  A new class of geometric statistics for analyzing galaxy catalogs is
presented. Filament statistics quantify filamentarity and planarity in large
scale structure in a manner consistent with catalog visualizations. These
statistics are based on sequences of spatial links which follow local
high-density structures. From these link sequences we compute the discrete
curvature, planarity, and torsion. Filament statistics are applied to CDM and
CHDM ($\Omega_\nu = 0.3$) simulations of Klypin \etal (1996), the CfA1-like
mock redshift catalogs of Nolthenius, Klypin and Primack (1994, 1996), and the
CfA1 catalog. We also apply the moment-based shape statistics developed by
Babul \& Starkman (1992), Luo \& Vishniac (1995), and Robinson \& Albrecht
(1996) to these same catalogs, and compare their robustness and discriminatory
power versus filament statistics. For 100 Mpc periodic simulation boxes ($H_0 =
50$ km s$^{-1}$ Mpc$^{-1}$), we find discrimination of $\sim 4\sigma$ (where
$\sigma$ represents resampling errors) between CHDM and CDM for selected
filament statistics and shape statistics, including variations in the galaxy
identification scheme. Comparing the CfA1 data versus the models does not yield
a conclusively favored model; no model is excluded at more than a $\sim
2\sigma$ level for any statistic, not including cosmic variance which could
further degrade the discriminatory power. We find that CfA1 discriminates
between models poorly mainly due to its sparseness and small number of
galaxies, not due to redshift distortion, magnitude limiting, or geometrical
effects. We anticipate that the proliferation of large redshift surveys and
simulations will enable the statistics presented here to provide robust
discrimination between large-scale structure in various cosmological models.
