We study the level spacing distribution p(s) in the spectrum of random networks. According to our numerical results, the shape of p(s) in the Erdős-Rényi (E-R) random graph is determined by the average degree k and p(s) undergoes a dramatic change when k is varied around the critical point of the percolation transition, k = 1. When k 1, the p(s) is described by the statistics of the Gaussian orthogonal ensemble (GOE), one of the major statistical ensembles in Random Matrix Theory, whereas at k = 1 it follows the Poisson level spacing distribution. Closely above the critical point, p(s) can be described in terms of an intermediate distribution between Poisson and the GOE, the Brodydistribution. Furthermore, below the critical point p(s) can be given with the help of the regularized Gamma-function. Motivated by these results, we analyse the behaviour of p(s) in real networks such as the internet, a word association network and a protein-protein interaction network as well. When the giant component of these networks is destroyed in a node deletion process simulating the networks subjected to intentional attack, their level spacing distribution undergoes a similar transition to that of the E-R graph.

We develop a model of the behavior of a potential investor (under uncertainty and in a fiscal environment) who wishes to invest into a project in the real sector of an economy and faces a timing problem. We find an optimal solution within this model and examine the dependence of the tax revenue from the newly created firm on the depreciation policy. It is shown that there exists a domain in the space of the parameters of the investment project where both the tax revenue and the incentives can be increased by using the depreciation policy.The paper analyzes the problem of creating investment incentives by using tax mechanisms. Suppose the government is interested in the implementation of an investment project aimed at establishing a new enterprise (firm). The absence of such an enterprise might lead, for example, to the necessity of importing goods, which otherwise could be produced by the firm, or to the necessity of paying unemployment benefits, etc. Furthermore, the creation of the enterprise can increase the tax revenues obtained by the state budget. Generally, the government may have many reasons to be interested in establishing the investment project as soon as possible, especially, taking into account the possibility of increasing tax revenues.The government can use certain fiscal mechanisms that can influence the investor's behavior. Important roles are played, in particular by two such mechanisms: (i) changing tax rates and (ii) decreasing the tax base by an appropriate choice of a depreciation policy.We emphasize that such fiscal benefits should be assigned to a project and not to a specific investor (which could lead to corruption). The information about the fiscal benefits is provided for potential investors as part of "the rules of the game."Depreciation, as it is well known, is the economic mechanism for the transfer of the cost of assets to the cost of the product. The depreciation policy mechanism seems to be more

We study a problem when a solution to optimal stopping problem for one-dimensional diffusion will generate by threshold strategy. Namely, we give necessary and sufficient conditions under which an optimal stopping time can be specified as the first time when the process exceeds some level (threshold), and a continuation set is a semi-interval. We give also second-order conditions, which allow to discard such solutions to free-boundary problem that are not the solutions to optimal stopping problem.

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