Development and application of the network weight matrix to predict traffic flow for congested and uncongested conditions

Abstract

To capture network dependence between traffic links, we introduce two distinct network weight matrices ( $W_{j, i}$ ), which replace spatial weight matrices used in traffic forecasting methods. The first stands on the notion of betweenness centrality and link vulnerability in traffic networks. To derive this matrix, we use an unweighted betweenness method and assume all traffic flow is assigned to the shortest path. The other relies on flow rate change in traffic links. For forming this matrix, we use the flow information of traffic links and employ user equilibrium assignment and the method of successive averages algorithm to solve the network. The components of the network weight matrices are a function not simply of adjacency, but of network topology, network structure, and demand configuration. We test and compare the network weight matrices in different traffic conditions using the Nguyen–Dupuis network. The results lead to a conclusion that the network weight matrices operate better than traditional spatial weight matrices. Comparing the unweighted and flow-weighted network weight matrices, we also reveal that the assigned flow network weight matrices perform two times better than a betweenness network weight matrix, particularly in congested traffic conditions.

Keywords

Traffic flow spatial weight matrix vulnerability traffic forecasting network analysis competitive links

Introduction

For decades, transport analysts tackled traffic forecasting, while focusing on time series approaches (Du et al., 2012; Smith et al., 2002). Following the emergence of spatial analysis in traffic studies, a growing interest has aimed to embed spatial components in forecasting methods (Jenelius and Koutsopoulos, 2013; Ji and Geroliminis, 2012). At the core is the idea that traffic links in a network have spatial dependence. As far as spatial dependence is concerned, traffic links on a network are spatially related and the intensity of this association declines with distance. The methods for capturing this dependence have essentially remained unchanged since the emergence of spatial weight matrices. While now acknowledged in transport science, its roots are found in geography and spatial econometrics.

Although embedding the spatial component in forecasting methods acts as a catalyst, its functioning is hindered by the constraints of spatial weight matrices. Spatial weight matrices require positivity of components, postulating that all traffic links have a positive spatial dependency. This hypothesis successfully represents complementary (upstream and downstream) traffic links. The complementary nature demonstrates that traffic streams are alike to fluid streams, and hence vehicles observed at upstream at one time point will be observed at downstream at a later time point. For simple single facility corridors, this may be sufficient. However, this misses the competitive nature of traffic links. Competitive links bear a significant proportion of diverted vehicles, for instance when one of them is saturated or closed or affected by an incident or event. Short-term forecasting of traffic was initially confined to scrutinizing complementary links. In consequence, the competitive nature of traffic links has been overlooked in the spatial weight matrix configuration (Ermagun and Levinson, 2017a).

We introduce two network weight matrices to fill the lacuna. These matrices are a function not simply of adjacent traffic links, but of network infrastructure, topology, and demand matrices. They have the potential of superseding the spatial weight matrices in traffic flow forecasting and, we suspect, other network applications.

The remainder of the paper is structured as follows. First, we review the traffic forecasting methods embedding spatial components, along with spatial weight matrices used in traffic analysis. Second, we discuss the concepts of betweenness centrality and vulnerability, as they are fundamental to derive network weight matrices. Third, we introduce two distinct network weight matrices. Fourth, for pedagogical purposes, we delve into the deriving process of network weight matrices for a toy network, which is followed by validity assessment of network weight matrices in different traffic conditions. We finally conclude by broaching a number of arguments and suggestions for future studies.

Traffic forecasting: A spatial analysis perspective

“Everything is related to everything else, but near things are more related than distant things” (Tobler, 1970). Tobler’s “law” opened a new gateway and the “near” and “related” concept has spread to broader research disciplines. The footprint is seen in transport science. Utilizing the “near” and “related” concept in traffic forecasting methods has had an active history. However, it is still as crude as it is perplexing (Ermagun and Levinson, 2016). This concept emerged from two strands of thought in two discrete time spans. The origin of the first strand dates back to 1984, in which Okutani and Stephanedes (1984) used the traffic flow on both the study link and its feeder links to predict traffic flow during the day. This so-called state-space approach subsequently burgeoned and developed in the literature. While this strand of traffic forecasting interested transport scientists, the contemporary theoretical movement is drawing equal, if not more, attention. This strand was noticed by Kamarianakis and Prastacos (2003), who tested the performance of the space-time autoregressive integrated moving average (STARIMA) model for relative velocity forecasting on major arterials of Athens, Greece. Since then, a whole host of studies have been carried out trying to test the space-time model class in forecasting traffic conditions. Comparable in concept to the state-space approach, the space-time approach benefits from the relationship between traffic links to augment the forecasting performance. Table 1 summarizes research conducted in these strands. Studies are chosen such that they cover a range of study locations, study years, and method of analysis.

Table 1.

Summary of selected studies on spatial network analysis.

Study	Location	Scale	Method	Variable	Dependency
Spatial weight matrix approach
Cheng et al. (2014)	London	Freeway	LSTARIMA	Flow	Spatial weight
Cheng et al. (2012)	London	Freeway	Moran’s I	Time	Spatial weight
Zou et al. (2010)	China	Arterial	Moran’s I	Speed	Spatial weight
Min et al. (2010)	China	Arterial	GSTARIMA	Flow	Spatial weight
Ma et al. (2010)	China	Freeway	Moran’s I	Speed	Spatial weight
Kamarianakis and Prastacos (2003)	Greece	Arterial	STARIMA	Flow	Spatial weight
State-space approach
Dong et al. (2015)	China	Freeway	Spatial-temporal	Flow	Adjacent links
Van Lint (2006)	Netherlands	Freeway	Neural network	Time	Up- and down-stream
Stathopoulos and Karlaftis (2003)	Greece	Arterial	Kalman filtering	Flow	Upstream
Okutani and Stephanedes (1984)	Japan	Arterial	Kalman filtering	Flow	Adjacent links

The state-space approach acknowledges the positive dependency between upstream and downstream traffic links by weighting the adjacent links. This weight is either equal among adjacent links or is a function of distance. The weighted information of adjacent traffic links is then embedded in the modeling framework as a set of independent variables. A fundamental challenge of this approach is the definition of “nearness” and distance metric. In spatiotemporal models, however, this dependency is captured by the spatial weight matrix borrowed from the field of spatial science.

A spatial weight matrix W is a square l × l matrix, where l is the number of links in the network, and the matrix components i and j are indexes of links, and $w_{i, j}$ , regularly satisfy the three following major rules. These rules represent non-negative, normalized, asymmetric, and nonzero diagonal spatial weight matrices.

$w_{i, j} \geq 0$

$w_{i, i} = 0$ , and

$\sum_{j = 1}^{l} w_{i, j} = 1$ , for all $i = 1, 2, \dots, l$ (Bavaud, 1998).

The weight matrix W defines the relative weight of spatial dependence between traffic links. While the spatial weight matrix can capture the self-influence of traffic link i upon itself, the matrix is typically considered to have a zero diagonal matrix. Non-diagonal elements are determined by a number of theoretical methods, which like the state-space model, their roots are in the definition of “nearness.”

In the remainder of this section, we delve into synthesizing spatial weight matrices used in traffic forecasting models, as their close connection with the contribution of the current research. In traffic analysis, the components of spatial weight matrices have typically been determined by two approaches: adjacency weights and distance weights.

The adjacency weights method assumes the spatial dependence only exists between adjacent traffic links, and the amount of this dependency is equal for all adjacent traffic links. This commonly leads to a spatial matrix with binary elements, in which zero and one values indicate spatial independence and spatial dependence, respectively. Some analyses apply the row normalization to follow the third rule represented above.

The distance weights method seeks more spatial dependence between traffic links as it allows us to capture more spatial information spread across the network. Unlike the adjacency weights, the distance weights approach assumes the spatial dependency is a function not simply of adjacency, but of the distance between traffic links and takes more spatial dependence between them into consideration. This idea is consistent with Tobler’s first “law” of geography (Tobler, 1970). Subsequently, methods have been introduced to explore the effective distance threshold in traffic networks. We categorize them in two general methods:

k^th-order neighbor weights: For a traffic network consisting of L connected traffic links, assume the order of connectivity between link i and all links $j \neq i$ is ranked as $k = {1, 2, \dots, l - 1}$ . In this definition, the first-order neighbor links are those that share a start or end node with link i, while the second-order neighbor links are those links that are connected to link i indirectly with the first-order neighbor links in middle. Then for each k, the set $M_{k} (i) = {j (1), j (2), \dots, j (k)}$ contains the k^th-order neighbor links to link i. The binary spatial weight components are then measured by equation (1) for each given k.

\begin{matrix} w_{i, j} = {\begin{matrix} 1 & j \in M_{k} (i) \\ 0 & j \notin M_{k} (i) \end{matrix} \end{matrix}

(1)

Cheng et al. (2012), for instance, employed the first-order neighbor matrix to explore the spatiotemporal autocorrelation structure of road networks of London, England. Likewise, Kamarianakis and Prastacos (2003) explored the spatial dependence between the relative velocity of traffic links in the city of Athens, Greece. They used the first- and second-order neighbors matrix to embed a spatial component in traffic forecasting models.

Power distance weights: This method assumes the spatial dependence reduces by distance as per equation (2). In this equation, κ represents any positive value, which commonly equals one or two, and r is a measure of distance (or travel time).

w_{i, j} = r_{i, j}^{- κ}

(2)

Notwithstanding the prevalence of this method in regional and transport science and its widespread acceptance among transport analysts for many spatial interaction problems (Anas, 1983; Ji and Geroliminis, 2012; Lam and Huang, 1992), to the best of our knowledge, there is no study using this method for short-term traffic forecasting.

The methods for deriving spatial weight matrices used in traffic analysis have room to grow. The spatial weight matrix:

prejudges spatial dependence between traffic links;

overlooks the competitive nature of traffic links in a road network;

is usually a fixed matrix for a network structure, with only a few examples of dynamic spatial weight matrices that change in fixed intervals or due to traffic conditions (Cheng et al., 2014).

These drawbacks motivated the authors to introduce two distinct network weight matrices to capture the network dependence between traffic links. Subsequently, we test whether embedding the network weight matrices in traffic forecasting models improves the accuracy of predictions.

Network topology and structure

Spatial networks have been employed to study the activities, locations, and flows of individuals and goods in quantitative geography for a long time (Barthélemy, 2011). Their roots are found in books such as “Networks Analysis in Geography” (Haggett and Chorley, 2015) and “Models in Geography” (Haggett, 1967). As emblematic real-world spatial networks, transport networks have been represented topologically using network models. This enabled researchers to explore the salient topological properties of road (Xie and Levinson, 2009), airline (Barrat et al., 2004), bus (Sienkiewicz and Hołyst, 2005), subway (Latora and Marchiori, 2002), and commuter (Chowell et al., 2003) networks.

Network topology stands on the foundation of the graph theory. A graph $G = (N, L)$ is a collection of n nodes ( $N = {1, 2, \dots, n}$ ), which are interconnected by l links ( $L = {1, 2, \dots, l}$ ). The topology of a network is the arrangement and connectivity of links and nodes, which is represented by a square adjacency matrix. Its components demonstrate whether two nodes are attached by a link. Contingent on the type of interaction between nodes, a network is classified as directed or undirected. An undirected network encompasses two-way connecting links, resulting in a symmetric adjacency matrix. One-way connecting links produce a directed network and generally asymmetric adjacency matrix. Contingent on the strength of interaction between nodes, a network is categorized as weighted or unweighted. A weighted network comprises weighted links that signify some distinguishing trait such as cost, capacity, and length. For an unweighted network, however, all links have the same weight.

Measures are utilized to characterize the topology, and thereby its ability to withstand link failures. Among the measures, we intend to introduce the concepts of link betweenness and vulnerability, as they are fundamental to understand the derivation of network weight matrices. In the following subsections, we expound on these concepts.

Betweenness

In transport networks, not all links are equivalent. Traffic flow on a network is highly concentrated on relatively few links. Hence, a fundamental question from the network science perspective is: How do you examine and measure the importance of a link in a network? The concept of centrality originating from social network science has been used to determine the importance of traffic links. In network science, the betweenness centrality measure is mostly used as measure of centrality for nodes. To evaluate the importance of a link, however, we measure the betweenness centrality of links. More precisely, we focus on the “dual” network perspective where links are connected through nodes, and thereby we represent the betweenness centrality measures associated with links, rather than nodes.

Freeman (1977) first introduced the betweenness centrality measurement (hereafter betweenness) for a node in a network. It is formulated in light of the hypothesis that flow is passed from an origin to a destination only along a network path linking them. The betweenness of node o, by Freeman’s definition, is the ratio of the shortest paths between each pair of nodes that pass through the given node o to all the shortest paths between node pairs. Girvan and Newman (2002) generalized this definition for measuring betweenness of a link. In a network with n nodes, the betweenness of link i is formally defined by equation (3). In this equation, P_od stands for the number of shortest paths between nodes o and d, and P_oid represents the number of shortest paths between nodes o and d that pass through link i.

B (i) = \sum_{o < d}^{n} \sum^{n} \frac{P_{oid}}{P_{od}}

(3)

One of the advantages of the betweenness measurement is taking the “cut-link” role into consideration. For better understanding of this role, Figure 1 depicts two regions R_A and R_B, which are connected with traffic link i. This link has a small connectivity compared with other links, but removing link i detaches region R_A from region R_B. Link i plays a “bridge” or “cut-link” role in this traffic network, and its importance is measured by betweenness.

Figure 1.

The “bridge'' role in traffic network.

The betweenness measurement, however, assumes traffic flow is passed from one node to another only along the shortest path. In most traffic networks, however, traffic flow does not stream only across shortest paths. To advance the betweenness index, Freeman et al. (1991) proposed the flow betweenness measurement for a node, which is generalizable to a link and considers all possible paths between an OD pair, not only geodesics. The flow betweenness of link i measures the proportion of flow through link i when the maximum flow is transmitted from origin o to destination d, averaged over all OD pairs. Formally, the flow betweenness for link i is defined by equation (4). In this equation, Max_oid is the maximum flow from node o to node d passing through link i and Max_od is the total flow through all links different from link i between nodes o and d.

FB (i) = \sum_{o < d}^{n} \sum^{n} \frac{{max}_{oid}}{{max}_{od}}

(4)

In traffic (route) assignment, the flow on a link is the sum for all OD pairs of the flow between each OD pair that uses that link. Different assignment methods have different assumptions about traffic (e.g. shortest path (Dijkstra, 1959), user equilibrium (Dial, 1971; Wardrop, 1952), stochastic user equilibrium (SUE) (Daganzo and Sheffi, 1977), boundedly rational user equilibrium (Mahmassani and Chang, 1987), behavioral user equilibrium (Zhang, 2011), and so on), and transit (e.g. optimal strategies (Spiess and Florian, 1989)). In transport assignment practice, this flow is not normalized, as the flow itself is the object of interest, though it would be easy enough to do so.

Vulnerability

The performance of traffic links depends on one another in a traffic network. This high level of interdependency has the potential of cascading failure. As a consequence, traffic network analysts ask:

What are the most critical links of a network?

How, and to what extent, does the degradation of traffic links affect network performance?

How to assess the risk of transport systems?

The concept of vulnerability attempts to answer these questions. Broadly defined, vulnerability refers to system performance following insecure conditions. More precisely, Berdica (2002) defines vulnerability as “a susceptibility to incidents that can result in considerable reductions in road network serviceability.” In transport science, the concept of vulnerability is largely used to assess network performance resulting from degradation or disruption of nodes and links in a traffic network. Ducruet et al. (2010) scrutinize the vulnerability of nodes in liner shipping networks. They conclude a node is vulnerable when it possesses low centrality and connection, but high dependency. Jenelius et al. (2006) measure the vulnerability of links in road networks as a change in travel cost stemming from a link failure. Taylor and D’Este (2007) broadly discuss the concept of vulnerability, and define vulnerability by using the notion of accessibility, the ease of reaching valued destinations, in the following terms:

“A network node is vulnerable if loss (or substantial degradation) of a small number of links significantly diminishes the accessibility of the node, as measured by a standard index of accessibility.”

“A network link is critical if loss (or substantial degradation) of the link significantly diminishes the accessibility of the network or of particular nodes, as measured by a standard index of accessibility.”

Following the recapitulation of vulnerability concept in transport literature and in parallel with existing definitions, we intend to define the criticality of the link by using the notion of betweenness in the following term: “The criticality of the link is determined by the change in the betweenness of other links upon the elimination of the link.”

We use this definition to extract the network dependence between traffic links and to form components of network weight matrices.

Network weight matrix

In this section, we derive the network weight matrices. We use topological and structural properties of the network in order to determine the network dependence between traffic links. We borrow and entangle the concepts of betweenness and vulnerability to introduce two distinct network weight matrices. One relies on the betweenness, and the other is inspired by the definition of flow betweenness.

Unlike the spatial weight matrix, which is unable to capture the competitive nature of links, the network weight matrices can deal with both competitive and complementary traffic links. To shed some light on the notion of competitive and complementary links in a network, we depict a simple three-link network in Figure 2, which consists of two different paths between an OD pair. By definition, link i is complementary to link j and link h, as an increase in the cost of link i not only decreases the flow of link i, but it also diminishes the flow of link j and link h. However, link j and link h are competitive, as an increase in the cost of link j decreases the flow of link j, but increases the flow of link h.

Figure 2.

A schematic example for complementary and competitive nature of traffic links.

Levinson and Karamalaputi (2003) propose an algorithm, which physically detects competitive or parallel links in a road network. The algorithm acknowledges four attributes: (1) Angular difference between two links, (2) perpendicular distance, (3) sum of the distance between the start and end nodes of two links, and (4) the ratio of lengths of two links. However, we let the data speak for themselves. We postulate three major hypotheses to form network weight matrices:

The network dependence of competitive links is negative, and more vulnerable competitive links have more network dependence weight in a network.

The network dependence of complementary links is positive, and more vulnerable complementary links have a greater network dependence weight in a network.

The components of network weight matrices are a function not simply of adjacency, but of the demand configuration and network topology.

To form the components of the network weight matrix, we propose two distinct, yet comparable methods. Although the two methods follow the same structure to form the network weight matrices, they employ different concepts to create the matrix components. Consider again $G = (N, L)$ as a network with n nodes and l links. The betweenness and assigned flow methods are formally defined as follows:

Unweighted betweenness: This method benefits from the (unweighted) betweenness centrality measurement and the concept of vulnerability discussed in the preceding sections. Let $B (i, G (N, L))$ denotes the betweenness centrality of link i in network G(N, L) and $B (i, G (N, L - {j}))$ denotes the betweenness centrality of link i following the elimination of link j from network G(N, L). By our definition, the change in betweenness centrality of link i, stemming from the elimination of link j, indicates the network influence of link j on link i and forms $W_{Bj, i}$ component of the unweighted network weight matrix. We label this method “unweighted betweenness,” as all links are equally weighted to one.

Assigned flow: Unlike the unweighted betweenness method that assumes all links are equally weighted to one, the assigned flow method takes the traffic flow that passes through each link into account. Let define $Q (i, G (N, L))$ as the traffic flow that passes through link i in network G(N, L). Accordingly, we define $Q (i, G (N, L - {j}))$ as the traffic flow that passes through link i following the elimination of link j from network G(N, L). By our definition, the change in the traffic flow of link i due to the removal of link j indicates the network influence of link j on link i and forms $W_{Qj, i}$ component of the assigned flow network weight matrix. This method is computationally more expensive than the unweighted betweenness method as we need to measure the flow of each link considering travel costs of links and travel demands between OD pairs in each iteration.

The algorithm of the unweighted betweenness method is illustrated in Figure 3. The assigned flow method follows the equivalent algorithm depicted in Figure 3, except for the weighting method.

Figure 3.

The flowchart of network weight matrix algorithm.

Network weight matrix calculation: A toy network problem

For pedagogical purposes, we derive the network weight matrices for a simple example. The toy network, depicted in Figure 4, consists of four nodes and five links. We extract the network weight matrices introduced in the preceding section for two disparate OD demands. It reveals the effects of demand configuration on network dependence between traffic links.

Figure 4.

A toy network and its link path graph.

We adopt the standard Bureau of Public Roads (BPR) link performance function as per equation (5). In this equation, t_i is the link travel time, $t_{i}^{0}$ stands for free-flow link travel time, v_i represents the assigned traffic flow, and c_i is the capacity of link i.

t_{i} = t_{i}^{0} [1 + α (\frac{v_{i}}{c_{i}})^{β}]

(5)

We outline the parameter values of each link in Table 2, which is extracted from Suwansirikul et al. (1987). However, we slightly changed the free-flow link travel times of links (1,2) and (3,4) to have a unique shortest path between given OD pairs. In the following examples, we point out the derivation of network weight matrices in detail.

Table 2.

Parameter values of the toy network.

Link	$t_{i}^{0}$	c_i	α	β
(1,2)	3.9	40.0	0.15	4.0
(1,3)	6.0	40.0	0.15	4.0
(2,3)	2.0	60.0	0.15	4.0
(2,4)	5.0	40.0	0.15	4.0
(3,4)	3.1	40.0	0.15	4.0

Example 1: The first instance passes 100 vehicles from node 1 to node 4 through three different paths. For the sake of understanding, we separately explain the process of deriving network weight matrices for two unweighted betweenness and two assigned flow methods in this example.

With the unweighted betweenness method, we are able to calculate betweenness based on either the free-flow travel time or the ultimate link travel time derived from equation (3) for each scenario. For simplicity, we employ the free-flow travel time to measure betweenness in this example. However, we later test both cases for evaluating network weight matrices. To find the shortest path, we employ Dijkstra’s algorithm. We represent the results in Table 3. For a network with all links scenario, looking at the betweenness indicates that the value of links (1,2) and (2,4) equal one, as path 1 is the shortest path between 1 and 4 OD pair. Following the removal of link (1,2), path 1 is no longer the shortest path and path 2 absorbs the traffic, as it is the only path in the network. Consequently, the values of links (1,3) and (3,4), which belong to path 2, take on the value of one. Differentiation of the betweenness of links when the network possesses all links and when link (1,2) is removed results in revealing the network dependence between link (1,2) and other links. The derived values form the first row of the network weight matrix depicted in Figure 5. The values disclose both complimentary and competitive nature of traffic links.

Table 3.

Weight matrix calculation for Example 1.

Network scenario	Unweighted betweenness					Assigned flow
Network scenario	(1,2)	(1,3)	(2,3)	(2,4)	(3,4)	(1,2)	(1,3)	(2,3)	(2,4)	(3,4)
All links	1	0	0	1	0	52	48	6	46	54
Without (1,2)	0	1	0	0	1	0	100	0	0	100
Without (1,3)	1	0	0	1	0	100	0	52.38	47.62	52.38
Without (2,3)	1	0	0	1	1	50	50	0	50	50
Without (2,4)	1	0	1	0	1	52	48	52	0	100
Without (3,4)	1	0	0	1	0	100	0	0	100	0

Figure 5.

Network weight matrices for Example 1.

As expected, link (1,2) is intensely competitive with links (1,3) and (3,4), as removing link (1,2) shifts the traffic flow to link (1,3) and (3,4), which is the only existing path in the network. However, link (1,2) does not have any network dependence with links (1,3) and (3,4), and thereby their corresponding components in the network weight matrix are zero. This is illustrated by the hypothesis of shortest path selection, which is the backbone of unweighted betweenness measurement. Link (1,2) belongs to the shortest path 1, which is selected by all users. Removal of link (1,3) does not change the path of flow in the network, and as a result, the betweenness of the links is similar to the network with all links scenario. It is the shortcoming of unweighted betweenness measurement, which assumes all network users choose the shortest path.

Link (1,2) is complementary to link (2,4), as its removal paralyzes link (2,4). However, as shown in Figure 5, there is no network influence from link (2,4) on link (1,2). The reason is the studied links are directed and traffic link (1,2) is upstream of traffic link (2,4). Hence, flow streams from link (1,2) to link (2,4), and in the free-flow condition link (2,4) does not have any network influence on link (1,2). While in the congested situation, we might imagine the shockwave stemming from link (2,4) affects link (1,2). However, it is neither the case in this example nor is it measurable by betweenness as defined here.

To implement the assigned flow method, the SUE and the method of successive averages (MSAs) solution algorithm are adopted to assign vehicular trip rates to the network (Sheffi and Powell, 1981). User equilibrium models are used to assign traffic flows to each link in a network with a given O-D matrix. The basic models are built on Wardrop’s First Principle, in which system users choose their path in order to achieve equal average travel cost in all used paths. The user equilibrium point is achieved if traffic flows follow this principle. In this case, no user can improve his or her travel costs from the equilibrium point by shifting from one route to another. The SUE takes into account the errors in user’s perceptions. This method represents travel times of links as random variables distributed across the system users. As these probabilities depend on traffic flows, an iterative procedure is needed to find the user equilibrium assignment. The MSA is an iterative procedure that consists of four steps, namely initialization, network loading, step size, and update, which is widely used in traffic assignment. This method can be computationally expensive due to the predetermined sequence of step size. Therefore, the slow convergence speed of MSA is one of its disadvantages. We refer the reader to Sheffi and Powell (1981) and Liu et al. (2009) for more detailed information.

The results of the assignment are depicted in Table 3. Removal of link (1,2) paralyzes path 1 and path 3, and consequently the flow on links (1,2), (2,3), and (2,4) equals zero. Calculating the change rate in traffic flow of each link following the link (1,2) removal results in revealing network dependence between link (1,2) and other traffic links. The first row of network weight matrix depicted in Figure 5 discloses this dependency. For illustration, the components of the first row are calculated as follows

\begin{matrix} w_{(1, 2) (1, 2)} = 52 - 0 = 52 \\ w_{(1, 2) (1, 3)} = 48 - 100 = - 52 \\ w_{(1, 2) (2, 3)} = 6 - 0 = 6 \\ w_{(1, 2) (2, 4)} = 46 - 0 = 46 \\ w_{(1, 2) (3, 4)} = 54 - 100 = - 46 \end{matrix}

Similar to the betweenness method, the values of the network weight matrix formed by the assigned flow method reveal both complementary and competitive nature of traffic links.

Link (1,2) is directly competitive with links (1,3) and (3,4), in line with our hypotheses and results of the betweenness method. In contrast with unweighted betweenness method, the assigned flow method acknowledges the reciprocal spatial dependence between link (1,2) and links (1,3) and (3,4). The reason is the assigned flow method is not simply a function of shortest path, but of user equilibrium assignment. However, the unweighted betweenness approach stands on the foundation of an all-or-nothing assumption. Interestingly, looking at the second row of the network weight matrix indicates that link (1,3) is competitive with links (2,3) and (2,4), but with different magnitudes. Although link (1,3) is highly correlated with link (2,3), there is a low correlation between link (1,3) and (2,4). It is empirically true as a significant amount of flow shifts to link (2,3) by removing link (1,3) in comparison with the network with all links scenario. However, the traffic flow of link (2,4) does not witness a remarkable change.

Unlike the unweighted betweenness approach, link (1,2) is complementary not only to link (2,4), but also to link (2,3). It is not surprising, given link (1,2) is a feeder of both links. However, there is no reciprocal network dependence between links (1,2) and (2,4), since traffic links are directed in this example and link (1,2) is the only feeder of link (2,4). When a link absorbs traffic from more than one feeder, the reciprocal network dependence shows up in the network weight matrix. The instance of such dependence is link (3,4), which is fed by links (1,3) and (2,3). Consequently, not just links (1,3) and (2,3) spatially affect link (3,4), but they are affected by link (3,4) as well.

Comparing two network weight matrices demonstrates that the network weight matrix built on the assigned flow method captures the more realistic network dependence between links than the unweighted betweenness method. It is then hypothesized that the assigned flow network weight matrix performs better than unweighted betweenness network weight matrix, particularly in congested traffic conditions. We later test this hypothesis.

Example 2: Similar to Example 1, this instance passes 100 vehicles, but from node 1 to node 3 through two different paths. Table 4 depicts the results of assigned flow and unweighted betweenness methods for different scenarios. Figure 6 also represents the corresponding network weight matrices. The derivation of network weight matrices for unweighted and assigned flow is akin to Example 1. The dissimilarity between network weight matrices is emphasized in two examples stemming from the change in demand configuration.

Table 4.

Weight matrix calculation for Example 2.

Network scenario	Unweighted betweenness					Assigned flow
Network scenario	(1,2)	(1,3)	(2,3)	(2,4)	(3,4)	(1,2)	(1,3)	(2,3)	(2,4)	(3,4)
All links	1	0	1	0	0	51.85	48.15	51.85	0	0
Without (1,2)	0	1	0	0	0	0	100	0	0	0
Without (1,3)	1	0	1	0	0	100	0	100	0	0
Without (2,3)	0	1	0	0	0	0	100	0	0	0
Without (2,4)	1	0	1	0	0	51.85	48.15	51.85	0	0
Without (3,4)	1	0	1	0	0	51.85	48.15	51.85	0	0

Figure 6.

Network weight matrices for Example 2.

Not surprisingly, the corresponding components to link (2,4) and (3,4) in network weight matrices are zero, as they pass no flow from node 1 to node 3. In this example, path 1 competes with path 2, and thereby links (1,2) and (2,3) are competitive with link (1,3). The negative sign of the components discloses this competitive nature. Comparing network weight matrices in two examples emphasizes the remarkable role of link (2,3) in 1–3 OD configuration. It is indeed true, as path 2 is the only substitute for path 1. The change in value of the components of the network weight matrices in two examples reveals that network dependence between traffic links is not only related to the topology of the network, but it is also defined by the demand configuration in traffic networks.

Network weight matrix verification

In order to exhibit the validity of the network weight matrices and comparing the efficacy of each, the Nguyen and Dupuis network (Nguyen and Dupuis, 1984) is adopted as depicted in Figure 7. The network consists of 13 nodes, 19 directed links and 4 OD pairs. The characteristics of each link are represented in Table 5, which are extracted from Xu et al. (2011). The network weight matrices are tested for three different demand scenarios:

Table 5.

Link characteristics of the Nguyen–Dupuis network.

Link	$t_{i}^{0}$	c_i	α	β	Link	$t_{i}^{0}$	c_i	α	β
1	7	300	0.15	4	11	9	500	0.15	4
2	9	200	0.15	4	12	10	550	0.15	4
3	9	200	0.15	4	13	9	200	0.15	4
4	12	200	0.15	4	14	6	400	0.15	4
5	3	350	0.15	4	15	9	300	0.15	4
6	9	400	0.15	4	16	8	300	0.15	4
7	5	500	0.15	4	17	7	200	0.15	4
8	13	250	0.15	4	18	14	300	0.15	4
9	5	250	0.15	4	19	11	200	0.15	4
10	9	300	0.15	4

Figure 7.

The Nguyen–Dupuis network.

$q_{12} = 20, q_{13} = 40, q_{42} = 30$ , and $q_{43} = 10$ ,

$q_{12} = 400, q_{13} = 800, q_{42} = 600$ , and $q_{43} = 200$ , and

$q_{12} = 800, q_{13} = 1, 600, q_{42} = 1, 200$ , and $q_{43} = 400$ .

The second scenario, which was used by Nguyen and Dupuis (1984), gives a semi-congested traffic condition. In this scenario, links 6, 8, 12, 15, and 17 have not reached their capacity. The first and third demand scenarios are designed to assess the network weight matrices in uncongested and congested traffic regimes, respectively.

Five distinct models are developed to test the validity of network weight matrices. The models intend to predict the traffic flow of each link using the travel cost of links as predictors. The functional form of the models assumes a simple linear relationship between the traffic flow and the travel cost variables. This is a naïve assumption. However, this does not jeopardize the results, as this study aims to judge whether and to what extent the network weight matrices have the potential of advancing the traffic flow forecasting. The first model simply considers a linear relationship between traffic flow in each link and its corresponding travel cost. The other models capture both direct and spatial relationships between traffic flow and travel cost. The models are unique in the method of measuring spatial dependence between traffic links. Table 6 depicts the information of the models. In these models:

Q_t is a vector of traffic flow representing dependent variables.

X_t is a vector of travel costs standing for independent variables or predictors.

W is a l × l spatial weight matrix derived from the binary adjacency weights approach.

$W$ is a l × l network weight matrix derived from the betweenness ( $W_{B}$ ) and assigned flow ( $W_{Q}$ ) approaches.

$ε_{t}$ denotes a normal distribution of error terms.

γ, μ_t, and $ϕ_{t}$ are unknown coefficients that are estimated in the modeling procedure.

Table 6.

Specification of models used in this study.

Models	Formula	Method of spatial dependence measurement
Model 1	$Q_{t} = γ + μ_{t} X_{t} + ε_{t}$	No spatial component
Model 2	$Q_{t} = γ + μ_{t} X_{t} + ϕ_{t} {WX}_{t} + ε_{t}$	Spatial Weight matrix: First-order neighbor
Model 3	$Q_{t} = γ + ϕ_{t} W_{B} X_{t} + ε_{t}$	Network weight matrix: unweighted betweenness based on $t_{i}^{0}$
Model 4	$Q_{t} = γ + ϕ_{t} W_{B} X_{t} + ε_{t}$	Network weight matrix: unweighted betweenness based on t_i
Model 5	$Q_{t} = γ + ϕ_{t} W_{Q} X_{t} + ε_{t}$	Network weight matrix: Assigned flow

For estimation, the ordinary least squares (OLS) estimation method is used, which is the best linear unbiased estimator (BLUE) of the coefficients for both bivariate linear regression and spatial cross-regressive models. Table 7 summarizes the results of the models. The student’s t-test is reported in parentheses for exogenous variables.

Table 7.

Results of the models in different demand scenarios.

Models	Scenario 1	Scenario 2	Scenario 3
Model 1
Constant	87.90 (4.29)	417.29 (3.01)	793.01 (4.07)
$μ_{t}$	−6.94 (−3.16)	1.44 (0.21)	2.66 (0.61)
R²	0.37	0.002	0.02
Model 2
Constant	99.62 (3.77)	423.96 (2.49)	804.47 (3.32)
$μ_{t}$	−7.48 (−3.18)	1.40 (0.20)	2.67 (0.59)
$ϕ_{t}$	−0.60 (−0.72)	−0.26 (−0.07)	−0.27 (−0.08)
R²	0.39	0.003	0.02
Model 3
Constant	1.62 (0.25)	428.51 (8.41)	903.57 (10.14)
$ϕ_{t}$	−9.41 (−5.11)	−41.99 (−1.07)	−7.21 (−0.25)
R²	0.60	0.06	0.003
Model 4
Constant	2.05 (0.31)	358.52 (6.49)	778.89 (7.81)
$ϕ_{t}$	−8.69 (−5.03)	−78.07 (−2.49)	−49.04 (−1.99)
R²	0.59	0.26	0.18
Model 5
Constant	8.59 (4.63)	356.54 (10.06)	808.95 (10.63)
$ϕ_{t}$	−0.05 (−18.09)	−4.29 (−5.21)	−4.52 (−3.11)
R²	0.95	0.61	0.36

Note: Student’s t-test is reported in the parentheses.

As for the significance of variables, the spatial component of Model 5 is significant constantly. It demonstrates that the network weight matrix deriving from the assigned flow approach is able to capture the network dependence significantly in all traffic conditions. It is also true for Model 4. The spatial component of Model 2, however, is not statistically significant. This discloses that the traditional spatial weight matrix is unable to measure the realistic spatial dependence between traffic links. Finally, the cost component of Model 1 statistically defines the traffic flow in free-flow traffic condition. When the traffic condition transits from free-flow to congested flow, there is no significant linear correlation between traffic cost and traffic flow. This is generalizable to the coefficient of the spatial component in Model 3. It was expected, as the network weight matrix used in Model 3 derived from the free-flow cost function, and thereby performs better in the free-flow traffic condition.

As for the fit of the models, the Adjusted R² measure of goodness of fit is computed for all models in whole scenarios. Figure 8 shows the results. Model 5 performs far better than the other models in all traffic conditions. There is not a significant difference between the performance of Model 2, which embeds the spatial weight matrix, and Model 1. In the uncongested traffic condition, Model 3 and Model 4 reach the same result, and both of them perform 1.6 times better than Model 1. Model 5 performs 1.5 times better than Models 3 and 4, and thereby 2.5 times better than Model 1. In the semi-congested and congested traffic conditions, Model 3 loses its ability to capture the realistic spatial dependence between traffic links, while it yet performs far better than both Model 1 and Model 2. Although the prediction power of Model 4 and Model 5 declines in the semi-congested and congested traffic conditions, their relative power increases significantly. In the congested traffic condition, for example, the Adjusted R² measure of Model 4 and Model 5 is 8.7 and 16.7 times Model 1, respectively. The results lead to the conclusion that the network weight matrices measure network dependency between traffic links more realistically.

Figure 8.

The goodness of fit comparison of the models in all demand scenarios.

Closing remarks

Despite the fact that forecasting traffic conditions is sophisticated, it is still tractable and predictable with a deep understanding of relationships between traffic components. Two strands have emerged that embed the spatial component in a traffic forecasting framework: state-space and spatiotemporal approaches. However, the evolution of spatial traffic forecasting models was mainly based on spatial weight matrices, which may not accurately reflect the spatial dependence between traffic links.

We introduced two distinct network weight matrices. The first is built on the notion of betweenness and link vulnerability in traffic networks. To derive this matrix, we assumed all traffic flow is assigned to the shortest path, and hence we used Dijkstra’s algorithm to find the shortest path. The second relies on flow rate change in traffic links. For forming this matrix, we employed user equilibrium assignment and MSA algorithm to solve the network. This approach enabled us to capture more realistic traffic flow distribution, especially in the congested traffic conditions. Both network weight matrices acknowledged the network topology and demand configuration. If topological and hierarchical attributes correctly capture the substitutive effects on the network, we are able to better predict how traffic flow would redistribute on the network in cases of major network changes.

We have tested and compared the network weight matrices in different traffic conditions. Such a comparison exemplifies the capability of network weight matrices to advance traffic forecasting. The models with network weight matrices perform better than both the model with spatial weight matrix and without the spatial component. This reveals that traditional spatial weight matrices cannot capture the realistic spatial dependence between traffic links. Drilling down further, the key findings include:

The spatial dependency that is captured by spatial weight matrix is unsuccessful in explaining the spatial relationship between traffic links.

The network weight matrix deriving from the unweighted betweenness method performed well in free-flow traffic conditions. Measuring betweenness by the ultimate travel cost instead of free-flow travel time enhanced the capability of this matrix in congested traffic conditions.

The assigned flow network weight matrix significantly operated better than the betweenness network weight matrix to measure realistic network dependence between traffic links, particularly in congested traffic conditions. The results disclosed that this superiority is more than two times in traffic congested situations.

Although this study formed and validated the network weight matrices in a simulation-based problem, the proposed method has the potential to be applied to real-world problems (Ermagun and Levinson, 2017b; Ermagun et al., 2017). One avenue for future research is to test the accuracy and performance of network weight matrices in a real-world problem. To form the betweenness and assigned flow network weight matrices for a real-world network, the trip tables and traffic demand matrices of the real-world network are employed to assign traffic flow to each link of the network, using either shortest path or user equilibrium assignment. In a simulation-based environment, one link of the real-world network is eliminated and the traffic flow is reassigned to the remaining links of the network. This step would be repeated for all traffic links, and the betweenness and assigned flow network weight matrices are formed using the proposed algorithm in this study.

In the study of real-world networks, one might benefit from observing traffic flow directly, when the system is temporarily modified (Larcom et al., 2015) or traffic links are temporarily shut down (Zhu and Levinson, 2015). However, for planning purposes, there will not always be “after” data. In real-world problems, route choices depend on driver knowledge (Zhang and Levinson, 2008) and this knowledge is embedded in the correlation coefficients.

Another avenue for future research is to test the method on more complex real-world networks and to explore spatial and temporal scales using the continuous streams of traffic data that are now collected by road agencies.

The methodologies proposed in the current research have room to grow within the more recent perspective of multilayer networks to take into account random walks (Gallotti et al., 2016) and the multilayer nature of the network, which is expected to induce further congestion (Solé-Ribalta et al., 2016).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Alireza Ermagun is a Postdoctoral Fellow in the Department of Civil and Environmental Engineering at Northwestern University. His academic work draws on different disciplines, including engineering, planning, economics, and geography. His current research focuses on understanding the process of network growth, evaluating transportation technology and policy, and modeling travel behavior.

David M Levinson teaches at the School of Civil Engineering at the University of Sydney, where he leads the Network Design Lab and the Transport Engineering group. He is an honorary affiliate of the Institute of Transport and Logistics Studies, where he is also a member of the Board of Advice. From 1999 to 2016, he served on the faculty of the University of Minnesota where he held the Richard P. Braun/CTS Chair in Transportation (2006–2016). Levinson has authored or edited several books, including Spontaneous Access, The Transportation Experience, and Planning for Place and Plexus, as well as numerous peer reviewed articles. He is the editor of the Journal of Transport and Land Use.

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