I'm trying to group the graph by maximizing the modularity

Where modularity is given by:

$ Q = frac {1} {2m} sum limits_ {i, j} left (A_ {ij} – frac {d_i d_j} {2m} right) delta (c_i, c_j) $

or form of equality

$ Q = frac {1} {4m} sum_ {i, j} (A_ {ij} – frac {d_id_j} {2m}) x_ix_j x_i, x_j in left {- 1,1 right } $

I'm trying optimization using SDP relaxation + hyperplane rounding, like Goemans and Williamson's approach for maximum cutting

but I read an article here

On page 5, he states that:

"

One of the reasons why modular grouping resists ap

approach, such as semi-definite rounding is that

matrix $ B = sum_ {i, j} (A_ {ij} – frac {d_id_j} {2m}) $ It contains both negative and non-negative entries. "

How is it happening?