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Costa et al. Graph theory for species persistence
A new interpretation of graph theory measures
in evaluating marine metapopulations persistence:
The study case of soft-bottom polychaetes in
the Gulf of Lion.
Andrea Costa1,2,*, Andrea M. Doglioli1,2, Katell Guizien3, Anne. A.Petrenko1,2
1 - Aix Marseille Universite, CNRS/INSU, IRD, Mediterranean Instituteof Oceanography (MIO), UM 110, 13288 Marseille
2 - Universite de Toulon, CNRS/INSU, IRD, Mediterranean Instituteof Oceanography (MIO), UM 110, 83957 La Garde
3 - Laboratoire d’Ecogeochimie des Environnements Benthique, CNRS,Universite Paris VI, UMR8222, Av. du Fontaule - F-66651 Banyuls-sur-Mer (France)
keywords
Connectivity, Persistence, Graph Theory, Shortest Cycles, Betweenness,Metapopulation Model, Directed Weighted Bridging Centrality, Modularity,Clustering
December 22, 2015
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Costa et al. Graph theory for species persistence
Abstract1
Herein we test graph theory analysis of hydrological connectivity for the2
demographical persistence of a soft-bottom polychaetes metapopulation in3
the Gulf of Lion (NW Mediterranean Sea). As a preliminary step of the4
graph theory analysis we introduce a metric for the node-to-node distance5
in graphs based on connectivity matrices containing larval transfer probabilities.6
This metric ensures a physically meaningful interpretation of shortest paths7
and, consequently, of betweenness. Then, we assess and -eventually- re-8
evaluate the interpretation of two classical graph theory centrality measures9
(betweenness and modularity) in the context of species persistence. New10
measures (directed weighted bridging centrality, minimum cycles identification)11
are also derived and evaluated. In particular, modularity and bridging12
centrality are shown to characterize clusters of interconnected nodes (i.e.,13
subpopulations), to highlight rescuing mechanisms and its source sites.14
Further, we show that shortest cycles indicate the sites ensuring species’15
regional persistence, whereas betweenness appears less relevant for persistence16
than in previous literature. Our new interpretation of graph theory proposed17
here is supported by a detailed comparison with a metapopulation model.18
Introduction19
Losses of biodiversity at sea due to deleterious effects of natural phenomena20
and human activities (e.g., global warming, habitat destruction, overfishing,21
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Costa et al. Graph theory for species persistence
etc.) are currently expected to be mitigated by the implementation of Marine22
Protected Areas (MPAs); see LeCorre et al. (2012) and Duraiappah and23
Shahid (2005) for comprehensive discussions on the argument. The basic24
assumption of this approach is that, if a carefully-chosen portion of the25
whole marine biological network is protected, the network would not be26
subject to breakdowns generating critical losses of both biodiversity and27
individual abundances in the marine ecosystem. Indeed, if a subset of populations28
are sufficiently interconnected, forming a persistent sub-network and alimenting29
other populations, the regional maintenance of the species is ensured. Thus,30
the challenge is to identify the sub-networks that could permit the species31
persistence in a whole habitat by minimizing the mortality only in some32
areas within it. Equivalently we can say that the problem is to identify33
the minimal sub-network that can maximise the connectivity of the whole34
network. In this way it will be possible to protect the network by minimizing35
the costs of MPAs implementation (see Andrello et al., 2014, for example).36
37
However, there are two major problems in identifying key sites for the38
conservation of a set of species distributed in disjunct sites between which39
species may disperse. First, great dissimilarities in dispersing ability among40
species translates into very different connection patterns (see Siegel et al.,41
2003, for example). Second, species interactions with the environment or42
between themselves in the various sites also affect differently the persistence43
of species in the network (McArthur and Levins, 1967). These difficulties44
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Costa et al. Graph theory for species persistence
led to different methodologies for tackling the problem, each with their45
own advantages and shortcomings (see Kool et al., 2013 and Lagabrielle46
et al., 2014). Among these techniques, graph theory has been increasingly47
employed for conservation studies (Moilanen, 2011) due to its ability to48
capture the essential features of a network. Due to this ability, it has been49
adapted to a vast range of contexts. It was first introduced in ecology by50
Urban and Keitt (2001) in a study of landscape connectivity, then Schick51
and Lindley (2007) extended its use to riverine networks. Treml et al. (2008)52
applied it to the study of the connection between marine reefs. Rozenfeld53
et al. (2008) used it to infer gene flux in marine populations networks.54
Jacobi et al. (2012) exploited graph theory for the identification of marine55
sub-populations.Andrello et al. (2013) applied it for the estimation of connectivity56
among marine MPAs.57
In particular, many different graph theory measures have been proposed58
to highlight nodes of interest in different networks in different contexts (see59
Rayfield et al., 2010; and Galpern et al., 2011; for exhaustive reviews).60
Because graph theory identifies well connected networks with networks61
that have an efficient transfer within them, in the literature, graph theory62
measures (e.g., betweenness) highlighting nodes important for an efficient63
transfer have been proposed as relevant for conservation (e.g., Treml et al.,64
2008). This point is not bereft of controversies. For example, the equivalence65
between efficient transfer and conservation relevance remains unproven66
(sensu Moilanen, 2011; and Lagabrielle et al., 2014).67
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Costa et al. Graph theory for species persistence
68
Herein we want to revise the interpretation of some graph theory69
measures from a conservation point of view in the framework of marine70
connectivity. For doing this, graph theory measures were compared with71
outputs from a metapopulation model. This latter model links local demography72
and regional dispersal. Specifically, metapopulation models have been extensively73
used to investigate conditions of species persistence (Caswel, 2001; Hastings74
and Botsford, 2005). In particular, in metapopulation models, well connected75
networks are networks ensuring the regional persistence of the species. It76
follows that, if we can find which graph theory measure best reproduces77
the information provided by metapopulation models, we can clarify the78
relevance of the different graph theory measures for conservation issues.79
80
Our study case consists in the identification of the sites important81
for the persistence of a soft-bottom polychaetes metapopulation in the82
Gulf of Lion (GoL), see Figure 1. The Gulf of Lion was selected as study83
case because of the numerous studies, both physical and biological, already84
performed in this area that can be used to interpret and validate our results.85
Moreover, recent studies (Rossi et al., 2014) support the choice of spatial86
scale of the size of the GoL as appropriate to study the hydrodynamical87
connectivity properties of month-long larvae dispersals. The GoL is located88
in the north-western Mediterranean Sea and is characterized by a large89
continental margin dominated by a soft bottom forming the habitat of90
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Costa et al. Graph theory for species persistence
uniform polychaetes assemblages in the 10 to 30 m bathymetric depth91
range. Its hydrodynamics is complex and highly variable (Millot, 1990).92
Depending on wind forcing, currents in the study zone can be either eastward93
or south-westward (Estournel et al., 2003; Petrenko et al., 2008). The circulation94
is strongly influenced by the Northern Current, which constitutes an effective95
dynamical barrier blocking coastal waters on the continental shelf (Petrenko,96
2003) and delimits the regional scale of hydrodynamic connectivity. Exchanges97
between the GoL and offshore waters are mainly induced by processes98
associated with the Northern Current (Petrenko et al., 2005).99
100
As reference for the metapopulation model analysis we use the study101
by Guizien et al. (2014). Some essential details on the model can be found102
in the Supplementary Materials A. Hydrodynamic connectivity was quantified103
by larval transfer probability between 32 reproductive sites along the shore104
of the GoL (Figure 1). These sites cover a substantial part of the available105
habitat in the GoL for soft-bottom polychaetes. The metapopulation model106
parameters were tuned for polychaetes on the base of the review by McHugh107
and Fong (2002). Ad hoc metapopulation simulations based on a threat108
scenario have been used to hierarchize four targeted sites in a metapopulation109
model of the Gulf of Lion (NW Mediterranean Sea). The scenario aimed at110
quantifying the resilience of the metapopulation to habitat losses due to111
anthropic pressure around each of the four principal commercial ports in112
the GoL. Vulnerability was thus hierarchized by determining the number113
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Costa et al. Graph theory for species persistence
of unsuitable sites, starting from each port and proceeding symmetrically114
around each port, necessary to cause the metapopulation crash. Evidence115
of a rescue mechanism from the sites located in the western part of the116
studied area by the sites in the eastern part was also provided by studying117
the spatial distribution of polychaetes population density.118
119
In the present study we focus on the relevance to the way of applying120
graph theory. Our principal concern is the validity of some choices for the121
node-to-node metric in previous literature. Hereby we propose a metric122
that provides physically meaningful results from graph theory analysis123
when analysing connectivity matrices based on larval transfer probabilities.124
125
The paper is organized as follows. In the Materials Section we discuss126
the main characteristics of the common input for graph theory and metapopulation127
model: 20 connectivity matrices issued from Lagrangian dispersal simulations.128
In the Procedures Section we summarize the graph theory measures we129
tested and introduce a new metric for the node-to-node distance in graphs130
built on current-based connectivity matrices. In the Assessment section we131
present the systematic analysis of the hydrological connectivity matrices132
with graph theory measures and its interpretation and explanation in the133
light of oceanographic structures.134
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Costa et al. Graph theory for species persistence
Materials135
In order to compare with the results of the metapopulation modelling study136
of Guizien et al. (2014), the same input connectivity matrices obtained137
from Lagrangian dispersal simulations were used in the present study.138
The Lagrangian simulations used a three-dimensional circulation139
model (see Marsaleix et al., 2006) with a horizontal resolution of 750 m.140
Spawning was simulated by releasing 30 particles in the center of each of141
32 sites alongshore the GoL, on the 30 m isobath, every hour from January142
5 until May 16 in 2004 and 2006. The final positions of larvae after three,143
four and five weeks were processed to compute the proportion of larvae144
coming from an origin site and arriving at a settlement site. Connectivity145
matrices were then built for ten consecutive 10-day spawning periods in146
each year and for each of three different pelagic larval durations (3, 4 and 5147
week), for a total of 20 matrices (numbered from #1 to #20).148
149
It is important to note that the connectivity matrices’ values depend150
strongly on the circulation present in the Gulf during the period of the151
dispersal simulation. The typical circulation of the Gulf of Lion is a westward152
current regime (Figure 1). This was the case of matrices #7,#11,#12,#15,#17.153
In this study, other types of circulation were also present. In particular154
matrix #1 was obtained after a period of reversed (eastward) circulation.155
Indeed this case of circulation is less frequent than the westward circulation156
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Costa et al. Graph theory for species persistence
(Petrenko et al., 2008). Matrices #14, #10 and #13 correspond to a circulation157
pattern with an enhanced recirculation in the centre of the gulf. Finally,158
matrices #2, #3, #5, #6, #8, #9, #14, #16, #18, #19, #20 correspond159
to a rather mixed circulation with no clear patterns. A geographic representation160
of some connectivity matrices is in Figure 2. Therein we can see simultaneously:161
(i) the geographic distribution of the sites, (ii) the geographic direction of162
the connectivity by advection aij (with the arrows pointing in the i → j163
direction) and (iii), for each couple of sites, the difference between the164
probability to go from one site to the other, or vice versa, by looking at165
the different colors of the arrows. For clarity the connectivity values lower166
than 2/3 of the maximum are not represented. When both probabilities in167
i → j and j → i directions are high and hence plotted, the arrows reach168
only the mid-distance between the nodes. This representation captures169
the circulation patterns: in Figure 2a (matrix #7), there are more arrows170
in the east-to-west direction and these are almost always stronger than171
the corresponding west-to-east ones, like we expect in a case of westward172
circulation. The opposite case is represented in Figure 2b (matrix #1),173
dominated by an eastward circulation pattern. As we can infer from the174
high number of arrows without a predominant direction in Figure 2c, matrix175
#10 is characterized by a recirculation pattern in the center of the GoL.176
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Costa et al. Graph theory for species persistence
Procedures177
Mathematically speaking, a graph G is a couple of sets (V,E) where V is178
the set of nodes and E is the set of edges. The set V represents the collection179
of objects under study that are pair-wise linked by an edge representing a180
relation of interest between these two objects. When the relation is symmetric,181
the graph is said to be ‘undirected’, otherwise it is ‘directed’. An example182
of an undirected graph in the context of biological networks is the genetic183
distance among populations used in Rozenfeld et al. (2008), while an example184
of directed graph is the probability of connections due to the current field185
between two zones of the sea as in Rossi et al. (2014). If every existing186
edge has the same importance as the others, the graph is said to be ‘binary’,187
that is the edges can exist or not. If each edge has a specific relative importance,188
a weight can be associated to each of them and the graph will then be189
called ‘weighted’. The total weight of the connections of a node i ∈ V is190
called strength k(i). In an undirected graph, this is equal to the number of191
edges incident on the node. In a directed graph, it is possible to distinguish192
between in-strength and out-strength. The first one is the sum of the values193
of the edges terminating in the node kin(i) =∑
j aji, while the second194
is kout(i) =∑
j aij with j ∈ V and i 6= j. Here the values aij are the195
terms of the connectivity matrix where all the values of the edges from196
node i to node j are stored. The density or connectance ρ of a graph can197
be defined as the ratio between the number of existing edges E and its198
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Costa et al. Graph theory for species persistence
maximum possible value N(N − 1), where N is the number of nodes. For199
a directed graph we have: ρ = EN ·(N−1)
. The network strength is defined as200
the sum of all the elements of a connectivity matrix. The number of not-201
null elements of a connectivity matrix is the number of matrix entries with202
a value greater than zero.203
204
In the present study, we deal with directed weighted graphs. The205
nodes of our graphs represent the 32 sites used in the metapopulation model,206
while the edges represent the not null probability that a Lagrangian particle207
released in one site is transported to another site, after a certain amount of208
time corresponding to the larval duration period.209
210
In a connected directed-unweighted graph (i.e., directed-unweighted211
with no disconnected parts), it is possible to define the shortest path σl,j212
connecting two nodes l ∈ V and j ∈ V as the shortest possible alternating213
sequence of nodes and edges, beginning with node l and ending with node214
j, such as each edge connects the preceding node with the succeeding one.215
The definition can be extended to directed weighted graphs: the shortest216
path has the lowest cost between two nodes. The most frequent choice217
to define the cost of a path is the sum of its edges’ weights. Nonetheless,218
other alternatives are possible and will be discussed in more detail at the219
end of this section (see Subsection ‘A new metric for node-to-node distance’).220
The definition of a widely used centrality measure called betweenness BC(i),221
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Costa et al. Graph theory for species persistence
i ∈ V , is based on the concept of shortest path. The betweenness estimates222
the relative importance of a node i within a graph by counting the fraction223
of existing shortest paths σl,j that effectively pass through this node σl,j(i):224
BC(i) =
∑l,j σlj(i)∑l,j σl,j
(1)
225
The concept of cycle (see, for example, Barrat et al., 2008, for an226
introduction), despite its simplicity, turns out to be useful in the study of227
species multi-generational persistence. Cycles are defined as those paths228
that, starting from node i ∈ V , end up to the node i itself, after a certain229
number L of steps. In order to neglect the effect of the particles remaining230
at the same site with respect to the effect of the ones leaving the site and231
coming back, we only consider cycles with L > 2. One of the essential232
requisites for ensuring the persistence of a species in a given zone is the233
high probability to see the larvae returning home after a certain number of234
generations (see Hastings and Botsford, 2005, for details). This means that235
the shorter the cycle starting from a given node, the more likely the site is236
important for persistence. In fact, in this case, the site survival would be237
quite independent from the import of larvae from other sites. Thus it can238
act as a source in our network (Hastings and Botsford, 2005).239
The main practical problem of this kind of analysis is the generally240
overwhelming computational power required. We used an algorithm that241
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Costa et al. Graph theory for species persistence
recursively finds all the possible cycles for every node of the network, thus242
involving a (N − 1)L − 1 complexity. Indeed, our analysis was doable only243
because the number of nodes (N = 32) in our network is small enough to244
make the problem treatable with easily accessible computational facilities.245
Nonetheless, we were constrained to limit L to 5; hence L is between 2 and246
5.247
248
A widely used method for identifying clusters in physical networks249
is the maximum modularity criterion first introduced by Newman and250
Girwan (2004). Modularity Q is defined, up to a multiplicative constant, as251
the difference between the number of edges falling within given groups of252
nodes and the expected value in a network that conserves the degree values253
but with randomly placed edges (further details can be found in Newman,254
2006). The values of modularity can be either positive or negative, with255
positive values indicating the possible presence of community structures.256
Therefore we are able to investigate the community structure of a network257
by looking for the divisions of the network associated with a maximum258
value of modularity. Given a network, let ci be the community in which259
node i is assigned. For a directed weighted graph the modularity assumes260
the form (see Nicosia et al., 2009, for details):261
Q =1
m
∑i,j∈V
[aij −
kouti kinjm
]δ(ci, cj) (2)
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Costa et al. Graph theory for species persistence
where ki and kj are the degrees of the nodes i and j, m =∑
i ki and262
δ(ci, cj) is the Kronecker δ-function.263
Exploiting a reformulation of modularity in matrix formalism, it is264
possible to recursively explore all the possible divisions of a network in265
order to identify the one that maximizes the modularity value of the network266
without exceedingly high computational power (see Newman, 2006, for267
details). One drawback of the algorithm is an intrinsic variability that268
eventually makes the results not completely compatible between different269
runs of the analysis. For example certain nodes could be assigned to different270
clusters without changing the maximum value of Q. This inconvenience271
can be bypassed by running the analysis multiple times and taking, as a272
best division, the one that is the most frequently found. In the present273
work we ran the analysis 1 · 104 times on the 20 different variant matrices,274
hence a total of 2 · 105 runs.275
276
In order to extract all possible information from the connectivity277
matrices about the role played by the different sites, we also used the bridging278
centrality CBR measure. This measure was first proposed by Hwang et al.279
(2008) for undirected unweighted graphs. For our analysis we reformulated280
it in order to extend its use to directed weighted graphs.281
Bridging centrality highlights those nodes that connect different clusters282
of a network (see Hwang et al., 2008). It is derived both from the betweenness283
value of a node and from the bridging coefficient that accounts for the284
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Costa et al. Graph theory for species persistence
probability of leaving the direct neighbourhood of the node by starting285
from one of the nodes composing it. Intuitively, nodes with a high number286
of such edges fall on the boundary of clusters. In Hwang et al. (2008), for a287
node i ∈ V , the bridging coefficient is defined as:288
Ψuu(i) =1
k(i)
∑v∈N(i)
∆(v)
k(v)− 1(3)
where k(i) is the strength of the node i ∈ V and N(i) the direct289
neighbourhood of i: that is the set of nodes reachable from i in one step.290
∆(v) is the out-strength of nodes v ∈ N once deleted the edges going from291
v to other nodes in N(i).292
We propose the way to generalize the bridging coefficient to directed293
weighted graphs by accounting for the weight of the edges and by checking294
which edges are effectively leaving the neighbourhood of the node. Then,295
we correct the out-strength of i via the term −avi and the strength of v296
via the term −(aiv + avi). Note that, for this calculation, all the terms297
avv on the diagonal of the connectivity matrix are irrelevant. Hence, in the298
directed weighted case, we redefine the bridging coefficient as:299
Ψdw(i) =1
ktot(i)
∑v∈N(i)
∆(v)− aviktot(v)− (aiv + avi)
(4)
where ktot(i) = kin(i) + kout(i) is the strength of the node i ∈ V . In300
this way, we retain both the information on the flux of information through301
a node (given by the betweenness) and the topological information on the302
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Costa et al. Graph theory for species persistence
position of this node relative to clusters (given by the bridging coefficient).303
In fact, a node falling on the border of a cluster and channelling a high flux304
of information will have both high bridging coefficient and high betweenness305
values. As a result, the removal of such a high bridging centrality node306
would have a much more disruptive effect than the removal of a node having307
only either a high betweenness value or a high bridging coefficient (see308
Hwang et al., 2008, for an analysis and discussion of this phenomenon309
in the undirected case). An important aspect to pay attention to, when310
calculating the betweenness centrality and the bridging coefficient of a311
node, is the different orders of magnitude in play. While the first is normalized312
to one, the second is not: its value depends upon the particular metric313
used to define the distance between the nodes. In order to give to the two314
parameters equal importance in characterizing the centrality of a node, we315
follow the suggestions of Hwang et al. (2008), and (i) calculate the betweenness316
centrality and the bridging coefficient for each node, (ii) calculate the rank317
vector of the nodes on the base of their value of betweenness and bridging318
values, and (iii) calculate the bridging centrality as:319
HBR(i) = ΓBC(i) · ΓΨ(i) (5)
where ΓBR(i) is the rank of a node i in the betweenness vector and320
ΓΨ(i) is the rank of a node i in the bridging coefficient vector. To summarize,321
bridging centrality allows us to identify the nodes which are likely to be on322
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Costa et al. Graph theory for species persistence
the boundaries of the clusters and hence able to prevent the fragmentation323
of the network in sub-networks.324
A new metric for node-to-node distance325
An essential aspect in analysing biological network stability and structure326
with graph theory is the choice of the metric used to define the distance327
between the nodes of the corresponding graph. Above all, this choice has328
important consequences on the physical interpretation of the results. In329
principle, many choices are possible: the genetic distance was used by Rozenfeld330
et al. (2008); the connection time between sites by Treml et al. (2008); the331
larval transfer probability by, for example, Andrello et al. (2013). One can332
refer to Rayfield et al. (2010) and Galpern et al. (2011) for reviews on the333
different metrics.334
Here we propose the use of a new metric to define the distance between335
nodes when dealing with larval transfer probabilities, in order to ensure336
that larger larval transfer probability between two nodes corresponds to337
smaller node-to-node distance. Consider that: (i) larval transfer probabilities338
are calculated by considering the position of the Lagrangian particles only339
at the beginning and at the end of the advection period; (ii) we are discarding340
the information on the effective path taken by a particle (i.e., the probability341
to go from i to j does not depend on the zone the particle is coming from342
before arriving in i); and (iii) the calculation of the shortest paths implies343
the summation of a variable number of these connectivity values (that is,344
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Costa et al. Graph theory for species persistence
in the calculation of betweenness, we are considering paths whose values345
are calculated from different number of generations). Thus it is clear that346
the probabilities we calculated by Lagrangian simulations are intrinsically347
independent from each other. Nevertheless, the classical algorithms calculate348
the shortest paths as the summation of the edges composing them (e.g.,349
the Dijkstra algorithm, Dijkstra, 1959). These algorithms, if directly applied350
to the probabilities at play here, are incompatible with their independence.351
So we define the distance between two nodes i and j as:352
dij = ln
(1
aij
)(6)
where aij is the connectivity probability given by the connectivity353
matrices used in the metapopulation model. This definition is the composition354
of two functions: h(x) = 1/x and f(x) = ln(x). The use of h(x) =355
1/x allows one to exchange the ordering of the metric in order to make356
the most probable path the shortest. The use of f(x) = ln(x), thanks to357
the basic property of logarithms, allows the use of classical shortest-path358
finding algorithms while dealing correctly with the independence of the359
connectivity values. In fact, we are de facto calculating the value of a path360
as the product of the values of its edges. It is worth mentioning that the361
values dij = ∞, resulting from the values aij = 0, do not influence the362
calculation of betweenness values via the Dijkstra algorithm.363
Note that Equation (6) is additive and homogeneous (see Supplementary364
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Costa et al. Graph theory for species persistence
Materials B for a detailed demonstration).365
This new metric combines previous distance definitions attempts366
(dij = 1/x, Gao et al., 2010; and dij = ln(x), Brockmann and Helbing,367
2013). Above all, it is consistent with the summation of probabilities and it368
respects the three mathematical properties of a distance (see Supplementary369
Material B).370
371
Note also that this node-to-node metric does not apply for the calculation372
of all the measures we presented in this section. It is of interest in the373
only case in which a probability reversal is needed: betweenness, minimum374
cycles and network strength calculation. Modularity, bridging centrality375
and number of not null elements are still calculated using the original larval376
transfer probability aij.377
378
For the interested reader, please note that the graph theory toolbox379
we developed for the present study can be freely downloaded from the web-380
page http://www.mio.univ-amu.fr/~costa.a.381
Assessment382
To the best of our knowledge, it is the first time that graph theory is applied383
to connectivity matrices obtained from Lagrangian trajectories based on a384
fully 3-D circulation model. Moreover the dispersal time of the numerical385
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Costa et al. Graph theory for species persistence
particles is set in order to mimic the principal biological characteristics386
of polychaetes. Both these aspects significantly complicate the dispersion387
dynamics (Siegel et al., 2003). As far as we can tell, it is also the first time388
that graph analysis is used on such a restricted coastal spatial domain for389
conservation aims. These facts add difficulty in the analysis since they390
result in dense connectivity matrices. Nonetheless the analysis provided391
meaningful results, supporting the validity of the application of graph theory392
in connectivity estimation problems at different spatial scales.393
Betweenness values and their variability394
Figure 2, showING a geographical representation of the connectivity matrices395
together with betweenness values, highlights the strong dependency of396
betweenness on the circulation pattern present in the gulf. Figure 2d displays397
the connectivity values of the mean of the 20 variant matrices. The betweenness398
values here are the mean of the betweenness values obtained for each site399
with the 20 connectivity matrices. Note that this calculation is in principle400
different from calculating betweenness from the mean matrix. In our case401
betweenness values differed by one order of magnitude when comparing402
the two calculations (data not shown). In order to evidence the influence403
of different circulation patterns on betweenness, one has to analyse the404
betweenness values issued from each single matrice. In our study, central405
retention, mixed and westward circulation patterns appear to induce a high406
betweenness value for site 21. One case of reversed circulation (matrix #1)407
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Costa et al. Graph theory for species persistence
was associated to a low betweenness value for all the sites (Figure 3).408
409
The number of not null elements of the connectivity matrices (Figure410
4a) can be used in order to clarify the influence of the circulation on the411
connectivity matrices and, consequently, on betweenness values; especially412
if we cross this information with the network strength (given in Figure 4b).413
Note that, in order to avoid infinite network strength values, we substituted414
the infinities coming from the null aij values in Equation 6, with a constant415
that was set to 1000 times the maximum value of dij in the different matrices416
after a sensitivity analysis (not shown here). We can see that matrix #1417
has a lot more connections between the 32 sites compared to almost all418
the other connectivity matrices (892 not null elements out of 1024). This419
implies that this kind of circulation retains many particles alongshore (at420
least during the final parts of their larval period). Moreover the fact that421
the network strength value (1.67 × 106) is much lower than the mean one422
(4.90 × 106) tells us that these numerous connections generally have small423
values. So that there are many paths sharing a limited amount of network424
strength. This agrees with the low betweenness values for matrix #1 at all425
the nodes.426
Case #11 (westward circulation) is also peculiar. It has the lowest427
number of existing connections (376, Figure 4a) but the highest network428
strength (7.5 × 106, Figure 4b). Thus this circulation pattern disperses a429
lot of particles offshore and only very few paths remain. Considering the430
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Costa et al. Graph theory for species persistence
network strength value, we know that these paths have a high probability.431
Hence, in this situation, only the predominant paths are left. Therefore,432
the fact that node 21 still has a significantly high betweenness value in433
matrix #11 (see Figure 3) is a strong indication about its importance in434
the dynamics of the network.435
Moreover, the number of not null elements provides information on436
the effect of a specific circulation pattern on the spatial distribution of437
the species. In fact, a circulation pattern characterized by a connectivity438
matrix with many not null elements favours an exchange between all the439
sites, even if the connections are weak (matrix #1). Therefore, the species440
will tend to be more homogenised by the action of the circulation. Conversely,441
a circulation pattern creating few but intense connections (matrix #11)442
will tend to form predominant migration fluxes and thus spatially structure443
the species distribution.444
Note that all the other matrices are a composition of an intermediate445
number of not-null elements with an intermediate value of network strength,446
thus they cannot be interpreted as easily as cases 1 and 11 presented above.447
448
In general, from the results in Figure 3, we see that in all these cases449
only node 21 happens to have a much greater value of betweenness (roughly450
one order of magnitude) than the other ones. It corresponds to the site in451
front of Port Camargue, approximately in the center of the Gulf of Lion.452
From an oceanographic point of view the presence of a westward alongshore453
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Costa et al. Graph theory for species persistence
circulation that is predominantly present in the GoL (Millot, 1990), is454
relevant to clarify the high betweenness of site 21. This kind of circulation455
induces a transport of particles alongshore that determines the high value456
of betweenness of the sites in the centre of the gulf. The occasional recirculation457
that forms offshore site 21 (Petrenko et al., 2008) can enhance the importance458
of this particular site. Note that if some off-shore sites had been used, the459
betweenness values may have had their maximum elsewhere. Nonetheless,460
here, site 21 is the most important for polychaetes because the 32 sites461
considered cover all its habitat which is coastal.462
Our results highlight the sensitivity of graph metrics to flow variability:463
different circulation patterns correspond to networks with different connection464
patterns (as already noted by Mitarai et al., 2009) and different centrality465
measure values. These results suggest that other than adapting the size of466
the MPAs on the basis of seasonal movements of a species (Meyer et al.,467
2010), it is possible to envision the implementation of adaptative MPA468
management, possibly on a meteorological time scale.469
470
One aspect of the analysis needs to be clarified. The values of betweenness471
were calculated for each variant matrix. This means that a specific value of472
betweenness relies on the hypothesis that a particular circulation pattern473
is maintained constantly throughout a multi-generational migration. This474
is an unrealistic assumption. Nonetheless, the constancy with which we475
find a high betweenness at node 21 means that this node is likely to have476
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Costa et al. Graph theory for species persistence
a high betweenness even with more realistic circulation patterns. This477
could be not true in other case studies. In that case a solution to this issue478
could come from the implementation of time-dependent networks (see for479
example Ser-Giacomi et al., 2015, and references therein).480
Modularity and identification of sub-populations and481
rescue mechanisms in the Gulf of Lion482
Cluster analysis shows a fairly simple division of the network into two clusters483
(see Figure 5): a western one (sites 1 to 18) and a central-eastern one (sites484
19 to 32). This division of the network is the one found more frequently:485
40% of the times against 10% for other possible divisions. In the majority486
of the other cases only sites 18, 19 and, in some cases, 20 are assigned differently.487
The modularity value (Q = 0.16) gives us information about how important488
the exchange of larvae between the two clusters is. In general, there is no489
absolute threshold to discriminate between low and high values of modularity.490
Considering that, by definition, −1 < Q < 1, we are confident in stating491
two things: (i) as our value of Q is positive there is a cluster structure and492
(ii) as Q is less than a fifth of the maximum possible value (that is 0.2),493
we can define it as low. This means that the clusters exist but are not494
separated in a sharp way. Our leading hypothesis for the oceanographic495
mechanism at the base of this separation is the presence of recirculations in496
the central part of the GoL (Estournel et al., 2003) that can dynamically497
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Costa et al. Graph theory for species persistence
separate the two parts of the gulf while still permitting a considerable communication498
between them. Thus within the gulf there is a considerable migration flow499
of polychaetes that is not, at least spatially, highly organized. This is very500
likely a characteristic of a coastal environment where all the sites are alongshore501
with no considerable physical and/or dynamical barrier between them.502
Noticeably, the modularity analysis result is in agreement with the503
results of the metapopulation model. Exactly like in Guizien et al. (2014),504
modularity analysis showed the presence of a rescue mechanism of the505
sites in the western part of the gulf by the eastern sites. This result is506
consistent with the organization of the metapopulation in two big inter-507
communicating sub-populations as evidenced by graph theory modularity508
clustering. The presence of a rescue mechanism is mirrored by the low509
value of positive modularity. This division into two clusters of polychaetes510
assemblages is also in line with the division into an eastern and a western511
cluster evidenced by Labrune et al. (2007) when studying sedimentary512
differentiation of the GoL seabed. Given these facts we are confident that513
modularity results are a reliable tool to identify subpopulations and rescue514
mechanisms.515
516
One possible objection to the modularity method is the validity (see517
Procedures) of a random model as null hypothesis (as pointed out, for518
example, by Thomas et al., 2014). Regarding this aspect we are convinced519
that a random null model is the best choice to model the effects of the520
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Costa et al. Graph theory for species persistence
current field, main forcing of the biological system’s dynamic that -although521
deterministic- is chaotic due to its turbulent behaviour. Indeed, subpopulation522
structures in marine biological networks are likely to arise due to the effects523
of currents. This hypothesis is also backed by the mean assortativity values524
(a measure quantifying the ‘preference’ of a node to establish links with525
nodes of similar strength). For random networks we expect this value to526
be zero (see Barrat et al., 2008, for example). In our case, it is indeed very527
small (−0.04).528
The reader must also be aware that research on modularity has been529
largely developed since its introduction by Newman and Girwan (2004)530
and various shortcomings of this quantity are nowadays well known (see531
for example Fortunato and Barthelemy, 2006; and Kehagias and Pitsoulis,532
2013). For example, resolution limit problems (i.e., identifying clusters533
under a certain size) appear in the presence of quite peculiar network structures534
(see Kehagias and Pitsoulis, 2013, for example) but are not present in our535
case, mostly due to the randomness of our network.536
Bridging centrality and the preservation of network537
integrity538
The three nodes that are characterized by the higher value of bridging539
centrality are the nodes 11, 12 and 16 (see Figure 6). These nodes have540
bridging centrality values of 600, 572 and 513 respectively. Following Hwang541
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Costa et al. Graph theory for species persistence
et al. (2008), these three nodes, representing the top ten percent of the 32542
nodes, are the nodes that are expected to prevent the network from easily543
breaking into separated sub-networks. Nevertheless, here, the removal of544
solely one of these nodes does not create sub-networks. The removal of545
all three high-bridging centrality nodes do not break the network either.546
Indeed, isolated sub-networks appear only after removing at least eight547
nodes. This result is due to the high average edge density of the 20 variant548
matrices (ρ = 0.604). Whereas the networks used by Hwang et al. (2008)549
or in other social sciences applications of graph theory have lower ρ. But550
one has to note that a lot of weak connections between the 32 sites are551
present (data not shown). The solidity of the biological network should552
be assessed independently from these many weak connections. In order to553
highlight the predominant connections, a threshold was set on the base of554
the following argument. Given a probability-based connectivity matrix,555
we can expect that a transfer rate T between the minimum not-null value556
Cm of the matrix and 1 is necessary for the maintenance of an overall good557
connectivity. We can estimate T as the geometric mean of Cm and 1: T =558
√Cm · 1. The geometric mean has the advantage of considering a vast559
range of values for the variables at play in determining an unknown quantity,560
while not being biased by the choice of too large/small extreme values.561
Dealing with living organisms, one also has to account for the survivorship562
of the propagulae: in an efficient network, we can expect that a percentage563
S of propagulae between T and at least Te
is likely needed for the maintenance564
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Costa et al. Graph theory for species persistence
of the persistence of the species, where e is the Nepero constant. Thus:565
S =√
Te· T . As a last step, we also account for the percentage of the566
surviving particles that successfully reproduce. A percentage R between S567
and Se
is likely needed for a good persistence of the species in the habitat.568
Thus: R =√
Se· S. In our case R = 0.0041, which we round to the value569
0.001 used in our analysis. A qualitative test of this estimation showed570
that, for a threshold equal to 0.01, we obtained an almost completely disconnected571
network. While, when retaining all values above 0.0001, nothing hardly572
changed. Thus a threshold equal to 0.001 seems to be exactly the threshold573
we looked for in order to have a minimally-connected network. In particular,574
this threshold eliminated, on average, 36% of the connections from the 20575
connectivity matrices.576
After applying the threshold, the deletion of both sites 11 and 12 led577
to nine sub-networks - sets of nodes disjoint from other portions of the578
graph - mainly consisting of isolated single nodes or couples of nodes. In579
contrast the removal of other triplets of nodes created, on average, only580
seven separated sub-networks (data not shown). Noticeably, the joint deletion581
of node 16 did not enhance the effect of the deletion of only nodes 11 and582
12. This fact is not surprising because node 16 is the node, among high-583
bridging centrality nodes, with the lowest value of bridging centrality. In584
fact, the indication by Hwang et al. (2008) to consider as crucial the top585
10% of high-bridging centrality nodes is just a guideline and as such has to586
be taken.587
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Costa et al. Graph theory for species persistence
588
The oceanographic explanation for a relevant value of bridging centrality589
for sites 11 and 12 is the separation into two clusters (east and west, as590
said above) and the presence of eastward and westward currents in the591
middle of the GoL (Petrenko, 2003). We think that these currents can592
justify the important communication between the clusters, as indicated593
by a low value of modularity and a high value of bridging centrality of two594
sites at the gulf’s mid height, such as sites 11 and 12.595
Remarkably the removal of node 21 had no particularly important596
effects on the fragmentation of the network. It is thus clear that bridging597
centrality adds more information on the structure of the network than598
what betweenness alone can provide.599
Minimum cycles identify retention loops induced by600
currents recirculation601
Here we direct the analysis to the inspection of the spatial scales at which602
multi-generational flows form retention loops.603
Minimum cycles provided evidence that the nodes with the greater604
probability to see their particles returning home, after a period ranging605
from 2 to 5 generations, are the nodes 13 to 16 (Figure 7). According to606
Hastings and Botsford (2005) these nodes are likely crucial for the persistence607
of polychaetes. Furthermore the nodes that are travelled through the most608
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Costa et al. Graph theory for species persistence
frequently during the minimum cycles are also the nodes 13 to 16 (data not609
shown).610
Indeed the zone of the gulf spanning from nodes 13 to 16, where the611
cycles are shortest, corresponds to an area where the currents often recirculate,612
due to the presence of eddies (Hu et al., 2011a; and Hu et al., 2011b).613
Relevance for conservation614
With the results so far we can compare the hierarchy of the four sites (1,615
10, 18 and 32) established with respect to their importance for species616
persistence (as determined by metapopulation analysis), and to their hierarchy617
in terms of betweenness, shortest cycles and bridging centrality values618
(Table 1). Shortest cycles analysis correctly points to site 18 as most important619
among these four sites. Thus, shortest cycles are able to identify the nodes620
sustaining persistence as Hastings and Botsford (2005) established on a621
theoretical basis. However, the shortest cycles do not agree with the hierarchy622
of the metapopulation model for the other three nodes. This pinpoints623
a methodological discrepancy between the site removal procedure in the624
metapopulation model and the shortest cycles identification procedure625
when cycles becomes long (as clarified in the Comments and Recommendations626
Section).627
628
Betweenness fails to reproduce the hierarchy from the metapopulation629
model concerning sites’ importance for demographic persistence.630
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Costa et al. Graph theory for species persistence
Herein, in almost all cases of circulation patterns in spring 2004 and631
2006 in the Gulf of Lion, site 21 offshore of Port Camargue has the highest632
value of betweenness (see Figure 3). This means that site 21 is the site633
through which the majority of the larvae pass, suggesting the importance634
of site 21 as a “gateway” for multi-generational migration. This role has635
been demonstrated to be crucial for preserving gene flow continuity across636
the GoL (Padron and Guizien, 2015) and confirms the utility of the betweenness637
measure for identifying relays of species spreading. But “gateways” through638
which the majority of the larvae pass during multi-generational migrations639
may not be sufficient for persistence, unlike previously put forward (Treml640
et al., 2008; Andrello et al., 2013). Persistence implies continuity in population641
cycles. Thus, an efficient transfer, although the most intuitive requirement642
for connectivity, is probably not the most important. This fact highlights a643
risk of failure of conservation polices when only high-betweenness sites are644
preserved while persistence cycles are not maintained.645
646
We introduced a new formulation for bridging centrality to adapt it647
to directed-weighted graphs. It fails to reproduce the hierarchy resulting648
from the metapopulation model concerning demographic persistence. However,649
bridging centrality is focused on the network integrity. Thus it indicates650
sites important for species spreading at the regional scale, that are secondary651
sources for species expansion.652
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Costa et al. Graph theory for species persistence
Comments and Recommendations653
Firstly, we want to further highlight the importance of the difficulty that654
arises when using matrices containing larval transfer probabilities in graph655
theory analysis. In this case, the most immediate choice for edge definition656
would be the probabilities themselves (e.g., Rossi et al., 2014). But, with657
this choice, one obtains conceptually wrong results when dealing with concepts658
relying on the calculation of the shortest paths as, for instance, when calculating659
betweenness (as in Andrello et al., 2013, for example). In fact, with this660
metric, the shortest path is the most improbable one and the high-betweennes661
sites indicate the less frequented ones. The node-to-node metric derived662
from larval transfer probability we propose solves this inconsistency, making663
the use of shortest paths, betweenness and shortest cycles meaningful for664
the analysis of networks based on transfer probabilities.665
666
The discrepancy between shortest paths analysis and metapopulation667
modelling could be due to the particular site-removal procedure used in668
the habitat loss scenario of the metapopulation model analysis. Firstly,669
consistent with the fact that anthropic pressure decreases as distance from670
the harbours increases, scenarios included only the four harbours sites.671
Secondly, under the hypothesis that the effect of anthropic pressure acts672
predominantly on neighbouring sites, the habitat removal procedure was673
done by progressively eliminating neighbouring sites. Shortest path analysis674
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Costa et al. Graph theory for species persistence
supports this point of view. In fact it points out that the nodes composing675
the shortest cycles are all close to each other. That is, the hypothesis that676
the survivorship of a site depends on the survivorship of the neighbouring677
ones is reasonable. Nevertheless shortest path analysis does not include678
any assumption on geographical proximity within the path. Consequently,679
shortest path analysis’ results would be fully comparable only with a more680
general removal procedure than the one used in Guizien et al. (2014).681
682
The general scope of the paper was to verify if the expected conservation683
interpretation of some graph theory measures was backed by an analysis684
with a more solid conservation interpretation like metapopulation model.685
We could not present a metapopulation equivalent of bridging centrality686
because the available metapopulation simulations did not test the role687
of isolated sites in rescuing the metapopulation. Nonetheless we think688
that the importance of high-bridging centrality nodes in the integrity of689
the network is a powerful indication of their importance for the regional690
spreading of a species.691
692
Overall we showed how some graph theory measures can be employed693
to obtain a considerable part of the information provided by metapopulation694
modelling exploiting only connectivity matrices and avoiding the difficult695
estimation of demographic parameters. A summary of the conservation696
interpretation of the graph theory measures analysed in this study can be697
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found in Table 2.698
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Acknowledgements699
This study was partially funded by the European CoCoNet Project and by700
the Ministere de l’Education Nationale, de l’Enseignement Superieur et de701
la Recherche.702
The authors want to thank Dr.Rubao Ji, Dr.Lucio Bellomo, Dr.Leo703
Berline and Dr.Jean-Christophe Poggiale for helpful discussions. The first704
author also thanks Dr.Alain de Verneil for helping correcting the manuscript.705
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Costa et al. Graph theory for species persistence
Supplementary Materials706
A Metapopualtion model707
Population density at a given time at a given site results from spatially708
structured local survivorship and reproductive success inputs potentially709
depending on all the other sites in the system. The model used by Guizien710
et al. (2014) accounts for both (i) recruitment limitation due to space availability711
at the destination site (computed as the proportion of free space based712
on the saturating density of adults, and (ii) the variability in propagule713
transfer rate.714
In particular the model can be written in matrix form as follows:715
P (t+ ∆t) = min(G(t)P (t), Pmax)
with a time step of ∆t and the growth transfer matrix G defined as716
Gij = liaij(t)bj + sjjδ(i, j)
where: Pmax = 1/αA where αA is the mean cross-sectional area717
of one adult, P (t) ∈ RN contains the spatial density of adults in each718
site i ∈ [1, 32] at time t, li [number of larva per adult] is the propagulae719
production rate at site j, bj [number of adult per larva] is the recruitment720
success at site j, sjj [no units] is the adult survivorship rate at site i, δ(i, j)721
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Costa et al. Graph theory for species persistence
is the Kronecker δ-function, Pmax is the site carrying capacity and aij is the722
propagulae transfer rate from site i to site j. The larvae production rate723
bj is equal to the number of larvae produced by each adult female FSRf724
where F is the fecundity rate, SR is the sex ratio in the adult population,725
and f is the probability of an egg being fertilized. The recruitment success726
bj accounts for all mortality losses since egg release until the first reproduction727
of new recruits, and includes mortality during larval dispersal, settlement728
and juveniles stages. Notice that adult survivorship rate can be related to729
species life expectancy LE as sjj = eln(0.01) ∆t
LE , where life expectancy LE is730
the age at which 99% of individuals of the same generation have died.731
B Node-to-node metric properties732
We verify that the new metric we propose for measuring the distance between733
nodes dij = ln( 1aij
), aij being the probability of advection from site i to site734
j in a given time, satisfies both homogeneity and additivity:735
1. Homogeneity:736
αdij = α ln
(1
aij
)= ln
(1
aαij
)
that is, a distance multiplied by a scalar is still a distance.737
2. Additivity:738
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Costa et al. Graph theory for species persistence
dil + dlj = ln
(1
ail
)+ ln
(1
alj
)= ln
(1
ail · alj
)= ln
(1
aij
)= dij
that is, the sum of two distances is still a distance and, in particular,739
the final distance is obtained by the multiplication of the partial740
original metric values. This aspect adapts particularly well to the741
problem addressed in the paper. In fact the length of a path can742
be calculated as the product of the probabilities associated with its743
components.744
Note that as the elements aij have no units, also the elements dij745
have no units.746
List of captions747
FIGURE 1. Schematic representation of the typical circulation in the Gulf748
of Lion. The thick arrow represents the dominant alongshore Northern749
Current. The thinner arrow represents the eastward current that can be750
detected in stratified conditions or under particular wind field conditions.751
The positions of the 32 studied sites are plotted. The sites 3, 10, 18 and752
32, used for the habitat loss scenario in the metapopulation model, are753
highlighted by bigger grey dots. Node 21 is the smallest of the grey dots.754
The grey lines correspond to the 100 m, 200 m, 1000 m and 2000 m isobaths.755
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Costa et al. Graph theory for species persistence
FIGURE 2. Spatial representation of the connectivity matrices and756
betweenness values in different circulation situations. (a) Westward drift757
(matrix #7), (b) eastward drift (matrix #1), (c) central retention (matrix758
#10). In the panel (d) the mean connectivity matrix values and the mean759
of the betweenness values are represented. In all the figures, a threshold on760
the value of connectivity was imposed for clarity: connectivity values lower761
than the 2/3 of the maximum one are not plotted.762
FIGURE 3. Values of betweenness for the 32 sites using the 20 variant763
connectivity matrices. The normalization is done on the maximum value of764
betweenness obtained using the different variant matrices.765
FIGURE 4. (a) Number of not null elements in the 20 variant matrices.766
Different circulation patterns have different effects on the connectivity767
inside the Gulf of Lion. (b) Network strength of the 20 variant matrices.768
FIGURE 5. Clusters identified with a criteria of maximization of769
modularity. The result is the average assignation of a node to one of the770
two clusters after 2·105 code runs with the 20 variant connectivity matrices.771
Two colors separate the two clusters.772
FIGURE 6. Bridging centrality values for the 32 sites. Geographical773
representation. Nodes 11 and 12 have the highest values: 600 and 572 respectively.774
FIGURE 7. Sum of the products of the weights of all the cycles (length775
from 2 to 5 steps) that start from each of the 32 sites. The minima correspond776
to the nodes for which the probability of particles returning home (in a 2777
to 5 generations time span) is higher.778
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Costa et al. Graph theory for species persistence
References779
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Costa et al. Graph theory for species persistence
Figure 1: Schematic representation of the typical circulation inthe Gulf of Lion. The thick arrow represents the dominant alongshoreNorthern Current. The thinner arrow represents the eastward currentthat can be detected in stratified conditions or under particular wind fieldconditions. The positions of the 32 studied sites are plotted. The sites 3,10, 18 and 32, used for the habitat loss scenario in the metapopulationmodel, are highlighted by bigger grey dots. Node 21 is the smallest of thegrey dots. The grey lines correspond to the 100 m, 200 m, 1000 m and2000 m isobaths.
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Costa et al. Graph theory for species persistence
(a) (b)
3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
Con
nect
ivity
8.5
9
9.5
10
10.5
11
11.5
12
Betweenness0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
Con
nect
ivity
8.5
9
9.5
10
10.5
11
11.5
12
12.5
Betweenness0 0.01 0.02 0.03 0.04 0.05 0.06
(c) (d)
3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
Con
nect
ivity
8.5
9
9.5
10
10.5
11
11.5
12
Betweenness0 0.05 0.1 0.15 0.2
3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
Con
nect
ivity
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
Betweenness0 0.05 0.1 0.15 0.2
Figure 2: Spatial representation of the connectivity matrices andbetweenness values in different circulation situations. (a) Westward drift(matrix #7), (b) eastward drift (matrix #1), (c) central retention (matrix#10). In the panel (d) the mean connectivity matrix values and the meanof the betweenness values are represented. In all the figures, a threshold onthe value of connectivity was imposed for clarity: connectivity values lowerthan the 2/3 of the maximum one are not plotted.
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Costa et al. Graph theory for species persistence
Sites
Var
iant
Mat
rices
5 10 15 20 25 30
2
4
6
8
10
12
14
16
18
200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 3: Values of betweenness for the 32 sites using the 20 variantconnectivity matrices. The normalization is done on the maximum value ofbetweenness obtained using the different variant matrices.
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0 5 10 15 200
100
200
300
400
500
600
700
800
900
Matrices
Num
ber
of N
ot N
ull E
lem
ents
1 5 10 15 200
1
2
3
4
5
6
7
8x 10
6
Matrices
Net
wor
k S
tren
gth
(a) (b)
Figure 4: a) Number of not null elements in the 20 variant matrices.Different circulation patterns have different effects on the connectivityinside the Gulf of Lion. (b) Network strength of the 20 variant matrices.
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3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
12
34
567
8910
1112
13 1415
1617 1819
2021 2223 2425
2627 28 29303132
Longitude
Latit
ude
Figure 5: Clusters identified with a criteria of maximization ofmodularity. The result is the average assignation of a node to one ofthe two clusters after 2 · 105 code runs with the 20 variant connectivitymatrices. Two colors separate the two clusters.
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Costa et al. Graph theory for species persistence
3oE 30’ 4oE 30’ 5oE
15’
30’
45’
43oN
15’
30’
12
34
567
8910
1112
13 1415
1617 1819
2021 22 23 24 25
2627 2829
303132
Brid
ging
Cen
tral
ity
0
100
200
300
400
500
600
Figure 6: Bridging centrality values for the 32 sites. Geographicalrepresentation. Nodes 11 and 12 have the highest values: 600 and 572respectively.
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Costa et al. Graph theory for species persistence
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3x 10
5
Sites
Cyc
les
Leng
th
Figure 7: Sum of the products of the weights of all the cycles (lengthfrom 2 to 5 steps) that start from each of the 32 sites. The minimacorrespond to the nodes for which the probability of particles returninghome (in a 2 to 5 generations time span) is higher.
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Costa et al. Graph theory for species persistence
Sites testedin metapopulationmodel analysis
Hierarchy basedon metapopulationmodel
Hierarchy basedon Betweenness
Hierarchy basedon ShortestCycles
Hierarchy basedon BridgingCentrality
1 4◦ 1◦ 3◦ 4◦
10 3◦ 2◦ 2◦ 1◦
18 1◦ 3◦ 1◦ 2◦
32 2◦ 4◦ 4◦ 2◦
Table 1: Comparison of the hierarchy based on metapopulationmodel analysis of the four nodes tested in Guizien et al. (2014) with thehierarchies issued from different graph theory measures.
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Costa et al. Graph theory for species persistence
Measure Scope InterpretationMinimum Cycles Identifying sites with high probability Important for persistence.
of returning home of the larvaespawning from them.
Betweenness Nodes through which the highest Nodes maintaining thepercentage of most probable paths gene flux at the whole
pass through them. network scale.Modularity Find sets of nodes more Sub-populations.
than randomly connected Indicates presence ofrescue mechanisms.
Bridging Centrality Find nodes leading the Nodes preventing thecommunication between clusters. fragmentation of the network.
Table 2: Recapitulation of the four main measures we use in theframework of this study. For each one we indicate scope and physical-biological interpretation.
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