735 research outputs found
Parallel Chip Firing Game associated with n-cube orientations
We study the cycles generated by the chip firing game associated with n-cube
orientations. We show the existence of the cycles generated by parallel
evolutions of even lengths from 2 to on (n >= 1), and of odd
lengths different from 3 and ranging from 1 to on (n >= 4)
The approach to criticality in sandpiles
A popular theory of self-organized criticality relates the critical behavior
of driven dissipative systems to that of systems with conservation. In
particular, this theory predicts that the stationary density of the abelian
sandpile model should be equal to the threshold density of the corresponding
fixed-energy sandpile. This "density conjecture" has been proved for the
underlying graph Z. We show (by simulation or by proof) that the density
conjecture is false when the underlying graph is any of Z^2, the complete graph
K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower
graph. Driven dissipative sandpiles continue to evolve even after a constant
fraction of the sand has been lost at the sink. These results cast doubt on the
validity of using fixed-energy sandpiles to explore the critical behavior of
the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320
Lock-in Problem for Parallel Rotor-router Walks
The rotor-router model, also called the Propp machine, was introduced as a
deterministic alternative to the random walk. In this model, a group of
identical tokens are initially placed at nodes of the graph. Each node
maintains a cyclic ordering of the outgoing arcs, and during consecutive turns
the tokens are propagated along arcs chosen according to this ordering in
round-robin fashion. The behavior of the model is fully deterministic. Yanovski
et al.(2003) proved that a single rotor-router walk on any graph with m edges
and diameter stabilizes to a traversal of an Eulerian circuit on the set of
all 2m directed arcs on the edge set of the graph, and that such periodic
behaviour of the system is achieved after an initial transient phase of at most
2mD steps. The case of multiple parallel rotor-routers was studied
experimentally, leading Yanovski et al. to the conjecture that a system of k
\textgreater{} 1 parallel walks also stabilizes with a period of length at
most steps. In this work we disprove this conjecture, showing that the
period of parallel rotor-router walks can in fact, be superpolynomial in the
size of graph. On the positive side, we provide a characterization of the
periodic behavior of parallel router walks, in terms of a structural property
of stable states called a subcycle decomposition. This property provides us the
tools to efficiently detect whether a given system configuration corresponds to
the transient or to the limit behavior of the system. Moreover, we provide
polynomial upper bounds of and on the
number of steps it takes for the system to stabilize. Thus, we are able to
predict any future behavior of the system using an algorithm that takes
polynomial time and space. In addition, we show that there exists a separation
between the stabilization time of the single-walk and multiple-walk
rotor-router systems, and that for some graphs the latter can be asymptotically
larger even for the case of walks
Abelian networks IV. Dynamics of nonhalting networks
An abelian network is a collection of communicating automata whose state
transitions and message passing each satisfy a local commutativity condition.
This paper is a continuation of the abelian networks series of Bond and Levine
(2016), for which we extend the theory of abelian networks that halt on all
inputs to networks that can run forever. A nonhalting abelian network can be
realized as a discrete dynamical system in many different ways, depending on
the update order. We show that certain features of the dynamics, such as
minimal period length, have intrinsic definitions that do not require
specifying an update order.
We give an intrinsic definition of the \emph{torsion group} of a finite
irreducible (halting or nonhalting) abelian network, and show that it coincides
with the critical group of Bond and Levine (2016) if the network is halting. We
show that the torsion group acts freely on the set of invertible recurrent
components of the trajectory digraph, and identify when this action is
transitive.
This perspective leads to new results even in the classical case of sinkless
rotor networks (deterministic analogues of random walks). In Holroyd et. al
(2008) it was shown that the recurrent configurations of a sinkless rotor
network with just one chip are precisely the unicycles (spanning subgraphs with
a unique oriented cycle, with the chip on the cycle). We generalize this result
to abelian mobile agent networks with any number of chips. We give formulas for
generating series such as where is the number of recurrent chip-and-rotor configurations with
chips; is the diagonal matrix of outdegrees, and is the adjacency
matrix. A consequence is that the sequence completely
determines the spectrum of the simple random walk on the network.Comment: 95 pages, 21 figure
On the effects of firing memory in the dynamics of conjunctive networks
Boolean networks are one of the most studied discrete models in the context
of the study of gene expression. In order to define the dynamics associated to
a Boolean network, there are several \emph{update schemes} that range from
parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each
possible dynamics defined by different update schemes might not be efficient.
In this context, considering some type of temporal delay in the dynamics of
Boolean networks emerges as an alternative approach. In this paper, we focus in
studying the effect of a particular type of delay called \emph{firing memory}
in the dynamics of Boolean networks. Particularly, we focus in symmetric
(non-directed) conjunctive networks and we show that there exist examples that
exhibit attractors of non-polynomial period. In addition, we study the
prediction problem consisting in determinate if some vertex will eventually
change its state, given an initial condition. We prove that this problem is
{\bf PSPACE}-complete
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