ar X iv : c on d - m at / 0 50 11 69 v 2 1 8 O ct 2 00 5 Towards a Theory of Scale - Free Graphs
نویسندگان
چکیده
Although the “scale-free” literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.
منابع مشابه
ar X iv : m at h . C O / 0 11 02 40 v 1 2 2 O ct 2 00 1 Kalai ’ s squeezed 3 - spheres are polytopal
In 1988, Kalai [5] extended a construction of Billera and Lee to produce many triangulated (d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack [2, 3], he derived that for every dimension d ≥ 5, most of these (d− 1)-spheres are not polytopal. However, for d = 4, this reasoning fails. We can now show that, as already conjectured by Kalai,...
متن کاملar X iv : h ep - l at / 0 11 00 06 v 1 2 O ct 2 00 1 1 Matrix elements of ∆ S = 2 operators with Wilson fermions
متن کامل
ar X iv : h ep - l at / 0 11 00 06 v 2 3 O ct 2 00 1 1 Matrix elements of ∆ S = 2 operators with Wilson fermions
متن کامل
ar X iv : m at h / 05 10 24 8 v 1 [ m at h . O A ] 1 2 O ct 2 00 5 THE CORONA FACTORIZATION PROPERTY
The corona factorization property is a property with connections to extension theory, K-theory and the structure of C *-algebras. This paper is a short survey of the subject, together with some new results and open questions.
متن کاملar X iv : m at h / 04 10 21 7 v 1 [ m at h . C O ] 8 O ct 2 00 4 Joints in graphs
In 1969 Erd˝ os proved that if r ≥ 2 and n > n0 (r) , every graph G of order n and e (G) > tr (n) has an edge that is contained in at least n r−1 / (10r) 6r (r + 1)-cliques. In this note we improve this bound to n r−1 /r r+5. We also prove a corresponding stability result.
متن کامل