2019 International Conference on Computational Modeling, Simulation and Optimization (CMSO 2019) ISBN: 978-1-60595-659-6

**Topological Connectivity Analysis on Freight Transportation Networks **

### Xi-sheng SHEN

1_{, Xiao-fang WANG}

2,*_{, Lian-yi ZHANG}

3_{ and Xi-fu WANG}

1
1_{School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China }

2_{School of Business, Renmin University of China, Beijing 100872, China }

3_{Science and Technology on Space System Simulation Laboratory, Beijing Simulation Center, }

Beijing 100854, China *Corresponding author

**Keywords:** Topological connectivity, Degree distribution, Betweenness, Freight transportation
networks.

**Abstract.** The connections between nodes in Freight Transportation Networks (FTNs) are highly
complex. With the topological modelling of real world FTNs, this paper introduces quantitative
indicators, such as degree distribution and edge/node betweenness centrality, to analyze the
topological connectivity. The exponential laws of degree distribution and edge/node betweenness
centrality are shown by numerical results on the coal transportation network in Shanxi, China.

**Introduction **

Freight transportation networks (FTNs) consists of a mass of nodes (i.e., demanders, providers, railway stations, highway transport hubs and ports et al), and the connections between nodes are seriously complex. Because there are many kinds of transportation, such as railways, waterways, highways et al, and goods are distributed from origins to destinations, going through different types of nodes and transport ways. Thus, it is very important to reveal the topological connectivity and its strength of FTNs, since properties such as robustness against damages are nontrivial consequences of their topological structure.

Recently, research on topological structure of complex networks has focused on many areas, such as electric power grids, social networks, biological networks, urban traffic networks, maritime freight networks and airline networks. Crucitti et al [1] and Albert et al [2] investigate the topological properties of electric power grid, and the structural vulnerability of the Italian GRTN power grid and the North American power grid respectively. Chasman and Siahpirani et al [3] consider complex biological systems of cellular function under changing environmental conditions and propose network-based computational approaches to gain a global understanding of cellular function under changing environmental conditions. Andrade and Espindola et al [4] apply complex networks indexes to the topologies to reveal structural patterns of interactions between marine plankton organisms. Atay and Koc et al [5] analyze community detection from biological and social networks by several methods containing a modularity optimization approach. For a complex weighted network of travel routes on the Singapore rail and bus transportation systems, Soh and Lim et al [6] show that the dynamical associativity of the bus networks differ from its topological counterpart. DβLima and Medda [7] propose a quantitative measure of resilience using a mean-reverting stochastic model and apply this model to the case of the London Underground. Fremont [8] presents a number of different strategies for linking the different regions of the world and study the route structure, port situation and development trend of Maersk's shipping network. Lee and Choo [9] present a bi-level modeling approach for capturing hierarchical relationships among major carriers and finding carrier decision-making processes in maritime freight networks.

transportation network in Shanxi, China are given to testify the exponential laws of degree distribution and edge/node betweenness centrality.

**Topological Model of Freight Transportation Networks **

In this section, we model the topological networks of real world FTNs by graph theory. There are many kinds of nodes in FTNs, such as demanders, providers, highway transport hubs, railway stations and ports et al. They are denoted by π£πs, and the collection of these nodes is π = {π£1, π£2, β¦ , π£π}.

The connections between nodes in FTNs are highly complex, such as railways, highways,
waterways and other means of transportation. As shown in Fig.1, the travelling of goods from the
origin to the destination, generally needs to go through different types of nodes and the coordination
among different transport ways. Let π_{ππ} be the connection between nodes π£_{π} and π£_{π}. The collection of
these edges is πΈ = {π_{ππ}| There is an edge between the pair of nodes (π£_{π}, π£_{π}) β π Γ π, π β π}.

Origin Port Port Destination

Railway station

Highway Waterway Railway Highway

Figure 1. The travelling of goods from the origin to the destination.

Weight π€_{ππ} of an edge π_{ππ} represents the attributes of transportation between the pair of nodes

(π£_{π}, π£π), such as mileage, vehicle speed, passing capacity, shipping cost and delivery time et al. For

simplicity, this paper only considers the mileage attribute. The collection of weights is denoted by

π = {π€_{ππ}|π_{ππ}β πΈ}. Thus, FTNs can be modelled as a graph πΊ = (π, πΈ, π). Let π Γ π matrix π΄ be the
adjacency matrix of the graph πΊ, with elements πππ= 1 only if there is some ways of transportation

between the pair of nodes (π£_{π}, π£_{π}), otherwise π_{ππ} = 0. If the graph πΊ is undirected, the existence of a
link implies that π_{ππ}= π_{ππ}, and the adjacency matrix π΄ = π΄π is a real symmetric matrix. It is assumed
that the graph πΊ does not contain self-loops, that is, π_{ππ} = 0.

**Topological Connectivity Indicators **

In this section, we introduce some quantitative indicators to analyze the topological connectivity of FTNs, including degree distribution and betweenness centrality as follows.

**Degree Distribution. **For a node in the FTN, the number of its links with other nodes is its basic
structural property, since the larger the number is, the more important the node is in the sense of
topological structure. We can analyze the topological connectivity of FTNs by the number of these
links between its nodes. For a node π£_{π} β π, the total number of its adjacent edges is called its degree

ππ, that is, ππ = βππ=1πππ. And the average of all node degrees is called the average degree < π >,

which is equal to βπ_{π=1}π_{π}/π. The probability that a randomly selected node has exactly π adjacent
edges is called degree distribution function π(π), which can be evaluated by the fraction of nodes in
the FTN that have degree π, that is, π(π) = π_{π}/π, where π_{π} is the number of nodes with degree π. In
order to reduce the statistical fluctuation, we can use the cumulative degree distribution function π(π)
instead, which is defined by the probability that the degree is greater than or equal to π,

π(π) = β π(π)

πβ₯π

.

For an edge π_{ππ}β πΈ, the fraction of shortest paths that pass through π_{ππ} is called edge betweenness
centrality π΅_{e}(π_{ππ}), that is, π΅_{e}(π_{ππ}) = β_{πβ πβπ}π_{ππ}(π_{ππ})/π_{ππ}, where π_{ππ} is the total number of shortest
paths from node π to node π, and πππ(πππ) is the number of those shortest paths that pass through πππ.

The probability that the betweenness centrality of a randomly selected edge is greater than or equal to

π, is called cumulative edge-betweenness distribution function πe(π), which can be calculated by the

fraction of the number of π΅e(πππ) β₯ π,

π_{e}(π) = πe(π)
π_{e} ,

where πe is the number of all π΅e(πππ), and πe(π) is the number of π΅e(πππ) β₯ π.

Similarly, for a node π£_{π}β π, the fraction of shortest paths that pass through π£_{π} is called node
betweenness centrality π΅_{n}(π£_{π}), that is, π΅_{n}(π£_{π}) = β_{πβ πβπ}π_{ππ}(π£_{π})/π_{ππ}, where π_{ππ}(π£_{π}) is the number of
those shortest paths that pass through π£_{π}. The cumulative node-betweenness distribution function

π_{n}(π) is

πn(π) =

πn(π)

πn ,

where π_{n} is the number of all π΅_{n}(π£_{π}), and π_{n}(π) is the number of π΅_{n}(π£_{π}) β₯ π.

**Numerical Results **

In this section, numerical results are presented to illustrate the effectiveness of the introduced quantitative indicators to evaluate the topological connectivity of FTNs. We conduct numerical experiments by R based on the data of the coal transportation network in Shanxi, China (CTNS) [10]. We first investigate the topological connectivity of the CTNS by its cumulative degree distribution

π(π). The degree ππ for all nodes and the degree distribution π(π) are shown in Fig.2. The average

degree < π >= 4.22. Fig.2 (b) shown that the degree distribution π(π) contains a single peak at value < π > and declines sharply away from the peak value.

(a) Degree ππ of all nodes π£πβ π (b) Degree distribution π(π)

Figure 2. Degree ππ and degree distribution function π(π) for the CTNS.

The cumulative degree distribution π(π) for the CTNS are shown in Fig.3 in the single/double logarithmic coordinates. Fig.3 (a) shows that in the linear-log plot, the cumulative degree distribution falls on a straight line nearly. This indicates that the cumulative degree distribution decreases exponentially, since the full line is an exponential fit to the data. In contrast, Fig.3 (b) shows that in the log-log plot, the cumulative degree distribution rejects to fall on a straight line, which is a power law fit to the data, and the tail of the cumulative degree distribution decreases even faster than a power law would. The cumulative degree distribution for the CTNS satisfies an exponential law rather than a power law, that is, π(π) β πβππ, where π is the scaling exponent. Thus, the CTNS is a single scale network and the scaling exponent is π β 0.386.

0 10 20 30 40 50 60

2

4

6

8

10

Index

D

π£π

ππ

k

p

k

0.00

0.05

0.10

0.15

0.20

0.25

0 2 4 6 8 10 12 14 16 π

π(

(a) Linear-log plot of π(π) (b) Log-log plot of π(π)

Figure 3. The cumulative degree distribution π(π) for the CTNS in the single/double logarithmic coordinates.

Next, we study the property of the connectivity strength of the CTNS by its edge/node betweenness
distribution. The cumulative edge/node-betweenness distribution π_{e}(π) and π_{n}(π) for the CTNS are
shown in Fig.4. Fig.4 (a) and (c) show that in the linear-log plot, the cumulative
edge/node-betweenness distributions scatter around a straight line. That is, both the cumulative
edge/node-betweenness distributions seem to decay exponentially. On the contrary, Fig.4 (b) and (d)
show that in the log-log plot, the cumulative edge/node-betweenness distributions deviate
significantly from a power law one, and their tails appear to decay faster than a power law would.
Thus, the cumulative edge/node-betweenness distribution for the CTNS also satisfy an exponential
law rather than a power law. For the CTNS, the edge/node scaling exponents are πe β 0.028 and

πn β 0.010.

(a) Linear-log plot of πe(π) (b) Log-log plot of πe(π)

(c) Linear-log plot of πn(π) (d) Log-log plot of πn(π)

Figure 4. The cumulative edge/node-betweenness distribution πe(π) and πn(π) for the

CTNS in the single/double logarithmic coordinates.

2 4 6 8 10 12

-4
-3
-2
-1
0
*k*
*log*
* P(k)*
π
log
π
(π
)
log
π
(π
)

0.0 0.5 1.0 1.5 2.0 2.5

-4

-3

-2

-1

0

θηΉηεΊ¦D

η΄―

η§―εΊ¦εεΈ

log

(DFC)

log π

0 50 100 150

-5 -4 -3 -2 -1 0 Edge Betweenness log (Edg e Be twee nn ess Distri bu tion ) π log πe (π )

0 1 2 3 4 5

-5 -4 -3 -2 -1 0 log(Edge Betweenness) log (Edg e Be twee nn ess Distri bu tion ) log π log πe (π )

0 50 100 150 200 250 300

-4 -3 -2 -1 Node Betweenness log (Nod e Be twee nn ess Distri bu tion ) π log πn (π )

1 2 3 4 5

**Conclusion **

In this paper, we investigate the topological connectivity for FTNs on the base of the topological
modelling. Quantitative indicators, such as degree distribution and edge/node-betweenness centrality,
are introduced. Numerical results on the coal transportation network in Shanxi, China are given to
show that: (i) The cumulative degree distribution for the CTNS satisfies an exponential law rather
than a power law with the scaling exponent π β 0.386, and the CTNS is a single scale network. (ii)
The cumulative edge/node-betweenness distribution for the CTNS also satisfy an exponential law
rather than a power law, with the edge/node scaling exponents π_{e} β 0.028 and π_{n} β 0.010.

**Acknowledgement **

Xiao-fang Wang is the corresponding author. This work was supported in part by the following grants: National key R&D program of China No. 2017YFC0820100.

**References **

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