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

156 7 Extracting and Analyzing Web Social Networks• Updating A: vectorizing X and R, and stacking them in an array such that the optimizationcriterion of Eq.7.8 is changed to⎛ ⎞ ⎛⎞Vec(X 1 ) R 1 ⊗ AD 1f (R)=⎝··· ⎠ − ⎝ ··· ⎠Vec(R)(7.10)∥ Vec(X m ) AD m ⊗ AD m∥where Vec(R) is the normalized ∑ m k=1 Vec( D k A T )X k AD k .• Updating D: by alternating over various slices D k , and holding A and R, the followingminimization condition is reachedm ∥∥ minD k∑ ∥X k − AD k RD k A T 2∥ (7.11)k=1FThe algorithm runs iteratively until a convergence is reached, i.e. f (A,R,D)/‖X‖ 2 F does notchange any more according to a threshold value or the maximum number of iteration is executed.in the other words, the iterative execution of the algorithm makes the optimizationcriterion satisfactory and converged to a stable value.7.2.3 Examples of Formed CommunityIn [15], two applications are investigated to demonstrate the temporal analysis of the socialnetwork structure. The first is a small case of international trade data and the second one is arelatively large email communications within the Enron corporation that are publicly availableduring the federal investigation.The world trade data consists of import/export data among 18 nations in European, NorthAmerican, and the Pacific Rim countries from 1981 to 1990. A semantic graph of this data iscorresponding to a densely connected adjacency array X of size 18×18×10. The Enron emailcommunication data contains 34,427 emails communicated between 184 email addresses during44 months from 13-Nov-1998 to 21-Jun-2002. The final processed data is a graph correspondingto a sparse adjacency array X of size 184×184×44. On these two semantic graphs,the authors employed the proposed algorithm to analyze the semantic network structures andthe temporal shifts of the networks.First, let’s see the aggregated networks derived from the international trade dataset. Figure7.9 depicts the aggregated trade patterns (or trade networks) in terms of R matrix. It is seenfrom Fig.7.9 that three major latent components corresponding to mostly three geographicalregions, namely North America, Europe and Japan, are revealed from the analysis of Rand A matrices. There are dense self cycle connections within North American or Europeancountries, indicating the higher international trade amongst the countries within these tworegions. And the trade patterns amongst different regions are asymmetric. For example, theexport trade amount of North America to Europe is bigger than that in reverse direction. Inaddition, Fig.7.10 gives the scales in D for different trade regions, which indicate the temporalchanges of commerce over time. Interestingly, the increase of scale of Japan during the middle80’s provides evidence of Japanese economic booming during that period. For Enron dataset,Fig.7.11 indicates the four latent user roles are captured from the communications’ frequenciesand directional modes (i.e. sender/recipients). During the period, the email conversationswithin legal parties or Executive and Government Affairs. And again, the communication isdramatically asymmetric in that the r 23 is significantly larger than r 32 . Figure 7.12 talks aboutthe comparisons of communication patterns among the four roles between October 2000 to