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The simplified self-consistent probabilities method for percolation and its application to interdependent networks

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The simplified self-consistent probabilities method for percolation and its application to interdependent networks
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Interdependent networks in areas ranging from infrastructure to economics are ubiquitous in our society, and the study of their cascading behaviors using percolation theory has attracted much attention in recent years. To analyze the percolation phenomena of these systems, different mathematical frameworks have been proposed, including generating functions and eigenvalues, and others. These different frameworks approach phase transition behaviors from different angles and have been very successful in shaping the different quantities of interest, including critical threshold, size of the giant component, order of phase transition, and the dynamics of cascading. These methods also vary in their mathematical complexity in dealing with interdependent networks that have additional complexity in terms of the correlation among different layers of networks or links. In this work, we review a particular approach of simple, self-consistent probability equations, and we illustrate that this approach can greatly simplify the mathematical analysis for systems ranging from single-layer network to various different interdependent networks. We give an overview of the detailed framework to study the nature of the critical phase transition, the value of the critical threshold, and the size of the giant component for these different systems.
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this video I'm going to give a short presentation about our recent paper a simplified self-consistent probity methods of evaluation and its application to independent networks that
started with a single layer network multiple you measure network around in a P a link which labeled direct users can alter it's the
that means it has 3 other links and maybe 1 of them this to the giant cost which has invaded in in this
case you can enable the probability of finding a link that research on cost us X so is a case for the initial link in rat also x is the probability of finding the upper middle link as well
now we right around the cell causes the equation for X so the probability of finding around and link it into a jackass is the accord to 1st the probability of finding a bank that is to no be the Greek K modified spoke orbited that and this 1 of the other k minus one Inc's made into the giant cost since we talking about multiple collection there is a probability of PE that's the mission mold found is still present so we need to modify the right hand side that P and that gives us the complete equation for x the problem definition of X
you can write it on the poor radio finding rent an old in the giant cast the equivalent this is the relative size of the John cost scumbag to the whole network that's defined there's this new infinity and that it goes to 1st the probability of finding an old 50 Greek modified by the probability that at least 1 both is k links leading to the John cost and finally you need to modify the right hand side by P which is a probability that some old really exist after the initial attack knowledge of two-layer
intuitive and the network in this case there are 2 layers of network a and B every molding a has a intuitive and the mode in B 1 a fellows is interdependent noting being fails as well let's say they the new peak link network in mislabeling brat iron users to an altered degree K you could for so it has to other makes 1 of them is to the mutually connected giant class sciences initially connected giant Costa a man's this giant any is interdependent with a giant cost in the and since it's in interdependent network the 1st note found the has a
interdependent note UBS well let's say this knowing B. C-Brief all this fall links and let's assume that wall physics canasta Muturi giant command connector cost as well and that's assume that 1 link this to the military connected giant cluster the probability why and the probability of finding around the link network Indonesia Muturi giant a master is X so look self comes as the equation for all x is simply the probability of finding note from the link with the Greek modified by the probability that areas while all of this the other came warnings leading into the M C G C modified by the fact that this nose into abandoned all the net the news to being the M C G C a spell and that is given by the probability that that is 1 of his Kay Prize links Mr. Dmitri giant John Costa and finally you need to modify everything on the right by probability P which is a probability that the null to exist after the initial attack so naturally you're right on the probability why ends the probability of finding a random multiday that belongs to a mutually connected jackass so this simple framework can be extended to other more complicated the cases of interdependent networks thank you