Degree distributions of bipartite networks and their projections

Demival Vasques Filho and Dion R. J. O'Neale

Phys. Rev. E 98, 022307 – Published 9 August 2018

Abstract

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.

Received 21 February 2018

DOI:https://doi.org/10.1103/PhysRevE.98.022307

Physics Subject Headings (PhySH)

Degree distributions Network structure

Networks

Authors & Affiliations

Demival Vasques Filho^* and Dion R. J. O'Neale^†

Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand and Te Pūnaha Matatini, University of Auckland, Private Bag 92019, Auckland, New Zealand

^*d.vasques@auckland.ac.nz
^†d.oneale@auckland.ac.nz

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Issue

Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
Schematic drawing of three projection methods for bipartite networks: simple graph, multigraph, and weighted graph projections.
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Figure 2
Degree distributions of projected networks with a delta function as the top distributions. (a) Delta function over delta function: top and bottom degrees are $d^{*} = k^{*} = 5$ . (b) Delta function over Poisson: $d^{*} = 〈 k 〉 = 5$ . (c) Delta function over exponential: $d^{*} = 〈 k 〉 = 5$ . CCD in the inset graph stands for complementary cumulative distribution. (d) Delta function over power law: $d^{*} = 5$ and $γ_{b} = 2$ ( $〈 k 〉 ≃ 5.2$ ). Similar to the delta over Poisson case, the delta over exponential and delta over power law projected distributions, follow the distribution of their respective bottom degree distributions, modulated by the delta function to give node degrees with multiples of $q = 4$ , with some broadening of the delta function, due to a small number of nodes with less than their target degree, due to frustration in the edge adding process; see Fig. 3.
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Figure 3
Schematic drawing of a minimal example of the frustration for adding links. Edges have been numbered to indicate the order in which they were added. Although each node has been assigned a target degree of 2, once the first five edges have been added, there is no possible way to added the sixth edge without connecting a pair on nodes that are already linked.
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Figure 4
Degree distributions of projected networks from bipartite networks with a range of top distributions (Poisson: $〈 d 〉 = 5$ ; exponential: $〈 d 〉 = 5$ ; and power law: $γ_{t} = 2$ ) and fixed bottom distributions. (a) Over delta function: $k^{*} = 5$ . (b) Over Poisson: $〈 k 〉 = 5$ . (c) Over exponential: $〈 k 〉 = 5$ . (d) Over power law: $γ_{b} = 2$ . The loss of information for the simple graph projections increases, in comparison to the multigraph projections, as the tail of the bottom distribution becomes heavier. The top distributions also play a role in increasing the loss of information when they are heavy-tailed.
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Figure 5
Degree distributions of projected networks from bipartite networks with power law distribution for bottom and top nodes. Theoretical power law distributions are also shown. Projections were created using simple graphs and multigraphs in all four cases. In the plots above the diagonal ( $γ_{t} > γ_{b}$ ), the projected degree distributions follow the degree distribution of the bottom node set. For the case of $γ_{t} \leq γ_{b}$ , where the bottom degree distribution is no longer more heavy-tailed than the top, the distributions show the same flattening of the distribution that was observed for the results in Fig. 4. This is due to large cliques in the projected network. The turning point in the distributions is due to the finite size of the node sets. Increasing the size of the network causes the turning point to be shifted to the right.
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