In many real networks the number of connections per node follows a power law, meaning most nodes have few links while a small number of hubs have enormously many. There is no typical or average degree that characterizes the system, so such networks are called scale-free.
Scale-free structure can emerge from a growth rule where new nodes attach preferentially to nodes that already have many connections (the rich get richer). This simple dynamic, rather than design, produces the heavy-tailed degree distribution seen in networks like the web and citations.
Despite containing huge numbers of nodes, many networks have a very small average number of steps separating any two nodes, often illustrated by the six degrees of separation in social ties. A few long-range shortcuts dramatically cut the distance across an otherwise locally clustered network.
Scale-free networks tolerate random failures well because most nodes are unimportant, so removing random nodes rarely breaks connectivity. The same hub structure makes them acutely vulnerable to targeted attack: removing the few high-degree hubs fragments the network.
Treating cells, ecosystems, the internet, economies, and societies all as networks of nodes and links reveals that systems with unrelated substance often share the same statistical architecture. The organizing patterns, such as hubs and short paths, recur regardless of what the nodes physically are.
How diseases, information, or failures propagate depends heavily on network structure, not just on transmission rate. In scale-free networks the usual epidemic threshold can vanish, so even weakly contagious items can spread widely because hubs act as powerful amplifiers.