Modeling, Analysis and Validation of Friendship Paradox in Evolving Networks

Friendship paradox states that individuals are likely to have fewer friends than their friends do, on average. Despite of its wide existence and appealing applications in real social networks, the mathematical understanding of friendship paradox is very limited. Only few works provide theoretical evidence of single-step and multi-step friendship paradoxes, given that the neighbors of interest are one-hop and multi-hop away from the target node. However, they consider non-evolving networks, as opposed to the topology of real social networks that are constantly growing over time. We are thus motivated to present a first look into friendship paradox in evolving networks, where newly added nodes preferentially attach themselves to those with higher degrees. Our analytical verification of both single-step and multi-step friendship paradoxes in evolving networks, along with comparison to the non-evolving counterparts, discloses that “friendship paradox is even more paradoxical in evolving networks”, primarily from three aspects: 1) we demonstrate a strengthened effect of single-step friendship paradox in evolving networks,with a larger probability（more than 0.8） of a random node's neighbors having higher average degree than the random node itself; 2) we unravel higher effectiveness of multi-step friendship paradox in seeking for influential nodes in evolving networks, as the rate of reaching the max degree node can be improved by a factor of at least ■(t2/3) with t being the network size; 3) we empirically verify our findings through both synthetic and real datasets, which suggest high agreements of results and consolidate the reasonability of evolving model for real social networks.