The emergence of nestedness

Nestedness implies a hierarchy in the linking rules of the networked system so there is heterogeneity in the number of interactions among its elements. The biological processes giving rise to nestedness can be multiple and non-exclusive, and certainly specific of each system [24,27–30]. In theory, these processes may impose constraints to the number of interactions an element of the system can have (leading to specificity) and to how these elements interact with each other (leading to affinity). Nestedness implies a gradient of specificity, and also particular affinity patterns, such that the highly specific elements tend to interact with (i.e. have greater affinity for) elements of low specificity. Therefore, in a nested network the highly connected core of nodes represents low-specificity elements, whereas the nodes in the periphery of the network represent high-specificity elements that interact with the highly connected core.

We can illustrate this theoretical scenario with well-studied two-mode networks depicting animal-plant mutualisms. Pollination by animals is a case in point. Successful pollination requires that species of plant visitors and flowering plants occur at the same place and make physical contact. While the probability of interaction is directly proportional to the local abundances of plant and pollinator species [29], flower morphology can impose species-specific constraints on number of animal species that can visit that flower. For instance, pollinators’ traits such as body length must be physically compatible with flowers’ traits, such as nectar holder depth and width [30]. Therefore, match in morphology (trait complementarity) and space (co-occurrence and abundance) define how species interact with each other (affinity) and so the number of interactions they can have (specificity). In a nested two-mode pollination network, this resultant interaction asymmetry includes a gradient of specificity in which the rare and/or more specialist species have more affinity for the more abundant and/or more generalist species.

Analogously, we hypothesize that the existence of constraints that determine specificity and affinity can be thought as an overarching feature of nested one-mode networks. Any one-mode biological system may contain elements with different specificities and affinities; but when there is a gradient of specificity among elements that have different affinities for each other in a way that highly specific elements have greater affinity for less specific elements, the system would be nested. Here, we investigated the emergence of nestedness in 18 one-mode biological systems across six different levels of biological organization (S1 Table). In the following section, we explore for each of these systems possible mechanisms generating constraints in specificity and defining affinity patterns that lead the emergence of nestedness. As these biological processes are system-dependent, often unknown and likely complex, some of our propositions are necessarily speculative at the moment. Pinpointing the definitive mechanisms for each system is beyond the scope of this study, but we hope these hypotheses help to point out research directions.

Nestedness across biological organization levels

Caveats and the way forward

Combined, these study cases suggest that system-specific interaction constraints may make nestedness a recurrent in biological networks at multiple levels of organization when both strong and weak interactions are taken into account. While common in bipartite biological networks (e.g. [9,23,24]), the recent search for nestedness in unipartite systems suggest it may not be as widespread as earlier thought [20,21]. Here, we temper these recent findings by reopening the possibility for nestedness to be present in very disparate systems throughout biological scales. There are, however, three caveats that leave the definitive assessment of pervasiveness of nestedness in nature an open question.

First, in our study we used an adapted version of the most used nestedness metrics (NODF, [5]) for one-mode networks [19]. But there are different analytical methods to compute nestedness [20,21,25,27]) and despite recent reviews (e.g. [10]) there is no consensus in the literature on which metric best describes nestedness. Second, our dataset is extensive and diverse, yet not comprehensive. We aimed to represent very different natural systems with which we are familiarized, so to discuss what nestedness reveals about their organization and function, but an exhaustive analysis of nestedness within scales was beyond the scope of this study. This leads to the final caveat. Since specificity and affinity among elements are intrinsic features of each system, the proposed general mechanisms assembling interactions into a nested architecture remain tentative. A wider investigation is warranted to reveal whether nested architectures are counter-shaded in other biological one-mode networks. We invite researchers to investigate proximate and ultimate processes that could generate nestedness in their biological networked systems of interest. These, once illuminated, will cast more light on the contribution of specificity and affinity as overarching mechanisms generating nestedness.

Empirical data

We analyzed 18 biological systems representing six levels of biological organization (S1 Table). At the (i) molecular level, we investigated interactions among proteins and among genes. We explored two protein-protein networks in the budding yeast Saccharomyces cerevisiae: first, the network formed by proteins related to the macromolecular complex spliceosome [58] and, second, the exonucleases complex exosome. The spliceosome is a multi-megadalton machinery composed of RNAs and more than 100 proteins. Most of these proteins are conserved from yeasts to humans [59,60]. We retrieved 103 spliceosome proteins from the STRING database [61], to build a weighted one-mode network, in which individual proteins are nodes linked by the level of support for the occurrence of interaction between them (i.e. probability that interaction is correct). This reliability score comes from the combination of different experimental evidences [62], and goes from zero (no evidence) to one (strong support for interaction) (additional details in [28]). Likewise, we built a weighted one-mode network with 44 proteins from the yeast nuclear exosome, retrieved from the IMEx database [63,64]. A third subcellular network depicted interactions among the genes of the worm C. elegans, a very reliable whole-animal model from which the largest metazoan genetic interaction network has been decoded [33]. Using the systematic interaction analysis (GSI), 11 query genes mutations (most of which is involved in signaling pathways) were tested for genetic interactions with 454 target genes [33,65]. The one-mode genetic interaction network contained the target genes as nodes linked by the interactions they shared with query genes, quantified by the Simpson index (as recommended in [66]).

At the (ii) individual level, we investigated network representations of a complex morphological structure, the human cranium. In this morphological network, nodes represented anatomically defined measurements between landmarks points, and links represented the Pearson correlation coefficients among the measurements between the landmarks [67]. Therefore, the network represent how integrated is cranial morphology and development. We analyzed 44 cranial measurements (obtained by [68]) from a sample of 1367 male and 1823 female individuals from 30 recent modern human populations distributed worldwide [68]. We explored three cases: the interpopulation morphology for females, for males, and within-population morphology. In the first and second networks, we averaged the 44 cranial measurements for each of the 30 populations (two 44 by 30 matrices); in the third network, the 44 measurements were correlated among 164 individual males from Europe (44 x 164 matrix) (Norse: Medieval samples from Oslo; Zalavar: recent samples from Hungary; and Berg: recent samples from Carinthia, Austria).

At the (iii) population level, we investigated social interactions among individuals of three species that live in societies with fission-fusion dynamics: spotted hyenas Crocuta crocuta [40,69], Guiana dolphins Sotalia guianensis [39], and bottlenose dolphins Tursiops truncatus [41]. These animal societies were represented as social networks in which nodes depicting identified individuals were linked by their social relationships, the strength of which was inferred based on co-occurrence in groups. Individual hyenas were identified by their unique spot patterns [40]. Individual dolphins were identified by natural markings on the dorsal fin using standard photo-identification protocols [39,41]. Dyadic associations were estimated as the proportion of times individuals were observed together [38], using the twice-weight index (TWI) for hyenas [40] and half-weight index (HWI) for dolphins [39,41]. We avoided spurious associations among hyenas by analyzing only individuals observed more than five times, all from a single, well-sampled clan [40]. For dolphins, we analyzed only individuals seen using the study area during the study period more than three times [39,40].

At the (iv) meta-population level, we investigated gene flow among 62 human populations worldwide (43 populations with more than 10 unrelated individuals from [47,70]; 19 populations with more than 10 unrelated individuals from [71]), among house sparrow populations from 15 Norwegian islands [43,72] and among 27 pond-frogs populations in continental and insular China [44,73]. In these networks, nodes represent populations and are linked by their genetic similarity inferred through variance of microsatellites alleles size, measured by 1-R_ST [74]. The human dataset consists of 678 autosomal microsatellite loci; the house sparrow dataset contains 13 microsatellite loci, all nuclear and selectively neutral; and the frog dataset consisted of nine neutral microsatellite loci.

At the (v) ecological community level, we investigated food webs representing trophic interaction among species. Food webs are networks of species depicted as nodes, which are connected by trophic interactions [75]. Since species can be both consumers and resource at the same time, food webs are better described as directed networks where nodes may be connected via in- and out-links. Here we focus on 3 food webs of coastal species and energy flux between them, which were chosen for varying in size (small: 29, medium: 77, large: 109 nodes) (S1 Table) and for being widely used in the literature because of their good resolution and sampling.

Finally, representing the (vi) meta-community level, we investigated the taxonomic and functional diversity of reef fishes along the Western Atlantic, including communities in the North-western Atlantic, Caribbean and the South-western Atlantic (2, 5, and 17 communities respectively). Underwater visual surveys were conducted using strip transects (20x2m) on shallow reefs (depth<20m), where all individuals (n = 536,378) were counted (total length estimated in 10-cm bins) and identified to the species level [76]. Species were grouped taxonomically at the genus level (161 genera) and using two functional grouping (FG) schemes, i.e. combination of traits: Diet x Body Size Class (40 FGs) and Diet x Body Size Class x Mobility x School size (122 FGs) (see [77] for full description and relevance of the chosen functional traits in reef fishes). Genera or functional groups were depicted as nodes connected by co-occurrence in the sampled communities given by the quantitative asymmetric Bray-Curtis index of similarity. The index was corrected using size-corrected body mass [78] since it accounts for biological turnover among individuals of different mass and differences in size structure among different communities [79].

Unipartite nestedness

Biological systems can be depicted as networks in which nodes represent elements of the system and the links between nodes describe interactions among these elements. A network can also be represented as an m × m adjacency matrix A, where m is the number of elements in the system. Each matrix cell a_ij is filled with 1 if the element represented in column j interacts (or is related to) with the element represented in row i, and is filled with 0 otherwise. The nature of interactions defines whether a network is directed or undirected (i.e., symmetric or asymmetric interactions), weighted or binary (i.e., quantitative or qualitative interactions), one- or two-mode (i.e., the network comprises a single set of interacting nodes, or two distinct sets) [80]. Nestedness had been traditionally assessed in binary two-mode networks (e.g. [8,9]), commonly via the Nestedness metric based on Overlap and Decreasing Fill (NODF [5]). Only recently, methods for assessing nestedness in direct and undirect one-mode networks have been introduced [19–21]. However, each metric reveals different properties of nestedness, for instance node overlap or segregation [20], and heterogeneous degree distribution or assortativity [21]. Given how widely used NODF is, we understand that an explicit generalization of this metric for one-mode networks is warranted. We present here the UNODF, the Unipartite version of the NODF metric.

NODF is based on two properties: paired overlap and decreasing fill [5]. The algorithm to compute NODF considers an incidence matrix with m rows and n columns (i.e. defining a two-mode network). The overlap in interactions of any two nodes, w and z, is computed depending on the relative position of the rows or columns and the number of interactions (i.e. degree) of each node, k_w and k_z. Assuming w < z (i.e. w represents a row located at an upper position from row z or a column located at a left position from column z) and k_w > k_z the pair conforms to the notion of decreasing fill and DF_wz = 100; however, if k_w ≤ k_z then DF_wz = 0. The paired overlap between w and z (PO_wz) is the proportion of interactions the row/column z shares with w. The degree of paired nestedness N_wz = 0 if DF_wz = 0 and N_wz = PO_wz if DF_wz = 100. NODF is then computed as the average of N_wz considering all pairs of rows and columns w and z.

Because NODF ultimately depends on pairwise comparisons there is no reason to restrict the metric to two-mode networks. Based on the algorithm proposed by Almeida Neto et al. [5], Lee et al. [19] adapted the NODF metric to be used in one-mode networks by defining: (1) where N is the total number of nodes of the network; A is a N × N matrix with all elements (a_ij) represented in the rows and columns; is the degree of node i. We noticed that S considers overlap between all nodes, even when these have the same degree (k_i = k_j). However, a given set may only be a proper subset in a larger set. Thus, we adapted the metric S so that the overlap is always unidirectional, computing the proportion of the interactions of the node with the smaller degree that are also present in the set of interactions of the node with larger degree (i.e. node pairs with same degree are not taken into account). Therefore, we redefine the metric S as the Unipartite Nestedness based on Overlap and Decreasing Fill (UNODF): (2) where is the Kronecker delta which means that if k_i = k_j, and otherwise.

Properties of unipartite nestedness

In completely non-nested one-mode networks UNODF = 0, while in perfectly nested networks UNODF tends towards 1 (Fig 1, S2 Fig). Because interactions of an element i with itself (a_ii = 0) are not considered, UNODF will only approach 1 for large matrices that are highly nested. Since UNODF is based on pairwise nestedness, each pair of nodes in a perfectly nested matrix is nested within each other (a staircase-like pattern); and the calculation of the nestedness value is independent of the ordering of the network adjacency matrix (different than the original NODF [5,25]).

In an adjacency matrix representing one-mode networks, all elements are represented as rows and also as columns. Thus, if such a matrix is symmetrical with respect to its main diagonal, which is the case for undirected networks, computing nestedness among rows or columns will result in the same value. Yet, the interactions depicted in matrix elements a_ij and a_ji represent different things in directed networks. In a food web, for instance, a presence in a_ij could represent the consumption of the species j by the species i, and a_ji depict the consumption of i by j. For this reason, directed networks will have two different UNODF values (and interpretations) if UNODF is obtained by computing the pairwise overlap among columns or among rows. For example, in a food web where species in rows consume the species in columns, nestedness among rows, UNODF_r, measures nestedness in resource use overlap among consumers; conversely, nestedness among columns, UNODF_c, describes the extent to which resources or prey share common consumers in a nested fashion. Additionally, food webs are the only case studied here in which a node (species) could interact with itself (cannibalism: a_ii = 1). However, because cannibalism was absent in our data sets, here we disregarded self-interactions a_ii = 1 for the sake of generality.

The choice of how to consider link weight in nestedness computation for weighted networks depends on network type and the biological meaning of the weights. As link weights are context-dependent and have very different meanings across systems (S1 Table), we focused on binary networks. We performed a sensitivity analysis to evaluate the effect of weighted links in the detection of nestedness. We evaluated the behaviour of UNODF under successive link weight thresholds (cut-off values) used to define a binary matrix (as in [26,28,67]). To allow for cross-system comparisons, we first standardized the link weight distribution of all weighted networks by dividing link weights by the maximum weight of each system, so that a_ij ϵ [0,1]. We subsequently computed UNODF for 10 binary networks for each system (S1 Table), one for each of the 10 weight cut-offs defined at 0.1 intervals (note that the cut-offs have no units). For example, for a cut-off of 0.3, all links with weight below this limit were filtered off (a_ij < 0.3 → a_ij = 0) or otherwise maintained (a_ij ≥ 0.3 → a_ij = 1) (S5 Fig). If nestedness is present regardless of value used as a threshold to build binary networks, the system can be regarded as nested despite link weight heterogeneity.

To evaluate if UNODF would capture distinctive topological features that add to other commonly analyzed properties of one-mode networks, we tested the correlation between the UNODF of all empirical networks with six network metrics (mean degree, betweeness centralization, closeness centralization, eigenvector centralization, mean clustering coefficient, mean shortest path length) (S3 Fig). To further explore this, we fitted a simple linear model to UNODF and the first principal component (from principal component analysis) summarizing the six centralization metrics of all 18 networks (S4 Fig).

Significance tests

We assessed nestedness significance with a null model, comparing the degree of nestedness computed for each empirical network to a benchmark distribution of nestedness values calculated for theoretical networks. We used a theoretical model widely used for examining the significance of structural patterns in two-mode biological networks (null model 2 in [9]), in which the probability of a link connecting two nodes is proportional to the number of links observed for both nodes. This model restrains the empirical network size (number of nodes), connectance (proportion of realized links) and the heterogeneity in the degree distribution (without fixing the degree of each node). We adapted this probabilistic null model for one-mode networks, whose adjacency matrix is squared, symmetric and the diagonal is zeroed. For each empirical network, we created 1,000 theoretical networks for each of the binary adjacency matrices resultant from the 10 weight cut-offs.

A network was considered significantly nested whenever the value of the observed network lied outside of the 95% confidence intervals of the benchmark distribution. Since the null model relies on the empirical data to build the theoretical networks it attempts to account for passive sampling artefacts and disentangle nestedness from sampling biases. Finally, since the effect of connectance on nestedness is often undesirable [20], we accounted for the relationship between UNODF and connectance (S1A Fig) with this null model that replicates the connectance of empirical networks in the theoretical networks (see also [20]).

Data and code availability

The nestedness metric and null model are available as the R package UNODF 1.2 deposited at https://bitbucket.org/maucantor/unodf. All data sets and the code to reproduce the analyses and figures are available for download in the repository https://bitbucket.org/maucantor/unodf_analyses/src.