Initial attractiveness

The Barabási–Albert model generates scale-free networks where the degree distribution can be described by a pure power law.

However, the degree distribution of most real life networks cannot be described by a power law solely.

The inclusion of initial attractiveness in the Barabási–Albert model addresses the low-degree saturation phenomenon.

But in the Barabási–Albert model a node that has degree zero has probability 0 of garnering new connections.

The Barabási–Albert model defines the following linear preferential attachment rule:

The preferential attachment function of the Barabási–Albert model can be modified as follows:

have a chance to obtain connections with the newly arriving nodes.

The presence of initial attractiveness results in two important consequences one is the small degree cut-off (or small degree saturation).

This increased attachment probability becomes marginal as the node obtains more connections – it does not affect the right tail of the distribution.

The degree distribution of a model with initial attractiveness can be described by the following:

The following list offers some examples: Importantly, in case of the Barabási–Albert model the exponent of the degree distribution, here denoted by

In case of the Barabási–Albert model with initial attractiveness the degree exponent is simply

denotes the initial number of links in the network.

is higher than 3 it follows that the network is in the random network regime and as the number of initial nodes is higher it converges to the scale-free regime.

This means that the number of nodes with relatively high degrees will be lower than it would be in the Barabási–Albert model.

The higher degree exponent generally implies that the network is more homogeneous – most nodes are average linked.