Enumeration of graphs with a heavy-tailed degree sequence

In this paper, we asymptotically enumerate graphs with a given degree sequence d = (d1, . . . , d(n)) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general result requires upper bounds on functions of Mk = an_i[dd k for a few small integers k >= 1. Note that M-k is simply the total degree of the graphs. As special cases, we asymptotically enumerate graphs with (i) degree sequences satisfying M-2 = o(M1(9/8))/8); (ii) degree sequences following a power law with parameter-gamma > 5/2; (iii) power -law degree sequences that mimic independent power -law "degrees" with parameter gamma > 1 -I- root 3. approximate to 2.732; (iv) degree sequences following a certain "long-tailed" power law; (v) certain bivalued sequences. A previous result on sparse graphs by McKay and the second author applies to a wide range of degree sequences but requires i = o(M,1/3), where Delta is the maximum degree. Our new result applies in some cases when A is only barely o(M-1(3/5)). Case (i) above generalises a result of Janson which requires M-2 = O(M-1) (and hence M-1 = O(n) and Delta = O(n(1/2))). Cases (ii) and (iii) provide the first asymptotic enumeration results applicable to degree sequences of real-world networks following a power law, for which it has been empirically observed that 2 < gamma < 3. (C) 2015 Elsevier Inc. All rights reserved.