The modularity of an undirected graph with node types ti,i=1,,n is defined as 2m1i,j(Aij2mkikj)(ti,tj) where A is the adjacency matrix, m is the number of edges, and (ti,tj)=1 if ti=tj and is equal to 0 if ti=tj. A few things to note: - The definition of modularity is a little abstract, but the main thing to note here is that the expected number of edges between a node i with degree ki and a node j with degree kj in a random graph with m edges according to the configuration model (to be explained in a later lecture) is equal to 2mkik1. - For a given pair of nodes i,j, a positive value of (Aij2nkikj)(ti,tj) indicates that nodes i,j have an affinity that is more than the expected affinity that they would otherwise have in a truly random graph obtained according to the configuration model with given node types and m edges. - Lilaewise, a negative value of (Aij2mkkk)(tj,t3) indicates that nodes i1j have lesser than expected affinity when i cominared to a randomigraph with the same characteristics. Modularity of a Small Graph 0/1 point (graded) Consider the following adjacency matrix of an undirected graph: A=0101010000101110000001000000001100010110010000001110010000000000000001000100000000011000010000101010 A=01010000111000000000000000010110000000110100000000000001010000000110000100101010 Raw matrix Assume that the nodes are numbered 0,,9. Let there be two node types: nodes 0,2,4,6,8 are of type 1 and noutse 1,3,5,7,9 are of type 2 . Compute the modularity of the graph. Provide an answer accurate to at least three significint figures (graded to 1% tolerance)