(30pt) Let G = (V, E) be an undirected (unweighted) graph and let S = (C1, C2, community structure of G with l communities. Let A be the adjacency matrix of G, i.e., Aj -1 if (i,j) E E and A ij = 0, otherwise. Prove that the following two formulas to compute modularity of S are equivalent 2. , Cl) be a kik ij 2m iji where E(C) is the number of edges with both ends inside C,; volC) is the total degree of nodes inside G; m is the number of edges in G; and (q,9)-1 if the two nodes i, and j are in the same community and (Ci, c)) = 0, otherwise (30pt) Let G = (V, E) be an undirected (unweighted) graph and let S = (C1, C2, community structure of G with l communities. Let A be the adjacency matrix of G, i.e., Aj -1 if (i,j) E E and A ij = 0, otherwise. Prove that the following two formulas to compute modularity of S are equivalent 2. , Cl) be a kik ij 2m iji where E(C) is the number of edges with both ends inside C,; volC) is the total degree of nodes inside G; m is the number of edges in G; and (q,9)-1 if the two nodes i, and j are in the same community and (Ci, c)) = 0, otherwise