Web Mining and Social Networking: Techniques and ... - tud.ttu.ee

160 7 Extracting and Analyzing Web Social Networkscommunity k involves nodes v i and v j , respectively. Written in a form of matrix, it changes toW ≈ XΛX T , where X ∈ R n×m is a non-negative matrix with x ik = p k→i and ∑ i x ik = 1. Λ is am × m nonnegative matrix with λ k = p k [267]. As such, the interaction (i.e. similarity) matrixW is approximated via the community structures.Based on above formulations, the snapshot cost CS is defined as the error introduced bysuch an approximation, i.e.,(∥) CS = D W ∥XΛX T (7.13)where D(A‖B) is the KL-divergence between distribution A and B.Temporal CostAs discussed above, the community structure is formulated by XΛ. The temporal cost meansthe community structure change from time stamp t − 1tot, thus it is measured by the differencebetween the community structures at two states, i.e.,A Combined CostCT = D(X t−1 Λ t−1 ‖XΛ ) (7.14)By combining the snapshot cost and temporal cost into a unified scheme, we eventually formulatethe analysis of communities and their evolutions as an optimization problem of bestcommunity structure at time stamp t, which is expressed by X and Λ. Apparently, the minimizationof the following equation leads to the final solution.(∥) cost = α · D W ∥XΛX T +(1 − α) · D(X t−1 Λ t−1 ‖XΛ ) (7.15)7.3.3 AlgorithmTo solve the above optimization, the authors used an iterative algorithm by updating the valuesof X and λ alternatively until the Eq.7.15 converges. The algorithm works as follows:• Updating X given W ,X and λthen normalize such that ∑ ix ik = 1,∀k• Updating λ given updated Xw ij· λ k · x jkx ik ← x ik · 2α ·∑j (XΛX T +(1 − α) · y) k (7.16)ijw ij· λ k · x jkλ k ← λ k · α ·∑j(XΛX T ) ij+(1 − α) ·∑iy ik (7.17)then normalize such that ∑λ k = 1kIt is proved that the iterative running of the algorithm that updating X and Λ alternativelyresults in the monotonic decrease of the cost function. More detailed regarding the proof isreferred to [164].