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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
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