Mauricio Soto

Treewidth and Hyperbolicity of the Internet

We study the measurement of the Internet according to two graph parameters: tree width and hyper bolicity. Both tell how far from a tree a graph is. They are computed from snapshots of the Internet released by CAIDA, DIMES, AQUALAB, UCLA, Rocket fuel and Strasbourg University, at the AS or at the router level. On the one hand, the tree width of the Internet appears to be quite large and being far from a tree with that respect, reflecting some high degree of connectivity. This proves the existence of a well linked core in the Internet. On the other hand, the hyper bolicity (as a graph parameter) appears to be very low, reflecting a tree-like structure with respect to distances. Additionally, we compute the tree width and hyper bolicity obtained for classical Internet models and compare with the snapshots.

Asymptotic modularity of some graph classes

Modularity has been introduced as a quality measure for graph partitioning. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study the modularity of a variety of graph classes. We first consider simple graph classes such as tori and hypercubes. We show that these regular graph families have asymptotic modularity 1 (that is the maximum possible). We extend this result to the general class of unit ball graphs of bounded growth metrics. Our most striking result concerns trees with bounded degree which also appear to have asymptotic modularity 1. This last result can be extended to graphs with constant average degree and to some power-law graphs.

Embedding signed graphs in the line

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388–399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

X-RAY STUDIES AND ANTIBACTERIAL ACTIVITY IN COPPER AND COBALT COMPLEXES WITH IMIDAZOLE DERIVATIVE LIGANDS

The complexes Diaqua-bis(5-nitroimidazole)-copper(II)-dinitrate (1); Tetrakis(4- phenylimidazole)-copper(II)-dinitrate, ethanol solvate (2); Bis(4phenylimidazole)-bis(acetate)-copper(II) (3); Hexakis(4-phenylimidazole)-cobalt(II)-acetate (4) and Bis(2- phenylimidazole)-bis(acetate)-cobalt(II) (5), have been synthesized. Their structures were determined by X-ray analysis. The antimicrobial activities of these complexes have been screened in vitro against the microorganisms Escherichia coli, Klebsiella pneumoniae, Shigella flexneri, Salmonella typhi, Salmonella enteritidis, Salmonella enterica, Staphylococcus aureus , Listeria monocytogenes. Complexes (1) and (3) showed antimicrobial activity over Gram positive and negative bacteria by a bacteriostatic effect. On the other hand, complexes (4) and (5) showed antibacterial activity with a bacteriolytic effect over Gram positive and negative bacteria. The cytotoxicity of complexes (3) and (4) was tested in vitro. These complexes presented cytotoxic effect to values near the MIC.

Adversarial queuing theory with setups

We look at routing and scheduling problems on Kelly type networks where the injection process is under the control of an adversary. The novelty of the model we consider is that the adversary injects requests of distinct types. Resources are subject to switch-over delays or setups when they begin servicing a new request class. In this new setting, we study the behavior of sensible policies as introduced by Dai and Jennings [J. Dai, O. Jennings, Stabilizing queueing networks with setups, Math. Oper. Res. (2004) 891-922]. We first show that the model is robust in the sense that under some mild conditions universal stability of work conserving packet routing protocols is preserved for natural variants of the underlying model. Also, the models equivalence to so called token networks is established. We adapt to the multi-type request and setup setting, standard arguments for proving stability. Nevertheless, we provide counterexamples that show that for several reasonable adaptations of contention resolution protocols to the multi-type case, stability results do not carry over from the single-type scenario. This motivates us to explore fluid model based arguments that could be used for proving stability for a given network. Specifically we show analogues of results obtained by Gamarnik [D. Gamarnik, Stability of adversarial queues via fluid model, in: Proc. of the 39th Annual Symposium on Foundations of Computer Science, 1998, pp. 60-70] but in the multi-type request with setups scenario.

Beyond Perfect Phylogeny: Multisample Phylogeny Reconstruction via ILP

Most of the evolutionary history reconstruction approaches are based on the infinite site assumption which is underlying the Perfect Phylogeny model. This is one of the most used models in cancer genomics. Recent results gives a strong evidence that recurrent and back mutations are present in the evolutionary history of tumors[19], thus showing that more general models then the Perfect phylogeny are required. To address this problem we propose a framework based on the notion of Incomplete Perfect Phylogeny. Our framework incorporates losing and gaining mutations, hence including the Dollo and the Camin-Sokal models, and is described with an Integer Linear Programming (ILP) formulation. Our approach generalizes the notion of persistent phylogeny[1] and the ILP approach[14,15] proposed to solve the corresponding phylogeny reconstruction problem on character data. The final goal of our paper is to integrate our approach into an ILP formulation of the problem of reconstructing trees on mixed populations, where the input data consists of the fraction of cells in a set of samples that have a certain mutation. This is a fundamental problem in cancer genomics, where the goal is to study the evolutionary history of a tumor. An experimental analysis shows that our ILP approach is able to explain data that do not fit the perfect phylogeny assumption, thereby allowing (1) multiple losses and gains of mutations, and (2) a number of subpopulations that is smaller than the number of input mutations.

Character-based phylogeny construction and its application to tumor evolution

Character-based Phylogeny Construction is a well-known combinatorial problem whose input is a matrix M and we want to compute a phylogeny that is compatible with the actual species encoded by M.