Topology Discovery of Sparse Random Graphs With Few Participants

We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is scheduled to appear in J. on Random Structures and Algorithm

The envirome and the connectome: exploring the structural noise in the human brain associated with socioeconomic deprivation

Complex cognitive functions are widely recognized to be the result of a number of brain regions working together as large-scale networks. Recently, complex network analysis has been used to characterize various structural properties of the large scale network organization of the brain. For example, the human brain has been found to have a modular architecture i.e. regions within the network form communities (modules) with more connections between regions within the community compared to regions outside it. The aim of this study was to examine the modular and overlapping modular architecture of the brain networks using complex network analysis. We also examined the association between neighborhood level deprivation and brain network structure – modularity and grey nodes. We compared network structure derived from anatomical MRI scans of 42 middle-aged neurologically healthy men from the least (LD) and the most deprived (MD) neighborhoods of Glasgow with their corresponding random networks. Cortical morphological covariance networks were constructed from the cortical thickness derived from the MRI scans of the brain. For a given modularity threshold, networks derived from the MD group showed similar number of modules compared to their corresponding random networks, while networks derived from the LD group had more modules compared to their corresponding random networks. The MD group also had fewer grey nodes – a measure of overlapping modular structure. These results suggest that apparent structural difference in brain networks may be driven by differences in cortical thicknesses between groups. This demonstrates a structural organization that is consistent with a system that is less robust and less efficient in information processing. These findings provide some evidence of the relationship between socioeconomic deprivation and brain network topology

Evolutionary constraints on the complexity of genetic regulatory networks allow predictions of the total number of genetic interactions

Genetic regulatory networks (GRNs) have been widely studied, yet there is a lack of understanding with regards to the final size and properties of these networks, mainly due to no network currently being complete. In this study, we analyzed the distribution of GRN structural properties across a large set of distinct prokaryotic organisms and found a set of constrained characteristics such as network density and number of regulators. Our results allowed us to estimate the number of interactions that complete networks would have, a valuable insight that could aid in the daunting task of network curation, prediction, and validation. Using state-of-the-art statistical approaches, we also provided new evidence to settle a previously stated controversy that raised the possibility of complete biological networks being random and therefore attributing the observed scale-free properties to an artifact emerging from the sampling process during network discovery. Furthermore, we identified a set of properties that enabled us to assess the consistency of the connectivity distribution for various GRNs against different alternative statistical distributions. Our results favor the hypothesis that highly connected nodes (hubs) are not a consequence of network incompleteness. Finally, an interaction coverage computed for the GRNs as a proxy for completeness revealed that high-throughput based reconstructions of GRNs could yield biased networks with a low average clustering coefficient, showing that classical targeted discovery of interactions is still needed.Comment: 28 pages, 5 figures, 12 pages supplementary informatio