17,857 research outputs found
Kasteleyn cokernels
We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in
enumerating matchings of planar graphs, up to matrix operations on their rows
and columns. If such a matrix is defined over a principal ideal domain, this is
equivalent to considering its Smith normal form or its cokernel. Many
variations of the enumeration methods result in equivalent matrices. In
particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus
matrices.
We apply these ideas to plane partitions and related planar of tilings. We
list a number of conjectures, supported by experiments in Maple, about the
forms of matrices associated to enumerations of plane partitions and other
lozenge tilings of planar regions and their symmetry classes. We focus on the
case where the enumerations are round or -round, and we conjecture that
cokernels remain round or -round for related ``impossible enumerations'' in
which there are no tilings. Our conjectures provide a new view of the topic of
enumerating symmetry classes of plane partitions and their generalizations. In
particular we conjecture that a -specialization of a Jacobi-Trudi matrix has
a Smith normal form. If so it could be an interesting structure associated to
the corresponding irreducible representation of \SL(n,\C). Finally we find,
with proof, the normal form of the matrix that appears in the enumeration of
domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction
-Colored Graphs - a Review of Sundry Properties
We review the combinatorial, topological, algebraic and metric properties
supported by -colored graphs, with a focus on those that are pertinent
to the study of tensor model theories. We show how to extract a limiting
continuum metric space from this set of graphs and detail properties of this
limit through the calculation of exponents at criticality
MultiAspect Graphs: Algebraic representation and algorithms
We present the algebraic representation and basic algorithms for MultiAspect
Graphs (MAGs). A MAG is a structure capable of representing multilayer and
time-varying networks, as well as higher-order networks, while also having the
property of being isomorphic to a directed graph. In particular, we show that,
as a consequence of the properties associated with the MAG structure, a MAG can
be represented in matrix form. Moreover, we also show that any possible MAG
function (algorithm) can be obtained from this matrix-based representation.
This is an important theoretical result since it paves the way for adapting
well-known graph algorithms for application in MAGs. We present a set of basic
MAG algorithms, constructed from well-known graph algorithms, such as degree
computing, Breadth First Search (BFS), and Depth First Search (DFS). These
algorithms adapted to the MAG context can be used as primitives for building
other more sophisticated MAG algorithms. Therefore, such examples can be seen
as guidelines on how to properly derive MAG algorithms from basic algorithms on
directed graph. We also make available Python implementations of all the
algorithms presented in this paper.Comment: 59 pages, 6 figure
Detecting rich-club ordering in complex networks
Uncovering the hidden regularities and organizational principles of networks
arising in physical systems ranging from the molecular level to the scale of
large communication infrastructures is the key issue for the understanding of
their fabric and dynamical properties [1-5]. The ``rich-club'' phenomenon
refers to the tendency of nodes with high centrality, the dominant elements of
the system, to form tightly interconnected communities and it is one of the
crucial properties accounting for the formation of dominant communities in both
computer and social sciences [4-8]. Here we provide the analytical expression
and the correct null models which allow for a quantitative discussion of the
rich-club phenomenon. The presented analysis enables the measurement of the
rich-club ordering and its relation with the function and dynamics of networks
in examples drawn from the biological, social and technological domains.Comment: 1 table, 3 figure
Statistical Analysis of Bus Networks in India
Through the past decade the field of network science has established itself
as a common ground for the cross-fertilization of exciting inter-disciplinary
studies which has motivated researchers to model almost every physical system
as an interacting network consisting of nodes and links. Although public
transport networks such as airline and railway networks have been extensively
studied, the status of bus networks still remains in obscurity. In developing
countries like India, where bus networks play an important role in day-to-day
commutation, it is of significant interest to analyze its topological structure
and answer some of the basic questions on its evolution, growth, robustness and
resiliency. In this paper, we model the bus networks of major Indian cities as
graphs in \textit{L}-space, and evaluate their various statistical properties
using concepts from network science. Our analysis reveals a wide spectrum of
network topology with the common underlying feature of small-world property. We
observe that the networks although, robust and resilient to random attacks are
particularly degree-sensitive. Unlike real-world networks, like Internet, WWW
and airline, which are virtual, bus networks are physically constrained. The
presence of various geographical and economic constraints allow these networks
to evolve over time. Our findings therefore, throw light on the evolution of
such geographically and socio-economically constrained networks which will help
us in designing more efficient networks in the future.Comment: Submitted to PLOS ON
First Passage Properties of the Erdos-Renyi Random Graph
We study the mean time for a random walk to traverse between two arbitrary
sites of the Erdos-Renyi random graph. We develop an effective medium
approximation that predicts that the mean first-passage time between pairs of
nodes, as well as all moments of this first-passage time, are insensitive to
the fraction p of occupied links. This prediction qualitatively agrees with
numerical simulations away from the percolation threshold. Near the percolation
threshold, the statistically meaningful quantity is the mean transit rate,
namely, the inverse of the first-passage time. This rate varies
non-monotonically with p near the percolation transition. Much of this behavior
can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma
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