Unbiased metrics of friends’ influence in multi-level networks

The most straightforward and quite common approach to quantify influence of friends in real data is based on measuring the probability that a user i collects an item α, given that f i α <img class=”mathimg” src=”http://www.epjdatascience.com/content/inline/s13688-015-0057-x-i4.gif” alt=”View MathML”/> of i’s friends have already collected it [3], [32]–[36]. In the rest of the paper, we refer to f i α <img class=”mathimg” src=”http://www.epjdatascience.com/content/inline/s13688-015-0057-x-i5.gif” alt=”View MathML”/> as the exposure of user i to item α (assuming that a user is exposed to an item whenever one of his friends collects it). However, this measurement is strongly influenced by the heterogeneity of item popularity. Indeed, if we randomly distribute an item α to k α <img class=”mathimg” src=”http://www.epjdatascience.com/content/inline/s13688-015-0057-x-i6.gif” alt=”View MathML”/> different users, each user collects item α with probability k α / U <img…

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