Threshold phenomena in random graphs

In the 1950s, random graphs appeared for the first time in a result of the prolific hungarian mathematician Pál Erd\H{o}s. Since then, interest in random graph theory has only grown up until now. In its first stages, the basis of its theory were set, while they were mainly used in probability and combinatorics theory. However, with the new century and the boom of technologies like the World Wide Web, random graphs are even more important since they are extremely useful to handle problems in fields like network and communication theory. Because of this fact, nowadays random graphs are widely studied by the mathematical community around the world and new promising results have been recently achieved, showing an exciting future for this field. In this bachelor thesis, we focus our study on the threshold phenomena for graph properties within random graphs

SANA NetGO: A combinatorial approach to using Gene Ontology (GO) terms to score network alignments

Gene Ontology (GO) terms are frequently used to score alignments between protein-protein interaction (PPI) networks. Methods exist to measure the GO similarity between two proteins in isolation, but pairs of proteins in a network alignment are not isolated: each pairing is implicitly dependent upon every other pairing via the alignment itself. Current methods fail to take into account the frequency of GO terms across the networks, and attempt to account for common GO terms in an ad hoc fashion by imposing arbitrary rules on when to "allow" GO terms based on their location in the GO hierarchy, rather than using readily available frequency information in the PPI networks themselves. Here we develop a new measure, NetGO, that naturally weighs infrequent, informative GO terms more heavily than frequent, less informative GO terms, without requiring arbitrary cutoffs. In particular, NetGO down-weights the score of frequent GO terms according to their frequency in the networks being aligned. This is a global measure applicable only to alignments, independent of pairwise GO measures, in the same sense that the edge-based EC or S3 scores are global measures of topological similarity independent of pairwise topological similarities. We demonstrate the superiority of NetGO by creating alignments of predetermined quality based on homologous pairs of nodes and show that NetGO correlates with alignment quality much better than any existing GO-based alignment measures. We also demonstrate that NetGO provides a measure of taxonomic similarity between species, consistent with existing taxonomic measures--a feature not shared with existing GO-based network alignment measures. Finally, we re-score alignments produced by almost a dozen aligners from a previous study and show that NetGO does a better job than existing measures at separating good alignments from bad ones

Optimal Precoders for Tracking the AoD and AoA of a mm-Wave Path

In millimeter-wave channels, most of the received energy is carried by a few paths. Traditional precoders sweep the angle-of-departure (AoD) and angle-of-arrival (AoA) space with directional precoders to identify directions with largest power. Such precoders are heuristic and lead to sub-optimal AoD/AoA estimation. We derive optimal precoders, minimizing the Cram\'{e}r-Rao bound (CRB) of the AoD/AoA, assuming a fully digital architecture at the transmitter and spatial filtering of a single path. The precoders are found by solving a suitable convex optimization problem. We demonstrate that the accuracy can be improved by at least a factor of two over traditional precoders, and show that there is an optimal number of distinct precoders beyond which the CRB does not improve.Comment: Resubmission to IEEE Trans. on Signal Processing. 12 pages and 9 figure