Abstract
We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.
Highlights
One popular and important type of random graph is given by the uniformly distributed random graph with a given degree sequence, defined as follows
We let G(n, d) be a random graph with degree sequence d, uniformly chosen among all possibilities. It is well-known that it is often simpler to study the corresponding random multigraph G∗(n, d) with given degree sequence d =1n, defined for every sequence d with i di even by the configuration model: take a set of di half-edges for each vertex i, and combine the halfedges into pairs by a uniformly random matching of the set of all half-edges; each pair of half-edges is joined to form an edge of G∗(n, d)
We will in this paper study the random multigraph G∗(n, d); the reader can think of doing this either for its own sake or as a tool for studying G(n, d)
Summary
One popular and important type of random graph is given by the uniformly distributed random graph with a given degree sequence, defined as follows. If π = (πd )∞ 0 is a given sequence of probabilities πd ∈ [0, 1], let Gπ,v be the random graph obtained by deleting vertices independently of each other, with vertex v ∈ G deleted with probability 1 − πd(v) where d(v) is the degree of v in G. The more general case when we are given a sequence π = (πd )∞ 0 is handled in the same way: Site percolation, general For each vertex i, replace it with probability 1 − πdi by di new vertices of degree 1. We instead explode each half-edge with probability 1 − π, independently of all other half-edges; to explode a half-edge means that we disconnect it from its vertex and transfer it to a new, red vertex of degree 1 This does not change the number of half-edges, and there is a one-to-one correspondence between configurations before and after the explosions. We sometimes use G∗(n, d)π to denote any of the percolation models G∗(n, d)π,v, G∗(n, d)π,v or G∗(n, d)π,e
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