106 research outputs found
A Discrete Evolutionary Model for Chess Players' Ratings
The Elo system for rating chess players, also used in other games and sports,
was adopted by the World Chess Federation over four decades ago. Although not
without controversy, it is accepted as generally reliable and provides a method
for assessing players' strengths and ranking them in official tournaments.
It is generally accepted that the distribution of players' rating data is
approximately normal but, to date, no stochastic model of how the distribution
might have arisen has been proposed. We propose such an evolutionary stochastic
model, which models the arrival of players into the rating pool, the games they
play against each other, and how the results of these games affect their
ratings. Using a continuous approximation to the discrete model, we derive the
distribution for players' ratings at time as a normal distribution, where
the variance increases in time as a logarithmic function of . We validate
the model using published rating data from 2007 to 2010, showing that the
parameters obtained from the data can be recovered through simulations of the
stochastic model.
The distribution of players' ratings is only approximately normal and has
been shown to have a small negative skew. We show how to modify our
evolutionary stochastic model to take this skewness into account, and we
validate the modified model using the published official rating data.Comment: 17 pages, 4 figure
A bibliometric index based on the complete list of cited Publications
We propose a new index, the j-index, which is defined for an author as the sum of the square roots of the numbers of citations to each of the author’s publications. The idea behind the j-index it to remedy a drawback of the h-index - that the h-index does not take into account the full citation record of a researcher. The square root function is motivated by our desire to avoid the possible bias that may occur with a simple sum when an author has several very highly cited papers. We compare the j-index to the h-index, the g-index and the total citation count for three subject areas using several association measures. Our results indicate that that the association between the j-index and the other indices varies according to the subject area. One explanation of this variation may be due to the proportion of citations to publications of the researcher that are in the h-core. The j-index is not an h-index variant, and as such is intended to complement rather than necessarily replace the h-index and other bibliometric indicators, thus providing a more complete picture of a researcher’s achievements
A Stochastic Evolutionary Growth Model for Social Networks
We present a stochastic model for a social network, where new actors may join
the network, existing actors may become inactive and, at a later stage,
reactivate themselves. Our model captures the evolution of the network,
assuming that actors attain new relations or become active according to the
preferential attachment rule. We derive the mean-field equations for this
stochastic model and show that, asymptotically, the distribution of actors
obeys a power-law distribution. In particular, the model applies to social
networks such as wireless local area networks, where users connect to
access-points, and peer-to-peer networks where users connect to each other. As
a proof of concept, we demonstrate the validity of our model empirically by
analysing a public log containing traces from a wireless network at Dartmouth
College over a period of three years. Analysing the data processed according to
our model, we demonstrate that the distribution of user accesses is
asymptotically a power-law distribution.Comment: 15 pages, 1 figur
Knot Graphs
We consider the equivalence classes of graphs induced by the unsigned
versions of the Reidemeister moves on knot diagrams.
Any graph which is
reducible by some finite sequence of these moves, to a graph with no
edges is called a knot graph. We show that the class of knot graphs
strictly contains the set of delta-wye graphs. We prove that the
dimension of the intersection of the cycle and cocycle spaces is an
effective numerical invariant of these classes
A new approach to graph reconstruction using supercards
The vertex-deleted subgraph G - v, obtained from the graph G
by deleting the vertex v and all edges incident to v, is called a
card of G. The deck of G is the multiset of its unlabelled vertex-
deleted subgraphs. The number of common cards of G and H is
the cardinality of a maximum multiset of common cards, i.e., the
multiset intersection of the decks of G and H. We introduce a new
approach to the study of common cards using supercards, where
we define a supercard G+ of G and H to be a graph that has at
least one vertex-deleted subgraph isomorphic to G, and at least
one isomorphic to H. We show how maximum sets of common
cards of G and H correspond to certain sets of permutations of
the vertices of a supercard, which we call maximum saturating
sets. We then show how to construct supercards of various pairs
of graphs for which there exists some maximum saturating set X
contained in Aut(G+). For certain other pairs of graphs, we show
that it is possible to construct G+ and a maximum saturating set
X such that the elements of X that are not in Aut(G+) are in one-
to-one correspondence with a set of automorphisms of a different
supercard G+_lambda ďż˝ of G and H. Our constructions cover nearly all of
the published families of pairs of graphs that have a large number
of common cards
Distance and consensus for preference relations corresponding to ordered partitions
Ranking is an important part of several areas of contemporary research, including social sciences, decision theory, data analysis and information retrieval. The goal of this paper is to align developments in quantitative social sciences and decision theory with the current thought in Computer Science, including a few novel results. Specifically, we consider binary preference relations, the so-called weak orders that are in one-to-one correspondence with rankings. We show that the conventional symmetric difference distance between weak orders, considered as sets of ordered pairs, coincides with the celebrated Kemeny distance between the corresponding rankings, despite the seemingly much simpler structure of the former. Based on this, we review several properties of the geometric space of weak orders involving the ternary relation “between”, and contingency tables for cross-partitions. Next, we reformulate the consensus ranking problem as a variant of finding an optimal linear ordering, given a correspondingly defined consensus matrix. The difference is in a subtracted term, the partition concentration, that depends only on the distribution of the objects in the individual parts. We apply our results to the conventional Likert scale to show that the Kemeny consensus rule is rather insensitive to the data under consideration and, therefore, should be supplemented with more sensitive consensus schemes
Fast Generation of Unlabelled Free Trees using Weight Sequences
In this paper, we introduce a new representation for ordered trees, the
weight sequence representation. We then use this to construct new
representations for both rooted trees and free trees, namely the canonical
weight sequence representation. We construct algorithms for generating the
weight sequence representations for all rooted and free trees of order n, and
then add a number of modifications to improve the efficiency of the algorithms.
Python implementations of the algorithms incorporate further improvements by
using generators to avoid having to store the long lists of trees returned by
the recursive calls, as well as caching the lists for rooted trees of small
order, thereby eliminating many of the recursive calls. We further show how the
algorithm can be modifed to generate adjacency list and adjacency matrix
representations for free trees. We compared the run-times of our Python
implementation for generating free trees with the Python implementation of the
well-known WROM algorithm taken from NetworkX. The implementation of our
algorithm is over four times as fast as the implementation of the WROM
algorithm. The run-times for generating adjacency lists and matrices are
somewhat longer than those for weight sequences, but are still over three times
as fast as the corresponding implementations of the WROM algorithm.Comment: 21 pages, 1 table and 8 figure
- …