Robustness of Trans-European Gas Networks

Abstract

Here, we uncover the load and fault-tolerant backbones of the trans-European gas pipeline network. Combining topological data with information on intercountry flows, we estimate the global load of the network and its tolerance to failures. To do this, we apply two complementary methods generalized from the betweenness centrality and the maximum flow. We find that the gas pipeline network has grown to satisfy a dual purpose. On one hand, the major pipelines are crossed by a large number of shortest paths thereby increasing the efficiency of the network; on the other hand, a nonoperational pipeline causes only a minimal impact on network capacity, implying that the network is error tolerant. These findings suggest that the trans-European gas pipeline network is robust, i.e., error tolerant to failures of high load links.

Full text

Robustness of trans-European gas networks
Rui Carvalho,
1,
*
Lubos Buzna,
2,3,†
Flavio Bono,
4,‡
Eugenio Gutiérrez,
4
Wolfram Just,
1
and David Arrowsmith
1
1
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
2
ETH Zurich, UNO C 14, Universitätstrasse 41, 8092 Zurich, Switzerland
3
University of Zilina, Univerzitna 8215/5, 01026 Zilina, Slovakia
4
European Laboratory for Structural Assessment, Joint Research Centre, Institute for the Protection and Security of the Citizen (IPSC),
Via. E. Fermi, 1 TP 480, Ispra, 21027 Varese, Italy
共Received 2 March 2009; published 10 July 2009
兲
Here, we uncover the load and fault-tolerant backbones of the trans-European gas pipeline network. Com-
bining topological data with information on intercountry flows, we estimate the global load of the network and
its tolerance to failures. To do this, we apply two complementary methods generalized from the betweenness
centrality and the maximum flow. We find that the gas pipeline network has grown to satisfy a dual purpose.
On one hand, the major pipelines are crossed by a large number of shortest paths thereby increasing the
efficiency of the network; on the other hand, a nonoperational pipeline causes only a minimal impact on
network capacity, implying that the network is error tolerant. These findings suggest that the trans-European
gas pipeline network is robust, i.e., error tolerant to failures of high load links.
DOI: 10.1103/PhysRevE.80.016106 PACS number共s兲: 89.75.Fb, 05.10.⫺a
I. INTRODUCTION
The world is going through a period when research in
energy is overarching 关1,2兴. Oil and gas prices are volatile
because of geopolitical and financial crises. The rate of
world-wide energy consumption has been accelerating, while
gas resources are dwindling fast. Concerns about national
supply and security of energy are on top of the political
agenda, and global climate changes are now believed to be
caused by the release of greenhouse gases into the atmo-
sphere 关1兴.
Although physicists have recently made substantial
progress in the understanding of electrical power grids
关3–10兴, surprisingly little attention has been paid to the struc-
ture of gas pipeline networks. Yet, natural gas is often the
energy of choice for home heating, and it is increasingly
being used instead of oil for transportation 关2,11兴. Although
renewable energy sources offer the best cuts in overall CO
2
emissions, the generation of electricity from natural gas in-
stead of coal can significantly reduce the release of carbon
dioxide to the atmosphere. As the demand for natural gas
rises in Europe, it becomes more important to gain insights
into the global transportation properties of the European gas
network. Unlike electricity, with virtually instantaneous
transmission, the time taken for natural gas to cross Europe
is measured in days. This implies that the coordination
among transport operators is less critical than for power
grids. Therefore, commercial interests of competing opera-
tors often lead to incomplete or incorrect network informa-
tion, even at the topological level. Until now, modeling has
typically been made in small systems by the respective op-
erators, who have detailed knowledge of their own infra-
structures. Nevertheless, Ukraine alone transits approxi-
mately 80% of Russian gas exports to Europe 关12兴,
suggesting the presence of a strong transportation backbone
crossing several European countries.
Historically, critical infrastructure networks have evolved
under the pressure to minimize local rather than global fail-
ures 关13兴. However, little is known on how this local optimi-
zation impacts network robustness and security of supply on
a global scale. The failure of a few important links may
cause major disruption to supply in the network not because
these links connect to degree hubs but because they are part
of major transportation routes that are critical to the opera-
tion of the whole network. Here, we adopt the view that a
robust infrastructure network is one which has evolved to be
error tolerant to failures of high load links. Our method is
slightly different from previous work on real world critical
infrastructure networks with percolation theory 关6,10,14,15兴,
which assume the simultaneous loss of many unrelated net-
work components. The absence of historical records on the
simultaneous failure of a significant percentage of compo-
nents in natural gas networks implies that the methods of
percolation theory are of little practical relevance in our case.
Hence, our approach to the challenge of characterizing the
robustness of global transport on the European gas network
was to characterize the hot transportation backbone which
emerges when measuring network load and error tolerance.
II. TRANS-EUROPEAN GAS NETWORKS
We have extracted the European gas pipeline network
from the Platts Natural Gas geospatial data 关16兴. The data set
cover all European countries 共including non-EU countries
such as Norway and Switzerland兲, North Africa 共main pipe-
lines from Morocco and Tunisia兲, Eastern Europe 共Belarus,
Ukraine, Lithuania, Latvia, Estonia, and Turkey兲 and West-
ern Russia 共see Fig. 1兲.
Similarly to electrical power grids, gas pipeline networks
have two main layers: transmission and distribution. The
*
rui@maths.qmul.ac.uk
†
lbuzna@ethz.ch
‡
flavio.bono@jrc.it
PHYSICAL REVIEW E 80, 016106 共2009兲
1539-3755/2009/80共1兲/016106共9兲 ©2009 The American Physical Society016106-1

transmission network transports natural gas over long dis-
tances 共typically across different countries兲, whereas pipe-
lines at the distribution level cover urban areas and deliver
gas directly to end consumers. We extracted the gas pipeline
transmission network from the complete natural gas network,
as the connected component composed of all the important
pipelines with diameter dⱖ15 inches. To finalize the net-
work, we added all other pipelines interconnecting major
branches 关18兴. We treated the resulting network as undirected
due to the lack of information on the direction of flows.
However, network links are weighted according to pipeline
diameter and length.
The European gas pipeline infrastructure is a continent-
wide sparse network which crosses 38 countries has about
2.4⫻10
4
nodes 关compressor stations, city gate stations, liq-
uefied natural gas 共LNG兲 terminals, storage facilities, etc.兴
connected by approximately 2.5⫻10
4
pipelines 共including
urban pipelines兲, spanning more than 4.3⫻10
5
km 共see
Table I兲. The trans-European gas pipeline network is, in fact,
a union of national infrastructure networks for the transport
and delivery of natural gas over Europe. These networks
have grown under different historical, political, economic,
technological, and geographical constraints and might be
very different from each other from a topological point of
view. The Platts data set did not include volume or direction-
ality of flows. Hence, we assessed the global structure of the
European gas network under the availability of incomplete
information on flows. To reduce uncertainty on flows, we
combined the physical infrastructure network with the net-
work of international natural gas trade movements by pipe-
line for 2007 关19兴共see Fig. 2兲.
To investigate similarities among the national gas net-
works, we first plotted in Fig. 3共a兲 the number N of nodes
versus the number L of links for each country. Figure 3共a兲
suggests that both the transmission and the complete 共i.e.,
transmission and distribution兲 networks have approximately
the same average degree because all points fall approxi-
FIG. 1. 共Color online兲 European gas pipeline network. We show the transmission network 关blue 共dark gray兲 pipelines兴 overlaid with the
distribution network 关brown 共light gray兲 pipelines兴. Link thickness is proportional to the pipeline diameter. We projected the data with the
Lambert azimuthal equal area projection 关17兴. Background colors identify EU member states.
TABLE I. Basic network statistics for the transmission and com-
plete European gas pipeline networks. The complete network is the
union of the transmission and distribution networks.
Statistics
Gas network
共transmission兲
Gas network
共complete network兲
Number of nodes 2207 24010
Number of edges 2696 25554
Total length 共km兲 119961 436289
CARVALHO et al. PHYSICAL REVIEW E 80, 016106 共2009兲
016106-2

mately along a straight line. Indeed, we found 具k
transmission
典
=2.4 and 具k
complete
典= 2.1 关20兴. Surprisingly, the size of the
complete European national gas networks ranges over three
orders of magnitude from two nodes 共former Yugoslav Re-
public of Macedonia兲 to 10334 nodes 共Germany兲. Further,
the German transmission network is considerably larger than
the Italian network. Germany has a long history of industrial
usage of gas and is a major hub for imports from Russia and
the North Sea 关11,21兴.
Given that the national networks have very different sizes
but approximately the same average degree, we looked for
regularities in the probability distribution of degree of the
European gas networks 关see Fig. 3共b兲兴. In accordance with
previous studies of electrical transmission networks 关10,22兴,
the complementary cumulative degree distribution of the
transmission network decays exponentially as P共K⬎ k兲
⬇exp共−k/ ␭兲, with ␭ = 1.44. Unexpectedly, we found that the
degree distribution of the complete gas network is heavy
tailed, as can be seen in the inset of Fig. 3共b兲. This suggests
that the gas network may be approximated by an exponential
network at transmission level but not when the distribution
level is considered as well. The distribution network is
mainly composed of trees which attach to nodes in the trans-
mission network, thus, forming the complete network.
Hence, the fat tails are a combined effect of increasing the
degree of existing transmission nodes in the complete net-
work and adding distribution nodes with lower degrees.
The July 2007 release of the Platts data set, which we
analyzed, did not include information on the capacity of
pipelines. To estimate the pipeline capacity, we compared
cross-border flows based on capacity estimates for an incom-
pressible fluid, where the capacity c can shown to scale as d
␥
with pipeline diameter d 共see the Appendix兲, to reported
cross-border flows extracted from the digitized Gas Trans-
mission Europe 共GTE兲 map 关18兴. Figure 4 is a plot of the
average pipeline capacity versus the pipeline diameter in a
double-logarithmic scale for pipelines in the GTE data set.
We found a good match between the theoretical prediction of
c⬃d
␥
with
␥
⯝2.6, and the capacity of major pipelines as
made evident by the regression to the data c ⬃d
␣
with
␣
=2.46. Hence, we used the exponent
␥
=2.5 as a trade off
between the theoretical prediction and the numerical regres-
sion and approximated c⬃d
2.5
.
To understand the national structure of the network, we
investigated the tendency of highly connected nodes to link
to each other over high capacity pipelines. Figure 5共a兲 shows
the Pearson correlation coefficient between the product of
the degrees of two nodes connected by a pipeline and the
capacity of the pipeline,
FIG. 2. 共Color online兲 Network of international gas trade move-
ments by pipeline 关19兴. Link thickness is proportional to the annual
volume of gas traded.
2 4 6 8 10 12 14 16 18 20
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
k
P(K>k)
10
0
10
1
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Europe
France
Germany
Italy
Slovakia
United Kingdom
Europe (complete)
10
0
10
1
10
2
10
3
10
4
10
0
10
1
10
2
10
3
10
4
Number of Nodes
Number of Links
Gas Network (complete)
Gas Network (transmission)
UK
transmission
Germany
transmission
Italy
complete
UK
complete
Germany
complete
Italy
transmission
(a)
(b)
FIG. 3. 共Color online兲共a兲 Plot of the number L of links versus
the number N of nodes for the transmission and complete 共i.e.,
transmission and distribution兲 natural gas networks of the countries
analyzed. The three largest national transmission and complete net-
works are also labeled 共Germany, UK, and Italy兲. 共b兲 Plot of the
complementary cumulative degree distribution of the European gas
networks, together with national transmission gas networks larger
than 100 pipelines. The inset shows the degree distribution of the
European complete and transmission networks on a double-
logarithmic scale, highlighting the presence of fat tails on the de-
gree distribution of the complete gas pipeline network.
ROBUSTNESS OF TRANS-EUROPEAN GAS NETWORKS PHYSICAL REVIEW E 80, 016106 共2009兲
016106-3

r =
兺
e
ij
共k
i
k
j
− k
i
k
j
兲共c
e
ij
− c
e
ij
兲
冑
兺
e
ij
共k
i
k
j
− k
i
k
j
兲
2
冑
兺
e
ij
共c
e
ij
− c
e
ij
兲
2
, 共1兲
where k
i
and k
j
are the degrees of the nodes at the ends of
pipeline e
ij
, and c
e
ij
is the capacity of the pipeline. Countries
with high values of r have a gas pipeline network where
degree hubs are interconnected by several parallel pipelines.
Further, we plotted the percentage q of capacity on parallel
pipelines for each national network. Typically, countries with
high values of r also have high values of q.
Figure 5共a兲 shows relatively large differences among
countries: Austria, the Czech Republic, Italy and Slovakia
have both high values of r and q. A visual inspection of the
network in these countries uncovers the presence of many
parallel pipelines organized along high capacity corridors
关see Fig. 5共b兲兴. Taken together, these results suggest that
some European national networks have grown structures
characterized by chains of high capacity 共parallel兲 pipelines
over-bridging long distances. This has the consequence of
improving the local error tolerance because the failure of one
pipeline implies only a decrease in flow.
Motivated by the finding of error tolerance in the gas
pipeline networks, we then asked the question of whether
there are global topological properties of the European net-
work which could characterize the network robustness.
III. ANALYSIS OF MAN-MADE DISTRIBUTION NET-
WORKS WITH INCOMPLETE FLOW INFORMATION
To gain insights into the overlaid infrastructure and aggre-
gate flow networks, we propose two complementary ap-
proaches which aim at identifying the backbones of the over-
laid networks in Figs. 1 and 2. In both approaches, the flow
network allows an approximate estimate of the volume of
directed flows as we will detail in Sec. III A.
In the first approach, we assume that transport occurs
along the shortest paths in geographical space. We search for
a global backbone characterized by the presence of flow cor-
ridors where individual components were designed to sustain
high loads.
In the second approach, we look at fault tolerance when
single components fail. Recent studies of network vulner-
ability in infrastructure networks suggest that, although these
networks have exponential degree distributions, under ran-
dom errors or attacks the size of the percolation cluster de-
creases in a way which is reminiscent of scale-free networks
关10,23兴. Typically, these studies presume that a large percent-
age of nodes or links may become nonoperational simulta-
neously, i.e., the time scale of node or link failure is much
faster than the time scale of repair. Whereas the underlying
scenario of a hacker or terrorist attack on an infrastructure
network causing large damage is certainly worth studying
关24兴, the consequences of such attacks may not be assessed
properly when measuring damage by the relative size of the
largest component. Here, we estimate the loss of flow when a
single link is nonoperational. We search for a global back-
bone characterized by corridors of interconnected nodes,
where the removal of one single link causes a high loss of
flow from source to sink nodes.
FIG. 5. 共Color online兲共a兲 Pearson correlation coefficient r for
the degrees of two linked nodes and the capacity of the linking
pipeline and percentage of capacity on parallel pipelines q. We con-
sidered only countries with more than 50 pipelines. 共b兲 Network
layout for the Czech Republic. Link thickness is proportional to the
pipeline capacity.
10
1
10
2
10
−1
10
0
10
1
Pi
p
eline diameter
(
inch
)
Average Capacity (Million m
3
/h)
α = 2.46
FIG. 4. 共Color online兲 Plot of the digitized GTE pipeline capac-
ity versus the pipeline diameter on a double-logarithmic scale. We
digitized the European gas network map from GTE and assigned
the GTE capacities to pipelines in the Platts data set. The straight
line is a regression to the data, which corresponds to c =ad
␣
with
␣
=2.46.
CARVALHO et al. PHYSICAL REVIEW E 80, 016106 共2009兲
016106-4

The maximum flow problem and the shortest paths prob-
lem are complementary, as they capture different aspects of
minimum cost flow. Shortest path problems capture link
lengths but not capacities; maximum flow problems model
link capacities but not lengths.
A. Generalized betweenness centrality
Many networks are in fact substrates, where goods, prod-
ucts, substances, or materials flow from sinks to sources
through components laid out heterogeneously in geographi-
cal space. Examples range from supply networks 关25兴, spa-
tial distribution networks 关26兴, and energy networks 关6兴 to
communication networks 关27兴. Node and link stress in these
networks is often characterized by the betweenness central-
ity. Consider a substrate network G
S
=共V
S
,E
S
兲 with node-set
V
S
and link-set E
S
. The betweenness centrality of link e
ij
苸 E
S
is defined as the relative number of shortest paths be-
tween all pairs of nodes which pass through e
ij
,
g共e
ij
兲 =
兺
s,t苸V
S
s⫽t
␴
s,t
共e
ij
兲
␴
s,t
, 共2兲
where
␴
s,t
is the number of shortest paths from node s to
node t and
␴
s,t
共e
ij
兲 is the number of these paths passing
through link e
ij
. The concept of betweenness centrality was
originally developed to characterize the influence of nodes in
social networks 关28,29兴 and, to our knowledge, was used in
the physics literature in the context of social networks by
Newman 关30兴 and in the context of communication networks
by Goh et al. 关31,32兴.
Betweenness centrality is relevant in man-made networks
which deliver products, substances, or materials as cost con-
straints on these networks condition transportation to occur
along shortest paths. However, nodes and links with high
betweenness in spatial networks are often near the network
barycenter 关33兴, whose location is given by x
G
=兺
i
x
i
/ N,
whereas the most important infrastructure elements are fre-
quently along the periphery, close to either the sources or the
sinks. Although flows are conditioned by a specific set of
sources and sinks, the traffic between these nodes may be
highly heterogeneous and one may have only access to ag-
gregate transport data but not to the detailed flows between
individual sources and sinks 共e.g., competition between op-
erators may prevent the release of detailed data兲. Here, we
propose a generalization of betweenness centrality in the
context of flows taking place on a substrate network, but
where flow data are available only at an aggregate level. We
then show in the next section how the generalized between-
ness centrality can help us to gain insights into the structure
of trans-European gas pipeline networks.
The substrate network is often composed of sets of nodes
which act like aggregate sources and sinks. The aggregation
can be geographical 共e.g., countries, regions, or cities兲 or
organizational 共e.g., companies or institutions兲. If the flow
information is only available at aggregate level then a pos-
sible extension of the betweenness centrality for these net-
works is to weight the number of shortest paths between
pairs of source and sink nodes by the amount of flow which
is known to go through the network between aggregated
pairs of sources and sinks. To do this, we must first create a
flow network by partitioning the substrate network G
S
=共V
S
,E
S
兲, into a set of disjoint subgraphs V
F
=兵共V
S
1
,E
S
1
兲, ...,共V
S
M
,E
M
兲其. The flow network G
F
=共V
F
,E
F
兲
is then defined as the directed network of flows among the
subgraphs in V
F
, where the links E
F
are weighted by the
value of aggregate flow among the V
F
. For our purposes, the
substrate network is the trans-European gas pipeline network
represented in Fig. 1 and the flow network is the network of
international gas trade movements by pipeline in Fig. 2.
The generalized betweenness centrality 共generalized be-
tweenness兲 of link e
ij
苸 E
S
is defined as follows. Let T
K,L
be
the flow from source subgraph K = 共V
K
,E
K
兲苸 V
F
to sink sub-
graph L=共V
L
,E
L
兲苸 V
F
. Take each link e
KL
苸 E
F
and compute
the betweenness centrality from Eq. 共2兲 of e
ij
苸 E
S
restricted
to source nodes s 苸 V
K
and sink nodes t 苸 V
L
. The contribu-
tion of that flow network link is then weighted by T
K,L
and
normalized by the number of links in a complete bipartite
graph between nodes in V
K
and V
L
,
G
ij
=
兺
e
KL
苸E
F
兺
s苸V
K
,t苸V
L
T
K,L
兩V
K
兩兩V
L
兩
␴
st
共e
ij
兲
␴
st
. 共3兲
B. Generalized max-flow betweenness vitality
The maximum flow problem can be stated as follows. In a
network with link capacities, we wish to send as much flow
as possible between two particular nodes, a source and a
sink, without exceeding the capacity of any link 关34兴. For-
mally, an s-t flow network G
F
=共V
F
,E
F
,s , t , c兲 is a digraph
关35兴 with node-set V
F
, link-set E
F
, two distinguished nodes, a
source s and a sink t, and a capacity function c: E
F
→ R
0
+
.A
feasible flow is a function f :E
F
→ R
0
+
satisfying the following
two conditions: 0ⱕ f共e
ij
兲ⱕ c共e
ij
兲, ∀e
ij
苸 E
F
共capacity con-
straints兲; and 兺
j:e
ji
苸E
F
f共e
ji
兲= 兺
j:e
ij
苸E
F
f共e
ij
兲, ∀i苸 V \ 兵s , t其
共flow conservation constraints兲. The maximum s-t flow is de-
fined as the maximum flow into the sink f
st
共G
F
兲
=max关兺
i:e
it
苸E
F
f共e
it
兲兴 subject to the conditions that the flow is
feasible 关34,36,37兴.
We are now interested in the answer to the question. How
does the maximum flow between all sources and sinks
change, if we remove a link e
ij
from the network? In the
absence of a detailed flow model, we calculated the flow that
is lost when a link e
ij
becomes nonoperational assuming that
the network is working at maximum capacity. In agreement
with Eq. 共3兲, we define the generalized max-flow between-
ness vitality 关29,38兴共generalized vitality兲,
V
ij
=
兺
e
KL
苸E
F
兺
s苸V
K
,t苸V
L
T
K,L
兩V
K
兩兩V
L
兩
⌬
st
G
F
共e
ij
兲
f
st
共G
F
兲
, 共4兲
where the amount of flow which must go through link e
ij
when the network is operating at maximum capacity is given
by the vitality of the link 关29兴 ⌬
st
G
F
共e
ij
兲= f
st
共G
F
兲− f
st
共G
F
\e
ij
兲
and f
st
共G
F
兲 is the maximum s-t flow in G
F
.
ROBUSTNESS OF TRANS-EUROPEAN GAS NETWORKS PHYSICAL REVIEW E 80, 016106 共2009兲
016106-5

C. Generalized betweenness centrality
versus max-flow betweenness vitality
A close inspection of Eq. 共3兲, generalized betweenness,
and Eq. 共4兲, generalized vitality, reveals that both measures
have the physical units of gas flow given by T
K,L
. Further, the
relative number of shortest paths crossing a link e
ij
is
bounded by 0 ⱕ
␴
st
共e
ij
兲
␴
st
ⱕ1, and the relative quantity of flow
which must go through the same link e
ij
is also bounded by
0ⱕ
⌬共e
ij
兲
f
st
ⱕ1. Thus, the generalized betweenness 共3兲 and gen-
eralized vitality 共4兲 can be compared for each link.
To examine the relationship between these two quantities,
we considered three simplified illustrative networks: a rooted
tree where the root is the source node and all other nodes are
sinks, the same rooted tree with additional links intercon-
necting children nodes at a selected level, and two commu-
nities of source and sink nodes connected by one single link.
We chose these particular examples because they resemble
subgraphs which appear frequently on the European gas
pipeline network and thus they may help us to gain insights
into the structure of the real world network.
Both the generalized betweenness and vitality have the
same values on the links of trees where the root is the source
node and the other nodes are sinks. To see this, consider
without loss of generality the case when T
K,L
=1. Then, the
generalized betweenness of a link is the proportion of sinks
reachable 共along shortest paths兲 over the link, and the gener-
alized vitality is the proportion of sinks fed by the link. The
two quantities have the same value on the links of a tree and
we illustrated this in Fig. 6共a兲, where we drew link thickness
proportional to the generalized betweenness 共gray兲 and gen-
eralized vitality 共black兲.
Figure 6共b兲 shows a modified tree network where we have
connected child nodes at a chosen level. Here, the shortest
paths between the root and any other node are unchanged
from the example of the tree, but removing a link e
ij
situated
above the lateral interconnection does not cut all connections
between the source 共root兲 and sink nodes. As a consequence,
the values of the generalized vitality are significantly smaller
in the upper part of the graphs. Figure 6共c兲 shows two com-
munities connected by one link, where source nodes are on
one community and sink nodes on the other. This example is
interesting for two reasons. First, the arguments used to ex-
plain why generalized betweenness and vitality take the
same values on trees are also valid in this example. Second,
the link connecting the two communities has a much higher
value of generalized betweenness and vitality than the links
inside the communities, which led us to expect that these two
measures could hint at the presence of modular structure in
the real world network.
IV. NETWORK ROBUSTNESS
The generalized betweenness measure defined by Eq. 共3兲
assumes that gas is transported from sinks to sources along
the shortest paths. To investigate whether this hypothesis is
correct, we plotted the European gas pipeline network and
drew the thickness of each pipeline proportional to the value
of its generalized betweenness 共see Fig. 7兲. We found that
major loads predicted by the generalized betweenness cen-
trality, were on the well-known high capacity transmission
interconnections such as the “Transit system” in the Czech
Republic, the “Eustream” in Slovakia, the “Yammal-Europe”
crossing Belarus and Poland, the “Interconnector” connect-
ing the UK with Belgium, or the “Trans-Mediterranean”
pipeline linking Algeria to mainland Italy through Tunisia
and Sicily. The dramatic difference between the values of
generalized betweenness of all the major European pipelines
and the rest of the network suggests that the network has
grown to some extent to transport natural gas with minimal
losses along the shortest available routes between the sources
and end consumers. These major pipelines are the transpor-
tation backbone of the European natural gas network.
During the winter season, cross-border pipelines are used
close to their full capacity 关18兴. In this situation, the gener-
alized vitality of a pipeline 关Eq. 共4兲兴 can be interpreted as the
network capacity drop or the amount of flow that cannot be
delivered if that pipeline becomes nonoperational. The obvi-
ous drawback of the generalized vitality is that it takes into
account the overall existing network capacity without con-
sidering the length of paths. Conversely, the generalized be-
tweenness considers the length of shortest paths but not the
capacity of pipelines. Since we assess the network from two
(a)
(b)
(c)
K
L
K
K
L
L
FIG. 6. 共Color online兲 Generalized betweenness 共gray兲 and vi-
tality 共black兲 measures on 共a兲 a rooted tree, 共b兲 a modified rooted
tree with interconnections at a chosen level, and 共c兲 two communi-
ties connected by one link. Nodes are shaped according to their
function: source nodes are squares and sink nodes are circles. Both
generalized betweenness and vitality depend on T
K,L
, which is a
constant for all examples. The smaller value of the two quantities is
always drawn on the foreground so that both measures are visible.
CARVALHO et al. PHYSICAL REVIEW E 80, 016106 共2009兲
016106-6

complementary viewpoints, we expect that the results will
allow us to get a more complete picture of the general prop-
erties of the European gas pipeline network.
Figure 8 shows the values of the generalized vitality in the
European gas pipeline network. We found several relatively
isolated segments with a high-generalized vitality located in
Eastern Europe, close to the Spanish-French border, as well
as on the south of Italy. The high values of the generalized
vitality in Eastern Europe can be explained by two factors.
On one hand, the generalized vitality of pipelines close to the
sources is higher than elsewhere simply because these pipe-
lines are the bottleneck of the network. On the other hand,
our approximation that a directed link in the flow matrix
implies gas flowing from all nodes in the source to all nodes
in the sink countries was clearly coarse grained for flows
between Russia and the Baltic states, as it would imply that
pipelines in southern Russia would also supply the Baltic
countries. This highlights boundary effects on the calculation
of betweenness vitality, as the data set excluded most of the
Russian gas pipeline network. The case of the Spanish-
French border was different though. The link with high vi-
tality separates the Iberian Peninsula from the rest of main-
land Europe. If this link was to be cut then Portugal and
Spain would only be linked to the pipeline network through
Morocco. Finally, the south of Italy highlights an interesting
example of two communities 共Europe and North Africa兲
separated by the Trans-Mediterranean pipeline, which is
reminiscent of the example in Fig. 6共c兲.
Perhaps surprisingly, we found that the generalized vital-
ity is more or less homogeneous in most of mainland Europe.
This result suggests that the EU gas pipeline network has
grown to be error tolerant and robust to the loss of single
links.
Distribution networks originate from the need for an ef-
fective connectivity among sources and sinks 关39,40兴. For
example, a spanning tree is highly efficient as it transports
goods from sinks to sources in a way that shortens the total
length of the network, thereby, increasing its efficiency and
viability. If the European gas pipeline network had been built
as a spanning tree, its links would have very similar values
of generalized betweenness and max-flow vitality 共see Fig.
6兲.
The values of the generalized betweenness are consider-
ably higher than the corresponding values of generalized vi-
tality for the most important pipelines in the European
Union. In other words, the major pipelines are crossed by
many shortest paths, but a nonoperational pipeline causes
only a minor capacity drop in the network. This dramatic
contrast between the two measures reveals a hot backbone
关41兴 showing that the trans-European gas pipeline network is
robust, i.e., error tolerant to failures of high load links.
V. CONCLUSIONS
We analyzed the trans-European gas pipeline network
from a topological point of view. We found that the European
national gas pipeline networks have approximately the same
value of average node degree, even if their sizes vary over
three orders of magnitude. Like the electrical power grid, the
degree distribution of the European gas transmission network
decays exponentially. Unexpectedly, the degree distribution
of the complete 共transmission and distribution兲 gas pipeline
network is heavy tailed. In some countries, which are crucial
for the transit of gas in Europe 共Austria, the Czech Republic,
Italy, and Slovakia兲, we found that the main gas pipelines are
organized along high capacity corridors, where capacity is
split among two or more pipelines which run in parallel over-
bridging long distances. This implies that the network is er-
ror tolerant because the failure of one pipeline causes only a
decrease in flow. Motivated by the finding of error tolerance
in national networks, we then addressed the problem of cap-
turing the topological structure of the European gas network.
FIG. 7. 共Color online兲 Trans-
European natural gas network.
Link thickness is proportional to
the generalized betweenness cen-
trality 关see Eq. 共3兲, where the sets
K and L are countries and the val-
ues of T
K,L
are taken from the data
in Fig. 2兴. We labeled several ma-
jor EU pipeline connections. The
large difference between the gen-
eralized betweenness on these
pipelines and the rest of the net-
work suggests that the network
has grown, to some extent, to
transport natural gas with minimal
losses along the shortest available
routes.
ROBUSTNESS OF TRANS-EUROPEAN GAS NETWORKS PHYSICAL REVIEW E 80, 016106 共2009兲
016106-7

At a global scale, the growth of the European gas pipeline
network has been determined by two competing mecha-
nisms. First, the network has grown under cost and efficiency
constraints to minimize the length of transport routes and
maximize transported volumes. Second, the network has de-
veloped error tolerance by adding redundant links. The com-
bination of the two mechanisms guarantees that the Euro-
pean gas pipeline network is robust, i.e., error tolerant to
failures of high load links. To reveal the network robustness,
we analyzed two measures—the generalized betweenness
and generalized vitality—which highlight global backbones
of transport efficiency and error tolerance, respectively. Fi-
nally, we proposed that the hot backbone of the network is
the skeleton of major transport routes where the network is
robust, in other words, where values of generalized between-
ness are high and values of generalized vitality are low. Our
method is of potential interest as it provides a detailed geo-
graphical analysis of engineered distribution networks.
Further research in continent-wide distribution networks
could proceed along several directions. The optimality of
existing networks and the existence of scaling laws could be
approached from a theoretical perspective 关42,43兴. Planned
and under-construction pipelines may change the robustness
of the network, in particular, within their geographical vicin-
ity. LNG, which is nowadays transported at low cost between
continents, is increasingly supplying the pipeline network.
The combined effect of LNG and storage facilities through-
out the European coastline has the potential to reduce the
dependency on one single exporting country such as Russia.
Last, but not the least, the dispute between Russia and
Ukraine in January 2009 has brought supply security to the
top of the European political agenda and highlighted how the
European gas network is robust to engineering failures yet
fragile to geopolitical crises.
ACKNOWLEDGMENTS
We wish to thank Dirk Helbing, Sergi Lozano, Amin Ma-
zloumian, and Russel Pride for valuable comments and
gratefully acknowledge the support of EU projects MAN-
MADE 共Grant No. 043363兲 and IRRIIS 共Grant No. 027568兲.
APPENDIX: PIPELINE CAPACITY
The capacity of a pipeline can be schematically derived as
follows: It is known that the flow of an incompressible vis-
cous fluid in a circular pipe can be described in the laminar
regime 共with a parabolic velocity profile兲 by the Hagen-
FIG. 8. 共Color online兲 Trans-European natural gas network. Link thickness is proportional to the generalized max-flow betweenness
vitality 关see Eq. 共4兲, where the sets K and L are countries and the values of T
K,L
are taken from the data in Fig. 2兴. Pipelines close to the major
sources tend to have high values of generalized vitality because this is where the network bottleneck is located. Pipelines along sparse
interconnections between larger parts of the network 共e.g., the Spanish-French border兲 also tend to have high values of generalized vitality,
when compared to neighboring pipelines.
CARVALHO et al. PHYSICAL REVIEW E 80, 016106 共2009兲
016106-8

Poiseuille equation 关44兴, which states that the volume of
fluid passing per unit time is
dV/dt =
␲
⌬pr
4
/共8
␩
l兲, 共A1兲
where ⌬p is the pressure difference between the two ends of
the pipeline, l is the length of the pipeline 共thus, −⌬p / l is the
pressure gradient兲,
␩
is the dynamic viscosity, and r is the
radius of the pipeline. However, the gas network operates in
the turbulent regime and the Hagen- Poiseuille equation is no
longer valid. Therefore, we apply the Darcy-Weisbach equa-
tion for the pipeline head loss h
f
. This is a phenomenological
equation which describes the loss of energy due to friction
within the pipeline and is valid in the laminar and turbulent
regimes 关45兴,
h
f
= f
l
v
2
2gd
, 共A2兲
where f is called the Darcy friction factor, d is the pipeline
diameter,
v
is the average velocity, and g is the acceleration
of gravity. Equation 共A2兲 can be written as a function of the
volumetric flow rate dV / dt=
␲
共
d
2
兲
2
v
共which is the capacity of
the pipeline 关46兴兲 as
h
f
=
16fl共dV/dt兲
2
␲
2
d
5
. 共A3兲
In general, the friction factor f and the pipeline loss h
f
de-
pend on the pipeline diameter d, so the capacity of the pipe-
line is given by c = dV/ dt =
␲
4
共
h
f
fl
兲
1/2
d
5/2
⬃d
␥
, where typically
␥
⯝2.6 for gas pipelines 关46兴.
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