Crux, space constraints and subdivisions

For a given graph $H$, its subdivisions carry the same topological structure. The existence of $H$-subdivisions within a graph $G$ has deep connections with topological, structural and extremal properties of $G$. One prominent examples of such connections, due to Bollob\'{a}s and Thomason and independently Koml\'os and Szemer\'edi, asserts that the average degree of $G$ being $d$ ensures a $K_{\Omega(\sqrt{d})}$-subdivision in $G$. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph from many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux. Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph $G$, which is determined by both its average degree and its crux size. As corollaries, we obtain (1) a characterisation of extremal graphs for which the square-root bound above is tight: they are essentially disjoint union of graphs each of which has the crux size linear in $d$; (2) a unifying approach to find a clique subdivision of almost optimal size in graphs which does not contain a fixed bipartite graph as a subgraph; (3) and that the clique subdivision size in random graphs $G(n,p)$ witnesses a dichotomy: when $p = \omega(n^{-1/2})$, the barrier is the space, while when $p=o( n^{-1/2})$, the bottleneck is the density.Comment: 37 pages, 2 figure

SLICER: inferring branched, nonlinear cellular trajectories from single cell RNA-seq data

Accuracy of trajectory reconstruction using a subset of cells. (a) Graph showing how similar the SLICER trajectory is when computed using a random subset of lung cells. The blue bars show the similarity in cell ordering (units are percent sorted with respect to the trajectory constructed from all cells). The orange bars show the similarity in branch assignments (percentage of cells assigned to the same branch as the trajectory constructed from all cells). The values shown were obtained by averaging the results from five subsampled datasets for each percentage (80 %, 60 %, 40 %, and 20 %). (b) Order preservation and branch identity values computed as in panel (a), but for datasets sampled from the neural stem cell dataset. (PDF 106 kb

Large-Scale Networks: Algorithms, Complexity and Real Applications

Networks have broad applicability to real-world systems, due to their ability to model and represent complex relationships. The discovery and forecasting of insightful patterns from networks are at the core of analytical intelligence in government, industry, and science. Discoveries and forecasts, especially from large-scale networks commonly available in the big-data era, strongly rely on fast and efficient network algorithms. Algorithms for dealing with large-scale networks are the first topic of research we focus on in this thesis. We design, theoretically analyze and implement efficient algorithms and parallel algorithms, rigorously proving their worst-case time and space complexities. Our main contributions in this area are novel, parallel algorithms to detect k-clique communities, special network groups which are widely used to understand complex phenomena. The proposed algorithms have a space complexity which is the square root of that of the current state-of-the-art. Time complexity achieved is optimal, since it is inversely proportional to the number of processing units available. Extensive experiments were conducted to confirm the efficiency of the proposed algorithms, even in comparison to the state-of-the-art. We experimentally measured a linear speedup, substantiating the optimal performances attained. The second focus of this thesis is the application of networks to discover insights from real-world systems. We introduce novel methodologies to capture cross correlations in evolving networks. We instantiate these methodologies to study the Internet, one of the most, if not the most, pervasive modern technological system. We investigate the dynamics of connectivity among Internet companies, those which interconnect to ensure global Internet access. We then combine connectivity dynamics with historical worldwide stock markets data, and produce graphical representations to visually identify high correlations. We find that geographically close Internet companies offering similar services are driven by common economic factors. We also provide evidence on the existence and nature of hidden factors governing the dynamics of Internet connectivity. Finally, we propose network models to effectively study the Internet Domain Name System (DNS) traffic, and leverage these models to obtain rankings of Internet domains as well as to identify malicious activities