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Markov property of correlated random networks and its application to the analysis of the internet topologies

We propose a framework for modeling higher-order (beyond two point) degree correlation among nodes, depending on their mutual connectivity. Our focus is on the introduction of the Markov property to find maximally unbiased networks under the constraint of a prescribed two-point degree correlation. The topological features of the Markovian networks -- networks satisfying the Markov property -- are fully characterized solely by the two-point degree correlation. We theoretically investigate the topological characteristics of Markovian networks and derive the analytical formulas for their graph theoretical metrics. We present a comparative analysis of AS- and router-level topologies in terms of whether they are Markovian or not. The results of the analysis show that the studied AS- and router-level topologies are not Markovian. This finding indicates that it is rather difficult to capture the topological characteristics of the Internet either at AS- or at router-level solely by the input of the two-point degree correlation.