Efficient network generation under general preferential attachment

James Atwood,Bruno Ribeiro,Don Towsley

doi:10.1186/s40649-015-0012-9

Abstract

Preferential attachment (PA) models of network structure are widely used due to their explanatory power and conceptual simplicity. PA models are able to account for the scale-free degree distributions observed in many real-world large networks by sequentially introducing nodes that attach preferentially to existing nodes with high degree. The ability to efficiently generate instances from PA models is a key asset in understanding both the models themselves and the real networks that they represent. Surprisingly, little attention has been paid to the problem of efficient instance generation. In this paper, we show that the complexity of generating network instances from a PA model depends on the preference function of the model, provides efficient data structures that work under any preference function, and presents empirical results from an implementation based on these data structures. We demonstrate that, by indexing growing networks with a simple augmented heap, we can implement a network generator which scales many orders of magnitude beyond existing capabilities (106 to 108 nodes). We show the utility of an efficient and general PA network generator by investigating the consequences of varying the preference functions of an existing model. We also provide ‘quicknet,’ a freely available open-source implementation of the methods described in this work.

Highlights

There is a clear need for scalable network generators, as the ability to efficiently generate instances from models of network structure is central to understanding both the models and the real networks that they represent
Complexity we provide a formal definition of a preferential attachment (PA) model, describe two existing PA models as examples
Definitions we provide a framework for representing general preferential attachment models

Summary

Introduction

There is a clear need for scalable network generators, as the ability to efficiently generate instances from models of network structure is central to understanding both the models and the real networks that they represent. In the case of preferential attachment (PA), arguably the most widely used generative model of networks, a nonlocal distribution over node degrees must be both sampled. If we naively index this distribution, we will need to update every node at every time-step, which implies that generating a network will have complexity of at least O |V |2. Recall that the salient problem in generating networks from a PA model is indexing the network’s nodes in such a way that sampling, insertion, and incrementation can be accomplished efficiently. For a star structured network in the limit, every iteration will be constant time and the generation of a network with |V | nodes will be (|V |) This behavior is apparent for finite |V |; we have observed that the runtime of the generator tends to decrease as α increases

Related work

Findings

Conclusion