Mediation-driven attachment model
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 [1] 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.
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.
The new node then connect with one of the neighbors of the mediator which is also picked at random.
the IHM value of each node fluctuate so wildly that the mean of the IHM values over the entire network bears no meaning.
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
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.
the MFA is no longer valid rather the attachment probability
The idea of MDA rule can be found in the growth process of the weighted planar stochastic lattice (WPSL).
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.
[3][4] Degree distribution: The two factors that the mean of the IHM is meaningful and it is independent of
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.
They are special because their existence make the mean distance, measured in units of the number of links, between nodes incredibly small thereby playing the key role in spreading rumors, opinions, diseases, computer viruses etc.
[5] It is, therefore, important to know the properties of the largest hub, which we regard as the leader.
An interesting question is: how long does the leader retain this leadership property as the network evolves?
To find an answer to this question, we define the leadership persistence probability
Persistence probability has been of interest in many different systems ranging from coarsening dynamics to fluctuating interfaces or polymer chains.
The basic idea of the MDA rule is, however not completely new as either this or models similar to this can be found in a few earlier works, albeit their approach, ensuing analysis and their results are different from ours.
For instance, Saramaki and Kaski presented a random-walk based model.
[6] Another model proposed by Boccaletti et al. may appear similar to ours, but it markedly differs on closer look.
[8] However, the nature of their expressions are significantly different from the one studied by Hassan et al..
Yet another closely related model is the Growing Network with Redirection (GNR) model presented by Gabel, Krapivsky and Redner where at each time step a new node either attaches to a randomly chosen target node with probability
One more difference is that, in the MDA model new node may join the existing network with
