PHYSICAL REVIEW E 98, 022307 (2018) Degree distributions of bipartite networks and their projections Demival Vasques Filho * and Dion R. J. O’Neale † Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand and Te P¯ unaha Matatini, University of Auckland, Private Bag 92019, Auckland, New Zealand (Received 21 February 2018; published 9 August 2018) Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections is well understood for only a few cases, or for networks that satisfy a restrictive set of assumptions. Here we show a broader analysis in order to fill the gap left by previous studies. We use the formalism of generating functions to prove that the degree distributions of both node types in the original bipartite network affect the degree distribution in the projected version. To support our analysis, we simulate several types of synthetic bipartite networks using a configuration model where node degrees are assigned from specific probability distributions, ranging from peaked to heavy-tailed distributions. Our findings show that when projecting a bipartite network onto a particular set of nodes, the degree distribution for the resulting one-mode network follows the distribution of the nodes being projected on to, but only so long as the degree distribution for the opposite set of nodes does not have a heavier tail. Furthermore, we show that bipartite degree distributions are not the only feature driving topology formation of projected networks, in contrast to what is commonly described in the literature. DOI: 10.1103/PhysRevE.98.022307 I. INTRODUCTION Bipartite structures are of great importance in the analysis of social and economic networks. They can be used to rep- resent conceptual relations—such as membership, affiliation, collaboration, employment, ownership and others—between two different types of entities within a system, namely, bottom and top nodes, or agents and artifacts [1–3]. Often, we are particularly interested in one of the types of nodes of a bipartite network (e.g., the agents) and, in order to investigate the relationships among them, we create a new network with only these nodes. This new graph is a projection of the original bipartite network. Connections in this projected network exist only if a pair of nodes share a common neighbor in the original bipartite structure. As an example, consider a bipartite network in which executives are connected to companies when they sit on a company’s board of directors. From this network, it is possible to construct a projection, which is a network of company directors [4,5], where two agents (i.e., directors) are connected if they sit in the same board. Projections of bipartite networks are frequently used in social and economic systems analysis but, as we will see in Secs. II and IV, there is an inherent loss of information when creating a one-mode network from a bipartite structure [6,7]. Moreover, the resulting network * d.vasques@auckland.ac.nz † d.oneale@auckland.ac.nz topology and the dynamics that can take place on a projected network are particularly sensitive to the degree distributions of the underlying bipartite graph [8,9]. Hence, the edges in a projected network are obviously a consequence of the edges between agents and artifacts from the original sets. In the present paper, we give a more general view of how different degree distributions in bipartite networks affect the distributions of their projections. Studies regarding bipartite structures have shed new light on the topic [5,8–11], however several of these results are applicable for only a few specific cases and with particular assumptions about the degree distri- butions of the underlying network. In Ref. [5] only one projection is built, where the bipartite network has Poisson degree distributions for both node types. Other works [10,11] investigate projections of networks with an exponential degree distribution projected on to nodes with a power law degree degree distribution. In Ref. [9] nodes with power law degree distribution are projected onto nodes with power law and exponential degree distributions. Finally, in Ref. [8] projections were created using several probability distributions (delta, normal, exponential, and power law) pro- jecting onto a β distribution [12,13]. Similarly to the latter, we use in this work four probability distributions—namely, delta, Poisson, exponential, and power law distributions—as the degree distributions of top nodes of the bipartite networks. However, we also use them as degree distributions of bottom nodes. This way it is possible to analyze their combinations and cover a range from very peaked distributions to heavy-tailed distributions for both sets of nodes in the network. 2470-0045/2018/98(2)/022307(13) 022307-1 ©2018 American Physical Society