A Random Walk Among Random Graphs
by Nicolas Curien · Université Paris-Saclay (Orsay) / arXiv
Master's-level lecture notes from Orsay whose central part gives three independent proofs that a giant component emerges in the Erdős–Rényi graph, approached via one-dimensional random walks, the cycle lemma and Bienaymé–Galton–Watson trees.
More resources on Random Graphs
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Random Graphs and Complex Networks Notes
Free lecture notes by Remco van der Hofstad covering random graphs comprehensively.
Networks (2nd Edition)
Comprehensive graduate textbook on network science covering how network data are gathered, centrality and structural measures, algorithms for large networks, random graph and configuration models, community detection, percolation and robustness, and epidemic and dynamical processes, with the underlying mathematics derived in full.
Network Science (full online edition)
Interactive textbook, free from the author, whose chapters build the random network model and then break it: degree distributions, scale-free networks, the small-world property, robustness and percolation, each checked against real biological, social and technological network data.
On the Evolution of Random Graphs (Erdős & Rényi, 1960)
The 1960 paper that founded the field, tracking how G(n,M) changes as edges accumulate and identifying the sharp threshold at n/2 edges where a giant component of order n appears. Scanned original from Erdős's own archive.
Introduction to Random Graphs
Cambridge graduate text covering G(n,m) and G(n,p), thresholds, the phase transition and giant component, connectivity, Hamilton cycles, random regular graphs, and algorithmic applications. The authors host a continuously updated full PDF free at Carnegie Mellon.