2. • A scale-free network is a network whose degree distribution follows a power law, at least
asymptotically.
• That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for
large values of k as
• where γ is a parameter whose value is typically in the range 2 < γ < 3 (wherein the second
moment (scale parameter) of is infinite but the first moment is finite), although occasionally it
may lie outside these bounds.
• The name "scale-free" means that some moments of the degree distribution are not defined, so that
the network does not have a characteristic scale or "size".
Scale-free networks
3. • A common feature of real world networks is the
presence of hubs, or a few nodes that are highly
connected to other nodes in the network.
• The presence of hubs will give the degree distribution a
long tail, indicating the presence of nodes with a much
higher degree than most other nodes.
4. • Scale-free networks are a type of network characterized by the presence of large hubs.
• A scale-free network is one with a power-law degree distribution.
• For an undirected network, we can just write the degree distribution as
5.
6. • The above figure illustrates the degree distribution of a scale-free network of N=10,000 nodes and
power-law exponent γ=2.
• The average degree is about 7, but 3/4 of the nodes have a degree of 3 or less.
• In the first bar plot, you cannot see that there are nodes with degree larger than 100, but plotting the
bar heights with a logarithmic scale (second bar plot) reveals the long tail of the degree distribution.
• Although most nodes have a very small degree, there are a few nodes with a degree above 500.
• These presence of hubs that are orders of magnitude larger in degree than most nodes is a
characteristic of power law networks.
7. Degree distribution for a network with 150000 vertices and mean degree = 6 created using
the Barabási–Albert model (blue dots).
The distribution follows an analytical form given by the ratio of two gamma functions (black line)
which approximates as a power-law.
8. • One can recognize that a degree distribution has a power-law form by plotting it on a log-log scale.
• As shown in the above scatter plot, the points will tend to fall along a line.
• The line get pretty messy, though, for large degree, as there are few points to average out the noise.
• One could use larger bins at the larger degrees in order to make the graph turn out nicer.
• To create the above plots, we didn't actually generate any networks
• One way to generate scale-free networks is using a preferential attachment algorithm.
• If you add new nodes to a network and preferentially attach them to the nodes with high degrees,
the “rich get richer” and you end up with hubs of very high degree.
• Another way to generate scale-free networks is to use the models that generate networks with given
degree distributions
