k-core organization of complex networks

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures -- k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birth points -- the bootstrap percolation thresholds. We show that in networks with a finite mean number z_2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if z_2 diverges, the networks contain an infinite sequence of k-cores which are ultra-robust against random damage.Comment: 5 pages, 3 figure

k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects

We develop the theory of the k-core (bootstrap) percolation on uncorrelated random networks with arbitrary degree distributions. We show that the k-core percolation is an unusual, hybrid phase transition with a jump emergence of the k-core as at a first order phase transition but also with a critical singularity as at a continuous transition. We describe the properties of the k-core, explain the meaning of the order parameter for the k-core percolation, and reveal the origin of the specific critical phenomena. We demonstrate that a so-called ``corona'' of the k-core plays a crucial role (corona is a subset of vertices in the k-core which have exactly k neighbors in the k-core). It turns out that the k-core percolation threshold is at the same time the percolation threshold of finite corona clusters. The mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point. We show that a random removal of even one vertex from the k-core may result in the collapse of a vast region of the k-core around the removed vertex. The mean size of this region diverges at the critical point. We find an exact mapping of the k-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment.Comment: 11 pages, 8 figure

Numerical models for the seismic assessment of an old masonry tower

The present paper describes the seismic assessment of the Qutb Minar in Delhi, India. Three models with different levels of complexity and simplifications were developed. The use of these models allows to overcome the complexity on the study of the seismic behavior of ancient masonry structures; by combining the results of the different models it is possible to obtain a better and more comprehensive interpretation of the seismic behavior. The models were used for non-linear static (pushover) and non-linear dynamic analyses. The static and dynamic analyses give different behaviors, indicating that push-over analysis should be used carefully in the seismic assessment of masonry structures. For the static analysis, the base of the tower is the most vulnerable part; while according to the dynamic analysis, it is the upper part of the tower. This last behavior is according to the historical damage suffered by the tower due to earthquakes. The different behaviors can be explained by the influence of the higher modes of vibration.This research was part of the activities of the EU-India Economic Cross Cultural Programme "Improving the Seismic Resistance of Cultural Heritage Buildings", Contract ALA-95-23-2003-077-122. F. Pefia acknowledges funding from the FCT grant contract SFRH/BPD/17449/2004, as well as the partial support of the project PAPIIT IN105409

Bootstrap Percolation on Complex Networks

We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any $f>0$ and $p>0$, and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure

Microsporogenesis in Brachiaria brizantha (Poaceae) as a selection tool for breeding.

The genus Brachiaria comprises more than 100 species and is the single most important genus of forage grass in the tropics. Brachiaria brizantha , widely used in Brazilian pastures for beef and dairy production, is native to tropical Africa. As a subsidy to the breeding program underway in Brazil, cytological studies were employed to determine the chromosome number and to evaluate microsporogenesis in 46 accessions of this species available at Embrapa Beef Cattle (Brazil)

Análise de estruturas antigas e novas em alvenaria: possibilidades e aplicações

A avaliação do desempenho sísmico de estruturas representa ainda um desafio complexo. A presente comunicação apresenta uma revisão dos métodos de análise sísmica de estruturas de alvenaria, com e sem diafragmas rígidos. Adicionalmente, inclui-se uma comparação entre a análise elástica, os métodos “pushover” e a análise não-linear dinâmica com integração no tempo, utilizando diferentes casos de estudo. Os resultados permitem concluir que: (a) a recente regulamentação europeia para o dimensionamento sísmico de estruturas modernas de alvenaria necessita melhorias; (b) as análises “pushover” tradicionais, adaptativas ou modais não reproduzem fielmente a análise não-linear dinâmica e os resultados experimentais

Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks

We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-$k$-core process: the giant heterogeneous-$k$-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-$k$-core and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$ when the degree distribution tends to a power law $P(q) \sim q^{-\gamma}$ with $\gamma < 3$.Comment: 10 pages, 4 figure

Seleção de indicadores químicos de qualidade para solos da região do entorno do lago de Sobradinho - BA, utilizando análise multivariada.

A conversão da condição natural do solo para a agricultura impõe mudanças drásticas nos atributos químicos do solo, podendo trazer sérias consequências ambientais para a região de entorno das áreas alteradas. Com o objetivo de identificar a origem dos fatores que determinam a qualidade dos solos sob caatinga e uso agrícola, nos municípios de Sobradinho, Casa Nova, Remanso e Sento Sé, região de entorno do Lago de Sobradinho, aplicou-se o modelo de estatística multivariada por meio da Análise Fatorial. Foram analisadas 12 variáveis químicas e identificou-se que o modelo de melhor ajuste para expressar a qualidade dos solos agrícolas na região foi aquele composto por três fatores, explicando cerca de 80% da variância total. Tanto para área de caatinga como para solos sob uso agrícola as componentes dos fatores foram as mesmas, mas a ordem de importância foi alterada, sendo a disponibilidade de nutrientes, considerada como componente de antropização mais importante para explicar a qualidade dos solos sob uso agrícola