A general critical condition for the emergence of a giant component in random graphs with given degrees

In this contribution, we investigate the giant component problem in random graphs with a given degree sequence. We generalize the critical condition of Molloy and Reed [Molloy, M., and B. Reed, A critical point for random graphs with given degree sequence, Random Structures Algorithms 6 (1995), 161–179], which determines the existence of a giant component in such a random graph, in order to include degree sequences with heavy tails. We show that the quantity which determines the existence of a giant component is the value of the smallest fixed point inside the interval [0, 1] of the generating function F ( s ) = ∑ i ⩾ 1 δ i s i − 1 , where δ i is the asymptotic proportion of the total degree contained in vertices of degree i. Moreover, we show that this quantity also determines the existence of a core (i.e., the maximal subgraph of minimum degree at least 2) that has linear total degree.