Social network models predict movement and connectivity in ecological landscapes

Abstract

Network analysis is on the rise across scientific disciplines because of its ability to reveal complex, and often emergent, patterns and dynamics. Nonetheless, a growing concern in network analysis is the use of limited data for constructing networks. This concern is strikingly relevant to ecology and conservation biology, where network analysis is used to infer connectivity across landscapes. In this context, movement among patches is the crucial parameter for interpreting connectivity but because of the difficulty of collecting reliable movement data, most network analysis proceeds with only indirect information on movement across landscapes rather than using observed movement to construct networks. Statistical models developed for social networks provide promising alternatives for landscape network construction because they can leverage limited movement information to predict linkages. Using two mark-recapture datasets on individual movement and connectivity across landscapes, we test whether commonly used network constructions for interpreting connectivity can predict actual linkages and network structure, and we contrast these approaches to social network models. We find that currently applied network constructions for assessing connectivity consistently, and substantially, overpredict actual connectivity, resulting in considerable overestimation of metapopulation lifetime. Furthermore, social network models provide accurate predictions of network structure, and can do so with remarkably limited data on movement. Social network models offer a flexible and powerful way for not only understanding the factors influencing connectivity but also for providing more reliable estimates of connectivity and metapopulation persistence in the face of limited data.

Network analysis has recently exploded across scientific disciplines, including the social sciences, physics, cellular biology, and ecology (1–4). Topics as divergent as the stability of the Internet and the structure of metabolic reactions can be depicted through network analysis (1, 3). Such analysis is beneficial because it can facilitate the identification of complex, and often emergent, patterns, and can provide hypotheses for relationships between structure and function in many systems (2, 3). Nonetheless, a growing, widespread concern in the topic of network analysis is the reliability of data used in constructing networks (4–8).

In ecology and conservation, network analysis is increasingly being used to assess population connectivity across landscapes (9–13). Because of the importance of connectivity in conservation and its relevance to population and community ecology (14–16), network analysis and the accompanying use of graph theory are often emphasized as powerful approaches that have modest data requirements for assessing connectivity (10, 11, 13). In this spatial context, resource patches are considered nodes (or vertices) and movements and/or flows between patches are described as links (or edges/arcs), such that the network is a landscape representation of patches that potentially interact (17).

Although movement is a critical component of the dynamics of populations and communities across landscapes (15, 16), movement is a notoriously difficult parameter to estimate at a landscape scale (13, 18). As a consequence, spatial network analysis is predominantly applied based on assumptions or indirect information regarding movement rather than on estimates of movements among resource patches. Links are typically assumed to occur if individuals can potentially move between patches based on maximum known dispersal distances of species or, less commonly, through assumptions regarding dispersal kernels (11, 13, 19). Given the difficulty to empirically quantify connectivity via experiments or observed movements, network constructions that use limited information on movement could provide valuable, cost-effective approaches for assessing connectivity (13). Despite this potential, direct tests of whether network analysis can provide meaningful estimates of actual connectivity have yet to emerge (17).

Statistical models developed for social networks (20, 21) provide a promising alternative approach for predicting links and assessing connectivity, although such models have not been applied in a landscape or conservation setting. These models were originally developed to test for factors influencing social relationships among individuals, but their generality offers applicability beyond the social sciences. Not only can these models predict different types of linkages, they can address formal hypotheses regarding network structure (20, 21). Social network models may be particularly relevant to connectivity assessments because they can be applied in situations where only limited data are collected on a network (20), which could be useful given the difficulties of measuring movements at landscape scales.

We consider two types of social network models, a sender–receiver model and a latent space model, to predict landscape connectivity (Materials and Methods) (20, 21). Both models come from stochastic social network models termed “latent position, random graphs,” which allow for different types of link responses (e.g., binary, Poisson) and can account for spatial dependencies in connections across networks (20–22). For connectivity assessments, a sender–receiver model allows for unobserved movement heterogeneity and directional movement among patches (i.e., movement from patch i to j is not equal to movement from patch j to i). In contrast, a latent space model estimates an unknown or “latent connectivity space” by leveraging the observed similarities in movement across patches within networks (20). Given the potential for highly directional movement across landscapes in many circumstances (23, 24) and the difficulty to estimate the complex nature of factors influencing connectivity (18, 25), these models may prove highly useful for connectivity assessments.

We test whether network constructions derived from social network models and recent approaches in spatial ecology can quantify observed connectivity and predict actual linkages in real-world, landscape networks. We contrast these network constructions using two empirical, mark-recapture datasets that span several orders of spatial magnitude: within-field movements of a cactus-feeding insect (Chelinidea vittiger) on patchy Opuntia cactus and breeding-season movements of the endangered Everglades snail kite (Rostrhamus sociabilis plumbeus) across wetlands in peninsular Florida, USA. These examples provide a rare opportunity to quantify the extent to which network constructions can predict linkages and whether constructions can recover observed landscape connectivity. We further contrast the ability of these approaches to predict linkages and recover landscape structure based on limited data by randomly removing an increasing proportion of the observed movement data and requantifying network metrics and link prediction accuracy (Materials and Methods) (cf. 6, 8).

Results

Both of these empirical examples are notable for assessing the utility of network analysis for ecological landscapes because in both situations, all patches (or nodes) were sampled in each time step, the networks captured spatial scales relevant to movements of each species (26, 27), and movement data were collected at fine temporal resolutions within each network (Materials and Methods). Based on these mark-recapture data, we observed 70 movements of C. vittiger and 108 movements of R. sociabilis plumbeus. The frequency of movements declined exponentially with distance for both species (Fig. S1). C. vittiger moved locally across cactus patches (median distance moved = 4.5 m) (cf. 26), whereas R. sociabilis plumbeus moved widely across its geographic range within breeding seasons (median distance moved = 43 km) (cf. 27). The resulting portrayal of these observed landscape networks showed that observed movements tended to be geographically localized (Fig. 1A). Movements were also highly directional: The fraction of links between patches that were reciprocal (i.e., movement in both directions) was extremely low (0.09 for C. vittiger and 0.07 for R. sociabilis plumbeus). These observed networks provided a powerful means to assess the ability of different network constructions to predict observed linkages.

Fig. 1.

The observed movement networks and network constructions for within-field movements of C. vittiger (Upper) on Opuntia cactus (n = 56 patches) and within-breeding season movements of R. sociabilis plumbeus (Lower) across wetlands within its geographic range in Florida (n = 15 wetlands). (A) The observed network, (B) maximum distance, (C) theoretical kernel, (D) empirical kernel, (E) latent space, and (F) sender–receiver constructions for each species. Node size proportional to the log(patch size) and link grayscale denotes increased relative intensity of observed and predicted movements. For kernel-based constructions, links greater than the average link prediction are shown; for latent space and sender–receiver models, link predictions greater than the threshold used for truncation are shown (based on maximum κ). Light-colored nodes highlight patches with no links to other patches.

We first constructed networks based on the maximum distance of movements observed in the literature (26, 28), which is the most common approach for constructing landscape networks when movement data are not available (9, 10, 19). This approach consistently had the lowest accuracy in predicting linkages compared with other methods, and predicted no better than chance for C. vittiger (Fig. 2 and Fig. S2). This low predictive accuracy arose because it consistently had high false-positive error rates, where it overpredicted linkages (Fig. 1B and Fig. S2).

Fig. 2.

The performance of network constructions, based on Cohen's κ, at predicting unknown linkages as a function of the amount of data used to construct the network. See Fig. S2 for other accuracy metrics.

An alternative to a maximum distance construction is to take information on a dispersal kernel (i.e., a movement probability density function based on geographic distance) to construct a landscape network (11, 12). We contrasted two kernel-based constructions for spatial networks (Materials and Methods). First, we constructed networks assuming a negative exponential kernel, which is frequently used in metapopulation ecology (16) and has been suggested as a useful model for spatial networks in other disciplines (29, 30). We contrasted this theoretical kernel construction (Fig. 1C) that does not require movement data to using the empirical movement kernel (Fig. 1D) that we estimated with mark-recapture data to construct landscape networks (Fig. S1). Both of these constructions were moderate in predictive accuracy, with each kernel-based method performing similarly (Fig. 2 and Fig. S2). This comparison suggests that, in the absence of movement data, using a theoretical kernel based on a negative exponential distribution can capture empirical kernel constructions and improve inference on connectivity over using the maximum known movement distance.

We then applied social network models to construct networks, which were generally best at predicting linkages (Fig. 1 E and F), with the sender–receiver model providing the highest predictive accuracy in both networks (Fig. 2 and Fig. S2). Interestingly, the sender–receiver models were generally more accurate at predicting linkages than other constructions, even when up to 50–80% of the data were missing when constructing the network (Fig. 2 and Fig. S2). The sender–receiver model was the only construction considered that could account for the observed directionality in movement, although it still predicted more reciprocity in movement than what was observed (the predicted fraction of reciprocal links between patches when no data were removed was 0.32 for C. vittiger and 0.18 for R. sociabilis plumbeus). In comparison with the sender–receiver model, the latent space model provided similar trends in predictive accuracy for C. vittiger, but for R. sociabilis plumbeus the latent space model suffered lower accuracy and it became the worst network construction when only very limited data were available (Fig. 2 and Fig. S2).

We also determined whether these network constructions could recover the observed landscape structure by contrasting several network metrics (Table S1) based on the predicted network relative to the observed network (Materials and Methods). Sender–receiver models consistently recovered the observed landscape structure and could do so with remarkably little data on movement, whereas the maximum distance construction was consistently the worst approach for recovering observed landscape structure (Fig. 3). For some network metrics, such as connectance (i.e., link density), kernel-based methods recovered the observed structure. We further calculated the metapopulation lifetime (31) based on these network constructions and found that the maximum distance construction predicted much higher metapopulation lifetime for both species than the other network constructions (Fig. 3).

Fig. 3.

Standardized network and metapopulation metrics based on different network constructions as a function of the amount of data used to construct the network. Shown for each metric is a measure of relative deviation of the predicted metric to the observed as (x_pred – x_obs)/x_obs. The number of clusters provides a measure of the number of units within the network above the scale of patches. Clustering coefficients capture clustering of movement in networks. Betweenness describes the average number of shortest paths traveling through a patch, whereas connectance measures the observed number of links relative to the total possible number of links in the network. Metapopulation lifetime provides a measure of the predicted persistence of the network (Table S1).

Although the utility of network constructions for assessing connectivity lies in their ability to predict linkages in landscapes (Fig. 2) and reconstruct known connectivity (Fig. 3), the network constructions we considered differ substantially in model complexity. Consequently, we also compared network constructions using model selection criteria (SI Text), which explicitly penalize for model complexity. Using this approach (Table S2), we found the same rank support for network constructions as we found when predicting links (Fig. 2), with the one exception being that the latent space model was the best fit for C. vittiger.

Discussion

Understanding and predicting connectivity are crucial for several questions in ecology and problems in conservation (14, 16). Network analysis and associated use of graph theory have been emphasized as offering a promising approach to connectivity analysis (10, 11, 13), yet we found that the way in which networks are constructed profoundly influences the resulting network and inferences on connectivity and metapopulation persistence. Although there have been several calls to revisit the foundations of network construction in other disciplines (5), this issue is extremely relevant to the use of network analysis for assessing connectivity in ecology.

The primary approach currently being used for problems in connectivity conservation—what we term the maximum distance construction—consistently overpredicted connectivity in our examples, suffered from the lowest accuracy in predicting linkages, and could not recover observed connectivity structure. This approach is frequently used because of the difficulty of collecting data on movement (e.g., 9, 10, 19). Although estimates from the maximum distance construction are often referred to as “potential connectivity” to acknowledge that actual connectivity is not measured (13), we found that this potential bears little resemblance to actual connectivity in real landscapes. This approach was worse in predicting linkages and recovering connectivity structure for C. vittiger than for R. sociabilis plumbeus, which was likely due to the differences in the prevalence of observed movements relative to the number of potential links in each landscape. This overprediction of connectivity is particularly relevant from a conservation perspective, because it translates to misleadingly high estimates of metapopulation lifetime in landscape networks. Consequently, the use of maximum distance constructions for connectivity assessment may overestimate the viability of metapopulations.

Even in the absence of empirical data on movement, our results suggest that using a theoretical kernel-based construction can provide a more viable alternative to a maximum distance construction. Theoretical kernels, such as the negative exponential distribution used here (11, 16), provide a more realistic assessment of connectivity by providing less weight to infrequent, long-distance movements. Other kernel distributions could also be used to emphasize different types of movement (32).

We found that social network models more accurately predicted linkages and recovered network structure than other constructions. Although these models required empirical movement data, the models were still more parsimonious (in terms of model selection criteria) (Table S2) than other constructions, and the sender–receiver model was accurate even when remarkably limited data on movement were available to construct landscape networks. The utility of the sender–receiver model for both networks was likely due to its ability to capture the highly directional aspect of observed connectivity and unmeasured heterogeneity of movements to and from different patches, which is likely common in many spatial networks in ecology (23, 24). These models also provide a means to test hypotheses regarding connectivity, such as the role of patch size or matrix resistance in connectivity. For instance, by accounting for the potential effects of patch size on movement, the predictive accuracy of the sender–receiver models increased for both species (Table S3). Nonetheless, link prediction accuracy from social network models could still be improved considerably. Future applications of such models may improve predictions with the addition of other relevant factors (e.g., matrix resistance, path redundancy) that may influence movement among patches (e.g., 33).

Social network models may also be useful for other types of ecological network problems (4, 34), because they provide a general and flexible framework for modeling network data. For example, Chiu and Westveld (35) recently used such models to understand the role of phylogeny in food web structure. Such models may not only advance our understanding of the causes and consequences of ecological network structure but may also improve conservation and management strategies that rely on network inferences.

Materials and Methods

Focal Species and Movement Data.

The cactus bug C. vittiger (Hemiptera: Coreidae) is dependent upon prickly pear cactus (Opuntia spp.), where it feeds, breeds, and aggregates throughout its life. We conducted a mark-recapture study in central Florida (29.4°N, 82.0°W), USA. In this area, C. vittiger uses O. humifusa, and has two or three generations per year. Adults are winged but rarely fly; instead, adult C. vittiger typically walk between cactus patches through an unsuitable matrix (26). Movements of adults are thus tractable and localized (Fig. S1) (26).

From September 2008 to December 2009, we censused all cactus patches (n = 56) in a 30 × 30-m network every 2–3 wk (except winter; 21 total visits), individually marking all adults using a nontoxic marker; these mark-recapture data allow estimation of movements among cactus patches. This network was not a closed system, but additional recapture rates surrounding this area were low. We mapped cactus patches (26) within the plot using a Trimble Global Positioning System (error <0.5 m).

The Everglades snail kite is dependent upon shallow freshwater ecosystems dominated by sparsely emergent vegetation, with its current range including suitable marsh and lake littoral habitats within a fragmented network of geographically distinct wetland units. Although often described as a panmictic population (36), short-term movement probabilities among wetlands are heterogeneous and within-breeding season movements appear more limited. We focused on within-breeding season movements of kites, because habitat suitability and prey availability are closely linked to local hydrology and can change dramatically within breeding seasons (37). Low water levels can cause kites to move in search of suitable habitat (37); these movements are associated with survival costs for younger, inexperienced individuals (27).

As a part of a long-term capture-mark-resight study, the entire breeding range of the snail kite in Florida was systematically searched via airboat during six intra-annual survey occasions, 2005–2009. Much of the population is banded with individually identifiable color-coded leg bands. During surveys, each detected individual was scanned to determine whether it was banded and the wetland location of each resighted individual was recorded, which allows for assessing movements among wetlands (for more details on field methods, see refs. 27 and 36). All techniques were approved by the Institutional Animal Care and Use Committee (approval F#149).

Maximum Distance Construction.

Most applications of network analysis to connectivity conservation use a binary matrix (with elements p_ij) that describes pairwise connections between patches. Frequently, pairwise connections are assumed if Euclidean or other distance measures (e.g., least-cost distance) are within the known maximum dispersal distances of a species of interest (9, 10, 19), what we refer to as a maximum distance construction. For both datasets, we constructed binary networks using this formulation based on the maximum observed movement distance taken from the literature (26, 28), such that when patches i and j were within this maximum distance they were considered connected (p_ij = 1), whereas patches beyond this distance were considered unconnected (p_ij = 0). Using the maximum distances observed in these datasets (rather than those taken from the literature) resulted in similar conclusions for C. vittiger and less accurate predictions for R. sociabilis plumbeus.

Kernel-Based Constructions.

We contrasted two kernel-based constructions for spatial networks. First, we constructed networks assuming a negative exponential kernel, which is frequently used in metapopulation ecology (16). We refer to this construction as a theoretical kernel (Eq. 1),

where p_ij is the probability of a link between patches i and j, d_ij is the Euclidean distance between patches, and α is a scaling factor based on the average assumed movement distance (Fig. S1 shows movement kernels). Second, we contrasted theoretical kernels to the distribution of observed movement distances, what we term an empirical kernel construction. To allow for seamless comparisons of kernel constructions to the maximum distance construction, we drew 100 samples from kernels (with replacement) and for each sample we constructed a binary network based on that distance, where patches within the distance were considered connected and patches greater than that distance were considered unconnected.

Social Network Constructions.

We modeled movement on spatial networks using social network models for three primary reasons. First, such models can predict linkages on relatively large, sparse networks compared with multistate models developed specifically for mark-recapture data (38). Second, the social network models we used allow for different types of movement data of the exponential family (e.g., binary, Poisson, etc.), which can be directed or undirected, and covariates regarding both nodes and links. Third, social network models have been developed to account for a wide variety of link dependencies that can occur within networks (22) and to leverage such dependencies to improve predictions. Link dependencies may be common on landscape networks, such as movements across stepping stones and movement variation in source-sink dynamics.

Hoff and colleagues (20, 22) have advanced a general latent position, random graph model for analyzing social networks that is highly relevant for network analysis in ecology and conservation. We considered two types of latent position, random graph models: a latent space model and a sender–receiver model. In the context of landscape connectivity, latent space models leverage observed similarities in movements within networks to predict links based on an unobservable or latent connectivity space (“social space” in social networks) (20). We modeled the probability of observed movement between patches as a Bernoulli distribution with a mean μ_ij and included distance, d_ij, between patches (pairwise distances) as the only fixed covariate in the model to provide direct comparisons with other constructions. We used a Euclidean distance measure based on similarities of between-patch movements to estimate latent connectivity space, such that our model formulation is y_ij ∼ Bernoulli(p_ij) and (Eq. 2)

where y_ij is the observed presence or absence of a link (movement) between patches i and j, α is an intercept, β is the coefficient for distance, and (Eq. 3)

where K is the number of dimensions in the Euclidean latent space (in our examples, we set K = 2) (20, 21). As a consequence, when high rates of movement occur for patches i and j, Euclidean distances in the latent space will be small compared with patches with low movement rates. This formulation of latent space is inherently symmetrical (21). Hoff (21) shows that in a generalized linear mixed-model context, the latent space variable can be described as a random effect regarding the relational data, z ∼ N(0, I_k × σ²).

A sender–receiver model can be formulated in a similar way, where random effects are specified regarding within-patch dependencies and movement heterogeneity of source and destination patches (sender and receiver patches) (21) (Eq. 4):

where δ and γ represent random effects of patches regarding emigration (δ_i) and immigration (γ_j) and δ ∼ N(0, σ_γ²) and γ ∼N(0, σ_γ²). The sender–receiver formulation allows for estimating directed movements in networks. The latent space and sender–receiver models could be integrated into a single model, but we treated these formulations separately for parsimony in model building. We also note that a logistic regression model that did not include these random effects (sender–receiver or latent space effects) was less accurate at predicting linkages than all other constructions for R. sociabilis plumbeus and was less accurate than both social network models for C. vittiger. These models were fit through Bayesian Markov chain Monte Carlo; see SI Text for a description of priors and assessment of model convergence.

Comparing Link Predictions.

To assess the ability of different network constructions to predict linkages with limited data, we used cross-validation where we randomly removed 10–80% of the relational data from each dataset and constructed networks with the remaining data. For each removal amount, we used 100 randomly sampled replicates. We then asked whether constructions accurately predicted the withheld data. The network constructions we considered make different types of predictions for links, including binary predictions and proportion/probability predictions. To make accuracy assessment comparable among network constructions, we reduced constructions to binary predictions, which is the most common type of network used in spatial ecology (9, 10, 19). Maximum distance constructions are binary networks, whereas in kernel-based constructions, each of 100 random draws from the kernel produced a binary network; we report average estimates for predicting network linkages and structure based on these samples. Because the latent space and random-effects models provide predictions of link probability, we truncated these predictions to [0, 1] data, based on the threshold that maximized Cohen's κ (39) (SI Text).

We then used Cohen's κ (κ hereafter), the true skill statistic (TSS), and the area under the receiver-operating curve (AUC) to assess predictive accuracy of network constructions (40). κ And TSS each range from −1 to 1, with values of zero indicating performance no better than random (40). AUC is a threshold-independent metric that ranges from 0 to 1, with values of 0.5 indicating performance no better than random. We also calculated false-positive and false-negative error rates for each construction to better interpret the sources of errors when predicting network linkages (Fig. S2).

Comparing the Recovery of Landscape Structure and Metapopulation Measures.

We assessed the potential for various network constructions to recover observed landscape structure and predict indices of metapopulation persistence. We calculated several metrics that describe connectivity and spatial structuring at different scales: number of clusters (components), clustering coefficient, average degree, average betweenness, average shortest path, and connectance (see Table S1 for definitions and examples of these metrics). We also calculated two measures that reflect metapopulation viability: metapopulation mean lifetime (31) and metapopulation capacity (41) (SI Text). We do not report average degree, average shortest path, or metapopulation capacity, because these metrics were redundant with other metrics. The network metrics capture spatial structuring (clustering coefficient, number of clusters) and connectivity at different scales (average betweenness, connectance). Clustering coefficients capture clustering of movement within networks (29), whereas the number of clusters provides a measure of the number of units within the network above the scale of patches (17). We chose betweenness as a measure of local-scale connectivity, and connectance as a network-scale measure of connectivity. For each metric, x, we calculate a measure of relative deviation of the predicted value to the observed value as (x_pred – x_obs)/x_obs (6). Consequently, network constructions with deviations close to 0 capture observed network structure.