  We present a new technique for proving logarithmic upper bounds for diameters
of evolving random graph models, which is based on defining a coupling between
random graphs and variants of random recursive trees. The advantage of the
technique is three-fold: it is quite simple and provides short proofs, it is
applicable to a broad variety of models including those incorporating
preferential attachment, and it provides bounds with small constants. We
illustrate this by proving, for the first time, logarithmic upper bounds for
the diameters of the following well known models: the forest fire model, the
copying model, the PageRank-based selection model, the Aiello-Chung-Lu models,
the generalized linear preference model, directed scale-free graphs, the
Cooper-Frieze model, and random unordered increasing $k$-trees. Our results
shed light on why the small-world phenomenon is observed in so many real-world
graphs.
