130 research outputs found
Stratification and enumeration of Boolean functions by canalizing depth
Boolean network models have gained popularity in computational systems
biology over the last dozen years. Many of these networks use canalizing
Boolean functions, which has led to increased interest in the study of these
functions. The canalizing depth of a function describes how many canalizing
variables can be recursively picked off, until a non-canalizing function
remains. In this paper, we show how every Boolean function has a unique
algebraic form involving extended monomial layers and a well-defined core
polynomial. This generalizes recent work on the algebraic structure of nested
canalizing functions, and it yields a stratification of all Boolean functions
by their canalizing depth. As a result, we obtain closed formulas for the
number of n-variable Boolean functions with depth k, which simultaneously
generalizes enumeration formulas for canalizing, and nested canalizing
functions
On Enumeration of Conjugacy Classes of Coxeter Elements
In this paper we study the equivalence relation on the set of acyclic
orientations of a graph Y that arises through source-to-sink conversions. This
source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a
Coxeter group. We give a direct proof of a recursion for the number of
equivalence classes of this relation for an arbitrary graph Y using edge
deletion and edge contraction of non-bridge edges. We conclude by showing how
this result may also be obtained through an evaluation of the Tutte polynomial
as T(Y,1,0), and we provide bijections to two other classes of acyclic
orientations that are known to be counted in the same way. A transversal of the
set of equivalence classes is given.Comment: Added a few results about connections to the Tutte polynomia
Equivalences on Acyclic Orientations
The cyclic and dihedral groups can be made to act on the set Acyc(Y) of
acyclic orientations of an undirected graph Y, and this gives rise to the
equivalence relations ~kappa and ~delta, respectively. These two actions and
their corresponding equivalence classes are closely related to combinatorial
problems arising in the context of Coxeter groups, sequential dynamical
systems, the chip-firing game, and representations of quivers.
In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y)
and whose connected components encode the equivalence classes. The number of
connected components of these graphs are denoted kappa(Y) and delta(Y),
respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y)
can be derived from kappa(Y), and give enumeration results for kappa(Y).
Moreover, we show how to associate a poset structure to each kappa-equivalence
class, and we characterize these posets. This allows us to create a bijection
from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y'
and Y'' denote edge deletion and edge contraction for a cycle-edge in Y,
respectively, which in turn shows that kappa(Y) may be obtained by an
evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two
papers (see also arXiv:0802.4412
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
Coxeter Groups and Asynchronous Cellular Automata
The dynamics group of an asynchronous cellular automaton (ACA) relates
properties of its long term dynamics to the structure of Coxeter groups. The
key mathematical feature connecting these diverse fields is involutions.
Group-theoretic results in the latter domain may lead to insight about the
dynamics in the former, and vice-versa. In this article, we highlight some
central themes and common structures, and discuss novel approaches to some open
and open-ended problems. We introduce the state automaton of an ACA, and show
how the root automaton of a Coxeter group is essentially part of the state
automaton of a related ACA.Comment: 10 pages, 4 figure
Nested canalyzing depth and network stability
We introduce the nested canalyzing depth of a function, which measures the
extent to which it retains a nested canalyzing structure. We characterize the
structure of functions with a given depth and compute the expected activities
and sensitivities of the variables. This analysis quantifies how canalyzation
leads to higher stability in Boolean networks. It generalizes the notion of
nested canalyzing functions (NCFs), which are precisely the functions with
maximum depth. NCFs have been proposed as gene regulatory network models, but
their structure is frequently too restrictive and they are extremely sparse. We
find that functions become decreasingly sensitive to input perturbations as the
canalyzing depth increases, but exhibit rapidly diminishing returns in
stability. Additionally, we show that as depth increases, the dynamics of
networks using these functions quickly approach the critical regime, suggesting
that real networks exhibit some degree of canalyzing depth, and that NCFs are
not significantly better than functions of sufficient depth for many
applications of the modeling and reverse engineering of biological networks.Comment: 13 pages, 2 figure
A Novel Proof of the Heine-Borel Theorem
Every beginning real analysis student learns the classic Heine-Borel theorem,
that the interval [0,1] is compact. In this article, we present a proof of this
result that doesn't involve the standard techniques such as constructing a
sequence and appealing to the completeness of the reals. We put a metric on the
space of infinite binary sequences and prove that compactness of this space
follows from a simple combinatorial lemma. The Heine-Borel theorem is an
immediate corollary
Braids and Juggling Patterns
There are several ways to describe juggling patterns mathematically using combinatorics and algebra. In my thesis I use these ideas to build a new system using braid groups. A new kind of graph arises that helps describe all braids that can be juggled
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