The architecture of bipartite networks linking two classes of constituents is affected by the interactions within each class. For the bipartite networks representing the mutualistic relationship between pollinating animals and plants, it has been known that their degree distributions are broad but often deviate from power-law form, more significantly for plants than animals. Here we consider a model for the evolution of the mutualistic networks and find that their topology is strongly dependent on the asymmetry and non-linearity of the preferential selection of mutualistic partners. Real-world mutualistic networks analyzed in the framework of the model show that a new animal species determines its partners not only by their attractiveness but also as a result of the competition with pre-existing animals, which leads to the stretched-exponential degree distributions of plant species.
Diverse interactions and dependencies among nonidentical elements are characteristic of complex systems [1][2][3]. Ecological systems are a prototypical example, in which numerous species interact via predation, herbivory, mutualistic support, competition, cooperation, and so on, and their network structure and function have attracted much attention [4][5][6][7][8]. The mutualistic relation between two species is beneficial for the survival and reproduction of both of them such as animal pollinators and flowering plants. Given numerous species of plants and insects, producing nectar with different composition and flavor and carrying a wide range of preference and capacities, respectively, the establishment of individual mutualistic relationship depends on specific needs and qualification. Nevertheless, the global organization of the mutualistic plant-pollinator networks exhibits common features [9][10][11][12][13][14][15]. The degrees of plant and animal species are distributed broadly in general, but their distributions often deviate from a power-law form, more significantly for plant species: While the degree distribution of animals are close to power laws, those of plants are of truncated power-law, exponential, or stretchedexponential form [10][11][12]. Biological matching, species abundance, and the difference between the numbers of animals and plants may bring such deviation from scale invariance [10,11,14], which, however, remains to be addressed further [15]. Interestingly, mutualistic networks have their topological features distinguished from trophic networks particularly in their nestedness and modularity, which is related to the stable architecture varying with the type of interaction [16].
To gain insight into the underlying mechanism of mutualistic community formation, here we propose and study a simple growing bipartite network model and apply it to analyze real-world mutualistic networks. The model is based on a generalized preferential-selection rule, being an extension of the model in Ref. [11]. From the nestedness and broad degree distributions identified in lots of mutualistic communities, the preferential selection [17] has been expected to play a role in their evolution [10,11,15,16]. Our model provides a unified picture of evolving mutualistic communities. The nonlinear preferential selection reflects the impact of individual characteristics and interspecific interactions on determining symbionts. Fitting the model prediction to empirical data, we discover a pattern of interspecific interaction affecting the architecture of mutualistic networks. While a new animal species is attracted to a plant species with high abundance and thus with many pollinators already, it should compete with the existing pollinators. As a result, the chance to find a plant species with a large number of pollinators is not as high as expected, which is shown to cause the degree distribution of plants to take a stretched-exponential form.
To be specific, we consider plant-pollinator mutualistic networks as in Fig. 1 (a), in which each node represents a species of either animal (A) type or plant (B) type and some pairs of nodes of the opposite types are connected representing their mutualistic relationship. A growing bipartite network (GBN) model defined below and also sketched in Fig. 1(b) [11,[18][19][20] illustrates the topological evolution of the plant-pollinator networks. Initially there are A nodes of type A and B nodes of type B, all pairs of nodes of the opposite types being connected. At each time step, a new node of type A (B) arrives with probability P A (P B ), where P A + P B = 1. The new node of type A (B) are linked to A ( B ) partners of type B (A) that are selected with the probability proportional to the λ B (λ A ) th power of their degrees. Iterating these procedures up to time N , one obtains a bipartite network of N P A nodes of type A and N P B nodes of type B on average.
The degree-based preferential selection by a new species is assumed in this model and can be motivated as follows. In case that a new species appears by speciation from an existing one, it can be assumed to in- herit the mutualistic interactions of the ancestor species, as the protein interactions are inherited by duplicated genes [21]. Then, the more partners an existing species has, the higher the chance to have a new mutualistic partner speciated from one of its old partners is. Also, if we assume that a new species in a community, appearing by either speciation or migration, selects randomly its partner organisms, the new species will be more likely to form a mutualistic relationship with a more abundant species due to the existence of more organisms of the species. Given that the number of mutualistic partners -degree -of a species is positively correlated with the abundance of the species, the degree-based preferential selection is expected to work in this case, too.
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