Exercise 2. Recall that the graph sequence (G)nzi is called sparse if limor all k0 n) P is the proportion for some...
Exercise 2. Recall that the graph sequence (G)nzi is called sparse if limor all k0 n) P is the proportion for some (deterministie) probability distribution (Pa)izo: Here of vertices in Gn-0%, En) with degree k (a) Suppose that the limiting probability distribution (Pk)k2 is a Poisson distribution with parameter λ 100. Instead of just assuming convergence to (Pr), suppose that in fact for all n greater than 1000, Pfor all k 2 0. Explain (using both words and calculations) how the adjective "sparse makes sense in describing the resulting graph sequence. (b) (This is optional, and won't count toward your grade) Now go back to the definition of sparsity and explain, using both words and calculations, why "sparse" is a reasonable adjective to use.
Exercise 2. Recall that the graph sequence (G)nzi is called sparse if limor all k0 n) P is the proportion for some (deterministie) probability distribution (Pa)izo: Here of vertices in Gn-0%, En) with degree k (a) Suppose that the limiting probability distribution (Pk)k2 is a Poisson distribution with parameter λ 100. Instead of just assuming convergence to (Pr), suppose that in fact for all n greater than 1000, Pfor all k 2 0. Explain (using both words and calculations) how the adjective "sparse makes sense in describing the resulting graph sequence. (b) (This is optional, and won't count toward your grade) Now go back to the definition of sparsity and explain, using both words and calculations, why "sparse" is a reasonable adjective to use.