Diophantine networks

We introduce a new class of deterministic networks by associating networks with Diophantine equations, thus relating network topology to algebraic properties. The network is formed by rep- resenting integers as vertices and by drawing cliques between M vertices every time that M dis- tinct integers satisfy the equation. We analyse the network generated by the Pythagorean equation x2+y2 = z2 showing that its degree distribution is well approximated by a power law with exponen- tial cut-o®. We also show that the properties of this network di®er considerably from the features of scale-free networks generated through preferential attachment. Remarkably we also recover a power law for the clustering coe±cient. We then study the network associated with the equation x2 + y2 = z showing that the degree distribution is consistent with a power-law for several decades of values of k and that, after having reached a minimum, the distribution begins rising again. The power law exponent, in this case, is given by ° » 4:5 We then analyse clustering and ageing and compare our results to the ones obtained in the Pythagorean case

Performance of Dairy Goats to Alfalfa Silage Based Diets Supplemented with Different Sources of Carbohydrates

Lactating Saanen dairy goats fed alfalfa silage (AS) based diets in four 4x4 Latin Square designed experiment were studied for the effects of supplementation of three different type of carbohydrates (wheat grain, (W); sorghum grain, (SG) and dry citrus pulp, (DCP)) on milk yield, composition and chewing activities. Sixteen does (45± 10 DIM and 2.016 kg ± 0.48 4% FCM) housed indoors in individual pens in a four 4x4 experiment were fed four diets 1) AS (33.9%DM, 19.9%CP, 44.01%NDF); 2) ASW (52.4%DM, 19.8%CP, 33.7%NDF); 3) ASSG (50.9%DM, 17.9%CP, 37%NDF), 4) ASDCP (52.5%DM, 16.12%CP, 39.1%NDF) with forage-to-concentrate ratios of 100:0 or 65:35, 67:33 and 64:36 respectively. Intake of AS DM (2.78%BW) was different (P\u3c 0.05) from the other treatments (average 3.53 ±0.07 %BW). Chewing efficiency (min/g NDF per kg BW 0.75) decrease (P\u3c 0.05) as a result of AS substitution or concentrate supplementation without effect (P\u3e 0.05) of carbohydrate type or dietary level of NDF. Milk, 4%FCM and fat-protein-corrected milk yield was affected (P\u3c 0.05) by concentrate supplementation. Either milk protein content (g/l) or yield (g/day) were not affected by treatments. Body weight changes appeared related to concentrate supplementation. Supplementation increase total DM intake, decrease forage DM intake and chewing efficiency and increase producing performance without changing milk composition

Topologies and Laplacian spectra of a deterministic uniform recursive tree

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.Comment: 7 pages, 1 figures, definitive version accepted for publication in EPJ

Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications

The success of new scientific areas can be assessed by their potential for contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling, after an introduction to the main concepts and models. A diversity of phenomena are surveyed, which may be classified into no less than 22 areas, providing a clear indication of the impact of the field of complex networks.Comment: 103 pages, 3 figures and 7 tables. A working manuscript, suggestions are welcome