# Unsupervised Learning for Topological Classification of Transportation
Networks

Sina Sabzekar
Department of Civil Engineering
Sharif University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
& Mohammad Reza Valipour Malakshah
Department of Civil Engineering
Sharif University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
& Zahra Amini

Department of Civil Engineering
Sharif University of Technology
Tehran, Iran
<EMAIL_ADDRESS>
Corresponding Author

###### Abstract

With increasing urbanization, transportation plays an increasingly critical
role in city development. The number of studies on modeling, optimization,
simulation, and data analysis of transportation systems is on the rise. Many
of these studies utilize transportation test networks to represent real-world
transportation systems in urban areas, examining the efficacy of their
proposed approaches. Each of these networks exhibits unique characteristics in
their topology, making their applications distinct for various study
objectives. Despite their widespread use in research, there is a lack of
comprehensive study addressing the classification of these networks based on
their topological characteristics. This study aims to fill this gap by
employing unsupervised learning methods, particularly clustering. We present a
comprehensive framework for evaluating various topological network
characteristics. Additionally, we employ two dimensionality reduction
techniques, namely Principal Component Analysis (PCA) and Isometric Feature
Mapping (ISOMAP), to reduce overlaps of highly correlated features and enhance
the interpretability of the subsequent classification results. We then utilize
two clustering algorithms, K-means and HDBSCAN, to classify 14 transportation
networks. The PCA method, followed by the K-means clustering approach,
outperforms other alternatives with a Silhouette score of $0.510$, enabling
the classification of transportation networks into five clusters. We also
provide a detailed discussion on the resulting classification.

_Keywords_ Unsupervised Learning, Transportation Networks, Classification,
Clustering, K-means, HDBSCAN

##  1 Introduction

In the last century, the rapid urbanization process has drawn significant
attention to the crucial role of transportation as a fundamental aspect of
urban development (Fancello et al., 2014; Badhrudeen et al., 2022).
Transportation systems play a vital role in facilitating the movement of
people and goods within cities, directly impacting the economic prosperity of
societies. This has led to substantial efforts from both academic and
industrial sectors to enhance the efficiency and safety of transportation
systems within urban environments. As cities expand, the complexity of urban
road networks has increased due to the growth of roads and streets (Sharifi,
2019).

Mathematical models of urban transportation networks have provided valuable
insights and solutions to various core issues associated with transportation
systems. These include travel demand modeling (Hassanzadeh and Amini, 2023),
traffic flow prediction (Sabzekar et al., 2023a), traffic flow assignment
(Rahman and Hasan, 2023), traffic congestion detection (Lu et al., 2019), and
traffic signal control (Amini, 2018; Amini et al., 2018, 2016). With the
evolution of research methodologies and the emergence of new challenges in the
transportation domain, an increasing number of studies aim to model and
address problems concerning transportation networks. Consequently, several
transportation networks have been extensively employed by researchers to
simulate real-world urban transportation networks, differing significantly in
size, topology, and geometry. Among these networks, SiouxFalls, a widely used
transportation test network, resembles the road network of SiouxFalls, South
Dakota (LeBlanc et al., 1975). Given the extensive utilization of SiouxFalls
and similar transportation networks in various ongoing research projects
within the transportation domain (Jayakrishnan et al., 1994; Bar-Gera, 2002;
Bar-Gera and Boyce, 2003; Boyce and Bar-Gera, 2003; Bar-Gera and Boyce, 2006;
Bar-Gera and Luzon, 2007a, b; Bar-Gera, 2010; Bar-Gera et al., 2012, 2013; Rey
et al., 2019; Yu et al., 2020; Rahman and Hasan, 2023; Liu and Meidani, 2023),
there arises a critical need to classify these networks. For instance, when
researchers test a new approach on different transportation networks and claim
its superiority, it is essential to determine the distinctiveness of these
transportation networks. Questions such as whether the differences lie in the
network size or what specific features contribute to their similarities or
differences need to be addressed. Moreover, the application of centrality
indices is constrained by the substantial computational expenses involved,
particularly when dealing with large-scale networks such as urban networks.
For instance, in study (Badhrudeen et al., 2022), the authors chose to solely
utilize degree centrality, omitting the computation of other centrality
indices due to their computational demands. Several studies in the existing
literature seek to identify specific correlations between certain centralities
and network features. Consequently, it becomes imperative to assist these
researchers in selecting the most appropriate networks based on the needs and
objectives of their studies.

This paper aims to provide answers to these questions and address the
mentioned challenges by employing unsupervised learning to classify
transportation networks. In contemporary studies, machine learning has
demonstrated remarkable effectiveness across various fields(Sadatnya et al.,
2023). Unsupervised learning, a subset of machine learning, operates with
unlabeled data, making it ideal for identifying hidden patterns (Barlow,
1989). Clustering, a key task in unsupervised learning, involves identifying
hidden patterns and relationships within data, ultimately grouping data
records with similar characteristics (Diday and Simon, 1976). Various
clustering methods, such as K-means (MacQueen et al., 1967), mini-batch
K-means (Sculley, 2010), DBSCAN (Ester et al., 1996), HDBSCAN (Campello et
al., 2015), and hierarchical clustering (Johnson, 1967), have been extensively
utilized. By clustering transportation networks, we aim to classify them based
on shared characteristics. This classification will provide valuable guidance
for future researchers in selecting appropriate transportation networks for
testing their methodologies. To address the complexity and interrelation of
multiple features associated with transportation networks, we employ
techniques from unsupervised learning for more effective representation of
network groups.

The main contributions of this study are as follows:

  * •

The incorporation of critical topological features of transportation networks,
primarily derived from graph theory, focusing on network structure.

  * •

The utilization of two different dimensionality reduction techniques to manage
the large number of topological features and potential correlations among
them.

  * •

The adoption of two distinct unsupervised learning clustering methods to
classify transportation networks, with a comparison of results using various
clustering metrics.

  * •

A detailed discussion of the topological features and their influence on the
resulting classification.

##  2 Background

###  2.1 Network Science

Network science is a field that focuses on modeling and analyzing various
networks (Müller et al., 1995; Barabási, 2013), spanning social networks
(Mitchell, 1974), molecular networks (Bray, 2003), communication networks
(Monge and Contractor, 2003), and road networks (Godfrey, 1969). Urban road
networks represent the spatial and geometrical relationships of roads and
streets within cities (Strano et al., 2013). The study of urban networks has
gained significant attention in recent years, thanks to advancements in
geographic information systems (GIS), the increased availability of data from
digital devices and sensors, and the development of more efficient and robust
computational systems (Badhrudeen et al., 2022). Given the growing utilization
of transportation networks in recent studies, the classification of these
networks has become a crucial task to aid scientists in selecting the most
suitable networks for their research. In this context, network topology serves
as a valuable source of information for this classification.

Network topology pertains to the arrangement of elements (nodes and links)
within a network (Casali and Heinimann, 2019). The topological structure of
networks typically encompasses essential information describing the network,
including the number of links connected to nodes, the available paths between
arbitrary nodes, and the significance of different nodes in the overall
resilience of the network (Wang et al., 2020). Various measures have been
employed in network analysis studies to gain insights into the topology of
networks (Spadon et al., 2018).

Fundamental topological characteristics of networks are related to the number
of nodes and links in the network, and the length of links (Porta et al.,
2006). Additionally, another important group of characteristics includes
centrality indices, which indicate the criticality of nodes within the network
(Koschützki et al., 2005). Several centrality indices have been proposed in
graph theory and widely used in network studies, such as degree centrality
(Merchan et al., 2020), closeness centrality (Shang et al., 2020), betweenness
centrality (Lin and Ban, 2017), and PageRank centrality (Page, 1998). Research
has shown strong correlations between these centrality indices and network
attributes, including disaster resilience, traffic propagation, and the
efficiency of mobility. For instance, the betweenness centrality of a node
measures the prevalence of shortest paths in the network that pass through the
node between arbitrary nodes. Higher values of betweenness centrality indicate
that the node plays a crucial role within the network (Barthelemy, 2004).
Leveraging these centrality features can assist in categorizing networks into
groups with similar characteristics. However, due to the challenges in
interpreting the associated values with these network indices, coupled with
the diversity of these indices, manual network classification is not feasible.
A comprehensive approach with the capability to perform this task
automatically with self-supervision is imperative.

###  2.2 Unsupervised Learning

In the past two decades, machine learning (ML) techniques have demonstrated
their effectiveness across various domains of knowledge. In the field of
transportation, ML methods have proven to be superior to conventional
approaches in tasks such as traffic forecasting (Li and Shahabi, 2018;
Schimbinschi et al., 2015), travel demand prediction (Chu et al., 2018;
Koushik et al., 2020), and autonomous vehicle navigation (Sabzekar et al.,
2023b; Mehditabrizi et al., 2023). ML is generally divided into three sub-
areas: supervised learning, unsupervised learning, and reinforcement learning.
While supervised learning deals with labeled data, unsupervised learning
operates with unlabeled data, aiming to derive insights from this raw data. In
the transportation domain, unsupervised learning methods have exhibited their
effectiveness in clustering various forms of public transportation big data
(Galba et al., 2013), GPS trajectories (Reyes et al., 2020), electric vehicle
charging stations (Straka and Buzna, 2019), and demand patterns (Liu et al.,
2019).

##  3 Problem Description

The transportation network is represented as a directed graph, denoted as
$G(V,E,A)$, where $V=\\{v_{1},v_{2},\dots,v_{N}\\}$ is the set of nodes (i.e.,
intersections), $|V|=N$, $E$ is the set of links (i.e., roads connecting
intersections) defined in Equation 1, and $A$ is the adjacency matrix of the
graph, defined in Equation 2:

$E=\\{(v_{i},v_{j})\mid\text{road connecting intersection }v_{i}\text{ to
intersection }v_{j}\\}$ (1) $A_{ij}=\begin{cases}1&\text{if }(v_{i},v_{j})\in
E\\\ 0&\text{otherwise}\end{cases}$ (2)

The problem of transportation networks classification involves identifying a
set of clusters, denoted as $C=\\{c_{1},c_{2},\dots,c_{m}\\}$, where $m$ is
the number of clusters, and each cluster $i$ ($i\in\\{1,2,\dots,m\\}$), is
defined as a set of networks, $c_{i}=\\{G_{1},G_{2},\dots,G_{r}\\}$, where
$G_{j}$ belongs to cluster $i$. As the number of items in each cluster may
vary, the value of $r$ is not consistent across different clusters. Clustering
techniques aim to assign each network, $G_{j}$, to the most appropriate
cluster, $c_{i}$, based on its similarities with other networks in $c_{i}$. We
define a feature matrix, $F\in R^{k\times s}$, where $k$ is the number of
networks to be clustered, and $s$ is the number of features per network. These
features are computed through topological analysis of networks and are
explained in detail in the following section.

##  4 Methodology

###  4.1 Network characteristics

Networks are represented as graph-structured data, and graph theory
encompasses a range of problems related to graphs. Within this framework,
various characteristics have been defined to represent different properties of
graphs. This paper focuses on topological characteristics, which are explained
in the following sections, including general features and centrality indices.

####  4.1.1 General Features

Number of nodes/links: Measure the total number of nodes and links in the
transportation network. This provides a basic understanding of the network’s
scale.

Average clustering coefficient (ACC): Determines the clustering coefficient
for the network. This measures the extent to which nodes tend to cluster
together. High clustering coefficients suggest the presence of tightly
interconnected subgroups within the network. The average clustering
coefficient for the graph $G$ is determined by

$C=\frac{1}{n}\sum_{v\in G}c_{v}$ (3)

where $n$ is the number of nodes in $G$.

Average shortest path length (ASPL): Calculates the ASPL between nodes in the
network. It indicates the average number of steps required to travel between
any two nodes. Shorter average path lengths imply better accessibility and
connectivity within the transportation network. The average shortest path
length is determined by

$a=\sum_{\begin{subarray}{c}s,t\in V\\\ s\neq
t\end{subarray}}\frac{d(s,t)}{n(n-1)}$ (4)

where $V$ is the set of nodes in $G$, $d(s,t)$ is the shortest path from $s$
to $t$, and $n$ is the number of nodes in $G$.

Diameter (longest shortest path): Determines the diameter of the network,
which is the longest shortest path between any pair of nodes. It indicates the
maximum distance one would need to travel to reach any other node in the
transportation network. In order to calculate diameter, we recall that
diameter is the maximum eccentricity, and the eccentricity of a node $v$ is
the maximum distance from $v$ to all other nodes in graph $G$.

Radius (shortest shortest path): Network radius is a concept that measures the
extent of reach or influence from a central point within a network. The
network radius indicates the shortest distance from a central node to the
farthest node within the network. In transportation terms, we can visualize
the network radius as the distance a traveler starting from a central hub
would need to cover to reach the outermost point in the network. It gives us
an idea of how widely the influence or connectivity of that central hub
extends within the network. In another definition, the radius is the minimum
eccentricity.

Density: Calculates the network density, which measures the proportion of
actual connections to the total number of possible connections in the network.
Dense networks indicate a high level of connectivity. Network density is
determined by

$d=\frac{m}{n(n-1)}$ (5)

where $n$ is the number of nodes, and $m$ is the number of edges in Network
$G$.

Number of weakly and strongly connected components (WCCs / SCCs): Counts the
number of weakly and strongly connected components. In a directed graph, a
weakly connected component is a subset of nodes wherethere is a path between
every pair of nodes in the subset, while ignoring the directions of the edges.
In a directed graph, a strongly connected component is a subset of nodes where
there is a directed path between every pair of nodes in the subset.

Size of the giant weakly and strongly connected components (GWCC / GSCC):
Determines the number of nodes in largest weakly and strongly connected
components.

Average global efficiency (AGE): Determines the average global efficiency of
the network. The efficiency of a pair of nodes in a network is the
multiplicative inverse of the shortest path distance between the nodes. The
average global efficiency of a network is the average efficiency of all pairs
of nodes. Global efficiency measures how efficiently information or resources
flow across the entire network

Average local efficiency (ALE): Local efficiency focuses on the efficiency of
communication within local neighborhoods or clusters. The local efficiency of
a node in the network is the average global efficiency of the sub-network
induced by the neighbors of the node. The average local efficiency is the
average of the local efficiencies of each node.

Reciprocity: The ratio of mutual connections (connections where both nodes
interact with each other) to the total number of connections in the network.
The reciprocity coefficient can range from $0$ to $1$. Network reciprocity is
determined by

$r=\frac{|{(u,v)\in G|(v,u)\in G}|}{|{(u,v)\in G}|}$ (6)

Transitivity: Network transitivity quantifies the tendency for triangles to
form within the network. In simpler terms, network transitivity examines how
interconnected a node’s immediate connections are with each other.
Transitivity measures the ratio of closed triangles to all connected triples
in a network. A closed triangle consists of three nodes that are
interconnected in a triangular formation, meaning each node is connected to
the other two. Network transitivity is defined by

$T=3\frac{\\#triangles}{\\#triads}$ (7)

Degree assortativity coefficient (DAC): Assortativity measures the similarity
of connections in the graph with respect to the node degree. In other words,
assortativity measures the tendency of nodes to connect to others with similar
or dissimilar characteristics. In the context of transportation networks,
assortativity analysis can reveal patterns of connections based on attributes
such as geographical location, transportation mode, or capacity.

####  4.1.2 Centrality Indices

In-degree centrality (IC): The in-degree centrality for a node $v$ is the
fraction of nodes its incoming edges are connected to.

Out-degree centrality (OC): The out-degree centrality for a node $v$ is the
fraction of nodes its outgoing edges are connected to.

Closeness centrality (CC): Closeness centrality of a node $u$ is the
reciprocal of the average shortest path distance to $u$ over all $n-1$
reachable nodes. Closeness centrality is defined by

$CC(u)=\frac{n-1}{\sum_{v=1}^{n-1}d(v,u)},$ (8)

where $d(v,u)$ is the shortest-path distance between $v$ and $u$, and $n-1$ is
the number of nodes reachable from $u$.

Betweenness centrality (BC): Betweenness centrality of a node $v$ is the sum
of the fraction of all-pairs shortest paths that pass through $v$ and is
defined by

$BC(v)=\sum_{s,t\in V}\frac{\sigma(s,t|v)}{\sigma(s,t)}$ (9)

where $V$ is the set of nodes, $\sigma(s,t)$ is the number of shortest (s,
t)-paths, and $\sigma(s,t|v)$ is the number of those paths passing through
some node $v$ other than $s,t$. If $s=t$, then $\sigma(s,t)=1$, and if
$v\in{s,t}$, then $\sigma(s,t|v)=0$.

Eigenvector centrality (EC): Eigenvector centrality is a measure used to
determine the importance of a node in a network. In the context of a
transportation network, eigenvector centrality helps identify the most
critical transportation hubs. It considers not only the number of connections
a node has but also the importance of the nodes to which it is connected.
Mathematically, the eigenvector centrality $EC_{i}$ of a node $i$ in a
transportation network can be calculated by

$EC_{i}=\frac{1}{\lambda}\sum_{j}A_{ij}x_{j}$ (10)

where $EC_{i}$ represents the eigenvector centrality of node $i$, $\lambda$ is
the dominant eigenvalue of the adjacency matrix, and $A_{ij}$ is the element
of the adjacency matrix that represents the connection between node $i$ and
node $j$. The eigenvector centrality helps identify critical nodes that are
not only well-connected but are also connected to other important nodes,
making them vital for the overall transportation flow in the network.

PageRank centrality (PC): PageRank is a method used to evaluate the importance
of nodes based on the concept that important nodes are likely to be linked to
by other important nodes. It helps in identifying critical transportation hubs
that are pivotal for the efficient flow of traffic within the network. The
formula for PageRank in a transportation domain can be represented as follows:

$PC(p_{i})=\frac{1-d}{N}+d\sum_{p_{j}\in M(p_{i})}\frac{PC(p_{j})}{L(p_{j})}$
(11)

where $PC(p_{i})$ represents the PageRank of node $p_{i}$, $N$ is the total
number of nodes in the network, $d$ is the damping factor that represents the
probability of following a link, $M(p_{i})$ is the set of nodes that link to
$p_{i}$, and $L(p_{j})$ is the number of outbound links from node $p_{j}$.

###  4.2 Scaling

Features calculated based on the network characteristics and centrality
indices exhibit varying scales. Continuing with this scenario, the accuracy of
downstream tasks, such as dimensionality reduction and clustering, may be
adversely affected (Ahsan et al., 2021). To mitigate potential inaccuracies,
we employ a scaling method. In this context, we opt for min-max scaling with a
target range of $(0,1)$ for all features, as calculated by the following
equations:

$X_{\text{std}}=\frac{X-X.\text{min}}{X.\text{max}-X.\text{min}}$ (12)

$X_{\text{scaled}}=X_{\text{std}}\times(\text{max}-\text{min})+\text{min}$
(13)

where $X.\text{min}$ and $X.\text{max}$ represent the minimum and maximum
values in the original range of the data, respectively. Similarly, min and max
indicate the minimum and maximum values in the scaled range. Finally,
$X_{\text{scaled}}$ denote the scaled value of the feature. This scaling
method linearly transforms the features into a fixed range, ensuring that the
largest occurring data point corresponds to the maximum value and the smallest
one corresponds to the minimum value.

###  4.3 Dimensionality Reduction

The evolution of data collection systems has granted access to larger
datasets, often featuring numerous features per data record. While these
features may contain valuable information, some may exhibit high correlation,
leading to potential redundancy (Fan et al., 2014). Additionally, the
abundance of features incurs high computational costs which is an undesirable
aspect. Furthermore, in clustering tasks, obtaining a visual representation of
data points in a two-dimensional space proves beneficial for determining an
optimal cluster count and interpreting clustering results (Davidson, 2002). To
address these considerations, we employ two distinct dimensionality reduction
methods: Principal Component Analysis (PCA) (Pearson, 1901) and Isometric
Feature Mapping (ISOMAP) (Tenenbaum et al., 2000), aiming to project our
features into a two-dimensional feature set.

####  4.3.1 Principal Component Analysis (PCA)

PCA stands as one of the most widely used linear dimension reduction
algorithms (Jolliffe and Cadima, 2016). It operates through a projection-based
approach, transforming data by projecting it onto a set of orthogonal axes.
The fundamental premise of PCA lies in maximizing the variance or spread of
data in the lower-dimensional space while mapping data from a higher-
dimensional space. The principal components, constituting linear combinations
of the original variables, are determined by eigenvectors satisfying the
principle of least squares (Abdi and Williams, 2010).

PCA is computed through the following steps:

  1. 1.

Compute the covariance matrix of the data matrix.

  2. 2.

Calculate the eigenvalues and eigenvectors of the covariance matrix.

  3. 3.

Sort the eigenvalues in decreasing order.

  4. 4.

Select the top $n$ principal components, capturing the most variance in the
data.

  5. 5.

Project the data onto the selected principal components.

While PCA is effective, interpreting principal components becomes challenging
when dealing with a large number of variables. It is most suitable when
variables exhibit a linear relationship, and susceptibility to significant
outliers should be noted.

####  4.3.2 Isometric Feature Mapping (ISOMAP)

In contrast to PCA, ISOMAP represents a non-linear dimensionality reduction
method. It offers a straightforward technique for estimating the intrinsic
geometry of a data manifold and embedding the data in a lower-dimensional
space. The algorithm derives an estimate of each data point’s neighbors on the
manifold, facilitating an efficient and widely applicable approach to various
data sources and dimensionalities (Van Der Maaten et al., 2009).

ISOMAP is computed through the following steps:

  1. 1.

Build a neighborhood graph: Identify the k-nearest neighbors for each data
point and create edges between points that are mutual k-nearest neighbors.

  2. 2.

Compute geodesic distances between all data points: Geodesic distance, the
shortest path on the neighborhood graph between two points, is calculated.

  3. 3.

Compute the embedding matrix: Utilize classical multidimensional scaling (MDS)
to compute an embedding matrix, representing each data point in a lower-
dimensional space.

###  4.4 Clustering Methods

In this paper, we employ two distinct clustering approaches: K-means and
Hierarchical Density-Based Spatial Clustering of Applications with Noise
(HDBSCAN).

####  4.4.1 K-means

The K-means algorithm clusters data by attempting to group samples into $n$
clusters with equal variance, minimizing a criterion known as inertia or
within-cluster sum-of-squares. This algorithm requires specifying the number
of clusters and performs effectively on large sample sizes, finding
application across various fields. The K-means algorithm partitions a set of
$N$ samples, $X$, into $K$ disjoint clusters, $C$, each characterized by the
mean, $\mu_{i}$, of the samples in the cluster. These means are commonly
referred to as cluster "centroids"; note that they are not necessarily points
from $X$, although they exist in the same space. Minimization of inertia is
defined by

$\sum_{i=0}^{n}\min_{\mu_{j}\in C}(||x_{i}-\mu_{j}||^{2})$ (14)

####  4.4.2 Hierarchical Density-Based Spatial Clustering of Applications with
Noise (HDBSCAN)

HDBSCAN performs DBSCAN over varying epsilon values and integrates the result
to identify a clustering that provides the best stability over epsilon. This
enables HDBSCAN to identify clusters of varying densities, unlike DBSCAN,
making it more robust to parameter selection. DBSCAN views clusters as regions
of high density separated by areas of low density, allowing clusters found by
DBSCAN to take any shape, in contrast to K-means, which assumes clusters are
convex. The central concept in DBSCAN is core samples, located in areas of
high density. A cluster is a set of core samples, each close to one another
(measured by some distance), and a set of non-core samples that are close to a
core sample but are not core samples themselves. The algorithm has two
parameters, min samples and eps, which formally define what is considered
dense. Higher min samples or lower eps indicate higher density required to
form a cluster.

###  4.5 Evaluation Metrics

In this section, we examine three widely used evaluation metrics for
clustering tasks.

Silhouette Coefficient

A higher Silhouette Coefficient score indicates a model with better-defined
clusters. The Silhouette Coefficient is computed for each sample and comprises
two scores (Rousseeuw, 1987):

$a$: The mean distance between a sample and all other points in the same
class.
$b$: The mean distance between a sample and all other points in the next
nearest cluster.

The Silhouette Coefficient $s$ for a single sample is then given by:

$s=\frac{b-a}{\max(a,b)}$ (15)

The Silhouette Coefficient for a set of samples is the mean of the Silhouette
Coefficient for each sample.

Calinski and Harabasz Score

Also known as the Variance Ratio Criterion, the score is defined as the ratio
of the sum of between-cluster dispersion to within-cluster dispersion. For a
dataset $E$ of size $n_{E}$ clustered into $K$ clusters, the Calinski-Harabasz
score $s$ is defined as the ratio of the mean between-clusters dispersion and
within-cluster dispersion (Caliński and Harabasz, 1974):

$s=\frac{\mathrm{tr}(B_{k})}{\mathrm{tr}(W_{k})}\times\frac{n_{E}-k}{k-1}$
(16)

where $\mathrm{tr}(B_{k})$ is the trace of the between-group dispersion
matrix, and $\mathrm{tr}(W_{k})$ is the trace of the within-cluster dispersion
matrix defined by:

$W_{k}=\sum_{q=1}^{k}\sum_{x\in C_{q}}(x-c_{q})(x-c_{q})^{T}$ (17)
$B_{k}=\sum_{q=1}^{k}n_{q}(c_{q}-c_{E})(c_{q}-c_{E})^{T}$ (18)

with $C_{q}$ being the set of points in cluster $q$, $c_{q}$ the center of
cluster $q$, $c_{E}$ the center of $E$, and $n_{q}$ the number of points in
cluster $q$.

Davies-Bouldin Index

A lower Davies-Bouldin index indicates a model with better separation between
clusters. This index represents the average similarity between clusters, where
similarity is a measure comparing the distance between clusters with the size
of the clusters. Zero is the lowest possible score, and values closer to zero
indicate a better partition (Davies and Bouldin, 1979).

The index is defined as the average similarity between each cluster $C_{i}$
for $i=1,\dots,k$ and its most similar one $C_{j}$. In the context of this
index, similarity is defined as a measure $R_{ij}$ that balances:

$s_{i}$: The average distance between each point of cluster $i$ and the
centroid of that cluster (also known as cluster diameter).

$d_{ij}$: The distance between cluster centroids $i$ and $j$.

A simple choice to construct $R_{ij}$ so that it is non-negative and symmetric
is:

$R_{ij}=\frac{s_{i}+s_{j}}{d_{ij}}$ (19)

Then the Davies-Bouldin index is defined as:

$DB=\frac{1}{k}\sum_{i=1}^{k}\max_{i\neq j}R_{ij}$ (20)

###  4.6 Data

The dataset of transportation networks is provided by the Transportation
Networks for Research Core Team (Stabler et al., 2023). This collection
includes transportation networks for 14 renowned cities worldwide (Figure 1).
The datasets consist of network structures (node-link relations), node
locations, origin-destination trip data, and, for some of them, traffic
assignment data. These networks include Anaheim (Ban and Jayakrishnan, 1992),
Austin (Xie, 2005), Barcelona (Stabler et al., 2023), Berlin (Jahn et al.,
2005), Birmingham (Vuren and Noekel, 2016), Chicago Sketch and Regional (Eash
et al., 1979; Boyce et al., 1985), Eastern Massachusetts (Zhang et al., 2016),
Golden Coast (Bliemer, 2016a), Philadelphia (Walker, 2016), SiouxFalls
(LeBlanc et al., 1975), Sydney (Bliemer, 2016b), and Winnipeg (Codina, 2016).
Chicago-Sketch is a fairly realistic yet aggregated representation of the
Chicago-Region network.

Figure 1: Transportation networks projected on a world map.

##  5 Results and Discussion

###  5.1 Network Characteristics

Table 1 and Table 2 present the general network characteristics for the test
networks used in this study. In Table 3, we present centrality indices, each
of which denotes the average values of centrality indices across all nodes in
the network. Additionally, we established a linear regression model to
estimate the number of links in the network based on the number of nodes.
Figure 2 illustrates the resulting line, expressed by $y=2.32x+1165$, where
$x$ and $y$ are the number of nodes and links, respectively. This line, with
an $R^{2}$ of $0.98$, demonstrates a high level of accuracy. The coefficient
of $2.32$ in the formula indicates that, on average, each node is connected to
$2.32$ links. In a similar study (Badhrudeen et al., 2022), the formula for
the fitted line was $y=1.33x+2907$, suggesting that, on average, each node is
connected to $1.33$ links. The distinction lies in the fact that the authors
in that study considered urban road networks, which typically include all
classifications of roads from freeways and highways to secondary roads and
even alleys. In contrast, the transportation networks used in this study
specifically represent primary roads in urban regions, excluding other road
types.

Table 1: Network Characteristics (General Features) Network | Nodes | Links | Link Length | Density | Diameter | Radius | Reciprocity | Transitivity
---|---|---|---|---|---|---|---|---
Anaheim | 416 | 914 | 0.510 | 0.005 | 31 | 16 | 0.613 | 0.035
Austin | 7388 | 18961 | 0.593 | 0 | 118 | 73 | 0.883 | 0.014
Barcelona | 1020 | 2522 | 0.645 | 0.002 | 31 | 20 | 0.574 | 0.069
Berlin | 12981 | 28376 | 0.052 | 0 | 116 | 72 | 0.486 | 0.216
Birmingham | 14639 | 33937 | 1.205 | 0 | 139 | 69 | 0.765 | 0.089
Chicago-Regional | 12982 | 39018 | 0.693 | 0 | 111 | 58 | 0.943 | 0.046
Chicago-Sketch | 933 | 2950 | 2.778 | 0.003 | 32 | 18 | 1 | 0.066
Eastern-Massachusetts | 74 | 258 | 8.55 | 0.048 | 9 | 5 | 1 | 0.223
Gold Coast | 4807 | 11140 | 0.243 | 0 | 135 | 70 | 0.931 | 0.043
Munich | 742 | 1872 | 0.502 | 0.003 | 56 | 29 | 1 | 0.028
Philadelphia | 13389 | 40003 | 0.457 | 0 | 98 | 59 | 0.938 | 0.013
SiouxFalls | 24 | 76 | 4.132 | 0.138 | 6 | 4 | 0.966 | 0.052
Sydney | 33837 | 75379 | 0.246 | 0 | 24 | 13 | 0.578 | 0.006
Winnipeg | 1052 | 2836 | 0.748 | 0.003 | 40 | 29 | 0.578 | 0.138
Table 2: Network Characteristics (General Features-Continued) Network | GSCC | GWCC | SCCs | WCCs | ACC | DAC | ASPL | AGE | ALE
---|---|---|---|---|---|---|---|---|---
Anaheim | 416 | 416 | 1 | 1 | 0.049 | 0.28 | 11.868 | 0.11 | 0.065
Austin | 7381 | 7388 | 8 | 1 | 0.01 | 0.245 | 48.627 | 0.028 | 0.011
Barcelona | 929 | 930 | 92 | 91 | 0.058 | -0.005 | 14.009 | 0.077 | 0.062
Berlin | 12842 | 12981 | 140 | 1 | 0.152 | 0.169 | 50.649 | 0.024 | 0.158
Birmingham | 14560 | 14578 | 41 | 28 | 0.088 | 0.258 | 44.243 | 0.027 | 0.093
Chicago-Regional | 12978 | 12979 | 5 | 4 | 0.037 | -0.013 | 44.711 | 0.029 | 0.037
Chicago-Sketch | 933 | 933 | 1 | 1 | 0.034 | -0.061 | 12.676 | 0.105 | 0.037
Eastern-Massachusetts | 74 | 74 | 1 | 1 | 0.287 | -0.169 | 4.465 | 0.288 | 0.308
Gold Coast | 4783 | 4783 | 25 | 25 | 0.031 | 0.247 | 54.081 | 0.027 | 0.031
Munich | 742 | 742 | 1 | 1 | 0.017 | 0.209 | 19.716 | 0.073 | 0.017
Philadelphia | 13389 | 13389 | 1 | 1 | 0.012 | 0.162 | 43.173 | 0.029 | 0.013
SiouxFalls | 24 | 24 | 12 | 12 | 0.052 | 0.162 | 3.011 | 0.427 | 0.053
Sydney | 32956 | 32956 | 14 | 14 | 0.005 | 0.216 | 81.574 | 0.016 | 0.005
Winnipeg | 1040 | 1040 | 1057 | 109 | 0.013 | -0.028 | 18.848 | 0.069 | 0.085
Table 3: Network Characteristics (Centrality Indices) Network | IC | OC | CC | BC | EC | PC
---|---|---|---|---|---|---
Anaheim | 0.00530 | 0.00530 | 0.08680 | 0.03090 | 0.01970 | 0.00240
Austin | 0.00030 | 0.00030 | 0.02090 | 0.00670 | 0.00130 | 0.00010
Barcelona | 0.00240 | 0.00240 | 0.06550 | 0.01220 | 0.00600 | 0.00100
Berlin | 0.00017 | 0.00017 | 0.02092 | 0.00394 | 0.00033 | 0.00008
Birmingham | 0.00020 | 0.00020 | 0.02680 | 0.00310 | 0.00050 | 0.00010
Chicago-Regional | 0.00023 | 0.00023 | 0.02284 | 0.00352 | 0.00075 | 0.00008
Chicago-Sketch | 0.00339 | 0.00339 | 0.08003 | 0.01466 | 0.01162 | 0.00107
Eastern-Massachusetts | 0.04776 | 0.04776 | 0.22951 | 0.07386 | 0.08752 | 0.01351
GoldCoast | 0.00048 | 0.00048 | 0.01873 | 0.01134 | 0.00097 | 0.00021
Munich | 0.00340 | 0.00340 | 0.05288 | 0.02792 | 0.01164 | 0.00135
Philadelphia | 0.00022 | 0.00022 | 0.02347 | 0.00330 | 0.00078 | 0.00007
SiouxFalls | 0.13768 | 0.13768 | 0.33630 | 0.16712 | 0.18337 | 0.04167
Sydney | 0.00007 | 0.00007 | 0.01411 | 0.00247 | 0.00015 | 0.00003
Winnipeg | 0.00257 | 0.00257 | 0.05280 | 0.01844 | 0.00307 | 0.00095
Figure 2: Relationship between number of nodes and number of links in the
transportation networks.

###  5.2 Dimensionality Reduction

This section presents the results of adopting dimensionality reduction
methods. Figure 3 displays the outcome of implementing PCA. The principal
components are ordered based on the variance they explain, with the first
principal component accounting for the most variance. The explained variance
graph of PCA illustrates the percentage of the total variance explained by
each principal component. In this study, the first two principal components
explain most of the variance, suggesting that using only these two components
may be sufficient for subsequent analysis.

Figure 3: Explained variance of each principal component in PCA method.

Figure 4 illustrates the outcomes of the dimensionality reduction methods,
PCA, and ISOMAP, as applied to transportation networks characteristics.
Examining the projected space generated by PCA and ISOMAP, it is clear that
clusters are more intuitively recognizable in the results obtained from PCA
than in those from ISOMAP. While this intuitive assessment is useful, our
determination of which dimensionality reduction method is superior must rely
on numerical analysis, thoroughly explored in the following subsection.

Figure 4: Projected space of dimensionality reduction methods: a) PCA, and b)
ISOMAP

###  5.3 Clustering Performance

This section evaluates the clustering methods and the upstream dimensionality
reduction approaches. Given that two dimensionality reduction approaches and
two clustering methods are utilized, four combinations are evaluated using the
metrics described in subsection 4.5, with the results presented in Table 4.
Larger values in Silhouette and Calinski-Harabasz scores indicate better
performance, while the Davies-Bouldin score indicates the best models with the
lowest values. Based on the results in this Table, the combination of PCA
followed by the K-means clustering method outperforms the other three
combinations. Consequently, this combination is selected for further analysis
of results.

The associated results of the K-means / PCA clustering are depicted in Figure
5. The figure shows that five clusters are chosen, and data points in each
cluster are closely grouped. Cluster 2 and Cluster 5 include only one data
point, indicating that these data points are unique and cannot be placed in
other clusters.

Table 5 lists the transportation networks classified in each cluster obtained
from K-means / PCA. Cluster 1 comprises the transportation networks of large
urban areas, characterized by a large number of nodes and links. These
networks exhibit higher resilience when facing congestions or disruptions in
critical nodes, as they offer more alternative links and paths. Cluster 2
exclusively accommodates the SiouxFalls dataset, a publicly known and widely
used transportation network for various purposes. While SiouxFalls is
valuable, it represents a rough approximation of the city, making it less
suitable for real-world applications. Barcelona and Winnipeg are classified in
Cluster 3, representing medium-sized transportation networks with virtually
identical features. Cluster 4 encompasses small-sized transportation networks,
including Anaheim, Chicago-Sketch, and Munich. Chicago-Sketch is widely
adopted in traffic assignment studies as a test network. These networks in
Cluster 4 are suitable choices as test networks, offering affordable
computational costs while preserving real-world characteristics. Lastly,
Cluster 5 is solely occupied by the Eastern-Massachusetts network, similar to
SiouxFalls in its wide usage as a test network. Eastern-Massachusetts is
slightly larger than SiouxFalls and differs in appearance, leading our
clustering method to categorize them differently.

Table 4: Clustering evaluation metrics. Method | Silhouette score | Calinski Harabasz score | Davies Bouldin score
---|---|---|---
K-means / PCA | 0.510 | 37.674 | 0.297
K-means / ISOMAP | 0.461 | 27.655 | 0.579
HDBSCAN / PCA | 0.393 | 8.747 | 1.066
HDBSCAN / ISOMAP | 0.155 | 0.931 | 9.211
Figure 5: Visualization of clustering outcome of using method K-means / PCA. Table 5: Classification of transportation networks. Cluster | Networks
---|---
Cluster 1 | Austin, Berlin, Birmingham, Chicago-Regional, Gold Coast, Philadelphia, Sydney
Cluster 2 | SiouxFalls
Cluster 3 | Barcelona, Winnipeg
Cluster 4 | Anaheim, Chicago-Sketch, Munich
Cluster 5 | Eastern-Massachusetts

Figure 6 illustrates the average values of network characteristics for each
cluster. Sub-figures a) and b) depict network features and centrality indices,
respectively. As shown in the figure, features, including the number of nodes
and links, diameter, and radius, follow the same pattern, indicating that
Cluster 1 and Cluster 2 have the highest and lowest values, respectively, with
other clusters ordered similarly. For features such as link length and
density, Cluster 1 shows the lowest values. However, the highest values in
these two features do not follow the same pattern. While Cluster 5 shows the
highest value of link length, Cluster 2 has the highest value of density.
Based on mathematical formulations, diameter and radius are dependent on the
number of links. The results of clustering also highlight this fact,
indicating that Cluster 1, with the highest number of links, has the highest
values for diameter and radius. Moreover, diameter and radius are much similar
in definition, which is shown in the figure.

Furthermore, the results of centrality indices, shown in Figure 6-b, validate
the outcomes of clustering. If our clustering method didn’t work well, we
would expect to see values of a specific centrality index to be similar among
two different colors. However, this discrepancy is not observed in the
results. Similar to Figure 6-a, Figure 6-b follows the pattern that centrality
indices have their lowest values for networks in Cluster 1 and the highest
values for Cluster 2. Centrality values of cluster 1 are low compared to other
clusters, indicating that there are fewer nodes that are placed on the
shortest paths between two other nodes. In other words, nodes in the networks
of cluster 1 are less critical compared to nodes in other clusters, since the
resiliency of the networks in cluster 1 is higher, meaning that there are
several alternative paths to move from one node to another. On the other hand,
centrality indices for cluster 2 (SiouxFalls) have their highest values,
indicating that some nodes in this network are very critical and play a key
role in the total resiliency of the network. If one of those nodes becomes
congested or fails, the overall performance of the network will be severely
affected. The key observation in both of these sub-figures in Figure 6 is that
values of each network characteristic differ among clusters, confirming the
effectiveness of the clustering method used.

The obtained results of this study can assist authors of future studies. Our
Cluster 1 of transportation networks is composed of large-scale networks with
high resiliency characteristics among nodes. Studies that focus on adopting
high computational centrality indices, such as betweenness and closeness
centrality, are suggested to use transportation networks of Clusters 3 or 4 to
mitigate the unbearable long run-times. Also, if a study aims to assess the
relationship between a centrality measure and a characteristic of the network,
such as traffic congestion, and investigates this relationship for different
transportation networks, the authors of the study should choose their test
networks from different clusters obtained in this study. In other words,
choosing two networks such as Barcelona and Winnipeg and claiming that a
proposed method is effective on different transportation networks is not a
valid claim, as both of these networks are classified in identical cluster
(Cluster 3) and have similar characteristics.

A general recommendation is that authors of future studies investigate the
effectiveness of their proposed approaches on at least three transportation
networks: one on SiouxFalls or Eastern-Massachusetts, which are very helpful
for debugging the methods due to their low computational demands; one on a
medium-scale network from Cluster 3 or 4, and finally, one on a network from
Cluster 1 to demonstrate the effectiveness of their proposed method on a
network close to the real world.

Figure 6: Network characteristics in each cluster of transportation networks:
a) Network Feature, and b) Centrality Index

##  6 Conclusion and Future work

In this study, we present a comprehensive framework for the classification of
transportation networks based on their topological features. We calculate
various network characteristics derived from the topological structure of the
network, encompassing both network features and centrality indices.
Subsequently, we employ two dimensionality reduction methods, PCA and ISOMAP.
These methods contribute to reducing dimensions while retaining the most
valuable features and combining highly correlated ones.

Following dimensionality reduction, we apply two clustering approaches, namely
K-means and HDBSCAN, to achieve an unsupervised classification of fourteen
transportation networks. Various metrics are employed to assess the accuracy
of clustering methods. K-means / PCA outperforms the other three combinations
of dimensionality reduction and clustering methods, achieving a Silhouette
metric score of $0.510$. This method results in the identification of $5$
clusters within the networks. An analysis of the topological characteristics
of the networks in each cluster validates the effectiveness of this
classification for the fourteen networks.

Future works may involve expanding the number of transportation networks for
classification, incorporating additional network features, and exploring
alternative clustering approaches that may yield improved results.

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