where N(i) is the set of proteins that have connections with the i-th protein, 0 ≤ Wij ≤ 1 is the (i, j)-th entry of W , and I is an n × n identity matrix. The first term in Eq. (1) enforces the prediction to be close to the initial annotation of the l proteins, and it is often viewed as the empirical loss term. The minimization of the second term enforces that the functions assigned to an unlabeled protein j are determined by the functions of its connected proteins in W; as such the second term acts as a smoothness loss term [33]. Eq. (1) is motivated by the observation that interacting proteins are more likely to share similar functions [31], and two proteins with similar amino acids often have similar functions [17]. The above equation can be rewritten in matrix notation as: