Planar growth generates scale free networks

In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in $\mathbb{R}^2$, forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of the two defining features of the network formation procedure; growth and planarity conservation. We demonstrate that the model can be understood as a variant of random Apollonian growth and further propose a one parameter family of models with the Random Apollonian Network and the Deterministic Apollonian Network as extreme cases and our model as a midpoint between them. We then relax the planarity constraint by allowing edge crossings with some probability and find a smooth crossover from power law to exponential degree distributions when this probability is increased.Comment: 27 pages, 9 figure

Nutritionally Enhanced Staple Food Crops

Crop biofortification is a sustainable and cost-effective strategy to address malnutrition in developing countries. This review synthesizes the progress toward developing seed micronutrient-dense cereals and legumes cultivars by exploiting natural genetic variation using conventional breeding and/or transgenic technology, and discusses the associated issues to strengthen crop biofortification research and development. Some major QTL for seed iron and zinc, seed phosphorus, and seed phytate in common bean, rice,J;md wheat have been mapped. An iron reductase QTL associated with seed-iron ~QTL is found in common bean where the genes coding for candidate enzymes involved in phytic acid synthesis have also been mapped. Candidate genes for Ipa co segregate with mutant phenotypes identified in rice and soybean. The Gpe-B1 locus in wild emmer wheat accelerates senescence and increases nutrient remobilization from leaves to developing seeds, and another gene named TtNAM-B1 affecting these traits has been cloned. Seed iron-dense common bean and rice in Latin America; seed iron-dense common bean in eastern and southern Africa;....

Planarity as a driver of Spatial Network structure

In this paper we introduce a new model of spatial network growth in which nodes are placed at randomly selected locations in space over time, forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of two features of our mechanism, growth and planarity conservation. We further propose that our model can be understood as a variant of random Apollonian growth. We then investigate the robustness of our findings by relaxing the planarity. Specifically, we allow edges to cross with a defined probability. Varying this probability demonstrates a smooth transition from a power law to an exponential degree distribution

Spatial networks: Growth models and dynamical processes

Many inherently spatial systems have been represented using networks. This thesis contributes to the understanding of such networks by investigating the effect of imposing spatial constraints upon both the process network formation and dynamics that occur upon the network. Degree heterogeneity is a feature of several real world networks. However, edge length is typically constrained in a spatial network, preventing the formation of the high degree nodes that are characteristic of degree heterogeneity. We instead constrain the network to be planar, producing networks that have a scale-free degree distribution. This model turns out to be a variant of random Apollonian growth and a one parameter family of models which incorporates the planar model alongside existing Apollonian models is proposed. We identify the REDS model as a spatial model that does constrain edge length and exhibits a form of degree heterogeneity, albeit a weaker form than the scale-free distribution. REDS seeks to model social network formation by conceiving its nodes as agents who disburse a personal budget in order to maintain social bonds. We strengthen the model’s plausibility by introducing uncertainty into the agents’ budget expenditure decisions. The degree heterogeneity that was readily observed in the original model is now recovered only where decisions are subject to high levels of uncertainty. An evolutionary game is a process that lends itself to simulation upon a spatial network. This is due to the fact that a spatially constrained population is more likely to exhibit network reciprocity known to result in increased levels of co operation. We find those experiments within existing literature to be unsatisfactory in that network connectivity is assumed a priori. We address this issue by further extending the REDS model such that its nodes play prisoner’s dilemma with their network neighbours. The budget with which agents form connections is now earned by accumulating payoffs from the dilemma game. This allows for a network topology that is now endogenous to the model. This model is further distinguished from prior coevolutinary models by its agents’ ignorance of the details of their individual strategic interactions

Planar growth generates scale-free networks

In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in R 2 , forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of the two defining features of the network formation procedure; growth and planarity conservation. We demonstrate that the model can be understood as a variant of random Apollonian growth and further propose a one parameter family of models with the Random Apollonian Network and the Deterministic Apollonian Network as extreme cases and our model as a midpoint between them. We then relax the planarity constraint by allowing edge crossings with some probability and find a smooth crossover from power law to exponential degree distributions when this probability is increased

Data used to create plots in "Planar growth generates scale-free networks"

Data used to generate plots in &quot;Planar growth generates scale-free networks&quot;.</span