This study concerns the decentralised prediction and reconstruction problems in a network. First of all, we propose a decentralised prediction algorithm in the framework of network consensus problem. It allows any individual to compute the consensus value of the whole network in finite time using only the minimal number of successive values of its own history. We further prove that the minimal number of steps can be characterised using other algebraic and graph theoretical notions: minimal external equitable partition (mEEP) that can be directly computed from the Laplacian matrix of the graph and from the underlying network structure. Later, we consider a number of possible theoretical extensions of the proposed algorithm to issues arising from practical applications, e.g., time-delays, noise, external inputs, nonlinearities in the network, and analyse how the proposed algorithm should be changed to incorporate such constraints. For the decentralised reconstruction problem, we firstly define a new presentation: dynamical structure functions encoding structural information and explore the properties of such a representation for the purpose of solving the reconstruction problem. We have studied a number of theoretical problems: identification, realisation, reduction, etc. for dynamical structure functions and showed that how these theoretical can be used in solving decentralised network reconstruction problems. We later illustrate the results on a number of in-silico examples. We conclude the thesis with some ideas and future perspectives to continue based on the research of decentralised prediction and reconstruction problems.