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A formal definition of random graphs is introduced which is applicable to graphical pattern recognition problems. The definition is used to formulate rigorously the structural-contextual dichotomy of random graphs. The probability of outcome graphs is expressed as the product of two terms, one due to the statistical variability of structure among the outcome graphs and the other due to their contextual variability. Expressions are obtained to estimate the various probability, typicality, and entropy measures. The members in an ensemble of signed digraphs are interpreted as outcome graphs of a random graph. The synthesized random graph is used to quantify the structural, contextual, and overall typicality of the outcome graphs with respect to the random graph.