For any finite colored graph we define the empirical neighborhood measure, which counts the number of vertices of a given color connected to a given number of vertices of each color, and the empirical pair measure, which counts the number of edges connecting each pair of colors. For a class of models of sparse colored random graphs, we prove large deviation principles for these empirical measures in the weak topology. The rate functions governing our large deviation principles can be expressed explicitly in terms of relative entropies. We derive a large deviation principle for the degree distribution of Erdős–Rényi graphs near criticality.

We prove asymptotic equipartition properties for simple hierarchical structures (modelled as multitype Galton-Watson trees) and networked structures (modelled as randomly coloured random graphs). For example, for large n, a networked data structure consisting of n units connected by an average number of links of order n / log n can be coded by about H × n bits, where H is an explicitly defined entropy. The main technique in our proofs are large deviation principles for suitably defined empirical measures.

This paper was aimed at investigating the volatility and conditional relationship among inflation rates, exchange rates and interest rates as well as to construct a model using multivariate GARCH DCC and BEKK models using Ghana data from January 1990 to December 2013. The study revealed that the cumulative depreciation of the cedi to the US dollar from 1990 to 2013 is 7,010.2% and the yearly weighted depreciation of the cedi to the US dollar for the period is 20.4%. There was evidence that, the fact that inflation rate was stable, does not mean that exchange rates and interest rates are expected to be stable. Rather, when the cedi performs well on the forex, inflation rates and interest rates react positively and become stable in the long run. The BEKK model is robust to modelling and forecasting volatility of inflation rates, exchange rates and interest rates. The DCC model is robust to model the conditional and unconditional correlation among inflation rates, exchange rates and interest rates. The BEKK model, which forecasted high exchange rate volatility for the year 2014, is very robust for modelling the exchange rates in Ghana. The mean equation of the DCC model is also robust to forecast inflation rates in Ghana.

Given devices space D, an intensity measure λm ϵ (0, ∞), a transition kernel Q from the space D to positive real numbers ℝ₊, a path-loss function (which depends on the Euclidean distance between the devices and a positive constant a), we define a Marked Poisson Point process (MPPP). For a given MPPP and technical constants , we define a Marked Signal-to- Interference and Noise Ratio (SINR) graph, and associate with it two empirical measures; the empirical marked measure and the empirical connectivity measure. For a class of marked SINR graphs, we prove a joint large deviation principle(LDP) for these empirical measures, with speed λ in the τ-topology. From the joint large deviation principle for the empirical marked measure and the empirical connectivity measure, we obtain an Asymptotic Equipartition Property(AEP) for network structured data modelled as a marked SINR graph. Specifically, we show that for large dense marked SINR graph one requires approximately about λ ² H(Q × Q)/log 2 bits to transmit the information contained in the network with high probability, where H(Q × Q) is a properly defined entropy for the exponential transition kernel with parameter c. Further, we prove a local large deviation principle (LLDP) for the class of marked SINR graphs on D, where , with speed λ from a spectral potential point. From the LLDP we derive a conditional LDP for the marked SINR graphs. Note that, while the joint LDP is established in the τ-topology, the LLDP assume no topological restriction on the space of marked SINR graphs. Observe also that all our rate functions are expressed in terms of the relative entropy or the kullback action or divergence function of the marked SINR on the devices space D.

In this article we find exponential good approximation of the empirical neigbourhood distribution of symbolled random graphs conditioned to a given empirical symbol distribution and empirical pair distribution. Using this approximation we shorten or simplify the proof of (Doku-Amponsah \& Morters, 2010, Theorem~2.5); the large deviation principle (LDP) for empirical neigbourhood distribution of symbolled random graphs. We also show that the LDP for the empirical degree measure of the classical Erd\H{o}s-R\'{e}nyi graph is a special case of (Doku-Amponsah \& Moerters, 2010, Theorem~2.5). From the LDP for the empirical degree measure, we derive an LDP for the the proportion of isolated vertices in the classical Erd\H{o}s-R\'{e}nyi graph.