Cture of numerous realworld networks creates conditions for the “majority illusion
Cture of numerous realworld networks creates situations for the “majority illusion” paradox.Components and MethodsWe used the configuration model [32, 33], as implemented by the SNAP library (https:snap. stanford.edudata) to make a scalefree network using a specified degree sequence. We generated a degree sequence from a energy law of the form p(k)k. Here, pk is the fraction of nodes which have k halfedges. The configuration model proceeded by linking a pair of randomly selected halfedges to kind an edge. The linking procedure was repeated until all halfedges 
have been utilised up or there were no more techniques to type an edge. To create ErdsR yitype networks, we started with N 0,000 nodes and linked pairs at random with some fixed probability. These probabilities have been selected to create average degree equivalent to the typical degree from the scalefree networks.PLOS A single DOI:0.37journal.pone.04767 February 7,3 Majority IllusionTable . Network properties. Size of networks F 11440 site studied within this paper, along with their average degree hki and degree assortativity coefficient rkk. network HepTh Reactome Digg Enron Twitter Political blogs nodes 9,877 6,327 27,567 36,692 23,025 ,490 edges 25,998 47,547 75,892 367,662 336,262 9,090 hki 5.26 46.64 2.76 20.04 29.two 25.62 rkk 0.2679 0.249 0.660 0.08 0.375 0.doi:0.37journal.pone.04767.tThe statistics of realworld networks we studied, such as the collaboration network of high power physicist (HepTh), Human protein rotein interactions network from Reactome project (http:reactome.orgpagesdownloaddata), Digg follower graph (DOI:0.6084 m9.figshare.2062467), Enron e-mail network (http:cs.cmu.eduenron), Twitter user voting graph [34], and also a network of political blogs (http:wwwpersonal.umich.edumejn netdata) are summarized in Table .ResultsA network’s structure is partly specified by its degree distribution p(k), which gives the probability that a randomly chosen node in an undirected network has k neighbors (i.e degree k). This quantity also impacts the probability that a randomly selected edge is connected to a node of degree k, otherwise called neighbor degree distribution q(k). Because highdegree PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/23139739 nodes have more edges, they’re going to be overrepresented inside the neighbor degree distribution by a aspect proportional to their degree; hence, q(k) kp(k)hki, exactly where hki is definitely the typical node degree. Networks usually have structure beyond that specified by their degree distribution: for instance, nodes might preferentially link to other people with a similar (or extremely diverse) degree. Such degree correlation is captured by the joint degree distribution e(k, k0 ), the probability to find nodes of degrees k and k0 at either end of a randomly chosen edge in an undirected network [35]. This quantity obeys normalization conditions kk0 e(k, k0 ) and k0 e(k, k0 ) q(k). Globally, degree correlation in an undirected network is quantified by the assortativity coefficient, which is simply the Pearson correlation amongst degrees of connected nodes: ” ! X X 0 2 0 0 0 0 kk ; k q two kk e ; k hkiq : r kk two sq k;k0 sq k;k0 P P 2 Here, s2 k k2 q k kq . In assortative networks (rkk 0), nodes possess a tendency q link to similar nodes, e.g highdegree nodes to other highdegree nodes. In disassortative networks (rkk 0), however, they choose to link to dissimilar nodes. A star composed of a central hub and nodes linked only towards the hub is an example of a disassortative network. We are able to use Newman’s edge rewiring process [35] to change a network’s degree assort.