royalsocietypublishing.org/journal/rstb
Research
Cite this article: Niquil N, Haraldsson M,
Sime-Ngando T, Huneman P, Borrett SR. 2020
Shifting levels of ecological network
’s analysis
reveals different system properties. Phil.
Trans. R. Soc. B 375: 20190326.
http://dx.doi.org/10.1098/rstb.2019.0326
Accepted: 25 October 2019
One contribution of 11 to a theme issue
‘Unifying the essential concepts of biological
networks: biological insights and philosophical
foundations
’.
Subject Areas:
ecology
Keywords:
networks, food web, hierarchy, levels,
interdisciplinary
Author for correspondence:
Nathalie Niquil
e-mail: nathalie.niquil@unicaen.fr
Electronic supplementary material is available
online at https://doi.org/10.6084/m9.figshare.
c.4824033.
Shifting levels of ecological network
’s
analysis reveals different system properties
Nathalie Niquil
1, Matilda Haraldsson
1,2,3, Télesphore Sime-Ngando
4,
Philippe Huneman
5and Stuart R. Borrett
6,71CNRS/Normandie Université, Research Unit BOREA (Biology of Aquatic Organisms and Ecosystems), MNHN,
CNRS 7208, IRD 207, Sorbonne Université, Université de Caen Normandie, Université des Antilles, team EcoFunc, CS 14032, 14000 Caen, France
2Department of Marine Sciences, University of Gothenburg, Box 461, 405 30 Göteborg, Sweden
3Sorbonne Université, Université Paris Est Créteil, Université Paris Diderot, CNRS, INRA, IRD, Institute of Ecology
and Environmental Sciences-Paris, IEES-Paris, 75005 Paris, France
4LMGE, Laboratoire Microorganismes: Génome et Environnement, Université Clermont Auvergne, UMR CNRS
6023, Aubière, France
5Institut d’Histoire et de Philosophie des Sciences et des Techniques, CNRS/Université Paris I Sorbonne,
13 rue du Four, 75 006 Paris, France
6University of North Carolina, Wilmington, Wilmington, NC 28403, USA
7Duke Network Analysis Center, Duke University, Durham, NC 27708, USA
NN, 0000-0002-0772-754X; PH, 0000-0002-7789-6197
Network analyses applied to models of complex systems generally contain at least three levels of analyses. Whole-network metrics summarize general organizational features ( properties or relationships) of the entire network, while node-level metrics summarize similar organization features but consider individual nodes. The network- and node-level metrics build upon the primary pairwise relationships in the model. As with many ana-lyses, sometimes there are interesting differences at one level that disappear in the summary at another level of analysis. We illustrate this phenomenon with ecosystem network models, where nodes are trophic compartments and pairwise relationships are flows of organic carbon, such as when a predator eats a prey. For this demonstration, we analysed a time-series of 16 models of a lake planktonic food web that describes carbon exchanges within an autumn cyanobacteria bloom and compared the ecological conclusions drawn from the three levels of analysis based on inter-time-step comparisons. A general pattern in our analyses was that the closer the levels are in hierarchy (node versus network, or flow versus node level), the more they tend to align in their conclusions. Our analyses suggest that selecting the appropriate level of analysis, and above all regularly using multiple levels, may be a critical analytical decision.
This article is part of the theme issue‘Unifying the essential concepts of
biological networks: biological insights and philosophical foundations’.
1. Introduction
Networks are a type of relational model used by scientists of numerous disci-plines to address a variety of innovative questions. One useful feature of network models is that they can be analysed at multiple nested hierarchical levels [1,2]. Here, we focus on this hierarchy in ecological networks, based on three levels of analysis. The ecological networks we consider are composed of individuals, species, or functional groups of species, represented by nodes, which are linked by interactions, such as competition, mutualism, facilitation and predation. We further view the ecological networks from three levels of
perspectives, the flow (pairwise interaction), node and whole-network levels
(figure 1). Recent work in ecology has applied a multi-layered network approach [3,4] that acknowledges multiple layers of interactions with different properties that coexist and co-create the observed multiplexes [5], i.e. the net-work made up by multiple layers of different nature of interaction. However,
© 2020 The Author(s) Published by the Royal Society. All rights reserved.
because interactions are difficult to quantify in the field, most studies focus on only one type of interaction at a time. Community and ecosystem ecologists often focus on preda-tor–prey interactions that define a food web or trophic network. To understand how food webs function in terms of processes or attributes that contribute to self-maintenance (e.g. energy flows, nutrient cycling, level of omnivory, etc.) [6], a field of research has arisen to describe the functioning of trophic networks and the emergent properties linked with species interactions (e.g. [7–9]). The domain is called ecological network analysis (ENA) and is used to answer a
wide range of questions like‘What is the impact of fishing
on the marine food web? Which species control the flux of nitrogen in an estuary? What is the ecological relationship among species in the food web when direct and indirect influences are considered? Would a proposed regulation make a city more sustainable?’ [10]. Applications of ENA have been used to identify properties of ecosystem function-ing (e.g. [11]), definfunction-ing typologies of functionfunction-ing (e.g. [12]), identification of key components (e.g. [13]), before–after ana-lyses of sudden events [14], stress characterization [15] and definition of indicators of ecosystem health [16,17]. In many of these applications, scientists compare ENA results between models, typically representing different systems, locations or time points.
Hierarchical notions in ecological systems are tightly linked with that of emergence, which is used to describe properties of ensembles that cannot be either explained by, or reduced to, or deduced from the properties of their com-ponents [18]. Here, is how Mayr defines emergence [18, p.
63]: ‘Systems almost always have the peculiarity that the
characteristics of the whole cannot (not even in theory) be deduced from the most complete knowledge of’ the com-ponents, taken separately or in other partial combinations. This appearance of new characteristics in wholes has been designated as emergence’.
Most generally considered, properties or items of a given
level are emergent if and only if they are bothdependent upon
properties of another level, and somehowautonomous from
those properties [19]. Thus, Bedau [20, p. 445] states that ‘Emergence always involves a certain kind of relationship between global or macro-phenomena and local or
micro-phenomena. Specifically, emergent macro-phenomena
somehow both depend on, and are autonomous from, micro-phenomena. Dependence and autonomy can be given
different interpretations, and different interpretations lead to different conceptions of emergence, including different conceptions of weak emergence’. Whether ‘autonomy’ and ‘dependence’ are to be understood in epistemic or ontological terms, and to what extent causation comes into play, and according to which concept of causation—are questions debated by metaphysicians and philosophers of science who theorize emergence (e.g. [21]). Yet for us here, this gen-eral characterization used in a pragmatic way suffices to say that some properties of ecological networks are likely to be emergent.
Ecological interaction networks can generate emergent system properties. Given the characterization of emergence, a typical example of an emergent property in food webs is recycling: it can be quantified by the percentage of energy or matter flowing within circular pathways. Thus, it depends upon those flows; yet is not always practically computable on the basis of those flows (as seen below); hence, it is auton-omous [22]. This emergent property is most often reported
at thewhole-network level (figure 1), which is referred to as
the Finn Cycling Index. Recycling can also be expressed at the node level (figure 1), which is the percentage of energy or matter that leaves a given compartment that then returns to the same compartment [22,23]. In addition, we can con-sider particular flows that are strongly related to recycling,
i.e. the flow level (figure 1)—for example, flows that close
these circular pathways. In a food-web tracing the flow of organic carbon, this could be flows linked with the pro-duction or the consumption of non-living organic carbon (e.g. particulate detritus or dissolved organic carbon). These three levels of analysis (figure 1) can be considered for most of the properties studied with ENA. Given these three common hierarchical levels of analysis, it is helpful for prac-titioners to consider the degree of alignment among the three levels for any particular analysis, and to what degree the analytical results at one level predict another.
Hines & Borrett [24] built a conceptual framework to organize ecological network analysis at multiple levels. Here, we applied this framework to 16 time-series models of a plank-tonic food web [25] to test the hypothesis that the information of selected ecological properties would be different across the three levels of the hierarchy. To test this hypothesis, we con-structed trinomials of processes assessed at the three levels (figure 1), for three possibly emergent properties:
(i) the recycling property (which, at the whole-network level, is both dependent upon the flows and possibly autonomous from them, hence emergent);
(ii) the property of being at a trophic level (TL),1which is
the position a species (individual or functional group) occupies in the food web (the primary producers ( plants) being on the lowest TL, and top-predators being on highest levels); and
(iii) the property of omnivory, which occurs when species (individuals or functional groups) feed on several food sources and on different TLs (see below), which increases the complexity of a network.
For each trinomial, based on the results of statistical tests comparing all time steps two-by-two for each level separately, we addressed the question whether the conclusions from these tests were the same for each hierarchical level of analyses. node
level
flow level network level
Figure 1. Illustration of the three hierarchical levels within a simple ecological
network composed of five nodes connected by eight directed pairwise
inter-actions or flows of organic carbon. The three hierarchical levels studied are
the flow level, the node level and the network level. (Online version in colour.)
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22. Material and methods
Our analyses of trinomials were based on a series of analyses (figure 2). Firstly, based on field measurements we estimated the flows of carbon in the 16 food-web models using linear inverse modelling using a Monte Carlo Markov chain approach (LIM-MCMC; §2a). Then, based on these flows, we calculated selected ENA indices for the network and node levels (recycling, TL and omnivory), which were part of our trinomials (§2b). The particular elements of the flow level were selected because the flows are relevant to the ENA index in question (defined in §2c). In the last step, we calculated the pairwise statistical com-parison between all possible time-steps (models) and for the three levels, which we illustrated in biplots for our three tri-nomials (§2c).
(a) Linear inverse modelling-Monte Carlo Markov chain
models and flow estimations (flow level)
Our analyses were based on 16 models describing the dynamics among organisms living in the water column composing the plankton, in Lake Aydat, France, during an autumn event called a bloom of cyanobacteria [25]. Cyanobacteria is a photosynthesiz-ing bacteria typically thrivphotosynthesiz-ing in nutrient-rich water where they can grow into huge quantities in a short time, often referred to as‘blooms’. These models were based on an extensive dataset cov-ering a complete period of bloom, from the pre-bloom to the post-bloom period, covering a two-month period sampled every 3 days. This dataset allowed the description of 17 time-steps (of which 16 were used here). From these models, LIM was used to quantify values for all flows. The models were built based on the samea priori model (as it is named in the technical literature) or model scheme, i.e. the compartments (17 compartments) and the possible
flows (83 flows) are the same (figure 3). These models captured the description of the dynamics of the bloom similar to a stop-motion animation flipbook.
An LIM is a topological food-web model (e.g. figure 3) described as a linear function of the flows, where the model par-ameters are derived from empirical data [26]. In order to estimate the unknown flows (vector x), they obey given constraints, which, in the LIM-MCMC are linear equations (for mass balance) and linear inequalities, respectively:
Ax ¼ b and Gx h,
where matricesA and G are the linear equations, and the vectors b andh are the thresholds considering processes and stocks from the field or defined from the literature. We estimated a sample of 200 000 flow values using the Xsample() function in the LIM library (forming the probability density function, pdf), which is based on the R limSolve library [27]. In this study, we used the resulting flow values based on mass balanced models because some ENA indices depend on the assumption of equal in- and out-flows of each compartment (steady-state), compared to Haraldssonet al. [25] whom assumed biomass accumulation for the two major phytoplankton compartments. The mean values and 5–95% credibility intervals (given in mgC m−2d−1) based
on the 200 000 solutions used in this study are presented in the electronic supplementary material, table SA1. Time-step 16 was not included in this study because the constraints were not compa-tible with a mass balanced model. The flow levels analysed in his study are specific flows directly taken from the LIM results, relevant to the emergent properties in our three trinomials.
(b) Ecological network analysis at the node and
network levels
From the results of the 16 models obtained from the LIM calcu-lation, each one composed of 200 000 vectors of 83 flows, we ran the calculation of the ENA at the node and network level. The format of the flow values from the LIM output was adapted for the use of the enaR library [28] using a script by Tecchio (electronic supplementary material, SA2).
(i) Recycling
Node-level recycling was characterized using the matrix of the coefficients of contribution (also known as the output-oriented integral flow intensity matrix,N ¼ ½nij,ði,j ¼ 1,2, ,vÞ, where v
is the number of compartments). The elements of this matrix indicate the flow intensity from one compartment to another over direct and indirect pathways as generated by the system boundary inputs. The diagonal elements express the total amount of carbon flowing through the node (Ti) that exits and
follow a cyclic pathway within the system, and then comes back to the initial compartment considered. Thus, we find the amount of nodei throughflow generated by cycling as
TiCycled¼ðNii 1Þ
Nii Ti:
This node-level property is emergent because it is generated by the energy or matter flowing (flow level) through the entire network, and is thus dependent on the whole-system structure; at the same time, changes in flow values may leave it unchanged, thus it is autonomous from the flow level. ENA scientists most often consider the Finn cycling index (FCI) [29], which is a time-step LIM MCMC models, with all
flows quantified (pdf) flow level cyanobacteria abundance cyanobacteria (10 6 cells l –1) infected cells (%) 30 8 4 0 10 0 infected cells
calculation of ENA indices on the different trophic compartments
node level
calculation of whole-system ENA indices
network level
Figure 2. Methods used in the present article, graphically summarized. The top
panel shows the abundance of cyanobacteria Dolichospermum macrosporum,
during the two months sampled every 3 days, and the prevalence of infected
cyanobacteria cells by parasites (chytrids). An extensive dataset including these
data was used to estimate the individual carbon flows using the linear inverse
modelling-Monte Carlo Markov chain method (LIM-MCMC). These flows were in
turn used to calculate ecological network analysis (ENA) indices on the
node-and network level. pdf, probability density function. (Online version in colour.)
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3network-level summation of the node cycling that is calculated as FCI¼ Pv i¼1ðððnii 1ÞÞ=niiÞ Ti Pv i¼1Ti ¼ Pv i¼1T Cycled i Pv i¼1Ti :
Again, FCI is the proportion of energy or matter (e.g. organic carbon) exchange that is created by cycling. While the network metrics is analytically a summation of the node-level metrics, considering recycling at the node and at the network levels can provide different insights into the system functioning [23].
(ii) Trophic level
Historically, TLs were considered as integer values, with one for autotrophs, two for herbivores, three for the carnivores eating them and so on [30]. As ecologists shifted their emphasis from simple food chains to more reticulated food webs, they began to estimate non-integer TLs [31]. The most common algorithm defines the TL as a weighted mean of the prey TLs plus one. This defines the TL characterization at the node level, and the property again emerges from the flow-level structure. Then, a whole-network level TL2is often considered as the mean trophic level (MTL): MTL¼ P iPTLi Bi iBi ,
where TLiis the TL of consumer compartments (node trophic
level greater than 2) and Bi its biomass. It is the weighted
mean of the TL of all consumer compartments.
(iii) Omnivory
Omnivory occurs when a species feeds on multiple TLs. ENA often quantifies an omnivory index (OI) for each node as the variance of the TL of the prey:
OIi¼
Xn j¼1
ð½TLj ðTLi 1Þ2 DCijÞ,
where TLjis the TL of compartmentj and DCijis the diet composition
(fraction) of preyj in the diet of i. Then the network-level index called the system omnivory index (SOI) was defined as the weighted log mean of omnivory of the consumers of the system [32,33]:
SOI¼ Pn
i¼1P[OIi log (Qi)] n
i¼1log(Qi)
,
whereQiis the consumption of compartmenti. As with cycling and
TL, the network-level indicator for omnivory is an analytic summary of the node-level metric.
The enaR R package was used to calculate the recycling and trophic-level metrics, while the NetIndices R package was used to determine omnivory.
(c) Time-step comparisons for the hierarchy analysis
Because of the high number of samples obtained from the LIM-MCMC (200 000 for each time-step model) and their non-independence, it is not possible to compare time-step models with a t-test. Instead, we used the non-parametric effect size COP
external compartments: GPP—gross primary production RES—respiration
LOS—loss
non-living compartments: DOC—dissolved organic carbon DET—detritus living compartments: PH1—picophytoplankton PH2—nanophytoplankton PH3—microphytoplankton BAC—bacteria VIR—virus SPG—chytrid sporangia ZSP—chytrid zoospore HNF—heterotrophic nanoflagellates CIL—ciliates ROT—rotifer CLA—cladocera COP—copepod CLA ROT CIL HNF VIR BAC DOC PH1 PH2 PH3 SPG ZSP DET LOS RES GPP
Figure 3. A priori food-web model (model scheme) of Lake Aydat
’s food web. Respiration flows (RES, not indicated) are present in all living compartments except
viruses. The dotted lines are flows of detritus to the DET or DOC compartment. Chytrid sporangia are parasites of microphytoplankton (PH3), and chytrid zoospores
an asexual reproductive stage. The figure is modified from Haraldsson et al. [25]. (Online version in colour.)
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4statistic Cliff’s δ introduced by Cliff [34] (J. Lequesne and V. Girardin 2015, personal communication [14]), to test the differ-ence between all unique pairwise comparisons between time-steps (n = 120). Cliff’s δ estimates the probability that a randomly selected value in one sample is higher than a randomly selected value in the second sample minus the reverse probability. Invert-ing the two samples changes the sign of the statistic. As in a t-test, the null hypothesis is an absence of difference of the studied variable between the two time steps and the alternative hypoth-esis the fact that the difference is significant. This test is not associated with a probability of rejection of the null hypothesis, but it uses threshold values for |δ| to indicate the magnitude of difference [35]: negligible for less than 0.147, small for less than 0.33, medium for less than 0.474 and otherwise large. We calculated Cliff’s δ using the effsize library in R.
To compare the results generated at the flow, node and system levels, we created bivariate plots (example in figure 4). Each dot represents one comparison between two time-steps, based on two levels. The tendency of dots to align to the diagonal 1 : 1 line is further illustrated by a fitted regression line (red line in figures 5–7) going through the origin. The closer the slope was to 1, the more well aligned were the conclusions of the two levels considered over the compared time-steps. In addition, we calcu-lated the closest distance between each pairwise comparison (dot) and the diagonal 1 : 1 line (black diagonal line). The lower the distance, the larger alignment between the conclusions of the two levels. We used the dist2Line() function in the geosphere R library to find the coordinates along the 1 : 1 line (x2,y2) closest
to each respective dot (x1,y1), which in turn was used to calculate
the distanced ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2 x1Þ2þ ðy2 y1Þ2
q
.
We applied these analyses on our selected trinomials, based on the three hierarchical levels, which represents features of food-web functioning. For one of the (i) recycling trinomial corresponding to the emergent property of recycling, we defined the network level as‘the percentage of carbon recycled within the whole system’, the node level as ‘the percentage of carbon leaving detritus (DET) that loops back to the compartment’ and the flow level as, for example,‘flow of dissolved organic carbon (DOC) uptake by heterotrophic bacteria’. (ii) The TL trinomials were defined as ‘MTL within the whole system’ for the network level, ‘the trophic level of a given compartment (the zooplankton compartments copepods, ciliates or cladocerans)’ for the node level, and, for example, ‘ingestion of heterotrophic nanoflagellates (small zoo-plankton which are important grazers of bacteria) by a given consumer (copepods, ciliates or cladocerans)’ for the flow level. Finally, the (iii) omnivory trinomials were defined as ‘system omnivory’ for the network level, ‘omnivory of a given consumer (copepods, ciliates or cladocerans)’ for the node level, and, for example, ‘ingestion of heterotrophic nanoflagellates by a given consumer (copepods, ciliates or cladocerans)’ for the flow level.
We tested different combinations within the different levels. For recycling, we compared the network level with the node-level recycling for all secondary producers (excluding the parasites, i.e. the chytrids), DOC and DET. For the flow-level comparisons with node and network levels, we tested the recycling with a selec-tion of flows concerning DOC inputs and bacteria (BAC; heterotrophic, i.e. non-photosynthetic, bacteria) outputs. For the omnivory and TL, we compared the network level with the node level for all consumers including virus (electronic sup-plementary material, appendix tables SA4 and SA5). For the flow-level comparison, we tested the difference of conclusions between network level and node level with all flows of consump-tion, by the larger consumers (cladocerans, copepods and ciliates). We present hereafter results illustrating the variety of situations observed, focusing on the DOC and DET compartment for recy-cling, and the larger consumers (ciliates, cladocerans and copepods) for the omnivory and TL. All comparisons and statistics are detailed in the electronic supplementary material, SA3–SA6.
3. Results
For the first trinomial going via DOC (figure 5a), there is a high correlation between all three levels (slope = 0.82, 1.23 and 1.37, distance = 0.10, 0.19 and 0.21, respectively, for the flow- versus node, flow- versus network and node- versus network-level comparisons), signifying a strong agreement in the results from the three levels. This seems to indicate that the flow from bacteria to DOC, corresponding to the DOC released during lysis of bacteria (the breaking down of the bacterial cell membrane) owing to viral infections, is highly determinant in the consideration of the recycling for DOC and for the recycling of the system. The flow is involved in a loop based of two flows: from DOC to bacteria and then back to DOC.
For the second recycling trinomial going via DET, i.e. particulate detritus (figure 5b), the node- versus network-comparison shows a stronger similarity in the conclusion (slope = 0.85, distance = 0.21) opposed to the flow level compared to the two other levels (slope = 0.61 and 0.75, distance = 0.29 and 0.20, for flow versus node level and flow versus network level, respectively). We can conclude that for this recycling trinomial, the conclusions drawn from the node and network level are more similar than when considering the flow level. Hence, the information on the flow level, brought by the flow from detritus to DOC, seems more independent from the information provided by the node or network level.
For the TL trinomial, we looked at thee consumers on the node level (ciliates, cladocerans and copepods), related to the ingestion flow of heterotrophic nanoflagellates by the same consumer on the flow level (figure 6). The flow, node and net-work level concurred quite well in the case of the cladocerans and ciliates (figure 6a,b). The slope of the regression line was close to 1 in these cases (slope range 0.64–0.99, and 1.11), and the distances were lower (distance range 0.16–0.36)
node level network level −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 1 2 3 4
Figure 4. Biplot comparing the
‘conclusion’ of differences or similarities
between food-web models for the network and node level. The x-axis and
y-axis show the Cliff
’s delta (δ) statistic for each unique pairwise comparisons
between the 16 food web models (e.g. model T1 versus T2, T1 versus T3,
etc.). Values close to the black diagonal line, points 1 or 2, for example, indicate
that the model comparisons (and
δ) were similar between the network- and
node-level indices. For illustration purpose, points 1 and 2 show the same
results, but while point 1 shows the T1 versus T2 comparison, point 2 shows
the T2 versus T1 comparison (note, in the analyses here, only one of these
com-parisons would be included). Values further away from the black diagonal line,
points 3 or 4, for example, indicate that the conclusion between system levels
was different. In the point 3 example, the Cliff
’s δ results from the network level
(
δ = 0.8) indicated a large magnitude of difference between models, while the
results from the node level (
δ = 0.2) indicated a small magnitude.
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5flow (BACDOC) node (fciDOC) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 (a) flow (BACDOC) netw or k (FCI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (fciDOC) netw or k (FCI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.25 0.50 0 2.5 5.0 flow (DETDOC) node (fciDET) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 (b) flow (DETDOC) netw or k (FCI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (fciDET) netw or k (FCI) −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2
Figure 5. FCI trinomial (Finn cycling index): biplots comparing the differences or similarities between food-web models for the flow and node levels (grey, on the
left), flow and network levels (green, in the middle), and node and network levels (blue, on the right). The node level was defined as the recycling of (a) DOC
compartment with the flow bacteria to DOC (BAC to DOC) for the flow level, and (b) recycling of detritus (DET) with the flow DET to DOC for the flow level. The
rightmost plots show the density distributions of the shortest distance between the dots in the biplots and the black diagonal line (1 : 1 line, where the two
compared levels are perfectly aligned). The red line is the regression line. (Online version in colour.)
flow (HNFCIL) node (mtlCIL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCIL) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (mtlCIL) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1.5 3.0 flow (HNFCLA) node (mtlCLA) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCLA) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (mtlCLA) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2 flow (HNFCOP) node (mtlCOP) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCOP) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (mtlCOP) netw or k (MTL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2 (a) (b) (c)
Figure 6. MTL trinomial (mean trophic level index): biplots comparing the
‘conclusions’ of difference or similarities between food-web models for the flow and node levels
(grey, on the left), flow and network levels (green, in the middle), and node and network levels (blue, on the right). The node level was defined as the TL of (a) ciliates (CIL)
compartment with the flow heterotrophic nanoflagellates to CIL (HNF to CIL) for the flow level, (b) the TL of cladocera (CLA) with the flow HNF to CLA for the flow level and (c)
the TL of copepods (COP) with the flow HNF to COP for the flow level. The rightmost plots show the density distributions of the shortest distance between the dots in the
biplots and the black diagonal line (1 : 1 line, where the two compared levels are perfectly aligned). The red line is the regression line. (Online version in colour.)
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6than for copepods (slope range 0.53–0.90, distance range 0.25–0.44). For copepods (figure 6c), the node and network levels concurred slightly better (slope = 0.53, distance = 0.44), compared to the flow-level comparisons.
For the omnivory trinomials (figure 7), the best alignment between levels was found for the node versus network level and flow- versus network-level comparisons going via cili-ates (slope 1.06, 0.95, distance range 0.19, 0.20). A high density of dots aggregated in the extremity of the line 1 : 1, showing the same conclusions for the system and compart-ment level (high significance of the difference). However, for cladocerans and copepods in particular, many of the
dots of δ values close to 1 or −1 for the network levels
seemed to present anyδ-value for the node or flow level, as
indicated by the large distance (0.33–0.51). By contrast, for these two compartments, the flow level when compared with the node level showed generally a better alignment of the regression line to the 1 : 1 line (slope range 0.70–0.94), and with smaller distances (0.23–0.24).
Altogether, there was a larger alignment between levels for the FCI trinomials (distance ranged from 0.10 to 0.29 for all comparisons), compared to the MTL (distance range 0.16– 0.44) and SOI (distance range 0.19–0.55). Also, when compar-ing over all trinomials presented in the main article, there
was on average a larger alignment between the flow and node level (average distance 0.23) and node and network levels (average distance 0.25) comparisons, and less so for the flow and network level (average distance 0.32, electronic sup-plementary material, table SA6). However, when looking over all trinomials compared in this study, the node- and net-work-level comparison concurred the most (average distance 0.42, compared to flow versus node level 0.47, flow versus net-work level 0.53, electronic supplementary material, table SA6).
4. Discussion
Comparisons of selected ENA results at three hierarchical levels (network, node and flow) for the Lake Aydat model presented in this paper illustrate a range of possible results. A general trend for the hierarchical trinomials considered is a stronger alignment in conclusions (i.e. low distance value) between the comparison including the node (flow versus node, node versus network) relative to the flow and network comparison (electronic supplementary material, table A6B). Hence, the con-clusions will tend to be more similar the closer the levels are in the hierarchy, indicating the interconnection of the node-level index to the other levels.
flow (HNFCIL) node (soiCIL) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCIL) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (soiCIL) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2 flow (HNFCLA) node (soiCLA) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCLA) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (soiCLA) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2 flow (HNFCOP) node (soiCOP) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 flow (HNFCOP) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 node (soiCOP) netw or k (SOI) −1.0 −0.5 0 0.5 1.0 −1.0 −0.5 0 0.5 1.0 distance density 0 0.5 1.0 0 1 2 (a) (b) (c)
Figure 7. SOI trinomial: biplots comparing the
‘conclusions’ of difference or similarities between food-web models for the flow and node levels (grey, on the left),
flow and network levels (green, in the middle), and node and network levels (blue, on the right). The node level was defined as the omnivory level of (a) ciliates
(CIL) compartment with the flow heterotrophic nanoflagellates to CIL (HNF to CIL) for the flow level, (b) the omnivory level of cladocera CLA with the flow HNF to
CLA for the flow level and (c) the omnivory level for copepods (COP) with the flow HNF to COP for the flow level. The rightmost plots show the density distributions
of the shortest distance between the dots in the biplots and the black diagonal line (1 : 1 line, where the two compared levels are perfectly aligned). The red line is
the regression line. (Online version in colour.)
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7Another general trend was that the strength of alignment depended on the trinomials and indices considered. Recy-cling showed slightly contrasting results. In figure 5a, when node-level concerned DOC, there was a strong alignment (and small distance) between the conclusions of all level com-parisons, implying that the same decision would be taken to announce a difference between the time-step models. On the contrary, in figure 5b, where node-level concerned detritus, the distance between dots and the 1 : 1 line indicated less con-curring between time-steps. The same discrepancies in the decisions at the network and node levels would have been observed if, for example, cladocerans or copepods had been considered (electronic supplementary material, table SA3 and SA6). All of the model compartments exhibit some degree of carbon recycling, but cycling is more important in some than in others, and some nodes contribute more to the network-level cycling index (electronic supplementary material, table SA3). The relative importance of DOC and DET cycling in the network cycling index explains this differ-ence between figure 5a,b. Concerning the flow level, the flow from bacteria to DOC showed a high degree of consistency with the two other levels, DOC node level and network. We hypothesize that this result is owing to the fact that this flow is embedded in a loop that is composed of only two flows (from DOC to bacteria and back to DOC through viral lysis) and that these flows have faster turnover rates than other, longer, loops. This importance of taking this loop between heterotrophic bacteria and DOC into account in food-web models suggests that the recycling is highly underestimated in models where viral infections are not considered, which is the case in almost all planktonic models published up to now (see the review [36]).
Concerning TL and omnivory, we also had situations where all the levels provided the same information and situ-ations where this was not true. Such results illustrate the importance of simultaneously considering ENA features at multiple hierarchical levels. The flows from heterotrophic nanoflagellates (HNF) to consumers showed slopes close to 1, indicating that the flow from HNF was well aligned with the network level and the node level. This suggests that the feeding flow from HNF was a key structuring flow located in the median TLs. This can be related to the microbial loop context, in which the HNF node plays a key role in defining the type of planktonic food web [37,38].
Our results present a limited number of trinomials, which we chose based on ecological reasoning, and we prioritized showing results with the strongest relations in conclusions between levels (electronic supplementary material, tables SA3–SA5). We therefore anticipate that further analysis might have discovered more disagreements among the levels of analysis. However, as shown with these results, the alignment and mismatch of results at the various levels and depending on the ENA index considered can be informa-tive about system functioning, and may prove useful for future applications of ENA. This general conclusion is con-sistent with previous attempts to compare ENA results at different hierarchical levels of analysis [24,39]. Further, the emergence of system properties is more or less influenced by the different levels hierarchically below the focal level, but a strong description of the ecosystem functioning would include all three levels in parallel. This is especially important in the present context where ENA indices are pro-posed to managers as indicators of ecosystem health [16,17].
The merit of this recommendation was supported by the association of the ENA indices with theories on maturity, stability and stress that were established based on the net-work level (e.g. [40–42]). However, a change of ENA at the system level can sometimes be mapped with a change in the different compartments, or no change in the system level can be owing to several changes at the compartment level that compensate each other. Hence, looking at all three levels may be essential so as not to draw faulty
con-clusions. Indeed, Whipple et al. [39] found that signals at
the node level tended to be smoothed at the network level. The potential misalignment of information provided by the three levels is epistemologically striking. In the nested hierar-chy case considered in this paper, we might consider the three levels of analysis as alternative approaches that provide distinct perspectives: the network-level metrics are the most general, the node-level analysis acknowledges some realism, and the flows are the object of a precision analysis. Flows are indeed quantified precisely and computing on those quantities may allow local predictions for a given network. At the whole-network level, many systems which differ at the level of flows may feature a similar dynamic, and therefore, information gathered about this dynamic will apply generally to this whole class of systems. At the level of nodes, various flows with distinct quantitative measures may concur in a specific pattern of input–output functions, which, among such general class of networks with similar dynamics, would characterize more realistically a specific system in terms of its local profile of functioning. Combining these three levels of analysis into a trinomial perspective fits well with the vertical explanatory mode proposed in the article by Kostic in this issue [43].
Emphasizing the diversity of epistemic values across levels also highlights the importance of using different levels to answer different types of questions. For example, FCI tells us how important cycling is in contributing to the total system activity (power, productivity, etc.), while the node-level cycling can tell us the importance of cycling to the throughflow of any node, but it also provides us a way to compare the relative importance of each node in contributing to the total system cycling. Looking at all levels will not answer all questions, but by including them all, it is possible to get further insight and knowledge about the food-web functioning. Scientists can then formulate better hypotheses about which mechanisms are in play. For example, a strong alignment between the network and a specific node-level ENA could indicate that this node plays a key role (as the DOC for the FCI in our results). In gen-eral, the system manifests a nested dependence: node-level analyses depend directly on the pairwise flow level, as do the network metrics. Summarizing at the node or network level lets the investigator focus the model information on features or properties that emerge at the different levels. If some ecological properties are emergent, they should prove to be autonomous in addition to being (nestedly) dependent, thus there should be a sort of discrepancy or mismatch between the levels. Sometimes, features at the flow level wash out or are smoothed at the other levels and it seems possible they might also amplify a signal. Those possibilities of coupling between levels are similar to
the ‘inter-level loops that constitute the phenomenon of life’
addressed in this issue by Chavalarias [44] and we hypothesize that such loops could see some of their dynamics modelled in a framework similar to ENA.
The inconsistency of results at different hierarchical levels of analysis requires additional reflection. The situation could be
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8interestingly related to problems met in community ecology about the explanations of biodiversity dynamics through inter-scale dynamics, and especially at the occasion of discussion of the neutral theory in ecology and biogeography [45,46]. Hub-bell’s theory foiled previous ideas about species interactions shaping biodiversity patterns, and argued that species stand neutral towards each other and it is rather random processes (in terms of birth, deaths, immigration and extinction) deter-mining biodiversity. Arguably, which process dominates seems therefore to be a question of scale. In our case, a hypoth-esis explaining the non-concurring between what goes on in compartments and the net result of the dynamics of the system could be that effects at the level of compartments add up in a way that they finally average away at the higher level of the system. What is notable is the similarity between the articulation of a regional and a global scale, on the one hand, and a system level and a lower level such as nodes, clusters or flows, on the other hand. Thus, ENA may benefit from similar explanations as diversity patterns in community ecology.
Characterizing this multiplicity of standpoints is another issue. Some authors have questioned whether there are levels,
or whether the notion of‘levels’, which usually grounds the
idea of a hierarchy, should be kept or replaced by the notion of
scale [1]. Thus, is the notion of‘level’ adequate to characterize
what is going on in our analyses? We assert that the level concept is useful in this case because it is a nested hierarchy. Each level grounds the next one, and each level provides specific kinds of information on the system. For example, the network level lets users quickly compare whole systems. While at the node level, one still can compare across systems, but it is more difficult. However, the node level lets users differently understand the key roles different nodes are playing to generate the dynamics. Finally, the flow level—provides all the detail. Such a multi-plicity of standpoints is also illustrated in this issue by Serban [47], in the case of networks used as exploratory models. In the most general perspective, the example of ENA considered
in this paper invites one to question what‘level’ and ‘hierarchy’
mean. The hierarchy addressed here embraces three levels, flows, node and network, based on the initial difference between general network topological properties and node-centred prop-erties. However, another kind of network called multiplexes can feature additional hierarchy such as when the timescales of each layer of the multiplex differ (often depending upon the nature of the dominant interaction in each layer) (e.g. [5]). Neuroscience [48] as well as ecology feature such networks. Thus, networks can be comprised of multiple distinct kinds of hierarchies.
5. Conclusion
This paper evaluates the value of simultaneously considering ecological network results at three different hierarchical levels: flows, nodes and whole network. While focusing on selected ENA features (cycling, TL and omnivory), we show that the alignment of information provided at the
different levels can vary, despite the nested nature of this hier-archy. However, a general pattern suggests that the closer the levels are in hierarchy (node- versus network, or flow- versus node), the more they tend to align in their conclusions. Further, our key findings agree with the observation that eco-systems features are often emergent properties, namely properties that are both dependent upon and (from the view-points of explanation and prediction) autonomous from their components. We argue that ecologists can use the infor-mation alignment and misalignment of the hierarchical levels to better understand the ecosystems under investi-gation, as well as their emergent ecosystem properties. This conclusion plausibly extends towards networks of all natures that display flows of matter and information: functional net-works in neuroscience (e.g. [48]), or protein–protein interactions networks or software systems networks [49]. In each case, looking for alignment or misalignment between the flows, the node level and the whole-network level should be informative regarding the dynamics of the system. It is likely that the distance between those levels will provide in general a proxy for the degree of misalign-ment between them, therefore, the degree of autonomy between levels to be expected in the system.
Data accessibility.All the software used are freely available online. The
code supporting this article has been uploaded as part of the elec-tronic supplementary material. All data included are from previously published work, and the combined dataset used here has been uploaded as part of the electronic supplementary material.
Authors’ contributions.N.N. and S.R.B. proposed the ideas about
hierar-chy. All authors developed the ideas and participated to the writing. N.N. was the lead author. M.H. and S.R.B. performed the analyses.
Competing interests.We declare we have no competing interests.
Funding. M.H. is funded by FORMAS mobility grant (reference
number 2017-01610). P.H. is funded by the GDR Sapienv CNRS3770. The data and models on which the article is based are issued from a grant from the French ANR Programme Blanc ROME (Rare and Overlooked Microbial Eukaryotes in aquatic ecosystems) coordinated by T.S.-N.
Acknowledgements.Samuele Tecchio for making available the code for
connecting LIM-MCMC results to the enaR package (R-Cran project). Mélanie Gerphagnon and Jonathan Colombet for the field and lab-oratory work and advice for the LIM model building. Justine Lequesne and Valerie Girardin from the laboratory of mathematics. Nicolas Oresme in Caen for statistical advice on the Cliff’s delta. Daniel Kostic for useful discussions about the integration in the special issue. Quentin Noguès for help on the ENA formulae.
Endnotes
1
Note that‘trophic levels’ is a technical term in functional ecology; it is not levels of the ENA (node level, flow level, whole-network level) that we are considering here. All issues regarding hierarchies and emergence in this paper are of course about the latter set of levels.
2
See note 1 to disentangle those two meanings of‘level’. Because this is the term used in the literature of trophic chain and in the methodolo-gical and philosophical literature on hierarchies, we were compelled to use it in both contexts.
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