In a scale-free network the degree distribution follows a power law function.
[2] The empirical degree-distribution typically deviates downward from the power-law function fitted on higher order nodes, which means low-degree nodes are less frequent in real data than what is predicted by the Barabási–Albert model.
[3] One of the key assumptions of the BA model is preferential attachment.
In other words, every new entrant favors to connect to higher-degree nodes.
is the probability of acquiring a link by a node with degree
With a slight modification of this rule low-degree saturation can be predicted easily, by adding a term called initial attractiveness (
With this modified attachment rule a low-degree node (with low
) has a higher probability to acquire new links compared to the original set-up.
Therefore, this handicap makes less likely the existence of small degree-nodes as it is observed in real data.
As a side effect it also increases the exponent relative to the original BA model.
It is called initial attractiveness because in the BA framework every node grows in degree by time.
goes large the significance of this fixed additive term
All the distinctive features of scale-free networks are due to the existence of extremely high degree nodes, often called "hubs".
Their existence is predicted by the power-law distribution of the degrees.
Low-degree saturation is a deviation from this theoretical degree distribution, since it characterize the low end of the degree distribution, it does not deny the existence of hubs.
Therefore, a scale-free network with low-degree saturation can produce all the following characteristics: small-world characteristic, robustness, low attack tolerance, spreading behavior.