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In this chapter, a combinatorial model for cosmology is analyzed. We consider each universe as a path in a graph, and the set of all such paths can be made into a finite probability space. We can then consider the probabilities for different kinds of behavior and under certain circumstances argue that a scenario where the behavior of the entropy is monotonic, either increasing or decreasing, should be much more likely than a scenario where the behavior is symmetric with respect to time. In this way we can attempt to construct a model for a multiverse which is completely time symmetric but where the individual universes tend to be time asymmetric, i.e., have an arrow of time. One of the main points with this approach is that this kind of broken symmetry can be studied in very small models using exact mathematical methods from, e.g., combinatorics. Even if the amount of computations needed increases very rapidly with the size of the model, we can still hope for valuable information about what properties more realistic models should have. Some suggestions for further research are pointed out.

**Keywords:** cosmology, multiverse, graph theory, entropy, time's arrow
