Affiliation Networks Skip to main content Explore our many areas of focus Explore all research areas Applied AI & sciences Earth AI Health AI Science AI Sustainability & crisis resilience Foundational ML & algorithms Algorithms & theory Information retrieval Machine intelligence Machine perception Natural language processing People, systems & quantum AI Human-computer interaction and visualization Networking Quantum AI Responsible AI Anti abuse Software engineering Software systems Learn More Publications Projects Building a collaborative ecosystem Datasets Access high-quality datasets to accelerate your research. Tools & services Explore our latest AI models and products. Open source Discover open-source code and collaborate with the community. Shaping the future together See all programs Faculty programs Participating in the academic research community through meaningful engagement with university faculty. Student programs Supporting the next generation of researchers through a wide range of programming. Locations Find your place in our global offices and research labs. Translating discovery into real-world impact People Our researchers drive advancements in computer science through both fundamental and applied research. Teams Collaborative groups tackling the world's most challenging AI problems. Research Explore our many areas of focus Explore all research areas Applied AI & sciences Earth AI Health AI Science AI Sustainability & crisis resilience Foundational ML & algorithms Algorithms & theory Information retrieval Machine intelligence Machine perception Natural language processing People, systems & quantum AI Human-computer interaction and visualization Networking Quantum AI Responsible AI Anti abuse Software engineering Software systems Learn More Publications Projects Resources Building a collaborative ecosystem Datasets Access high-quality datasets to accelerate your research. Tools & services Explore our latest AI models and products. Open source Discover open-source code and collaborate with the community. Conferences & events Careers Shaping the future together See all programs Faculty programs Participating in the academic research community through meaningful engagement with university faculty. Student programs Supporting the next generation of researchers through a wide range of programming. Locations Find your place in our global offices and research labs. Blog About Translating discovery into real-world impact People Our researchers drive advancements in computer science through both fundamental and applied research. Teams Collaborative groups tackling the world's most challenging AI problems. Google Research Google AI Learn about all our AI Google DeepMind Explore the frontier of AI Google Labs Try our AI experiments Research Resources Conferences & events Careers Blog About Search Home Publications Affiliation Networks Silvio Lattanzi D. Sivakumar Proceedings of the 41st Annual ACM Symposium on Theory of Computing, ACM (2009), pp. 427-434 Download Google Scholar Copy Bibtex Abstract In the last decade, structural properties of several naturally arising networks (the Internet, social networks, the web graph, etc.) have been studied intensively with a view to understanding their evolution. In recent empirical work, Leskovec, Kleinberg, and Faloutsos identify two new and surprising properties of the evolution of many real-world networks: densification (the ratio of edges to vertices grows over time), and shrinking diameter (the diameter reduces over time to a constant). These properties run counter to conventional wisdom, and are certainly inconsistent with graph models prior to their work. In this paper, we present the first model that provides a simple, realistic, and mathematically tractable generative model that intrinsically explains all the well-known properties of the social networks, as well as densification and shrinking diameter. Our model is based on ideas studied empirically in the social sciences, primarily on the groundbreaking work of Breiger (1973) on bipartite models of social networks that capture the affiliation of agents to societies. We also present algorithms that harness the structural consequences of our model. Specifically, we show how to overcome the bottleneck of densification in computing shortest paths between vertices by producing sparse subgraphs that preserve or approximate shortest distances to all or a distinguished subset of vertices. This is a rare example of an algorithmic benefit derived from a realistic graph model. Finally, our work also presents a modular approach to connecting random graph paradigms (preferential attachment, edge-copying, etc.) to structural consequences (heavy-tailed degree distributions, shrinking diameter, etc.) Research Areas Algorithms and theory Meet the teams driving innovation Our teams advance the state of the art through research, systems engineering, and collaboration across Google. See our teams Follow us Explore our other initiatives Google AI Discover how Google AI is committed to enriching knowledge and solving complex challenges Products Build Research Responsibility Societal Impact About Google Cloud High-performance infrastructure for cloud computing, data analytics & machine learning Overview Solutions Products Pricing Resources Google DeepMind Our mission is to build AI responsibly to benefit humanity Models Research Science About Google Labs Explore the future of AI responsibly with Google Labs About Experiments Stay connected About Google Google Products Privacy Terms Cookies management controls ×