Matthew Macauley

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.

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.

Cycle equivalence of graph dynamical systems

Graph dynamical systems (GDSs) generalize concepts such as cellular automata and Boolean networks and can describe a wide range of distributed, nonlinear phenomena. Two GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs, which captures the notion of having comparable long-term dynamics. In this paper, we study cycle equivalence of GDS si n which the vertex functions are applied sequentially through an update sequence. The main result is a general characterization of cycle equivalence based on the underlying graph Y and the update sequences. We construct and analyse two graphs C(Y) and D(Y ) whose connected components contain update sequences that induce cycle equivalent dynamical system maps. The number of components in these graphs, denoted κ(Y) and δ(Y) , bound the number of possible long-term behaviour that can be generated by varying the update sequence. We give a recursion relation for κ(Y) which in turn allows us to enumerate δ(Y) . The components of C(Y) and D(Y ) characterize dynamical neutrality, their sizes represent structural stability of periodic orbits and the number of components can be viewed as a system complexity measure. We conclude with a computational result demonstrating the impact on complexity that results when passing from radius-1 to radius-2 rules in asynchronous cellular automata.

Dangerous Reference Graphs and Semantic Paradoxes

The semantic paradoxes are often associated with self-reference or referential circularity. Yablo (Analysis 53(4):251–252, 1993), however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook (J Symb Log 69(3):767–774, 2004) and Yablo (2006) with respect to more restricted sentences systems, which we call

Dynamics of Complex Boolean Networks: Canalization, Stability, and Criticality

\mathcal{F}

An atlas of limit set dynamics for asynchronous elementary cellular automata

-systems.

Coxeter groups and asynchronous cellular automata

Abstract Boolean networks play an important role in modern biology as popular alternatives to traditional continuous models. After a brief introduction, we will analyze two main features that influence the dynamics: network topology and the Boolean functions. Of particular interest are the canalizing functions, which model a biological robustness concept proposed by geneticist C.H. Waddington in 1942. After an in-depth look at canalizing functions, we will conclude with an analysis of the stability of Boolean network dynamics. Loosely speaking, Boolean networks fall into one of two dynamical regimes, ordered and critical, which are characterized by whether small perturbations tend to die out or propagate through the network. These regimes are separated by the narrow critical threshold, where many real-world networks are believed to lie. Critical networks optimize the trade-off of being robust enough to withstand external perturbations yet flexible enough to exhibit complex dynamics and evolve.

RNA Secondary Structures: Combinatorial Models and Folding Algorithms

In this paper we provide an overview of the possible @w-limit set structures for 104 of the 256 asynchronous elementary cellular automata over the circle graph on n vertices. We consider only fixed, sequential updates where the update sequence is given by a permutation of the vertices, that is, the class of sequential dynamical systems. The ECA rules covered are precisely the @p-invariant rules, that is, the rules for which the set of periodic points does not depend on the permutation update sequence. This paper reviews existing work on @p-invariance and cycle-equivalence, and provides and atlas of the possible limit set structures up to topological conjugation.

Dynamics groups of 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.

Locational market power in network constrained markets

Ribonucleic acid (RNA) is a long macromolecule built from nucleotides strung together along a sugar-phosphate backbone. Unlike its double-stranded cousin deoxyribonucleic acid, which twists into a double-helix structure, RNA folds and bonds to itself. Because the three-dimensional structure into which RNA folds determines its cellular function, scientists are very interested in understanding how it folds. The challenge is to predict an RNA’s three-dimensional structure given only its raw primary sequence. We begin this chapter by interpreting this problem in a combinatorial framework, using partial matchings and noncrossing arc diagrams called secondary structures. Next, we take an in-depth tour of two very different approaches to the secondary structure prediction problem. The first method attempts to minimize the free energy using a recursive technique called dynamic programming. The second method arises from the field of computational linguistics—RNA secondary structures are generated using a stochastic context-free grammar. Finally, the chapter concludes with a brief overview of how to extend the RNA combinatorial framework to more complicated structures called pseudoknots.Abstract Ribonucleic acid (RNA) is a long macromolecule built from nucleotides strung together along a sugar-phosphate backbone. Unlike its double-stranded cousin deoxyribonucleic acid, which twists into a double-helix structure, RNA folds and bonds to itself. Because the three-dimensional structure into which RNA folds determines its cellular function, scientists are very interested in understanding how it folds. The challenge is to predict an RNA’s three-dimensional structure given only its raw primary sequence. We begin this chapter by interpreting this problem in a combinatorial framework, using partial matchings and noncrossing arc diagrams called secondary structures. Next, we take an in-depth tour of two very different approaches to the secondary structure prediction problem. The first method attempts to minimize the free energy using a recursive technique called dynamic programming. The second method arises from the field of computational linguistics—RNA secondary structures are generated using a stochastic context-free grammar. Finally, the chapter concludes with a brief overview of how to extend the RNA combinatorial framework to more complicated structures called pseudoknots.