Division of labor as a bipartite network

Division of labor as a bipartite network Abstract Bipartite ecological networks are increasingly used to describe and model relationships between interacting species (e.g., plant-pollinator or host parasite). Here, we apply network methods developed in community ecology to quantify division of labor in insect societies. We consider 2 quantitative indices (d’ and H2’) derived from information theory that inform on how much the actual patterns of task performance deviates from the null expectation that workers perform tasks randomly. In addition, we computed network modularity to identify clusters of specialized individuals that are preferentially engaged in the completion of subset of available tasks. We analyzed both simple synthetic networks, varying in size and degree of specialization, and published datasets to introduce the metrics and to show that a bipartite approach provides useful insights into task allocation. Considering division of labor as a bipartite network offers a conceptual framework that could substantially increase our understanding of division of labor in animal societies. INTRODUCTION Division of labor is a pervasive property of group living and has a long research history in social insects (Beshers and Fewell 2001; Jeanson and Weidenmüller 2014). Division of labor can be broadly defined as the existence of “any behavioral pattern that results in some individuals in a colony performing different functions from others” (Michener 1974). Division of labor is a colony or group attribute while task specialization, defined as the existence of any worker performing one task disproportionately with respect to other available tasks, is considered an individual trait. Several metrics have previously been employed to assess specialization such as the proportion of total task performance of an individual in a given task (Dornhaus 2008), the proportion of a worker’s transition to different tasks over a given period of observation (Gautrais et al. 2002) or the computation of Shannon diversity index (O’Donnell and Jeanne 1990; Thomas and Elgar 2003). At the colony level, Gorelick et al. (2004) proposed a combination of indices, also deriving from Shannon theory, that proved fruitful to quantify division of labor in theoretical and experimental studies (Jeanson et al. 2005, 2007; Dornhaus et al. 2009; Holbrook et al. 2014). Knowing whether the completion of a subset of available tasks involves more specialized workers or whether workers differ in their degree of specialization can however provide additional and valuable information to the understanding of task allocation. Quantifying task allocation in insect societies closely matches the issue of measuring specialization in interaction networks in ecology. In pollination networks for instance, a central question is to characterize the patterns of specialization versus generalization of animal species visiting plants and, reciprocally, of plants hosting pollinators (Blüthgen et al. 2006, 2008). Pollination networks can be modeled as bipartite networks, also known as 2-mode networks. In bipartite networks, the links are established between nodes belonging to 2 different classes (e.g., plants vs. animals) and interactions strictly occur between, but not within, those parties (Latapy et al. 2008). As workers and tasks belong to 2 different classes, a bipartite network approach is appropriate to describe the patterns of division of labor: workers are connected to the tasks they are engaged in and, reciprocally, tasks are linked to the subgroup of workers that perform them (Figure 1). Whereas earlier work that examined insect societies under a network perspective has already focused on interactions between colony members (i.e., one mode network) (O’Donnell and Bulova 2007; Jeanson 2012; Pinter-Wollman et al. 2014; Mersch 2016), considering division of labor as bipartite networks still remains to be explored (Charbonneau et al. 2013). Figure 1 View largeDownload slide Matrix representations of a bipartite network. Gray scale represents interaction frequencies between workers (columns) and tasks (rows) (white: no worker activity on the task, black: activity of the worker completely focused on the task). Figure 1 View largeDownload slide Matrix representations of a bipartite network. Gray scale represents interaction frequencies between workers (columns) and tasks (rows) (white: no worker activity on the task, black: activity of the worker completely focused on the task). In this context, we aim at highlighting that the tools and metrics developed in community ecology to characterize bipartite networks are relevant to quantify task allocation in animal societies. Using simple synthetic networks, we first introduce 2 complementary indices (d’ and H2’) deriving from Shannon theory that informs on the degree of interaction specificity at the individual (task or worker) and group levels, respectively (Blüthgen et al. 2006, 2008). We also show that the quantification of division of labor benefits from methods developed to detect modules in bipartite networks. Indeed, ecological networks are often partitioned into densely connected subgroups and the existence of such modular organization is a signature of specialization (Figure 1). Several algorithms are now available to detect modules in quantitative bipartite networks and to identify subsets of preferentially interacting partners. Finally, we compared bipartite metrics to statistics commonly used to quantify specialization and we reanalyzed published datasets to show that a bipartite network approach offers a comprehensive picture of division of labor that cannot be captured using previous statistics. METHODS Networks metrics: information theory approach Several indices are available to characterize bipartite networks (Blüthgen et al. 2008; Dormann 2011; Poisot et al. 2012). We focused on 2 of these metrics, d’ and H’2, proposed by Blüthgen et al. (2006) to quantify specialization between partners in plant-pollinator webs. Transposed to division of labor, these indices inform on how much the actual patterns of task performance (e.g., the number of behavioral acts performed by each individual on each task) measured at the individual (i.e., worker) and group (i.e., colony) level deviate from the null expectation that workers perform tasks randomly depending on task need and workers’ activity. All functions introduced below are implemented in the package “bipartite” in R (Dormann et al. 2008, 2009; Dormann 2011). “All analyses were performed using R 3.3 (R Development Core Team 2017)”. The index d’ quantifies for each task (or worker) the deviation of the empirical distribution of task performance from a null model which assumes task allocation in proportion to the availability of workers and tasks (for mathematical and methodological details, see Blüthgen et al. 2006). The values of d’ are provided by the function dfun in the package “bipartite” (Dormann et al. 2008). In short, it is first required to calculate d for each task j as:  dj=∑i=1N(p'ijlnp'ijqi) (1) with N the number of workers, p’ij the number of occurrences of worker i performing task j divided by the total performance for task j and qi the total number of acts performed by worker i divided by the total number of acts in the association matrix. The index d’ is computed as:  d'i=di−dmindmax−dmin (2) where dmin and dmax are the theoretical minimal and maximal values (for mathematical details see Blüthgen et al. 2006). The index d’ ranges between 0 (no specialization) and 1 (full specialization). Hence, the values of d’ informs on the exclusiveness of interactions and, importantly, they are not independent from the performances of other individuals. Hereafter, we refer to d’ for tasks as d’task. Transposing the association matrix allows the computation of d’ for workers, defined as d’indiv. Overall, the computation of d’ provides complementary information on the patterns of specialization at the task and worker level and such approach is particularly appropriate for comparisons among workers (or tasks) within a network. For the entire network, the degree of specialization considering both parties (e.g., tasks and workers) can be determined with the index H2, which is calculated as:  H2=−∑i=1N∑j=1M(pijlnpij) (3) with N is the number of individuals, M the number of tasks and pij the number of acts performed by individual i on task j divided by the total number of acts in the association matrix. H2 is the 2-dimensional Shannon entropy, which corresponds to the Shannon diversity of links in the entire network (Blüthgen et al. 2006, 2008). The index H’2 is obtained as:  H2'=H2max−H2H2max−H2min (4) with H2min and H2max being the minimal and maximal theoretical values of H2 (Blüthgen et al. 2006). H’2 is a measure of niche partitioning that informs on the selectivity in the use of resources at the network level. The degree of specialization of the entire network increases with the deviation from a neutral configuration of worker-task interactions and equals the weighted sum of the specialization of its elements, that is, <di> = H2max-H2 (Blüthgen et al. 2006, 2008; Blüthgen 2010). The metric H’2 ranges between 0 (no specialization) and 1 (specialization) and it is appropriate for statistical comparisons across networks (Blüthgen et al. 2006). Networks metrics: modularity approach Network modularity has received considerable attention across biological scales (metabolic networks: Ravasz et al. 2002, epidemiological networks: Sah et al. 2017, ecological networks: Fletcher et al. 2013). Different algorithms are now available to detect modules in quantitative bipartite networks (Dormann and Strauss 2014; Beckett 2016). Here we employed the DIRTLPAwb+ algorithm recently developed by Beckett (2016) to search for modules in weighted networks, implemented in the R package “bipartite.” The quality of partitioning into different modules is given by a modularity measure Q (Newman and Girvan 2002). This measure Q is calculated using the computeModules function implemented in the R package “bipartite.” Although Q ranges between 0 (community structure not different from random) and 1 (complete separation between modules), the comparison of modularity between networks requires some forms of normalization because the magnitude of modularity depends on network configuration (e.g., number of partners) (Supplementary Figure S2c, Dormann and Strauss 2014; Beckett 2016). For each network, we normalized modularity as  Qnorm=QQmax where Qmax is the modularity in a rearranged network that maximizes the number of modules while preserving the marginal sums of the observed network. Indeed the maximal intensity of division of labor is expected when each individual tackles a limited number of tasks and, reciprocally, when each task is performed by a reduced set of workers (Figures 1 and 3, Supplementary Materials). Whereas Beckett (2016) proposed a method to obtain Qmax while maintaining the original modular structure of the network, his approach does not maximize the number of potential modules (see Supplementary Materials for details). To this end, we created a specific function to reorganize the association matrix into a new one presenting the highest possible network modularity (see Supplementary Materials). Note that the number of modules in a bipartite network is at most the minimum between the number of tasks and individuals. Although H’2 and Q are usually correlated (see below and also Dormann and Strauss 2014), modularity is particularly useful to provide quantitative evidence for community membership (i.e., which workers and tasks cluster together). Comparison with random networks For a given network, the values of H’2 and d’ inform on the deviations of the realized interaction frequencies from the expected values of a neutral configuration of associations (Blüthgen et al. 2006). In addition, it is relevant to determine the likelihood that the actual network configuration could have arisen by chance given network’s size and connectivity. Therefore, the different metrics (H’2, d’ and Q) can be tested against null models based on random networks. Different algorithms are available to generate random networks (Dormann et al. 2009). In the context of division of labor, we recommend using Patefield’s algorithm, which generates random networks preserving marginal sums (i.e., worker performance and task need are maintained) but where links are randomly assigned between workers and tasks. In the case studies described below, we concluded that the metrics for experimental networks differed statistically from random if they lie outside the 95-percentile boundary of the corresponding values obtained for random networks (N = 100). Simple bipartite networks We analyzed one synthetic dataset representing the performance of 12 workers across 8 available tasks (A to H) (Table 1, Figure 2a). Each cell gives individual task performance, which is the number of times each individual tackled each task. This network was deliberately simple to highlight metrics’ properties but it is not intended to mimic real patterns of task allocation. Table 1 Association matrices of one synthetic network and one corresponding random network Synthetic network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  50  50  0  0  0  0  0  0  0  0  0  0  100  B  50  50  0  0  0  0  0  0  0  0  0  0  100  C  0  0  100  0  0  0  0  0  0  0  0  0  100  D  0  0  0  50  0  0  0  0  0  0  0  0  50  E  0  0  0  50  0  0  0  0  0  0  0  0  50  F  0  0  0  0  100  100  0  0  0  0  0  0  200  G  0  0  0  0  0  0  100  100  100  0  0  0  300  H  0  0  0  0  0  0  0  0  0  100  25  25  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Random network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  7  8  12  8  11  7  11  13  12  6  2  3  100  B  8  17  10  9  11  6  7  9  12  9  2  0  100  C  12  7  6  7  10  10  16  8  12  8  3  1  100  D  6  4  3  5  8  5  1  6  3  7  1  1  50  E  4  5  2  4  6  3  3  4  8  10  1  0  50  F  25  15  20  19  24  23  14  15  17  17  4  7  200  G  24  25  29  37  15  33  30  32  19  35  8  13  300  H  14  19  18  11  15  13  18  13  17  8  4  0  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Synthetic network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  50  50  0  0  0  0  0  0  0  0  0  0  100  B  50  50  0  0  0  0  0  0  0  0  0  0  100  C  0  0  100  0  0  0  0  0  0  0  0  0  100  D  0  0  0  50  0  0  0  0  0  0  0  0  50  E  0  0  0  50  0  0  0  0  0  0  0  0  50  F  0  0  0  0  100  100  0  0  0  0  0  0  200  G  0  0  0  0  0  0  100  100  100  0  0  0  300  H  0  0  0  0  0  0  0  0  0  100  25  25  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Random network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  7  8  12  8  11  7  11  13  12  6  2  3  100  B  8  17  10  9  11  6  7  9  12  9  2  0  100  C  12  7  6  7  10  10  16  8  12  8  3  1  100  D  6  4  3  5  8  5  1  6  3  7  1  1  50  E  4  5  2  4  6  3  3  4  8  10  1  0  50  F  25  15  20  19  24  23  14  15  17  17  4  7  200  G  24  25  29  37  15  33  30  32  19  35  8  13  300  H  14  19  18  11  15  13  18  13  17  8  4  0  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Rows and columns represent tasks and workers. Each cell gives the performance of worker j on task i. For example, worker 1 performed 50 acts on Task A and 50 acts on Task B. Ʃworkers and Ʃtasks are the total number of acts performed by each worker and the total performance of each task, respectively. View Large Figure 2 View largeDownload slide Worker-task bipartite graphs for (a) a synthetic and (b) one corresponding random network (obtained from data reported in Table 1). Upper and lower rectangles represent workers and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and workers. For each network, numbers in upper rectangles represent worker identities. Values of d’ and H for each worker and each task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are given. The different modules of workers and tasks are identified in different colors. Figure 2 View largeDownload slide Worker-task bipartite graphs for (a) a synthetic and (b) one corresponding random network (obtained from data reported in Table 1). Upper and lower rectangles represent workers and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and workers. For each network, numbers in upper rectangles represent worker identities. Values of d’ and H for each worker and each task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are given. The different modules of workers and tasks are identified in different colors. Variation in the proportion of specialists versus generalists We examined how the proportion of generalists and specialists influenced the values of H’2, d’indiv, and d’task. We composed groups of 10 individuals in presence of 5 tasks. In each group, individuals were either full specialists (i.e., workers performed only one task) or generalists (workers divided their activity evenly between all tasks). The proportion of generalists and specialists varied between groups (all specialists, 4/5 specialists + 1/5 generalists, 1/2 specialists + 1/2 generalists, 1/5 specialists + 4/5 generalists, all generalists) (Figure 3). Each individual (generalist or specialist) performed the same number of acts (100, arbitrarily chosen). An analysis of the influence of group size and task number on the different metrics is provided in the Supplementary Materials. Figure 3 View largeDownload slide Synthetic bipartite networks of 10 workers and 5 tasks with different proportions of generalists and specialists. Each worker has an equal task performance (100, arbitrarily chosen) that was either shared evenly between tasks (generalists, yellow rectangles) or directed at a single task (specialists, red rectangles). Figure 3 View largeDownload slide Synthetic bipartite networks of 10 workers and 5 tasks with different proportions of generalists and specialists. Each worker has an equal task performance (100, arbitrarily chosen) that was either shared evenly between tasks (generalists, yellow rectangles) or directed at a single task (specialists, red rectangles). Comparison with other metrics We compared bipartite metrics to statistics commonly used to quantify specialization and division of labor in previous studies. For each individual, we computed the Shannon’s diversity index as  Hindiv=−∑j=1M(pjln(pj)) where pj is the proportion of acts performed by the focal worker on task j and M represents the number of tasks. Reciprocally, for each task, we computed  Htask=−∑i=1N(piln(pi)) where pi is the proportion of acts performed by a worker i on the focal task and N is the number of individuals. In contrast to d’, low values of H indicate high level of specialization. We compared Hindiv and Htask to d’indiv and d’task. To quantify division of labor, Gorelick and Bertram (2007, 2010) recommended the use of metrics based on normalized mutual entropy (i.e., Shannon’s mutual entropy divided by marginal entropy, see below). Two metrics (hereafter, DOL statistics) proposed by Gorelick et al. (2004) have been often used in empirical and theoretical studies (e.g., Jeanson et al. 2005; Jeanson et al. 2007, Dornhaus et al. 2009, Holbrook et al. 2011, 2014). To compute the DOL statistics, it is first required to normalize association matrix so that the sum of all entries equals 1. From the normalized matrix, the marginal entropy of tasks is given by  HY=−∑y=1N(pyln(py)) where py is the probability that any worker engaged in the yth task and N is the number of workers. Reciprocally, the marginal entropy of individuals is given by  HX=−∑x=1M(pxln(px)) where px is the probability that the xth individual performed any task and M is the number of tasks. The mutual entropy is calculated as  Iindiv,tasks=−∑y=1N∑x=1M(pxylnpxypxpy) where pxy is the joint probability that the xth individual engaged in task y. Finally, DOLindiv and DOLtasks are defined as  DOLindiv=Iindiv,tasksHY  DOLtasks=Iindiv,tasksHX The 2 indices range between 0 (no division of labor) and 1. DOLindiv is invariant over the number of individuals and DOLtasks is insensitive to changes in the number of tasks. For mathematical details, see Gorelick et al. (2004) but note that the definitions of DOLindiv and DOLtasks were mistakenly inverted in the original paper. We provide the R code to compute DOL statistics in the Supplementary Materials. We ran simulations to compare the values of DOLindiv,DOLtasks, H2’ and Qnorm under different network configurations. We built synthetic networks of different sizes by varying group size (20, 50, and 100 individuals) and task numbers (2, 5, 10, 15, or 20 tasks). We implemented variation in task performance between individuals: the total number of acts performed by each worker was determined by extracting a random number from a uniform distribution between 0 and 200 (arbitrarily chosen). Because we aimed at exploring the relationship between the different metrics over the whole range of specialization, we varied the proportion of specialists and generalists within groups (from 0% specialists to 100% specialists). A specialist performed a single task, which was determined randomly among all tasks. A generalist divided up its performance equally between tasks. Experimental case studies We reanalyzed 3 published datasets to show the relevance of using a bipartite approach to characterize the degree of specialization and division of labor. Case study #1 In several ant species, young queens can initiate colony either solitarily (haplometrosis) or in groups (pleometrosis). In the seed-harvester ant Pogonomyrmex californicus, populations show regional variation in their nest-founding strategies. Jeanson and Fewell (2008) showed that division of labor emerged in groups of haplometrotic or pleometrotic foundresses or in mixed groups of the 2 types of queens during the incipient stages of colony founding. Here, we quantified division of labor and specialization using H2’, d’indiv, d’task, and Qnorm from published datasets where the occurrences of brood care and excavation were recorded during 4 consecutive days (Jeanson and Fewell 2008). Case study #2 In many ant species, queens are inseminated by several males. In the leaf-cutting ant Acromyrmex echinatior, Waddington et al. (2010) investigated the influence of patriline on worker’s performance between foraging and waste management. We used the data plotted in Figure 1 in Waddington et al. (2010) to test whether patrilines differ in task performance in 2 colonies (Ae213 and Ae216). Case study #3 In the ant Pheidole dentata, workers belong to 2 physical classes differing in body size (minor and major workers). Wilson (1976) recorded the relative frequencies of behavioral acts performed by each physical caste. From the total number of observations and the frequency of behavioral acts (Table 1 in Wilson 1976), we calculated the number of acts for each task. For our analysis, we only considered tasks with relative frequencies greater than 1% and we excluded self-grooming. RESULTS Simple bipartite networks The bipartite graphs and the different metrics (d’, H, H’2, Q, and Qnorm) for a synthetic and one corresponding random networks are reported in Figure 2. The metric d’ informs on the exclusiveness of interactions between entities (Blüthgen 2010). Large values of d’task indicate that specialized workers mostly performed these tasks. Similarly, large values of d’indiv reveal that workers engaged in tasks that were not performed by other individuals. Interactions between worker #3 and Task C were exclusive, thereby yielding values of d’ equaling 1 for each entity. Worker #5 also engaged in a single task (Task F) but d’indiv equaled 0.71, not 1, because another individual also tackled Task F (i.e., there was no exclusive relation between worker #5 and Task F). Because Task F was performed by 2 individuals engaged in no other task, d’task equaled 1. In contrast, d’task equaled 0.7 for Task B because each of the 2 workers was also engaged in Task A. The value of d’indiv equaled 1 for worker #4 because it was the only worker that performed Task D and E. Reciprocally, d’task for Tasks D and E did not equal 1 because each of these tasks was performed by a single worker (#4) that was also engaged in the other task. The values of d’indiv were higher for worker #5 than for worker #7 although they both engaged in unique task with a similar performance. The reason is that an increase in the number of individuals engaged in a task reduces the exclusiveness between any of these workers and the task, which thereby yields lower values of d’indiv. In contrast, the Shannon diversity Hindiv for these 2 individuals equaled 0, which indicates full specialization, because the computation of this metric is independent from the performance of others. Although Task H was only performed by 3 workers that were not engaged in any other task, the values of d’indiv for workers #11 and #12 was lower than for worker #10 because worker #10 contributed more to task performance than did workers #11 and #12. For the synthetic network, the high value of H’2 (0.92) indicates that the observed pattern of specialization deviated from a neutral configuration that would only depend on the availability of workers and tasks. The normalized modularity Qnorm (0.99) significantly differed from the values of Qnorm (mean = 0.06) obtained from 100 random networks (P < 0.05). This means that the hypothetical network almost reached the highest degree of specialization expected given the total sum of interactions and the number of workers and tasks. We identified a total of 6 modules and each module comprises workers that were preferentially engaged in the same tasks (Figure 2a). For the random network, values of d’indiv and d’task logically approached 0 because both metrics evaluate the deviation of task performance from the null expectation that workers behave randomly (Figure 2b). Variation in the proportion of specialists versus generalists We explored the influence of variation in group size on the metrics used to characterize bipartite networks. We constructed synthetic networks that contained different proportions of specialists and generalists (Figure 3). In absence of specialists (network A, Figure 3), both d’task and d’indiv equaled 0. In presence of specialists (from 20% to 100%), d’indiv never equaled 1 because each task was performed by at least 2 individuals in each network, thereby reducing the exclusiveness between workers and tasks. The same argument is valid for network D where the values of d’indiv were higher for specialists engaged in only one task (d’indiv = 0.85) than for those performing 2 tasks (d’indiv = 0.62). In network E, d’task for each task equaled 1 because each task was exclusively performed by 2 specialists that did not contribute to any other tasks. In network D, d’task was larger (0.81) for each of the 3 tasks performed by one generalist and 2 specialists than for the 2 tasks tackled by one generalist and one specialist (d’task = 0.77) because the relative contribution of specialized workers in the total performance of these 3 tasks increased with the number of specialists, thereby enhancing exclusiveness. Logically, both H2’ and Qnorm increased with the proportion of specialists in the networks. Relation between H2’, Qnorm, and DOL indices We examined the relationship between H2’ and the DOL indices in synthetic networks of different sizes and different levels of specialization. Values of DOLindiv and H2’ collapsed on a single line of slope 1, which indicates that both metrics give identical values on the intensity of division of labor (Figure 4a). In contrast, the larger range of variation between DOLtasks and H2’ (Figure 4b) resulted from the fact that DOLtasks is sensitive to group size when there are more individuals than tasks (see Figure 2 in Gorelick et al. 2004), which was the case here where groups of 20 or more individuals were simulated in presence of 20 or less tasks. Finally there H2’ and Qnorm were positively correlated (Figure 4c). Overall, this indicates that DOLindiv, H2’, and Qnorm provide similar information on the patterns of task allocation at the network level. Figure 4 View largeDownload slide Relationship between H2’, DOLindiv, DOLtasks, and Qnorm in simulated networks differing in group size (20, 50, 100, 500, and 1000), tasks number (5, 10, 15, or 20) and the proportion of specialists. The red line has a slope of 1. Figure 4 View largeDownload slide Relationship between H2’, DOLindiv, DOLtasks, and Qnorm in simulated networks differing in group size (20, 50, 100, 500, and 1000), tasks number (5, 10, 15, or 20) and the proportion of specialists. The red line has a slope of 1. Experimental case studies Case study #1 In the ant P. californicus, the occurrences of brood care and excavation were recorded at the incipient stages of colony founding in groups of haplometrotic or pleometrotic foundresses or in mixed groups of the 2 types of queens (Jeanson and Fewell 2008) (Figure 5). At the group level, association type influenced specialization but not group size (2 vs. 6 foundresses) (Figure 5a). The values of modularity and H2’ were positively correlated (Figure 5b,c). At the individual level, d’indiv was higher in mixed pairs than in haplometrotic or pleometrotic pair, which indicates that the degree of specialization depended on the social context (Figure 6). Such effect was not detected in groups of 6 queens. In pairs, the values of d’task did not differ between excavation and brood care and they were higher in mixed pairs than in pure associations. In groups of 6 queens, the higher values of d’task for excavation indicates that this task was performed by more specialized foundresses. Figure 5 View largeDownload slide (a) Boxplots of H2’ in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. (b) Relationship between H2’ and Q in pairs and groups of 6 foundresses. (c) Relationship between H2’ and Qnorm in pairs and groups of 6 foundresses. Figure 5 View largeDownload slide (a) Boxplots of H2’ in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. (b) Relationship between H2’ and Q in pairs and groups of 6 foundresses. (c) Relationship between H2’ and Qnorm in pairs and groups of 6 foundresses. Figure 6 View largeDownload slide Boxplots of d’task (a) and d’indiv (b) in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. Figure 6 View largeDownload slide Boxplots of d’task (a) and d’indiv (b) in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. Case study #2 In the ant A. echinatior, Waddington et al. (2010) recorded the performance of workers from different patrilines on waste management and foraging. We found that the modularity (Qnorm = 0.48) in the bipartite network of colony Ae213 was significantly higher than the average modularity obtained in random networks (mean ± SD = 0.28 ± 0.09, N = 100, P < 0.05) (Figure 7a). In contrast, no modular organization was found in colony Ae216 where the experimental modularity (Qnorm = 0.19) did not differ significantly from random (mean ± SD = 0.16 ± 0.06, N = 100, P > 0.05) (Figure 7b). This reveals that patrilines differed in task performance in colony Ae213 but not in colony Ae216. Figure 7 View largeDownload slide Worker-task bipartite graphs for 2 colonies (a: colony Ae213, b: colony Ae216) of the ant A. echinatior (data from Waddington et al. 2010). Upper and lower rectangles represent patrilines and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and patrilines. For each network, numbers in upper rectangles represent patrilines identities. Values of d’ for each patriline and task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are indicated. The different modules of workers and tasks are identified in different colors. Figure 7 View largeDownload slide Worker-task bipartite graphs for 2 colonies (a: colony Ae213, b: colony Ae216) of the ant A. echinatior (data from Waddington et al. 2010). Upper and lower rectangles represent patrilines and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and patrilines. For each network, numbers in upper rectangles represent patrilines identities. Values of d’ for each patriline and task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are indicated. The different modules of workers and tasks are identified in different colors. Case study #3 In the ant Atta dentata, Wilson (1976) recorded the number of behavioral acts performed by workers belonging to 2 physical castes. The value of modularity in this experimental network was low (Qnorm = 0.62) but it was significantly greater than the average normalized modularity computed in random networks (mean ± SD = 0.11 ± 0.03, N = 100, P < 0.05). Although total task performance was lower for major than for minor workers, this indicates that majors tend to specialize on a subset of 3 tasks (Figure 8). Figure 8 View largeDownload slide Caste-task bipartite graph in the ant P. dendata (data from Wilson 1976). Upper and lower rectangles represent castes (minor and major) and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between castes and tasks. Values of d’ for each caste and task are indicated above upper rectangles and below lower rectangles, respectively. The value of H’2, Q, and Qnorm are indicated. The different modules of castes and tasks are identified in different colors. Task 1: Feeding outside the nest, Task 2: Carrying food, Task 3: Feeding inside the nest, Task 4: Allogrooming, Task 5: Foraging, Task 6: Brood care, Task 8: Carrying corpse, Task 8: Regurgitating, Task 9: Eating corpse. Figure 8 View largeDownload slide Caste-task bipartite graph in the ant P. dendata (data from Wilson 1976). Upper and lower rectangles represent castes (minor and major) and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between castes and tasks. Values of d’ for each caste and task are indicated above upper rectangles and below lower rectangles, respectively. The value of H’2, Q, and Qnorm are indicated. The different modules of castes and tasks are identified in different colors. Task 1: Feeding outside the nest, Task 2: Carrying food, Task 3: Feeding inside the nest, Task 4: Allogrooming, Task 5: Foraging, Task 6: Brood care, Task 8: Carrying corpse, Task 8: Regurgitating, Task 9: Eating corpse. DISCUSSION In this paper, we showed that the metrics originally developed to characterize interactions between lower and higher levels of interacting parties in an ecological context are useful to study task allocation in insect societies. At the individual level, values of d’indiv or d’task provide different but complementary information the Shannon diversity index H (e.g., Thomas and Elgar 2003). The computation of H for individuals only depends on the proportional use of resources (i.e., the proportion of time spent performing each task), irrespective of coworkers performance or task needs. This metric thus informs on how workers partition their work effort among the different tasks, which matches the traditional definition of specialization (e.g., Gautrais et al. 2002; Dornhaus 2008; Duarte et al. 2011). In contrast, the computation of d’indiv takes into account the performance of co-occurring workers, which implies that d’indiv will generally be lower for individuals engaged in tasks that are already performed by many others. In Figure 2 for instance, workers #3 and #8 tackled a single task (Task C and Task G, respectively) but d’indiv for worker # 8 was half the value of d’indiv for worker #3. The index d’indiv and other related metrics were introduced in ecology because it was considered that a measure of specialization should not only take into account the proportional utilization of resources (as H does) but also their availability (Blüthgen et al. 2006). Thus, species using resources in proportion of their abundance can be seen as more opportunistic than species exploiting rare resources disproportionately more (Hurlbert 1978; Feinsinger et al. 1981). Such consideration might help refining our concept of specialization. What is a specialist: a worker engaged in a unique task or the only worker to perform a task (e.g., worker #4 vs. worker #5 in Figure 2)? Computing the metric H informs on the frequency of task performance and answers the question: is this individual a specialist? The score of d’indiv tells us to what extent an individual’s performance is essential for task completion and answers the question: is this individual a specialist for this task? In measuring the exclusiveness of interactions between workers and tasks, d’indiv provides insights into colony’s vulnerability to the removal of workers and allows identifying individuals that are critical to the maintenance of colony homeostasis. Although the metrics H and d’indiv provide useful and complementary information, they ignore the patterns of task choice (Duarte et al. 2011). Indeed, it might be argued that a worker alternating randomly or repeatedly between tasks is less specialized than a worker performing one task then switching to another one and so on (Gautrais et al. 2002; Duarte et al. 2011). Overall, as pointed out by other authors (Gorelick and Bertram 2007; Duarte et al. 2011), no single metric can capture all facets of specialization and the combination of several indices is required to provide an accurate picture of task allocation at the individual level. At the network level, the index H2’ informs on the complementary specialization between workers and tasks and increases with increasing values of d’indiv and d’task (Blüthgen et al. 2006; Blüthgen 2010). We have shown that H2’ is largely invariant over changes in the number of individuals or tasks, which indicates that this metric is appropriate for comparisons across systems. We found that the metrics Q1norm, DOLindivi, and H2’ were highly positively correlated and thus provided similar information about the intensity of division of labor (Figure 5). Such correlation was expected because the presence of modules in bipartite networks relies on the existence of exclusive interactions between workers and tasks (Figure 1). However, the use of modularity offers the great advantage of providing a quantitative method to identify modules of preferentially interacting workers and tasks and, to our knowledge, no such method has been proposed so far to study division of labor. The identification of modules allows investigating additional issues to refine our understanding of task allocation. For instance it can be asked whether individuals from the same cluster share some common phenotypic traits or whether tasks from different modules are spatially segregated within nests. Overall, a bipartite approach offers a unified conceptual framework to capture simultaneously specialization at the individual (tasks and workers) and colony level. It is worth emphasizing that a bipartite approach can also be used to study patterns of interactions between tasks and any other relevant entities, such as patrilines or physical castes as illustrated here with different case studies. In addition, the different metrics can be very easily calculated from simple association matrices thanks to the availability of dedicated R packages (Dormann et al. 2008). To conclude, we focused here on the quantification of specialization and division of labor in animal groups. It is important to emphasize that although a bipartite approach assumes that interactions only occur between workers and tasks, this by no means implies that interactions among workers are not involved in task allocation. Considering how social interactions among workers shape task performance is out of the scope of the present study. However, over the past years, there have been increasing efforts to model social systems as interaction networks where actors are connected with links representing a single type of relationship. The current tendency is to describe social systems under a more holistic approach by constructing multiple networks combining different categories of interactions among the same set of individuals. Interestingly, an analysis of bipartite networks can be overlapped with the analysis of pairwise interactions among individuals using mathematical algorithms derived from multilayer network analysis (Kivelä et al. 2014). Such approach opens promising avenues to provide new insights into the study of division of labor in insect societies. SUPPLEMENTARY MATERIAL Supplementary data are available at Behavioral Ecology online. FUNDING This work was supported by the Centre National de la Recherche Scientifique (CNRS). C.P. was funded by a grant of the Federal University of Toulouse (IDEX UNITI). Data accessibility: Analyses reported in this article can be reproduced using the data provided by Pasquaretta and Jeanson (2017). We thank 2 reviewers for helpful comments and suggestions that greatly improved the manuscript. REFERENCES Beckett SJ. 2016. Improved community detection in weighted bipartite networks. R Soc Open Sci . 3: 140536. Google Scholar CrossRef Search ADS PubMed  Beshers SN, Fewell JH. 2001. Models of division of labor in social insects. Annu Rev Entomol . 46: 413– 440. Google Scholar CrossRef Search ADS PubMed  Blüthgen N. 2010. 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Division of labor as a bipartite network

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Abstract

Abstract Bipartite ecological networks are increasingly used to describe and model relationships between interacting species (e.g., plant-pollinator or host parasite). Here, we apply network methods developed in community ecology to quantify division of labor in insect societies. We consider 2 quantitative indices (d’ and H2’) derived from information theory that inform on how much the actual patterns of task performance deviates from the null expectation that workers perform tasks randomly. In addition, we computed network modularity to identify clusters of specialized individuals that are preferentially engaged in the completion of subset of available tasks. We analyzed both simple synthetic networks, varying in size and degree of specialization, and published datasets to introduce the metrics and to show that a bipartite approach provides useful insights into task allocation. Considering division of labor as a bipartite network offers a conceptual framework that could substantially increase our understanding of division of labor in animal societies. INTRODUCTION Division of labor is a pervasive property of group living and has a long research history in social insects (Beshers and Fewell 2001; Jeanson and Weidenmüller 2014). Division of labor can be broadly defined as the existence of “any behavioral pattern that results in some individuals in a colony performing different functions from others” (Michener 1974). Division of labor is a colony or group attribute while task specialization, defined as the existence of any worker performing one task disproportionately with respect to other available tasks, is considered an individual trait. Several metrics have previously been employed to assess specialization such as the proportion of total task performance of an individual in a given task (Dornhaus 2008), the proportion of a worker’s transition to different tasks over a given period of observation (Gautrais et al. 2002) or the computation of Shannon diversity index (O’Donnell and Jeanne 1990; Thomas and Elgar 2003). At the colony level, Gorelick et al. (2004) proposed a combination of indices, also deriving from Shannon theory, that proved fruitful to quantify division of labor in theoretical and experimental studies (Jeanson et al. 2005, 2007; Dornhaus et al. 2009; Holbrook et al. 2014). Knowing whether the completion of a subset of available tasks involves more specialized workers or whether workers differ in their degree of specialization can however provide additional and valuable information to the understanding of task allocation. Quantifying task allocation in insect societies closely matches the issue of measuring specialization in interaction networks in ecology. In pollination networks for instance, a central question is to characterize the patterns of specialization versus generalization of animal species visiting plants and, reciprocally, of plants hosting pollinators (Blüthgen et al. 2006, 2008). Pollination networks can be modeled as bipartite networks, also known as 2-mode networks. In bipartite networks, the links are established between nodes belonging to 2 different classes (e.g., plants vs. animals) and interactions strictly occur between, but not within, those parties (Latapy et al. 2008). As workers and tasks belong to 2 different classes, a bipartite network approach is appropriate to describe the patterns of division of labor: workers are connected to the tasks they are engaged in and, reciprocally, tasks are linked to the subgroup of workers that perform them (Figure 1). Whereas earlier work that examined insect societies under a network perspective has already focused on interactions between colony members (i.e., one mode network) (O’Donnell and Bulova 2007; Jeanson 2012; Pinter-Wollman et al. 2014; Mersch 2016), considering division of labor as bipartite networks still remains to be explored (Charbonneau et al. 2013). Figure 1 View largeDownload slide Matrix representations of a bipartite network. Gray scale represents interaction frequencies between workers (columns) and tasks (rows) (white: no worker activity on the task, black: activity of the worker completely focused on the task). Figure 1 View largeDownload slide Matrix representations of a bipartite network. Gray scale represents interaction frequencies between workers (columns) and tasks (rows) (white: no worker activity on the task, black: activity of the worker completely focused on the task). In this context, we aim at highlighting that the tools and metrics developed in community ecology to characterize bipartite networks are relevant to quantify task allocation in animal societies. Using simple synthetic networks, we first introduce 2 complementary indices (d’ and H2’) deriving from Shannon theory that informs on the degree of interaction specificity at the individual (task or worker) and group levels, respectively (Blüthgen et al. 2006, 2008). We also show that the quantification of division of labor benefits from methods developed to detect modules in bipartite networks. Indeed, ecological networks are often partitioned into densely connected subgroups and the existence of such modular organization is a signature of specialization (Figure 1). Several algorithms are now available to detect modules in quantitative bipartite networks and to identify subsets of preferentially interacting partners. Finally, we compared bipartite metrics to statistics commonly used to quantify specialization and we reanalyzed published datasets to show that a bipartite network approach offers a comprehensive picture of division of labor that cannot be captured using previous statistics. METHODS Networks metrics: information theory approach Several indices are available to characterize bipartite networks (Blüthgen et al. 2008; Dormann 2011; Poisot et al. 2012). We focused on 2 of these metrics, d’ and H’2, proposed by Blüthgen et al. (2006) to quantify specialization between partners in plant-pollinator webs. Transposed to division of labor, these indices inform on how much the actual patterns of task performance (e.g., the number of behavioral acts performed by each individual on each task) measured at the individual (i.e., worker) and group (i.e., colony) level deviate from the null expectation that workers perform tasks randomly depending on task need and workers’ activity. All functions introduced below are implemented in the package “bipartite” in R (Dormann et al. 2008, 2009; Dormann 2011). “All analyses were performed using R 3.3 (R Development Core Team 2017)”. The index d’ quantifies for each task (or worker) the deviation of the empirical distribution of task performance from a null model which assumes task allocation in proportion to the availability of workers and tasks (for mathematical and methodological details, see Blüthgen et al. 2006). The values of d’ are provided by the function dfun in the package “bipartite” (Dormann et al. 2008). In short, it is first required to calculate d for each task j as:  dj=∑i=1N(p'ijlnp'ijqi) (1) with N the number of workers, p’ij the number of occurrences of worker i performing task j divided by the total performance for task j and qi the total number of acts performed by worker i divided by the total number of acts in the association matrix. The index d’ is computed as:  d'i=di−dmindmax−dmin (2) where dmin and dmax are the theoretical minimal and maximal values (for mathematical details see Blüthgen et al. 2006). The index d’ ranges between 0 (no specialization) and 1 (full specialization). Hence, the values of d’ informs on the exclusiveness of interactions and, importantly, they are not independent from the performances of other individuals. Hereafter, we refer to d’ for tasks as d’task. Transposing the association matrix allows the computation of d’ for workers, defined as d’indiv. Overall, the computation of d’ provides complementary information on the patterns of specialization at the task and worker level and such approach is particularly appropriate for comparisons among workers (or tasks) within a network. For the entire network, the degree of specialization considering both parties (e.g., tasks and workers) can be determined with the index H2, which is calculated as:  H2=−∑i=1N∑j=1M(pijlnpij) (3) with N is the number of individuals, M the number of tasks and pij the number of acts performed by individual i on task j divided by the total number of acts in the association matrix. H2 is the 2-dimensional Shannon entropy, which corresponds to the Shannon diversity of links in the entire network (Blüthgen et al. 2006, 2008). The index H’2 is obtained as:  H2'=H2max−H2H2max−H2min (4) with H2min and H2max being the minimal and maximal theoretical values of H2 (Blüthgen et al. 2006). H’2 is a measure of niche partitioning that informs on the selectivity in the use of resources at the network level. The degree of specialization of the entire network increases with the deviation from a neutral configuration of worker-task interactions and equals the weighted sum of the specialization of its elements, that is, <di> = H2max-H2 (Blüthgen et al. 2006, 2008; Blüthgen 2010). The metric H’2 ranges between 0 (no specialization) and 1 (specialization) and it is appropriate for statistical comparisons across networks (Blüthgen et al. 2006). Networks metrics: modularity approach Network modularity has received considerable attention across biological scales (metabolic networks: Ravasz et al. 2002, epidemiological networks: Sah et al. 2017, ecological networks: Fletcher et al. 2013). Different algorithms are now available to detect modules in quantitative bipartite networks (Dormann and Strauss 2014; Beckett 2016). Here we employed the DIRTLPAwb+ algorithm recently developed by Beckett (2016) to search for modules in weighted networks, implemented in the R package “bipartite.” The quality of partitioning into different modules is given by a modularity measure Q (Newman and Girvan 2002). This measure Q is calculated using the computeModules function implemented in the R package “bipartite.” Although Q ranges between 0 (community structure not different from random) and 1 (complete separation between modules), the comparison of modularity between networks requires some forms of normalization because the magnitude of modularity depends on network configuration (e.g., number of partners) (Supplementary Figure S2c, Dormann and Strauss 2014; Beckett 2016). For each network, we normalized modularity as  Qnorm=QQmax where Qmax is the modularity in a rearranged network that maximizes the number of modules while preserving the marginal sums of the observed network. Indeed the maximal intensity of division of labor is expected when each individual tackles a limited number of tasks and, reciprocally, when each task is performed by a reduced set of workers (Figures 1 and 3, Supplementary Materials). Whereas Beckett (2016) proposed a method to obtain Qmax while maintaining the original modular structure of the network, his approach does not maximize the number of potential modules (see Supplementary Materials for details). To this end, we created a specific function to reorganize the association matrix into a new one presenting the highest possible network modularity (see Supplementary Materials). Note that the number of modules in a bipartite network is at most the minimum between the number of tasks and individuals. Although H’2 and Q are usually correlated (see below and also Dormann and Strauss 2014), modularity is particularly useful to provide quantitative evidence for community membership (i.e., which workers and tasks cluster together). Comparison with random networks For a given network, the values of H’2 and d’ inform on the deviations of the realized interaction frequencies from the expected values of a neutral configuration of associations (Blüthgen et al. 2006). In addition, it is relevant to determine the likelihood that the actual network configuration could have arisen by chance given network’s size and connectivity. Therefore, the different metrics (H’2, d’ and Q) can be tested against null models based on random networks. Different algorithms are available to generate random networks (Dormann et al. 2009). In the context of division of labor, we recommend using Patefield’s algorithm, which generates random networks preserving marginal sums (i.e., worker performance and task need are maintained) but where links are randomly assigned between workers and tasks. In the case studies described below, we concluded that the metrics for experimental networks differed statistically from random if they lie outside the 95-percentile boundary of the corresponding values obtained for random networks (N = 100). Simple bipartite networks We analyzed one synthetic dataset representing the performance of 12 workers across 8 available tasks (A to H) (Table 1, Figure 2a). Each cell gives individual task performance, which is the number of times each individual tackled each task. This network was deliberately simple to highlight metrics’ properties but it is not intended to mimic real patterns of task allocation. Table 1 Association matrices of one synthetic network and one corresponding random network Synthetic network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  50  50  0  0  0  0  0  0  0  0  0  0  100  B  50  50  0  0  0  0  0  0  0  0  0  0  100  C  0  0  100  0  0  0  0  0  0  0  0  0  100  D  0  0  0  50  0  0  0  0  0  0  0  0  50  E  0  0  0  50  0  0  0  0  0  0  0  0  50  F  0  0  0  0  100  100  0  0  0  0  0  0  200  G  0  0  0  0  0  0  100  100  100  0  0  0  300  H  0  0  0  0  0  0  0  0  0  100  25  25  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Random network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  7  8  12  8  11  7  11  13  12  6  2  3  100  B  8  17  10  9  11  6  7  9  12  9  2  0  100  C  12  7  6  7  10  10  16  8  12  8  3  1  100  D  6  4  3  5  8  5  1  6  3  7  1  1  50  E  4  5  2  4  6  3  3  4  8  10  1  0  50  F  25  15  20  19  24  23  14  15  17  17  4  7  200  G  24  25  29  37  15  33  30  32  19  35  8  13  300  H  14  19  18  11  15  13  18  13  17  8  4  0  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Synthetic network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  50  50  0  0  0  0  0  0  0  0  0  0  100  B  50  50  0  0  0  0  0  0  0  0  0  0  100  C  0  0  100  0  0  0  0  0  0  0  0  0  100  D  0  0  0  50  0  0  0  0  0  0  0  0  50  E  0  0  0  50  0  0  0  0  0  0  0  0  50  F  0  0  0  0  100  100  0  0  0  0  0  0  200  G  0  0  0  0  0  0  100  100  100  0  0  0  300  H  0  0  0  0  0  0  0  0  0  100  25  25  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Random network    1  2  3  4  5  6  7  8  9  10  11  12  Ʃtasks  A  7  8  12  8  11  7  11  13  12  6  2  3  100  B  8  17  10  9  11  6  7  9  12  9  2  0  100  C  12  7  6  7  10  10  16  8  12  8  3  1  100  D  6  4  3  5  8  5  1  6  3  7  1  1  50  E  4  5  2  4  6  3  3  4  8  10  1  0  50  F  25  15  20  19  24  23  14  15  17  17  4  7  200  G  24  25  29  37  15  33  30  32  19  35  8  13  300  H  14  19  18  11  15  13  18  13  17  8  4  0  150  Ʃworkers  100  100  100  100  100  100  100  100  100  100  25  25    Rows and columns represent tasks and workers. Each cell gives the performance of worker j on task i. For example, worker 1 performed 50 acts on Task A and 50 acts on Task B. Ʃworkers and Ʃtasks are the total number of acts performed by each worker and the total performance of each task, respectively. View Large Figure 2 View largeDownload slide Worker-task bipartite graphs for (a) a synthetic and (b) one corresponding random network (obtained from data reported in Table 1). Upper and lower rectangles represent workers and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and workers. For each network, numbers in upper rectangles represent worker identities. Values of d’ and H for each worker and each task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are given. The different modules of workers and tasks are identified in different colors. Figure 2 View largeDownload slide Worker-task bipartite graphs for (a) a synthetic and (b) one corresponding random network (obtained from data reported in Table 1). Upper and lower rectangles represent workers and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and workers. For each network, numbers in upper rectangles represent worker identities. Values of d’ and H for each worker and each task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are given. The different modules of workers and tasks are identified in different colors. Variation in the proportion of specialists versus generalists We examined how the proportion of generalists and specialists influenced the values of H’2, d’indiv, and d’task. We composed groups of 10 individuals in presence of 5 tasks. In each group, individuals were either full specialists (i.e., workers performed only one task) or generalists (workers divided their activity evenly between all tasks). The proportion of generalists and specialists varied between groups (all specialists, 4/5 specialists + 1/5 generalists, 1/2 specialists + 1/2 generalists, 1/5 specialists + 4/5 generalists, all generalists) (Figure 3). Each individual (generalist or specialist) performed the same number of acts (100, arbitrarily chosen). An analysis of the influence of group size and task number on the different metrics is provided in the Supplementary Materials. Figure 3 View largeDownload slide Synthetic bipartite networks of 10 workers and 5 tasks with different proportions of generalists and specialists. Each worker has an equal task performance (100, arbitrarily chosen) that was either shared evenly between tasks (generalists, yellow rectangles) or directed at a single task (specialists, red rectangles). Figure 3 View largeDownload slide Synthetic bipartite networks of 10 workers and 5 tasks with different proportions of generalists and specialists. Each worker has an equal task performance (100, arbitrarily chosen) that was either shared evenly between tasks (generalists, yellow rectangles) or directed at a single task (specialists, red rectangles). Comparison with other metrics We compared bipartite metrics to statistics commonly used to quantify specialization and division of labor in previous studies. For each individual, we computed the Shannon’s diversity index as  Hindiv=−∑j=1M(pjln(pj)) where pj is the proportion of acts performed by the focal worker on task j and M represents the number of tasks. Reciprocally, for each task, we computed  Htask=−∑i=1N(piln(pi)) where pi is the proportion of acts performed by a worker i on the focal task and N is the number of individuals. In contrast to d’, low values of H indicate high level of specialization. We compared Hindiv and Htask to d’indiv and d’task. To quantify division of labor, Gorelick and Bertram (2007, 2010) recommended the use of metrics based on normalized mutual entropy (i.e., Shannon’s mutual entropy divided by marginal entropy, see below). Two metrics (hereafter, DOL statistics) proposed by Gorelick et al. (2004) have been often used in empirical and theoretical studies (e.g., Jeanson et al. 2005; Jeanson et al. 2007, Dornhaus et al. 2009, Holbrook et al. 2011, 2014). To compute the DOL statistics, it is first required to normalize association matrix so that the sum of all entries equals 1. From the normalized matrix, the marginal entropy of tasks is given by  HY=−∑y=1N(pyln(py)) where py is the probability that any worker engaged in the yth task and N is the number of workers. Reciprocally, the marginal entropy of individuals is given by  HX=−∑x=1M(pxln(px)) where px is the probability that the xth individual performed any task and M is the number of tasks. The mutual entropy is calculated as  Iindiv,tasks=−∑y=1N∑x=1M(pxylnpxypxpy) where pxy is the joint probability that the xth individual engaged in task y. Finally, DOLindiv and DOLtasks are defined as  DOLindiv=Iindiv,tasksHY  DOLtasks=Iindiv,tasksHX The 2 indices range between 0 (no division of labor) and 1. DOLindiv is invariant over the number of individuals and DOLtasks is insensitive to changes in the number of tasks. For mathematical details, see Gorelick et al. (2004) but note that the definitions of DOLindiv and DOLtasks were mistakenly inverted in the original paper. We provide the R code to compute DOL statistics in the Supplementary Materials. We ran simulations to compare the values of DOLindiv,DOLtasks, H2’ and Qnorm under different network configurations. We built synthetic networks of different sizes by varying group size (20, 50, and 100 individuals) and task numbers (2, 5, 10, 15, or 20 tasks). We implemented variation in task performance between individuals: the total number of acts performed by each worker was determined by extracting a random number from a uniform distribution between 0 and 200 (arbitrarily chosen). Because we aimed at exploring the relationship between the different metrics over the whole range of specialization, we varied the proportion of specialists and generalists within groups (from 0% specialists to 100% specialists). A specialist performed a single task, which was determined randomly among all tasks. A generalist divided up its performance equally between tasks. Experimental case studies We reanalyzed 3 published datasets to show the relevance of using a bipartite approach to characterize the degree of specialization and division of labor. Case study #1 In several ant species, young queens can initiate colony either solitarily (haplometrosis) or in groups (pleometrosis). In the seed-harvester ant Pogonomyrmex californicus, populations show regional variation in their nest-founding strategies. Jeanson and Fewell (2008) showed that division of labor emerged in groups of haplometrotic or pleometrotic foundresses or in mixed groups of the 2 types of queens during the incipient stages of colony founding. Here, we quantified division of labor and specialization using H2’, d’indiv, d’task, and Qnorm from published datasets where the occurrences of brood care and excavation were recorded during 4 consecutive days (Jeanson and Fewell 2008). Case study #2 In many ant species, queens are inseminated by several males. In the leaf-cutting ant Acromyrmex echinatior, Waddington et al. (2010) investigated the influence of patriline on worker’s performance between foraging and waste management. We used the data plotted in Figure 1 in Waddington et al. (2010) to test whether patrilines differ in task performance in 2 colonies (Ae213 and Ae216). Case study #3 In the ant Pheidole dentata, workers belong to 2 physical classes differing in body size (minor and major workers). Wilson (1976) recorded the relative frequencies of behavioral acts performed by each physical caste. From the total number of observations and the frequency of behavioral acts (Table 1 in Wilson 1976), we calculated the number of acts for each task. For our analysis, we only considered tasks with relative frequencies greater than 1% and we excluded self-grooming. RESULTS Simple bipartite networks The bipartite graphs and the different metrics (d’, H, H’2, Q, and Qnorm) for a synthetic and one corresponding random networks are reported in Figure 2. The metric d’ informs on the exclusiveness of interactions between entities (Blüthgen 2010). Large values of d’task indicate that specialized workers mostly performed these tasks. Similarly, large values of d’indiv reveal that workers engaged in tasks that were not performed by other individuals. Interactions between worker #3 and Task C were exclusive, thereby yielding values of d’ equaling 1 for each entity. Worker #5 also engaged in a single task (Task F) but d’indiv equaled 0.71, not 1, because another individual also tackled Task F (i.e., there was no exclusive relation between worker #5 and Task F). Because Task F was performed by 2 individuals engaged in no other task, d’task equaled 1. In contrast, d’task equaled 0.7 for Task B because each of the 2 workers was also engaged in Task A. The value of d’indiv equaled 1 for worker #4 because it was the only worker that performed Task D and E. Reciprocally, d’task for Tasks D and E did not equal 1 because each of these tasks was performed by a single worker (#4) that was also engaged in the other task. The values of d’indiv were higher for worker #5 than for worker #7 although they both engaged in unique task with a similar performance. The reason is that an increase in the number of individuals engaged in a task reduces the exclusiveness between any of these workers and the task, which thereby yields lower values of d’indiv. In contrast, the Shannon diversity Hindiv for these 2 individuals equaled 0, which indicates full specialization, because the computation of this metric is independent from the performance of others. Although Task H was only performed by 3 workers that were not engaged in any other task, the values of d’indiv for workers #11 and #12 was lower than for worker #10 because worker #10 contributed more to task performance than did workers #11 and #12. For the synthetic network, the high value of H’2 (0.92) indicates that the observed pattern of specialization deviated from a neutral configuration that would only depend on the availability of workers and tasks. The normalized modularity Qnorm (0.99) significantly differed from the values of Qnorm (mean = 0.06) obtained from 100 random networks (P < 0.05). This means that the hypothetical network almost reached the highest degree of specialization expected given the total sum of interactions and the number of workers and tasks. We identified a total of 6 modules and each module comprises workers that were preferentially engaged in the same tasks (Figure 2a). For the random network, values of d’indiv and d’task logically approached 0 because both metrics evaluate the deviation of task performance from the null expectation that workers behave randomly (Figure 2b). Variation in the proportion of specialists versus generalists We explored the influence of variation in group size on the metrics used to characterize bipartite networks. We constructed synthetic networks that contained different proportions of specialists and generalists (Figure 3). In absence of specialists (network A, Figure 3), both d’task and d’indiv equaled 0. In presence of specialists (from 20% to 100%), d’indiv never equaled 1 because each task was performed by at least 2 individuals in each network, thereby reducing the exclusiveness between workers and tasks. The same argument is valid for network D where the values of d’indiv were higher for specialists engaged in only one task (d’indiv = 0.85) than for those performing 2 tasks (d’indiv = 0.62). In network E, d’task for each task equaled 1 because each task was exclusively performed by 2 specialists that did not contribute to any other tasks. In network D, d’task was larger (0.81) for each of the 3 tasks performed by one generalist and 2 specialists than for the 2 tasks tackled by one generalist and one specialist (d’task = 0.77) because the relative contribution of specialized workers in the total performance of these 3 tasks increased with the number of specialists, thereby enhancing exclusiveness. Logically, both H2’ and Qnorm increased with the proportion of specialists in the networks. Relation between H2’, Qnorm, and DOL indices We examined the relationship between H2’ and the DOL indices in synthetic networks of different sizes and different levels of specialization. Values of DOLindiv and H2’ collapsed on a single line of slope 1, which indicates that both metrics give identical values on the intensity of division of labor (Figure 4a). In contrast, the larger range of variation between DOLtasks and H2’ (Figure 4b) resulted from the fact that DOLtasks is sensitive to group size when there are more individuals than tasks (see Figure 2 in Gorelick et al. 2004), which was the case here where groups of 20 or more individuals were simulated in presence of 20 or less tasks. Finally there H2’ and Qnorm were positively correlated (Figure 4c). Overall, this indicates that DOLindiv, H2’, and Qnorm provide similar information on the patterns of task allocation at the network level. Figure 4 View largeDownload slide Relationship between H2’, DOLindiv, DOLtasks, and Qnorm in simulated networks differing in group size (20, 50, 100, 500, and 1000), tasks number (5, 10, 15, or 20) and the proportion of specialists. The red line has a slope of 1. Figure 4 View largeDownload slide Relationship between H2’, DOLindiv, DOLtasks, and Qnorm in simulated networks differing in group size (20, 50, 100, 500, and 1000), tasks number (5, 10, 15, or 20) and the proportion of specialists. The red line has a slope of 1. Experimental case studies Case study #1 In the ant P. californicus, the occurrences of brood care and excavation were recorded at the incipient stages of colony founding in groups of haplometrotic or pleometrotic foundresses or in mixed groups of the 2 types of queens (Jeanson and Fewell 2008) (Figure 5). At the group level, association type influenced specialization but not group size (2 vs. 6 foundresses) (Figure 5a). The values of modularity and H2’ were positively correlated (Figure 5b,c). At the individual level, d’indiv was higher in mixed pairs than in haplometrotic or pleometrotic pair, which indicates that the degree of specialization depended on the social context (Figure 6). Such effect was not detected in groups of 6 queens. In pairs, the values of d’task did not differ between excavation and brood care and they were higher in mixed pairs than in pure associations. In groups of 6 queens, the higher values of d’task for excavation indicates that this task was performed by more specialized foundresses. Figure 5 View largeDownload slide (a) Boxplots of H2’ in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. (b) Relationship between H2’ and Q in pairs and groups of 6 foundresses. (c) Relationship between H2’ and Qnorm in pairs and groups of 6 foundresses. Figure 5 View largeDownload slide (a) Boxplots of H2’ in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. (b) Relationship between H2’ and Q in pairs and groups of 6 foundresses. (c) Relationship between H2’ and Qnorm in pairs and groups of 6 foundresses. Figure 6 View largeDownload slide Boxplots of d’task (a) and d’indiv (b) in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. Figure 6 View largeDownload slide Boxplots of d’task (a) and d’indiv (b) in associations of haplometrotic or pleometrotic foundresses and in mixed associations of the 2 types of founding queens of the ant P. californicus. Horizontal line in each box represents the median, and the lower and upper hinges indicate the first and third quartiles. Lower (or higher) whisker extends to the most extreme value within 1.5 interquartile ranges from the first (or third) quartile. Black dots give individual values. Case study #2 In the ant A. echinatior, Waddington et al. (2010) recorded the performance of workers from different patrilines on waste management and foraging. We found that the modularity (Qnorm = 0.48) in the bipartite network of colony Ae213 was significantly higher than the average modularity obtained in random networks (mean ± SD = 0.28 ± 0.09, N = 100, P < 0.05) (Figure 7a). In contrast, no modular organization was found in colony Ae216 where the experimental modularity (Qnorm = 0.19) did not differ significantly from random (mean ± SD = 0.16 ± 0.06, N = 100, P > 0.05) (Figure 7b). This reveals that patrilines differed in task performance in colony Ae213 but not in colony Ae216. Figure 7 View largeDownload slide Worker-task bipartite graphs for 2 colonies (a: colony Ae213, b: colony Ae216) of the ant A. echinatior (data from Waddington et al. 2010). Upper and lower rectangles represent patrilines and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and patrilines. For each network, numbers in upper rectangles represent patrilines identities. Values of d’ for each patriline and task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are indicated. The different modules of workers and tasks are identified in different colors. Figure 7 View largeDownload slide Worker-task bipartite graphs for 2 colonies (a: colony Ae213, b: colony Ae216) of the ant A. echinatior (data from Waddington et al. 2010). Upper and lower rectangles represent patrilines and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between tasks and patrilines. For each network, numbers in upper rectangles represent patrilines identities. Values of d’ for each patriline and task are indicated above upper rectangles and below lower rectangles, respectively. For each network, the value of H’2, Q, and Qnorm are indicated. The different modules of workers and tasks are identified in different colors. Case study #3 In the ant Atta dentata, Wilson (1976) recorded the number of behavioral acts performed by workers belonging to 2 physical castes. The value of modularity in this experimental network was low (Qnorm = 0.62) but it was significantly greater than the average normalized modularity computed in random networks (mean ± SD = 0.11 ± 0.03, N = 100, P < 0.05). Although total task performance was lower for major than for minor workers, this indicates that majors tend to specialize on a subset of 3 tasks (Figure 8). Figure 8 View largeDownload slide Caste-task bipartite graph in the ant P. dendata (data from Wilson 1976). Upper and lower rectangles represent castes (minor and major) and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between castes and tasks. Values of d’ for each caste and task are indicated above upper rectangles and below lower rectangles, respectively. The value of H’2, Q, and Qnorm are indicated. The different modules of castes and tasks are identified in different colors. Task 1: Feeding outside the nest, Task 2: Carrying food, Task 3: Feeding inside the nest, Task 4: Allogrooming, Task 5: Foraging, Task 6: Brood care, Task 8: Carrying corpse, Task 8: Regurgitating, Task 9: Eating corpse. Figure 8 View largeDownload slide Caste-task bipartite graph in the ant P. dendata (data from Wilson 1976). Upper and lower rectangles represent castes (minor and major) and tasks, respectively. The width of each rectangle is proportional to the number of acts and the width of link indicates the frequency of interactions between castes and tasks. Values of d’ for each caste and task are indicated above upper rectangles and below lower rectangles, respectively. The value of H’2, Q, and Qnorm are indicated. The different modules of castes and tasks are identified in different colors. Task 1: Feeding outside the nest, Task 2: Carrying food, Task 3: Feeding inside the nest, Task 4: Allogrooming, Task 5: Foraging, Task 6: Brood care, Task 8: Carrying corpse, Task 8: Regurgitating, Task 9: Eating corpse. DISCUSSION In this paper, we showed that the metrics originally developed to characterize interactions between lower and higher levels of interacting parties in an ecological context are useful to study task allocation in insect societies. At the individual level, values of d’indiv or d’task provide different but complementary information the Shannon diversity index H (e.g., Thomas and Elgar 2003). The computation of H for individuals only depends on the proportional use of resources (i.e., the proportion of time spent performing each task), irrespective of coworkers performance or task needs. This metric thus informs on how workers partition their work effort among the different tasks, which matches the traditional definition of specialization (e.g., Gautrais et al. 2002; Dornhaus 2008; Duarte et al. 2011). In contrast, the computation of d’indiv takes into account the performance of co-occurring workers, which implies that d’indiv will generally be lower for individuals engaged in tasks that are already performed by many others. In Figure 2 for instance, workers #3 and #8 tackled a single task (Task C and Task G, respectively) but d’indiv for worker # 8 was half the value of d’indiv for worker #3. The index d’indiv and other related metrics were introduced in ecology because it was considered that a measure of specialization should not only take into account the proportional utilization of resources (as H does) but also their availability (Blüthgen et al. 2006). Thus, species using resources in proportion of their abundance can be seen as more opportunistic than species exploiting rare resources disproportionately more (Hurlbert 1978; Feinsinger et al. 1981). Such consideration might help refining our concept of specialization. What is a specialist: a worker engaged in a unique task or the only worker to perform a task (e.g., worker #4 vs. worker #5 in Figure 2)? Computing the metric H informs on the frequency of task performance and answers the question: is this individual a specialist? The score of d’indiv tells us to what extent an individual’s performance is essential for task completion and answers the question: is this individual a specialist for this task? In measuring the exclusiveness of interactions between workers and tasks, d’indiv provides insights into colony’s vulnerability to the removal of workers and allows identifying individuals that are critical to the maintenance of colony homeostasis. Although the metrics H and d’indiv provide useful and complementary information, they ignore the patterns of task choice (Duarte et al. 2011). Indeed, it might be argued that a worker alternating randomly or repeatedly between tasks is less specialized than a worker performing one task then switching to another one and so on (Gautrais et al. 2002; Duarte et al. 2011). Overall, as pointed out by other authors (Gorelick and Bertram 2007; Duarte et al. 2011), no single metric can capture all facets of specialization and the combination of several indices is required to provide an accurate picture of task allocation at the individual level. At the network level, the index H2’ informs on the complementary specialization between workers and tasks and increases with increasing values of d’indiv and d’task (Blüthgen et al. 2006; Blüthgen 2010). We have shown that H2’ is largely invariant over changes in the number of individuals or tasks, which indicates that this metric is appropriate for comparisons across systems. We found that the metrics Q1norm, DOLindivi, and H2’ were highly positively correlated and thus provided similar information about the intensity of division of labor (Figure 5). Such correlation was expected because the presence of modules in bipartite networks relies on the existence of exclusive interactions between workers and tasks (Figure 1). However, the use of modularity offers the great advantage of providing a quantitative method to identify modules of preferentially interacting workers and tasks and, to our knowledge, no such method has been proposed so far to study division of labor. The identification of modules allows investigating additional issues to refine our understanding of task allocation. For instance it can be asked whether individuals from the same cluster share some common phenotypic traits or whether tasks from different modules are spatially segregated within nests. Overall, a bipartite approach offers a unified conceptual framework to capture simultaneously specialization at the individual (tasks and workers) and colony level. It is worth emphasizing that a bipartite approach can also be used to study patterns of interactions between tasks and any other relevant entities, such as patrilines or physical castes as illustrated here with different case studies. In addition, the different metrics can be very easily calculated from simple association matrices thanks to the availability of dedicated R packages (Dormann et al. 2008). To conclude, we focused here on the quantification of specialization and division of labor in animal groups. It is important to emphasize that although a bipartite approach assumes that interactions only occur between workers and tasks, this by no means implies that interactions among workers are not involved in task allocation. Considering how social interactions among workers shape task performance is out of the scope of the present study. However, over the past years, there have been increasing efforts to model social systems as interaction networks where actors are connected with links representing a single type of relationship. The current tendency is to describe social systems under a more holistic approach by constructing multiple networks combining different categories of interactions among the same set of individuals. Interestingly, an analysis of bipartite networks can be overlapped with the analysis of pairwise interactions among individuals using mathematical algorithms derived from multilayer network analysis (Kivelä et al. 2014). 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Behavioral EcologyOxford University Press

Published: Mar 1, 2018

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