January 1, 2011.
The “commute time” between two nodes i and j in a network is a sophisticated distance metric which measures the expected time for a random walker to start at i, arrive at j and return back to i. Currently, the most efficient method for calculating commute time involves summing the squared difference of the ith and jth entries of all eigenvectors of the Graph Laplacian, weighting each difference by the reciprocal of the corresponding eigenvalue, and weighting the entire sum by the number of edges in the graph. However, this method is still prohibitively expensive for large (million or billion node) networks. The objective of this study was to observe the how accurately a commute time found by incorporating only a percentage of the eigenpairs could approximate the actual commute time. I tested the accuracy of the commute time approximation as more and more eigenpairs were added in addition to which specific eigenpairs contributed most to the overall commute time. Finally, the approximated commute times (at various levels of truncation) were measured against the actual commute time for accurate detection of Top-k neighbors. Results showed that, depending on the degree of nodes involved in the commute time, drastically different eigenpairs are the main “contributors” to the commute time. This was confirmed as a significant amount of eigenpairs were needed to accurately predict the Top-k neighbors.
Van Emden Henson
Lawrence Livermore National Laboratory (LLNL)
This material is based upon work supported by the S.D. Bechtel, Jr. Foundation and by the National Science Foundation under Grant No. 0952013. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the S.D. Bechtel, Jr. Foundation or the National Science Foundation.