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The Multilayer Nature of Ecological Networks
Shai Pilosof, Mason A. Porter, Mercedes Pascual, Sonia Kéfi
TL;DR
Ecological networks often omit multiple interaction types, changing interactions, and interconnected systems. This paper formalizes multilayer ecological networks, applies them to existing data, and finds that temporal structure shapes species’ functional groupings.
Problem
Ecological networks do not systematically capture multiple interaction types, spatial and temporal variation, or connections among networks, limiting representations of multifaceted ecological interactions.
Method
The paper reviews related approaches, formally defines ecological multilayer networks, and illustrates their analysis with existing ecological data.
Results
About 47% of hosts and 35% of parasites changed module affiliation over time, while multilayer tests rejected hypotheses attributing modularity to random or temporally independent associations.
Takeaways & Limitations
Multilayer networks provide a framework for analyzing ecological processes within and across layers and addressing questions unavailable to monolayer representations.
Takeaways & Limitations
Defining and measuring interlayer edges, including their relative weights to intralayer edges, remains an active research challenge.
Abstract
from arXiv · showhide
Although networks provide a powerful approach to study a large variety of ecological systems, their formulation does not typically account for multiple interaction types, interactions that vary in space and time, and interconnected systems such as networks of networks. The emergent field of `multilayer networks' provides a natural framework for extending analyses of ecological systems to include such multiple layers of complexity, as it specifically allows one to differentiate and model `intralayer' and `interlayer' connectivity. The framework provides a set of concepts and tools that can be adapted and applied to ecology, facilitating research on high-dimensional, heterogeneous systems in nature. Here, we formally define ecological multilayer networks based on a review of previous and related approaches, illustrate their application and potential with analyses of existing data, and discuss limitations, challenges, and future applications. The integration of multilayer network theory into ecology offers largely untapped potential to further address ecological complexity, to ultimately provide new theoretical and empirical insights into the architecture and dynamics of ecological systems.
Introduction
Ecological networks have generated insights into ecological systems but are usually studied as isolated, single-point, or spatially and temporally aggregated structures. Recent multilayer-network theory offers a promising mathematical framework for encoding different interactions or entities together.
- Ecological networks have advanced understanding of ecological systems’ structure, function, and dynamics.
- Most ecological networks have been studied in isolation, at a single spatial and temporal point, or aggregated across locations and times.
- Recent multilayer-network theory provides a promising approach for analyzing structures that encode different interaction types or entities as one mathematical object.Although the mathematical framework was developed only recently, multilayer structures have a longer history in sociology and engineering.
Ecological Multilayer Networks
Ecological multilayer networks extend conventional network analyses by representing multiple interaction types, communities, spatial or temporal contexts, and organizational levels through intralayer and interlayer connectivity. The framework formalizes these systems and supports analyses of ecological complexity across interconnected layers.
- Definition: Multilayer networks contain physical nodes, layers, state nodes, and weighted or unweighted edges, including intralayer and interlayer connections.State nodes represent physical entities on specific layers, while interlayer edges connect their manifestations across layers.
- Prior approaches: Earlier ecological studies predominantly used multiple independent networks of the same system, calculating diagnostics separately because interlayer edges were formally absent.One example represented a plant–pollinator system sampled over 12 years as 12 individual networks.
- Spatial and temporal networks: Spatial and temporal variability can be represented by defining a monolayer network at each location or time and connecting corresponding nodes with interlayer edges.Temporal food webs can use ordinal connections, whereas spatial food webs can connect nodes across all layers.
- Multiple interaction types: Multilayer networks simultaneously represent multiple interaction types, including diagonally coupled networks for shared species and node-aligned networks in which all entities occur on every layer.In diagonally coupled networks, each interaction type occupies a layer and interlayer edges connect common species; node-aligned networks can represent trophic and non-trophic interactions.
- Interconnected and multilevel networks: Interconnected and multilevel networks model dependencies among populations, species, food webs, and organizational levels through interlayer relationships.These representations can combine within-population social structure with disease transmission among populations, or connect processes across biological levels.
Analyses of Ecological Multilayer Networks
The analyses use synthetic and empirical ecological multilayer networks to examine maximum modularity and extinction cascades while accounting for explicitly connected layers. They show that interlayer connectivity, temporal structure, and multilayer representation affect inferred community organization and robustness.
- Framework: The framework represents entities across layered contexts as node-layer tuples connected by intralayer and interlayer edges, with edge weights encoding connection strength.A multilayer network is defined as M = (V_M, E_M, V, L), where layers may represent interaction types, space, or time.
- Maximum modularity: Community organization depends strongly on how ecological processes in different layers affect one another through interlayer connectivity.Intermediate interlayer weights can identify plants whose layer-specific instances occupy different modules and may buffer perturbations.
- Maximum modularity: About 47% of hosts change module affiliation at least once in the temporal network, showing that modules vary across time.State nodes can belong to different modules at different times, while module sizes also change over time.
- Maximum modularity: The observed network has higher maximum modularity than shuffled networks, with Q_shuffled B ≈0.21 and P < 0.001.It also has about 6 (15) times fewer modules than networks with shuffled interlayer host edges (parasite edges).
- Maximum modularity: Analyzing layers separately is incomplete because independent layers cannot capture interlayer effects, while the aggregated network provides only about 52% of multilayer module-affiliation information.The analysis finds that all six layers are necessary to describe the system’s entire complexity.
Limitations and Challenges
Multilayer ecological analyses are not universally preferable and can require resource-intensive data collection across places, times, methods, and interaction types. Key challenges include defining and measuring interlayer edges and developing methods and diagnostics suited specifically to ecological applications.
- Data and scope: Multilayer approaches are appropriate only for research questions that require their additional complexity.The choice between multilayer and monolayer formulations depends on the specific research question.
- Data and scope: Collecting multilayer data can be resource-intensive because observations may span multiple places, times, methods, and interaction types.Measuring interlayer edge weights may also require additional sampling efforts different from those used for other data.
- Interlayer representation: Defining and measuring interlayer edges is challenging, and their definition can significantly affect subsequent analyses.Interlayer edges may represent species dispersal or changes in species state, such as abundance.
- Interlayer representation: Intralayer and interlayer edges may represent ecological processes at different scales, making their relative weighting difficult to define.The supplied passage identifies scale differences as a source of uncertainty in assigning interlayer weights.
- Methodological development: Researchers can adapt existing multilayer-network techniques, but must determine which methods fit ecology and develop ecology-specific diagnostics and models.Relevant developments include multilayer vulnerability and generality measures, null models, and mathematical models for dynamical processes.
Future Directions and Applications
Future applications extend multilayer networks across biogeography, disease ecology, metacommunity theory, food-web parasitism, animal behaviour, movement ecology, and coupled ecological–social systems. Developing theoretical models remains critical for comparing multilayer frameworks with data and advancing ecological theory.
- Biogeography: In biogeography, dispersal or colonization/extinction dynamics represented as interlayer edges can clarify how spatial dynamics affect community structure and stability.The framework can also address correlations between geographic distance and beta diversity of species and interactions.
- Theory and disease ecology: Theoretical models are critical because they provide a basis for comparison with data and can stimulate further developments in multilayer ecological theory.Disease ecology is one application involving different host–parasite interaction types in transmission.
- Disease ecology: A three-layer model of contact, trophic, and host–vector interactions found that transmission occurring only on the trophic layer hinders infection of host populations.The model varied coupling between the trophic and vector layers to assess vector importance.
- Metacommunity theory: Interlayer edges support spatially explicit metacommunity models of species movement among local communities and resource and species flow across interconnected food webs.Examples include interconnected lakes or ponds sharing common sets of animals.
- Food-web theory: Coupling host–parasite networks with food webs can investigate how parasitism affects trophic interactions, food-web structure, and stability.Parasitized hosts may be more susceptible to predation, while the structural effects of host–parasite networks on trophic interactions remain unclear.
- Animal behaviour and movement ecology: Multilayer networks can link social interactions, movement, reproduction, and gene flow to study genetic relatedness and moving decisions.In animal behaviour, intralayer edges can represent reproduction and interlayer edges movement; movement ecology can connect social-interaction and movement-pattern networks.
- Ecological–social systems: In a multiplex network of three indigenous Alaskan communities, changes in social relations had greater effects on network robustness than depletion of ecological resources.The layers combined ecological resources with social relations, including marine species used as food.
Conclusions
The expanding availability of ecological data and multilayer-network analysis tools creates a timely opportunity to investigate ecological networks as multilayer systems. Multilayer approaches explicitly incorporate processes operating within and across layers, including interactions between those processes.
- Conclusions: Expanding ecological data and multilayer-network analysis tools provide a timely opportunity to explore the multilayer nature of ecological networks.The passage identifies this expansion as valuable for ecologists.
- Conclusions: Multilayer approaches formulate and analyse complex ecological systems by incorporating processes operating within layers and across layers.This capability is presented as a central strength of the multilayer framework.
- Conclusions: The framework also incorporates interactions between processes operating within and across layers.
Data and Code
The example temporal network’s raw data are provided in the Supplementary Information, while analysis code is available in R and Matlab repositories.
- Raw data for the example temporal network are provided in the Supplementary Information as Data set 1.
- R code supports data preparation, network manipulation, modularity-maximization post-processing, and network-robustness analysis.Available at 10.6084/m9.figshare.3472664.
- Matlab code supports examination of modular structure.Available at 10.6084/m9.figshare.3472679.
Supplementary Information
The supplementary information illustrates ecological multilayer networks through examples of multilayer structure, temporal variation, robustness, and reducibility. It also catalogs diagnostics, software, and methodological details supporting multilayer modularity analyses.
- Multilayer examples: Supplementary examples depict aquatic food-web and host–parasite layers connected by interlayer edges, including a parasite that functions as prey and/or pelican parasite.The supra-adjacency representation formalizes the same multilayer network.
- Temporal structure: Species change module affiliation across years, producing variation in module size across temporal layers of a host–parasite network.One example shows module 1 persisting across years while module 4 appears in only four of six layers.
- Temporal structure: Interlayer edge weights in the host–parasite temporal network represent relative abundance changes between consecutive time windows across six layers.The supplementary figure shows distributions for the five resulting time windows.
- Robustness: Robustness curves assess surviving plants and plants plus parasitoids as flower visitors are removed from a diagonally coupled multilayer network.Visitors are removed in increasing intralayer node-degree order.
- Diagnostics: Reducibility uses Jensen–Shannon distances between layers, hierarchical clustering, and successive aggregation to identify layers needed to represent the ecological system distinctly.The procedure merges clustered layers beginning with the smallest distance.
- Diagnostics and software: Supplementary materials list multilayer diagnostics and software for visualization, correlations, centrality, reducibility, modularity, motifs, community structure, and basic network analysis.The generalized Louvain analysis averages maximized modularity, module number, and host and parasite adjustability over 100 instantiations.
1 Supplementary Note 1: Multilayer Modularity Maximization
The method maximizes a multilayer modularity quality function for diagonally coupled networks, adapting the null model for host–parasite bipartite structure. A generalized Louvain-like algorithm performs the optimization, but its stochastic heuristic nature can produce different local-maximal partitions and modularity values.
- Quality function: Multilayer modularity is defined for weighted or unweighted unipartite networks whose interlayer edges connect each node only to its counterparts across layers.The formulation uses the standard Newman–Girvan null model and distinguishes intralayer from diagonal interlayer connectivity.
- Quality function: In ordinal multilayer networks, interlayer edge weights can be nonzero only between consecutive layers.This applies to the usual multilayer representation of a temporal network.
- Bipartite adaptation: The null-model contribution is modified to account for host–parasite bipartite structure, allowing adjacency only between hosts and parasites.The adaptation uses host and parasite strengths and modifies the GenLouvain code by changing Pijs; the default resolution parameter is γ = 1.
- Optimization: A generalized Louvain-like locally greedy algorithm maximizes the modularity quality function and determines module affiliation.The algorithm receives a modularity matrix calculated from the network’s supra-adjacency matrix.
- Optimization: The Louvain optimization is heuristic and stochastic, yielding local modularity maxima with potentially different module assignments and somewhat different modularity values.The modularity landscape contains a very large number of “good” local maxima.
2 Supplementary Note 2: Analysis of a Synthetic Network with Two Interaction Types
The synthetic network has a planted structure of three modules in each layer and uses binary intralayer edges with uniformly weighted interlayer edges. Optimization results are reported as mean values averaged over 100 modularity-maximization instantiations.
- The network contains a planted modular structure with 3 modules in each layer.
- Intralayer edges are binary, taking value 1 when present and 0 otherwise.
- All interlayer edges are assumed to have the same weight.
- Optimization results are reported as mean values obtained by averaging over 100 modularity-maximization instantiations.
3 Supplementary Note 3: Analysis of a Temporal Network
The temporal host–parasite network spans six Siberian summers and uses abundance-based intralayer and interlayer edge weights to represent ecological change across time. Analyses show that temporal ordering and multilayer structure are essential for accurately describing modular organization.
- Temporal network construction: The network records infections among 22 small mammalian host species and 56 ectoparasite species across 6 layers representing consecutive summers from 1982–1987.Not all species occur in every year.
- Temporal network construction: Intralayer and interlayer weights reflect species abundances and their relative changes between consecutive time points.Interlayer links are omitted when a species is absent from either consecutive layer.
- Temporal network construction: Interlayer edge weights range from 0.02 to 10.33, enabling comparison with the extreme cases ω = 0 and ω = ∞.The calculations assume undirected intralayer and interlayer edges.
- Modular structure and aggregation: The most modular observed partition has approximately 6 times fewer modules than host-order shuffles and approximately 15 times fewer than parasite-order shuffles.Almost all hosts and parasites switch modules at least once in the shuffled networks.
- Extreme interlayer-weight scenarios: With all interlayer weights set to 0, both QB and n vary among the 6 layers, while cross-layer module-composition changes cannot be defined unambiguously.Modules are defined separately within each layer in this limit.
- Modular structure and aggregation: Across layers, the mean NMI is 0.35±0.12, showing that analyzing each layer separately provides limited information about multilayer species-to-module assignments.The reported range is 0.19−−0.51 across M = 6 layers.
- Modular structure and aggregation: All 6 layers are needed to accurately describe the temporal network because aggregating layers removes interaction ordering and can produce qualitatively incorrect conclusions.Aggregated networks retain only about 0.521 ± 0.029 or 0.517 ± 0.032 correlation in information with individual-layer assignments, depending on the aggregation method.
4 Supplementary Note 4: Further Details on Robustness Analysis
The robustness analysis uses a two-layer, diagonally coupled subset of a 10-layer ecological network to examine cascading extinctions and compare monolayer with multilayer robustness. It also frames these calculations as percolation processes that support more nuanced analyses of species robustness.
- Network construction: The empirical network contains 10 interaction layers, from which the analysis selects two layers to form a diagonally coupled multilayer network.The original layers include plant interactions with birds, rodents, butterflies and flower visitors, aphids, granivorous insects, and leaf-miner parasitoids.
- Cascading perturbations: The analysis examines how extinction cascades across interconnected species, including pollinator–plant and plant–leaf-miner parasitoid co-extinctions.Earlier work studied plant-removal co-extinctions but did not examine disturbance percolation through plants or the multilayer network’s role in cascading perturbations.
- Robustness comparison: Robustness is compared between a plant–parasitoid monolayer and a flower visitor–plant–parasitoid multilayer network after removing the same flower-visitor nodes in the same order.The analysis tracks the total proportion of species remaining in each network.
- Percolation framework: The calculations have the flavour of a percolation process, enabling richer and more nuanced analyses of species robustness in multilayer ecological networks.Multilayer networks permit a wider variety of percolation processes than monolayer networks.