Etworks can also be substantially skewed. When the attribute represents an
Etworks may also be substantially skewed. When the attribute represents an opinion, beneath some situations, even a minority opinion can seem to become exceptionally common locally.PLOS A single DOI:0.37journal.pone.04767 February 7,7 Majority IllusionQuantifying the “Majority Illusion” in NetworksHaving demonstrated empirically several of the relationships in between “majority illusion” and network structure, we subsequent develop a model that involves network properties inside the calculation of paradox strength. Just like the friendship paradox, the “majority illusion” is rooted in variations in between degrees of nodes and their neighbors [22, 4]. These variations result in nodes observing that, not only are their neighbors get (-)-Indolactam V improved connected [22] on typical, but that they also have extra of some attribute than they themselves have [28]. The latter paradox, which can be referred to as the generalized friendship paradox, is enhanced by correlations in between node degrees and attribute values kx [27]. In binary attribute networks, exactly where nodes is usually either active or inactive, a configuration in which larger degree nodes tend to become active causes the remaining nodes to observe that their neighbors are a lot more active than they’re (S File). Though heterogeneous degree distribution and degree ttribute correlations give rise to friendship paradoxes even in random networks, other components of network structure, which include degree assortativity rkk [42], may possibly also have an effect on observations nodes make of their neighbors. To understand why, we have to have a extra detailed model of network structure that involves correlation between degrees of connected nodes e(k, k0 ). Consider 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 Inside the equation above, e(k, k0 ) may 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 all the identical degree distribution p(k), their neighbor degree distribution q(k) is going to be precisely the same even when they have distinct degree correlations e(k, k0 ). For the identical configuration of active nodes, the probability that a node in each and every network observes an active neighbor P(x0 ) is often a function of k,k0 (k0 k)e(k, k0 ). Due to the fact 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 techniques. This suggests that the probability of obtaining an active PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/19119969 neighbor increases as degree assortativity decreases and vice versa. Therefore, 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 greater 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:Right here P(x0 k) will be the conditional probability of having an active neighbor, given a node with degree k, and is specified by Eq (3). While the threshold in Eq (four) may very well be any fraction, within this paper we concentrate on , which represents a straight majority. As a result, the fraction of all nodesPLOS One DOI:0.37journal.pone.04767 February 7,8 Majority Illusionmost of whose neighbors are active is P 2 Xkp k Xk nk n! P 0 jk P 0 jkn k:Using Eq (five), we are able to calculate the strength with the “majority illusion” paradox for any network whose degree sequence, joint degree distribution e(k, k0 ), and con.