The Multilayer Nature of Ecological Networks

Introduction

• Ecological networks have advanced understanding of system structure, function, and dynamics1–7, but are often isolated or aggregated across space and time.Natural interactions can vary by season, and a plant species may interact with both pollinators and herbivores8–10.Source: Introduction · p. 3
• Extending monolayer analyses within a unified framework remains challenging, including detecting community structure across interaction types10,12 and analyzing resource flow in temporal networks13.Source: Introduction · p. 3
• Recent multilayer-network theory14,15, alongside growing ecological data sets, creates an opportunity to integrate this framework into theoretical and practical network ecology.Multilayer structures represent different interaction types and/or entities as one mathematical object.Source: Introduction · p. 3
• The paper defines ecological multilayer networks, illustrates the insights they can enable, and discusses challenges and future applications.Source: Introduction · p. 3

Ecological Multilayer Networks

• A multilayer network represents entities as physical nodes across layers, with state nodes joined by intralayer edges and interlayer edges that connect manifestations across layers.Layers can represent interaction types, communities, or times, and ecological processes such as dispersal can affect multiple layers.Source: Ecological Multilayer Networks · pp. 3–4

Layers Defined Across Space or Time

• Spatial and temporal networks connect layer-specific counterparts, while a network of communities can use interlayer edges to represent species extinction and colonization.In a network of layers, communities with higher betweenness were also more nested, with potential consequences for local stability.Source: Layers Defined Across Space or Time · p. 4, Figure 1 · p. 5

Layers Defined by Interaction Type

• In a node-aligned Chilean community network, non-trophic edges were distributed differently from chance expectations, indicating an association between interaction-type layers.Diagonally coupled networks link shared species across interaction types; combining interaction types can affect coexistence, biomass, and extinction robustness.Source: Layers Defined by Interaction Type · p. 6

Layers Defined by Different Group Identity

• Interconnected networks can represent individual-level social ties within populations alongside disease transmission between populations.Nodes appear in only one layer in this representation.Source: Layers Defined by Different Group… · p. 6

Layers Defined by Levels of Organization

• Multilevel networks represent dependencies across organizational levels, where lower-level interactions entail interactions at upper levels.In a three-level population, community, and metacommunity network, population processes such as social dynamics cascaded across levels and affected the metacommunity.Source: Layers Defined by Levels of Organization · p. 7

Analyses of Ecological Multilayer Networks

• The formal definition represents entities as state nodes on combinations of elementary layers, with weighted intralayer and interlayer edges connecting them.Layers can encode multiple aspects, such as interaction type and space; the corresponding adjacency tensor can be flattened into a supra-adjacency matrix with intralayer and interlayer edges in diagonal and off-diagonal blocks.Source: Analyses of Ecological Multilayer… · p. 8

Modularity

• With comparable interlayer and intralayer weights, a plant’s two state nodes can belong to different modules, identifying plants that may buffer perturbations.Modularity depends on how strongly processes in different layers affect one another; intermediate coupling allows plant instances to connect different ecological groupings.Source: Modularity · pp. 9–10
• About 47% of hosts and 35% of parasites changed module affiliation at least once across six summers, as modules also changed in size over time.Abundance fluctuations encoded by interlayer edges and changing interactions encoded by intralayer edges can alter a species’ functional role across time.Source: Modularity · p. 12
• Observed temporal networks had Q_shuffled ≈0.21 (P < 0.001) and about 6 (15) times fewer modules than networks with shuffled host (parasite) interlayer-edge order.The results reject the tested hypotheses that modular structure arose from random temporal abundance changes, random within-layer host–parasite associations, or independence from temporal order.Source: Modularity · p. 14
• All six host–parasite time layers were necessary; assignments from disconnected layers contained only about 35% of multilayer information, and aggregation retained about 52%.These comparisons show that examining layers independently or aggregating them can omit information about species’ module affiliations in the interconnected temporal network.Source: Modularity · pp. 14–15

Robustness to Perturbations

• Parasitoid extinctions occur more slowly in the multilayer than in the plant–parasitoid monolayer network, and including plant extinctions yields nontrivial tertiary-extinction patterns.The analysis considers direct parasitoid extinctions after plant removal and tertiary extinctions after pollinator removal causes plant extinctions; relaxing infinite interlayer coupling could model abundance changes across layers.Source: Robustness to Perturbations · pp. 15–16

Limitations and Challenges

• Whether multilayer analysis is warranted depends on the research question, while collecting interaction data across places, times, and methods—and measuring interlayer weights—can be resource-intensive.Relevant datasets are becoming available but remain scattered in the literature and need curation.Source: Limitations and Challenges · p. 16
• Ecology lacks established principles for defining and weighting interlayer edges, which may represent dispersal or state changes and can involve interaction currencies measured on different scales.The authors favor direct measurement where possible (61), or systematic sweeps of interlayer weights to assess robustness and how conclusions change.Source: Limitations and Challenges · p. 16
• Ecology needs to identify suitable borrowed multilayer methods and develop ecology-specific diagnostics, null models, dynamical models, and accessible, well-described software.Software packages are already available, but are less mature than those for monolayer ecological networks.Source: Limitations and Challenges · p. 18

• Multilayer models can represent spatial connectivity and repeated species across locations to investigate how dispersal, colonization, and habitat alteration affect community structure and stability.They can also examine mechanisms behind correlations between geographic distance and beta diversity of species and interactions.Source: Future Directions and Applications · p. 18
• In disease models, interlayer coupling can encode the relative importance of interaction types; Stella et al.41 found that trophic-only transmission hinders infection of host populations.Their three-layer model represented contact, trophic, and host–vector interactions.Source: Future Directions and Applications · pp. 18–19
• Multilayer networks can support spatially explicit metacommunity models of species movement and resource flows, and temporal food webs can represent changes in biomass across time.Interlayer values can represent dispersal or biomass flows among patches; bioenergetic flows across food webs remain largely unexplored.Source: Future Directions and Applications · p. 19
• Coupling host–parasite networks with food webs could clarify how parasitism affects food-web structure, stability, and hosts’ trophic interactions.Multilayer networks also offer an approach to disease transmission involving multiple hosts and parasites, a problem that has been difficult to study [18,42].Source: Future Directions and Applications · p. 19
• Animal-behaviour and movement models can link social interactions, reproduction, movement, and gene flow to study relatedness and movement decisions.Representing different species’ movement patterns can help assess land-use designs for conservation, while centrality measures incorporating interaction types and spatial or temporal dependencies can refine keystone-species identification.Source: Future Directions and Applications · p. 19
• Multilayer networks can represent reciprocal ecological and social effects; Baggio et al.74 found social-relation changes affected network robustness more than depletion of ecological resources.Their multiplex represented three Indigenous Alaskan communities through layers combining ecological resources and social relations.Source: Future Directions and Applications · pp. 19–20

Conclusions

• Multilayer networks explicitly represent processes within and across layers, including interactions between those processes, and can address questions beyond monolayer representations.The passage identifies this capacity as a strength of the multilayer approach.Source: Conclusions · p. 20
• A shared multilayer framework supports comparisons across ecological systems and network types through consistent terminology and methods, and can encourage interdisciplinary collaboration.Source: Conclusions · p. 20
• Ecology’s integration of multilayer network theory may yield new theoretical and empirical insights into ecological-system architecture and dynamics, while other fields may benefit from ecological multilayer methods and theory.The passage notes that other fields have benefited from monolayer ecological networks [76].Source: Conclusions · p. 20

Data and Code

• Raw temporal-network data are in Supplementary Information (Data set 1); R code covers data and network preparation, modularity post-processing, and robustness, while Matlab code examines modular structure.R code is available at 10.6084/m9.figshare.3472664; Matlab code is available at 10.6084/m9.figshare.3472679.Source: Data and Code · p. 20

Multilayer representation

• A two-layer example represents aquatic food-web and host–parasite interactions separately, linking shared species across layers while representing them as distinct state nodes in the supra-adjacency matrix.A parasite present in both layers can act as prey and/or parasite for the pelican.Source: Supplementary Information · p. 27

Temporal modular structure

• Species can change module affiliation across years, while module sizes vary; one example shows module 1 remaining relatively large and module 4 appearing in only four of six layers.The examples show selected results among 100 generalized Louvain instantiations; the module-size example uses the instantiation with the maximum maximum modularity.Source: Figure 2 · p. 28, Supplementary Information · p. 28, Figure 3 · p. 29

Interlayer edge weights

• Temporal-network interlayer edge weights are calculated as relative changes in species abundance between consecutive time windows, yielding five windows across six layers.Supplementary Figure 4 shows the weight distributions for each window.Source: Figure 4 · p. 30

Network robustness

• Robustness analysis removes flower visitors in increasing order of intralayer degree and tracks surviving plants alone or together with parasitoids as visitors are removed.The example concerns a diagonally coupled plant–flower-visitor and plant–leaf-miner parasitoid network.Source: Supplementary Information · p. 31

Community-structure calculations

• For host–parasite modularity calculations, reported values average 100 generalized Louvain runs, use a bipartite-adjusted null model, and set interlayer weight ω = 1000 as an approximation to ω = ∞.The passage notes that monolayer networks cannot compare modules across layers without an ad hoc approach.Source: Supplementary Information · p. 36

• Modularity, module counts, and host/parasite adjustability are averaged over 100 Louvain runs, with significance assessed against 100 shuffled networks under a bipartite-adjusted null model.The reported p-values appear in parentheses and compare the mean across 100 observed-network instantiations with 100 shuffled networks.Source: Supplementary Information · p. 37
• Supplementary Table 6 lists algorithms for examining network robustness to species extinctions.Source: Table 6 · p. 38

Supplementary Note 1: Multilayer Modularity Maximization

• The multilayer modularity function uses intralayer edges, diagonal interlayer coupling, and a Newman–Girvan null model to identify communities across layers.In ordinal networks, interlayer coupling can be restricted to consecutive layers.Source: §1 · p. 39
• For host–parasite networks, the method replaces the standard null-model contribution with a bipartite version that allows only host–parasite interactions.The implementation modifies GenLouvain by changing the null model and uses the default resolution parameter γ = 1.Source: §1 · p. 39
• The generalized Louvain-like heuristic finds local modularity maxima, and its stochastic runs can produce different partitions and modularity values.The authors average maximized modularity and module counts across 100 instantiations.Source: §1 · pp. 39–40
• The generalized Louvain algorithm takes a modularity matrix calculated from the network’s supra-adjacency matrix and returns modularity and a module assignment for each state node.Any supra-adjacency matrix representing a diagonally coupled multilayer network can serve as input.Source: §1 · p. 40

2 Supplementary Note 2: Analysis of a Synthetic Network with Two Interaction Types

• The network in Fig. 2 has a planted modular structure of three modules in each layer.The structure is analyzed in a binary, unweighted form.Source: §2 · p. 40
• Intralayer edges are binary, taking value 1 when present and 0 when absent, while all interlayer edges share the same weight.Source: §2 · p. 40
• Reported mean values for multilayer modularity QB and module count n average results over 100 modularity-maximization algorithm instantiations.Source: §2 · p. 40

3.1 Interlayer Edge Weights on a Comparable Scale as Intralayer Edges.

• Across six Siberian summers (1982–1987), the network represents 22 host and 56 ectoparasite species in six layers; interlayer weights reflect relative abundance changes, with links omitted if a species is absent in either adjacent year.Interlayer weights range from 0.02 to 10.33, and calculations assume undirected intralayer and interlayer edges.Source: §3.1 · pp. 40–41

3.1.1 Testing Hypotheses About the Modular Structure

• Observed networks differed significantly from intralayer-shuffled networks in module count and host/parasite adjustability; they were more modular than node-identity shuffles, with about 6 and 15 times fewer modules than host- and parasite-shuffled cases.For layer-order shuffles, observed modularity was not higher; observed partitions had about 2.5 times fewer modules and different proportions of hosts and parasites switching modules.Source: §3.1.1 · pp. 42–43

3.2 Interlayer Edge Weights of 0 or Infinity

• At ω=1000, QB≈1 and the maximum-modularity partition had 3 modules with no species switching; at ω=0, QB and module count varied across the six layers.With zero interlayer weights, modules are defined independently by layer, so changes in module composition and host and parasite adjustability cannot be calculated unambiguously.Source: §3.2 · pp. 43–44

3.3 Calculation of Normalized Mutual Information

• 0.35±0.12 mean layerwise NMI indicates that individual layers contain only about 35% of the multilayer species-to-module information; aggregated networks capture about 52%.Summed-edge and mean-edge aggregation yielded information correlations of 0.521 ± 0.029 and 0.517 ± 0.032, respectively.Source: §3.3 · p. 45

3.4 Reducibility of the Host–Parasite Network

• The six-layer network requires all six layers to describe its structure accurately; aggregating temporal layers can remove interaction ordering and lead to qualitatively incorrect conclusions.The analysis reports that temporal interaction dynamics can be ordered and bursty, making naive aggregation particularly lossy.Source: §3.4 · p. 45

4 Supplementary Note 4: Further Details on Robustness Analysis

• Using Pocock et al.38's data, the analysis selects two diagonally coupled mutualistic layers from a 10-layer network, with 189 flower visitors, 60 leaf-miner parasitoids, and 17 plants.The network is a binary version of the original, and mutualism in the leaf-miner parasitoid layer is indirect.Source: §4 · p. 46
• The modified co-extinction approach assumes infinite interlayer edge weights, so plant extinction in one layer entails extinction in the other, and cascading perturbations can be examined across layers.Pollinator extinction entails plant co-extinction, which in turn entails leaf-miner parasitoid extinction.Source: §4 · p. 46
• Under matched flower-visitor removals, the multilayer network retained a larger total proportion of species and experienced slower extinction than the plant–parasitoid monolayer (Supplementary Fig. 5).Both networks lost the same flower visitors in increasing-degree order; the monolayer tracked surviving plants, while the multilayer tracked plants and parasitoids.Source: §4 · p. 46