A new approach to the minimum cut problem
Journal of the ACM (JACM)
The small-world phenomenon: an algorithmic perspective
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Regular Article: The Diameter of Sparse Random Graphs
Advances in Applied Mathematics
Graph evolution: Densification and shrinking diameters
ACM Transactions on Knowledge Discovery from Data (TKDD)
Scalable modeling of real graphs using Kronecker multiplication
Proceedings of the 24th international conference on Machine learning
Realistic, mathematically tractable graph generation and evolution, using kronecker multiplication
PKDD'05 Proceedings of the 9th European conference on Principles and Practice of Knowledge Discovery in Databases
Proceedings of the forty-first annual ACM symposium on Theory of computing
News Posting by Strategic Users in a Social Network
WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Kronecker Graphs: An Approach to Modeling Networks
The Journal of Machine Learning Research
Deterministic decentralized search in random graphs
WAW'07 Proceedings of the 5th international conference on Algorithms and models for the web-graph
Generalizing Kronecker graphs in order to model searchable networks
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Giant components in Kronecker graphs
Random Structures & Algorithms
Approximate axial symmetries from continuous time quantum walks
SSPR'12/SPR'12 Proceedings of the 2012 Joint IAPR international conference on Structural, Syntactic, and Statistical Pattern Recognition
An in-depth analysis of stochastic Kronecker graphs
Journal of the ACM (JACM)
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A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large-scale real-world networks such as the web. This model simultaneously captures several well-known properties of real-world networks; in particular, it gives rise to a heavy-tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this paper, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real-world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm.