[pt] Nessa dissertação vamos introduzir, elaborar e combinar ideias da Teoria de martingais, a Teoria de grafos aleatórios e o Teorema Central do Limite. Em particular, veremos como martingais podem ser usados para representar desvios de contagem de subgrafos. Usando esta representação e o Teorema Central do Limite para martingais, conseguiremos demonstrar um Teorema Central do Limite para a contagem de subgrafos livres de triângulos no grafo aleatório Erdos-Rényi G(n,m) . Além disso, nossa demonstração também nos trará informação sobre a taxa de convergência, mostrando que a distribuição dos desvios converge rapidamente para a distribuição normal. / [en] In this dissertation we shall introduce, elaborate and combine ideas from martingale Theory, random graph Theory and the Central Limit Theorem. In particular, we will see how martingales can be used to represent deviations of subgraph counts. Using this representation and the Central Limit Theorem for martingales, we will be able to demonstrate a Central Limit Theorem for the triangle-free subgraph count in the Erdos-Rényi G(n,m) random graph. Furthermore, our proof also gives us information about the rate of convergence, showing that the distribution of deviations converges rapidly to the normal distribution.

The astonishing progress of molecular biology, engineering and computer science has resulted in mature technologies capable of examining multiple cellular components at a genome-wide scale. Protein-protein interactions are one example of such growing data. These data are often organised as networks with proteins as nodes and interactions as edges. Albeit still incomplete, there is now a substantial amount of data available and there is a need for biologically meaningful methods to analyse and interpret these interactions. In this thesis we focus on how to compare protein interaction networks (PINs) and on the rela- tionship between network architecture and the biological characteristics of proteins. The underlying theme throughout the dissertation is the use of small subgraphs – small interaction patterns between 2-5 proteins. We start by examining two popular scores that are used to compare PINs and network models. When comparing networks of the same model type we find that the typical scores are highly unstable and depend on the number of nodes and edges in the networks. This is unsatisfactory and we propose a method based on non-parametric statistics to make more meaningful comparisons. We also employ principal component analysis to judge model fit according to subgraph counts. From these analyses we show that no current model fits to the PINs; this may well reflect our lack of knowledge on the evolution of protein interactions. Thus, we use explanatory variables such as protein age and protein structural class to find patterns in the interactions and subgraphs we observe. We discover that the yeast PIN is highly heterogeneous and therefore no single model is likely to fit the network. Instead, we focus on ego-networks containing an initial protein plus its interacting partners and their interaction partners. In the final chapter we propose a new, alignment-free method for network comparison based on such ego-networks. The method compares subgraph counts in neighbourhoods within PINs in an averaging, many-to-many fashion. It clusters networks of the same model type and is able to successfully reconstruct species phylogenies solely based on PIN data providing exciting new directions for future research.