Both node degree and clustering coefficient seem to be ideal spatial explicit metrics to examine tree spatial patterns. We found that average node degree k showed a distinctive pattern without any overlap ranges among cluster processes, CSR, and Gibbs processes. Average clustering coefficient C also showed its ability to characterize different processes in specific ranges despite some minor overlaps (Fig. 3). According to the definitions of the cluster processes, a set of ‘parent’ points is randomly generated before each of them gives rise to a random pattern of ‘offspring’ points within a disc and dies26,40. Nodes in the networks are hence more likely to find close neighbors within a short distance, leading to higher a node degree. Clustering coefficient C analyzed local density of the network. ‘offspring’ points generated by cluster models tend to be closer compare to points generate by CSR, thus the division of the number of connected edges by maximum possible edges within local network is higher.
Despite of their excellent performance in distinguishing various processes, average node degree k and cluster coefficient C showed different degree of variations. For cluster models, huge variation of average node degree was found (Fig. 3A, B). One of the reasons might be variations of paired distance between clusters. Since ‘parent’ points are generated according to CSR, the clusters are randomly distributed. Isolated clusters can lead to low values of k. However, clustering coefficient calculates the number of connected edges between the neighbors of nodes16, thus diminishes the effects of cluster distance. We also found little variation of k value in Gibbs models (Fig. 3A, B). In Gibbs process, strong interactions (represented by closed distance) between trees are not allowed. Hence a uniformly distributed pattern would be generated26, which might reduce the variations of the number of neighbors. Consequently, we advised that average node degree k is plausible to identify regular patterns while cluster coefficient C is an ideal metrics to examine aggregated patterns.
Node degree distributions of the networks followed heavy-tailed distributions, similar to a wide range of previous studies in forest ecology based on network approaches16,35,39. Particularly, node degrees in cluster models were highly variable (Fig. 4A, B). Again, we suspected that it is caused by the uncertainty of paired distance between clusters. Isolated clusters tend to have fewer neighbors, leading to lower node degree. Mechanisms of Gibbs process and CSR are similar expect that the former deletes realizations that contain two points within a threshold distance26. Consequently, the shapes of node degree distribution curve showed similarities between among HC, Strass process, and CSR. The only difference was that node degrees were generally lower in HC and Strass process, compared to that in CSR.
Unfortunately, the density D and the average path length L of the network fail to distinguish the spatial models. Average path length L is a measure of global connectivity or spatial segregation of the network16,35,41. It calculates the shortest distance between two connected nodes, quantified by the number of edges between them35. According to the definition of CL and CS networks, two nodes are connected only if there are enough points or overlapping tree crowns between them, which introduces uncertainties to the calculation of L value. The same might also be true for density D.
By applying network-based metrics to real plantation dataset in Tuscany, our findings revealed regular and random distribution patterns, which indicates low intensity of tree competition (Table 1; Fig. 3). Currently, most plantations are established with one tree on each planting point. Pairwise distance between planting points is often constant, resulting in regular patterns42, which is detected by our network approach.
Network-based metrics is a powerful way to analyze complex interactions in forests16. Like the Ripley’s K and g(r) (Fig. 5), we found that network characteristics such as average node degree k and cluster coefficient C exhibited excellent performance in examining underlying processes, at least at small scales (Fig. 3). In addition, network-based metrics are highly flexible in order to examine complex tree interactions36. Network approaches can also help to reveal additional information of forest structures. Based on spline interpolation, attempts can be made to infer the regions with high competition values. Betweenness centrality has been applied in previous studies to identify the populations that have a huge impact on gene flow43. In this study, we found regions that largely affected the overall structure of competition networks indicated by high values of betweenness centrality (Fig. 6B and Supplementary Fig. S1). Management implications can be selective loggings on trees with high centrality scores for regeneration and seedling survival.
Nevertheless, a bit of caution is needed when designing the network-based spatial explicit metrics. Ecology networks are highly complex, with their general patterns and underlying causes still debated37. Network and graph theory provide a flexible conceptual model that can help identify the relationship between complex structures and processes44. Additional information like numerous competition kernel functions were applied in previous studies39. We cautioned that, however, introducing network-based metrics may increase the uncertainty and thus reducing its ability to identify underlying processes. In this study, we found that all of the network characteristics failed to distinguish different spatial processes in WCL (Fig. 3C). Neither weighted node degree distribution nor edge length distribution showed differences among various types of the spatial models (Fig. 4C, D). We provided three possible explanations for this: (1) For each pair of the nodes in bi-directed WCL, a high CI value from Tree-1 to Tree-2 is corresponding to a low CI value from Tree-2 to Tree-1. Hence the counteracting effects may occur for each pair of the CI values; (2) The CI value increases disproportionally with the decreasing distance, which adds uncertainty in the calculation of weighted node degree; (3) A fractional formula, despite its popularity in the design of competition index, may amplify the effects of crown sizes on the competition intensity of trees and, consequently, aggregated patterns are not necessarily translated into higher intensity of competition. We then suggested that trade-offs between complexity and uncertainty as well as a deeper understanding of tree competitions are important when designing network-based spatial explicit metrics.
Spatial network is a powerful tool to reveal ecological complexity and underlying processes16,35,36. Network approaches can examine fine-scale spatial distribution, connectivity, and intensity of tree interactions using a combination of geographic and crown size datasets. Network-based spatial metrics prompts us to rethink previous spatial explicit metrics (i.e. Ripley’s K, Nearest-neighborhood approach, etc.)29. However, validations of network-based metrics were often ignored in previous studies. We advised a workflow (Fig. 1) to characterizing tree spatial patterns and interactions using network approaches: (1) Collect data by remote sensing techniques; (2) Tree crown segmentation and validation; (3) Construct the network and define the weight; (4) Validate the network using spatial null models; (5) If accepted, the expression of weight can be applied to a specific research. We also provided spatio-temporal datasets in various forests based on UAV. The dataset is available at BioRS (http://bis.zju.edu.cn/biors).
Overall, the current study described the tree spatial patterns and interactions using three types of networks, namely competition for space (CS), competition for light (CL), and weighted competition for light (WCL). We applied four types of network-based metrics to detected underlying ecological processes using five hypothetic models: (1) CSR, (2) Matérn process, (3) Thomas process, (4) HC, (5) Strass process. We concluded that: (1) Both node degree distribution and clustering coefficient are ideal metrics in CS and CL networks to distinguish multiple processes, (2) Network approaches are powerful tools to describe fine-scale spatial variations of tree interactions. They can also help identify units that have a large impact on the overall structure of the network and thus giving management implications, (3) Despite its complexity, an important precaution for reducing the uncertainties of the network is careful validation using corresponding spatial null models.