Etworks may also be substantially skewed. When the attribute represents an
Etworks can also be substantially skewed. In the event the attribute represents an opinion, beneath some conditions, even a minority opinion can seem to become extremely well-liked locally.PLOS A single DOI:0.37journal.pone.04767 February 7,7 Majority IllusionQuantifying the “Majority Illusion” in NetworksHaving demonstrated 
empirically many of the relationships amongst “majority illusion” and network structure, we next create a model that consists of network properties within the calculation of paradox strength. Just like the friendship paradox, the “majority illusion” is rooted in variations involving degrees of nodes and their neighbors [22, 4]. These variations lead to nodes observing that, not only are their neighbors greater connected [22] on typical, but that in addition they have more of some attribute than they themselves have [28]. The latter paradox, which is referred to as the generalized friendship paradox, is enhanced by correlations involving node degrees and attribute values kx [27]. In binary attribute networks, where nodes may be either active or inactive, a Trans-(±)-ACP configuration in which higher degree nodes tend to become active causes the remaining nodes to observe that their neighbors are extra active than they’re (S File). Whilst heterogeneous degree distribution and degree ttribute correlations give rise to friendship paradoxes even in random networks, other elements of network structure, for instance degree assortativity rkk [42], may well also impact observations nodes make of their neighbors. To understand why, we have to have a much more detailed model of network structure that contains correlation among degrees of connected nodes e(k, k0 ). Contemplate a node with degree k which has a neighbor with degree k0 and attribute x0 . The probability that the neighbor is active is: P 0 jkXkP 0 jk0 0 jkXkP 0 jk0 e ; k0 : q In the equation above, e(k, k0 ) will be the joint degree distribution. Globally, the probability that any node has an active neighbor is P 0 XkP 0 jk XXk kP 0 jk0 e ; k0 p q X X P 0 ; k0 hki X P 0 ; k0 X k0 e ; k0 e ; k0 p 0 k q 0 k k k k0 kGiven two networks with the exact same degree distribution p(k), their neighbor degree distribution q(k) might be the exact same even after they have diverse degree correlations e(k, k0 ). For precisely the same configuration of active nodes, the probability that a node in every single network observes an active neighbor P(x0 ) is a function of k,k0 (k0 k)e(k, k0 ). Since degree assortativity rkk is really a function of k,k0 kk0 e(k, k0 ), the two expressions weigh the e(k, k0 ) term in opposite ways. This suggests that the probability of possessing an active PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/19119969 neighbor increases as degree assortativity decreases and vice versa. As a result, we count on stronger paradoxes in disassortative networks. To quantify the “majority illusion” paradox, we calculate the probability that a node of degree k has more than a fraction of active neighbors, i.e neighbors with attribute value x0 :k X nkP k n! P 0 jk P 0 jkn k:Here P(x0 k) may be the conditional probability of having an active neighbor, given a node with degree k, and is specified by Eq (3). Even though the threshold in Eq (four) might be any fraction, within this paper we concentrate on , which represents a straight majority. Thus, the fraction of all nodesPLOS 1 DOI:0.37journal.pone.04767 February 7,eight Majority Illusionmost of whose neighbors are active is P 2 Xkp k Xk nk n! P 0 jk P 0 jkn k:Employing Eq (5), we are able to calculate the strength of your “majority illusion” paradox for any network whose degree sequence, joint degree distribution e(k, k0 ), and con.