Etworks also can be substantially skewed. If the attribute represents an

Etworks may also be substantially skewed. If the attribute represents an opinion, beneath some situations, even a minority opinion can appear to be extremely preferred locally.PLOS 1 DOI:0.37journal.pone.04767 February 7,7 Majority IllusionQuantifying the “Majority Illusion” in NetworksHaving demonstrated empirically a few of the relationships amongst “majority illusion” and network structure, we next create a model that MedChemExpress Stattic involves network properties within the calculation of paradox strength. Just like the friendship paradox, the “majority illusion” is rooted in variations between degrees of nodes and their neighbors [22, 4]. These variations lead to nodes observing that, not simply are their neighbors improved connected [22] on typical, but that additionally they have additional 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, exactly where nodes might be either active or inactive, a configuration in which higher degree nodes tend to be active causes the remaining nodes to observe that their neighbors are much more active than they’re (S File). While 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 well also influence observations nodes make of their neighbors. To know why, we require a much more detailed model of network structure that consists of correlation between degrees of connected nodes e(k, k0 ). Look at 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 ) 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 the exact same degree distribution p(k), their neighbor degree distribution q(k) is going to be the same even when they have diverse degree correlations e(k, k0 ). For precisely the same configuration of active nodes, the probability that a node in each network observes an active neighbor P(x0 ) is actually a function of k,k0 (k0 k)e(k, k0 ). Since degree assortativity rkk is usually a function of k,k0 kk0 e(k, k0 ), the two expressions weigh the e(k, k0 ) term in opposite methods. 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. As a result, we anticipate 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:Here P(x0 k) is the conditional probability of possessing an active neighbor, given a node with degree k, and is specified by Eq (3). Even though the threshold in Eq (four) could possibly be any fraction, within this paper we focus on , which represents a straight majority. As a result, the fraction of all nodesPLOS A single DOI:0.37journal.pone.04767 February 7,eight Majority Illusionmost of whose neighbors are active is P two Xkp k Xk nk n! P 0 jk P 0 jkn k:Utilizing Eq (5), we are able to calculate the strength on the “majority illusion” paradox for any network whose degree sequence, joint degree distribution e(k, k0 ), and con.

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