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Ecology, 89(12), 2008, pp. 3387–3399
Ó2008 by the Ecological Society of America
WHAT DO INTERACTION NETWORK METRICS TELL US
ABOUT SPECIALIZATION AND BIOLOGICAL TRAITS?
NICO BLU
¨THGEN,
1,3
JOCHEN FRU
¨ND,
1,4
DIEGO P. VA
´ZQUEZ,
2
AND FLORIAN MENZEL
1
1
Department of Animal Ecology and Tropical Biology, Biozentrum, University of Wu
¨rzburg, Am Hubland, Wu
¨rzburg 97074 Germany
2
Instituto Argentino de Investigaciones de las Zonas A
´ridas, CONICET, CC 507, 5500 Mendoza, Argentina
Abstract. The structure of ecological interaction networks is often interpreted as a
product of meaningful ecological and evolutionary mechanisms that shape the degree of
specialization in community associations. However, here we show that both unweighted
network metrics (connectance, nestedness, and degree distribution) and weighted network
metrics (interaction evenness, interaction strength asymmetry) are strongly constrained and
biased by the number of observations. Rarely observed species are inevitably regarded as
‘‘specialists,’’ irrespective of their actual associations, leading to biased estimates of
specialization. Consequently, a skewed distribution of species observation records (such as
the lognormal), combined with a relatively low sampling density typical for ecological data,
already generates a ‘‘nested’’ and poorly ‘‘connected’’ network with ‘‘asymmetric interaction
strengths’’ when interactions are neutral. This is confirmed by null model simulations of
bipartite networks, assuming that partners associate randomly in the absence of any
specialization and any variation in the correspondence of biological traits between associated
species (trait matching). Variation in the skewness of the frequency distribution fundamentally
changes the outcome of network metrics. Therefore, interpretation of network metrics in terms
of fundamental specialization and trait matching requires an appropriate control for such
severe constraints imposed by information deficits. When using an alternative approach that
controls for these effects, most natural networks of mutualistic or antagonistic systems show a
significantly higher degree of reciprocal specialization (exclusiveness) than expected under
neutral conditions. A higher exclusiveness is coherent with a tighter coevolution and suggests a
lower ecological redundancy than implied by nested networks.
Key words: abundance distribution; biological traits; connectance; degree distribution; ecological
networks; interaction diversity; interaction strength; nestedness; null models; specialization.
INTRODUCTION
The analysis of ecological interaction networks is an
important tool in community ecology and is often used
to draw conclusions about such diverse topics as
community robustness, biodiversity maintenance, re-
source partitioning, and natural selection (May 1972,
Pimm 1982, Pimm et al. 1991, McCann et al. 1998,
Proulx et al. 2005, Bascompte et al. 2006, Beckerman et
al. 2006, Montoya et al. 2006, Santamarı
´aand
Rodrı
´guez-Girone
´s 2007). Ecological interaction net-
works are representations of associations (links) between
species (nodes). Network studies like those just cited
commonly select one or a few network metrics (Table 1)
that are based either on unweighted links (only
representing the presence or absence of a link, also
known as ‘‘qualitative’’ or binary metrics) or on
weighted links (‘‘quantitative’’ metrics). Important
unweighted metrics include connectance (May 1972,
Jordano 1987, Beckerman et al. 2006, Santamarı
´a and
Rodrı
´guez-Girone
´s 2007), nestedness (Bascompte et al.
2003, Santamarı
´a and Rodrı
´guez-Girone
´s 2007), and
degree distribution (Jordano et al. 2003, Va
´zquez 2005).
Among weighted metrics, dependence or interaction
strength (Bascompte et al. 2006, Va
´zquez et al. 2007) or
different measures of interaction diversity or evenness
(Bersier et al. 2002, Tylianakis et al. 2007) are commonly
used.
Describing network patterns based on these metrics
may help to unravel important patterns of community
organization and heterogeneity in trophic associations,
particularly if the underlying data represent an unbiased
estimate of realized interactions. However, network
patterns are often interpreted as a result of biological
trait matching and fundamental specialization or
generalization, which reflects the dependency between
species or redundancy of associations, respectively, and
thus contributes to the stable coexistence of species in
communities (Pimm et al. 1991, Montoya et al. 2006).
For instance, secondary extinctions of consumers due to
resource losses have been suggested to decrease with
increasing connectance (Dunne et al. 2002b, Estrada
2007), whereas nestedness and asymmetries in interac-
tion strength may promote community stability (Bas-
Manuscript received 2 December 2007; revised 27 March
2008; accepted 7 April 2008. Corresponding Editor: C. M.
Herrera.
3
E-mail: bluethgen@biozentrum.uni-wuerzburg.de
4
Present address: Agroecology, Georg-August-Universi-
ta
¨t, 37073 Go
¨ttingen, Germany.
3387
compte et al. 2003, 2006). In the present paper, we will
argue that these raw metrics as such, uncontrolled for
neutrality, may be substantially flawed regarding an
interpretation of trait matching or fundamental special-
ization. In this study we define trait matching as the
degree of interaction partitioning between species,
resulting from the correspondence of phenotypic traits
of interacting species (Fig. 1); these traits include body
size, morphology, chemical composition, physiological
abilities, temporal activity patterns, preferences, and
behavior. For instance, species-specific preferences may
strengthen some links, while structural barriers or an
avoidance of defenses may inhibit others. A strong
structural importance of trait matching leads to a high
level of specialization in the community (‘‘fundamental
specialization’’ sensu Va
´zquez and Aizen 2006), while
the absence of structuring by trait matching implies
maximum generalization. The latter scenario is often
simulated in null model networks, where partners
associate randomly (Va
´zquez and Aizen 2006).
TABLE 1. Definitions of common bipartite network metrics.
Metric Definition
A) Metrics based on unweighted links
Connectance The proportion of possible links actually observed in a web. The connectance (or connectivity) is:
C¼L/(IJ ) for bipartite networks, or C¼L/S
2
in unipartite networks (e.g., food webs),
respectively. Ldescribes the number of realized links; Iand Jare the number of species of each
party in bipartite networks, e.g., hosts vs. parasites; Sis the total number of species. The related
‘‘linkage density’’ is D¼L/(IþJ).
Generality and
vulnerability
In a network of Iconsumers and Jprey species, the mean number of prey species (links) per
consumer is termed ‘‘generality’’ (G¼L/I) and the mean links per prey ‘‘vulnerability’’:
(V¼L/J) (Schoener 1989).
Nestedness The nestedness ‘‘temperature’’ T(08–1008) (Atmar and Patterson 1993) measures the departure from
a perfectly nested interaction matrix (degree of disorder). T¼08is defined for maximum
nestedness: when rows and columns are ordered by decreasing number of links, links of each row
and column exactly represent a subset of the previous ones. For each species, the deviation from
the expected nested order accounts for a higher ‘‘idiosyncratic temperature’’ (Atmar and
Patterson 1993). Nestedness can be defined as N¼(1008T)/1008(Bascompte et al. 2003).
Degree distribution For each species, the number of links describes its ‘‘degree’’ k. A power law or ‘‘scale-free’’
distribution implies that the cumulative degree distribution (i.e., the proportion of species with k
or more links) is described by P(k);k
c
, where cis a constant. Many ecological networks are
better fitted by a truncated power law function P(k);k
c
exp(k/k
x
), assuming an exponential
decay of the power law distribution, where k
x
represents an additional constant (Jordano et al.
2003, Va
´zquez 2005).
B) Metrics based on weighted links
Interaction strength Interaction strength of species jon species i(b
ij
) can be defined by the proportion of interactions
between iand j(a
ij
) of the total interactions recorded for i; thus b
ij
¼a
ij
/RJ
j¼1a
ij
. For mutualistic
networks, Jordano (1987) and Bascompte et al. (2006) used b
ij
as a measure of dependence of
species ion its partner j. Asymmetries of interaction strength can be defined as AS
ij
¼(b
ij
b
ji
)/(b
ij
þb
ji
), where b
ji
is the reciprocal dependence of species jon species i(see Bascompte et al.
2006, Blu
¨thgen et al. 2007, Va
´zquez et al. 2007).
Interaction diversity The Shannon diversity of links is H
i
¼RJ
j¼1[(a
ij
/A
i
) ln(a
ij
/A
i
)] for species i, or for the whole web,
H
2
¼RI
i¼1RJ
j¼1[(a
ij
/m) ln(a
ij
/m)], with A
i
¼RJ
j¼1a
ij
and m¼RI
i¼1RJ
j¼1a
ij
. Their reciprocals eHi
and eH2express the equivalent ‘‘effective’’ number of links (see Bersier et al. 2002).
Interaction evenness Based on Shannon diversity, the interaction evenness is E
i
¼H
i
/ln L
i
for each species, or for the
whole web, E
2
¼H
2
/ln L, where L
i
is the number of links of species i, and Lis the number of all
links. First suggested by Bersier et al. (2002), these measures or other standard diversity metrics
have been applied to different interaction networks (e.g., Sahli and Conner 2006, Albrecht et al.
2007, Tylianakis et al. 2007).
Weighted generality and
vulnerability
The weighted analog of generality can be derived from Shannon diversity of links (H
i
), representing
the mean ‘‘effective’’ links per consumer G
q
¼(1/I)RI
i¼1eHt, or as the weighted mean G
qw
¼
RI
i¼1ðAi=mÞeHi(Bersier et al. 2002, with equations based on log
2
instead of ln). For weighted
vulnerability, replace iby jand Iby J.
Standardized interaction
diversity
Derived from Shannon diversity of links in the network (H
2
), the following specialization index (H0
2;
Blu
¨thgen et al. 2006) increases with the deviation of realized interaction frequencies from
expected values of a null distribution of interactions (H
2min
, corresponding to a perfectly
quantitatively ‘‘nested’’ matrix) based on the given species frequency distribution (fixed
interaction totals): H0
2¼(H
2max
H
2
)/(H
2max
H
2min
), where H
2
is defined as above. Note that
av
2
analysis of homogeneity is conceptually similar.
Standardized distance In a similar way, the nonconformity of a focal species ican be described in relation to total
interaction frequencies (standardized Kullback-Leibler distance d0
i, Blu
¨thgen et al. 2006). In
analogy to ‘‘idiosyncratic temperatures’’ defined for nestedness above, d0
iindicates the
exclusiveness of the interactions of a species, i.e., its deviation from a null distribution assuming
that interactions of irepresent a subset of the overall sample. Consequently, d0
iincreases with
reciprocal specialization between iand its partner.
Link weight is often defined as interaction frequency (a
ij
) between species iand species jfrom a total set of Ivs. Jspecies. The
sum of all interaction for species iis A
i
¼RJ
j¼1a
ij
, for jA
j
¼RI
i¼1a
ij
, and for the whole network m¼RI
i¼1RJ
j¼1a
ij
. Most equations in
the section ‘‘Metrics based on weighted links’’ are only valid if species lacking any interaction (A
i
¼0orA
j
¼0) are excluded.
NICO BLU
¨THGEN ET AL.3388 Ecology, Vol. 89, No. 12
In addition to trait matching or neutral interactions,
however, metrics depicting the network structure may
also be affected by variation in sampling intensity (Paine
1988, Va
´zquez and Aizen 2006). This impact was studied
using rarefaction techniques (Goldwasser and Rough-
garden 1997, Banas
ˇek-Richter et al. 2004, Herrera 2005,
Blu
¨thgen et al. 2006). It is a well-known fact that
ecological data are constructed of incomplete samples:
only a subset of the interactions naturally present in a
community is recorded. However, the consequences are
often ignored when these metrics are currently used for
characterizing specialization in ecological networks.
Most importantly, for rarely observed species (e.g., see
Plate 1) most metrics are necessarily biased, a simple fact
that is poorly considered in interpreting networks. For
example, species that are represented only by a single
observation (singletons) are inevitably assigned only a
single link and regarded as ‘‘specialists,’’ inflating the
FIG. 1. Conceptual framework of a quantitative interaction network, represented by an empirical pollination network of nine
plant species and 27 pollinator species (‘‘Safariland’’ in Va
´zquez and Simberloff 2003). The different levels are: abundance
distribution (A
1
and A
2
for plants and pollinators, respectively), total observation records per species (R
1
and R
2
), and the
distribution of interactions (I) in the core of the network. Numbers in the boxes are the number of observed interactions between a
particular plant species and a particular pollinator species: R
1
¼790 indicates that there were 790 observations of interactions
between one plant species and six different pollinator species, and R
2
¼673 indicates that there were 673 observations of one
pollinator species at a single plant species. The darkness of the shading increases with the number of observations and serves to
show the pattern of links in the matrix. General biological traits (B
1
and B
2
), overall sampling intensity, and sampling bias may
affect the relationship between Aand R. The interactions (I) are shaped by Rand by trait matching between associated partners (T).
The effect caused by Tcan be studied by comparing the distribution of neutral encounter probabilities based on R(null model) with
the actual distribution of interactions (I). This comparison may facilitate the search for biological processes (examples a–c), while
the scenarios (b) and (c) are not distinguished in network metrics when predictions based on Rare uncontrolled for. Note that A
1
represents an independent estimate of plant densities (mean number of individuals per 20-m
2
quadrat). Such independent measures
are unavailable for pollinators in this network; hence, A
2
represents the number of individual pollinators observed on flowers (A
2
¼
R
2
). The prediction based on the null model is displayed in Fig 2b.
December 2008 3389DETERMINANTS OF NETWORK METRICS
estimate of the prevalence of specialization, nestedness,
and strength asymmetry (if singletons are not associated
with exclusive partners), and deflating connectance and
interaction diversity. Because ecological networks usu-
ally include several rarely collected species with limited
observation records (including singletons; see Fig. 1), it
is crucial to know to what extent network patterns can
be attributed to biological processes such as trait
matching, in contrast to effects of sampling intensity,
sampling bias, or the underlying species abundances.
We therefore examined the sensitivity of commonly
used network metrics for variation in the number of
observations per species, given a realistic level of overall
sampling intensity and thus average observations per
species. In most ecological data sets, some species will be
much more commonly recorded than others, a conse-
quence of the abundance distribution that is typically
log-normal or alike (McGill et al. 2007), different
activity patterns, and possibly sampling bias. A typical
network thus involves a mixture of species at different
information levels, and the relative contribution of
frequently vs. poorly observed species may vary across
networks. Such variation in observation frequency per
species affects the detectability of links in unweighted
metrics, but also the relative strength of links in
weighted metrics. This combined impact of rarely and
frequently recorded species may need to be considered in
network comparisons, particularly if networks differ in
the shape of the frequency distribution.
The goals of the present study are threefold: (1) to
compare the degree of skewness of the frequency
distribution of observations per species across a set of
empirical networks; (2) to use simulations to evaluate
how much this skewness affects the metrics in the
absence of trait matching and fundamental specializa-
tion; and (3) to compare the biological conclusions
drawn from earlier network metrics with those derived
from an information-theoretical approach (Blu
¨thgen et
al. 2006) that controls for variable number of observa-
tions. We focused on bipartite networks, assuming that
all species of one party can potentially interact with all
partners of the other party and are completely oppor-
tunistic in their associations and consequently interact in
the most generalized way possible. This generalized
scenario was represented by simulating neutral interac-
tions between species in a null model approach. If a link
is absent in such a simulated network, we thus assume
that this is caused by the information deficit due to
limited sampling. Whereas some previous studies also
modeled the effect of the specific abundance or
interaction frequency on individual network metrics
(Va
´zquez and Aizen 2003, Va
´zquez 2005, Va
´zquez et al.
2005, 2007), we explicitly varied only the skewness of the
frequency distribution in the null model to examine its
structural consequences in the absence of other vari-
ables, and compared its effect across all commonly used
metrics for the first time.
SIMULATION METHODS
A bipartite ecological network describes the interac-
tions between two communities containing Iand J
species. Networks can be displayed as I3Jcontingency
tables, where each cell entry depicts the number of
interactions recorded between a specific pair of species
(Fig. 1). Our model of neutral encounter probabilities is
based on the marginal totals of this contingency table,
defining the total number of observation records per
species as the vectors R
1
and R
2
(Fig. 1). Henceforth, we
refer to R
1
and R
2
together as R
k
(for k¼[1, 2]).
Null models are useful to evaluate whether structural
patterns may be produced by stochastic processes in the
absence of particular mechanisms (Gotelli and Graves
1996). In this case, we aimed to investigate how network
metrics are determined by the frequency distribution in
the total observation records per species, in the absence
of any limitation by trait matching. We assume that the
observation records per species (R
k
) in natural networks
are driven mainly by sampling intensity, sampling bias,
species abundances, or variation in general species traits
that are independent of the specific distribution of the
interactions between the parties Iand J(Fig. 1). The null
model scenario thus implies that specialization and
abundance vary independently and do not interact,
thereby ignoring potential feedbacks from the interac-
tion level that might influence R
k
as emergent network
property. For given marginal totals (R
k
), the expected
value for the interactions between species iand species j
can be calculated as A
i
A
j
/m, where A
i
and A
j
represent
the total observation records of species iand j,
respectively, and mis the grand total number of
observed interactions in the network. This procedure is
analogous to calculating expected values in a chi square
test of homogeneity. Note that this theoretical expecta-
tion does not constrain interactions to integer values and
does not provide stochastic variation of interactions.
The null model algorithm applied here (Patefield 1981)
uses fixed marginal totals to distribute the interactions
and produce a set of networks where all species are
randomly associated. Patefield’s (1981) algorithm is
implemented as function ‘‘r2dtable(100, R
1
,R
2
)’’ in R
statistical software version 2.4.1 (R Development Core
Team 2006). We generated 100 randomizations for each
specific distribution of marginal totals. An example of a
null model web generated from R
k
of a plant–pollinator
network (Fig. 1) is displayed in Fig. 2b.
In addition to Patefield’s (1981) algorithm, two
alternative null models were used, in which the marginal
totals R
k
defined the probability of an interaction but
were not fixed. In the first alternative model, at least one
link was assigned to each species in order to maintain
the network size (as in Va
´zquez et al. 2007, but without
fixing connectance). This condition to maintain one link
per species was not implemented in the second
alternative null model, allowing the network to become
smaller when some species were assigned no interactions
and hence disappeared from the network (Fig. 3c).
NICO BLU
¨THGEN ET AL.3390 Ecology, Vol. 89, No. 12
FIG. 2. Four networks with the same distribution of total observation records (¼row and column totals) of 27 pollinator and
nine plant species, but increasing exclusiveness of interactions. Numbers and shading are as described for Fig. 1. (a) Hypothetical
network showing a perfectly nested distribution of interactions (H0
2¼0.0). (b) One network generated from randomized interactions
(H0
2¼0.03). (c) An empirical pollination network (as in Fig. 1; H0
2¼0.85). (d) A hypothetical network with the most exclusive
distribution of interactions possible for the same frequency distribution (H0
2¼1.0). Randomized interactions as in (b) yield an
average H0
2ran ¼0.016 60.007 (mean 6SD of 100 simulations).
FIG. 3. Networks (27 pollinator 39 plant species) displayed in their unweighted form, with links coded as present (black) or
absent (white). The empirical network (e), the extreme scenarios (a, f ), and a network based on Patefield’s (1981) algorithm (b) are
equivalent to the four webs shown in Fig. 2 and maintain the species observation records R
k
from (e). Networks derived from two
additional null models are shown: for (c), R
k
defined the probability of an interaction, but was not fixed; network (d) was generated
with the null model ‘‘CE’’ previously established for nestedness analysis based on the number of links. Note that (b) and (c) more
closely resemble the most nested scenario, whereas (d) strongly deviates from perfect nestedness based on R
k
. The original order of
rows and columns was maintained as in Fig. 2, except for (c) and (d), where some pollinator species and one plant species did not
receive a link and were rearranged.
December 2008 3391DETERMINANTS OF NETWORK METRICS
Because the three null models revealed very similar
results, only those generated by the Patefield (1981)
algorithm are presented in detail, and results based on
the least conservative third null model are shown in the
Appendix.
In order to manipulate the skewness of R
k
, we varied
the standard deviation (r) of a log normal distribution,
while maintaining the grand total number of observa-
tions (m). Both R
1
and R
2
were generated by drawing
series of Ior Jrandom numbers from a log normal
distribution, with l¼ln(M) – 0.5 r
2
, where lis the
mean, ris the standard deviation, and Mdescribes the
sampling density of the network (mean number of
observed interactions per cell), obtained from the grand
total number of interactions recorded (m)asM¼m/(IJ).
Each R
k
was the median distribution of 100 sorted
random drawings. Variation in rgenerated a different
heterogeneity of R
k
. We chose rvalues ranging from 0
to 10, and calculated laccordingly. In order to quantify
the heterogeneity of R
1
,E
R1
, we used the evenness index
based on Shannon diversity:
ER1 ¼X
I
i¼1
piln pi
ln I
where p
i
is the proportion of the interaction totals for
species i(A
i
) of all interactions in the community (m);
hence, p
i
¼A
i
/m. The heterogeneity of R
2
(E
R2
) was
defined accordingly (replace iby jand Iby J). Evenness
approaches 0 for the most heterogeneous distribution
and 1 for a perfectly homogenous distribution. R
1
and/or R
2
were often slightly readjusted in order to
allow the evenness for both parties to become as similar
as possible (E
R1
’E
R2
) and to ensure that R(R
1
)¼R
(R
2
)¼m.
We generated networks of a size and interaction
density typical for empirical studies. All networks shown
here thus contained I3J¼30 310 species and a total of
m¼600 interactions, equivalent to an average of two
interactions per cell, m/(IJ )¼2. These values for
network size and interaction density are similar to the
median found across 51 empirical mutualistic networks
(26 310 and 2.1, respectively; compiled in Blu
¨thgen et
al. 2007). Other network sizes and shapes, e.g., squared
networks (I¼J), were examined as well, but general
trends remained largely unaffected (see Appendix).
To calculate nestedness temperature (Table 1: Nested-
ness), we used the program Aninhado 2.0.2 (Guimara
˜es
and Guimara
˜es 2006), which complements the original
procedure by Atmar and Patterson (1993) with im-
proved null models and allows an efficient calculation of
a large number of matrices. We also compared our null
model simulations to the results of significant nestedness
inferred from a previously used null model based on
unweighted networks (null model ‘‘CE’’ in Aninhado
2.0.2, among other studies used in Bascompte et al.
[2003], Guimara
˜es et al. [2006, 2007]). This null model
PLATE 1. Many insect species are rare, rarely observed, or only occasionally forage on flowers such as this scorpion fly.
Conventional network metrics regard such flower visitors as ‘‘specialized’’—they are assigned only one or few links and high values
of interaction strength or dependence. They also contribute to low connectance and to a nested pattern of a network, if they interact
with commonly visited flowers, e.g., a hogweed shown here. Photo credit: Michael Werner.
NICO BLU
¨THGEN ET AL.3392 Ecology, Vol. 89, No. 12
assigns a link between each pair of species with a
probability deduced from the product of their total
number of links (species degree). This null model has
been considered conservative for nestedness analysis
compared to alternative unweighted null models (Bas-
compte et al. 2003, Guimara
˜es et al. 2006). A typical CE
null model web is shown in Fig. 3d.
We characterized the degree distribution (Jordano et
al. 2003) by fitting the observed distribution to two
models, the power law (PL), and the truncated power
law (TPL) (Table 1: Degree distribution). Because TPL
is an extension of PL and requires an additional
parameter, namely a constant defining the exponential
decay, the Akaike Information Criterion (AIC) was used
to evaluate which model provides the better fit. TPL was
considered superior to PL only if the difference in AIC
(i.e., DAIC) was greater than 2.
RESULTS AND DISCUSSION
Variation in natural networks
In order to maintain a constant sampling density M¼
m/(IJ) across different networks, the overall number of
observations (m) must increase proportionally to the size
of each network (IJ ). The slope is clearly lower for
empirical data sets. We analyzed 51 mutualistic plant–
animal networks, including pollinators, seed dispersers,
and ants from six continents (Blu
¨thgen et al. 2007):
many of them are available online.
5
All networks
comprised weighted links, represented by the number
of individuals of an animal species recorded on a
particular plant species. The networks differed strongly
in the number of species and observations (range 3–219
plant species, 4–679 animal species; 39 m19 946).
Whereas mincreases consistently with network size (IJ )
across the 51 networks (Spearman’s r
S
¼0.65, P,
0.0001), the increase is weak and much less than
proportional to IJ, since Mdeclines monotonically over
IJ (r
S
¼0.57, P,0.0001). Therefore, larger networks
typically contain fewer observations per potential link.
Total interaction frequencies per species in different
assemblages vary substantially in their heterogeneity,
causing a high variation in the skewness of marginal
totals (R
k
) in different networks. R
k
vectors of plants
and animals in the set of 51 mutualistic networks cover a
broad range of evenness values, ranging from 0.36 to
1.00 (mean 6SD: 0.75 60.13; evenness based on
Shannon diversity). Particularly homogenous R
k
vectors
were recorded for myrmecophytic plants (0.88 60.09, n
¼14 networks), while pollinators, on average, had the
lowest evenness (0.67 60.14, n¼21 networks), but
variation was high within each network type as well.
Given this broad variability of the skewness of R
k
,
network comparisons merit a closer examination of the
impact of R
k
alone.
Connectance
Connectance (C) is one of the simplest and most
commonly used metrics to describe the density of links
in interaction networks or food webs (Table 1), and is
usually interpreted as the degree of generalization or
redundancy in a system, with consequences for commu-
nity stability (May 1972, Dunne et al. 2002b, Estrada
2007). However, sampling intensity is crucial. In a
scenario where all species are completely generalized and
interactions are unlimited by trait matching, additional
sampling would increase the number of links between
these species and thus connectance, eventually leading to
a fully connected network. Therefore, in the absence of
sampling limitation, the expected connectance would be
C¼1. In our null-model analysis of generalized
associations based on limited sampling intensity, con-
nectance was much lower than 1 in networks with few
frequent and many rare species, and showed a substan-
tial decline with increasing skewness of the frequency
distribution (R
k
) (Fig. 4a). Therefore, variation in the
species sampling frequencies, or in the underlying
abundance distribution, additionally shapes Capart
from the previously documented effect of total species
numbers or total sampling effort (Goldwasser and
Roughgarden 1997, Olesen and Jordano 2002, Bana-
s
ˇek-Richter et al. 2004, Blu
¨thgen et al. 2006).
In addition, an increased information deficit in
larger networks may, at least partly, account for the
hyperbolic decline in Cover network size. If the same
absolute effort is taken to sample larger networks, the
number of observations per species will decline with
increasing number of possible links (IJ),andsowillC
(see Kenny and Loehle 1991). Alternative explanations
for this decay have been proposed, including an
increased specialization in larger networks, ‘‘forbidden
links,’’ or alternative trait-based models (Jordano et al.
2006, Santamarı
´a and Rodrı
´guez-Girone
´s 2007), but
should be evaluated against the contribution of
information deficit due to limited sampling, represent-
ing a more simple and parsimonious explanation. The
same effect applies to metrics such as generality,
vulnerability, or linkage density of unweighted links,
as they are directly related to Cfor a given network
size (Table 1).
Nestedness
Nestedness (Table 1) in ecological interaction net-
works is commonly interpreted as specialization asym-
metry: specialists (species with few links) interact with
generalists (species with many links). Consequently, a
nested pattern may suggest that reciprocal specializa-
tion, required for tight coevolution, is uncommon
(Bascompte et al. 2003, Va
´zquez and Aizen 2004,
Montoya et al. 2006). In most previous studies of
natural networks, the associations were described as
being highly nested, and significantly so in comparison
with null models (Bascompte et al. 2003, Guimara
˜es et
5
hhttp://www.nceas.ucsb.edu/interactionweb/i
December 2008 3393DETERMINANTS OF NETWORK METRICS
al. 2006, 2007, Ollerton et al. 2007). However, a
heterogeneous abundance distribution alone, which is
typical for community samples, generates a nested
pattern when partners are randomly associated (Fischer
and Lindenmayer 2002, Lewinsohn et al. 2006). Our
analysis confirmed this effect for simulated neutral
networks and additionally demonstrates that the even-
ness of observed species frequencies (R
k
)greatly
FIG. 4. Effect of the heterogeneity in species observation records (R
k
) on unweighted (a–d) and weighted (e–h) network metrics
(see Table 1 for definitions). Simulated networks contain 30 310 species that interact randomly in the most generalized way,
unconstrained by trait matching. Heterogeneity of R
k
is displayed as Shannon evenness. For a given R
k
, mean with 95%CI for 100
networks is shown. The dashed line in (b) separates the range of R
k
leading to significantly nested networks according to a null
model used elsewhere (‘‘CE’’ in Guimara
˜es and Guimara
˜es [2006]) in all cases (100%) vs. none (0%)(n¼100 networks for each R
k
).
(c) The fit (r
2
value) of the power law (black circles) and truncated power law (gray circles) degree distribution, which correspond to
(d) differences in Akaike Information Criterion (DAIC) between both distributions. For evenness values higher than the dashed
line, the truncated power law provides a sufficiently better fit (mean DAIC .2). The inset in panel (e) shows the pattern for a
squared network (20 320 species). In (g), generality is shown as a quantitative weighted (qw) mean for the party with 10 species.
NICO BLU
¨THGEN ET AL.3394 Ecology, Vol. 89, No. 12
decreased nestedness: matrix temperatures increase with
increasing evenness of R
k
(Fig. 4b). Although a potential
impact of the abundance distribution has been appreci-
ated earlier (Jordano 1987), appropriate controls have
been scarce, and nestedness has often been interpreted
directly in terms of specialization asymmetry.
As opposed to the original null models used in
community ecology and biogeography (Atmar and
Patterson 1993), most studies of nestedness in ecological
networks used null models that allowed a heterogeneous
distribution of the number of links among species and
were regarded as more conservative (Bascompte et al.
2003, Guimara
˜es et al. 2006, 2007, Ollerton et al. 2007).
Nestedness in many empirical networks was reported to
be significantly greater than expected by those earlier
null models. However, most of the networks generated
by our null model also lead to ‘‘significant nestedness’’ in
terms of previous null models, except randomized
associations with a nearly perfectly even R
k
(Fig. 4b).
This may be largely due to conceptual differences in the
null model algorithm (Fig. 3). Null models to evaluate
nestedness in ecological networks have so far been based
on the species degrees (unweighted links) rather than
frequency totals (but see Va
´zquez and Aizen 2003, Stang
et al. 2007); such marginal totals of species degrees are
typically less uneven than R
k
.
Therefore, nestedness is a common pattern of
empirical networks, but also of simulated networks
based on completely generalized interactions, where R
k
shapes the likelihood of interactions irrespective of trait
matching (Fig. 3). If partners associate randomly,
species are more likely to interact with common than
with rare partners. Singletons or rarely observed species
will be recorded interacting with common species,
producing a nested pattern. This should be considered
when comparing empirical data and interpreting nested-
ness as a biologically meaningful trait. For instance,
Santamarı
´a and Rodrı
´guez-Girone
´s (2007) used nested-
ness in order to test various trait models in plant–
pollinator interactions. These authors admit that null
models with random interactions alone explain the
variation in both nestedness and connectance over
network size when the underlying frequency distribution
was assumed to be log-normal, but downplay the value
of null models in general to describe plant–pollinator
interactions realistically. However, we suggest that if
sampling limitation under neutral conditions alone
explains a pattern, a more complex trait model is not
required as alternative explanation (parsimony princi-
ple). An insightful comparison between the explanatory
powers of morphological trait models vs. neutral models
in plant–pollinator networks has recently been per-
formed by Stang et al. (2007).
Degree distribution
In many ecological networks sampled so far, few
species have many links, while many species have only
few links, although this pattern may vary across
different networks and food webs. The resulting degree
distribution often decays more strongly than in ‘‘scale-
free’’ networks and can be better described by a
truncated power law function (Table 1). This particular
topology characterizes ecological networks in relation to
some other types of networks (Dunne et al. 2002a,
Jordano et al. 2003, Montoya et al. 2006). Ecological
traits of interacting partners have been suggested to
shape this particular degree distribution, such as the
non-matching of species attributes that may prevent
interactions between species (‘‘forbidden links’’; Jordano
et al. 2003). However, networks with completely
generalized interactions can also be very well character-
ized by truncated power law functions when observed
total frequencies of different species are sufficiently
uneven (often with r
2
’1.0) (Fig. 4c). This corresponds
to an earlier notion that the degree distribution can be
generated by a random interaction model (Va
´zquez
2005). The heterogeneity of R
k
has a strong influence on
the fit of the power law distribution, and the truncated
power law model provides a sufficiently better fit
(considering AIC) over a large range of R
k
except for
networks with a very low evenness (Fig. 4d).
Interaction strength
Quantitative interaction strength in interaction net-
works (b
ij
) is often defined as the proportion of the
interactions iwith a specific partner jof the total
interactions of this species i(Table 1) (‘‘dependence’’
sensu Jordano 1987, Bascompte et al. 2006). Thus, when
a plant is visited solely by a single pollinator species, its
interaction strength with that pollinator is assumed to be
1.0, while lower values indicate that this pollinator
contributes only a smaller fraction of the visits.
Asymmetric interaction strengths have been interpreted
as a quantitative cousin of qualitative specialization
asymmetries and nestedness: when a pollinator is highly
dependent on a plant, this plant often exhibits low
dependence on the pollinator in turn, and this asym-
metric dependence may enhance community stability
(Bascompte et al. 2006; but see comment by Holland et
al. [2006]). However, the same argument applies as for
nestedness above: when assuming neutral encounters
among interacting species suffices to reproduce observed
asymmetric interaction strengths (based on R
k
), it is not
imperative that trait matching is generating such
asymmetry.
Indeed, simulated random networks based on a
heterogenous distribution of interaction totals (low
evenness of R
k
) usually produce high asymmetries in
interaction strength between species. With an increas-
ingly homogenous abundance distribution, average
asymmetries decline (Fig. 4e). For squared randomized
networks (I¼J), a uniform interaction frequency
(perfectly even R
k
) leads to completely symmetric
interaction strengths (see inlet in Fig. 4e). An increased
imbalance in the number of associated partner species (I
December 2008 3395DETERMINANTS OF NETWORK METRICS
6¼ J, network asymmetry) generates higher interaction
strength asymmetries even for uniform abundances (Fig.
4e). This confirms that both sampling intensity and
network asymmetry (Blu
¨thgen et al. 2007) shape the
average interaction strength asymmetry for neutral
interactions without any effect of trait matching.
Correspondingly, Va
´zquez et al. (2007) showed that
the predominance of interaction strength asymmetries in
many mutualistic and antagonistic networks may reflect
their underlying abundance distribution.
Interaction strengths are highly constrained for
infrequently recorded species. Most notably, an inter-
action by a species with a single observation will
automatically be assigned an interaction strength of
1.0, two observation records will either lead to a value of
1.0 if they occur on the same link or 0.5 for each of two
links, and so forth. Thus, rarely observed species will
have, on average, higher interaction strengths than
frequently sampled species. In a given network, even the
majority of high interaction strengths assigned (e.g.,
values 0.5) may involve species with only one or two
total observation records (including those networks
analyzed by Bascompte et al. 2006). However, because
high values of interaction strength b
ij
often correspond
to poorly sampled and infrequent species, these may not
reflect particularly ‘‘strong’’ interactions as implied by
Bascompte et al. (2006). If there is an inverse relation-
ship between sampling intensity and interaction
strength, and if rarely observed species typically
associate with commonly observed partners, this com-
bination of features may produce interaction strength
asymmetry in networks. In order to account for this bias
in interaction strength for low number of observations
in species i(n
i
), interaction strength could be standard-
ized for n
i
.1as
b0
ij ¼ðbij bminÞ=ð1bmin Þ
where b
min
¼1/n
i
; excluding all cases where n
i
¼1. Such
standardized b0
ij vary between 0 and 1 irrespective of n
i
and may arguably underestimate the real interaction
strength, but may provide a useful way to test for effects
of potential bias due to the information deficit under
neutral conditions. Incorporating the modified equation
for b0
ij and the condition n
i
.1 in our simulation of
randomized networks often leads to more moderate
values of interaction strength asymmetries, but does not
fully compensate for their dependence on R
k
evenness
(see Appendix). Alternatively, interaction strengths may
be weighted according to the number of observations, as
the credibility given to interaction strengths calculated
for each species increases with n
i
.
Interaction diversity, evenness, and generality
Bersier et al. (2002) suggested several metrics based on
Shannon diversity (Table 1) to describe the heterogene-
ity in mass flows in food webs, which can be transferred
to characterize the diversity of associations based on
interaction frequencies. Correspondingly, other authors
suggested Simpson’s diversity metric to characterize
generalization in plant–pollinator networks (Sahli and
Conner 2006). In an extensive study of bee–parasite
associations, Tylianakis et al. (2007) showed that
Shannon diversity and evenness (i.e., Shannon diversi-
ty/log species richness) of interactions decreased as a
function of habitat disturbance. This pattern depicts the
heterogeneity of the interactions between trophic levels,
but not necessarily changes in specialization. Not
surprisingly, in our simulation interaction evenness
(and consequently interaction diversity) is almost
perfectly correlated with the evenness of the total
observation records (R
k
) (Fig. 4f ). Consequently, a
pattern reported for empirical networks (e.g., Tylianakis
et al. 2007) may be driven by a decrease in evenness of
the host or resource abundances alone, even when
preferences of consumers or parasites are unchanged.
The impact of R
k
is equally important for other metrics
derived from diversity measures such as generality or
vulnerability (Fig. 4g).
In order to characterize specialization, Hurlbert
(1978) emphasized in the analogous context of niche
breadth that appropriate measures should not only take
the distribution of consumers into account, but also
control for variation in resource availability. This
concept is implemented in the following metrics (H0
2
and d0
i) that are based on the Shannon diversity of
interactions as well, but are standardized against a null
expectation.
Quantitative metrics that control for R
k
The standardized two-dimensional Shannon diversity
H0
2and the related standardized Kullback-Leibler
distance d0
idescribe the degree of mutual specialization
among two parties in the entire network and for each
species, respectively (Table 1; Blu
¨thgen et al. 2006).
These metrics calculate the deviation of realized
associations from the null expectation based on the
frequency distribution of interaction totals R
k
. Conse-
quently, in randomized associations H0
2and d0
iremain
near zero and are largely unaffected by the evenness of
the observation records (Fig. 4f; Blu
¨thgen et al. 2006),
and these metrics are robust against variable species
frequencies or biased sampling intensity. Therefore, any
deviation from this null expectation may be driven by
processes at the interspecific level such as trait matching.
For instance, a higher number of interactions recorded
for a certain link than predicted by the null model may
suggest a preference of the animal species for this
resource, whereas a lower frequency may be indicative of
a morphological barrier, defenses, or other inhibitory
processes (Fig. 1). Without reference to the null model,
however, it may be difficult to distinguish whether
unobserved or underrepresented links are due to limited
sampling or due to biologically meaningful restrictions
(compare scenarios b and c in Fig. 1). In other words,
failure to record a link between two rarely observed
partners may not be surprising, even when an obvious
NICO BLU
¨THGEN ET AL.3396 Ecology, Vol. 89, No. 12
morphological barrier is lacking, whereas an unobserved
link between a common consumer and a common
resource more vigorously calls for a biological interpre-
tation.
The null model relates to the concept of nestedness in
a quantitative way, implying that each species opportu-
nistically visits its partners in the same proportions as
the other species do, so that all species conform to the
same overall R
k
. Consequently, interactions of each
species are a subset of the interactions of more
frequently observed species. A quantitatively ‘‘nested’’
network based on weighted links looks like the one
displayed in Fig. 2a. For a perfectly nested matrix, H0
2¼
0. Therefore,generalized interactions generate a pattern
of quantitative nestedness. The pattern opposite to
nestedness would be a scenario with strongly symmetric
(reciprocal) specialization between partners, i.e., highly
mutually exclusive interactions. This corresponds to an
interaction matrix in which mainly the diagonal
elements are realized when species are sorted by their
frequency (Fig. 2d), a scenario that yields H0
2¼1.
Consequently, H0
2is an index that describes specializa-
tion in terms of exclusiveness (or reciprocal specializa-
tion), where H0
2¼0 represents the most nested scenario
possible given the overall frequency distribution, and H0
2
¼1 the highest possible deviation from this nested
assemblage. In contrast, note that the concept of ‘‘anti-
nestedness’’ in unweighted networks is controversial and
has been used for several distinct network characteristics
(Almeida-Neto et al. 2007). Conventional nestedness
analyses based on unweighted links identify species that
particularly deviate from the expectations based on the
most nested pattern, shown by high ‘‘idiosyncratic
temperatures’’ (Atmar and Patterson 1993). Such
idiosyncratic temperatures in unweighted networks have
their counterpart in high d0
ivalues in networks with
weighted links. For species that exclusively interact with
their partners (i.e., reciprocal specialization) and have
thus nothing in common with their putative competitors,
d0
iapproaches 1. Species with high d0
iare, e.g.,
bellflowers (Campanula spp.) visited by a particularly
unconventional spectrum of bees (J. Fru
¨nd, unpublished
data), suggesting a symmetric mutual dependence of the
partners. These cases of reciprocal specialization may
represent targets that deserve particular attention for
conservation. In contrast, low d0
imay suggest a higher
ecological redundancy with members of the same guild.
Most mutualistic interaction networks show a higher
exclusiveness (H0
2) than expected by random interactions
based on the marginal totals (R
k
), thus they are
significantly less nested quantitatively than expected by
the null model (Blu
¨thgen et al. 2007). This significant
exclusiveness is also apparent for antagonistic networks.
For instance, in fish–parasite associations (seven net-
works from Va
´zquez et al. [2005]), H0
2ranged between
0.34 and 0.74, and all networks were significantly more
exclusive than expected by the null model (all P,
0.001). Therefore, empirical networks as well as ran-
domized associations with a skewed R
k
may appear
nested when analyzed as presence–absence matrices
(Fig. 3), whereas our analysis based on weighted links
reveals that empirical networks are usually not nested
and are even more exclusive than expected by chance
(Fig. 2).
Differences in the degree of specialization (H0
2) among
different network types may suggest structural differ-
ences in phenotypic matching of partners, morpholog-
ical or spatiotemporal constraints, and active behavioral
choices triggered by species preferences and avoidance.
High specialization also suggests a higher potential of
coevolutionary processes to shape interaction networks
than implied by qualitative nestedness (Thompson
2005). Antagonistic associations, e.g., host–parasite or
plant–herbivore systems, are strongly driven by host
defenses and subsequent consumer offenses, and this
evolutionary arms race may promote a high exclusive-
ness of interactions due to strong limitations and
barriers at the level of trait matching. Among mutual-
istic systems, benefits from specialization may differ. For
instance, the relatively high exclusiveness in pollinator
networks may reflect the plants’ interest in reliable
pollen vectors that transfer pollen to conspecific flowers,
compared to low H0
2in seed dispersal networks
(Blu
¨thgen et al. 2007).
In terms of community stability, a smaller overlap
between species of the same guild in terms of interaction
frequencies suggests a lower redundancy of species, and
thus a higher potential of negative effects of a
population decline of a host or resource species on its
associated consumer species. In addition, high levels of
exclusiveness in mutualistic networks may indicate a
pronounced reciprocal dependence between interacting
partners.
CONCLUSIONS
In this study we simulated associations at the highest
possible level of generalization, i.e., no limitation by
trait matching. In this null model scenario, every species
ican associate with every potential partner j, only
constrained by the total observations per species (R
k
vectors). Failure to observe a link between iand j,ora
relatively low interaction strength between iand j,is
therefore only an effect of insufficient sampling, not of
trait matching. In contrast to the expected maximum
degree of generalization in all simulations, most metrics
are fundamentally affected by the evenness of R
k
alone,
without intentionally changing the generalized character
of associations between partners. The impact of R
k
is
severe for both unweighted and weighted interaction
network metrics, and represents a realistic constraint in
empirical network analysis, considering the typical
sampling limitation in ecological data, and given that
many species in ecological samples are rare or rarely
observed (see Plate 1).
Conventional network metrics render all links equally
likely, irrespective of whether the absence or weakness of
December 2008 3397DETERMINANTS OF NETWORK METRICS
a link results from mismatch of traits or is solely due to
limited information. Therefore, any assessment of the
determinants structuring networks must consider to
what extent variation in number and distribution of
observed interactions may be solely enforced by limited
sampling intensity. The standardized indices H0
2and d0
i
(Blu
¨thgen et al. 2006) represent a solution to compare
networks directly by controlling for variable observation
records and abundances under neural conditions, and
are thus more conclusive about different structural
forces at the level of trait matching. The comparison
with the expectation by the neutral model facilitates an
interpretation of the presence, absence, or strength of a
link, particularly when external data about ecological
barriers (e.g., as in Stang et al. 2007) are lacking. An
absence of a link in a real network that was also
unexpected by the respective null model may represent
an information deficit (Fig. 1c). In contrast, other gaps
in the realized associations that are expected to occur
frequently under neutral conditions may suggest an
important underlying mechanism such as a morpholog-
ical or physiological barrier or phenological mismatch
(Fig. 1b). Without reference to the null model, all links
or interaction strengths are weighted equally.
The need to control for R
k
to evaluate trait matching
does not imply that uncontrolled metrics are meaning-
less. For instance, in order to describe energy or mass
flows between trophic levels in food webs (Bersier et al.
2002) or heterogeneity of interactions (Tylianakis et al.
2007), a measurement of the diversity of links (e.g.,
unstandardized Shannon diversity) may be useful. This
is also true if different impacts of consumers on their
host are to be evaluated and increase with their overall
frequency (e.g., pollination, predation, or herbivory)
(e.g., Sahli and Conner 2006). Asymmetric interaction
strengths may describe an important pattern in realized
associations if the recorded frequencies are representa-
tive for the species’ overall activity. Depending on the
scope of the analysis, variation in R
k
may represent a
structural property or emergent feature not to be
deducted from the overall pattern. However, the
uncontrolled metrics are problematic in their interpre-
tation of fundamental specialization among associated
species, i.e., a direct conclusion about processes that act
at the interguild level of trait matching. For each area of
ecological analysis it is therefore crucial to choose
appropriate metrics in order to avoid flawed or
misleading interpretation of network data. Use of
appropriate standardizations and/or comparisons with
null models is vital to understand and correctly interpret
the structuring elements of a pattern while removing
effects of limited or biased sampling. Rarefaction
techniques additionally help to unravel the contribution
of sampling intensity for quantifying specialization and
network patterns (Gotelli and Colwell 2001, Herrera
2005). Most importantly, the frequency distribution of
the species observations (R
k
) itself and its underlying
determinants such as the abundance distribution
(McGill et al. 2007), merits a more intensive focus in
future investigations.
ACKNOWLEDGMENTS
We thank Carsten Dormann, Thomas Hovestadt, Robert
Junker, and an anonymous referee for helpful comments that
greatly improved the clarity of this manuscript.
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APPENDIX
Effect of the heterogeneity in species observation records on network metrics for an alternative parameter setting (Ecological
Archives E089-194-A1).
December 2008 3399DETERMINANTS OF NETWORK METRICS
http://esapubs.org/Archive/ecol/E089/194/appendix-A.htm
Ecological Archives E089-194-A1
Nico Blüthgen, Jochen Fründ, Diego P. Vázquez, and Florian Menzel. 2008. What do interact ion network metrics tell us about specialization and
biological traits? Ecology 89:3387–3399.
Appendix A. Effect of the heterogeneity in species observation records on network metrics alternative parameter setting.
Here we provide the results of the simulation for a different parameter setting (Fig. A1). Generated networks contained I×J = 20×20 species and a total of m = 400 interactions, equivalent to an
average of one interaction per cell (m / [IJ] = 1). The most liberal alternative null model was used, where the total observation frequencies Rk defined the probability of an interaction but was
not fixed unlike in the Patefield algorithm, and the condition to maintain one link per species was not implemented, partly resulting in a reduced number of species.
FIG. A1. Effect of the heterogeneity in species observation records (Rk) on unweighted (a–d) and weighted (e–h) network
metrics (mean ± 95% C.I. for 10 networks). Heterogeneity of Rk displayed as Shannon evenness. Dashed line in (b)
separates the range of Rk leading to significantly nested networks according to null model ‘CE’ in all cases (100%) versus a
lower proportion (89%) (n = 100 networks for each Rk). In (c), the fit (r2 value) of the power law (black circles) and
truncated power law (gray circles) degree distribution shown, which correspond to differences in ΔAIC between both
distributions (d). For evenness values higher than the dashed line, the truncated power law provides a sufficiently better fit
(mean ΔAIC > 2). Inlet in (e) shows the pattern for networks based on the Patefield algorithm and a modified equation for
interaction strength, bij’ = (bij – bmin) / (1 – bmin), where bmin = 1/ni; all cases where ni = 1 have been excluded. For the
other null models, bij’ showed a similar decrease over Rk evenness as the unstandardized bij.
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