Remarks on spectral gaps on the Riemannian path space

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces

Social Structure and Opinion Formation

We present a dynamical theory of opinion formation that takes explicitly into account the structure of the social network in which in- dividuals are embedded. The theory predicts the evolution of a set of opinions through the social network and establishes the existence of a martingale property, i.e. that the expected weighted fraction of the population that holds a given opinion is constant in time. Most importantly, this weighted fraction is not either zero or one, but corresponds to a non-trivial distribution of opinions in the long time limit. This co-existence of opinions within a social network is in agreement with the often observed locality effect, in which an opinion or a fad is localized to given groups without infecting the whole society. We verified these predictions, as well as those concerning the fragility of opinions and the importance of highly connected individuals in opinion formation, by performing computer experiments on a number of social networks

Bootstrapping the Long Tail in Peer to Peer Systems

We describe an efficient incentive mechanism for P2P systems that generates a wide diversity of content offerings while responding adaptively to customer demand. Files are served and paid for through a parimutuel market similar to that commonly used for betting in horse races. An analysis of the performance of such a system shows that there exists an equilibrium with a long tail in the distribution of content offerings, which guarantees the real time provision of any content regardless of its popularity