In the analysis of social networks, the uniform-preferential-attachment model, or UPA model is a variation of the Barabási–Albert model in which the preferential attachment is perceived as having a double nature.
This behaviour can be noticed in some examples of social networks, such as the citation network of scientific publications.
, we define for an arriving node
This subset is called a window, which represents the w last nodes inserted into the network.
In the former case, the node probability distribution is uniform: each node has a probability
In the latter, node selection follows a preferential attachment rule, as in the Barabási–Albert model.
may be constant during the addition of new nodes, expressed by
is a discrete time variable.
It can also grow with time in accord to
The network keeps its asymptotic power law behavior in degree distribution for both cases.
[1] The degree distribution for an UPA network is, considering
is: The demonstration of these formulae involves the analysis of recursive functions and the Azuma-Hoeffding Inequality.
, the degree distribution follows a power law with exponent
, as expected for the equivalent Barabási–Albert model.
, the network asymptotically follows a power law and thus keeps its scale free behavior.
[1] An UPA network may be used to model Reddit positive votes (upvotes).
Consider each node represented by a post
and the links representing upvotes given by the author after posting
Whenever a user posts a comment, he or she usually look in the same topic for another post to comment on, which characterizes a uniform attachment.
However, this user may also find it more interesting to search for another topic to comment on, possibly a popular one.
The latter represent a preferential attachment in the UPA network model.
A citation network of scientific publications is usually represented by scientific papers as nodes and citations as links.
Considering a network of papers from the same field of knowledge, whenever a new node is inserted into this network, it either attaches itself to the latest publications (uniform attachment) or to the most important papers in its field of expertise (preferential attachment).
Thus, the general behavior of these networks can be described by an UPA model.