65,903 research outputs found
AKLT Models with Quantum Spin Glass Ground States
We study AKLT models on locally tree-like lattices of fixed connectivity and
find that they exhibit a variety of ground states depending upon the spin,
coordination and global (graph) topology. We find a) quantum paramagnetic or
valence bond solid ground states, b) critical and ordered N\'eel states on
bipartite infinite Cayley trees and c) critical and ordered quantum vector spin
glass states on random graphs of fixed connectivity. We argue, in consonance
with a previous analysis, that all phases are characterized by gaps to local
excitations. The spin glass states we report arise from random long ranged
loops which frustrate N\'eel ordering despite the lack of randomness in the
coupling strengths.Comment: 10 pages, 1 figur
Fast Distributed PageRank Computation
Over the last decade, PageRank has gained importance in a wide range of
applications and domains, ever since it first proved to be effective in
determining node importance in large graphs (and was a pioneering idea behind
Google's search engine). In distributed computing alone, PageRank vector, or
more generally random walk based quantities have been used for several
different applications ranging from determining important nodes, load
balancing, search, and identifying connectivity structures. Surprisingly,
however, there has been little work towards designing provably efficient
fully-distributed algorithms for computing PageRank. The difficulty is that
traditional matrix-vector multiplication style iterative methods may not always
adapt well to the distributed setting owing to communication bandwidth
restrictions and convergence rates.
In this paper, we present fast random walk-based distributed algorithms for
computing PageRanks in general graphs and prove strong bounds on the round
complexity. We first present a distributed algorithm that takes O\big(\log
n/\eps \big) rounds with high probability on any graph (directed or
undirected), where is the network size and \eps is the reset probability
used in the PageRank computation (typically \eps is a fixed constant). We
then present a faster algorithm that takes O\big(\sqrt{\log n}/\eps \big)
rounds in undirected graphs. Both of the above algorithms are scalable, as each
node sends only small (\polylog n) number of bits over each edge per round.
To the best of our knowledge, these are the first fully distributed algorithms
for computing PageRank vector with provably efficient running time.Comment: 14 page
On the complexity of the vector connectivity problem
We study a relaxation of the Vector Domination problem called Vector
Connectivity (VecCon). Given a graph with a requirement for each
vertex , VecCon asks for a minimum cardinality set of vertices such that
every vertex is connected to via disjoint paths.
In the paper introducing the problem, Boros et al. [Networks, 2014] gave
polynomial-time solutions for VecCon in trees, cographs, and split graphs, and
showed that the problem can be approximated in polynomial time on -vertex
graphs to within a factor of , leaving open the question of whether
the problem is NP-hard on general graphs. We show that VecCon is APX-hard in
general graphs, and NP-hard in planar bipartite graphs and in planar line
graphs. We also generalize the polynomial result for trees by solving the
problem for block graphs.Comment: 14 page
Graph-Based Analysis and Visualisation of Mobility Data
Urban mobility forecast and analysis can be addressed through grid-based and
graph-based models. However, graph-based representations have the advantage of
more realistically depicting the mobility networks and being more robust since
they allow the implementation of Graph Theory machinery, enhancing the analysis
and visualisation of mobility flows. We define two types of mobility graphs:
Region Adjacency graphs and Origin-Destination graphs. Several node centrality
metrics of graphs are applied to identify the most relevant nodes of the
network in terms of graph connectivity. Additionally, the Perron vector
associated with a strongly connected graph is applied to define a circulation
function on the mobility graph. Such node values are visualised in the
geographically embedded graphs, showing clustering patterns within the network.
Since mobility graphs can be directed or undirected, we define several Graph
Laplacian for both cases and show that these matrices and their spectral
properties provide insightful information for network analysis. The computation
of node centrality metrics and Perron-induced circulation functions for three
different geographical regions demonstrate that basic elements from Graph
Theory applied to mobility networks can lead to structure analysis for graphs
of different connectivity, size, and orientation properties.Comment: 19 pages, 7 figure
Representation and generation of plans using graph spectra
Numerical comparison of spaces with one another is often achieved with set scalar
measures such as global and local integration, connectivity, etc., which capture a
particular quality of the space but therefore lose much of the detail of its overall
structure. More detailed methods such as graph edit distance are difficult to calculate,
particularly for large plans. This paper proposes the use of the graph spectrum, or the
ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space
as a whole. The result is a vector of high dimensionality that can be easily measured
against others for detailed comparison.
Several graph types are investigated, including boundary and axial representations, as
are several methods for deriving the spectral vector. The effectiveness of these is
evaluated using a genetic algorithm optimisation to generate plans to match a given
spectrum, and evolution is seen to produce plans similar to the initial targets, even in
very large search spaces. Results indicate that boundary graphs alone can capture the
gross topological qualities of a space, but axial graphs are needed to indicate local
relationships. Methods of scaling the spectra are investigated in relation to both global
local changes to plan arrangement. For all graph types, the spectra were seen to
capture local patterns of spatial arrangement even as global size is varied
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