In this thesis, we present analytical analysis of key distribution schemes on wireless sensor networks. Since wireless sensor network is under unreliable environment, many random key pre-distribution based schemes have been developed to enhance security. Most of these schemes need to guarantee the existence of specific properties, such as disjoint secure paths or disjoint secure cliques, to achieve a secure cooperation among nodes. Two of the basic questions are as follows: 1. Under what conditions does a large-scale sensor network contain a certain structure? 2. How can one give a quantitative analysis behave as n grows to the infinity? However, analyzing such a structure or combinatorial problem is complicated in classical wireless network models such as percolation theories or random geometric graphs. Particularly, proofs in geometric setting models often blend stochastic geometric and combinatorial techniques and are more technically challenging. To overcome this problem, an approximative quasi-random graph is employed to eliminate some properties that are difficult to tackle. The most well-known solutions of this kind problems are probably Szemeredi's regularity lemma for embedding. The main difficulty from the fact that the above questions involve extremely small probabilities. These probabilities are too small to estimate by means of classical tools from probability theory, and thus a specific counting methods is inevitable.