congestion . Physical Review Letters . 2002 ; 89 : 248701 . doi: 10 .1103/PhysRevLett.89. 248701 PMID: 12484988 54. Zhao L , Lai Y C , Park K , Ye N. Onset of traffic congestion in complex networks . Physical Review E . 2005 ; 71 : 026125 . 55. Yan G , Zhou T , Hu B , Fu ZQ , Wang BH . Efficient routing on complex networks . Physical Review E . 2006 ; 73 : 046108 . 56. Du WB , Zhou XL , Lordan O , Wang Z , Zhao C , Zhu YB . Analysis of the Chinese airline network as multilayer networks . Transportation Research Part E: Logistics and Transportation Review . 2016 ; 89 : 108 ± 116 .


This is a preview of a remote PDF: http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0168241&type=printable

Bo Gao, Ronghui Zhang, Xiaoming Lou. Modeling Day-to-day Flow Dynamics on Degradable Transport Network, PLOS ONE, 2016, DOI: 10.1371/journal.pone.0168241