Henning S. Mortveit

Discrete, sequential dynamical systems

Abstract We study a class of discrete dynamical systems that consists of the following data: (a) a finite loop-free graph Y with vertex set {1, …, n} where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F i, Y : F 2 n → F 2 n ) i and (c) a permutation π∈Sn. The function F i, Y updates the state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F i, Y are applied. By composing the functions F i, Y in the order given by π we obtain the dynamical system [ F Y , π]=∏ i=1 n F π(i),Y : F 2 n → F 2 n , which we refer to as a sequential dynamical system (SDS). Among various basic results on SDS we will study their invertibility and analyze the set |{[ F Y , π] | π∈S n }| for fixed Y and (F i, Y ) i . Finally, we give an estimate for the number of non-isomorphic digraphs Γ[ F Y , π] (having vertex sets F 2 n and directed edges {(x, [ F Y , π](x)) | x∈ F 2 n } ) for a fixed graph Y and a fixed multi-set (F i, Y ) i .

Metropolitan road traffic simulation on FPGAs

This work demonstrates that road traffic simulation of entire metropolitan areas is possible with reconfigurable supercomputing that combines 64-bit microprocessors and FPGAs in a high bandwidth, low latency interconnect. Previously, traffic simulation on FPGAs was limited to very-short road segments or required a very large number of FPGAs. Our data streaming approach overcomes scaling issues associated with direct implementations and still allows for high-level parallelism by dividing the data sets between hardware and software across the reconfigurable supercomputer. Using one FPGA on the Cray XD1 supercomputer, we are able to achieve a 34.4/spl times/ speed up over the AMD microprocessor. System integration issues must be optimized to exploit this speedup in the overall simulation.

Partitioning Hardware and Software for Reconfigurable Supercomputing Applications: A Case Study

Often reconfigurable systems are reported to have 10× to 100× speedup over that of a software system. However, the reconfigurable hardware must usually be combined with software to form an entire system. This system integration presents a hardware/software co-design problem with many system engineering issues. Here, we present traffic acceleration on the Cray XD1 supercomputer and describe the costs involved in different hardware/software trade-offs.

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.

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.

Interactions among human behavior, social networks, and societal infrastructures: A Case Study in Computational Epidemiology

Human behavior, social networks, and the civil infrastructures are closely intertwined. Understanding their co-evolution is critical for designing public policies and decision support for disaster planning. For example, human behaviors and day to day activities of individuals create dense social interactions that are characteristic of modern urban societies. These dense social networks provide a perfect fabric for fast, uncontrolled disease propagation. Conversely, people’s behavior in response to public policies and their perception of how the crisis is unfolding as a result of disease outbreak can dramatically alter the normally stable social interactions. Effective planning and response strategies must take these complicated interactions into account. In this chapter, we describe a computer simulation based approach to study these issues using public health and computational epidemiology as an illustrative example. We also formulate game-theoretic and stochastic optimization problems that capture many of the problems that we study empirically.

ON ASYNCHRONOUS CELLULAR AUTOMATA

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map

Sequential dynamical systems and applications to simulations

[\mathfrak{F}_Y,w]=\prod_{i=1}^{k} F_{w_{i}}: K^n\rightarrow K^n

SimDL: a model ontology driven digital library for simulation systems

. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circn), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

A general-purpose graph dynamical system modeling framework

Computer simulations are extensively used for business and science applications. However a simulation generically generates a certain class of dynamical system whose properties are poorly understood. We address some theoretical issues of computer simulations and illustrate our concepts for the simulation of circular one-lane traffic. We propose a certain class of discrete dynamical systems (SDS) that captures key features of computer simulations and then show how SDS techniques can be applied to a case of infrastructure simulations.