Question

# (a) Consider the following table representing the adjacency matrix A of a digraph (di-rected graph) of a small simulated ecosystem. A value of 1 in position a¿¡ means that species j is a food source for species i. Answer the following questions: (i) Calculate the out degree and the in degree for each node and hence determine which species has more direct sources of food, and what species is the direct food source for most other species. (ii) If k is a food source of j, and j is a food source of i, we say that k is an indirect food source of i. Use operations on the adjacency matrix to rank the species in terms of their indirect food sources. Then determine the species that have more total food sources (direct and indirect foods combined). [Hint: Relate your answer with the number of walks of length one and two.] >) Consider the Network in Figure 9: (i) Find the Freeman's network centrality.11 (ii) Calculate the distance of the shortest path between the nodes (the distance matrix) and use this to determine the closeness centrality. (iii) Calculate the number of times a node appears in a geodesic path and use this to determine the between ness centrality. (c) (i) Calculate the stationary distribution of a simple random walk on Kn, that is on a complete graph with n vertices. (ii) Calculate the stationary distribution of a simple random graph on a regular-graph, that is on a graph where each vertex has the same number of neighbours or the same degree.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8  Fig: 9  Fig: 10  Fig: 11  Fig: 12