Introduction

Species interactions are variable and uncertain

As we try to navigate global biodiversity change, filling in knowledge gaps about biodiversity becomes instrumental to monitoring and mitigating those changes (Abrego et al. 2021; Gonzalez & Londoño 2022; Hortal et al. 2015). However, cataloging species, populations and, in particular, ecological interactions (e.g., predation, parasitism, and pollination) is a substantial challenge (Pascual et al. 2006; Polis 1991). There are methodological and biological constraints that hinder our ability to document species interactions, inevitably leading to uncertainty in our knowledge of interactions. For example, the spatial and temporal uncoupling of species (e.g., nocturnal and diurnal species coexisting in the same space with different daily activity timings, Jordano 1987) and the large number of rare and cryptic interactions in a community, contribute to these knowledge gaps by making it more difficult to observe interactions (Jordano 2016).

Several conditions must be satisfied for an interaction to be observed locally. First, both species must have overlapping geographic ranges, i.e. they must co-occur within the region of interest (Cazelles et al. 2016; Morales-Castilla et al. 2015). Second, they must have some probability of meeting (Poisot et al. 2015). Probabilities of interspecific encounters are typically low, especially for rare species with low abundances. While species’ absolute abundances may impact interaction frequencies (Vázquez et al. 2007), encounter probabilities are determined by their relative abundances (Canard et al. 2012; Canard et al. 2014). The probability that species meet also depends on their biology, such as their phenology (Olesen et al. 2010; Singer & McBride 2012) and discoverability (Broom & Ruxton 2005). Finally, when species do come into contact, an interaction occurs only if their traits, such as their phenotypes (Bolnick et al. 2011; Gravel et al. 2013; Stouffer et al. 2011) and behavior (Choh et al. 2012; Pulliam 1974), are locally compatible in that specific environment (Poisot et al. 2015). Because these conditions are not consistently met locally, there will inevitably be instances where interactions will be observed and others where they will not.

Documenting the location and timing of interactions is difficult when accounting for the spatiotemporal variability of ecological interactions (Poisot et al. 2012, 2015). Knowing the biological capacity of two species to interact directly (via e.g., trophic interactions) is necessary but not sufficient for inferring their interaction at a specific time and space. Environmental factors, such as temperature (Angilletta et al. 2004), drought (Woodward et al. 2012), climate change (Araujo et al. 2011; Gilman et al. 2010; Woodward et al. 2010), and habitat modifications (Tylianakis et al. 2007), contribute to this spatiotemporal variability by impacting species abundance and traits. Interactions may also be influenced by a third species (e.g., a more profitable prey species, Golubski & Abrams 2011; Sanders & van Veen 2012). Even under favorable circumstances, there remains a possibility that the interaction does not occur locally, either due to the intricate nature of the system or simply by chance. If it does occur, it might go undetected, particularly if it happens infrequently. In this context, it is unsurprising that our knowledge of ecological interactions remains limited (Hortal et al. 2015) despite extensive biodiversity data collection (Schmeller et al. 2015).

We distinguish the variability of interactions from their uncertainty. Interaction variability is defined as the variation of interactions along spatial, temporal, or environmental axes (Poisot et al. 2015). It is a property of interactions that should be quantified if we aim for a comprehensive understanding of ecological networks. Stochasticity is the inherent randomness or unpredictability of interactions that lead to this variability. Conversely, uncertainty is defined as a lack of knowledge about the occurrence of interactions. When using statistical models to infer interactions, uncertainty sources include input data, parameter, and model structure uncertainties (Simmonds et al. 2024). Input data uncertainty arises from our inability to empirically observe all interactions and from measurement errors in environmental and biological variables used for inference. Parameter uncertainty represents a plausible range of values for a parameter whose exact value is unknown. For example, we may calculate a range of plausible values for interaction variability (e.g., there could be a 50% certainty that an interaction occurs 50% of the time). Model structure uncertainty recognizes that different statistical models may adequately predict interactions. In contrast to variability, uncertainty can be reduced by sampling additional data. Simmonds et al. (2024) underscores the importance of quantifying and reporting these diverse sources of uncertainty, alongside ensuring their appropriate propagation to model output (such as predicted interactions) and higher-level measures (such as network structure). While recognizing that these definitions may not be universally accepted, clarifying the distinction between variability and uncertainty enables us to better comprehend the sources of our knowledge gaps about ecological interactions.

Species interactions as probabilistic objects

The recognition of the intrinsic variability and uncertainty of species interactions has led ecologists to expand their representation of ecological networks to include a probabilistic view of interactions (Dallas et al. 2017; Fu et al. 2021; Poisot et al. 2016). This allows filling in the Eltonian shortfall (i.e., the gap between our current knowledge and a comprehensive understanding of interactions, Hortal et al. 2015) by modeling the probability of occurrence of interactions (e.g., Gravel et al. 2019), which can be an important tool for directing efforts and taking action (Carlson et al. 2021), especially in places where access and resources for research are scarce. A probability is a measure of how likely a specific outcome is, based on both the uncertainty and variability of interactions. Interaction probabilities may be uncertain when there is a distribution of plausible probability values. The probabilistic representation of interactions has been applied to direct interactions, which are conceptually and mathematically analogous regardless of their biological type (e.g., predation and pollination). This is in contrast with indirect interactions (e.g., interspecific competition), which arise from distinct ecological processes and are often not directly observable (Kéfi et al. 2015, 2016). By accounting for the uncertainty and variability of direct interactions, networks of probabilistic interactions (which differ from probabilistic networks describing the uncertainty and variability of the whole network) may provide a more realistic portrait of species interactions.

Probabilistic interactions differ from binary interactions. Networks of probabilistic interactions, within a Bayesian perspective, express our degree of belief (or confidence) regarding the occurrence or observation of interactions. In contrast, interactions are simply regarded as either occurring or not in networks of deterministic binary interactions. Based on the scale at which they are estimated, interaction probabilities may reflect our level of confidence in whether interactions will be observed, realized locally, or biologically feasible. Our level of confidence should be more definitive (approaching either 0 or 1) as we extend our sampling to a broader area and over a longer duration, thereby diminishing the uncertainty of our knowledge of interactions (but not necessarily the estimation of their variability). In the broadest sense, binary interactions are also a type of probabilistic interaction, in which the numerical value of an interaction is restrained to 0 (non-occurring) or 1 (occurring). In networks of probabilistic interactions, only forbidden interactions (i.e., interactions prohibited by biological traits or species absence, Jordano et al. 2003; Olesen et al. 2010) have a probability value of zero, provided that intraspecific trait variability is considered (Gonzalez-Varo & Traveset 2016). Understanding the nuances between probabilistic and binary interactions is essential for accurately modeling and interpreting ecological networks.

The application and development of computational methods in network ecology, often based on a probabilistic representation of interactions, can alleviate (and guide) the sampling efforts required to document species interactions (Strydom et al. 2021). For example, statistical models can be used to estimate the uncertainty of pairwise interactions (Cirtwill et al. 2019) and the probability of missing (false negatives) and spurious (false positives) interactions (Guimerà & Sales-Pardo 2009), helping us identify places where sampling is most needed to reduce this uncertainty. Statistical models can also predict networks without prior knowledge of pairwise interactions. They may do so using body size (Caron et al. 2024; Gravel et al. 2013; Petchey et al. 2008), phylogeny (Elmasri et al. 2020; Strydom et al. 2022), or a combination of niche and neutral processes (Bartomeus et al. 2016; Pomeranz et al. 2019) for inference. Before being used to test ecological hypotheses, predicted networks must be validated against empirical data (Brimacombe et al. 2024), which could be sampled strategically to optimize the validation process. Topological null models, which generate networks of probabilistic interactions by preserving chosen characteristics of the adjacency matrix of binary interactions while intentionally omitting others (Bascompte et al. 2003; Fortuna & Bascompte 2006), are examples of common probabilistic interaction models. Null models can produce underlying distributions of network measures for null hypothesis significance testing. However, how the uncertainty of pairwise interactions propagates to network structure (i.e., community-level properties driving the functioning, dynamics, and resilience of ecosystems, McCann 2007; McCann 2011; Proulx et al. 2005; Rooney & McCann 2012) remains to be elucidated. Many measures have been developed to describe the structure (Poisot et al. 2016) and diversity (Godsoe et al. 2022; Ohlmann et al. 2019) of probabilistic interaction networks. These models and measures support the use of probabilistic interactions for the study of a wide range of ecological questions, from making better predictions of species distribution (Cazelles et al. 2016) to forecasting the impact of climate change on ecological networks (Gilman et al. 2010).

We lack a clear understanding of probabilistic species interactions

We still lack a precise definition of probabilistic interactions, which makes the estimation and use of these data more difficult. In this manuscript, we aim to take a step back by outlining different ways in which probabilistic interactions are defined and used in network ecology. We distinguish two broad categories of probabilistic interaction networks that necessitate distinct approaches: local networks describing probabilities of realized interactions, and regional networks (metawebs) describing probabilities of potential interactions. We highlight the distinctions in the ecological meaning of these two representations of interactions and examine their properties and relationships (particularly with space, time, and between each other).

The lack of clear guidelines on the use of probabilistic interaction data is worrisome, as it affects both data producers and re-users who generate and manipulate these numbers. This is concerning because sampling strategies and decisions regarding network construction can affect our understanding of network properties (Brimacombe et al. 2023). There is currently no reporting standard that could guide the documentation of all types of probabilistic interactions (Salim et al. 2022 discuss data standards for deterministic mutualistic networks). Clear reporting standards for probabilistic interactions would support more adequate manipulation and integration of interaction data from different sources and guard against possible misinterpretations arising from ambiguous definitions of probabilistic interaction networks. Data documentation should outline the nature (i.e., local or regional) and type (e.g., predatory or pollination) of interactions, provide information regarding the taxonomic level, identities, and characteristics (e.g., life stages) of the individuals involved in an interaction, present the mathematical formulation of probabilities, including clearly identified conditional variables (e.g., spatial and temporal scales), and describe the methods and contexts (e.g., location, time, environmental conditions) in which interactions were estimated. Inadequately documented probabilistic interaction data should be used with caution when analyzing ecological networks. These broad principles remain relevant and applicable across different types of direct interactions. In the following sections, we discuss the definitions, conditions, and estimation of probabilistic interactions as we scale up from pairwise interactions to interactions within local and regional networks.

Pairwise interactions: the building blocks of ecological networks

What are probabilistic interactions?

Consider a scenario where an avian predator has just established itself in a northern habitat home to a small rodent. Suppose these species have never co-occurred before, and as a result, their interaction has not been previously observed. What is the probability that the rodent is part of the diet of the avian predator, or put differently, what is the probability that they interact? Answering this question requires some clarification, as there are multiple ways to interpret and calculate interaction probabilities. We could calculate the probability that the traits of these species match, i.e. that the avian predator possesses the biological attributes to capture and consume the rodent. We could also calculate the probability that their traits support an interaction under the typical environmental conditions of the new habitat. For example, because avian predators hunt by sight, predation could be possible in the absence of snow but highly improbable when snow is present, as rodents may use it as a shelter to hide from predators. Finally, we could calculate the probability that the avian predator will consume the rodent at that particular location, for which the spatial and temporal boundaries need to be specified. The estimation of the probability of interaction between these two species, whether through predictive models or informative prior probabilities, hinges on our understanding of these probabilities and the specific ecological processes we aim to capture.

An important aspect to consider when estimating or using interaction probabilities is knowing if they describe the probability of potential or realized interactions, as these two types of interactions have distinct meanings and sources of uncertainty and variability. A potential (regional) interaction is defined as the biological or ecological capacity of two taxa to interact (i.e., the probability that they interact if they were to encounter each other, given sufficient time and appropriate environmental conditions) whereas a realized (local) interaction is the occurrence or observation of this interaction in a well-defined space and time (i.e., the probability that they interact locally). For two co-occurring taxa and over enough time, the probability of local interaction tends toward the probability of regional (potential) interaction. A longer duration increases the probability that species will eventually encounter each other and that local environmental conditions supporting an interaction will occur, provided that species have the biological capacity to interact. Recognizing the distinction between probabilistic regional and local interactions is crucial for accurately interpreting interaction probabilities in ecological networks.

We use the terms metaweb (Dunne 2006) to designate regional networks of potential interactions and local networks (Poisot et al. 2012) for those of realized interactions. Metawebs are the network analogs of the species pool, where local networks originate from a subset of both species (nodes) and interactions (edges) of the regional metaweb (Saravia et al. 2022). Without clear documentation, it can be challenging to know if published probabilistic interaction networks describe local or regional interactions. When probabilistic local interactions are used and interpreted incorrectly as regional interactions (and conversely), this may generate misleading findings during data analysis. A better understanding of probabilistic local and regional interaction networks would facilitate a more adequate use of interaction data (e.g., when studying network-area relationships in local networks and metawebs) and prevent misinterpretations of the biological meaning of probabilistic interactions.

What is the outcome of probabilistic interactions?

The outcome of probabilistic interactions is usually binary

Local networks and metawebs, like any type of network, are made of nodes and edges that may be represented at different levels of organization. The basic units of ecological networks are individuals that interact with each other (e.g., by predation in food webs, Elton 2001), forming individual-based networks (Melián et al. 2011). The aggregation of these individuals into more or less homogeneous groups (e.g., populations, species, families, feeding guilds) allows us to represent nodes at broader taxonomic scales, which affects our interpretation of the properties of these systems (Guimarães 2020; Hemprich-Bennett et al. 2021).

Ecologists have traditionally represented interactions (edges) as binary objects that were considered realized after observing at least one individual from group i interact with at least another individual from group j. In an adjacency matrix B of binary interactions, the presence or absence of an interaction B_i, j between two taxa can be viewed as the result of a Bernoulli trial $B_{i, j} \sim {\rm Bernoulli}(P(B_{i, j}))$, with P(B_i, j) being the probability of interaction. This interaction probability characterizes our limited ecological knowledge and/or the intrinsic spatiotemporal variability of the interaction. It may be estimated through predictive models (e.g., those based on biological traits and species abundances) or expert (prior) knowledge about the interaction. In networks of probabilistic interactions, the edge values P(B_i, j) are probabilistic events whose only two possible outcomes are the presence (B_i, j = 1) or absence (B_i, j = 0) of an interaction between each pair of nodes. Depending on the type of probabilistic interaction network (local network or metaweb), the mathematical formulation and interpretation of stochastic parameters like P(B_i, j) can be linked to environmental and biological factors such as species abundances, species traits, area, and time, for example using logistic regression with continuous explanatory variables. This allows us to model the probability that at least two individuals interact under these conditions.

The variability of an interaction determines the fraction of networks in which it occurs. This fraction can be predicted by using a Binomial distribution, assuming a constant interaction probability and independence between interactions in different networks (trials). When considering uncertainties around the estimation of P(B_i, j), a Beta distribution may be used to represent the relative likelihood of different probability values. For example, when calculating the probability of interaction between two taxa based on their local abundances, any uncertainty in their abundances would introduce uncertainty in the interaction probability at the local scale. If we take into account the uncertainty of the interaction probability, a Beta-Binomial distribution can be used to predict the number of networks in which the interaction occurs. Empirically observing an interaction between two taxa at a given location and time provides important information that can be used to update previous estimates of P(B_i, j), informing us of the conditions that enabled them to interact locally. By sampling binary interactions in different contexts, we can thus estimate their local variability more precisely.

The outcome of probabilistic interactions may also be quantitative

Even though binary interaction networks constitute a highly valuable source of ecological information (Pascual et al. 2006), they overlook interaction strengths. Represented in a quantitative adjacency matrix W, interaction strengths better describe the energy flows, demographic impacts or frequencies of interactions between nodes (Berlow et al. 2004; Borrett & Scharler 2019), with W_i, j being a natural ℕ or real ℝ number depending on the measure. For example, they may represent local interaction rates between pairs of taxa (e.g., the flower-visiting rates of pollinators in a mutualistic network, Herrera 1989). When interaction strengths characterize predation pressure on prey, they can serve as parameters in a Lotka-Volterra model (e.g., Emmerson & Raffaelli 2004). The extra amount of ecological information in quantitative networks typically comes at a cost of greater sampling effort and data volume (Strydom et al. 2021), especially when using predictive models that quantify the uncertainty and variability of quantitative interactions (Berlow et al. 2004).

Like binary interaction networks, the uncertainty and variability of interaction strengths can be represented probabilistically. Interaction strengths can follow many probability distributions depending on the measure. For instance, they can follow a Poisson distribution $W_{i, j} \sim {\rm Poisson}(\lambda_{i, j} t_0)$ when predicting the number of interactions between individuals during a time interval t₀, with λ_i, j being the expected rate at which individuals of taxa i and j interact (e.g., the expected number of prey j consumed by all predators i). The Poisson distribution can also be 0-inflated when taking into account non-interacting taxa (e.g., Boulangeat et al. 2012 employ a 0-inflated model to analyze species abundance following the modeling of species presence and absence), which constitute the majority of taxa pairs in most local networks (Jordano 2016). Regardless of the measure, representing interaction strengths probabilistically enables the propagation of uncertainty from pairwise interactions to the broader dynamics of ecological networks.

Because of the methodological difficulties typically encountered when building deterministic quantitative networks, binary interaction networks, which are easier to sample (Jordano 2016) and predict (Strydom et al. 2021), have been more frequently studied and used. Mathematical models such as Ecopath (Plagányi & Butterworth 2004) partially mitigate these difficulties, but the number of biological parameters required to make predictions hinders their application in many systems. Moreover, most published probabilistic interaction networks (e.g., Strydom et al. 2022) and methods (e.g., Poisot et al. 2016) involve probabilistic interactions whose outcome is binary. This underlines the need for better guidelines on the interpretation and manipulation of probabilistic interactions with binary outcomes first, to ensure the appropriate use of these networks and methods. For these reasons, the primary focus of the remainder of this manuscript is on the interpretation of interaction probabilities that determine the presence or absence of interactions, in both local networks and metawebs.

Local networks: communities interacting in space and time

What are local probabilistic interactions?

Local networks of probabilistic interactions describe how likely taxa are to interact in a local context. Local interactions are contingent upon the environmental conditions experienced by the community and the matching of taxa’s local biological traits. In local networks, edges commonly represent our degree of belief that two taxa interact in nature, but can also represent the probability of empirically observing this interaction (Catchen et al. 2023). Local interactions may thus arise from both the ecological (realized interactions) and sampling (observed interactions) processes taking place locally.

Local networks are delineated within a particular location and time. We define space as the collection of geographic coordinates (x,y,z), with (x,y) representing longitude and latitude coordinates, and z denoting either altitudes or depths. These point coordinates delineate the spatial boundaries of the system, which may be portrayed as a polyhedron. Ecological interactions may vary along latitudinal and altitudinal gradients, as evidenced by changes in hummingbird-plant interactions (Weinstein & Graham 2017a, b) and mosquito biting rates (e.g., Kulkarni et al. 2006) at different elevations. On the other hand, time is defined as the specific time period within which interactions were either observed or predicted. Even though space and time are continuous variables that should yield probability densities of interactions (i.e., relative likelihoods of interactions occurring at infinitesimal locations and instants in time), these definitions enable them to be conceptualized as distinct patches and time segments. Treating space and time as discrete dimensions aligns with the common sampling methods of ecological networks and provides probabilities of interactions, which can be obtained by integrating probability densities over space and time. We can quantify both an area A₀ and a duration t₀ with these definitions. By studying probabilistic local interaction networks, we may thus conduct spatiotemporal analyses of local interactions (Box 1), enhancing our understanding of interactions occurring in distinct environmental contexts.

What are local probabilistic interactions conditioned on?

Local interactions may be conditioned on co-occurrence

The probability that two taxa i and j interact in a local network L_{x, y, z, t} (spatial and temporal subscripts hereafter replaced by the shorter subscript k for clarity) can be conditioned on many environmental and biological factors. In addition to network area (or volume) A₀ and duration t₀, they may be conditioned on taxa co-occurrence X_i, j, k, which is usually Boolean, describing if the geographic distributions of both taxa overlap within the study area. As illustrated in Box 1, co-occurrence may be modeled probabilistically, in which case it may conform to a Bernoulli distribution $X_{i,j,k} \sim {\rm Bernoulli}(P(X_{i,j,k}))$. The probability of co-occurrence can be calculated using the individual (marginal) occurrence probabilities P(X_i, k) and P(X_j, k). Given that taxa occurrences are not independent of each other, the probability of co-occurrence can be calculated by multiplying the probability of occurrence of one taxon by the probability of occurrence of the other given that the first one is present:

P(X_i, j, k) = P(X_i, k,X_j, k) = P(X_i, k|X_j, k)P(X_j, k)   (1)

The probability of co-occurrence P(X_i, j, k) can be estimated through the application of joint species distribution models (e.g., Pollock et al. 2014), potentially taking into account biotic interactions (Staniczenko et al. 2017). Given that the probability that two non-co-occurring taxa interact locally is zero (i.e., P(L_i, j, k|X_i, j, k=0) = 0), the probability of local interaction can be obtained by multiplying the probability of interaction given co-occurrence with the probability of co-occurrence:

P(L_i, j, k) = P(L_i, j, k|X_i, j, k) × P(X_i, j, k).   (2)

Knowing that two taxa co-occur improves our estimation of the probability that they interact locally by mitigating a potential source of uncertainty.

Local interactions may be conditioned on different environmental and biological factors

Local interactions may also be conditioned on local environmental factors such as temperature (Angilletta et al. 2004), precipitation (Woodward et al. 2012), habitat structure (Klecka & Boukal 2014), and presence of other taxa in the network (Kéfi et al. 2012; Pilosof et al. 2017). We use the variable E_k to describe the local environmental context in which interaction probabilities were estimated. For example, in a mesocosm experiment estimating interaction probabilities between predators and prey with and without shelters, E_k would represent the presence or absence of these shelters. Like co-occurrence, E_k can also be modeled probabilistically when the variability or uncertainty of environmental factors is considered. E_k represents all environmental variables that were taken into consideration when measuring interaction probabilities; it is therefore a subset of all environmental factors acting on ecological interactions.

Other important factors that can impact interaction probabilities at the local scale are taxa local abundances N_i, k and N_j, k, which affect encounter probabilities (Canard et al. 2012), and local traits T_i, k and T_j, k (e.g., body mass, longevity, and habitat breadth, Caron et al. 2024), which determine the ability of individuals to interact after encountering each other (Poisot et al. 2015). Local interaction probabilities may also be conditioned on higher-level properties of the community, which we denote by f(L_k). Many topological null models (i.e., statistical models that randomize interactions by retaining certain properties of the network while excluding others) provide interaction probabilities from selected measures of network structure, such as connectance (Fortuna & Bascompte 2006) and the degree distribution (Bascompte et al. 2003). Biological factors, whether at the scale of individual taxa pairs or the community, may thus impact how we estimate and define interaction probabilities.

Local interactions may be conditioned on biological feasibility

Local interactions must be biologically feasible before occurring at a specific time and space. A local probability of interaction P(L_i, j, k) can be expressed as the product of the probability of local interaction given that the two taxa can potentially interact P(L_i, j, k|M_i, j=1), which we sometimes denote as P(L_i, j, k|M_i, j) for the sake of simplicity, with their probability of regional interaction P(M_i, j):

P(L_i, j, k) = P(L_i, j, k|M_i, j=1) × P(M_i, j=1).   (3)

Low values of P(L_i, j, k|M_i, j) indicate that feasible interactions rarely occur locally, intermediate values around 50% suggest considerable spatiotemporal variability, while high values indicate that regional interactions are nearly always realized locally. The local probability of interaction between a given pair of taxa is thus always equal to or below their probability of regional interaction. Taking into account biological feasibility in our estimation of local interaction probabilities leverages information from the metaweb to better predict the local occurrence of interactions (Dansereau et al. 2023; Strydom et al. 2021).

Conditional variables must be explicitly stated

The probability that two taxa i and j interact in a local network L_k can thus be conditioned on their co-occurrence X_i, j, k (or more explicitly on their occurrences X_i, k and X_j, k), local abundances N_i, k and N_j, k, local traits T_i, k and T_j, k, local environmental conditions E_k, network area (or volume) A₀, time interval t₀, network properties f(L_k), and biological feasibility M_i, j. The local probability of interaction is described by the following expression when all of these conditional variables are included:

P(L_i, j, k|X_i, k,X_j, k,N_i, k,N_j, k,T_i, k,T_j, k,E_k,A₀,t₀,f(L_k),M_i, j).   (4)

These conditional variables do not all need to be considered at all times. The representation of the local context in which probabilities are estimated and the variables that should be taken into consideration depend on the study system, the objectives of the study, and the resources available to the researchers. For example, Gravel et al. (2019) analyzed local European host-parasite networks of willow-galling sawflies and their natural enemies, all referenced in space and time, to infer probabilities of local interactions between co-occurring species. This was achieved by including temperature and precipitation as conditional variables in their models. In Box 2, we reuse these data to show the extent of variation among these local networks. We do so by measuring their dissimilarity with the regional network (metaweb aggregating all local interactions), both in terms of species composition and interactions. We built local probabilistic networks following eq. 3, showing that insufficient local variation (high probability of local interaction among potentially interacting species) results in an overestimation in both the number of interactions and connectance (i.e., the proportion of all of the non-forbidden links that are realized). This analysis was conducted for illustrative purposes, and other conditional variables could have been used to make these comparisons.

When accounted for, conditional variables should be clearly described in the documentation of the data (Brimacombe et al. 2023), preferentially in mathematical terms to avoid any confusion in their interpretation and to limit manipulation errors during their re-use. For instance, ecologists should be explicit about their consideration (P(L_i, j, k|X_i, j, k)) or not (P(L_i, j, k)) of co-occurrence in their estimation of local interaction probabilities, as this can change our interpretation of the data and understanding of potential uncertainty sources. Reporting the scale and level of aggregation of the data enables us to more accurately study the underlying ecological processes (Clark et al. 2011) and manipulate or propagate uncertainty to different aggregation levels (Simmonds et al. 2024). In Tbl. 1, we present examples of studies that used different expressions of probabilistic interactions with different conditional variables. We included in this table the probability of empirically observing an interaction that is realized locally P(O_i, j, k|L_i, j, k) to underscore the distinction between local observations and actual realizations of interactions.

How are local probabilistic interactions estimated?

Various statistical models can be used to estimate local interaction probabilities, some of which are presented in Tbl. 1. These models can be based on multiple conditional variables. Although these variables correspond to distinct ecological inquiries or mechanisms related to ecological interactions, they may covary with each other, such as the possible dependence of X_i, j, k and E_k on spatial and temporal scales. When estimating interaction probabilities using e.g. a generalized linear model with multiple explanatory variables that might not all be independent, it may become important to address collinearity. In such cases, it may be necessary to use variable selection techniques before fitting the model to data to mitigate this issue. Other challenges and opportunities associated with predictive models of species interactions are reviewed in Strydom et al. (2021).

When using multiple competing models to estimate local interaction probabilities, rather than selecting a single model that best fits the data, model averaging may enhance our estimations. Model weights represent the probability that each model is the most suitable for explaining the data, and may be measured using Akaike weights (Burnham & Anderson 2004; Wagenmakers & Farrell 2004). For instance, given two competing models mod₁ and mod₂ with respective probabilities (or weights) P(mod₁) and P(mod₂), the average probability of interaction P(L_i, j, k) can be calculated as follows:

P(L_i, j, k) = P(L_i, j, k|mod₁) × P(mod₁) + P(L_i, j, k|mod₂) × P(mod₂).   (5)

Model averaging takes into account the uncertainty of model structure in our estimation of local interaction probabilities. Regardless of the model used for prediction, it is crucial to quantify and disclose all sources of uncertainty to understand better the validity and limitations of our predictions (Simmonds et al. 2024).

Box 1: A spatiotemporally explicit model of interactions

Ecologists may resort to predictive models to reconstruct local networks across time and space. We introduce and develop a simple generative Bayesian model for probabilistic local interactions, which explicitly accounts for their spatiotemporal variability. Our model is not designed for regional interactions, which do not vary spatially nor temporally. Rather, it could prove valuable for generating new data on local interactions across time and space, following parameter inference.

As indicated by Eq. 2, the probability that two taxa i and j interact locally can be obtained by multiplying their probability of interaction given co-occurrence with their probability of co-occurrence. The probability of interaction given co-occurrence can be made temporally explicit by modeling it as a Poisson process with rate parameter λ_k. This parameter represents the local expected frequency of interaction between co-occurring taxa. The probability that two co-occurring taxa interact during a time interval t₀ can be given by:

P(L_i, j, k|X_i, j, k) = 1 − e^−λ_kt₀,   (6)

which tends toward 1 as t₀ → ∞ if λ_k > 0. In other words, two co-occurring taxa with a nonzero rate of interaction will inevitably interact in a sufficiently long time interval.

The occurrence of an interaction between i and j may be the result of a Bernoulli trial with probability P(L_i, j, k). A Bayesian model can be built using the preceding equations to generate new interaction data, following the inference of the λ_k and P(X_i, j, k) parameters.

L_i, j, k ∼ Bernoulli(P(L_i, j, k))   (7)

P(L_i, j, k) = P(X_i, j, k)(1−e^−λ_kt₀)   (8)

P(X_i, j, k) ∼ Beta(2,2)   (9)

λ_k ∼ Exponential(2)   (10)

In Fig. 1, we show the variation in the probability of interaction under different parameter values. In the right panel, we notice that the probability of interaction always converges toward the probability of co-occurrence P(X_i, j, k), for all positive values of the interaction rate.

This model can be customized in different ways, such as linking both parameters to specific environmental or biological variables. For instance, the probability of co-occurrence could be modeled as a function of climatic variables, while the interaction rate parameter could be modeled based on taxa abundances.

Box 2: Dissimilarity of local host-parasite networks

We present a way to assess local network variability and dissimilarity regarding species composition and interactions. We do so by comparing local tripartite host-parasite networks to the metaweb using data from Kopelke et al. (2017). This collection of networks consists of interactions between willows, willow-galling sawflies, and their natural enemies sampled across Europe. All data manipulation and methods are described in Appendix 1. All code and data to reproduce these analyses are available at the Open Science Framework (TBD).

In Fig. 2a-b, we show how the dissimilarity between the metaweb of binary interactions and aggregated local networks changes with the number of sampled local networks. We compared the metaweb and the aggregated local networks using the dissimilarity in species composition (β_S, Fig. 2a) and the dissimilarity of interactions between common species (β_OS, Fig. 2b) indices (Poisot et al. 2012). Expectedly, local networks are highly dissimilar from the metaweb in terms of species composition, especially when only a limited number of sites have been sampled. This is because few species from the metaweb (species pool) occur locally. Moreover, we observe a peak in the dissimilarity of interactions between common species at intermediate sampling levels. This suggests that species are collected faster than their interactions. With a limited number of sampled local networks, few regional interactions are observed locally. Adding more sites brings new species, but not always their interactions. Quadratic relationships of network properties with sampling effort were also observed by McLeod et al. (2021).

Next, we investigate how the number of local interactions and connectance scale with the number of sampled (aggregated) local networks of probabilistic or binary interactions (Fig. 2c-d). By comparing the scaling relationships observed in local networks of binary and probabilistic interactions, we observe that high values of P(L_i, j, k|M_i, j) lead to systematic overestimations in the number of interactions and connectance, especially when P(L_i, j, k|M_i, j) = 1 (i.e., when local and regional probabilities of interactions are equivalent). This suggests that high values of P(L_i, j, k|M_i, j) do not adequately capture the variability of local interactions. However, these biases tend to diminish as the number of sampled networks increases, indicating that most interactions are eventually captured when P(L_i, j, k|M_i, j) is high. In contrast, low values of P(L_i, j, k|M_i, j) lead to missing interactions, resulting in an underestimation of the number of interactions and connectance. These results underscore the importance of using the appropriate level of variability when estimating local interaction probabilities.

Metawebs: regional catalogs of interactions

What are regional probabilistic interactions?

Metawebs (Dunne 2006) are networks of potential interactions over broad spatial, temporal, and taxonomic scales (e.g., food webs at the continental scale). Potential interactions describe the biological capacity of taxa to interact under optimal or feasible environmental conditions, which is typically assessed at the regional scale. Metawebs of probabilistic interactions are particularly useful in situations where there is uncertainty in the ability of taxa to interact (Strydom et al. 2023). They may also be used as informative priors of local interactions. Therefore, building a metaweb of probabilistic interactions may be an important first step before predicting networks at finer scales.

In contrast to local networks, where interaction probabilities arise from the variability of interactions and the lack of information on the conditions, interaction probabilities in metawebs solely result from a lack of knowledge. This uncertainty arises due to insufficient interaction data, especially for taxa that have not yet been observed to co-occur, and uncertainties in trait-matching models. As data accumulates, interactions in metawebs should tend towards binarity, either taking a value of 1 (observing an interaction at least once) or approaching 0 (repeatedly failing to observe an interaction between co-occurring taxa). Confidently observing an interaction once confirms its biological feasibility, but failing to observe it (even on multiple occasions) does not ensure that it is non-feasible (e.g., due to false negatives, Catchen et al. 2023). While local interaction probabilities are irreducible because of local variability, the uncertainty of regional interactions reduces to 0 with the addition of information. Moreover, although neutrally forbidden interactions (i.e., forbidden interactions between rare species, Canard et al. 2012) have low probability values in local networks, they would have a probability of 1 in the metaweb (this is because the species’ traits could support an interaction if they were to encounter each other at high enough abundances). Likewise, non-co-occurring taxa may have a non-zero probability of interaction in the metaweb. Regional interaction probabilities are thus fundamentally different from local interaction probabilities, both in terms of uncertainty sources and probability values.

The extent of sampling effort influences our evaluation of probabilities of regional interactions, as sampling over a larger area or for a longer duration enables us to capture a greater number of regional interactions (McLeod et al. 2021). However, in contrast with local networks of probabilistic interactions, regional interactions are not evaluated for any particular local context, which impacts how they scale with space and time. In Box 3, we discuss the differences in spatial and temporal scaling of regional interactions compared to local interactions. We do so using the host-parasite networks of Kopelke et al. (2017) as an illustration of spatial scaling. Understanding the effect of spatial and temporal scales (including sampling effort) on local and regional interaction probabilities is important for effectively propagating uncertainty across scales and highlighting the fundamental differences between these two types of networks.

What are regional probabilistic interactions conditioned on?

Regional interactions describing biological feasibility are conditioned on traits

Potential interactions describe what we refer to as the biological feasibility of interactions, which is based solely on the regional traits distributions T_i and T_j of taxa i and j, respectively. We define regional traits distributions as the range of phenotypes that a taxon can express across various environments. Local traits T_i, k and T_j, k, which vary spatially and temporally because of phenotypic plasticity and local environmental variability (Berg & Ellers 2010), are a subset of regional traits. A probability of potential interaction in a metaweb M describing the biological feasibility of interactions may be expressed as:

P(M_i, j|T_i,T_j),   (11)

which, in contrast with local networks, is not conditioned on any spatial, temporal, co-occurrence or environmental variables (Tbl. 1). Because phylogenetically close species often share similar traits, we should expect that closely related species will have similar interacting partners. We can thus use phylogeny to predict species traits and infer regional interactions (Eklöf & Stouffer 2016; Stouffer et al. 2012; Strydom et al. 2022). The taxonomic level at which interactions are evaluated also influences the distribution of regional traits. However, as explained in Box 4, there is no fundamental difference in the taxonomic scaling of regional and local interactions (i.e., how interaction probabilities change with taxonomic level) because they both depend on trait aggregation.

The biological feasibility of interactions expresses our degree of belief that there exists at least one combination of phenotypes that could support an interaction if they were to encounter each other, assuming they had enough time to interact. Evaluating this probability is conducted without incorporating the environmental conditions under which they encounter each other into the model. It is the complement of the probability P(F_i, j|T_i,T_j) of forbidden interactions (i.e., the probability that their traits do not support an interaction), which is based uniquely on biological traits:

P(M_i, j|T_i,T_j) = 1 − P(F_i, j|T_i,T_j).   (12)

For example, let i be a western diamondback rattlesnake (Crotalus atrox) and j, a wood lemming (Myopus schisticolor). These two taxa never co-occur, the rattlesnake being adapted to warm regions of North America (Castoe et al. 2007) and the lemming, to northern habitats of Eurasia (Fedorov et al. 2008). As we lack direct observations of an interaction between these two species, we have to rely on expert knowledge or trait-matching models to estimate their probability of potential interaction. To accurately estimate this probability using trait-matching models, it is crucial to ensure that the set of traits considered reflects the overall traits distributions of both taxa. We could for instance consider their average body mass and the average phylogenetic distance of lemmings to rattlesnakes’ prey. Doing so, we might find a high probability of potential interaction based on these traits. This example illustrates how regional interactions describing biological feasibility may be estimated solely based on traits, without taking into account environmental conditions.

Regional interactions describing ecological feasibility are conditioned on traits and environmental conditions

The biological feasibility of interactions should not be confused with what we refer to as the ecological feasibility of interactions. A probability of potential interaction in a metaweb M^* describing the ecological feasibility of interactions may be expressed as:

P(M_i, j^*|T_i,T_j,E),   (13)

where E is the environmental conditions under which potential interactions are evaluated (Tbl. 1). Unlike E_k, these environmental conditions do not represent conditions occurring at specific locations. Ecological feasibility represents the probability that two taxa interact if they were to encounter each other under given environmental conditions, assuming they had enough time to interact. Incorporating environmental conditions into a trait-matching model may be important when there is high covariance between the environment and traits. For instance, in our example involving rattlesnakes and lemmings, the probability of potential interaction between these two species may be low in most environmental conditions. Western diamondback rattlesnakes may be unactive under low temperatures (Kissner et al. 1997), whereas wood lemmings may have low tolerance to high temperatures (Kausrud et al. 2008). The probability that an interaction is ecologically feasible is always lower than the probability that it is biologically feasible, even across all environmental conditions:

∫_EP(M_i, j^*|T_i,T_j,E)dE ≤ P(M_i, j|T_i,T_j).   (14)

This is because the biological feasibility of an interaction is a prerequisite for its ecological feasibility. In other words, biological feasibility is necessary but not sufficient for an interaction to be ecologically feasible. Our discussion of metawebs focuses on the biological feasibility of interactions since most methods developed for inferring probabilities of regional interactions do not explicitly take into account environmental conditions (e.g., Strydom et al. 2022).

How are regional probabilistic interactions estimated?

Starting from a selected set of taxa, which are usually distributed within a broad region of interest, metawebs can be built using different data sources, including literature review (e.g., Maiorano et al. 2020), aggregated interaction data (e.g., Gravel et al. 2019; Saravia et al. 2022), trait-matching models (e.g., Shaw et al. 2024; Strydom et al. 2022), and expert knowledge. Every pair of taxa that have confidently been observed to interact at least once can be given a probability of 1 (i.e., P(M_i, j) = 1) since we know that they can interact. This differs from local networks of probabilistic interactions, where interaction events may remain stochastic (i.e., P(L_i, j, k) < 1) even after empirically observing interactions due to their spatiotemporal variability. Interactions that were never observed typically have low probability values in local networks and vary from low to high values in metawebs, contingent upon taxa traits distributions (reaching 0 for forbidden links). The aggregation of model predictions and data from different sources thus tends to raise the number of potential interactions in metawebs.

When using local interaction data to estimate probabilities of regional interactions, repeatedly failing to observe an interaction between two co-occurring taxa should decrease the probability that the interaction is biologically feasible. Using Bayes’ theorem, the probability that the interaction is biologically feasible given that it was never observed locally, P(M_i, j=1|O_i, j, k=0), may be calculated as follows:

$$P(M_{i, j} = 1 | O_{i, j, k} = 0) = \frac{P(O_{i, j, k} = 0 |M_{i, j} = 1) \times P(M_{i, j} = 1)}{P(O_{i, j, k} = 0)}.\qquad{(15)}$$

The reduction in the probability of regional interaction after considering that it was never observed locally (i.e., P(M_i, j=1|O_i, j, k=0) < P(M_i, j=1)) occurs because P(O_i, j, k=0|M_i, j=1) must be lower than P(O_i, j, k=0), i.e. there is a higher chance of observing an interaction when it is biologically feasible.

Observations of interactions may be false positives because of observation errors due to taxonomic misidentifications and ecological misinterpretations, such as those involving phylogenetically close species or cryptic species and interactions (Pringle & Hutchinson 2020). Likewise, forbidden interactions may be false negatives, e.g. if they have been evaluated based on unrepresentative or incomplete traits distributions. Employing Bayesian models proves valuable when estimating interaction probabilities in metawebs (e.g., Bartomeus et al. 2016; Cirtwill et al. 2019). This improvement is achieved by updating prior information regarding the feasibility of interactions (e.g., experts’ prior assessments of interaction probabilities) with empirical data on interactions and traits. By improving our estimation of potential interaction probabilities, we may build more reliable metawebs that adequately reflect our uncertainty on the biological feasibility of interactions.

Box 3: Spatial and temporal scaling of interactions

Local networks and metawebs have distinct relationships with space (area or volume) and time (sampling effort or duration). Local probabilities of interaction scale both spatially and temporally, because local interactions have more opportunities to be realized in larger areas and longer durations. In a larger sampling area and duration, we increase the likelihood of sampling favorable conditions for interactions to occur. If a local network of probabilistic interactions L₁ with an area A₁ is compared to a larger network L₀ with an area A₀, and A₁ is entirely nested within A₀, interaction probabilities should be lower in the smaller network, i.e. P(L_i, j, 1|A₁<A₀) ≤ P(L_i, j, 0|A₀). However, if A₁ and A₀ are disjoint, interaction probabilities could be higher in the smaller area, contingent upon local environmental and biological conditions. In contrast, regional probabilities of interaction do not scale with space and time. The probability of two taxa potentially interacting should be the same in all metawebs in which they are present regardless of scale, provided that the data and methods used for estimation are consistent. This is because they depend solely on the biological capacity of two taxa to interact, regardless of co-occurrence and local environmental conditions. However, probabilities of regional interactions may change, tending to become more definitive, with increased sampling effort.

In Fig. 3, we show how the expected number of local host-parasite interactions scales with the spatial boundary of the network (represented by an expanding latitudinal window) in comparison with regional interactions. We do so using the host-parasite networks of Kopelke et al. (2017). The increase in the number of regional interactions is due to the inclusion of more species in a larger area. To ensure a conservative comparison between aggregated local and regional networks, we employed equal interaction probabilities (i.e., using P(L_i, j, k|M_i, j) = 1) in both types of network. This means that local interaction probabilities could not increase further when aggregating them. Despite this, we notice that the total number of regional interactions scales more rapidly than local interactions. This is because numerous regional interactions involve species that never co-occur, and as a result, these interactions are not captured in local networks. All data manipulation and methods are described in Appendix 1.

Box 4: Taxonomic scaling of interactions

Given that our interpretation of the properties of ecological networks depends on their taxonomic level (Guimarães 2020), investigating the taxonomic scaling of interactions (i.e., how interaction probabilities change with taxonomic level) is important. There are no inherent differences between the taxonomic scaling of local and regional interactions. The taxonomic level of interactions impacts the definition of nodes. Local and regional interaction probabilities are not directly conditioned on taxonomic scale. However, some conditional variables (e.g., trait distribution) may covary with taxonomic scale. In such cases, local and regional interaction probabilities would change taxonomically following the scaling of these variables.

In both types of interactions, transitioning to a broader level of organization (e.g., from a species-level network S to a genus-level network G) can be done using interaction probabilities from finer scales. For example, in a network with n₁ species of genus g₁ and n₂ species of genus g₂, one can calculate the probability that at least one species from genus g₁ interacts with at least one species from genus g₂ (i.e., the probability that the genus-level interaction occurs) as follows:

$$P(G_{g_1, g_2}) = 1 - \prod_{i = 1}^{n_1}\prod_{j = 1}^{n_2}(1 - P(S_{g_{1,i}, g_{2,j}})),\qquad{(16)}$$

where g_1, i and g_2, j are the species of the corresponding genus and assuming independence between species-level interactions. In contrast, a different approach is necessary when transitioning from a broader to a finer level of organization. This is because the knowledge of an interaction between two genera does not guarantee that all possible pairwise species combinations will also interact. One possible method is to build a finer-scale network by generating probabilities of interaction through random sampling from a beta distribution, parameterized by the broader-scale network.

Fundamentally, the taxonomic scaling of interactions involves aggregating interactions between individuals into larger groups. Interaction probabilities at broader taxonomic scales should thus conform to probabilities of interactions between individuals. For example, Canard et al. (2012) built a species-based network using simulated individual-based networks. In local individual-based food webs, the probability that two individuals interact reflects our degree of belief that one individual will consume the other. Likewise, in local species-based food webs, the probability that two species interact represents our degree of belief that at least one individual from the predator species will consume at least another individual from the prey species. In that regard, taxonomic scaling is analogous to the spatial and temporal scaling of interactions, as they all represent different ways to aggregate individuals into broader groups (either spatially, temporally, or taxonomically).

Box 5: Sampling for binary interaction networks

Networks of binary interactions may be predicted by performing independent Bernoulli trials for each probabilistic interaction. This is particularly useful when analyzing the structure of probabilistic interaction networks in the absence of specific analytical formulas (Poisot et al. 2016), even though it may introduce biases in our estimations when connectance is low (Chagnon 2015; Poisot & Gravel 2014). There are at least two techniques to sampling binary interaction networks across space, each predicting a binary interaction network for each location k within a given region. The first technique involves performing a single Bernoulli trial for each pair of taxa based on their regional probability of interaction:

$$M_{i, j} \sim {\rm Bernoulli}(P(M_{i, j})).$$

In employing this technique, we predict a single metaweb of binary interactions for each simulation. Every pair of taxa predicted to interact in this metaweb will be treated as interacting in all localized networks where they co-occur, i.e. L_i, j, k = M_i, j when X_i, j, k = 1. This will result in local pairwise interactions without spatial variation.

The second technique is to independently sample each local network of probabilistic interactions:

$$L_{i, j, k} \sim {\rm Bernoulli}(P(L_{i, j, k})).$$

This can be achieved by first generating distinct probabilistic interaction networks for each location. Because binary interactions are sampled independently for each location, this second technique captures network structure across space and time more effectively. When sampling binary interactions from local interaction probabilities, it is crucial to sample at the same spatial scale for which probabilities were estimated to prevent systematic biases in predictions.

In Fig. 4, we compare the average connectance of binary interaction networks resulting from these two sampling techniques. We sampled regional and local interactions from our host-parasite networks of probabilistic interactions (Kopelke et al. 2017), generating a number of binary interaction network realizations for each site in the dataset. These two sampling techniques yield different outcomes, particularly for intermediate values of P(L_i, j, k|M_i, j) of 0.50, which represent instances where regional interactions do not consistently manifest locally (i.e., with the largest local variability). As anticipated, we observe that sampling binary interactions from the metaweb tends to overestimate connectance on average compared to sampling them from local networks (Fig. 4). We also observe an increase in the variability of connectance when employing a single simulation (Fig. 4a-c, cross markers), which is a more tangible representation of the process leading to the realization of local interactions in nature. All data manipulation and methods are described in Appendix 1.

Both sampling techniques assume independence between interactions, which might not hold true in reality. Covariation among interactions could exist even if we do not explicitly condition interactions on others. For example, an interaction between two taxa could be more probable when another interaction occurs. The consequences of this assumption of independence on the prediction of network structure have yet to be empirically examined.

Future perspectives

In this contribution, we underline the importance of network documentation for adequately interpreting and manipulating probabilistic interaction data. The mathematical representation of probabilities and their statistical properties depend on the type of interactions (local or regional) and the conditions under which these interactions were evaluated. We show that local networks and metawebs of probabilistic interactions differ in their relationship to spatial and temporal scales (Box 3), with regional interactions remaining consistent across scales. In contrast with metawebs, local interactions are measured in a specific context (e.g., in a given area, time, and biological and environmental conditions) and depend on taxa co-occurrence. These differences bring to light the need to use probabilistic data with caution, for instance when generating network realizations of binary interactions across space (Box 5). Clear documentation describing the type of interaction and the variables used in their estimation are required to ensure adequate data manipulation. Sound data practices and foundations for probabilistic thinking in network ecology facilitate reliable assessments of the spatiotemporal variability and uncertainty of biotic interactions. Here we identify key research priorities for improving our understanding of probabilistic local and regional interactions.

Predicting local networks from metawebs

Metawebs are a valuable source of ecological information for predicting local networks across time and space. Local networks of binary interactions can be reconstructed by selecting a subset of taxa and interactions from the metaweb (Dunne 2006). Determining the list of taxa to select can be achieved empirically (e.g., observed occurrence data for a site) or numerically (e.g., species distribution models). As species composition is arguably easier to sample and predict than pairwise interactions, the primary challenge lies in deciding which interactions to select from the metaweb. Inferring the structure of local networks from the metaweb before predicting local pairwise interactions could hold promise (Strydom et al. 2021), considering that the structure of local networks is constrained by the metaweb (Saravia et al. 2022).

Inferring local networks of probabilistic interactions from a metaweb comes with its own set of challenges. For example, Dansereau et al. (2023) inferred spatially-explicit food webs from a metaweb of probabilistic trophic interactions between Canadian mammals. Their predicted localized food webs are downscaled versions of the metaweb (i.e., localized metawebs with the same interaction probabilities as those in the regional metaweb). To infer local networks as defined in this manuscript (i.e., describing local realizations of interactions), local interaction probabilities must be smaller than regional interaction probabilities. Inferring local networks from a metaweb by maintaining identical interaction probability values introduces systematic biases into the predictions, as discussed in Box 2 (unless networks are seen as downscaled metawebs).

As suggested by McLeod et al. (2021), metawebs establish an upper limit for local interactions (similarly for metawebs of probabilistic interactions, Strydom et al. 2023). In other words, the probability that two taxa interact at a specific location and time is consistently lower or equal to the probability of their regional interaction, regardless of the conditional variables considered:

P(L_i, j, k|...) ≤ P(M_i, j|T_i,T_j).   (17)

Moreover, the probability that two taxa possess the biological capacity to interact must be higher than the probability of them interacting at any location and time because they may never co-occur or encounter locally. Specifically, the cumulative probability of local interaction across all spatial, temporal, and environmental conditions must be less than the probability of regional interaction, i.e.

∫_Ek∫_A0∫_t0P(L_i, j, k|E_k,A₀,t₀) dt₀ dA₀ dE_k ≤ P(M_i, j|T_i,T_j).   (18)

Estimating more precisely the probability P(L_i, j, k|M_i, j) that two taxa interact locally if they can potentially interact allows for improved predictions of local networks from the metaweb of probabilistic interactions. This task is challenging due to the variability of this probability across space and time, as well as its variability across pairwise interactions within a network. Using simple models of P(L_i, j, k|M_i, j), as shown in Appendix 1, represents an initial step toward the overarching objective of reconstructing local networks from metawebs.

Quantifying and reducing interaction uncertainty

While sampling biological communities decreases the uncertainty of interactions by accumulating evidence for their feasibility and local realization, there is a limit to how much we can reduce uncertainty. In metawebs, probabilities reflect our limited knowledge of interactions, which is expected to improve with a larger volume of data. Regional interactions should become more definitive (with probabilities approaching 0 or 1) as we investigate various conditions, including different combinations of species traits.

In comparison, local interaction probabilities represent both our knowledge uncertainty and their spatiotemporal variability. Owing to environmental heterogeneity, there will invariably be instances in which an interaction occurs and others in which it does not, across different times and locations, irrespective of the extent to which we can improve our knowledge of its biological feasibility and the local conditions that facilitate its occurrence. When local networks describe probabilities of observing interactions rather than their realization, we must also consider observation uncertainty (sampling error) as an additional source of uncertainty. Quantifying and partitioning this uncertainty will enable us to make more accurate predictions about ecological interactions at various spatial and temporal scales, and to identify priority sampling locations to reduce this uncertainty. This will prove to be of vital importance as our time to understand nature runs out, especially at locations where the impacts of climate change and habitat loss hit harder.

Relaxing the independence assumption

Estimating local interaction probabilities independently for each taxa pair and assembling them into a network of probabilistic interactions comes with limitations. Predicting local networks of binary interactions based on these interaction probabilities assumes independence among interactions, a condition seldom respected in practice (Golubski & Abrams 2011). Relaxing this assumption is the next logical step in the stochastic representation of interactions.

A more accurate representation of the uncertainty and variability of ecological networks involves creating probabilistic networks (P(L_k) and P(M)), rather than networks of probabilistic interactions (P(L_i, j, k) and P(M_i, j)). Probabilistic networks describe the probability that a particular network of binary (or quantitative) interactions (its whole adjacency matrix) is realized. For example, Young et al. (2021) used a Bayesian approach to estimate the probability of different plant-pollinator network structures derived from imperfect observational data. A probability distribution of ecological networks may also be derived using the principle of maximum entropy given structural constrained (e.g., Cimini et al. 2019; Park & Newman 2004).

Regardless of the method used, generating probabilistic local networks could lead to more accurate predictions of local networks of binary interactions by bypassing the independence assumption. Probabilistic networks could serve as an alternative to null hypothesis significance testing when comparing the structure of a local network to some random expectations or, as done in Pellissier et al. (2018) and Box 2, to the metaweb. These random expectations are typically derived by performing a series of Bernoulli trials on probabilistic interactions, assuming independence, to generate a distribution of networks of binary interactions to calculate their structure (Poisot et al. 2016). One could instead compare the likelihood of an observed network to the one of the most likely network structure (according to the probabilistic network distribution), thereby directly obtaining a measure of discrepancy of the empirical network. Generating probabilistic ecological networks represents a tangible challenge, one that, in the coming years, promises to unlock doors to more advanced and adequate analyses of ecological networks.

Acknowledgment

We acknowledge that this study was conducted on land within the traditional unceded territory of the Saint Lawrence Iroquoian, Anishinabewaki, Mohawk, Huron-Wendat, and Omàmiwininiwak nations. This work was supported by the Institute for Data Valorisation (IVADO) and the Natural Sciences and Engineering Research Council of Canada (NSERC) Collaborative Research and Training Experience (CREATE) program, through the Computational Biodiversity Science and Services (BIOS²) program. A special thanks to all members of the Black Holes and Revelations working group (organized by BIOS²) and the Poisot Lab for their insightful discussions and valuable feedback on this manuscript.

References