Is there a formal relationship between graphs and probability distributions?

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Intuitively probability distributions and graphs are two different views on an underlying set of data. Is there a formal description of the relationship between these two concepts?

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I'm not quite sure what you mean by "two different vies on an underlying set of data," so if you give me a better sense of what you mean by each I might be able to describe different relationships.

One relationship between probability distributions and graph theory comes in the theory of finite Markov chains. The transition probabilities in such a chain give a weighted directed graph structure on the set of vertices.