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 (10 6 to 10 8 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.
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 through the remarkably simple mechanism of sequentially introducing nodes that attach preferentially to high-degree nodes. 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, provide efficient data structures that work under any preference function, and present 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 (10 6 -10 8 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.
INTRODUCTION If today's furor over jurisdictional conflicts was not born in the field of antitrust law, surely that is where it matured. National differences in attitude and policy toward cartels and antitrust are deep and widespread. What may be one country's chosen instrument to gain hard currency might be condemned by another as an exploitative monopoly or trust. As Lord Wilberforce observed, "[i]t is axiomatic that in antitrust matters the policy of one state may be to defend what it is the policy of another state to attack."' This symposium exposes the fascinating diversity ofjurisdictional conflicts found in today's world of complex and highly interdependent economies. In the face of that diversity, it is instructive that even the simple, classic case of the export cartel continues to defy clear legal rules. If legal systems, both domestic and international, are unable to develop workable and predictable guidelines for the application of national antitrust laws to export cartels, what are the chances that more complex and multifaceted disputes can be satisfactorily addressed by lawyers, courts, or even governments? The United States must assume principal responsibility for the unsettled state of the law on export cartels. United States courts, urged on by the Department ofJustice, took the first bold steps to have national antitrust laws applied to offshore consortia. Government enforcement actions such as Alcoa in the 1940's,2 Swiss Watchmakers in the 1960's, 3 and the Uranium Antitrust Litigation in the 1970's 4 raised jurisdictional problems in stark, practical terms. Met with diplomatic protests, the United States then took steps to retrench on the reach of its law, but the retrenchment has been so ad hoc and case-by-case
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