Literature recommendation on random graphs

Question

I'm looking for introductory references on random graphs (commonly mentioned as Erdős–Rényi graphs), having previous acquaintance with basic graph theory. I know that Bela Bollobas' book on random graphs is the used reference, as are all his books really, but I find the book too terse for an introduction and not very accessible for non-experts of the field. If it helps, two books that I have very much enjoyed reading are, Introduction to graph theory by Trudeau and A first Course in graph theory by Chartrand and Zhang, unfortunately none of them cover random graphs.

• Any suggestions, be they books, blogs or articles are welcome.

Answer 1

There are two nice introductions. First, as I mentioned in comments, it is

• Introduction to Random Graphs, a recent book on the classical theory of random graphs, which presupposes much milder prerequisites than, e.g., Bollobas' classic. (The book is not sold yet, but you can find a draft on one of the authors' webpages)
• I also very much like the monumental (and freely available) work of Remco van der Hofstad Lecture Notes Random Graphs and Complex Networks. It goes well beyond Erdos-Renyi random graphs.

Answer 2

I'm studying random geometric graphs (RGG's) in the context of ad-hoc wireless networks. I am not sure that I can help you but I will tell you what I know. Erdos-Renyi (or Bernoulli) random graphs are one example of a random graph but there are many others. Indeed, since the probability that a distinct pair of vertices share an edge is the same for all such pairs in the Erdos-Renyi graph, there is no spatial embedding of the vertices. This makes such graphs not so useful for the kind of things that I am interested in, where the probability that two vertices share an edge depends upon a random spatial embedding of the nodes (vertices), usually according to a spatial point process.

Anyway, if you are only interested in the Erdos-Renyi graph, do read their original paper - it is very accessible - called "On random graphs I".

If, alternatively, you think you may be interested in an overview of results primarily on the Gilbert graph and the $k$-nearest neighbour graph, then I can recommend "Random geometric graphs" in Surveys in Combinatorics 2011, authored by Mark Walters, as it provides both a very readable summary of some of the foundational results and simple proof techniques. It is important to note that both of these graph models involve a spatial embedding of the nodes in the plane, which you may not be interested in.

Answer 3

There are a few books that have multiple sections about random graphs, such as:

• Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Michael Mitzenmacher and Eli Upfal (I like this one)
• Networks: An Introduction, Mark Newman
• Network Science, Albert-László Barabási