Ativity devoid of altering its degree distribution p(k). The rewiring procedure
Ativity with no changing its degree distribution p(k). The rewiring procedure randomly chooses two pairs of connected nodes and swaps their edges if undertaking so adjustments their degree correlation. This can be repeated till preferred degree assortativity is accomplished. The configuration of attributes in a network is specified by the joint probability distribution P(x, k), the probability that node of degree k has an attribute x. Within this operate, we take into account binary attributes only, and refer to nodes with x as active and these with x 0 as inactive. ThePLOS One DOI:0.37journal.pone.04767 February 7,4 Majority Illusionjoint distribution could be made use of to compute kx, the correlation in between node degrees and attributes: X xk ; kP rkx sx sk x;k X P k ; kP kix hki: sx sk k sx sk Inside the equations above, k and x would be the common deviations of the degree and attribute distributions respectively, and hkix could be the typical degree of active nodes. Randomly activating nodes creates a configuration with kx close to zero. We are able to transform it by swapping attribute values amongst the nodes. For instance, to increase kx, we randomly select nodes v with x and 
v0 with x 0 and swap their attributes when the degree of v0 is greater than the degree of v. We are able to continue swapping attributes till preferred kx is achieved (or it no longer modifications).”Majority Illusion” in Synthetic and Realworld NetworksSynthetic networks permit us to systematically study how network structure impacts the strength on the “majority illusion” paradox. First, we looked at networks having a highly heterogeneous degree distribution, which contain a number of highdegree hubs and quite a few lowdegree nodes. Such networks are often modeled with a scalefree degree distribution of the form p(k)k. To make a heterogeneous network, we first sampled a degree sequence from a distribution with exponent , exactly where exponent took 3 unique values (2 two.four, and three.), and then used the configuration model to make an undirected network with N 0,000 nodes and that degree sequence. We employed the edge rewiring process described above to make a series of networks that have the same degree distribution p(k) but distinctive values degree assortativity rkk. Then, we activated a fraction P(x ) 0.05 of nodes and utilized the attribute swapping procedure to attain various values of degree ttribute correlation kx. Fig 2 shows the fraction of nodes with more than half of active neighbors in these scalefree networks as a function of the degree ttribute correlation kx. The fraction of nodes experiencing the “majority illusion” is usually very large. For KS176 25750535″ title=View Abstract(s)”>PubMed ID:https://www.ncbi.nlm.nih.gov/pubmed/25750535 two 60 0 with the nodes will observe that more than half of their neighbors are active, despite the fact that only 5 with the nodes are, in truth, active. The “majority illusion” is exacerbated by three factors: it becomes stronger as the degree ttribute correlation increases, and because the network becomes much more disassortative (i.e rkk decreases) and heaviertailed (i.e becomes smaller). Even so, even when 3 below some situations a substantial fraction of nodes will expertise the paradox. The lines inside the figure show show theoretical estimates of your paradox utilizing Eq (five), as described within the subsequent subsection. “Majority illusion” also can be observed in networks using a a lot more homogeneous, e.g Poisson, degree distribution. We applied the ErdsR yi model to generate networks with N 0,000 and average degrees hki five.two and hki two.5. We randomly activated 5 , 0 , and 20 in the nodes, and utilized edge rewiring.
