Fast construction on a restricted budget

We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve minimum degree $k$ at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to achieve it (slightly) after the threshold for minimum degree $k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal); (b) Builder has a strategy to create a Hamilton cycle if either $t\ge(1+\varepsilon)n\log{n}/2$ and $b\ge Cn$, or $t\ge Cn\log{n}$ and $b\ge(1+\varepsilon)n$, for some $C=C(\varepsilon)$; similar results hold for perfect matching; (c) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\{(n/t)^{k-2},1\}$, and this is optimal; and (d) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max\{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal.Comment: 20 pages, 2 figure

A general co-expression network-based approach to gene expression analysis: comparison and applications

<p>Abstract</p> <p>Background</p> <p>Co-expression network-based approaches have become popular in analyzing microarray data, such as for detecting functional gene modules. However, co-expression networks are often constructed by ad hoc methods, and network-based analyses have not been shown to outperform the conventional cluster analyses, partially due to the lack of an unbiased evaluation metric.</p> <p>Results</p> <p>Here, we develop a general co-expression network-based approach for analyzing both genes and samples in microarray data. Our approach consists of a simple but robust rank-based network construction method, a parameter-free module discovery algorithm and a novel reference network-based metric for module evaluation. We report some interesting topological properties of rank-based co-expression networks that are very different from that of value-based networks in the literature. Using a large set of synthetic and real microarray data, we demonstrate the superior performance of our approach over several popular existing algorithms. Applications of our approach to yeast, Arabidopsis and human cancer microarray data reveal many interesting modules, including a fatal subtype of lymphoma and a gene module regulating yeast telomere integrity, which were missed by the existing methods.</p> <p>Conclusions</p> <p>We demonstrated that our novel approach is very effective in discovering the modular structures in microarray data, both for genes and for samples. As the method is essentially parameter-free, it may be applied to large data sets where the number of clusters is difficult to estimate. The method is also very general and can be applied to other types of data. A MATLAB implementation of our algorithm can be downloaded from <url>http://cs.utsa.edu/~jruan/Software.html</url>.</p

The Erlang Weighted Tree, A New Branching Process

In this paper, we propose a new branching process which we call Erlang Weighted Tree(EWT). EWT appears as the local weak limit of a random graph model proposed by Richard La and Maya Kabkab. We derive the main properties of EWT such as the probability of extinction, the emergence of phase transition and growth rate

A Study of Phase Transition in New Random Graph Families

Random graphs are mathematical models for understanding real-world networks. Important properties can be captured, processes studied, and rigorous predictions made. Phase transitions (sudden changes in structural properties caused by varying an underlying parameter) are commonly observed in random graphs. Our work focuses on phase transitions in three models. We study emergence of cascades and impact of community structure on phase transition in threshold-based contagion models using modular random graphs generated by configuration model and differential equation method. Using local weak analysis, we study a new graph model generated by bilateral agreement of individuals and analyze when a giant component emerges. Using the objective method and motivated by particle tracking in physics and object tracking in videos, we study detectability threshold of a hidden planted matching in a complete bipartite randomly weighted graph.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155026/1/moharami_1.pd