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We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of $\kappa$-equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order (Macauley and Mortveit (2009)). We extend earlier work by Veliz-Cuba and Stigler (2011), Goles et al. (2014), and others by comparing long-term dynamics up to topological conjugation: rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to ($i$) the lac operon model for Escherichia coli proposed by Veliz-Cuba and Stigler (2011), and ($ii$) the regulatory network involved in the control of the cell cycle and cell differentiation in the Caenorhabditis elegans vulva precursor cells proposed by Weinstein et al. (2015). In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all $10! > 3.6 \cdot 10^6$ sequential update orders, we demonstrate that it is sufficient to consider $344$ representative update orders, and, more notably, that these $344$ representatives give rise to $4$ distinct attractor structures. A similar analysis performed for the C. elegans model demonstrates that it has precisely $125$ distinct attractor structures. We conclude with observations on the variety and distribution of the models' attractor structures and use the results to discuss their robustness.

In this paper, we present BESSIE (Behavior and Epidemic Simulator for Synthetic Information Environments), an open source, agent-based simulator for COVID-type epidemics. BESSIE uses a synthetic population where each person has demographic attributes, belong to a household, and has a base activity- and visit schedule covering seven days. The simulated disease spreads through contacts that arise from joint visits to the locations where activities take place. The simulation model has a plugin-type programmable behavioral model where, based on the dynamics and observables tracked by the simulator, agents decide on actions such as wearing a mask, engaging in social distancing, or refraining from certain activity types by staying at home instead. The plugins are supplied as Python code. To the best of our knowledge, BESSIE is a unique simulator supporting this feature set, and most certainly as open software. To illustrate the use of BESSIE, we provide a COVID-relevant example demonstrating some of its capabilities. The example uses a synthetic population for the City of Charlottesville, Virginia. Both this population and the Python plugin modules used in the example are made available. The Python implementation, which can run on anything from a laptop to a cluster, is made available under the Apache 2.0 license (https://www.apache.org/licenses/LICENSE-2.0.html). The example population accompanying this publication is made available under the CC BY 4.0 license (https://creativecommons.org/licenses/by/4.0/)., Comment: 24 pages; the git repository and accompanying data will be made openly available upon final journal publication

In this paper we analyze Garden-of-Eden (GoE) states and fixed points of monotone, sequential dynamical systems (SDS). For any monotone SDS and fixed update schedule, we identify a particular set of states, each state being either a GoE state or reaching a fixed point, while both determining if a state is a GoE state and finding out all fixed points are generally hard. As a result, we show that the maximum size of their limit cycles is strictly less than ${n\choose \lfloor n/2 \rfloor}$. We connect these results to the Knaster-Tarski theorem and the LYM inequality. Finally, we establish that there exist monotone, parallel dynamical systems (PDS) that cannot be expressed as monotone SDS, despite the fact that the converse is always true.

A broad range of nonlinear processes over networks are governed by threshold dynamics. So far, existing mathematical theory characterizing the behavior of such systems has largely been concerned with the case where the thresholds are static. In this paper we extend current theory of finite dynamical systems to cover dynamic thresholds. Three classes of parallel and sequential dynamic threshold systems are introduced and analyzed. Our main result, which is a complete characterization of their attractor structures, show that sequential systems may only have fixed points as limit sets whereas parallel systems may only have period orbits of size at most two as limit sets. The attractor states are characterized for general graphs and enumerated in the special case of paths and cycle graphs; a computational algorithm is outlined for determining the number of fixed points over a tree. We expect our results to be relevant for modeling a broad class of biological, behavioral and socio-technical systems where adaptive behavior is central., Comment: Submitted for publication

This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of classical threshold functions and have separate threshold values for the transitions 0 -> 1 (up-threshold) and 1 -> 0 (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation: when the difference \Delta of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for \Delta >= 2 they may have long periodic orbits. The limiting case of \Delta = 2 is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graph classes., Comment: 18 pages, 4 figures, Discrete Mathematics and Theoretical Computer Science 2011

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 figures

We study the equivalence relation on the set of acyclic orientations of an undirected graph G generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over G to those over G' and G", the graphs obtained from G by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: (i) A complete combinatorial invariant of the equivalence classes, and (ii) a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach., Comment: 16 pages, 4 figures. Several examples have been added

In this article, we study finite dynamical systems defined over graphs, where the functions are applied asynchronously. Our goal is to quantify and understand stability of the dynamics with respect to the update sequence, and to relate this to structural properties of the graph. We introduce and analyze three different notions of update sequence stability, each capturing different aspects of the dynamics. When compared to each other, these stability concepts yield vastly different conclusions regarding the relationship between stability and graph structure, painting a more complete picture of update sequence stability., Comment: 9 pages, 1 figure

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is pi-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 256 cellular automaton rules are pi-independent. In this article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research., Comment: Revised per referee's comments

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.

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 polynomial

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)

A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the special case where Y is a circular graph with n vertices and all of the local functions are identical. The 256 possible local functions are known as Wolfram rules and the resulting sequential dynamical systems are called finite asynchronous elementary cellular automata, or ACAs, since they resemble classical elementary cellular automata, but with the important distinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said to be pi-independent if the set of periodic states does not depend on the choice of pi, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005 Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with this property. In addition to reproving and extending this earlier result, our proofs of pi-independence also provide significant insight into the dynamics of these systems., Comment: 18 pages. New version distinguishes between functions that are pi-independent but not w-independent