Thomas I Treloar

We introduce a simple and natural iterative version of the well-known and widely studied Markov rating method. We show that this iterative Markov method converges to the usual global Markov rating, and shares a close relationship with the well-known Elo rating. Together with recent results on the relationship between the global Markov method and the maximum likelihood estimate of the rating vector in the Bradley–Terry (BT) model, we connect and explore the global and iterative Markov, Elo, and Bradley–Terry ratings on real and simulated data.

We present a one parameter family of ratings and rankings that includes the Markov method, as well as the methods of Colley and Massey as particular cases. The rankings are based on a natural network diffusion process that unites the methodologies above in a common framework and brings strong intuition to how and why they differ. We also explore the behavior of the ranking family using both real and simulated data.

We consider an evolutionary prisoner's dilemma on a random network. We introduce a simple quantitative network-based parameter and show that it effectively predicts the success of cooperation in simulations on the network. The criterion is shown to be accurate on a variety of networks with degree distributions ranging from regular to Poisson to scale free. The parameter allows for comparisons of random networks regardless of their underlying topology. Finally, we draw analogies between the criterion for the success of cooperation introduced here and existing criteria in other contexts.

We use the standardized variance (nu(st)) of the degree distribution of a random network as an analytic measure of its heterogeneity. We show that nu(st) accurately predicts, quantitatively, the success of cooperators in an evolutionary prisoner's dilemma. Moreover, we show how the generating functional expression for nu(st) suggests an intrinsic interpretation for the heterogeneity of the network that helps explain local mechanisms through which cooperators thrive in heterogeneous populations. Finally, we give a simple relationship between nu(st), the cooperation level, and the epidemic threshold of a random network that reveals an appealing connection between epidemic disease models and the evolutionary prisoner's dilemma.

We study the symplectic geometry of the moduli spaces $M_r=M_r(\s^3)$ of
closed n-gons with fixed side-lengths in the 3-sphere. We prove that these
moduli spaces have symplectic structures obtained by reduction of the fusion
product of $n$ conjugacy classes in SU(2), denoted $C_r^n$, by the diagonal
conjugation action of SU(2). Here $C_r^n$ is a quasi-Hamiltonian SU(2)-space.
An integrable Hamiltonian system is constructed on $M_r$ in which the
Hamiltonian flows are given by bending polygons along a maximal collection of
nonintersecting diagonals. Finally, we show the symplectic structure on $M_r$
relates to the symplectic structure obtained from gauge-theoretic description
of $M_r$. The results of this paper are analogues for the 3-sphere of results
obtained for $M_r(\h^3)$, the moduli space of n-gons with fixed side-lengths in
hyperbolic 3-space \cite{KMT}, and for $M_r(\E^3)$, the moduli space of n-gons
with fixed side-lengths in $\E^3$

We study the symplectic geometry of the moduli space of closed n-gons with
fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces
have a symplectic structure coming from Poisson Lie theory. We construct
completely integrable systems on these moduli spaces by bending n-gons along
their diagonals. The results of this paper are the analogues of the results of
the first two authors for n-gon linkages in Euclidean 3-space in JDG 44. We
conclude by proving by a deformation argument that the moduli spaces of n-gon
linkages in hyperbolic 3-space and Euclidean 3-space with the same set of
side-lengths are symplectomorphic.