Juan Manuel Pastor

Short communication: Weighted-Interaction Nestedness Estimator (WINE): A new estimator to calculate over frequency matrices

We propose a new nestedness estimator that takes into account the weight of the interactions, that is, it runs over frequency matrices. A nestedness measurement is calculated through the average distance from each matrix cell containing a link to the cell with the lowest marginal totals, in the packed matrix, using a weighted Manhattan distance. The significance of this nestedness measure is tested against a null model that constraints matrix fill to observed values and retains the distribution of number of events. This is the first methodological approach that allows for the characterization of weighted nestedness. We have developed a graphical user interface (GUI) running in Matlab to compute all these parameters. The software is also available as a script for R-package and in C++ version.

Rethinking the logistic approach for population dynamics of mutualistic interactions

Mutualistic communities have an internal structure that makes them resilient to external perturbations. Late research has focused on their stability and the topology of the relations between the different organisms to explain the reasons of the system robustness. Much less attention has been invested in analyzing the systems dynamics. The main population models in use are modifications of the r-K formulation of logistic equation with additional terms to account for the benefits produced by the interspecific interactions. These models have shortcomings as the so-called r-K formulation diverges under some conditions. In this work, we introduce a model for population dynamics under mutualism that preserves the original logistic formulation. It is mathematically simpler than the widely used type II models, although it shows similar complexity in terms of fixed points and stability of the dynamics. We perform an analytical stability analysis and numerical simulations to study the model behavior in general interaction scenarios including tests of the resilience of its dynamics under external perturbations. Despite its simplicity, our results indicate that the model dynamics shows an important richness that can be used to gain further insights in the dynamics of mutualistic communities.

Robustness of Alpine Pollination Networks: Effects of Network Structure and Consequences for Endemic Plants

Abstract Global threats to plant-pollinator interactions are potentially serious in alpine ecosystems, which combine great diversity with particular fragility. We utilized tools from complex network theory to assess the robustness to species extinction of two Spanish alpine pollination networks. A comparison with ten additional alpine and subalpine pollination (ASP) networks allowed us to give our assessment a broader scope and provide a general view of ASP network robustness. We found a broad range of robustness among ASP networks. The two Spanish pollination networks ranked intermediate to high in robustness. This could be due to two of their structural features, connectance (proportion of potential interactions actually observed) and asymmetry (normalized difference between pollinator and plant richness), which showed a positive relationship with network robustness. A finer-scale focus on the two Spanish networks did not reveal differences between endemic and nonendemic plants in their functional role within the network but indicated that they differed in their robustness to pollinator extinction. Contrasting patterns across networks suggested that endemic robustness depends on community particularities. To improve the utility of robustness assessment as a conservation tool, we should increase our understanding on (1) the order in which network species will get extinct, (2) how species rewire once they have lost their partners, and (3) how much species depend on their mutualistic interaction.

Effects of topology on robustness in ecological bipartite networks

High robustness of complex ecological systems in the face of species extinction has been hypothesized based on the redundancy in species. We explored how differences in network topology may affect robustness. Ecological bipartite networks used to be small, asymmetric and sparse matrices. We created synthetic networks to study the influence of the properties of network dimensions asymmetry, connectance and type of degree distribution on network robustness. We used two extinction strategies: node extinction and link extinction, and three extinction sequences differing in the order of species removal (least-to-most connected, random, most-to-least connected). We assessed robustness to extinction of simulated networks, which differed in one of the three topological features. Simulated networks indicated that robustness decreases when (a) extinction involved those nodes belonging to the most species-rich guild and (b) networks had lower connectance. We also compared simulated networks with different degree- distribution networks, and they showed important differences in robustness depending on the extinction scenario. In the link extinction strategy, the robustness of synthetic networks was clearly determined by the asymmetry in the network dimensions, while the variation in connectance produced negligible differences.

New dynamic scaling in increasing systems

We report a new dynamic scaling ansatz for systems whose system size is increasing with time. We apply this new hypothesis in the Eden model in two geometries. In strip geometry, we impose the system to increase with a power law, L ∼ ha. In increasing linear clusters, if a < 1/z, where z is the dynamic exponent, the correlation length reaches the whole system, and we find two regimes: the first, where the interface fluctuations initially grow with an exponent β = 0.3, and the second, where a crossover comes out and fluctuations evolve as haα. If a = 1/z, there is not a crossover and fluctuations keep on growing in a unique regimen with the same exponent β. In particular, in circular geometry, a = 1, we find this kind of regime and in consequence, a unique regime holds.

Ranking of critical species to preserve the functionality of mutualistic networks using the k-core decomposition

Background Network analysis has become a relevant approach to analyze cascading species extinctions resulting from perturbations on mutualistic interactions as a result of environmental change. In this context, it is essential to be able to point out key species, whose stability would prevent cascading extinctions, and the consequent loss of ecosystem function. In this study, we aim to explain how the k-core decomposition sheds light on the understanding the robustness of bipartite mutualistic networks. Methods We defined three k-magnitudes based on the k-core decomposition: k-radius, k-degree, and k-risk. The first one, k-radius, quantifies the distance from a node to the innermost shell of the partner guild, while k-degree provides a measure of centrality in the k-shell based decomposition. k-risk is a way to measure the vulnerability of a network to the loss of a particular species. Using these magnitudes we analyzed 89 mutualistic networks involving plant pollinators or seed dispersers. Two static extinction procedures were implemented in which k-degree and k-risk were compared against other commonly used ranking indexes, as for example MusRank, explained in detail in Material and Methods. Results When extinctions take place in both guilds, k-risk is the best ranking index if the goal is to identify the key species to preserve the giant component. When species are removed only in the primary class and cascading extinctions are measured in the secondary class, the most effective ranking index to identify the key species to preserve the giant component is k-degree. However, MusRank index was more effective when the goal is to identify the key species to preserve the greatest species richness in the second class. Discussion The k-core decomposition offers a new topological view of the structure of mutualistic networks. The new k-radius, k-degree and k-risk magnitudes take advantage of its properties and provide new insight into the structure of mutualistic networks. The k-risk and k-degree ranking indexes are especially effective approaches to identify key species to preserve when conservation practitioners focus on the preservation of ecosystem functionality over species richness.

A Structural Approach to Disentangle the Visualization of Bipartite Biological Networks

Interactions between two different guilds of entities are pervasive in biology. They may happen at molecular level, like in a diseasome, or amongst individuals linked by biotic relationships, such as mutualism or parasitism. These sets of interactions are complex bipartite networks. Visualization is a powerful tool to explore and analyze them, but the most common plots, the bipartite graph and the interaction matrix, become rather confusing when working with real biological networks. We have developed two new types of visualization which exploit the structural properties of these networks to improve readability. A technique called k-core decomposition identifies groups of nodes that share connectivity properties. With the results of this analysis it is possible to build a plot based on information reduction (polar plot) and another which takes the groups as elementary blocks for spatial distribution (ziggurat plot). We describe the applications of both plots and the software to create them.

BipartGraph: An interactive application to plot bipartite ecological networks

Biotic interactions among two different guilds of species are very common in nature and are modelled as bipartite networks. The usual ways to visualize them, the bipartite graph and the interaction matrix, become rather confusing when working with real ecological networks. We have developed two new types of visualization, using the observed structural properties of these networks. A technique called k-core decomposition identifies groups of species by their connectivity. With the results of this analysis we build a plot based on information reduction (Polar Plot) and another which takes the groups as elementary blocks for spatial distribution (Ziggurat plot). We describe the basic kcorebip package for analysis and static plotting and BipartGraph, the interactive tool to explore and visualize networks. Software Availability Name of software: BipartGraph Programming language: R Operating system: Windows, Linux and MacOS Availability: Source code and documentation can be accessed at https://github.com/jgalgarra/bipartgraph User interface: Web browser License: Free, under MIT LicenseInteractions between two different guilds of entities are pervasive in biology. They may happen at molecular level, like in a diseasome, or amongst individuals linked by biotic relationships, such as mutualism or parasitism. These sets of interactions are complex bipartite networks. Visualization is a powerful tool to explore and analyse them but the most common plots, the bipartite graph and the interaction matrix, become rather confusing when working with real biological networks. We have developed two new types of visualization that exploit the structural properties of these networks to improve readability. A technique called k-core decomposition identities groups of nodes that share connectivity properties. With the results of this analysis it is possible to build a plot based on information reduction (Polar Plot) and another which takes the groups as elementary blocks for spatial distribution (Ziggurat plot). We describe the applications of both plots and the software to create them.

Two-walks degree assortativity in graphs and networks

Degree assortativity is the tendency for nodes of high degree (resp. low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is usually quantified by the Pearson correlation coefficient of the degreedegree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degreedegree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortativeassortative and disassortativedisassortative graphs according to degree and two-walks degree, respectively. More surprisingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortativeassortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortativedisassortative structure.