Forest edges increase pollinator network robustness to extinction with declining area

Plant-pollinator community structure and network architecture on islands might differ from the mainland reference sites both as a result of fragmentation processes and as a result of biases in sampling effort. The purpose of the null model is to determine whether variation in plant-pollinator community structure and network architecture on islands is significantly greater (or less) than expected from a simple ‘passive sampling effect’ from the continuous mainland reference pool. To do this, we compiled plant-pollinator interaction data from all sampling transects on the mainland edge and interior, respectively (16 transects in the mainland edge: 52 plant species, 266 pollinator species, and 1098 individual interactions; and 16 transects in the mainland interior: 19 plant species, 92 pollinator species, and 152 individual interactions), then used this as our expected 'reference' pools. From the edge and interior reference pools, we use two methods to simulate 'null communities' and 'null networks': (1) Null model Ⅰ (S1 code): a random draw of the same number of transects from the mainland as observed on each of the 41 sampled islands (i.e., constraining the number of sampling transects used to acquire a null estimate of floral resources, plant richness, pollinator richness and pollinator abundance); (2) Null model Ⅱ (S2 code): a random draw of the same number of pairwise interactions, while ensuring the same numbers of plant and pollinator species were selected as those observed on each of the 41 sampled islands (i.e., constraining both network abundance and network size). The re-sampling process was repeated 1000 times, generating 1000 simulated null networks for each island. We repeated this process separately for the edge and interior networks of each island. The detailed steps are as follows: Null model Ⅰ: constraining the same number of transects For each of the 41 islands, we conducted a random draw of the same number of transects as sampled on the island (across the 20 sampling intervals), and then calculated cumulative floral resources, plant richness, pollinator abundance, and pollinator richness. This process was repeated 1000 times, generating the null expected mean value (± 95% confidence limits) for each variable from 1,000 null draws. Null model Ⅱ: Constraining both network abundance and network size In null model II we constrained null draws by both network abundance and network size. The simplest and most intuitive approach to null model II was a fully-random process (S2 code: step of “A. Null model Ⅱ”) in which potential plant and pollinator species were drawn at random until the number of species was equal to the observed network, after which potential interactions were drawn at random until equal to the observed network. Following the random draws, the resulting null network was evaluated to determine whether it was equal in size to the observed network. However, the probability of the null network meeting both network abundance and size criteria was low and the fully-random process (S2 code: step of “B. Null model Ⅱ”) was computationally slow, except for networks that contained only a small number of plant and pollinator species but had a high network abundance. Therefore, a more computationally efficient constrained 'stepwise-random' process was developed. First, following the random draw of potential plant species (network rows), the randomly-drawn pollinator species (network columns) were constrained to only those with known potential links in the reference pool, in order to avoid null network column totals that intrinsically sum to zero. Second, prior to the random draw of potential interactions, the null network was 'auto-populated' with the minimum number of randomly-drawn interactions to avoid network row and column totals stochastically summing to zero. Finally, after null network size was constrained to equal the observed network size, additional potential interactions were drawn at random to equal observed network abundance. Therefore, the fully-random process (S2 code: step of “B. Null model Ⅱ”) we eventually used to complete the Null model Ⅱ.

岛屿上的植物-传粉者群落结构与网络架构，可能因生境破碎化过程以及采样投入偏差，与大陆参照样地存在显著差异。本研究中的零模型（null model）旨在判定：岛屿上的植物-传粉者群落结构与网络架构的变异程度，是否显著高于（或低于）从连续大陆参照物种库中提取的简单"被动采样效应"所预期的水平。

为达成这一目标，我们分别从大陆边缘与大陆内部的所有样带（transect）中收集植物-传粉者互作数据：大陆边缘共设16条样带，涵盖52种植物、266种传粉者及1098次个体间互作；大陆内部共设16条样带，涵盖19种植物、92种传粉者及152次个体间互作。随后将上述数据作为我们的参照物种库。

基于边缘与内部参照物种库，我们采用两种方法模拟"零群落"与"零网络"：
(1) 零模型Ⅰ（S1代码）：从大陆种群中随机抽取与41个调查岛屿各自采样所得样带数量相等的样带（即限定用于估算花资源、植物丰富度、传粉者丰富度及传粉者多度的采样样带数目）；
(2) 零模型Ⅱ（S2代码）：随机抽取与观测网络数目相等的成对互作，同时确保选取的植物与传粉者物种数与41个调查岛屿各自观测所得的物种数一致（即同时限定网络多度与网络规模）。

重采样过程重复1000次，为每个岛屿生成1000个模拟零网络。我们针对每个岛屿的边缘与内部网络分别重复该流程。详细步骤如下：

#### 零模型Ⅰ：限定样带数目
针对41个岛屿中的每一个，我们随机抽取与该岛屿采样所得（覆盖20个采样间隔）样带数目相等的样带，随后计算累积花资源、植物丰富度、传粉者多度及传粉者丰富度。该过程重复1000次，由此从1000次零模型抽样中得到各变量的零期望均值（±95%置信限）。

#### 零模型Ⅱ：同时限定网络多度与网络规模
在零模型Ⅱ中，我们同时通过网络多度与网络规模对零抽样进行限定。零模型Ⅱ最简便直观的实现方式为完全随机过程（S2代码："A. 零模型Ⅱ"步骤）：先随机抽取潜在植物与传粉者物种，直至物种数与观测网络一致；随后随机抽取潜在互作，直至互作数目与观测网络一致。经随机抽样后，需验证生成的零网络规模是否与观测网络相等。然而，零网络同时满足网络多度与规模限定条件的概率较低，且完全随机过程（S2代码："B. 零模型Ⅱ"步骤）的计算效率较低，仅在植物与传粉者物种数少但网络多度高的网络中表现尚可。因此，我们开发了计算效率更高的约束性"逐步随机"流程：首先，在随机抽取潜在植物物种（网络行）后，将随机抽取的传粉者物种（网络列）限定为参照物种库中已知存在潜在互作的物种，以避免零网络列总和固有为零的情况；其次，在随机抽取潜在互作前，先以最少数量的随机抽取互作"自动填充"零网络，以避免网络行与列总和随机为零的情况；最后，在零网络规模约束至与观测网络规模一致后，再随机抽取额外潜在互作以匹配观测网络多度。最终，我们采用完全随机过程（S2代码："B. 零模型Ⅱ"步骤）完成零模型Ⅱ的构建。