Etworks also can be substantially skewed. If the attribute represents an
Etworks also can be substantially skewed. In the event the attribute represents an opinion, below some conditions, even a minority opinion can seem to be very common locally.PLOS A single DOI:0.37journal.pone.04767 February 7,7 Majority IllusionQuantifying the “Majority Illusion” in NetworksHaving demonstrated empirically a number of the relationships in between “majority illusion” and network structure, we next develop a model that involves network properties inside the calculation of paradox strength. Like the friendship paradox, the “majority illusion” is rooted in variations between degrees of nodes and their neighbors [22, 4]. These variations result in nodes observing that, not just are their neighbors greater connected [22] on average, but that additionally they have much more of some attribute than they themselves have [28]. The latter paradox, that is known as the generalized friendship paradox, is enhanced by correlations involving 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 have a tendency to be active CAY10505 site 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, such as degree assortativity rkk [42], may possibly also affect observations nodes make of their neighbors. To know why, we need a additional detailed model of network structure that incorporates correlation between degrees of connected nodes e(k, k0 ). Contemplate a node with degree k that 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 ) would 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 very same degree distribution p(k), their neighbor degree distribution q(k) might be the identical even when they have distinct degree correlations e(k, k0 ). For the same configuration of active nodes, the probability that a node in each and every network observes an active neighbor P(x0 ) is really a function of k,k0 (k0 k)e(k, k0 ). Considering that degree assortativity rkk can be a function of k,k0 kk0 e(k, k0 ), the two expressions weigh the e(k, k0 ) term in opposite approaches. 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. Thus, we anticipate 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 worth x0 :k X nkP k n! P 0 jk P 0 jkn k:Right here P(x0 k) would be the conditional probability of obtaining an active neighbor, provided a node with degree k, and is specified by Eq (three). While the threshold in Eq (4) may be any fraction, within this paper we focus on , which represents a straight majority. Thus, the fraction of all nodesPLOS A single 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:Working with Eq (5), we are able to calculate the strength in the “majority illusion” paradox for any network whose degree sequence, joint degree distribution e(k, k0 ), and con.
