Barabási and Albert's 1999 paper showed that many real-world networks — the web, citation networks, actor collaborations — aren't wired randomly. They grow over time, and new nodes preferentially connect to nodes that are already well-connected. These two ingredients, growth and preferential attachment, are enough to produce scale-free networks: a handful of highly connected hubs and a power-law degree distribution P(k) ~ k^-3, with no hub deliberately designed.

I built the simulation below to see this emerge myself — growing a network node by node under preferential attachment versus plain random attachment, and watching the degree distribution bend into a power law (a straight line on a log-log plot) only in the preferential-attachment case.

Watts and Strogatz's 1998 paper started from a regular ring lattice, where every node connects only to its nearest neighbors — highly clustered, but far apart. Rewiring each local edge to a random node with probability p interpolates toward a random graph. The surprise: even a tiny amount of rewiring collapses the average path length almost to the random-graph level while the clustering coefficient barely drops, so a network can be highly clustered and a "small world" at the same time.

I built the simulation below to see this myself — rewiring an illustrative ring lattice at a chosen p, and tracking where that p falls on the clustering-coefficient and path-length curves computed from a larger reference network, reproducing the shape of the paper's central figure.

Erdős and Rényi's 1959 paper asked three questions about a graph formed by connecting n labelled points at random: what is the probability it's completely connected, how big is its largest connected component, and how many connected components does it have? Answering these revealed one of the most influential results in network science — as the average number of connections per node (c) crosses 1, the graph undergoes a sudden phase transition from many small, disconnected fragments into a single "giant component" that swallows almost every node.

I built the simulation below to see this threshold myself — adding random edges to a network and watching the largest connected component suddenly take over as c passes 1, tracked against a reference curve computed from a larger network.