44 research outputs found
Simplicial Ricci Flow
We construct a discrete form of Hamilton's Ricci flow (RF) equations for a
d-dimensional piecewise flat simplicial geometry, S. These new algebraic
equations are derived using the discrete formulation of Einstein's theory of
general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation
is naturally associated to each edge, L, of a simplicial lattice. In defining
this equation, we find it convenient to utilize both the simplicial lattice, S,
and its circumcentric dual lattice, S*. In particular, the RRF equation
associated to L is naturally defined on a d-dimensional hybrid block connecting
with its (d-1)-dimensional circumcentric dual cell, L*. We show that
this equation is expressed as the proportionality between (1) the simplicial
Ricci tensor, Rc_L, associated with the edge L in S, and (2) a certain volume
weighted average of the fractional rate of change of the edges, lambda in L*,
of the circumcentric dual lattice, S*, that are in the dual of L. The inherent
orthogonality between elements of S and their duals in S* provide a simple
geometric representation of Hamilton's RF equations. In this paper we utilize
the well established theories of Regge calculus, or equivalently discrete
exterior calculus, to construct these equations. We solve these equations for a
few illustrative examples.Comment: 34 pages, 10 figures, minor revisions, DOI included: Commun. Math.
Phy
Ricci Curvature of the Internet Topology
Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201