1,362 research outputs found
Chromatic thresholds in dense random graphs
The chromatic threshold of a graph with respect to the
random graph is the infimum over such that the following holds
with high probability: the family of -free graphs with
minimum degree has bounded chromatic number. The study of
the parameter was initiated in 1973 by
Erd\H{o}s and Simonovits, and was recently determined for all graphs . In
this paper we show that for all fixed , but that typically if . We also make significant progress towards determining
for all graphs in the range . In sparser random graphs the
problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with
minor modifications from arXiv:1108.1746 for a self-contained proof of a
technical lemma; accepted to Random Structures and Algorithm
Sizing the length of complex networks
Among all characteristics exhibited by natural and man-made networks the
small-world phenomenon is surely the most relevant and popular. But despite its
significance, a reliable and comparable quantification of the question `how
small is a small-world network and how does it compare to others' has remained
a difficult challenge to answer. Here we establish a new synoptic
representation that allows for a complete and accurate interpretation of the
pathlength (and efficiency) of complex networks. We frame every network
individually, based on how its length deviates from the shortest and the
longest values it could possibly take. For that, we first had to uncover the
upper and the lower limits for the pathlength and efficiency, which indeed
depend on the specific number of nodes and links. These limits are given by
families of singular configurations that we name as ultra-short and ultra-long
networks. The representation here introduced frees network comparison from the
need to rely on the choice of reference graph models (e.g., random graphs and
ring lattices), a common practice that is prone to yield biased interpretations
as we show. Application to empirical examples of three categories (neural,
social and transportation) evidences that, while most real networks display a
pathlength comparable to that of random graphs, when contrasted against the
absolute boundaries, only the cortical connectomes prove to be ultra-short
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
Symmetry adapted Assur decompositions
Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited the orbit matrix as a symmetry
adapted form of the rigidity matrix. This paper explores how the decomposition
and analysis of symmetric frameworks and their symmetric motions can be
supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure
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