Emergence of a non-scaling degree distribution in bipartite networks: a numerical and analytical study

We study the growth of bipartite networks in which the number of nodes in one of the partitions is kept fixed while the other partition is allowed to grow. We study random and preferential attachment as well as combination of both. We derive the exact analytical expression for the degree-distribution of all these different types of attachments while assuming that edges are incorporated sequentially, i.e., a single edge is added to the growing network in a time step. We also provide an approximate expression for the case when more than one edge are added in a time step. We show that depending on the relative weight between random and preferential attachment, the degree-distribution of this type of network falls into one of four possible regimes which range from a binomial distribution for pure random attachment to an u-shaped distribution for dominant preferential attachment. A bipartite network is a graph which connects two distinct sets (or partitions) of nodes, which we will refer to as the top and the bottom set. An edge in the network runs between a pair of a top and a bottom node but never between a pair of top or a pair of bottom nodes (see Fig. 1). Typical examples of this type of networks include collaboration networks such as the movie-actor [1–6], articleauthor [7–11], and board-director [12, 13] network. In the movie-actor network, for instance, the movies and actors are the elements of the top and the bottom set respectively, and an edge between an actor a and a movie m indicates that a has acted in m. The actors a and a are collaborators if both have participated in the same movie, i.e., if both are connected to the same node m. The concept of collaboration can be extended to include so diverse phenomena represented by bipartite networks as the citypeople network [14], in which an edge between a person and a city indicates that the person has visited that particular city, the word-sentence [15,16], bank-company [17] or donor-acceptor network, which accounts for injection and merging of magnetic field lines [18]. Several models have been proposed to synthesize the structure of bipartite networks when both partitions grow unboundedly over time [1–4, 16]. It has been found that for such growth models when each incoming top node connects through preferential attachment to bottom nodes the emergent degree distribution of bottom nodes follows a power-law [1]. Another important property of bipartite networks is that the clustering coefficient cannot be measured in the standard way [2], and has to be measured as a cycle of four connections [19]. On the other hand, bipartite networks, where one of the partitions remains fixed over time (i.e., the number of bottom nodes are constant), have received comparatively much less attention. Recently it was shown through numerical simulations that restrictions in the growth rate of the partitions can lead to non-scaling degree distribution highly sensitive to the parameters of the growth model [20]. However, there is still no systematic and analytical study of this kind of networks. Realizations of this type of bipartite networks include numerous relevant systems such as the interaction between the codons and genes as well as amino acids and proteins in biology and elements and compounds in chemistry. We can also include in this group those networks in which one partition can be considered to be in a pseudo-steady state while the other one keeps on growing at a much faster rate. For instance, it is reasonable to assume that for the city-people network [14], the city growth rate is zero compared with the population growth rate. Other examples of this type