Bipartite graphs as models of complex networks

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Abstract

It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model which achieves the following challenges: it produces graphs which have the three main wanted properties (clustering, degree distribution, average distance), it is based on some real-world observations, and it is sufficiently simple to make it possible to prove its main properties. This model consists in sampling a random bipartite graph with prescribed degree distribution. Indeed, we show that any complex network may be viewed as a bipartite graph with some specific characteristics, and that its main properties may be viewed as consequences of this underlying structure. We also propose a growing model based on this observation.

Introduction

It has been shown recently that most real-world complex networks have some specific properties in common. These properties are not captured by the model generally used before this discovery, although they play a central role in many contexts like the robustness of the Internet [1], [2], [3], [4], [5], the spread of viruses or rumors over the Internet, the Web or other social networks [6], [7], [8], the performance of protocols and algorithms [9], [10], [11], and many others.

This is why a strong effort has been put in the realistic modeling of complex networks in the last few years, and much progress has been accomplished in this field. Some models achieve the aim of producing graphs which capture some, but not all of the main properties of real-world complex networks. Some models obtain all the wanted properties but rely on artificial methods which give unrealistic graphs (trees, graphs with uniform degrees, etc.). Others rely on construction processes which may induce some hidden properties, or are too difficult to analyze.

In this paper, we propose the random bipartite graph model as a general model for complex networks. We will show that this model produces graphs with many observed properties. It relies on real-world observations and gives realistic graphs. Finally, it is simple enough to make it possible to prove its main properties. We will also discuss some identified drawbacks of this model.

We will first present an overview of the context in which our work lies. In particular, we use some ideas introduced in previous papers, which we need to describe precisely. Then we show how all complex networks may be described as bipartite structures. After this, we present the random bipartite model and analyze it to show that the main properties of complex networks are somehow a consequence of their underlying bipartite structure. We also present a growing bipartite model based on the same ideas. Finally, we discuss the advantages and limitations of these models.

Section snippets

Context

Throughout our presentation, we will use a representative set of complex networks which have received much attention and span quite well the variety of contexts in which complex networks appear: an Internet graph at router level [12], [13], [14] consisting of physical links between routers; a web graph from Notre Dame university [15], [16] where web pages are linked by hyperlinks; a co-occurrence graph in which words are linked if they belong to a same sentence in a given text [17], [18]; the

Complex networks as bipartite graphs

A bipartite graph is a triple G=(,,E) where and are two disjoint sets of vertices, respectively, the top and bottom vertices, and E× is the set of edges. The difference with classical graphs lies in the fact that edges exist only between top vertices and bottom vertices.

Two degree distributions can naturally be associated with such a graph, namely the top degree distribution: k=|{t:d(t)=k}||| and the bottom degree distribution: k=|{t:d(t)=k}|||. These two distributions play a

The bipartite models

Our aim is now to use the new general property of real-world complex networks discovered in the previous section, namely their underlying bipartite structure, as a way to propose a model which captures the main wanted properties.

As discussed in the first section of this paper, there are basically two ways to achieve this goal. First, we may try to sample random bipartite graphs with prescribed (top and bottom) degree distributions. Second, we may try to propose a construction process similar to

Analysis of the models

Our aim in this section is to give formal proofs for the main properties of the -projection of a random bipartite graph with prescribed degree distributions. Some of these properties, and others, have been studied independently in Refs. [24], [35] with different techniques and a different point of view. We however believe that our proofs give new insight on these properties, therefore, we give them below. In particular, our proof techniques may be considered as more mathematically rigorous.

Experimental results

The formal results of the previous section give a precise intuition on how the random bipartite graph model with prescribed degree distributions behaves. We can also check its properties experimentally by generating graphs using this model and the same parameters as the ones measured on real-world complex networks. This is what we do in this section with our six examples, for the purely random bipartite model as well as for the one with preferential attachment.

More precisely, the networks are

Conclusion and discussion

In this paper, we propose bipartite graphs as a general tool for the modeling of real-world complex networks. They make it possible to achieve the following challenges:

  • the obtained networks have the three main wanted properties (logarithmic average distance, high clustering and power law degree distribution);

  • the models are based on a realistic construction process representative of what happens in some real-world cases, and

  • their definitions are simple enough to make it possible to give some

Acknowledgments

We thank Annick Lesne, Clémence Magnien and James Martin for careful reading of preliminary versions and useful comments. We also thank the anonymous referees for helpful comments.

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