Antimatroids and Balanced Pairs

We generalize the 1/3-2/3 conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between 1/3 and 2/3 of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure

Mathematical evolution in discrete networks

This paper provides a mathematical explanation for the phenomenon of \triadic closure" so often seen in social networks. It appears to be a natural consequence when network change is constrained to be continuous. The concept of chordless cycles in the network's \irreducible spine" is used in the analysis of the network's dynamic behavior. A surprising result is that as networks undergo random, but continuous, perturbations they tend to become more structured and less chaotic

Betweenness of partial orders

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the be-tweenness of a partial order