Degrees and distances in random and evolving Apollonian networks

This paper studies Random and Evolving Apollonian networks (RANs and EANs), in d dimension for any d>=2, i.e. dynamically evolving random d dimensional simplices looked as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power law tail with exponent tau=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the n-th step of the dynamics is q_n->0 and sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once the occupation parameter q->0. We also determine the asymptotic behavior of shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show that the shortest path between two uniformly chosen vertices (typical distance), the flooding time of a uniformly picked vertex and the diameter of the graph after n steps all scale as constant times log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar CLT for typical distances in EANs

Magyar Gyógypedagógia 15 (1927) 09-10

A Magyar Gyógypedagógiai Társaság folyóirata 15. évfolyam, 9-10. szám, Budapest, 1927. Havi folyóirat a fogyatékosok (siketnémák, vakok, szellemileg gyengék, beszédhibások, idegesek, epileptikusok és nyomorékok) ügyeinek tárgyalására. 1939-től beolvadt a Magyar gyógypedagógiai tanárok közlönyébe

Magyar Tanítóképző 53 (1940) 9

Magyar Tanítóképző A Tanítóképző-intézeti Tanárok Országos Egyesületének folyóirata 53. évfolyam, 9. szám Budapest, 1940. szeptembe

Magyar Tanítóképző 55 (1942) 11

Magyar Tanítóképző A Tanítóképző-intézeti Tanárok Országos Egyesületének folyóirata 55. évfolyam, 11. szám Budapest, 1942. novembe

Degrees and distances in random and evolving Apollonian networks

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d ≥ 2, i.e. dynamically evolving random ddimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ = (2d - 1)/(d - 1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d ≥ 2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs

Interference of <i>M</i> = 3 modules with different choice of scales.

<p><b>(a)</b> Top: Posterior densities for three modules with rational scale ratios. The overlap between the modules is shown in black, its height indicates the interference of the three modules as a function of distance from the origin. The representation becomes ambiguous only if all 3 modules interfere, as at distance 6. Bottom: Ambiguity in position coding quantified by the multi-modality of the combined posterior. <b>(b)</b> Posterior densities for three modules with pairwise optimal scale ratios. The scales are 1 (blue), <i>σ</i> (red), and <i>σ</i>² (olive), where <i>σ</i> is the golden ratio. As we have more modules (3) than the order of <i>σ</i> (2), wherever any two modules interfere with each other, then they interfere with the third as well: at distance 8 the three peaks almost coincide. <b>(c)</b> The same as in (b) for scales 1, 2^1/3, 2^2/3, powers of a third order algebraic number. Although pairwise interference can be very strong between any pairs (e.g. at distances 5, 6.2 and 8), the total interference is substantially lower than in panel b (bottom).</p