Modeling Day-to-day Flow Dynamics on Degradable Transport Network
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.
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
], 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 ([
],[22±28]), partial user equilibrium [
] and bounded rational user
]). Readers may refer to Cantarella  and Watling and  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. [
], Chen and Mahmassani [
] 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
Risk attitudes can be captured by several theories such as prospect theory (PT) [
] or its
cumulative representation (CPT) [
] 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
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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
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:
S~t1 a S_ tr
a S~tr a
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
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,
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dclt can be expressed as the difference between cl
xlt; ult and 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:
lLlr dclt; 8r 2 Rod:
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Then the perceived variation range
dS~tr1 can be achieved by applying the
exponentialsmoothing updating process:
dS~tr1 a dStr
a dS~tr a
a dS~tr; 8r 2 Rod:
Besides of the within-day variation dStr, travelers will also perceive day-to-day travel time
DStr from their past experiences. Mathematically, DStr can be represented as the
difference of the mean route travel time on two consecutive days:
DStr jS_ tr S_ tr 1j j lLlr cl
xlt; u_lt lLlr cl
xlt 1; u_lt 1j; 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:
DS~tr1 a j lLlr cl
xlt; u_lt lLlr cl
xlt 1; u_lt 1j
1 a DS~tr; 8r 2 Rod:
dS~tr1 and DS~tr1 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~tr1 and DS~tr1,
its updating equation can be easily achieved by combining Eqs 4 and 6 together:
dS~tr1 DS~tr1 a
lLlr dclt j lLlr cl
DS~tr; 8r 2Rod:
xlt 1; u_ lt 1j
2.3 Travelers' Perception of Systematic Disutility Associated to Every
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~tr1; the perceived mean route travel time
S~tr1 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
Assumption II: If the integrated variation range is zero, then the systematic disutility
is equal to the perceived mean route travel time
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~t1
dS~tr1 DS~tr1; S~tr1 0:5
2S~tr1 x and @m
2S~tr1 x=@x (the symmetrical distribution).
In addition, according to the nature of PDF, if dS~tr1 DS~tr1 0, then htr1 S~tr1, which
actually presents an anchor point of the applied disutility function, namely Z
S~tr1 S~tr1. On
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the other hand, if dS~tr1 DS~tr1 > 0, the systematic disutility htr1 can be represented as
Z S~t1 0:5
xdx; 8r 2 Rod:
Although the value of htr1 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 htr1 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 htr1.
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
x > 2S~tr1;
x 6 S~tr1:
With this inequality, the systematic disutility htr1 given by Eq 8 can be compared with the
perceived mean route travel time
S~tr1 as follows:
htr1 S~trr1 0:5
xdx > S~tr1; 8r 2 Rod:
In addition, when x 2
dS~tr1 DS~tr1; S~tr1, the convexity of the disutility
function also implies the increase velocity of Z
2S~tr1 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~tr1 DS~tr1 increases, the systematic disutility htr1 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
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.
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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 htr1
reflecting travelers' perception of route systematic disutility. In this study, htr1 is formulated as
the weighted sum of the perceived mean route travel time and its associated variation ranges:
htr1 S~tr1 rtod1
dS~tr1 DS~tr1; 8r 2 Rod;
Where rtod1 is defined as the risk attitude parameter, its value range is set as [−ρmax,ρmax].
According to Properties I~III, a positive value of rtod1 indicates risk aversion, while risk
proneness is associated with a negative rt1, and when travelers are risk neutral, rtod1 is equal to zero.
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 rtod1 in Eq 10 can be treated as an aggregated form of the realistic travelers'
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 [
] 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
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 rtod1 defined by Eq 11 is depicted in Fig 1.
In Fig 1, all possible values of rtod1 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 rtod1 > 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
rtod1 < rtod.
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Fig 1. Updatings of the risk attitude parameter ρtod1 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 htr1 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:
Prtod1 P expf y htr1g
k2Rod expf y htk1g ; 8r 2 Rod:
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
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in Section 2 is reformulated as below.
DS~tr1 a j lLlr cl
xlt; u_ lt
xlt; u_ lt
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~tr1 S~tr S~r ,
dS~tr1 dS~tr dS~r , DS~tr1 DS~tr DS~r , rtod1 rtod rod and xlt1 xlt xl , thus:
xl ; u_lt; 8r 2 Rod;
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
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
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 rt1 will change under any small fluctuation
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 ([
]). 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 rt1 will never return to the original value rtod by removing the fluctuation. Under
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 rtod1 rtod 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
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the irreversibility issue ([
]). 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 clfree
1 0:15 uxltlt 4
; l 1; 2;
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_lt, thus ult 0:8 u_lt, ult 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 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.
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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
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.
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4.2 The Effects of Travelers' Risk-taking Behaviors on Fluctuation and
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.
The fluctuation function Yt 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 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 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
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
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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
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
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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
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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.
S1 Information. Experimental data for Figs 3, 4 and 5.
S2 Information. The program source code for Figs 3, 4 and 5.
The comments provided by two anonymous referees are much appreciated.
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Conceptualization: BG RZ.
Data curation: BG RZ.
Formal analysis: XL RZ.
Investigation: RZ BG.
Methodology: RZ 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.
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