In the scale-free network theory (mathematical theory of networks or graph theory), a mediation-driven attachment (MDA) model appears to embody a preferential attachment rule tacitly rather than explicitly. According to MDA rule, a new node first picks a node from the existing network at random and connect itself not with that but with one of the neighbors also picked at random. where, is the probability that the new node picks a node from the labelled nodes of the existing network. It directly embodies the rich get richer mechanism. It can be re-written as
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| - In the scale-free network theory (mathematical theory of networks or graph theory), a mediation-driven attachment (MDA) model appears to embody a preferential attachment rule tacitly rather than explicitly. According to MDA rule, a new node first picks a node from the existing network at random and connect itself not with that but with one of the neighbors also picked at random. where, is the probability that the new node picks a node from the labelled nodes of the existing network. It directly embodies the rich get richer mechanism. It can be re-written as (en)
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| - In the scale-free network theory (mathematical theory of networks or graph theory), a mediation-driven attachment (MDA) model appears to embody a preferential attachment rule tacitly rather than explicitly. According to MDA rule, a new node first picks a node from the existing network at random and connect itself not with that but with one of the neighbors also picked at random. Barabasi and Albert in 1999 noted through their seminal paper noted that (i) most natural and man-made networks are not static, rather they grow with time and (ii) new nodes do not connect with an already connected one randomly rather preferentially with respect to their degrees. The later mechanism is called preferential attachment (PA) rule which embodies the rich get richer phenomena in economics. In their first model, known as the Barabási–Albert model, Barabási and Albert (BA model) choose where, is the probability that the new node picks a node from the labelled nodes of the existing network. It directly embodies the rich get richer mechanism. Recently, Hassan et al. proposed a mediation-driven attachment model which appears to embody the PA rule but not directly rather in disguise. In the MDA model, an incoming node choose an existing node to connect by first picking one of the existing nodes at random which is regarded as mediator. The new node then connect with one of the neighbors of the mediator which is also picked at random. Now the question is: What is the probability that an already existing node is finally picked to connect it with the new node? Say, the node has degree and hence it has neighbors. Consider that the neighbors of are labeled which have degrees respectively. One can reach the node from each of these nodes with probabilities inverse of their respective degrees, and each of the nodes are likely to be picked at random with probability . Thus the probability of the MDA model is: It can be re-written as where the factor is the inverse of the harmonic mean (IHM) of degrees of the neighbors of the node . Extensive numerical simulation suggest that for small the IHM value of each node fluctuate so wildly that the mean of the IHM values over the entire network bears no meaning. However, for large (specially approximately greater than 14) the distribution of IHM value of the entire network become left skewed Gaussian type and mean starts to have a meaning which becomes a constant value in the large limit. In this limit one finds that which is exactly the PA rule. It implies that the higher the links (degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways through mediators which essentially embodies the intuitive idea of rich get richer mechanism. Therefore, the MDA network can be seen to follow the PA rule but in disguise. Moreover, for small the MFA is no longer valid rather the attachment probability becomes super-preferential in character. The idea of MDA rule can be found in the growth process of the weighted planar stochastic lattice (WPSL). An existing node (the center of each block of the WPSL is regarded as nodes and the common border between blocks as the links between the corresponding nodes) during the process gain links only if one of its neighbor is picked not itself. It implies that the higher the links (or degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways. It essentially embodies the intuitive idea of PA rule. Therefore, the dual of the WPSL is a network which can be seen to follow preferential attachment rule but in disguise. Indeed, its degree distribution is found to exhibit power-law as underlined by Barabasi and Albert as one of the essential ingredients. The two factors that the mean of the IHM is meaningful and it is independent of implies that one can apply the mean-field approximation (MFA). That is, within this approximation one can replace the true IHM value of each node by their mean, where the factor that the number of edges the new nodes come with is introduced for latter convenience. The rate equation to solve then becomes exactly like that of the BA model and hence the network that emerges following MDA rule is also scale-free in nature. The only difference is that the exponent depends on where as in the BA model independent of . (en)
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