23 research outputs found
Voting in Agreeable Societies
No abstract provided in this article
Voting for Committees in Agreeable Societies
We examine the following voting situation. A committee of people is to be
formed from a pool of n candidates. The voters selecting the committee will
submit a list of candidates that they would prefer to be on the committee.
We assume that . For a chosen committee, a given voter is said to
be satisfied by that committee if her submitted list of candidates is a
subset of that committee. We examine how popular is the most popular committee.
In particular, we show there is always a committee that satisfies a certain
fraction of the voters and examine what characteristics of the voter data will
increase that fraction.Comment: 11 pages; to appear in Contemporary Mathematic
Voter Compatibility In Interval Societies
In an interval society, voters are represented by intervals on the real line, corresponding to their approval sets on a linear political spectrum. I imagine the society to be a representative democracy, and ask how to choose members of the society as representatives. Following work in mathematical psychology by Coombs and others, I develop a measure of the compatibility (political similarity) of two voters. I use this measure to determine the popularity of each voter as a candidate. I then establish local “agreeability” conditions and attempt to find a lower bound for the popularity of the best candidate. Other results about certain special societies are also obtaine
Interval voting technique for active and dormant state synchronization of the nodes in WSN
Energy conservation is one of the main issues of WSN research. To save energy and prolong sensor nodes lives, nodes operate in a duty-cycling work mode. Synchronous duty-cycling protocols require time synchronization in order to enable nodes to simultaneously sleep and wake up. In this paper, a method for nodes synchronization based on interval voting is proposed. The method allows to obtain consensus relation in a form of interval by means of sensor data fusion technique and apply it for nodes synchronization, thus solving a problem of choosing reference time node
On (2,3)-agreeable Box Societies
The notion of -agreeable society was introduced by Deborah Berg et
al.: a family of convex subsets of is called -agreeable if any
subfamily of size contains at least one non-empty -fold intersection. In
that paper, the -agreeability of a convex family was shown to imply the
existence of a subfamily of size with non-empty intersection, where
is the size of the original family and is an explicit
constant depending only on and . The quantity is called
the minimal \emph{agreement proportion} for a -agreeable family in
.
If we only assume that the sets are convex, simple examples show that
for -agreeable families in where . In this paper,
we introduce new techniques to find positive lower bounds when restricting our
attention to families of -boxes, i.e. cuboids with sides parallel to the
coordinates hyperplanes. We derive explicit formulas for the first non-trivial
case: the case of -agreeable families of -boxes with .Comment: 15 pages, 10 figure
Schedule (2007)
Sixteenth Conference of the Association of Christians in the Mathematical Science
Tur\'an and Ramsey Properties of Subcube Intersection Graphs
The discrete cube is a fundamental combinatorial structure. A
subcube of is a subset of of its points formed by fixing
coordinates and allowing the remaining to vary freely. The subcube
structure of the discrete cube is surprisingly complicated and there are many
open questions relating to it.
This paper is concerned with patterns of intersections among subcubes of the
discrete cube. Two sample questions along these lines are as follows: given a
family of subcubes in which no of them have non-empty intersection, how
many pairwise intersections can we have? How many subcubes can we have if among
them there are no which have non-empty intersection and no which are
pairwise disjoint? These questions are naturally expressed as Tur\'an and
Ramsey type questions in intersection graphs of subcubes where the intersection
graph of a family of sets has one vertex for each set in the family with two
vertices being adjacent if the corresponding subsets intersect.
Tur\'an and Ramsey type problems are at the heart of extremal combinatorics
and so these problems are mathematically natural. However, a second motivation
is a connection with some questions in social choice theory arising from a
simple model of agreement in a society. Specifically, if we have to make a
binary choice on each of separate issues then it is reasonable to assume
that the set of choices which are acceptable to an individual will be
represented by a subcube. Consequently, the pattern of intersections within a
family of subcubes will have implications for the level of agreement within a
society.
We pose a number of questions and conjectures relating directly to the
Tur\'an and Ramsey problems as well as raising some further directions for
study of subcube intersection graphs.Comment: 18 page
Random subcube intersection graphs I: cliques and covering
We study random subcube intersection graphs, that is, graphs obtained by
selecting a random collection of subcubes of a fixed hypercube to serve
as the vertices of the graph, and setting an edge between a pair of subcubes if
their intersection is non-empty. Our motivation for considering such graphs is
to model `random compatibility' between vertices in a large network. For both
of the models considered in this paper, we determine the thresholds for
covering the underlying hypercube and for the appearance of s-cliques. In
addition we pose some open problems.Comment: 38 pages, 1 figur