Call detail records (CDRs) have recently been used in studying different aspects of human mobility. While CDRs provide a means of sampling user locations at large population scales, they may not sample all locations proportionate to the visitation frequency of a user, owing to sparsity in time and space of voice-calls, thereby introducing a bias. Also, as the rate of sampling is inherently dependent on the calling frequencies of an individual, high voice-call activity users are often chosen for conducting a meaningful study. Such a selection process can, inadvertently, lead to a biased view as high frequency callers may not always be representative of an entire population. With the advent of 3G technology and wide adoption of smart-phones, cellular devices have become versatile end-hosts. As the data accessed on these devices does not always require human initiation, it affords us with an unprecedented opportunity to validate the utility of CDRs for studying human mobility. In this work, we investigate various metrics for human mobility studied in literature for over a million cellular users in the San Francisco bay-area, for over a month. Our findings reveal that although the voice-call process does well to sample significant locations, such as home and work, it may in some cases incur biases in capturing the overall spatio-temporal characteristics of individual human mobility. Additionally, we motivate an "artificially" imposed sampling process, vis-a-vis the voice-call process with the same average intensity. We observe that in many cases such an imposed sampling process yields better performance results based on the usual metrics like entropies and marginal distributions used often in literature.

The expected commute times for a strongly connected directed graph are related to an asymmetric Laplacian matrix as a direct extension to similar well known formulas for undirected graphs. We show the close relationships between the asymmetric Laplacian and the so-called Fundamental matrix. We give bounds for the commute times in terms of the stationary probabilities for a random walk over the graph together with the asymmetric Laplacian and show how this can be approximated by a symmetrized Laplacian derived from a related weighted undirected graph.

A divide-and-conquer based approach for computing the Moore-Penrose pseudo-inverse of the combinatorial Laplacian matrix (L + ) of a simple, undirected graph is proposed. The nature of the underlying sub-problems is studied in detail by means of an elegant interplay between L + and the effective resistance distance (Ω). Closed forms are provided for a novel two-stage process that helps compute the pseudo-inverse incrementally. Analogous scalar forms are obtained for the converse case, that of structural regress, which entails the breaking up of a graph into disjoint components through successive edge deletions. The scalar forms in both cases, show absolute element-wise independence at all stages, thus suggesting potential parallelizability. Analytical and experimental results are presented for dynamic (time-evolving) graphs as well as large graphs in general (representing real-world networks). An order of magnitude reduction in computational time is achieved for dynamic graphs; while in the general case, our approach performs better in practice than the standard methods, even though the worst case theoretical complexities may remain the same: an important contribution with consequences to the study of online social networks.

We explore the geometry of complex networks in terms of an n-dimensional Euclidean embedding represented by the Moore-Penrose pseudo-inverse of the graph Laplacian (L + ). The squared distance of a node i to the origin in this n-dimensional space (l + ii ), yields a topological centrality index, defined as C * (i) = 1/l + ii . In turn, the sum of reciprocals of individual node centralities, i 1/C * (i) = i l + ii , or the trace of L + , yields the well-known Kirchhoff index (K), an overall structural descriptor for the network. To put in context this geometric definition of centrality, we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics -first, as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network; and then as the average connectedness of i in all the bi-partitions of the graph. These interpretations respectively help establish the topological centrality (C * (i)) of node i as a measure of its overall position as well as its overall connectedness in the network; thus reflecting the robustness of i to random multiple edge failures. Through empirical evaluations using synthetic and real world networks, we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and, along with Kirchhoff index, is appropriately sensitive to perturbations/rewirings in the network.