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Recently, Bollob\'as, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with $\Theta(n)$ edges.
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B. Bollobás, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs, Random Structures and Algorithms
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C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós and K. Vesztergombi, Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing, preprint (2006) (revised 2007). arXiv:0702004
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F. Chung and R. Graham, Sparse quasi-random graphs, Combinatorica
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S. Gerke and A. Steger, The sparse regularity lemma and its applications, in Surveys in combinatorics 2005
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C. Borgs, J.T. Chayes, L. Lovász, V.T. Sós and K. Vesztergombi, Counting graph homomorphisms, in Topics in Discrete Mathematics
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B. Bollobás, S. Janson and O. Riordan, Sparse random graphs with clustering, preprint (2008). arXiv:0807:2040
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