ORIGINAL PAPER

Graph theoretic topology of the Great but small BarrierReef world

Stuart J. Kininmonth & Glenn De’ath &

Hugh P. Possingham

Received: 31 January 2009 /Accepted: 24 July 2009# Springer Science + Business Media B.V. 2009

Abstract The transport of larvae between coral reefs iscritical to the functioning of Australia’s Great Barrier Reef(GBR) because it determines recruitment rates and geneticexchange. One way of modelling the transport of larvaefrom one reef to another is to use information aboutcurrents. However the connectivity relationships of theentire system have not been fully examined. Graph theoryprovides a framework for the representation and analysis ofconnections via larval transport. In the past, the geometricarrangement (topology) of biological systems, such as foodwebs and neural networks, has revealed a common set ofcharacteristics known as the ‘small world’ property. We usegraph theory to examine and describe the topology andconnectivity of a species living in 321 reefs in the centralsection of the GBR over 32 years. This section of the GBRcan be described by a directional weighted graph, and wediscovered that it exhibits scale-free small-world character-istics. The conclusion that the GBR is a small-worldnetwork for biological organisms is robust to variation inboth the life history of the species modelled and yearlyvariation in hydrodynamics. The GBR is the first reportedmesoscale biological small-world network.

Keywords Small world . Coral reefs . Scale free . Graphtheory .Metapopulation

Introduction

The transport of larvae between coral reefs is critical totheir function because it determines recruitment rates andgenetic exchange (Veron 2002; van Oppen and Gates2006). Dispersal processes for marine organisms are diverse(Ninio and Meekan 2002; Mora et al. 2003) but two maintypes can be identified: the brooders and broadcasters. Eachof these strategies affects the fraction of larvae from onereef that arrives at another. For broadcasters in particular,the pre-competent larvae will drift in a relatively passivemanner with the currents (Ayre and Hughes 2000). Strongcurrents can transport larvae over long distances for severalweeks while they remain pre-competent (Ayre and Hughes2000). Both local and distant populations can be linked bythis transport of larvae.

The processes that shape the diversity of Australia’sGreat Barrier Reef (GBR) are complex and dependent on arange of biophysical factors (Hughes et al. 1999; Sale 1999;van Woesik 2000; Brinkman et al. 2001; Harriott andBanks 2002), including water currents that transport larvaeand warm water (Mora et al. 2003). The oceanic flow,called the Eastern Australian Current, interacts with thetidal cycles and wind conditions to provide transportmechanisms (Brinkman et al. 2001; James et al. 2002) forthe exchange of larvae between reefs throughout the centraland southern GBR (Ayre and Hughes 2000; Harriott andBanks 2002). While there is general acceptance of theimportance of these processes, their implications for thefunction and management of the GBR has not been fullyexamined.

While many studies describe the transport of larvae fromone reef to another (Swearer et al. 1999; James et al. 2002;Siegel et al. 2003; Nishikawa and Sakai 2005), the effect onthe entire system has not been fully examined (Mumby

S. J. Kininmonth :G. De’athAustralian Institute of Marine Science,PMB #3,Townsville 4810, Australia

S. J. Kininmonth (*) :H. P. PossinghamThe Ecology Centre and Centre for Applied EnvironmentalDecision Analysis, The University of Queensland,Brisbane 4072, Australiae-mail: s.kininmonth@aims.gov.au

Theor EcolDOI 10.1007/s12080-009-0055-3

1999). Over an extended period, larval transport linkagescreate a recruitment network, the structure of whichdetermines the nature of interactions between distant reefs.Matrices describing a single recruitment event have beencalculated (James et al. 2002; Bode et al. 2006). One wayof assessing the role each reef has within the entire networkis to use graph theory (Franc 2004; Proulx et al. 2005).

Graph theory models of disparate systems, like theWorld Wide Web, metabolic pathways, the neural networkof a worm, electronic circuit design and the professionalrelationships of film actors, have found networks that arehighly clustered and well connected (Hayes 2000; Ferrer etal. 2001; Wagner and Fell 2001). Biological systems, frommetabolic pathways (Jeong et al. 2000) to lake trophicstructures (Montoya and Sole 2002) and the neural net-works of a worm, all show a relatively new pattern called a‘small world’ (Watts and Strogatz 1998). This pattern isnamed after the universal exclamation of ‘what a smallworld!’ when strangers appear to be closely associated bymutual connections (Watts and Strogatz 1998). Firstdescribed in sociometry research (Milgram 1967; Traversand Milgram 1969), small worlds describe how communityformation can facilitate the effective interaction of distantindividuals across an entire population where there are onlyinteractions between small groups of neighbours (Radicchiet al. 2004).

Graph theory has been applied to ecological researchpreviously but with a focus on measuring simple landscapeindices such as dispersal pathway length (van Langevelde etal. 1998; Bunn et al. 2000; Urban and Keitt 2001). Theselandscape connectivity graphs that record the movement ofwildlife are relatively simple due to the difficulty in recordingthe movements of individual animals. While these graphsform a foundation for the use of graph theory in ecology,they lack the size and intricate structure for complex networkanalysis, especially when compared with the forest songbirdhabitat network of Minor and Urban (2008).

In this study, we use graph theory to examine and describethe network topology of species inhabiting 321 reefs in thecentral section of the GBR (Fig. 1) based on 32 years ofconnectivity matrices generated by James et al. (2002). Inparticular, we ask three questions: (1) Is the GBR a scale-freesmall-world network for marine organisms? (2) Do varia-tions in the dispersal model parameters (duration of the pre-competent period, the size of the settlement zone, predationand when larvae are released) influence the pattern? (3) Howinfluential is the connection strength on the network metrics?

Graph theory, small worlds and scale-free distributions

Graph theory is a branch of mathematics that describes thestatistical nature of simple (Harary 1969; Gross and Yellen

1999) and complex (Dorogovtsev and Mendes 2002;Proulx et al. 2005; Steuer and Lopez 2008) static networks.A graph G(V,E) can be defined as a finite set of vertices V,connected by edges E (Fig. 2). Geometric theorems of theproperties of simple graphs dominated the use of graphtheory until the advent of fast computers. Large graphswere modelled as randomly assembled graphs or as regularlattices (Albert and Barabasi 2002).

The degree (ki) of a vertex i defines the number ofconnecting edges and hence the number of neighbouringvertices (Fig. 2). All vertices in a regular lattice share acommon degree, <k>, and this configuration (often dis-played as a grid of equally sized cells) is commonly used asthe basis for neighbourhood interaction models (Bjornstadet al. 1999; Vuilleumier and Metzger 2006). In a randomgraph, the degree of all the vertices has a Poissondistribution and is characterised by a modal hump at themean degree (Proulx et al. 2005). The average degree forall the vertices in a graph is denoted as <k>.

The clustering coefficient, as defined by Montoya andSole (2002), is the sum of the number of triangular linkagesϑi i 2 V1; :::;Vnð Þ within the set of neighbours (Fig. 3) for aselected vertex divided by the maximum possible neigh-bour linkages where n is the number of vertices in thenetwork. For a unidirectional network, there are potentiallyk k � 1ð Þ=2 triangular linkages. The clustering coefficient isthus defined as:

CI ¼ 2

n

Xn

i¼1

ϑi= ki ki � 1ð Þð Þ: ð1Þ

The graphs modelling the GBR are directional weightedgraphs. For a directional network, there are potentiallytwice as many, k(k−1), triangular linkages and so theclustering coefficient is redefined from Eq. 1 as:

CI ¼ 1

n

Xn

i¼1

ϑi= ki ki � 1ð Þð Þ: ð2Þ

The average minimum path length L is the averagenumber of edges in the shortest path LMin(i, j) between allpairs V(i, j) in a graph averaged over n n� 1ð Þ=2 vertices(Montoya and Sole 2002)

L ¼ 2

n n� 1ð ÞXn

i¼1

Xn

j¼1

LMin i; jð Þ: ð3Þ

Calculation of the average minimum path lengthencompassed the directional structure of the networkby forcing the path tracing routine to travel in thedirection of lines. The assumption in Eq. 3 that Li,j=Lj,iis relaxed in an asymmetric graph, and thus, the summa-

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Fig. 1 Graph of 1998 connectivity with hydrodynamic parameters of release by new moon, 1 km pick up zone, 1 week pre-competent period,3 weeks competent period and no predation. The edges displayed are only those with weight greater than 0.02

Vertex

Edge

Fig. 2 A simple graph contain-ing five vertices and sevenedges. Four of the vertices havea degree of 3; one has a degreeof 2

Fig. 3 Neighbourhood triangle(dashed lines) forms the basis ofthe clustering coefficient. Aver-age minimum path length high-lights the number of steps(labelled) to reach the targetvertex

Fig. 1 Graph of 1998 connectivity with hydrodynamic parameters of release by new moon, 1 km pick up zone, 1 week pre-competent period,3 weeks competent period and no predation. The edges displayed are only those with weight greater than 0.02

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tion of path lengths is averaged over possible between-vertex connections, n(n−1):

L ¼ 1

n n� 1ð ÞXn

i¼1

Xn

j¼1

LMin i; jð Þ: ð4Þ

As a consequence of measuring the paths through thenetwork, three more metrics can be usefully derived:diameter, closeness and betweenness (Freeman 1978). Thediameter, D, is the longest minimum path length that existsbetween any vertices in a network. The closeness centralitymeasure, CCi, is the sum of the minimum path lengthsconnecting a vertex i to all other vertices j (i, j ∈ V1,...,Vn).The betweenness centrality measure, BCk, is the proportionof minimum paths connecting vertices i and j that passthrough vertex k (i, j, k ∈ V1,...,Vn). The centrality measuresrelate to the isolation (low CC), contagion (high CC) and‘stepping stone’ (high BC) landscape ecology concepts(Gustafson 1998).

Graphs that have a high level of clustering and short pathlengths compared with random or regular lattices areknown as small-world graphs (Watts and Strogatz 1998).However, there is no explicit definition of the properties ofsmall-world graphs (Schnettler 2009). Small-world topolo-gy increases the connectivity of vertices, effectively makingit smaller, in terms of interactions, than a random or regularone. Clusters connected by longer links enable any point inthe network to be located a small number of steps from anyother point, and this forms the basis of Stanley Milgram’ssix degrees of separation for the network describing humansocial interactions (Barabasi 2002).

An additional characteristic of some small-world networksin the real world is that the number of vertices with kconnections can be described by a power-law PðkÞ � k�g

where γ is a constant (Barabasi et al. 2000). This property,referred to as a scale-free degree distribution, defines a graphwhere the majority of vertices are weakly connected and asmall, but significant, number are very strongly connected(Albert et al. 2000). In contrast in a random graph, thenumber of vertices, n, with degree, k, shows a Poissondistribution where few, if any, vertices are very stronglyconnected. However Li et al. (2005) present a convincingargument that the power law used in many publications doesnot represent a scale-free network consistently, and instead,they suggest the use of the size-rank measure: RðkÞ ¼ cYk�g ,where R(k) is the rank of the degree yk subject to yk beingfrom a finite sequence such that y1 � y2 � :::: � yn and c isa fixed constant and γ is a scaling index.

While the number of connections reveals many of thetopological structures, the strength of each connection can beused to describe the exchanges between the vertices. In thecase where the strength of connection between two verticesis known, the value of the edge weight, wij, describes the

intensity of interaction occurring between vertex i to vertex j.The inbound and outgoing strength indices for vertex i aredefined (Barrat et al. 2004; Newman 2004) as:

sINi ¼X

j2vðiÞ;j2Eji

wji; sOUTi ¼X

j2vðiÞ;j2Eij

wij ð5Þ

where v(i) is the set of neighbours of vertex i, Eji is the set ofedges having the inbound direction to vertex i, and Eij is thecomplementary set of edges having an outbound origin fromvertex i. The average in- and outbound strengths for theentire graph are:

Sin ¼ 1

n

Xn

i¼1

sINi and Sout ¼ 1

n

Xn

i¼1

sOUTi : ð6Þ

Empirical studies of complex networks show them to besparsely connected, tightly clustered and with a relativelysmall diameter (Hayes 2000), which can be described by therespective indices of degree distribution, cluster coefficientand average minimum path length (Montoya and Sole 2002).Having explained the necessary graph theory, we nowdescribe the methodology used to create and analyse a largelarval transport network for a section of the GBR.

Methods

In this section, we will outline how the larval transportnetwork in the marine environment was constructed andanalysed. For many marine organisms, the mechanism ofdispersal is limited to the broadcasting of relatively passivelarvae into highly dynamic water bodies (Barber et al.2000; Shanks et al. 2003). Due to the difficulties inmeasuring the larval passages, connectivity research isheavily reliant on hydrodynamic simulations. Our studyused connectivity matrices generated by James et al. (2002)that simulate the release of larvae from 321 reefs into aLagrangian hydrodynamic model of the Cairns section ofthe GBR (Fig. 1). These matrices estimate the number ofsimulated fish larvae that manage to survive from thespawning reef to the settlement reef. The spawning reef canalso be the settlement reef (self-seeding) but these resultsare not used in the graph structure since the primary aim isto examine inter-reef connections. The strength of the pair-wise connections is a function of the size and shape of thedonor and recipient reefs since we assume that larvae arespawned with uniform density along the reef edge.Allowing for variation of factors such as mortality rates,release timing, distance of attachment, pre-competent andcompetent periods, the larvae are moved around in twodimensional spaces according to currents that are driven bywind and tidal forces. The simulations extend over the periodfrom 1967 to 1998 (Bode et al. 2006) and focus only on the

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summer months when dispersal is concentrated (James et al.2002). We consider the model to represent all larvae thathave the dispersal properties with the parameters used in theJames et al. (2002) model, and hence, this can also includecoral larvae. Due to lengthy processing time required to runthe hydrodynamic model, the capacity to conduct sensitivitytests on these dispersal parameters is limited.

We transformed the James et al. (2002) connectivitymatrices into graphs using tools created in both R statistics(especially utilising packages ‘igraph’ (Csardi and Nepusz2006) and ‘sna’ (Butts 2007)) and within an ArcGIS 9.3(Environmental Systems Research Institute, Redlands,USA) environment. Where the strength of connection isgreater than zero, a line is constructed between pointsrepresenting reefs. The line has the attributes of connectionstrength, distance and direction of flow. The reef centroidshave the spatial attributes of area, perimeter and location.

We addressed the initial questions in the following manner.

1. Is the GBR a scale-free small-world network? Thevalues of average minimum path length (L), averagedegree (<k>), connection strength (S), diameter (D),edge number (E) and cluster index (CI) were analysedfor the years 1967 to 1998 and compared with othernetworks of various topologies. The constructed net-works, used for comparison and described in Table 1,contained the same number of vertices and, wherepossible, the same average degree. To examine the roleof individual reefs in maintaining high larval flowthrough the network over time, the closeness andbetweenness centrality metrics were used.

2. How robust is our small-world emergent property to thelife history characteristic of species? The release timing,distance of attachment, pre-competent and competentperiods are altered using the hydrodynamics of 1998 toexamine the changes in average minimum path length (L),connection strength (S) and cluster index (CI).

3. How influential is the connection strength on the graphtopology? Using the 1998 graph, we removed theconnections in 15 equal steps based on three differentstrategies; strongest links removed first, weakest linksremoved first and randomly selected links removed. Weconsidered the impact on: average minimum pathlength (L), edge number (E) and cluster index (CI).We also compare the Euclidian distance and the weightderived from the hydrodynamic model for each edge.

Results

Is the GBR a small-world network? The averageminimum path lengths, L, (Fig. 4), for the GBR from1967 to 1998 (with parameters of day release, 1 km pickupzone, 1 week pre-competence, 3 weeks competent and nopredation (James et al. 2002)), are in the range 2.2 to 3.9links, and the average cluster coefficient, CI, is in therange of 0.26 to 0.31. The average degree <k> ranges from128 to 67 ( kh i ¼ 105:93; sk ¼ 14:04), while the edgecount E ranges from 20,554 to 10,860 (E ¼ 16; 950:2;sE ¼ 2; 247:2). These GBR hydrodynamic networks arecomparable to the constructed small-world networks(Table 1) of Watts and Strogatz (1998) and the ForestFire Network (Leskovec et al. 2007) and more clusteredthan the classic Barabasi–Albert (Barabasi and Albert1999) and Erdos–Renyi (Erdos and Renyi 1959) networks(Fig. 5). The 1998 network was rewired randomly (0.5probability) while preserving the original graph's degreedistribution, and this had the effect of reducing the clusteringcoefficient. Other models included in Fig. 5 were the Treemodel, 2D lattice (or planar graph) and Star graph (everysingle vertex is connected to the centre vertex only). Thecentrality measures showed a large diameter network

Table 1 Alternative networks for comparison

Model name Description Source

Watts-Strogatz network Firstly, a regular lattice is created, then the edges are rewired uniformlywith a specified probability

Watts and Strogatz 1998

Forest Fire network This network model resembles how a forest fire spreads by igniting treesclose by. Vertices are added sequentially, and edges are created with respectto the neighbouring configuration

Leskovec et al. 2007

Barabasi–Albert network In this model, one vertex is added in each time step, and edges are then createdto link existing vertices with a scale-free probability, PðkÞ � k�g

Barabasi and Albert 1999

Erdos–Renyi network This simple network creates the complete set of vertices and then adds edgeschosen uniformly randomly from the set of all possible edges

Erdos and Renyi 1959

Tree network This model is based on a regular tree with 2 edges connecting the ‘child’ neighbours Csardi and Nepusz 2006

2D lattice A lattice of two dimensions is composed of a set of four vertices joined in asquare, which is then replicated for the specified size of the network

Csardi and Nepusz 2006

Star graph In this simple network, every single vertex is connected to the centre vertex only Csardi and Nepusz 2006

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(D ¼ 8:68; sD ¼ 2:45; 5 � D � 14) with an exponentialdistribution of the betweenness measure (BC ¼ 17; 300;sBC ¼ 25; 176; 45:26 � BC � 178; 125), which highlight-ed that a small number of reefs were acting as steppingstones. The closeness measure (CC ¼ 16:99; sCC ¼1:90; 11:32 � CC � 21:61) showed a normal distribution

that describes that some reefs are isolated while others arestrongly connected. For each measure of high CC, low CCand high BC, the reefs with highest rank were identified andexamined through the 32 years (Fig. 6). The plots show thatthese reefs were consistent for the respective centralitymeasure through time.

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Fig. 4 The graph shows the graph theory indices for the Cairnssection of the GBR for the years 1967 to 1998, assuming thehydrodynamic model parameters of new moon release, 1 km sensorysettlement zone, 1 week pre-competence, 3 weeks competent and no

predation. Metrics used are edge number (E), vertex number (N),diameter (D) average minimum path length (L), average clustercoefficient (CI), average strength (S) and average degree (<k>). From1967 to 1998, the graphs consistently show small-world topologies

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Fig. 5 Comparison with alter-native models for the keysmall-world metrics of averageminimum path length (L) andcluster coefficient (CI). Thelower right area of the graph isconsidered to demonstratesmall-world properties. The al-ternative models are describedin Table 1

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The degree distribution was not found to be scale free(based on both degree frequency and size-rank metrics) forall years. This lack of scale-free degree distribution issimilar to other biological networks such as the Caeno-rhabditis elegans neural network and the polymer chainmodel reported by Amaral et al. (2000).

Is the GBR a small-world network for the larvaldispersal characteristics of different species? Varying thelarval dispersal parameters showed that the averageminimum path lengths, L, for the GBR from the 1998network are in the range 2.69 to 4.31 links, and the clustercoefficient, CI, is in the range of 0.21 to 0.34 (Fig. 7).These ranges are comparable to the small-world networksshown in Fig. 5. The life history characteristics influencedthe topology. Larvae released at dusk within 3 days of thenew moon created networks with less clustering comparedwith those networks with larvae released daily during thehigh tide. Other life history characteristics did not influencethe network consistently to the same extent; however, theaverage minimum path length was reduced when thecompetent period was shortened from 3 to 2 weeks. Oneparticular configuration consisting of new moon release,

1 km pickup zone, 2 weeks pre-competence, 3 weekscompetent and no predation produced a network withminimal average path length of L=2.69. This small networkwith a diameter of 9 became the largest network (diameter=15, L=4.31) with a simple change in the competency periodto 2 weeks. Altering the predation intensity did not changethe small-world properties but did alter the in- andoutbound average connection strength, S, substantiallyfrom 0.64 to 0.03.

Despite the large range in the number of edges, from 4,053to 25,319 (mean=12,222, σE=5,884.23), the average mini-mum path lengths and the cluster coefficient showed anarrow distribution (2:69 < L < 4:31; sL ¼ 0:37; 0:21< CI < 0:34; sCI ¼ 0:041) across the parameter range.Hence, we conclude that the GBR larval dispersal network,as modelled by James et al. (2002), is a small-world networkfor most species.

Our third question asks how influential the strength ofthe connections on the graph topology is. First, let usconsider the distribution of out- and inbound strengthvalues. The inbound connection weights for the 1994 graphshows a scale-free distribution (Fig. 8), which highlights

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Fig. 6 Box plots of the centrality metrics for the highest ranking reefscompared with all the reefs, highlighting the individual reefconsistency through the 32 years of model runs. The closenesscentrality (a) shows the five reefs with highest and lowest values

indicating the contagion and isolated reefs. The betweenness centrality(b) shows the top 10 reefs indicating the stepping stone or cut nodereefs

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the sensitivity of the graph to changes in strength thresh-olds. The inbound strength distribution (Fig. 8a) can bedescribed as RðkÞ ¼ cYk�g , where Yk=sk and γ=1.38 forthe 1994 graph. The distribution of the strength values(siIN; siOUT) shows that a significant number of reefs have ahigh inbound strength, indicating that these reefs wereacting as strong sinks of larvae. The inbound distributionpattern did vary between the years but the majority weredescribed as scale free. We acknowledge that the size of areef in the James et al. (2002) model will partiallydetermine the larval numbers released and so influencethe source capacity of each reef. As a consequence, the

outbound strength distribution across all years did notexhibit scale-free trends and were similar to the 1994network shown in Fig. 8b.

Edges (connections) in the 1998 graph were removedbased on a changing weight threshold using three differentstrategies (strongest, weakest and randomly removed), andthe edge number, average minimum path length (L) andcluster coefficient (CI) were recorded (Fig. 9). The weightthreshold was decreased from the maximum weight value,for the strongest removal first strategy, in 15 equal incre-ments. Similarly, for the weakest removal first strategy, thethreshold was increased from the minimum weight value.The random removal strategy selected 1,011 edges eachstep and deleted them. The strategy of removing thestrongest edges first shows that until the 13th step, wherethe weight threshold was equal to 0.14, only 178 edgeswere removed, reflecting that the 98% of edges had aweight less than 0.14 (min=1.915e−06, mean=0.002,max =0.608). This trend is also observed in the clustercoefficient (CI) and the average minimum path length (L),indicating that the small world character is robust for largereductions in the maximum weight values. The secondstrategy of removing the weakest edges initially shows arapid reduction in edge number and subsequent changes inthe small world properties. The cluster coefficient (CI)reduced rapidly to zero at step 6, highlighting that smallreductions in the minimum edge weight elicit large changeson community structure. Weaker edges tend to be longer(measured by Euclidean distance, Fig. 10), and consequent-ly, they act to bind the network together. Their removalleaves a fragmented network sparsely connected withstrong edges that have short Euclidean lengths. The averageminimum path length was less affected and initially rose,while edges with generally longer Euclidean lengths(Fig. 10) were removed. By step 6, the network hadfractured into isolated communities with a diameter of one.The random removal strategy shows only small changes inthe average minimum path length (L) until finally thenetwork consists of isolated pairs of connected reefs. The

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Day release on New Moon

Day release on High Tide

Fig. 7 A comparison of changes in the small-world metrics ofaverage minimum path length (L) and cluster coefficient (CI) for reefsconnected by different assumptions about hydrodynamics and speciesecology. The various 1998 networks are grouped according to dayrelease on high tide/day release on dusk with new moon and labelledwith sensory settlement zone of 1 or 4 km (1kmSettle, 4kmSettle), pre-competent period of 1 or 2 weeks (1wkPre, 2wkPre) and competentperiod of 3 or 2 weeks (3wkCom, 2wkCom)

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Fig. 8 Size-rank distributions of connection strength, s (Eq. 5), for the 1994 graph, a in- and b outbound showing the scale-free nature of theinbound strength distribution

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cluster coefficient (CI) did not change until approximatelyhalf the edges were removed (7,078 edges remaining from15,167 total edges).

Next, we consider how robust the small-world networkproperty is given changes in the edge length. The longeredges were removed (444, 333, 222, 111, 55 and 27 km)but this did not alter the topology of the graph (Fig. 11)until all the links were less than 55 km long. At this

distance, the topology of the graph altered rapidly with theaverage degree, <k>, and strength index, S, decreasing. At55 km, the average minimum path length (L) increased asedges that connected widely separated network sectionswere removed from the network. However, the averageminimum path length (L) decreased as the maximumEuclidean length was reduced to 27 km. In this case, onlythe smaller more consolidated network remained (V=258),

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and subsequent edge length reduction resulted in fragmen-tation of the network. The cluster coefficient (CI), whichmeasures the ratio of triangular linkages against themaximum possible, increased as the clusters joined byshort lengths remained. The GBR remained a small-worldnetwork despite the removal of edges with lengths longerthan 55 km but disintegrated into isolated clusters when thelengths were limited to less than 27 km.

Discussion

The Great Barrier Reef exhibits small-world patterns in theyears 1967 to 1998 across a range of larval dispersalproperties using the connectivity matrices generated byJames et al. (2002). This is the first mesoscale ecologicalsmall-world network reported, and the properties of the

network are similar to other published small-world net-works of biological systems. The average minimum pathlength of approximately L=3 links with an averagemaximum diameter of D=9 links (mean arc length≈90 km) distributed over 600 km of reef matrix shows thatthe reefs are well connected relative to other sorts of graph.For example, a random network connecting the 321 reefswith the same number of connections (D=7, L=2.23)would have links with a longer average Euclidean distance,while a planar or nearest neighbour network where only theneighbouring reefs are connected (D=34, L=12) wouldhave links with significantly shorter average Euclideandistances. The high value for the clustering index within thereef system indicates that a substantial proportion of inter-reef larvae settle on neighbouring local reefs with a smallernumber travelling longer distances between clusters. Thispattern is supported by the experimental evidence usingparticle dispersal around One Tree Island by Kingsford etal. (2002). This experiment highlighted that the majority ofparticles remained close to the donor reef despite the tidalinfluences, while a small number were swept away to moredistant locations. The volume of larvae circulating in such asystem need not be immense to strongly influencecolonisation patterns (Ayre and Hughes 2000; Andrefouetet al. 2002; Mumby 2006; Treml et al. 2008). The role ofeach reef within the network, in terms of influencing larvalflow, can be assessed using the centrality metrics. Theresults for the betweenness and closeness values (which isprimarily a function of their geographic location, size andshape) show that the reefs play a consistent role throughtime. Conservation measures should bias protection to-wards these consistently important reefs.

We found that the GBR remained a well-connected smallworld when the life history properties of the species being

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modelled were varied substantially. Despite altering themortality rates, release timing, distance of attachment andpre-competent and competent periods, the changes inaverage minimum path length and cluster coefficient(Fig. 7) remained within the range of other published smallworlds and those artificially generated (Fig. 5). This impliesthat the small-world structure is the most likely structure formarine species found in the GBR marine environmentnotwithstanding that this network model ignores othermigration processes such as adult movement that are partof the complete life cycle of marine organisms.

The GBR does not exhibit scale-free distributions of thenumber of connections; however, the distribution of reefswith respect to the number of larvae settling or migrating toanother reef (strength of inbound connection) can bedescribed as scale free (Fig. 8). The implications of thispattern are that the reefs in this system did not have adisproportionate number of connections, but that a substan-tial number of reefs were strongly connected. In contrast,weighted networks such as the world airline network andscientific authorship (Barrat et al. 2004) exhibited heavy-tailed distributions for weight. The weight of a connectionin these networks will be determined by the state of theadjoining nodes (i.e. reef health and airport weather) andso understanding the impact that the state will have on thenetwork topology and function will be essential inunderstanding the dynamics of the system.

Disturbances to the GBR in the form of cyclones, ElNiño events and climate change are likely to impact onthe way reefs are connected by larvae (Harriott andBanks 2002). The edge weight of the network, created bythe hydrodynamic model, is determined by a range ofcomplex interactions between reef topography, populationfecundity, water currents and neighbourhood configura-tion. Within ecological time frames, only the currents andfecundity are likely to be extensively disturbed (such as ElNiño events influencing the East Australian Current), andthis will impact on the connection strength. The removalof weaker or stronger edges from the network (Fig. 9)reveals that the small-world character is robust when thestronger edges are diminished but the removal of theweaker edges rapidly resulted in a fragmented sparselyconnected network. The more numerous weaker edges actto bind the network across the GBR, and this supports thehypothesis by Ayre and Hughes (2000) that infrequent andweak long distance dispersal is contributing to communitystructure across the GBR. There are strong larval con-nections between reefs located in relatively close proxim-ity (Fig. 10), and field studies highlight the high levels ofsimilarity of neighbouring populations (van Oppen et al.2008). Random removal of dispersal edges does not alterthe small world properties until only a small percentage ofedges are remaining. These results imply that the GBR

small-world character is robust against random distur-bance but any disturbance that results in a regionalreduction in the capacity of the larvae to successfullymigrate long distances will effectively and rapidly increasethe GBR network size (specifically the diameter). Recov-ery from large scale disturbances, through externalrecruitment, will then be diminished.

When dispersal distances were limited to less than27 km, the reef network broke into isolated clusters(Fig. 11), rather than a small-world pattern. Until fieldexperiments resolve the issue of dispersal curves and self-seeding proportions for a wide range of fish and coralspecies (Jones et al. 1999), the small-world pattern willremain speculative. In particular, the accurate parameter-isation of dispersing larvae behaviour is required for theproper use of the hydrodynamic model in predicting larvaldispersal (Kingsford et al. 2002) and hence the networkstructure.

Numerous authors have expressed the need to includeinformation about connectivity in planning for and manag-ing coral reef systems (Garcia-Charton and Perez-Ruzafa1999; Jones et al. 2005; Mumby 2006; Minor and Urban2008). With some justification, cross shelf and latitudinalgradients dominate the ecological classification of the GBR(for example Fabricius and De'ath 2001). These classifica-tions do not attempt to include metapopulation processessuch as the dispersal of larvae, yet biodiversity protectioninstruments such as the marine protected areas rely onpreserving these processes (Possingham et al. 2005). Graphtheory provides the capability to identify how individual reefsare contributing to community structures. Prioritising reefsbased on the functional role within a larval network couldassist with the design of conservation reserves (Garcia-Charton and Perez-Ruzafa 1999). Maintaining the networkas a strongly connected system across a diverse geographicspace (inshore to outer shelf) rather than as a series ofisolated communities will require the identification andprotection of reefs essential to enhancing larval flow. Thisincludes key isolated and highly connected reefs as well asstepping stone reefs such as Low Island Reef (Fig. 6).

If the GBR is a small-world network, how does this alterour perspective on the reef metapopulation? Is each reeffunctioning as an isolated community (i.e. Jones et al.2005) with occasional larval exchange (i.e. Ayre andHughes 2000)? Isolation occurs at different scales, and thehierarchical nature of small worlds describes how clustersexist within clusters and indeed some networks maybefractal in nature (Song et al. 2005). For example, the PalmPassage (Fig. 12) provides a barrier to larval transport andeffectively isolates the southern cluster from larval recruit-ment (Bode et al. 2006). This does not detract from thefindings of Jones et al. (1999) that many larvae circulatethen return to the reef of origin. Substantial evidence exists

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that planktonic larvae are found outside the local dispersalzone (Mora et al. 2003), and this could provide themechanism to link populations in the GBR (Ayre andHughes 2000). The contribution that each reef makes toGBR ecology needs to be evaluated in the context of thewider network. This network can be simulated by ahydrological model, such as James et al. (2002). Constrain-ing the relationships between reefs to latitudinal andlongitudinal descriptors will disregard the intergenerationalconnections that can serve to connect or, when absent,isolate populations (Barber et al. 2000; Bode et al. 2008).

Our conclusion that the GBR is a small-world systempresents the largest known ecological network thatexhibits high levels of connectivity combined with tightclustering. The implications for GBR conservation andscientific research are significant because understandingthe metapopulation structure underpins our understand-ing of coral reef function. Exactly how this newunderstanding would explicitly change where we placeno-take marine protected areas in any rezoning of theGreat Barrier Reef is an important avenue for futureresearch.

Fig. 12 Map showing connecting edges, with weights greater than 0.10, for the reefs near Hinchinbrook Island (−18 S, 146.5 E)

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Acknowledgements The authors would like to thank T. Hardy, L.Bode, L. Mason and M. James from the Marine Modelling Unit atJames Cook University for the provision of the hydrodynamic outputsused in this study. We also thank Liz Tynan for suggestions on thismanuscript. The comments from two anonymous reviewers signifi-cantly enhanced our manuscript on this application of networks tomarine ecology.

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