3,248 research outputs found
Path-tables of trees: a survey and some new results
The (vertex) path-table of a tree contains quantitative information about the paths in . The entry of this table gives the number of paths of length passing through vertex . The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree does not determine uniquely. We shall show that in trees, the number of paths passing through edge can only be expressed in terms of paths passing through vertices and up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables
Recent Developments of World-Line Monte Carlo Methods
World-line quantum Monte Carlo methods are reviewed with an emphasis on
breakthroughs made in recent years. In particular, three algorithms -- the loop
algorithm, the worm algorithm, and the directed-loop algorithm -- for updating
world-line configurations are presented in a unified perspective. Detailed
descriptions of the algorithms in specific cases are also given.Comment: To appear in Journal of Physical Society of Japa
Graphs, Friends and Acquaintances
As is well known, a graph is a mathematical object modeling the
existence of a certain relation between pairs of elements of a given set.
Therefore, it is not surprising that many of the first results concerning
graphs made reference to relationships between people or groups of
people. In this article, we comment on four results of this kind, which
are related to various general theories on graphs and their applications:
the Handshake lemma (related to graph colorings and Boolean
algebra), a lemma on known and unknown people at a cocktail party
(to Ramsey theory), a theorem on friends in common (to distanceregularity
and coding theory), and Hall’s Marriage theorem (to the
theory of networks). These four areas of graph theory, often with
problems which are easy to state but difficult to solve, are extensively
developed and currently give rise to much research work. As examples
of representative problems and results of these areas, which are
discussed in this paper, we may cite the following: the Four Colors
Theorem (4CTC), the Ramsey numbers, problems of the existence of
distance-regular graphs and completely regular codes, and finally the
study of topological proprieties of interconnection networks.Preprin
From Spectral Graph Convolutions to Large Scale Graph Convolutional Networks
Graph Convolutional Networks (GCNs) have been shown to be a powerful concept
that has been successfully applied to a large variety of tasks across many
domains over the past years. In this work we study the theory that paved the
way to the definition of GCN, including related parts of classical graph
theory. We also discuss and experimentally demonstrate key properties and
limitations of GCNs such as those caused by the statistical dependency of
samples, introduced by the edges of the graph, which causes the estimates of
the full gradient to be biased. Another limitation we discuss is the negative
impact of minibatch sampling on the model performance. As a consequence, during
parameter update, gradients are computed on the whole dataset, undermining
scalability to large graphs. To account for this, we research alternative
methods which allow to safely learn good parameters while sampling only a
subset of data per iteration. We reproduce the results reported in the work of
Kipf et al. and propose an implementation inspired to SIGN, which is a
sampling-free minibatch method. Eventually we compare the two implementations
on a benchmark dataset, proving that they are comparable in terms of prediction
accuracy for the task of semi-supervised node classification
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