UPA model
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. New nodes joining the network may either attach themselves with high-degree nodes or with most recently added nodes. This behaviour can be noticed in some examples of social networks, such as the citation network of scientific publications. Model description. For an UPA network with nodes formula_1, we define for an arriving node formula_2 a subset of nodes formula_3 with formula_4. This subset is  called a "window", which represents the "w" last nodes inserted into the network. A new node may link itself either with a node from the window subset, with probability "p", or with any other node from formula_1 with probability "1-p". In the former case, the node probability distribution is uniform: each node has a probability formula_6 of being chosen. In the latter, node selection follows a preferential attachment rule, as in the Barabási–Albert model. The window size formula_7 may be constant during the addition of new nodes, expressed by formula_8, where formula_9 is a discrete time variable. It  can also grow with time in accord to formula_10, where formula_11, which means that window size growth is linear with the size of the network. The network keeps its asymptotic power law behavior in degree distribution for both cases. Note that when formula_12 and formula_13, the UPA model reduces to the Barabási–Albert model. Degree distribution. The degree distribution for an UPA network is, considering formula_14 and formula_12: And for formula_17 we have: Where formula_19 is the Beta function and formula_20 is: The demonstration of these formulae involves the analysis of recursive functions and the Azuma-Hoeffding Inequality. It is  observable that for formula_22 and formula_23, the degree distribution follows a power law with exponent formula_24, as expected for the equivalent Barabási–Albert model. It is also proven that for every probability formula_25 and window size formula_7, the network asymptotically follows a power law and thus keeps its scale free behavior. Occurrences in real world. Reddit. An UPA network may be used to model Reddit positive votes (upvotes). Consider each node represented by a post formula_27 and the links representing upvotes given by the author after posting formula_27. 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. Citation network. 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.