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\markboth{Googling Food Webs}{Googling Food Webs}
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\begin{document}
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{\Large
\textbf{Googling food webs: can an eigenvector measure species' importance for coextinctions?}
}
% Insert Author names, affiliations and corresponding author email.
\\
Stefano Allesina$^{1}$,
Mercedes Pascual$^{2}$
\\
\bf{1} National Center for Ecological Analysis and Synthesis, 735
State Street, Santa Barbara, CA 93101, USA\\
\bf{2} Department of Ecology and Evolutionary Biology,
University of Michigan, the Santa Fe Institute and
the Howard Hughes Medical Institute.\\
$\ast$ E-mail: allesina@nceas.ucsb.edu
\end{flushleft}
\section*{Text S1}
\subsection*{Food web data \& Data issues}
To test our algorithm and the ones presented below, we used 12 published
food webs (Table 1). These food webs come from the so-called ``second
generation'' data set. The quality of data could impact the results
presented here. First, the data has been aggregated into
``trophospecies'' (i.e. species with the same set of predators and prey
are considered as one species). Of course the extinction of a whole
trophic species is not as likely as that of a single species. Second,
one aspect of food webs that could make the results depart from the ones
presented here is predator ``switching''. Because species are plastic,
we can expect that a predator with no available prey would ``switch'' to
a less palatable, but available, prey \cite{Kondoh03}. The quality of
the data dictates if this switching could alter the pattern presented
here. If the data has been gathered for a sufficiently long period of
time, we expect that most possible predator-prey interactions have been
recorded. In this case, we will find links between predator and their
preferred prey, but also with the less-palatable prey that the predator
could switch to. If the data, instead, is a simple ``snapshot'' of an
ecosystem at a given time, then some of the rare interactions could be
missing. Given the robustness of the results, we do not expect any of
the problems illustrated above to alter our conclusion. ``Third
generation'' food webs have started appearing in the literature. These
are huge datasets with hundreds of nodes resolved at the species level.
Once a considerable number of these networks is published, we will be
able to test if the eigenvector approach presented here is still the
best algorithm.
The published data sets have been modified by attaching a new node
(root) to each food web. The root points to all primary producers
($a_{ri}=1$ if $i$ has no incoming link). All nodes are connected to the
root ($a_{ir}=1$) to represent excretion, egestion, death and decay. In
this framework, a species that is not reachable from the root will
surely go extinct, as it lacks resources for growth.
\subsection*{Extinction area}
Extinction plots (Fig. 2) are generated in the following way: a) a
removal sequence is determined based on one of the methods; b) a species
in the sequence is removed (if present) at each step ($x$ axis); c) the
total number of species lost with respect to the original number is
computed ($y$ axis). These numbers are normalized by dividing them by
the total number of species $S$. The area is therefore 1 if all species
go extinct in response to the first removal (a case that is possible
only when a single producer is present). If there are no secondary
extinctions, then the area will be $(S-1)/2S$, and will tend to 1/2 for
large $S$. The root cannot be removed and is not considered in the
computation of the area.
\subsection*{Linear algebra remarks} The problem of assigning importance
recursively by using the eigenvector $v$ applies only to matrices that
are both irreducible and primitive. If a matrix $A$ is irreducible, then
for each $i$ and $j$ there is some power $z$ for which $a^z_{ij}>0$. A
matrix is primitive if there exists a power $z$ for which $a^z_{ij}>0\,
\forall \ i,j$. Clearly a primitive matrix is always irreducible (for
details on these linear algebra considerations in an ecological context,
see Caswell \cite{Caswell2001}). A stochastic matrix (i.e. where every
column sums to 1 and all entries are nonnegative) that is irreducible
and primitive, has a unique dominant eigenvalue $\lambda^*=1$
(Perron-Frobenius theorem). In this case, the eigenvector $v$ provides a
solution to the problem of assigning importance to the nodes. If a
matrix is irreducible, then its associated network is strongly
connected: there is a pathway to reach each node from any other node in
the network. By modifying the food webs as described above, we create
strongly connected networks and therefore irreducible matrices. Their
primitivity has been numerically tested, yielding positive results in
all cases.
\subsection*{Ranking Algorithms}
In the $D$ algorithm, we choose to remove the species that has the
greatest number of out-going connections (using the total number of
connections or the number of incoming connections does not improve the
algorithm). In the case of ties, all the nodes are considered equally
important: we therefore followed all the extinction sequences that are
originated by the tie. For example, if at some step two species $a$, $b$
are assigned the same importance, we follow the extinction sequence in
which we remove $a$ and that in which we remove $b$. Because this
branching could lead to very many extinction sequences, we followed the
branching until we found all possibilities or we reached $5\cdot 10^5$
solutions. We follow the branches up to this ceiling for all
algorithms. In the $CLOS$ and $BETW$ centrality we compute centralities
for the corresponding undirected graph with no root (this guarantees the
applicability of the algorithm). When all nodes had centrality 0 (when
we obtain a completely disconnected network), we considered species
importance inversely proportional to their trophic level. At each step
of the $EIG$ algorithm, we compute the eigenvector $v$ associated with
the dominant eigenvalue and remove the node with the maximum score in
the vector. To compute $v$, we take the adjacency matrix of the food
web modified as described above, normalize by columns, and compute
eigenvalues and eigenvectors. The dominant eigenvalue is unique and
equal to 1, and $v$ is the associated positive eigenvector whose entries
sum to 1. The $EIG 2$ algorithm works exactly as the $EIG$ one, with the
difference that $v$ is computed after simplifying the web by removing
``redundant'' connections \cite{Allesina2008a}. A detailed description
of the new algorithms is presented below.
We give here an example of the computation of the vector $v$ for the
food web depicted in Fig. 1. The network is composed of 6 species
($a,b,c,d,e,f$) connected by 10 feeding relations. The adjacency matrix
$Adj$, in which an entry of 1 means that the row-species is a prey of the
column-species is given by
\begin{equation}
Adj= \left[ \begin{array}{cccccc}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array} \right]
\end{equation}
We then add the ``root'' node to obtain a modified adjacency matrix $A$
in which the root node is represented by the first row and the first
column. For example, if the ``root'' points to the primary producers
$a$ and $b$ and receives a link from each species, we obtain
\begin{equation}
A= \left[
\begin{array}{ccccccc}
0 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right]
\end{equation}
We can then build the matrix $S$ obtained by dividing each species by
the column sum:
\begin{equation}
S= \left[
\begin{array}{ccccccc}
0 & 1 & 1 & 0 & 0 & 0 & 0 \\
\frac{1}{6} & 0 & 0 & \frac{1}{4} & 0 & 0 & 0 \\
\frac{1}{6} & 0 & 0 & \frac{1}{4} & 1 & 0 & 0 \\
\frac{1}{6} & 0 & 0 & \frac{1}{4} & 0 & 1 & \frac{1}{4} \\
\frac{1}{6} & 0 & 0 & \frac{1}{4} & 0 & 0 & \frac{1}{4} \\
\frac{1}{6} & 0 & 0 & 0 & 0 & 0 & \frac{1}{4} \\
\frac{1}{6} & 0 & 0 & 0 & 0 & 0 & \frac{1}{4}
\end{array}\right]
\end{equation}
\noindent
The entry $s_{ij}$ represents the fraction of importance of $i$ arising
from being a prey of $j$. We can write a system of equations that
describes the relation between the importance of the nodes:
\begin{equation}
\begin{cases}
R = a+b\\
a = \frac{1}{6}R+\frac{1}{4}c\\
b = \frac{1}{6}R+\frac{1}{4}c+d\\
c = \frac{1}{6}R+\frac{1}{4}c+e+\frac{1}{4}f\\
d = \frac{1}{6}R+\frac{1}{4}c+\frac{1}{4}f\\
e = \frac{1}{6}R+\frac{1}{4}f\\
f = \frac{1}{6}R+\frac{1}{4}f
\end{cases}
\end{equation}
\noindent
The system of equations has infinitely many solutions. We can however
express all the variables in terms of $R$:
\begin{equation}
\begin{cases}
R = R\\
a = \frac{17}{54}R\\
b = \frac{37}{54}R\\
c = \frac{16}{27}R\\
d = \frac{10}{27}R\\
e = \frac{2}{9}R\\
f = \frac{2}{9}R\\
\end{cases}
\end{equation}
Now we can set $R$ to any arbitrary value (e.g. 54) and recover a
solution, that will be directly proportional to the eigenvector $v$. In
our $EIG$ algorithm, we will therefore remove species $b$ and then solve
the new system of equations, choosing the species with the maximum value
(excluding $R$), and so on. Because all solutions are linearly dependent
on each other, we can set an arbitrary value for a node and recover a
solution that maintains the order of importance. We can also recover a
solution by considering the eigenvalues of matrix $S$: the dominant
eigenvalue will be $1$ and unique. The eigenvector associated with the
dominant eigenvalue solves the problem of assigning importance.
The $EIG2$ algorithm works in the same way but instead of assigning
importance using the matrices $Adj \rightarrow A \rightarrow S$, it
substitutes $Adj$ by the matrix $Adj'$ obtained by removing
``redundant'' connections from $D$ (Fig. 1 right). Redundant connections
are those that do not contribute to the flow-based robustness to
secondary extinctions: any sequence of removals yields the same
extinctions in $Adj$ and $Adj'$, but $Adj'$ contains only a subset of
the connections in $Adj$ \cite{Allesina2008a}. The simplest example of a
redundant connection is given by those that represent
cannibalism. Obviously, a species cannot sustain itself fully from
cannibalism, and therefore removing the cannibalistic link does not
alter robustness. It is interesting to note that the problem of
redundant connections is similar to that of ``collusion'', or nepotistic
linking, in ranking web-pages \cite{Collusion}. In the Web there are
currently several million pages created with the specific intent of
increasing the PageRank score of a given website. This is done by
artificially incrementing the incoming connections to a node through
ad-hoc pages. In our algorithm, the ``collusion links'' would be removed
from the analysis thanks to the pre-treatment that takes care of the
redundant connections.
Yonatan Zunger (personal communication) drew an interesting parallel
between the problem of secondary extinctions in food webs and some
well-known problems in computer science. For example, the literature
about multicuts in directed graphs is definitely relevant to the problem
studied here \cite{MultiCuts}. In this classical problem in
graph theory given a directed graph and a list of sources and sinks, one
wants to find the most efficient way of disconnecting the source from
the sinks, by removing the smallest possible set of links. It is clear
that if the source is the ``root'' node, the disconnection through cuts
is exactly equivalent to the notion of extinction we are examining
here. This means that some results that have been produced in computer
science could have a direct implication for ecology as well.
\subsection*{The biological significance of the eigenvector $v$}
To explain the biological significance of the eigenvector $v$, we start
from a system of differential equations describing the abundance of each
species in time:
\begin{equation}
\frac{dX_i}{dt}=f \left(X_1,X_2, \ldots, X_S \right)
\end{equation}
\noindent
where $X_i$ is the abundance of species $i$. The growth rate of $X_i$ is
a function of the density of the other species and of the value of
several constants (parameters). If the system is at a feasible fixed
point (where the growth rate is $0$ for all species, and all species
have positive abundance), we can measure the quantity of nutrients
flowing from one species to any other in the network. This can be
measured empirically, for example by measuring the grams of carbon
moving from one species to the others in a given interval of time and
for a given spatial area. At equilibrium, the flows entering a species
must match exactly the flows exiting it (steady state)
\cite{AllesinaB03,Ulanowicz2004}. Note that this consideration applies
also to limit cycles and other periodic or quasi-periodic attractors
(when the system is sampled for a period long enough). As an example, we
take the Lovinkhoeve Experimental Farm \cite{DeRuiter1995} for which the
flows at steady state have been measured as kilograms of biomass per
hectare per year. We can also attach to the food web the ``Root'' node
representing the external environment. From this node, we will draw
inputs to the ``Plants'' and ``Detritus'' compartments and all
compartments will return to ``Root'' to describe the losses experienced
from egestion, excretion, death and decay. Such a modified food web is
represented in Fig. S1. We can compute the $S$ matrix and the eigenvector
$v$ for this web by normalizing the flow matrix by column sums. The
procedure is exactly the one utilized for the $EIG$ algorithm, with the
difference that now links exiting a species are not equally important,
but rather reflect the diet preferences of the predators. We show in
Fig. S2 that the eigenvector $v$ in this case is proportional to the flow
through each species. We can recover $v$ by summing all the flows
entering (or exiting) a node and dividing this value by the total amount
of flows in the food web.
For quantitative networks, where link strengths is specified, the
eigenvector expresses the flows through each species. Another way of
interpreting this is to notice that $S$ can be read as a Markov Chain,
and that $v$ represents its stationary state. Therefore $v_i$ (in its
normalized form) represents the probability that a unit of matter is
found in species $i$ at a given time. With this in mind, we can extend
the definition to the qualitative networks we analyzed in the main text
in the absence of information on diet preferences. The $EIG$ algorithm
must consider by default that each prey contributes equally to the diet
of predators. The vector $v$ specifies the flows through each species if
it were to receive the same amount of nutrients from each of its prey.
Note that in this sense $v$ is a proxy for quantitative connectivity.
Nevertheless, by considering the whole structure of the network, the
performance of the $EIG$ algorithm is superior to that of the local,
unweighted measure given by the (local) number of links (in the $D$
algorithm). Also, $v$ bridges the gap between qualitative and
quantitative networks, as it expresses the flows for the most
parsimonious quantitative network that one could build using qualitative
information.
\subsection*{Supplementary results}
In Table S1 we report, for each food web ({\em FW}), the number of
sequences considered ({\em Nseq}), the number of sequences yielding the
maximum area ({\em Nbest}), and the {\em Max}, {\em Min}, {\em Mean} and
{\em Var} area for the sequences considered. Note that $D$ yields very
many ties, as testified by the great number of sequences generated by
the algorithm. Also $CLOS$ cannot discriminate easily among nodes,
generating therefore many ties.
\section*{References}
%\bibliography{BibGoogling}
\begin{thebibliography}{1}
\providecommand{\url}[1]{\texttt{#1}}
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\providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else
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\urlstyle{rm}\Url}\fi
\providecommand{\bibAnnoteFile}[1]{%
\IfFileExists{#1}{\begin{quotation}\noindent\textsc{Key:} #1\\
\textsc{Annotation:}\ \input{#1}\end{quotation}}{}}
\providecommand{\bibAnnote}[2]{%
\begin{quotation}\noindent\textsc{Key:} #1\\
\textsc{Annotation:}\ #2\end{quotation}}
\providecommand{\eprint}[2][]{\url{#2}}
\bibitem{Kondoh03}
Kondoh M (2003) Foraging adaptation and the relationship between food-web
complexity and stability.
\newblock Science 299: 1388-1391.
\bibAnnoteFile{Kondoh03}
\bibitem{Caswell2001}
Caswell H (2001) Matrix Population Models: Construction, Analysis, and
Interpretation.
\newblock Sinauer Associates.
\bibAnnoteFile{Caswell2001}
\bibitem{Allesina2008a}
Allesina S, Bodini A, Pascual M (2009) Functional links and robustness in food
webs.
\newblock Phil Trans Roy Soc B 364: {1701-1709}.
\bibAnnoteFile{Allesina2008a}
\bibitem{Collusion}
Baeza-Yates R, Castillo C, L\'{o}pez V Pagerank increase under different
collusion topologies.
\newblock In: Davison BD, editor, Proceedings of First International Workshop
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\bibAnnoteFile{Collusion}
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Chuzhoy J, Khanna S (2006) Hardness of cut problems in directed graphs.
\newblock In: STOC '06: Proceedings of the thirty-eighth annual ACM symposium
on Theory of computing. New York, NY, USA: ACM, pp. 527--536.
\newblock \doi{http://doi.acm.org/10.1145/1132516.1132593}.
\bibAnnoteFile{MultiCuts}
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Allesina S, Bondavalli C (2003) Steady state of ecosystem flow networks: a
comparison between balancing procedures.
\newblock Ecol Model 165: 221-229.
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Ulanowicz R ({2004}) {Quantitative methods for ecological network analysis}.
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\end{thebibliography}
%\section*{Figure Legends}
%\setcounter{figure}{3}
%\begin{figure}[!ht]
%\begin{center}
%\includegraphics[width=4in]{figure_name.2.eps}
%\end{center}
%\caption{
%{\bf The Lovinkhoeve Experimental Farm food web modified as
% described in the text.} The flows are expressed in kg of biomass per
% year per hectare. Red links represent biomass losses experienced by
% the species.
%}
%\label{fig:S1}
%\end{figure}
%\begin{figure}[!ht]
%\begin{center}
%\includegraphics[width=4in]{figure_name.2.eps}
%\end{center}
%\caption{
%{\bf Relationship between the size of flows in the food web (Fig.
% S1) and the values of the eigenvalue $v_i$.} The $y$ axis is the sum
% of all flows entering (or exiting) a species. The $x$ axis is the
% corresponding value in the eigenvector $v$. The logarithms of both
% values are shown in the graph to better discriminate the points.
%}
%\label{fig:S2}
%\end{figure}
\section*{Tables}
%\begin{table}[!ht]
%\caption{
%\bf{Table title}}
%\begin{tabular}{|c|c|c|}
\setcounter{table}{1}
\begin{sidewaystable}
%\begin{table}[htbp]
\caption{\bf{Full results.}}
\begin{center}
{\footnotesize
\begin{tabular}{|l|c|c|c|c|c|c|c|l|c|c|c|c|c|c|c|}
\hline
FW &Alg &Nseq &Nbest &Max &Min &Mean &Var &FW &Alg &Nseq &Nbest &Max &Min &Mean &Var\\ \hline
benguela &D &24 &4 &0.7943 &0.7812 &0.7878 &0.0000 &shelf &D &500000 &36 &0.6380 &0.6153 &0.6203 &0.0000\\
&CLOS &66 &1 &0.7539 &0.7206 &0.7300 &0.0001 & &CLOS &359424 &6912 &0.6561 &0.6461 &0.6494 &0.0000\\
&BETW &1 &1 &0.9025 &0.9025 &0.9025 &0.0000 & &BETW &1 &1 &0.8103 &0.8103 &0.8103 &0.0000\\
&DOM &1 &1 &0.9798 &0.9798 &0.9798 &0.0000 & &DOM &1 &1 &0.9885 &0.9885 &0.9885 &0.0000\\
&EIG &2 &1 &0.9798 &0.9679 &0.9738 &0.0000 & &EIG &2 &1 &0.9885 &0.9880 &0.9882 &0.0000\\
&EIG2 &2 &1 &0.9798 &0.9679 &0.9738 &0.0000 & &EIG2 &2 &1 &0.9885 &0.9880 &0.9882 &0.0000\\
&GA &1 &1 &0.9798 &0.9798 &0.9798 &0.0000 & &GA &1 &1 &0.9885 &0.9885 &0.9885 &0.0000\\ \hline
bridge &D &500000 &318 &0.6160 &0.5712 &0.5792 &0.0001 &skipwith &D &34784 &72 &0.6560 &0.5728 &0.5954 &0.0003\\
&CLOS &181440 &48 &0.7888 &0.7136 &0.7389 &0.0002 & &CLOS &273 &113 &0.6448 &0.5920 &0.6178 &0.0006\\
&BETW &40320 &12 &0.8384 &0.7664 &0.7900 &0.0002 & &BETW &2 &2 &0.6448 &0.6448 &0.6448 &0.0000\\
&DOM &500000 &12 &0.5904 &0.5584 &0.5595 &0.0000 & &DOM &1 &1 &1.0000 &1.0000 &1.0000 &0.0000\\
&EIG &24781 &12 &0.8384 &0.7664 &0.7919 &0.0002 & &EIG &1 &1 &1.0000 &1.0000 &1.0000 &0.0000\\
&EIG2 &26100 &10 &0.8384 &0.7664 &0.7899 &0.0002 & &EIG2 &1 &1 &1.0000 &1.0000 &1.0000 &0.0000\\
&GA &1 &1 &0.8384 &0.8384 &0.8384 &0.0000 & &GA &1 &1 &1.0000 &1.0000 &1.0000 &0.0000\\ \hline
chesapeake &D &12 &4 &0.8949 &0.8710 &0.8848 &0.0001 &stmarks &D &1758 &1 &0.8550 &0.8134 &0.8315 &0.0001\\
&CLOS &20 &2 &0.8273 &0.8169 &0.8219 &0.0000 & &CLOS &500000 &16 &0.6263 &0.5990 &0.6119 &0.0000\\
&BETW &2 &2 &0.8241 &0.8241 &0.8241 &0.0000 & &BETW &14400 &4 &0.6680 &0.6324 &0.6451 &0.0001\\
&DOM &4320 &24 &0.8720 &0.8606 &0.8626 &0.0000 & &DOM &5 &2 &0.9180 &0.9145 &0.9165 &0.0000\\
&EIG &98 &2 &0.9251 &0.8876 &0.9032 &0.0001 & &EIG &588 &2 &0.9197 &0.9036 &0.9123 &0.0000\\
&EIG2 &104 &2 &0.9251 &0.8876 &0.9032 &0.0001 & &EIG2 &612 &2 &0.9210 &0.9054 &0.9129 &0.0000\\
&GA &1 &1 &0.9251 &0.9251 &0.9251 &0.0000 & &GA &1 &1 &0.9210 &0.9210 &0.9210 &0.0000\\ \hline
coachella &D &96 &4 &0.8288 &0.7681 &0.7982 &0.0003 &stmartin &D &166464 &4 &0.8067 &0.7534 &0.7710 &0.0001\\
&CLOS &15 &3 &0.7979 &0.7931 &0.7950 &0.0000 & &CLOS &34560 &16 &0.8050 &0.7800 &0.7910 &0.0000\\
&BETW &1 &1 &0.8811 &0.8811 &0.8811 &0.0000 & &BETW &720 &2 &0.8571 &0.8350 &0.8449 &0.0000\\
&DOM &500000 &2 &0.9394 &0.6254 &0.6303 &0.0001 & &DOM &30 &4 &0.9036 &0.8906 &0.8954 &0.0000\\
&EIG &6 &2 &0.9394 &0.9346 &0.9370 &0.0000 & &EIG &648 &4 &0.9178 &0.8940 &0.9036 &0.0000\\
&EIG2 &6 &2 &0.9394 &0.9346 &0.9370 &0.0000 & &EIG2 &112 &2 &0.9178 &0.8980 &0.9069 &0.0000\\
&GA &1 &1 &0.9394 &0.9394 &0.9394 &0.0000 & &GA &1 &1 &0.9178 &0.9178 &0.9178 &0.0000\\ \hline
grass &D &500000 &120 &0.8995 &0.8836 &0.8861 &0.0000 &ythan91 &D &4 &2 &0.9505 &0.9492 &0.9498 &0.0000\\
&CLOS &2448 &48 &0.8866 &0.8742 &0.8762 &0.0000 & &CLOS &46080 &6 &0.9205 &0.8881 &0.9004 &0.0001\\
&BETW &1680 &24 &0.8804 &0.8750 &0.8767 &0.0000 & &BETW &120 &1 &0.9554 &0.9313 &0.9415 &0.0000\\
&DOM &33 &3 &0.9481 &0.9473 &0.9475 &0.0000 & &DOM &2 &1 &0.9772 &0.9763 &0.9768 &0.0000\\
&EIG &27228 &3 &0.9481 &0.9239 &0.9383 &0.0000 & &EIG &120 &1 &0.9772 &0.9538 &0.9639 &0.0000\\
&EIG2 &38736 &3 &0.9481 &0.9237 &0.9379 &0.0000 & &EIG2 &112 &1 &0.9772 &0.9538 &0.9636 &0.0000\\
&GA &1 &1 &0.9481 &0.9481 &0.9481 &0.0000 & &GA &1 &1 &0.9772 &0.9772 &0.9772 &0.0000\\ \hline
reef &D &451440 &56 &0.7632 &0.7272 &0.7392 &0.0001 &ythan96 &D &2 &2 &0.9349 &0.9349 &0.9349 &0.0000\\
&CLOS &672 &16 &0.7180 &0.7032 &0.7061 &0.0000 & &CLOS &46080 &6 &0.9448 &0.9240 &0.9315 &0.0000\\
&BETW &12 &2 &0.7700 &0.7672 &0.7684 &0.0000 & &BETW &120 &1 &0.9685 &0.9538 &0.9597 &0.0000\\
&DOM &1 &1 &0.9640 &0.9640 &0.9640 &0.0000 & &DOM &1 &1 &0.9781 &0.9781 &0.9781 &0.0000\\
&EIG &6 &1 &0.9640 &0.9612 &0.9624 &0.0000 & &EIG &120 &1 &0.9807 &0.9688 &0.9738 &0.0000\\
&EIG2 &6 &1 &0.9640 &0.9612 &0.9624 &0.0000 & &EIG2 &102 &1 &0.9807 &0.9688 &0.9743 &0.0000\\
&GA &1 &1 &0.9640 &0.9640 &0.9640 &0.0000 & &GA &1 &1 &0.9807 &0.9807 &0.9807 &0.0000\\ \hline
\end{tabular}}
\end{center}
%\end{tabular}
\begin{flushleft}See Supporting Information for a description of the data.
\end{flushleft}
%\label{tab:label}
% \end{table}
\end{sidewaystable}
\end{document}