A General Dependency Structure for Random Graphs and Its Effect on Monotone Properties
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We consider random graphs in which the edges are allowed to be dependent. In our model the edge dependence is quite general, we call it p-robust random graph. It means that every edge is present with probability at least p, regardless of the presence/absence of other edges. This is more general than independent edges with probability p, as we illustrate with examples. Our main result is that for any monotone graph property, the p-robust random graph has at least as high probability to have the property as an Erdos-Renyi random graph with edge probability p. This is very useful, as it allows the adaptation of many results from classical Erdos-Renyi random graphs to a non-independent setting, as lower bounds.
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