Problem 2 (Homophily) [ 3 points] Consider a graph G=(V,A) with m edges and n nodes of R different types: 1,,R. Let Aij=1 if there is an edge {i,j} and Aij=0 otherwise. Given a node i let di be its degree and ti{1,,R} be its type. For any type r{1,,R} let er:=2m1ijAij(ti,r)(tj,r) be the fraction of edges between nodes of type r, where (a,b)=1 if a=b and 0 otherwise. Similarly, let ar:=2m1idi(ti,r) be the fraction of ends of edges attached to vertices of type r. 1. Prove that modularity can be written in terms of only er and ar. Report the formula as well as the steps you used to derive it. (1p) Hint: You may want to use the relation (ti,tj)=r(ti,r)(tj,r). 2. In the fictitious city of Ithacaa there are 85 couples. Ithacaa has citizens of four ethnicity (A, B, C and D). Researchers counted the number of couples whose members were from each possible pairing of ethnic groups and found the following: where younger and older identify the youngest and oldest member in each couple. Compute the terms er and ar for each ethnicity and use your formula to compute modularity. Would you say there is homophily