In reality, foxes and rabbits don’t exist in a vacuum, they exist within a food web, in which nodes are organisms and directed edges indicate the flow of energy (an edge from A to B exists if B consumes A). In our network we will assume that there are plants, herbivores and carnivores. Moreover, we will assume perfect trophic coherence – herbivores only eat plants, and carnivores only eat herbivores. We consider a 9-organism network, with 4 plants (A, B, C, D), 4 herbivores (E, F, G, H) and 1 carnivore (I). Each herbivore consumes 2 plants, and the carnivore consumes 4 herbivores. (d) We first calculate the probability that randomly assigned edges, consistent with trophic coherence, that each herbivore consumes 2 plants, and the carnivore consumes 4 herbivores, would lead to an unconnected network. i Explain why the network of herbivores and carnivores, taken in isolation, must be connected. ii How many ways are there to assign edges between herbivores and plants consistent with the rules above? How many ways are there to assign edges so that plant A is not connected to the herbivores? How many ways are there to assign edges so that only two of the plants are connected? Use these numbers to calculate the probability that edges assigned randomly, but according to the rules above, would give an unconnected graph.
In reality, foxes and rabbits don’t exist in a vacuum, they exist within a food web, in which nodes are organisms and directed edges indicate the flow of energy (an edge from A to B exists if B consumes A). In our network we will assume that there are plants, herbivores and carnivores. Moreover, we will assume perfect trophic coherence – herbivores only eat plants, and carnivores only eat herbivores. We consider a 9-organism network, with 4 plants (A, B, C, D), 4 herbivores (E, F, G, H) and 1 carnivore (I). Each herbivore consumes 2 plants, and the carnivore consumes 4 herbivores.
(d) We first calculate the probability that randomly assigned edges, consistent with trophic coherence, that each herbivore consumes 2 plants, and the carnivore consumes 4 herbivores, would lead to an unconnected network.
i Explain why the network of herbivores and carnivores, taken in isolation, must be connected. ii How many ways are there to assign edges between herbivores and plants consistent with the rules above? How many ways are there to assign edges so that plant A is not connected to the herbivores? How many ways are there to assign edges so that only two of the plants are connected? Use these numbers to calculate the probability that edges assigned randomly, but according to the rules above, would give an unconnected graph.
(e) Calculating the probability of an Erd˝os-R´enyi network being unconnected is not easy. However, we can get a lower bound by estimating the probability that an Erd˝os-R´enyi network contains isolated nodes. For these purposes, we consider a directed Erd˝os-R´enyi network with 9 nodes and no self edges. We assume that each edge exists independently with a probability q that is consistent with the average out-degree ¯ kout identified in (b). Write code to generate 1000 Erd˝os-R´enyi networks to this specification, and identify the fraction that contain isolated nodes. Compare your answer to that obtained in (d)
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