Topological properties of P.A. random graphs with edge-step functions

In this work we investigate a preferential attachment model whose parameter is a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when $f$ is a regularly varying function at infinity with strictly negative index of regular variation $-\gamma$. For this particular class, we prove a characterization for the diameter that depends only on $-\gamma$. More specifically, we prove that the diameter of such graphs is of order $1/\gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved

Large Networks of Diameter Two Based on Cayley Graphs

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.Comment: 9 pages, Published in Cybernetics and Mathematics Applications in Intelligent System

The degree/diameter problem in maximal planar bipartite graphs

The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.Peer ReviewedPostprint (published version

Generating realistic scaled complex networks

Research on generative models is a central project in the emerging field of network science, and it studies how statistical patterns found in real networks could be generated by formal rules. Output from these generative models is then the basis for designing and evaluating computational methods on networks, and for verification and simulation studies. During the last two decades, a variety of models has been proposed with an ultimate goal of achieving comprehensive realism for the generated networks. In this study, we (a) introduce a new generator, termed ReCoN; (b) explore how ReCoN and some existing models can be fitted to an original network to produce a structurally similar replica, (c) use ReCoN to produce networks much larger than the original exemplar, and finally (d) discuss open problems and promising research directions. In a comparative experimental study, we find that ReCoN is often superior to many other state-of-the-art network generation methods. We argue that ReCoN is a scalable and effective tool for modeling a given network while preserving important properties at both micro- and macroscopic scales, and for scaling the exemplar data by orders of magnitude in size.Comment: 26 pages, 13 figures, extended version, a preliminary version of the paper was presented at the 5th International Workshop on Complex Networks and their Application