Diagrammatic perturbation methods in networks and sports ranking combinatorics

Abstract

Ranking a set of objects from the most dominant one to the least, based on the results of paired comparisons, proves to be useful in many contexts. Using the rankings of teams or individuals players in sports to seed tournaments is an example. The quality of a ranking is often evaluated by the number of violations, cases in which an object is ranked lower than another that it has dominated in a comparison, that it contains. A minimum violations ranking (MVR) method, as its name suggests, searches specifically for rankings that have the minimum possible number of violations which may or may not be zero. In this paper, we present a method based on statistical physics that overcomes conceptual and practical difficulties faced by earlier studies of the problem.

Systematic inequality and hierarchy in faculty hiring networks

“…If faculty placement overall followed a perfect social hierarchy, then no faculty would be hired at an institution more prestigious than their doctorate ( 6 ). The extent to which a particular hiring network exhibits this pattern may be determined by identifying the minimum violation ranking ( 14 , 15 ), which is a hierarchy that is maximally close to this extreme.…”

“…Both the inferred hierarchy π and its strength ρ are of interest. For large networks, there are typically many equally plausible rankings with the maximum ρ ( 15 ). To extract a consensus ranking, we sample optimal rankings by repeatedly choosing a random pair of vertices and swapping their ranks, if the resulting ρ is no smaller than for the current ranking.…”

Universal statistics of the knockout tournament

“…Conditional probability to progress only to the nth round for given competitiveness r [see equation(6)]. (a) Probability distribution of r at the 5th round when f (x) is the Heaviside step function, equation(2).…”

Bayesian Inference of Natural Rankings in Incomplete Competition Networks

“…We can also compute the expected variance of the final outdegree given as We note that the second term is non-vanishing, since 〈 σ ji σ li 〉 ≠ 〈 σ ji 〉 〈 σ li 〉 due to the shared index i ; we call σ ji σ li connected , analogous to connected Feynman diagrams in quantum field theory, which have also found some uses in network theory 16 17 . 〈 σ ji σ li 〉 is where is the incomplete gamma function.…”