681 research outputs found
Every property is testable on a natural class of scale-free multigraphs
In this paper, we introduce a natural class of multigraphs called
hierarchical-scale-free (HSF) multigraphs, and consider constant-time
testability on the class. We show that a very wide subclass, specifically, that
in which the power-law exponent is greater than two, of HSF is hyperfinite.
Based on this result, an algorithm for a deterministic partitioning oracle can
be constructed. We conclude by showing that every property is constant-time
testable on the above subclass of HSF. This algorithm utilizes findings by
Newman and Sohler of STOC'11. However, their algorithm is based on the
bounded-degree model, while it is known that actual scale-free networks usually
include hubs, which have a very large degree. HSF is based on scale-free
properties and includes such hubs. This is the first universal result of
constant-time testability on the general graph model, and it has the potential
to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and
SF become more general. Typos were fixe
Properties of Random Graphs with Hidden Color
We investigate in some detail a recently suggested general class of ensembles
of sparse undirected random graphs based on a hidden stub-coloring, with or
without the restriction to nondegenerate graphs. The calculability of local and
global structural properties of graphs from the resulting ensembles is
demonstrated. Cluster size statistics are derived with generating function
techniques, yielding a well-defined percolation threshold. Explicit rules are
derived for the enumeration of small subgraphs. Duality and redundancy is
discussed, and subclasses corresponding to commonly studied models are
identified.Comment: 14 pages, LaTeX, no figure
Energy flow polynomials: A complete linear basis for jet substructure
We introduce the energy flow polynomials: a complete set of jet substructure
observables which form a discrete linear basis for all infrared- and
collinear-safe observables. Energy flow polynomials are multiparticle energy
correlators with specific angular structures that are a direct consequence of
infrared and collinear safety. We establish a powerful graph-theoretic
representation of the energy flow polynomials which allows us to design
efficient algorithms for their computation. Many common jet observables are
exact linear combinations of energy flow polynomials, and we demonstrate the
linear spanning nature of the energy flow basis by performing regression for
several common jet observables. Using linear classification with energy flow
polynomials, we achieve excellent performance on three representative jet
tagging problems: quark/gluon discrimination, boosted W tagging, and boosted
top tagging. The energy flow basis provides a systematic framework for complete
investigations of jet substructure using linear methods.Comment: 41+15 pages, 13 figures, 5 tables; v2: updated to match JHEP versio
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