The rich get richer, and the resulting network has hubs that connect everything.
Most real networks grow rather than getting built all at once. New web pages link to popular pages, new scientists co-author with established collaborators, new airports open routes to busy hubs. Each newcomer prefers nodes that already have many connections.
This preferential attachment produces a power-law degree distribution: most nodes have few connections, a handful become massive hubs, and the gap spans orders of magnitude.
Each new node connects to existing nodes with probability proportional to their degree. A node with 10 links is twice as likely to attract the next connection as a node with 5. Early arrivals accumulate connections at an accelerating rate and become hubs.
Figure 1. Barabási-Albert preferential attachment. Each new node (orange flash) connects to two existing nodes with probability proportional to their current degree. Node radius scales with degree. Watch hubs emerge within the first 30 to 40 additions.
On log-log axes the degree distribution falls on a straight line: $P(k) \propto k^{-\gamma}$, with $\gamma$ typically between 2 and 3 for real networks.
An Erdős-Rényi random graph (edges placed uniformly) has a Poisson distribution that looks like a tight bell curve on the same axes — no hubs, every node roughly average.
Figure 2. Log-log plot of degree distributions. The scale-free network (blue) falls along a roughly straight line, the signature of a power law. The random network (red) clusters tightly around the mean. Change network size and regenerate to see how the distributions scale.
Random removals barely dent a scale-free network — they almost always hit low-degree peripheral nodes. This is why the Internet keeps working through daily router failures.
Targeted hub removal collapses the network quickly. Knocking out a few highest-degree nodes shatters the giant component. The structure that resists accidents is uniquely vulnerable to attacks.
Figure 3. Robustness test. Both panels start with identical scale-free networks. Random failure (left) removes nodes uniformly; the giant component barely shrinks. Targeted attack (right) removes the highest-degree hub at each step; the network fragments rapidly.
The Internet, airline route maps, protein interaction networks, and citation networks all show power-law degree distributions. This is why the Internet survives router failures daily but would collapse if a few key exchange points were knocked out simultaneously.