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In a scale-free network the degree distribution follows a power law function. In some empirical examples this power-law fits the degree distribution well only in the high degree region, however for small degree nodes the empirical degree-distribution deviates from it. This deviation of the observed degree-distribution from the theoretical prediction at the low-degree region is often referred as low-degree saturation.

Typically the empirical degree-distribution deviates downwards 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.

Theoretical background
One of the key asumptions of the Barabási–Albert model is preferential attachment. It states the probability of acquiring a new link from a new entrant node is proportional to the degree of each node. Thus every new entrant favors to connect to higher-degree nodes. Formally:

$$\Pi\left(k_i\right)=\frac{k_i}{\sum_j k_j}$$

Where $$\Pi\left(k_i\right)$$ is the probability of acquiring a link by a node with degree k. With a slight modification of this rule low-degree saturation can be predicted easily, by adding a term called initial attractiveness (<\math>k<\math>).

$$\Pi\left(k_i\right)=\frac{A + k_i}{A +\sum\limits_{j} k_j}$$

With this modified attachment rule a low-degree node (with low $$k$$) has a higher probability to acquire new links compared to the original set-up. Thus it is more attractive. It is called initial as in the Barabási-Albert framework every node grows in the degree by time, and as $$k$$ goes large the significance of this additive term diminishes.

Significance
All the distinctive features of scale-free networks are due to the existence of extremely high degree nodes, often referred as hubs. The existence of these hubs are predicted by the power-law distribution of the degrees. However small-degree saturation is a deviation from this theoretical degree distribution, since it characterize the low degree 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.