10-6-2014_The comple

first published online 19 January 2010, doi: 10.1098/rsif.2009.04957 2010 J. R. Soc. Interface

Pablo Kaluza, Andrea Klzsch, Michael T. Gastner and Bernd Blasius

The complex network of global cargo ship movements

Supplementary datal http://rsif.royalsocietypublishing.org/content/suppl/2010/01/19/rsif.2009.0495.DC1.htm

"Data Supplement"

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Electronic supplementary material is available at http://dx.doi.org/10.1098/rsif.2009.0495 or via http://rsif.royalsocietypublishing.org.

J. R. Soc. Interface (2010) 7, 10931103

on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from Received 11 NAccepted 21 DDUCTION

y to travel, trade commodities and sharearound the world with unprecedented ef-dening feature of the modern globalized

Among the different means of transport,ping stands out as the most energy efcientng-distance transport for large quantities ofrigue et al. 2006). According to estimates,s 90 per cent of world trade is hauled byrnational Maritime Organization 2006). Inbillion tons of goods were loaded at therts. The trade volume currently exceeds 30-miles and is growing at a rate faster thaneconomy (United Nations Conference onDevelopment 2007).ldwide maritime network also plays a crucialays spread of invasive species. Two majoror marine bioinvasion are discharged water ballast tanks (Ruiz et al. 2000) and hullake & Lodge 2007). Even terrestrial speciescts are sometimes inadvertently transportedcontainers (Lounibos 2002). In several partsld, invasive species have caused dramaticpecies extinction and landscape alteration,ging ecosystems and creating hazards for

human livelihoods, health and local economet al. 2000). The nancial loss owing to bioiestimated to be $120 billion per year inalone (Pimentel et al. 2005).

Despite affecting everybodys daily lives,ping industry is far less in the public eye tsectors of the global transport infrastructuringly, it has also received little attention inliterature on complex networks (Wei et al. 2Zhu 2009). This neglect is surprising consicurrent interest in networks (Albert & BaraNewman 2003a; Gross & Blasius 2008), espeport (Barrat et al. 2004; Guimera` & AmHufnagel et al. 2004; Guimera` et al. 2005), ret al. 2006; Barthelemy & Flammini 2008)networks (Latora & Marchiori 2002; Sen etIn the spirit of current network research, wea large-scale perspective on the global cnetwork (GCSN) as a complex system dennetwork of ports that are connected by lintrafc passes between them.

Similar research in the past had to maassumptions about ows on hypotheticalwith connections between all pairs of ports iapproximate ship movements (Drake & LoTatem et al. 2006). By contrast, our analyson comprehensive data of real ship journeyus to construct the actual network. We shoorrespondence ([email protected]).Keywords: complex network; cargo ships; bioinvasion; transportationship moPablo Kaluza, Andrea K

and Bern

Institute for Chemistry and Biology of thUniversitat, Carl-von-Ossietzky-Str

Transportation networks play a crucial rolethe spread of invasive species. With 90 pernetwork of merchant ships provides one ofHere, we use information about the itinerarto construct a network of links between portures that set it apart from other transportaclassied into three categories: bulk dry cathree categories do not only differ in the shmobility patterns and networks. Containerbulk dry carriers and oil tankers move lessship movements possesses a heavy-tailed dfor the loads transported on the links witThe data analysed in this paper improve cuship movements, an important step towardbioinvasion.ovember 2009ecember 2009 1093rk of global cargoementssch, Michael T. GastnerBlasius*

arine Environment, Carl von Ossietzky9-11, 26111 Oldenburg, Germany

human mobility, the exchange of goods andt of world trade carried by sea, the globale most important modes of transportation.of 16 363 cargo ships during the year 2007We show that the network has several fea-networks. In particular, most ships can bers, container ships and oil tankers. Theses physical characteristics, but also in theirps follow regularly repeating paths whereasdictably between ports. The network of allribution for the connectivity of ports andsystematic differences between ship types.nt assumptions based on gravity models ofnderstanding patterns of global trade and

doi:10.1098/rsif.2009.0495Published online 19 January 2010This journal is q 2010 The Royal Society

SNaloetw

1094 Complex network of ship movements P. Kaluza et al.

on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from gross tonnage (GT)) follows a heavy-tailed distribution.This capacity scales superlinearly with the number ofdirectly connected ports. We identify the most centralports in the network and nd several groups of highlyinterconnected ports showing the importance ofregional geopolitical and trading blocks.

A high-level description of the complete network,however, does not yet fully capture the networks com-plexity. Unlike previously studied transportationnetworks, the GCSN has a multi-layered structure.There are, broadly speaking, three classes of cargoshipscontainer ships, bulk dry carriers and oiltankersthat span distinct subnetworks. Ships indifferent categories tend to call at different ports andtravel in distinct patterns. We analyse the trajectoriesof individual ships in the GCSN and develop techniquesto extract quantitative information about characteristic

5000journeys

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the 20 most central ports1 Panama Canal2 Suez Canal3 Shanghai4 Singapore5 Antwerp6 Piraeus7 Terneuzen8 Plaquemines9 Houston10 Ijmuiden

11 Santos12 Tianjin13 New York and New Jersey14 Europoort15 Hamburg16 Le Havre17 St Petersburg18 Bremerhaven19 Las Palmas20 Barcelona

(b)

. (a) The trajectories of all cargo ships bigger than 10 000 GTng each route. Ships are assumed to travel along the shortesteenness centrality and a ranked list of the 20 most central ports.data are available, taken as representative of theglobal trafc and long distance trade between the 951ports equipped with AIS receivers (for details see elec-tronic supplementary material). For each ship, weobtain a trajectory from the database, i.e. a list ofports visited by the ship sorted by date. In 2007,there were 490 517 non-stop journeys linking 36 351 dis-tinct pairs of arrival and departure ports. The completeset of trajectories, each path representing the shortestroute at sea and coloured by the number of journeyspassing through it, is shown in gure 1a.

Each trajectory can be interpreted as a smalldirected network where the nodes are ports linkedtogether if the ship travelled directly between theports. Larger networks can be dened by merging tra-jectories of different ships. In this article, weaggregate trajectories in four different ways: the com-bined network of all available trajectories and thesubnetworks of container ships (3100 ships), bulk drycarriers (5498) and oil tankers (2628). These three sub-networks combinedly cover 74 per cent of the GCSNstotal GT. In all four networks, we assign a weight wijto the link from port i to j equal to the sum of the

the light of the network size (the WAN consists of3880 nodes), this difference becomes even more pro-

Complex network of ship movements P. Kaluza et al. 1095

on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from nounced, indicating that the GCSN is much moredensely connected. This redundancy of links gives thenetwork high structural robustness to the loss ofroutes for keeping up trade.available space on all ships that have travelled on thelink during 2007 measured in GT. If a ship made a jour-ney from i to j more than once, its capacity contributesmultiple times to wij.

3. THE GLOBAL CARGO SHIP NETWORK

The directed network of the entire cargo eet is notice-ably asymmetric, with 59 per cent of all linked pairs ofports being connected only in one direction. Still, thevast majority of ports (935 out of 951) belongs to onesingle strongly connected component, i.e. for any twoports in this component, there are routes in both direc-tions, though possibly visiting different intermediateports. The routes are intriguingly short: only few stepsin the network are needed to get from one port toanother. The shortest path length l between two portsis the minimum number of non-stop connections onemust take to travel between origin and destination. Inthe GCSN, the average over all pairs of ports is extre-mely small, kll 2:5. Even the maximum shortestpath between any two ports (e.g. from Skagway,Alaska, to the small Italian island of Lampedusa) isonly of length lmax 8. In fact, the majority of all poss-ible origindestination pairs (52%) can already beconnected by two steps or less.

Comparing these ndings to those reported for theworldwide airport network (WAN) shows interestingdifferences and similarities. The high asymmetry ofthe GCSN has not been found in the WAN, indicatingthat ship trafc is structurally very different from avia-tion. Rather than being formed by the accumulation ofback-and-forth trips, ship trafc seems to be governedby an optimal arrangement of unidirectional, often cir-cular routes. This optimality also shows in the GCSNssmall shortest path lengths. In comparison, in theWAN, the average and maximum shortest path lengthsare kll 4:4 and lmax 15, respectively (Guimera` et al.2005), i.e. about twice as long as in the GCSN. Similarto the WAN, the GCSN is highly clustered: if a port X islinked to ports Y and Z, there is a high probability thatthere is also a connection from Y to Z. We calculated aclustering coefcient C (Watts & Strogatz 1998) fordirected networks and found C 0.49, whereasrandom networks with the same number of nodes andlinks only yield C 0.04 on average. Degree-dependentclustering coefcients Ck reveal that clusteringdecreases with node degree (see electronic supplemen-tary material). Therefore, the GCSNlike theWANcan be regarded as a small-world network pos-sessing short path lengths despite substantialclustering (Watts & Strogatz 1998). However, the aver-age degree of the GCSN, i.e. the average number of linksarriving at and departing from a given port (in-degreeplus out-degree), kkl 76.5, is notably higher than inthe WAN, where kkl 19.4 (Barrat et al. 2004). InJ. R. Soc. Interface (2010)The degree distribution P(k) shows that most portshave few connections, but there are some ports linkedto hundreds of other ports (gure 2a). Similar right-skewed degree distributions have been observed inmany real-world networks (Barabasi & Albert 1999).While the GCSNs degree distribution is not exactlyscale-free, the distribution of link weights, P(w), fol-lows approximately a power law Pw/ wm withm 1:71+ 0:14 (95% CI for linear regression,gure 2b, see also electronic supplementary material).By averaging the sums of the link weights arrivingat and departing from port i, we obtain the nodestrength si (Barrat et al. 2004). The strength distri-bution can also be approximated by a power lawPs/ sh with h 1:02+ 0:17, meaning that asmall number of ports handle huge amounts of cargo(gure 2c). The determination of power law relation-ships by line tting has been strongly criticized (e.g.Newman 2005; Clauset et al. 2009), therefore, weanalysed the distributions with model selection byAkaike weights (Burnham & Anderson 1998). Ourresults conrm that a power law is a better t thanan exponential or a lognormal distribution for P(w)and P(s), but not P(k) (see electronic supplementarymaterial). These ndings agree well with the conceptof hubsspokes networks (Notteboom 2004) thatwere proposed for cargo trafc, for example, in Asia(Robinson 1998). There are a few large, highly con-nected ports through which all smaller ports transacttheir trade. This scale-free property makes the shiptrade network prone to the spreading and persistenceof bioinvasive organisms (e.g. Pastor-Satorras &Vespignani 2001). The average nearest-neighbourdegree, a measure of network assortativity, addition-ally underlines the hubsspokes property of cargoship trafc (see electronic supplementary material).

Strengths and degrees of the ports are related accord-ing to the scaling relation kskl/ k1:46+0:1 (95% CI forstandardized major axis regression; Warton et al. 2006).Hence, the strength of a port grows generally fasterthan its degree (gure 2d). In other words, highly con-nected ports not only have many links, but their linksalso have a higher than average weight. This obser-vation agrees with the fact that busy ports are betterequipped to handle large ships with large amounts ofcargo. A similar result, kskl/ k1:5+0:1, was found forairports (Barrat et al. 2004), which may hint at a gen-eral pattern in transportation networks. In the light ofbioinvasion, these results underline empirical ndingsthat big ports are more heavily invaded because ofincreased propagule pressure by ballast water of moreand larger ships (Williamson 1996; Mack et al. 2000;see Cohen & Carlton 1998).

A further indication of the importance of a node is itsbetweenness centrality (Freeman 1979; Newman 2004).The betweenness of a port is the number of topologi-cally shortest directed paths in the network that passthrough this port. In gure 1b, we plot and list themost central ports. Generally speaking, centrality anddegree are strongly correlated (Pearsons correlationcoefcient: 0.81), but in individual cases other factorscan also play a role. The Panama and Suez canals, forinstance, are shortcuts to avoid long passages around

degree, k


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103South America and Africa. Other ports have a high cen-trality because they are visited by a large number ofships (e.g. Shanghai), whereas others gain their statusprimarily by being connected to many different ports(e.g. Antwerp).

4. THENETWORK LAYERS OFDIFFERENTSHIP TYPES

To compare the movements of cargo ships of differenttypes, separate networks were generated for each ofthe three main ship types: container ships, bulk dry car-riers and oil tankers. Applying the network parametersintroduced in the previous section to these three subnet-works reveals some broad-scale differences (table 1).The network of container ships is densely clustered,C 0.52, has a rather low mean degree, kkl 32:44,and a large mean number of journeys (i.e. number oftimes any ship passes) per link, kJ l 24:26. The bulkdry carrier network, on the other hand, is less clustered,

10106107108109

1010

104 105 106 107 108

107106105104node strength, s

108

1010

109

108P

Figure 2. Degrees and weights in the GCSN. (a) The degree distrithe GCSN nor its subnetworks. The degree k is dened here as the sship types; square, container ships; circle, bulk dry carriers; trianclear power law relationships for the GCSN and the three subnetwterns of the different ship types (asterisk, m 1.71+0.14; square,0.25). (c) The node strength distributions P(s) are also heavy tailecalculated by linear regression with 95% condence intervals (similelectronic supplementary material) (asterisk, h 1.02+0.17; sq1.01+ 0.16). (d) The average strength of a node ks(k)l scales suhighly connected ports have, on average, links of higher weight.

J. R. Soc. Interface (2010)(b)

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link weight, w

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10410111010109

P(w

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(k)has a higher mean degree and fewer journeys per link(C 0:43, kkl 44:61, kJ l 4:65). For the oil tankers,we nd intermediate values (C 0:44, kkl 33:32,kJ l 5:07). Note that the mean degrees kkl of the sub-networks are substantially smaller than that of the fullGCSN, indicating that different ship types useessentially the same ports but different connections.

A similar tendency appears in the scaling of the linkweight distributions (gure 2b). P(w) can be approxi-mated as power laws for each network, but withdifferent exponents m. The container ships have thesmallest exponent (m 1.42) and bulk dry carriers thelargest (m 1.93) with oil tankers in between (m 1.73). In contrast, the exponents for the distribution ofnode strength P(s) are nearly identical in all three sub-networks, h 1.05, h 1.13 and h 1.01, respectively.

These numbers give a rst indication that differentship types move in distinctive patterns. Containerships typically follow set schedules visiting severalports in a xed sequence along their way, thus provid-ing regular services. Bulk dry carriers, by contrast,

104

105

106

101 100 1000degree, k

s

butions P(k) are right-skewed, but not power laws, neither forum of in- and out-degree, thus k 1 is rather rare (asterisk, allgle, oil tankers). (b) The link weight distributions P(w) revealorks, with exponents m characteristic for the movement pat-m 1.42+ 0.15; circle, m 1.93+0.11; triangle, m 1.73+d, showing power law relationships. The stated exponents arear results are obtained with maximum likelihood estimates, seeuare, h 1.05+ 0.13; circle, h 1.13+ 0.21; triangle, h perlinearly with its degree, ks(k)l/ k1.46+0.1, indicating that

ofreendno

kl l kJ l m h kN l kLl kS l kpl

2.2.2.2.


on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from appear less predictable as they frequently change theirroutes on short notice depending on the currentsupply and demand of the goods they carry. Thelarger variety of origins and destinations in the bulkdry carrier network (n 616 ports, compared withn 378 for container ships) explains the higher averagedegree and the smaller number of journeys for a givenlink. Oil tankers also follow short-term market trends,but, because they can only load oil and oil products,the number of possible destinations (n 505) is morelimited than for bulk dry carriers.

These differences are also underlined by the between-ness centralities of the three network layers (seeelectronic supplementary material). While some portsrank highly in all categories (e.g. Suez Canal,Shanghai), others are specialized on certain shiptypes. For example, the German port of Wilhelmshavenranks tenth in terms of its worldwide betweenness for oiltankers, but is only 241st for bulk dry carriers and324th for container ships.

We can gain further insight into the roles of the portsby examining their community structure. Communitiesare groups of ports with many links within the groupsbut few links between different groups. We calculatedthese communities for the three subnetworks with amodularity optimization method for directed networks(Leicht & Newman 2008) and found that they differsignicantly from modularities of correspondingErdosRenyi graphs (gure 3; Guimera` et al. 2004).The network of container trade shows 12 communities(gure 3a). The largest ones are located (i) on theArabian, Asian and South African coasts, (ii) on theNorth American east coast and in the Caribbean, (iii)in the Mediterranean, the Black Sea and on the Euro-pean west coast, (iv) in Northern Europe and (v) inthe Far East and on the American west coast. Thetransport of bulk dry goods reveals seven groups

Table 1. Characterization of different subnetworks. Numbersubnetwork; together with network characteristics: mean degmean journeys per link kJ l, power law exponents m and h; alinks kLl, port calls kS l per ship and regularity index kpl. Some

ship class ships MGT n kkl C

whole eet 16 363 664.7 951 76.4 0.49container ships 3100 116.8 378 32.4 0.52bulk dry carriers 5498 196.8 616 44.6 0.43oil tankers 2628 178.4 505 33.3 0.44(gure 3b). Some can be interpreted as geographicalentities (e.g. North American east coast, trans-Pacictrade), while others are dispersed on multiple continents.Especially interesting is the community structure of theoil transportation network that shows six groups(gure 3c): (i) the European, north and west Africanmarket, (ii) a large community comprising Asia,South Africa and Australia, (iii) three groups for theAtlantic market with trade between Venezuela, theGulf of Mexico, the American east coast and NorthernEurope and (iv) the American Pacic coast. It shouldbe noted that the network includes the transport ofcrude oil as well as commerce with already rened oil

J. R. Soc. Interface (2010)products so that oil-producing regions do not appearas separate communities. This may be due to thelimit in the detectability of smaller communities bymodularity optimization (Fortunato & Barthelemy2007), but does not affect the relevance of the revealedship trafc communities. Because of the, by denition,higher transport intensity within communities, bioinva-sive spread is expected to be heavier between ports ofthe same community. However, in gure 3, it becomesclear that there are no strict geographical barriersbetween communities. Thus, spread between commu-nities is very likely to occur even on small spatialscales by shipping or ocean currents between close-byports that belong to different communities.

Despite the differences between the three main cargoeets, there is one unifying feature: their motif distri-bution (Milo et al. 2002). Like most previous studies,we focus here on the occurrence of three-node motifsand present their normalized Z score, a measure fortheir abundance in a network (gure 4). Strikingly,the three eets have practically the same motif distri-bution. In fact, the Z scores closely resemble thosefound in the World Wide Web and different social net-works that were conjectured to form a superfamily ofnetworks (Milo et al. 2004). This superfamily displaysmany transitive triplet interactions (i.e. if X ! Y andY ! Z , then X ! Z); for example, the overrepresentedmotif 13 in gure 4, has six such interactions. Intransi-tive motifs, like motif 6, are comparably infrequent. Theabundance of transitive interactions in the ship net-works indicates that cargo can be transported bothdirectly between ports as well as via several intermedi-ate ports. Thus, the high clustering and redundancy oflinks (robustness to link failures) appear not only in theGCSN but also in the three subnetworks. The similarityof the motif distributions to other humanly optimizednetworks underlines that cargo trade, like social net-

5 13.57 1.71 1.02 10.4 15.6 31.8 0.6376 24.25 1.42 1.05 11.2 21.2 48.9 1.8457 4.65 1.93 1.13 8.9 10.4 12.2 0.0374 5.07 1.73 1.01 9.2 12.9 17.7 0.19ships, total GT (106 GT) and number of ports n in eachkkl, clustering coefcient C, mean shortest path length k l l,trajectory properties: average number of distinct ports kNl,table values are highlighted in bold.works and the World Wide Web, depends crucially onhuman interactions and information exchange. Whileadvantageous for the robustness of trade, the clusteringof links as triplets also has an unwanted side effect: ingeneral, the more clustered a network, the more vulner-able it becomes to the global spread of alien species,even for low invasion probabilities (Newman 2003b).

5. NETWORK TRAJECTORIES

Going beyond the network perspective, the databasealso provides information about the movement


on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from (a)characteristics per individual ship (table 1). The aver-age number of distinct ports per ship kNl does notdiffer much between different ship classes, but con-tainer ships call much more frequently at ports thanbulk dry carriers and oil tankers. This difference isexplained by the characteristics and operationalmode of these ships. Normally, container ships arefast (between 20 and 25 knots) and spend less time

(b)

(c)

Figure 3. Communities of ports in three cargo ship subnetworks. Tof links within the groups, as opposed to between the groups, inmap, the colours represent the c distinct trading communities fo0.605), (b) bulk dry carriers (c 7, Q 0.592) and (c) oil tankerworks differ signicantly from modularities in ErdosRenyi graphFor the networks corresponding to (a), (b) and (c), values are QE

J. R. Soc. Interface (2010)(1.9 days on average in our data) in the ports forcargo operations. By contrast, bulk dry carriers andoil tankers move more slowly (between 13 and 17knots) and stay longer in the ports (on average 5.6days for bulk dry carriers and 4.6 days for oiltankers).

The speed at sea and of cargo handling, however, isnot the only operational difference. The topology of

he communities are groups of ports that maximize the numberterms of the modularity Q (Leicht & Newman 2008). In eachr the goods transported by (a) container ships (c 12, Q s (c 6, Q 0.716). All modularities Q of the examined net-s of the same size and number of links (Guimera` et al. 2004).R 0.219, QER 0.182 and QER 0.220, respectively.

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on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from the trajectories also differs substantially. Characteristicsample trajectories for each ship type are presented ingure 5ac. The container ship (gure 5a) travels onsome of the links several times during the studyperiod, whereas the bulk dry carrier (gure 5b) passesalmost every link exactly once. The oil tanker(gure 5c) commutes a few times between some ports,but by and large also serves most links only once.

We can express these trends in terms of a regularityindex p that quanties how much the frequency with

0

0.25

norm

aliz

ed Z

s

0.50

0.75 1 2 3 4 5 6

Figure 4. Motif distributions of the three main cargo eets. A po(less) frequent in the real network than in random networks wscores of the World Wide Web and social networks. The agreemnetworks (Milo et al. 2004). The motif distributions of the eenections are removed. Line with open circles, www-1; line withtriangles, social-1; line with crosses, social-2; line with asterisksquares, container ships; line with black diamonds, oil tankers0.75

0.50

ewhich each link is used deviates from a random network.Consider the trajectory of a ship calling S times at N dis-tinct ports and travelling on L distinct links. Wecompare the mean number of journeys per linkfreal S=L to the average link usage fran in an ensembleof randomized trajectories with the same number ofnodes N and port calls S. To quantify the differencebetween real and random trajectories, we calculate theZ score p freal fran=s (where s is the standard devi-ation of f in the random ensemble). If p 0, the realtrajectory is indistinguishable from a random walk,whereas larger values of p indicate that the movementis more regular. Figure 5df present the distributionsof the regularity index p for the different eets. For con-tainer ships, p is distributed broadly around p 2, thussupporting our earlier observation that most containerships provide regular services between ports along theirway. Trajectories of bulk dry carriers and oil tankers,on the other hand, appear essentially random with thevast majority of ships near p 0.

6. APPROXIMATING TRAFFIC FLOWSUSING THE GRAVITY MODEL

In this article, we view global ship movements as a net-work based on detailed arrival and departure records.

J. R. Soc. Interface (2010)Until recently, surveys of seaborne trade had to relyon far fewer data: only the total number of arrivals atsome major ports was publicly accessible, but not theships actual paths (Zachcial & Heideloff 2001). Missinginformation about the frequency of journeys, thus, hadto be replaced by plausible assumptions, the gravitymodel being the most popular choice. It posits thattrips are, in general, more likely between nearby portsthan between ports far apart. If dij is the distancebetween ports i and j, the decline in mutual interaction

7

bgraph

8 9 10 11 12 13

ive (negative) normalized Z score indicates that a motif is morethe same degree sequence. For comparison, we overlay the Zsuggests that the ship networks fall in the same superfamily ofre maintained even when 25, 50 and 75% of the weakest con-n squares, www-2; line with open diamonds, www-3; line withocial-3; line with red circles, bulk dry carriers; line with blueis expressed in terms of a distance deterrence functionf(dij). The number of journeys from i to j then takesthe form Fij aibjOiIj f dij, where Oi is the totalnumber of departures from port i and Ij the numberof arrivals at j (Haynes & Fotheringham 1984). Thecoefcients ai and bj are needed to ensure

Pj Fij Oi

andP

i Fij Ij .How well can the gravity model approximate real

ship trafc? We choose a truncated power law for thedeterrence function, f dij dbij expdij=k. Thestrongest correlation between model and data isobtained for b 0:59 and k 4900 km (see electronicsupplementary material). At rst sight, the agreementbetween data and model appears indeed impressive.The predicted distribution of travelled distances(gure 6a) ts the data far better than a simpler non-spatial model that preserves the total number ofjourneys, but assumes completely random origins anddestinations.

A closer look at the gravity model, however, revealsits limitations. In gure 6b, we count how often linkswith an observed number of journeys Nij are predictedto be passed Fij times. Ideally, all data points wouldalign along the diagonal Fij Nij, but we nd thatthe data are substantially scattered. Although the par-ameters b and k were chosen to minimize the scatter,

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3the correlation between data and model is only moder-ate (Kendalls t 0.433). In some cases, the predictionis off by several thousand journeys per year.

Recent studies have used the gravity model to pin-point the ports and routes central to the spread ofinvasive species (Drake & Lodge 2004; Tatem et al.2006). The models shortcomings pose the question asto how reliable such predictions are. For this purpose,we investigated a dynamic model of ship-mediatedbioinvasion where the weights of the links are eitherthe observed trafc ows or the ows of the gravitymodel.

We follow previous epidemiological studies (Rvachev &Longini 1985; Flahault et al. 1988; Hufnagel et al. 2004;Colizza et al. 2006) in viewing the spread on the

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D(p

)

00 2 4

p6 8

bulk dr

0 2

Figure 5. Sample trajectories of (a) a container ship with a regularoil tanker, p 1.027. In the three trajectories, the numbers and th(d f ) Distribution of p for the three main eets.

J. R. Soc. Interface (2010)Mina Al Ahmadi4ideo

Jebel AliJubail

as Laffan

(c)

4

1

1

111

1

1

2

22

Jebel Dhanna/Ruwaisnetwork as a metapopulation process where the popu-lation dynamics on the nodes are coupled bytransport on the links. In our model, ships can transporta surviving population of an invasive species with only asmall probability ptrans 1% on each journey betweentwo successively visited ports. The transported popu-lation is only a tiny fraction s of the population at theport of origin. Immediately after arriving at a newport, the species experiences strong demographic uctu-ations that lead, in most cases, to the death of theimported population. If, however, the new immigrantsbeat the odds of this ecological roulette (Carlton &Geller 1993) and establish, the population P growsrapidly following the stochastic logistic equationdP=dt rP1 P Pp jt with growth rate r 1

Port Rashid

Santos

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YuzhnyPiraeuslgiers

El Dekheila

1

11

1

11

Singapore

( f )y carriers

4p

6 8

oil tankers

0 2 4p

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ity index p 2.09, (b) a bulk dry carrier, p 0.098 and (c) ane line thickness indicate the frequency of journeys on each link.

observed journeys, Nij0 10 100 1000

pred

icte

d jou

01

10

102

1

10

102

he GCSN (navigable distances around continents as indicated inThe gravity model (red), based on information about distancesa simpler model (blue) which only xes the total number of jour-andom trafc). (b) Count of port pairs with Nij observed and Fijty model (rounded to the nearest integer). Some of the worst out- 0 versus Fij 200); triangle, Hook of Holland to Europoort; square, Harwich to Hook of Holland (644 versus 0).

0

100

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0 10 20 30 40 500

200

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f inv

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(b)


on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from per year and Gaussian white noise j. For details of themodel, see the electronic supplementary material.

Starting from a single port at carrying capacity P 1,we model contacts between ports as Poisson processeswith rates Nij (empirical data) or Fij (gravity model).As shown in gure 7a, the gravity model systematicallyoverestimates the spreading rate, and the difference canbecome particularly pronounced for ports that are wellconnected, but not among the central hubs in the net-work (gure 7b). Comparing typical sequences ofinvaded ports, we nd that the invasions driven bythe real trafc ows tend to be initially conned tosmaller regional ports, whereas in the gravity model

0 5000 10 000 15 000 20 000distance (km)

1

10

102

103

104

105(a)

num

ber o

f journ

eys

Figure 6. (a) Histogram of port-to-port distances travelled in tgure 1). We overlay the predictions of two different models.between ports and total port calls, gives a much better t thanneys (black line, observed; red line, gravity model; blue line, rpredicted journeys. The ows Fij were calculated with the graviliers are highlighted in blue. Circle, Antwerp to Calais (Nij(16 versus 1895); diamond, Calais to Dover (4392 versus 443)the invasions quickly reach the hubs. The total out-and in-ows at the ship journeys origin and departureports, respectively, are indeed more strongly positivelycorrelated in reality than in the model (t 0.157versus 0.047). The gravity model thus erases toomany details of a hierarchical structure present in thereal network. That the gravity model eliminates mostcorrelations is also plausible from simple analytic argu-ments (see electronic supplementary material fordetails). The absence of strong correlations makes thegravity model a suitable null hypothesis if the corre-lations in the real network are unknown, but severalrecent studies have shown that correlations play animportant role in spreading processes on networks(e.g. Boguna & Pastor-Satorras 2002; Newman 2002).Hence, if the correlations are known, they should notbe ignored.

While we observed that the spreading rates for theAIS data were consistently slower than for the gravitymodel even when different parameters or populationmodels were considered, the time scale of the invasionis much less predictable. The assumption that only asmall fraction of invaders succeed outside their nativehabitat appears realistic (Mack et al. 2000). Further-more, the parameters in our model were adjusted so

J. R. Soc. Interface (2010)(b)

rneys

, Fij 103

103

104

number oflinksthat the per-ship-call probability of initiating invasionis approximately 4.4 1024, a rule-of-thumb valuestated by Drake & Lodge (2004). Still, too little isempirically known to pin down individual parameterswith sufcient accuracy to give more than a qualitativeimpression. It is especially difcult to predict how a

time (years)Figure 7. Results from a stochastic population model for thespread of an invasive species between ports. (a) The invasionstarts from one single, randomly chosen port. (b) The initialport is xed as Bergen (Norway), an example of a well-connected port (degree k 49) which is not among the centralhubs. The rate of journeys from port i to j per year is assumedto be Nij (real ows from the GCSN) or Fij (gravity model).Each journey has a small probability of transporting a tiny frac-tion of the population from origin to destination. Parameterswere adjusted (r 1 per year, ptrans 0.01 and s 4 1025)to yield a per-ship-call probability of initiating invasion ofapproximately 4.4 1024 (Drake & Lodge 2004; see electronicsupplementary material for details). Plotted are the cumulativenumbers of invaded ports (population number larger than halfthe carrying capacity) averaged over (a) 14 000 and (b) 1000simulation runs (standard error equal to line thickness).Black line, real trafc ows; grey line, gravity model.

potential invader reacts to the environmental con-

industry. An important characteristic of the network aredifferences in the movement patterns of different ship

(doi:10.1073/pnas.0400087101)Barthelemy, M. & Flammini, A. 2008 Modeling urban street

Burnham, K. P. & Anderson, D. R. 1998 Model selection and


on October 5, 2014rsif.royalsocietypublishing.orgDownloaded from patterns. Phys. Rev. Lett. 100, 138 702. (doi:10.1103/PhysRevLett.100.138702)

Boguna, M. & Pastor-Satorras, R. 2002 Epidemic spreading incorrelated complex networks. Phys. Rev. E 66, 047 104.(doi:10.1103/PhysRevE.66.047104)

Buhl, J., Gautrais, J., Reeves, N., Sole, R. V., Valverde, S.,Kuntz, P. & Theraulaz, G. 2006 Topological patterns instreet networks of self-organized urban settlements. Eur.Phys. J. B 49, 513522. (doi:10.1140/epjb/e2006-00085-1)types. Bulk dry carriers and oil tankers tend to move in aless regular manner between ports than container ships.This is an important result regarding the spread of invasivespecies because bulk dry carriers and oil tankers often sailempty and therefore exchange large quantities of ballastwater. The gravity model, which is the traditionalapproach to forecasting marine biological invasions, cap-tures some broad trends of global cargo trade, but formany applications its results are too crude. Future strat-egies to curb invasions will have to take more details intoaccount. The network structure presented in this articlecan be viewed as a rst step in this direction.

We thank B. Volk, K. H. Holocher, A. Wilkinson, J. M. Drakeand H. Rosenthal for stimulating discussions and helpfulsuggestions. We also thank Lloyds Register Fairplay forproviding their shipping database. This work was supportedby German VW-Stiftung and BMBF.

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The complex network of global cargo ship movementsIntroductionDataThe global cargo ship networkThe network layers of different ship typesNetwork trajectoriesApproximating traffic flows using the gravity modelConclusionWe thank B. Volk, K. H. Holocher, A. Wilkinson, J. M. Drake and H. Rosenthal for stimulating discussions and helpful suggestions. We also thank Lloyds Register Fairplay for providing their shipping database. This work was supported by German VW-Stiftung and BMBF.References

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