Googling food webs: can an eigenvector measure species' importance for coextinctions?

Abstract

A major challenge in ecology is forecasting the effects of species' extinctions, a pressing problem given current human impacts on the planet. Consequences of species losses such as secondary extinctions are difficult to forecast because species are not isolated, but interact instead in a complex network of ecological relationships. Because of their mutual dependence, the loss of a single species can cascade in multiple coextinctions. Here we show that an algorithm adapted from the one Google uses to rank web-pages can order species according to their importance for coextinctions, providing the sequence of losses that results in the fastest collapse of the network. Moreover, we use the algorithm to bridge the gap between qualitative (who eats whom) and quantitative (at what rate) descriptions of food webs. We show that our simple algorithm finds the best possible solution for the problem of assigning importance from the perspective of secondary extinctions in all analyzed networks. Our approach relies on network structure, but applies regardless of the specific dynamical model of species' interactions, because it identifies the subset of coextinctions common to all possible models, those that will happen with certainty given the complete loss of prey of a given predator. Results show that previous measures of importance based on the concept of "hubs" or number of connections, as well as centrality measures, do not identify the most effective extinction sequence. The proposed algorithm provides a basis for further developments in the analysis of extinction risk in ecosystems.

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Googling Food Webs: Can an Eigenvector Measure
Species’ Importance for Coextinctions?
Stefano Allesina1*, Mercedes Pascual2,3,4
1National Center for Ecological Analysis and Synthesis, Santa Barbara, California, United States of America, 2Department of Ecology and Evolutionary Biology, University
of Michigan, Ann Arbor, Michigan, United States of America, 3Santa Fe Institute, Santa Fe, New Mexico, United States of America, 4Howard Hughes Medical Institute
Abstract
A major challenge in ecology is forecasting the effects of species’ extinctions, a pressing problem given current human
impacts on the planet. Consequences of species losses such as secondary extinctions are difficult to forecast because
species are not isolated, but interact instead in a complex network of ecological relationships. Because of their mutual
dependence, the loss of a single species can cascade in multiple coextinctions. Here we show that an algorithm adapted
from the one Google uses to rank web-pages can order species according to their importance for coextinctions, providing
the sequence of losses that results in the fastest collapse of the network. Moreover, we use the algorithm to bridge the gap
between qualitative (who eats whom) and quantitative (at what rate) descriptions of food webs. We show that our simple
algorithm finds the best possible solution for the problem of assigning importance from the perspective of secondary
extinctions in all analyzed networks. Our approach relies on network structure, but applies regardless of the specific
dynamical model of species’ interactions, because it identifies the subset of coextinctions common to all possible models,
those that will happen with certainty given the complete loss of prey of a given predator. Results show that previous
measures of importance based on the concept of ‘‘hubs’’ or number of connections, as well as centrality measures, do not
identify the most effective extinction sequence. The proposed algorithm provides a basis for further developments in the
analysis of extinction risk in ecosystems.
Citation: Allesina S, Pascual M (2009) Googling Food Webs: Can an Eigenvector Measure Species’ Importance for Coextinctions? PLoS Comput Biol 5(9):
e1000494. doi:10.1371/journal.pcbi.1000494
Editor: Philip E. Bourne, University of California San Diego, United States of America
Received June 1, 2009; Accepted July 29, 2009; Published September 4, 2009
Copyright: ? 2009 Allesina, Pascual. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by a Centennial Fellowship of the James S. McDonnell Foundation to MP and by NSF grant EF-0827493. SA is a postdoctoral
fellow at the National Center for Ecological Analysis and Synthesis, funded by NSF DEB 0553768, the University of California, Santa Barbara, and the State of
California. MP is a Howard Hughes Medical Investigator. The funders had no role in study design, data collection and analysis, decision to publish, or preparation
of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: allesina@nceas.ucsb.edu
Introduction
The robustness of ecosystems to species losses is a central
question in ecology given the current pace of extinctions and the
many species threatened by human impacts [1–3]. The loss of
species in complex ecological networks can cascade into further
extinctions because of the mutual dependence of species. Of all the
possible causes leading to these ‘‘cascading’’ extinctions, the
simplest case to analyze is that of species left with no exploitable
resources [4–8]. These extinctions due to lack of nutrient flows
represent the most predictable subset of secondary losses and also
the best case scenario, since the addition of other effects [9,10],
related to the loss of dynamical regulation, will result in additional
losses. The former scenario is the simplest to analyze because the
extinction of consumers that are left with no resources will happen
with certainty, unless the consumers can switch to a different set of
resources. Because modern data sets are obtained by sampling
extensively a system over time, it is unlikely that potential
resources resulting from switching prey go unregistered. If these
potential interactions have been included in the prey of a given
predator, then the dynamics of extinction for this flow-based case
are completely described by the network structure. This simple
analysis also represents the best case scenario, since other causes of
extinction such as low population abundance can increase the loss
of species in response to the original disturbance, but cannot
prevent flow-based extinctions from happening. From the flow-
based perspective, the effects of a single species loss can be easily
analyzed [7], but those of multiple losses and sequences of
extinctions rapidly become an intractable problem.
Species’ importance in this context has been traditionally
measured using local network properties, such as the number of
species’ connections [4,5]. In particular, species with a large
number of links are considered keystones (or hubs [11]) for the
robustness of ecological networks [5,6,8,12]. A different take on
species’ importance in networks makes use of centrality measures:
species that are central mediate the interaction among those that
are more peripheral and therefore should be considered the most
important species [13–15].
Here we propose a new algorithm for assessing the importance of
species for food web robustness that takes into account the full
topology of the network. When species importance from the
perspective of robustness is correctly measured, the ordered removal
of species according to this ranking should lead to the fastest collapse
of the network. Our approach inspired by PageRankTM, the
algorithm at the heart of GoogleTM[16], uses a recursive definition:
a species is important if important species rely on it for their survival.
Results show that the algorithm outperforms all other measures of
species importance from the perspective of fastest route to collapse.
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Moreover, it performs as well as a genetic algorithm [17,18], an
evolutionary intensive search that can evaluate millions of solutions,
even if the eigenvector implementation is much simpler and faster. A
biological interpretation of speciesimportance followsnaturallyasthe
amountofmatterflowingthroughagivenspecies,forbothqualitative
networks constructed from the presence and absence of links, and
quantitative networks for which interaction strengths are explicitly
specified [19–21]. The proposed approach provides the basis for a
more comprehensive treatment of extinction risk in food webs.
Materials and Methods
The World Wide Web is a directed network in which web pages
(nodes) are connected with each other by hyper-links. We can
write a matrix A in which the presence and absence of a link from
the row-page i to the column-page j are represented as entries
aij~1 and 0, respectively. PageRankTMrates pages as important if
they receive links from pages that are in turn also rated as
important. The PageRankTMalgorithm solves this recursive
definition using a clever application of linear algebra [16]. Each
page i is assigned an importance, and each link aij(exiting page i
to enter page j) carries an equal fraction of the importance value.
The importance of a page is the sum of the importance assigned to
the incoming connections. The recursive problem can be solved by
building a matrix S in which each element represents the fraction
of importance assigned to a linkand given by sij~aij=
When matrix S satisfies two conditions (it is both irreducible and
primitive [16]), then the problem of assigning importance is solved
by computing a fundamental and well-known quantity in linear
algebra, the eigenvector v associated with the dominant eigenvalue
l?~1 of S. If the two conditions are met, the Perron-Frobenius
Theorem guarantees the existence of this dominant eigenvector
(Text S1).
One main problem, besides the numerical challenge of
computing the eigenvectors of a matrix with several billions rows
and columns, is that the World Wide Web is not irreducible [16].
For irreducible matrices, the associated network must be strongly
connected, with any two nodes connected by a directed pathway.
Because he WWW clearly does not meet this condition, the matrix
is modified by applying a ‘‘damping factor’’, d. A new matrix H is
constructed with entries hij~d:sijz(1{d)=N, where N is the
number of nodes in the network. The damping factor effectively
X
jaij.
mimics the probability (1{d) that a user browsing the web can
decide to move directly to another (random) page [16]. The
eigenvector is then computed for H.
Here we propose an algorithm to rank the importance of species
for food web robustness that uses a similar principle. Nutrients
move from one species to another in a food web through feeding
links. For their survival, species must be able to receive energy and
matter from primary producers through some pathway in the
network [7,22]. Thus, we define a species as important if it
supports (directly or indirectly) other species that are in turn
important. The problem is similar to that of ranking web pages,
with the difference that now importance moves in the opposite
direction than that of the links (i.e. a web page is important if
important pages point to it; species are important if they point to
important species). Also food webs are neither irreducible nor
primitive, but we can find a biologically sound solution to this
problem. A damping factor would be completely unrealistic since
nutrients cannot randomly ‘‘jump’’ among links in the food web.
We make instead two observations: first, all matter in the food web
must originate from primary producers who receive it from the
external environment and channel it through the food web to all
other species through feeding pathways [21,23]. We therefore
attach to the network a special node (a ‘‘root’’) that points to all the
primary producers [7,22]. Second, every species has an intrinsic
loss of matter which can be represented by adding a link from
every node to the root. This process represents the buildup of
detritus that in turn is partly recycled into the food web [21,23].
With these two modifications any food web becomes irreducible
and primitive (Fig. 1, Text S1) and we can now solve the problem
of assigning importance by computing the eigenvector v associated
with the dominant eigenvalue l?~1. For simplicity, we consider
the normalized eigenvector so that
Recent research on food web robustness has emphasized the
role of connectivity: species with a high number of connections are
likely to be essential for the survival of other species [4–8]. In-silico
extinction experiments also showed that random removal
sequences rarely cascade in the secondary loss of species, whereas
the removal of highly connected species is likely to generate many
secondary extinctions. Another line of research borrowed
measures of centrality from sociology. Central species mediate
the spread of disturbances through the network. In this sense,
species with high centrality would be considered ‘‘keystone’’ to the
maintenance of connectivity in networks [13–15].
To test our algorithm, we performed in-silico extinction
experiments in which a single species is removed at each step
and the number of secondary extinctions is recorded. We
compared several simple algorithms: a) the removal of the most
connected species at each step (D, where we measured the number
of connections coming out of each node); b) the removal of species
according to closeness centrality (CLOS): nodes are highly central
from this point of view if they have short distance to many nodes in
the network; c) the removal according to betweeness centrality
(BETW): a node has high betweeness if it lies on the shortest path
between many couples of nodes; d) removal according to
dominators (DOM): a x node dominates another y if all the
paths from ‘‘root’’ to y contain x - the removal of x will therefore
drive y extinct [7]; finally, e) we removed according to the
eigenvector-based algorithm outlined above (EIG).
All the algorithms are ‘‘greedy’’: at each step, we compute the
‘‘importance’’ of each node according to a particular algorithm,
and we remove the one with the highest importance. The
procedure is repeated until all the species have gone extinct or
have been removed. The algorithms are explained in detail in the
Text S1. For each extinction sequence, we measured the
X
ivi~1.
Author Summary
Predicting the consequences of species’ extinction is a
crucial problem in ecology. Species are not isolated, but
connected to each others in tangled networks of
relationships known as food webs. In this work we want
to determine which species are critical as they support
many other species. The fact that species are not
independent, however, makes the problem difficult to
solve. Moreover, the number of possible ‘‘importance’’’
rankings for species is too high to allow a solution by
enumeration. Here we take a ‘‘reverse engineering’’
approach: we study how we can make biodiversity
collapse in the most efficient way in order to investigate
which species cause the most damage if removed. We
show that adapting the algorithm Google uses for ranking
web pages always solves this seemingly intractable
problem, finding the most efficient route to collapse. The
algorithm works in this sense better than all the others
previously proposed and lays the foundation for a
complete analysis of extinction risk in ecosystems.
Googling Food Webs
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‘‘extinction area’’, a quantity that equals 1 when all species go
extinct after the first removal and tends to 1/2 when no secondary
extinction is observed (Fig. 2). In this way, we can assess the
performance of each algorithm with a single number. If important
species are removed early on, then the area will be larger.
The algorithms could yield ties - nodes with the same
importance. Whenever we encountered ties, we considered all
the possible sequences of extintions that may result exploring all
the ties. Therefore, algorithms with low ranking power (i.e.
yielding many ties) could produce very many extinction sequences.
We followed all extinction sequences generated by ties whenever
they were less than half a million. If there were more possible
solutions, we analyzed the first half million.
We applied all the algorithms to 12 published food webs
(Table 1). For each algorithm and network, we tracked the total
number of solutions produced by the algorithm, the minimum,
maximum and mean ‘‘extinction area’’ and the number of
solutions yielding the maximum area (Text S1).
We then evaluated the value of the maximal extinction area.
Because the number of possible removal sequences is N! where N
is the number of species in the network, the enumeration of all
possible cases is clearly unfeasible. We therefore programmed a
Genetic Algorithm [17] (GA) that seeks to find the best possible
sequence using an evolutionary search. This type of algorithm has
been shown to be effective for similar problems in food web theory
[18], even when computationally expensive and when its
performance declines with food web size. Here, the GA search
performs at least as well as the best among the other algorithms, as
expected for an effective search (Fig. 3).
Results
In all cases, the best solution for the degree-based algorithm (D)
and the closeness centrality (CLOS) did not match the genetic
algorithm (GA): these measures do not correctly identify the fastest
route to collapse (Fig. 3). Betweeness centrality yields an area as large
as that of the GA in only 1 case (benguela). The dominators-based
procedure findsthe bestsolution in 2/3 of the cases. The eigenvector-
based algorithm finds the best solution in 11 cases out of 12. To
improve the EIG algorithm, we build upon a previous approach of
ours [22], based on the observation that not all the links in a food web
contribute to robustness. The idea that more complex networks
would contain a multiplicity of pathways that would in turn render
the networks more robust was put forward by MacArthur more than
fifty years ago [24]. We recently showed that, while this is generally
true, somelinks do not contribute to robustness, whileothers dampen
the effects of species removal and increase robustness (Fig. 1) [22].
Thus links can be classified as ‘‘redundant’’ or ‘‘functional’’ from the
perspective of their effects on secondary extinctions. From this
classification, one can obtain a simplified food web by removing all
Figure 1. Modification of food webs from ecological considerations to satisfy the two constraints required for application of the EIG
algorithm. Left) A special node is added to the food web by connecting this ‘‘root’’ to the primary producers. Every species in turn connects to the
root to represent the buildup of detritus (dashed line). Right) The analysis can be improved by removing the ‘‘redundant’’ connections that do not
contribute to robustness (dashed, in red).
doi:10.1371/journal.pcbi.1000494.g001
Figure 2. The extinction area is the area described by the area
below the curves. The area can take values from1
extinctions in response to the removal of species) to 1 (all species go
extinct after the first removal). The x axis represents the fraction of
species removed in the numerical experiment, while the y axis is the
fraction of species that are extinct as the result of these removals. The
example uses the St. Mark’s food web [29] and the D (red) and EIG
(blue) algorithm.
doi:10.1371/journal.pcbi.1000494.g002
2(no secondary
Googling Food Webs
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redundant connections, that has exactly the same robustness
properties than the original network in terms of the secondary
extinctions. For the algorithm EIG2, then, we repeated the removal
sequence experiment but we computed v for the simplified food web
obtained by first removing redundant connections. The results
indicate that the algorithm is capable of finding the best solution
provided by the GA in all cases (Fig. 3, Text S1).
Discussion
We have developed two algorithms to rank species in food webs
according to their role in extinction cascades. We considered a flow-
based perspective in which species go extinct if they lack a
connectionthrough some pathways to primary producers. Although
it is evident that many other types of extinctions can increase total
species loss, the subset considered here provides a baseline and
corresponds to the best case scenario in which the minimum impact
to the network is taken into account. Species left with no resources
will go extinct,unlesstheycanswitchtheir choiceof prey sufficiently
fast. It is known that species can exhibit this type of adaptive
behavior in response to the relative abundance of prey, with
consequences for the stability of predator-prey systems [25].
Because the food webs we have analyzed are sampled in the field
over time and space, it is most likely that the links included in the
networks already reflect prey switching. An important source of
additional secondary extinctions will be related to the population
dynamicsofspecies. Thecomplete consideration of dynamics with a
system of nonlinear differential equations that simulates the
outcome of species losses, will only increase the number of species
predicted to go extinct by the simplest scenario. The analysis of
removal effects remains very challenging if not prohibitive for large
ecological networks (but see [9,10,26]), requiring information most
often unavailable on the functional form of a large number of
interactions and their associated parameters, the exploration of
different assumptions and a huge parameter space. The simple and
elegant solution for the flow-based case provides a baseline from
which additional impacts can be considered.
The results obtained here with a simple algorithm emphasize
that the position of a species in the food web, rather than its sheer
number of connections, is the main determinant of its impact on
extinction cascades. This contrasts with the emphasis given so far
to the number of connections and to the concept of ‘‘hubs’’ in
networks. We have shown that the performance of the D
algorithm, which considers only the neighbors of a given species,
is considerably worse than that of the eigenvector based algorithms
at finding the fastest route to collapse. The latter algorithms solve
the problem of importance by considering the full topology of the
network and the particular position that each species occupies. We
further showed that an algorithm that first removes ‘‘redundant’’
connections provides a valuable improvement, because it relies on
the functional role of connections in maintaining the flow of
nutrients through the food web. Interestingly, a parallel problem
has been analyzed in computer science (Text S1).
Srinivasan et al. [27] have shown that many realistic removal
sequences are not likely to cascade in massive species’ losses, with
the loss of threatened species not necessarily resulting in further
extinctions. It is therefore difficult to discriminate importance
among species whose removal has little direct effect on network
structure. The eigenvector approach provides a simple and
effective way of comparing species importance even when their
removal does not result in extinction cascades. This should help
assessing the relative importance of threatened species for network
robustness and from the perspective of network structure. Coll et al.
analyzed the effect of actual human-induced extinction in the
Mediterranean sea and found that removing commercially
valuable species had typically a higher impact than random
removals, but lower than maximum degree driven removals [28].
The dominant eigenvector has also a simple biological interpre-
tation. To show this, we assume for the moment that we can
fully describe the interacting community by means of differential
equations representing the dynamics of species’ abundance,
dXi=dt~f(X1,X2,...,XS). We further consider that the system is
at a feasible equilibrium point (dXi=dt~0 for all species, Xiw0). For
this case, we can measure the flow of biomass entering and exiting
each species (for example, in kilograms of biomass per year per
hectare) and the amount entering and exiting each node must be
equal given the equilibrium condition [19–21]. These quantities are
proportional to the eigenvector used here: specifically, v provides an
estimate of the flow through each species (Text S1, Fig. S1). In the
absence of available information on diet preferences, v measures the
flow that each species would receive if each of its prey provided equal
amounts of nutrients. When quantitative information on these inputs
is available, v and the flow-based description become exactly
equivalent (Text S1, Fig. S2).
Table 1. Extinction Area.
Food WebNum. Species Max. DMax. CLOSMax. BETW Max. DOM Max. EIG1Max. EIG2 GAReference
benguela 290.7943 0.7539 0.90250.9798 0.97980.9798 0.9798[31]
bridge25 0.61600.7888 0.83840.59040.8384 0.83840.8384[32]
chesapeake31 0.89490.8273 0.82410.87200.92510.9251 0.9251 [33]
coachella29 0.82880.79790.88110.9394 0.93940.9394 0.9394 [34]
grass61 0.89950.8866 0.88040.9481 0.94810.9481 0.9481 [35]
reef 500.7632 0.7180 0.77000.96400.96400.96400.9640[36]
shelf790.63800.65610.8103 0.98850.98850.9885 0.9885 [37]
skipwith25 0.65600.6448 0.64481.0000 1.0000 1.00001.0000 [30]
stmarks48 0.85500.62630.66800.91800.9197 0.92100.9210[29]
stmartin420.80670.80500.85710.9036 0.9178 0.91780.9178 [38]
ythan91 830.95050.9205 0.95540.9772 0.97720.9772 0.9772 [39]
ythan961240.93490.9448 0.9685 0.97810.98070.98070.9807[40]
For each food web, we report the number of nodes, and the maximum ‘‘extinction area’’ (Fig. 2) obtained using the algorithms presented in the text (Fig. 3).
doi:10.1371/journal.pcbi.1000494.t001
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The proposed algorithm further provides a measure of
eigenvector centrality in directed, rooted networks. Other centrality
measures have been proposed to evaluate species importance [13–
15], but they typically consider undirected networks and have not
been adapted to food webs. This is reflected in the poor
performance achieved by the centrality algorithms. Here we have
shown that consideration of ecological knowledge on food web
processescanimprovealgorithms that havebeen developedinother
branches of science. It should be possible to adapt the methods
presented here to other types of biological networks, especially
metabolic ones. For food webs, the next challenge is to add other
dynamical effects to this framework, to obtain a more complete
description of extinction risk in complex ecological networks.
Supporting Information
Text S1
Found at: doi:10.1371/journal.pcbi.1000494.s001 (0.03 MB TEX)
Supporting Information
Figure S1
modified as described in the text. The flows are expressed in kg of
The Lovinkhoeve Experimental Farm food web
Figure 3. Extinction areas for 12 published food webs (Table 1) according to the 7 algorithms presented in the text. The area is 1 (as in
the ‘‘skipwith’’ [30] food webs) only when there is a single primary producer. Because each algorithm can give raise to several solutions, we report the
minimum (red), mean (blue) and maximum (black) registered extinction area. We indicate with an asterisk ‘‘*’’ the algorithms that are able to match
the performance of the genetic algorithm (GA).
doi:10.1371/journal.pcbi.1000494.g003
Googling Food Webs
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