A Transportation Network Efficiency Measure that Captures Flows, Behavior, and Costs
with Applications to
Network Component Importance Identification and Vulnerability
Anna Nagurney and Qiang Qiang
Isenb erg School of Management
University of Massachusetts
Amherst, Massachusetts 01003
e-mail: nagurney@gbfin.umass.edu; qqiang@som.umass.edu
phone: 413-545-5635; 413-577-2761
Proceedings of the POMS 18th Annual Conference
Dallas, Texas, U.S.A.
May 4 to May 7, 2007
Abstract: In this paper, we propose a transportation network efficiency measure that can
be used to assess the performance of a transportation network and which differs from other
proposed measures, including complex network measures, in that it captures flows, costs,
and travel behavior information, along with the topology. The new transportation network
efficiency measure allows one to determine the criticality of various nodes (as well as links) as
we demonstrate through a network component importance definition, which is well-defined
even if the network becomes disconnected. Several illustrative transportation network ex-
amples are provided in which the efficiencies and importance of network components are
explicitly computed, and their rankings tabulated.
This framework can be utilized to assess the vulnerability of network components in terms
of their criticality to network efficiency/performance and to, ultimately, enhance security.
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1. Introduction
Networks and, in particular, complex networks, have been the subject of intense research
activity in recent years although the topic, which is based on graph theory, is centuries
old. Indeed, the subject of networks, with its rich applications has been tackled by oper-
ations researchers/management scientists, applied mathematicians, economists, engineers,
physicists, biologists, and sociologists; see, for some examples: Beckmann, McGuire, and
Winsten (1956), Sheffi (1985), Ahuja, Magnanti, and Orlin (1993), Nagurney (1993), Pa-
triksson (1994), Ran and Boyce (1996), Watts and Strogatz (1998), Barab´asi and Albert
(1999), Latora and Marchiori (2001), Newman (2003), Roughgarden (2005), and the refer-
ences therein. Three types of networks, in particular, have received recent intense attention,
especially in regards to the development of network measures, and we note, specifically, the
random network model (Erd¨os-R´enyi, 1960), the small-world model (Watts and Strogatz,
1998), and scale-free networks (Barab´asi and Albert, 1999).
The importance of studying and identifying the vulnerable components of a network,
in turn, has been linked to events such as 9/11 and to Hurricane Katrina, as well as to
the biggest blackout in North America that occurred on August 14, 2003 (cf. Sheffi, 2005;
Nagurney, 2006). In order to hedge against terrorism and natural disasters, a majority of
the associated complex network (sometimes also referred to as network science) literature
(cf. the survey by Newman, 2003) focuses on the graph characteristics (e.g. connectivity
between nodes) of the associated application in order to evaluate the network reliability and
vulnerability; see also, for example, Chassin and Posse (2005) and Holme et al. (2002).
However, in order to be able to evaluate the vulnerability and the reliability of a network,
a measure that can quantifiably capture the efficiency/performance of a network must be
developed. For example, in a series of papers, beginning in 2001, Latora and Marchiori
discussed the network performance issue by measuring the “global efficiency” in a weighted
network as compared to that of the simple non-weighted small-world network. In a weighted
network, the network is not only characterized by the edges that connect different nodes,
but also by the weights associated with different edges in order to capture the relationships
between different nodes. The network efficiency E of a network G is defined in the paper
of Latora and Marchiori (2001) as E =
1
n(n−1)
P
i6=j∈G
1
d
ij
, where n is the number of nodes
in G and d
ij
is the shortest path length (the geodesic distance) between nodes i and j.
This measure has been applied by the above authors to a variety of networks, including the
(MBTA) Boston subway transportation network and the Internet (cf. Latora and Marchiori
2002, 2004).
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Although the topological structure of a network obviously has an impact on network
performance and the vulnerability of the network, we believe that the flow on a network
is also an important indicator, as are the induced costs, and the behavior of users of the
network(s). Indeed, flows represent the usage of a network and which paths and links have
positive flows and the magnitude of these flows are relevant in the case of network disruptions.
Interestingly, although recently a few papers have appeared in the complex network literature
that emphasize flows on a transportation network, with a focus on airline networks (cf.
Barrat, Barth´elemy, and Vespignani, 2005, and Dall’Asta et al., 2006), the aforementioned
papers only consider the importance of nodes and not that of links and ignore the behavior
of users. It is well-known in the transportation literature that the users’ perception of the
travel costs will affect the traffic pattern on the network (see, e.g., Beckmann, McGuire,
and Winsten, 1956; Dafermos and Sparrow, 1969; Boyce et al., 1983; Ran and Boyce, 1996,
and Nagurney, 1993). Therefore, a network efficiency measure that captures flows, the
costs associated with “travel,” and user behavior, along with the network topology, is more
appropriate in evaluating networks such as transportation networks, which are the classical
critical infrastructure. Indeed, in the case of disruptions, which can affect either nodes,
or links, or both, we can expect travelers to readjust their behavior and the usage of the
network accordingly. Furthermore, as noted by Jenelius, Petersen, and Mattsson (2006), the
criticality of a network component, consisting of a node, link, or combination of nodes and
links, is related to the vulnerability of the network system in that the more critical (or, as
we consider, the more important) the component, the greater the damage to the network
system when this component is removed, be it through natural disasters, terrorist attacks,
structural failures, etc.
In this paper, we propose a new transportation network performance measure that can
be used to evaluate the efficiency of a transportation network as well as the importance of
its network comp onents. We also relate the new measure to the Latora and Marchiori (2001)
measure used in the “complex” network literature. In addition, we compare the resulting
network component importance definitions derived from our measure to those recently pro-
posed by Jenelius, Petersen, and Mattsson (2006) (see also Taylor and D’Este, 2004) and
also provide illustrative examples. Our measure has the additional notable feature that it
is applicable, as is our proposed importance definition of network components, even in the
case that the network becomes disconnected (after the removal of the component).
The paper is organized as follows. In Section 2, we present some preliminaries. The
new transportation network efficiency measure is introduced in Section 3, along with the
associated definition of the importance of network components. We also prove that the new
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measure contains, as a sp ecial case, an existing measure, due to Latora and Marchiori (2001),
that has been much studied and applied in the complex network literature. Section 4 then
presents three network examples for which the efficiency measures are computed and the node
and link importance rankings determined using the new transportation network efficiency
measure. Comparisons with the Latora and Marchiori (2001) measure are also provided, for
completeness, and with the Jenelius, Petersen, and Mattsson (2006) importance indicators
in the case of the link components. Section 5 then applies the new measure to a larger scale
network to further illustrate the applicability of the proposed measure. Section 6 summarizes
the results in this paper and provides suggestions for future research.
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2. Some Preliminaries
In this Section, we recall the transportation network equilibrium model with fixed de-
mands, due to Dafermos (1980) (see also Smith, 1979). We consider a network G with the
set of links L with n
L
elements and the set of nodes N with n elements. The set of ori-
gin/destination (O/D) pairs of nodes is denoted by W and consists of n
W
elements. We
denote the set of paths joining an O/D pair of nodes w by P
w
and the set of all paths by P .
Links are denoted by a, b, etc; paths by p, q, etc., and O/D pairs by w
1
, w
2
, etc. Links are
assumed to be directed and paths to be acyclic.
We denote the nonnegative flow on path p by x
p
and the flow on link a by f
a
. The link
flows are related to the path flows through the following conservation of flow equations:
f
a
=
X
p∈P
x
p
δ
ap
, ∀a ∈ L, (1)
where δ
ap
= 1, if link a is contained in path p, and δ
ap
= 0, otherwise. Hence, the flow on
a link is equal to the sum of the flows on paths that contain that link. We group the link
flows into the vector f ∈ R
n
L
+
and the path flows into the vector x ∈ R
n
P
+
.
The demand for O/D pair w is denoted by d
w
and is assumed to be positive. We assume
that the following conservation of flow equations hold:
X
p∈P
w
x
p
= d
w
, ∀w ∈ W, (2)
that is, the sum of flows on paths connecting each O/D pair w must be equal to the demand
for w.
The cost on a path p is denoted by C
p
and the cost on a link a by c
a
.
The costs on paths are related to costs on links through the following equations:
C
p
=
X
a∈L
c
a
δ
ap
, ∀p ∈ P, (3)
that is, the cost on a path is equal to the sum of costs on links that make up the path.
Furthermore, we allow the cost on a link to depend, in general, upon the flows on the
network links, so that
c
a
= c
a
(f), ∀a ∈ L, (4)
and we assume that the link cost functions are continuous and strictly monotonically in-
creasing (cf. Nagurney, 1993) so that the equilibrium link flows, defined below, will be
unique.
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