Ergodic properties of multi-agent iterated function systems with subjective probabilities

On the code space and Hutchinson measure for countable iterated function system consisting of cyclic [formula omitted]-contractions

An iterated function system (IFS) is a system of contractive mappings τi:Y→Y, i=1,…,N (finite), where Y is a complete metric space. Every such IFS has a unique (up to scale) equilibrium measure (also called the Hutchinson measure μ), and we study the Hilbert space L2(μ). In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory and from subband filter operators in signal processing. These Cuntz-like operator systems were used in recent papers on wavelet analysis by Baggett, Jorgensen, Merrill, and Packer [Contemp. Math. 345, 11–25 (2004)], where they serve as a first step to generating wavelet bases of Parseval type (alias normalized tight frames), i.e., wavelet bases with redundancy. Similarly, it was shown in work by Dutkay and Jorgensen [Rev. Mat. Iberoam. 22, 131–180 (2006)] that the iterative operator approach works well for generating wavelets on fractals from IFSs without overlap. But so far the more general and more difficult case of essential overlap has resisted previous attempts at a harmonic analysis and explicit basis constructions, in particular. The operators generating the appropriate Cuntz relations are composition operators, e.g., Fi:f→f∘τi, where (τi) is the given IFS. If the particular IFS is essentially nonoverlapping, it is relatively easy to compute the adjoint operators Si=Fi*, and the Si operators will be isometries in L2(μ) with orthogonal ranges. For the case of essential overlap, we can use the extra terms entering in the computation of the operators Fi* as a “measure” of the essential overlap for the particular IFS we study. Here the adjoint operators Fi* refer to the Hilbert space L2(μ), where μ is the equilibrium measure μ for the given IFS (τi).

Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.

The challenge we address in this work is the control of complex systems based on the multi-agent paradigm. Our focus is on identifying what specific questions arise when using the multi-agent paradigm in the control of complex systems context. We present here a solution under the form of an equation-free control architecture based on multi-agent model simulation.The results of the implementation on a free-riding problem in p2p networks demonstrate that the proposed architecture can control such a network. Our contribution is to identify key questions that rise when using the multi-agent paradigm in the context of control of complex systems, concerning the relationship between the model entities and the target system entities.

We study contraction conditions for an iterated function system (IFS) of continuous maps on a metric space which are chosen randomly, identically and independently. We investigate metric changes, preserving the topological structure of the space, which turn the IFS into one which is contracting on average. For the particular case of a system of C 1-diffeomorphisms of the circle which is proximal and does not have a probability measure simultaneously invariant by every map, we derive an equivalent metric which contracts on average and conclude that it carries a unique stationary probability measure. Here we strongly use local contraction properties established by Malicet in 2017, which then imply that the IFS has a negative random Lyapunov exponent and generalizes a result by Baxendale.

ABSTRACTWe consider iterated function systems (IFSs) acting on a phase space X that, whilst not necessarily uniformly contracting, do satisfy a ‘contraction on average’ condition. We introduce the notion of a coupled IFS acting on a new phase space formed by taking infinite (when X is compact) or finite (when X is not compact) products, in analogy with coupled map lattices. For appropriate couplings, we prove the existence of a unique invariant probability measure for the coupled system and show that it depends continuously on the coupling as the coupling tends to zero. We also prove an ergodic theorem and a central limit theorem for the coupled IFS. The methodology is to introduce a family of transfer operators acting quasi-compactly on an appropriate function space and use results of Keller and Liverani [Stability of the spectrum for transfer operators, Ann. Sc. Norm. Super. Pisa 28 (1999), pp. 141–152] to prove continuity of their spectral properties in the perturbation.

A multi-agent system is presented here. Agent decisions are based on economic partial utilities. It’s a new computational tool using cellular and agent-based models. Decision processes are represented through computational agents. Land-use and land cover changes are represented through a cellular model. It belongs to kind of models called Multi-Agent Systems model of Land-Use Cover Change. Decisions concern the estate development in the neighborhood of the agents. Space is discretized in heterogeneous cells where heterogeneous agents interact together (directly or via space). Based on partial utility probabilities, several stochastic simulations are achieved. These simulations are then summarized, replicated and a parameter sensitivity analysis provided. The main parameters influencing the decisions of an agent depend on his type (ecologist, etc.) and on the expected economic activities in his neighborhood. Little information is assumed to be known about the mechanisms determining the parameter values guiding the decisions of the agents, except their type. The whole system constitutes a general virtual model, mostly independent from initial conditions and real data. Sensitivity analysis are used to explore behavioral paths and influencing parameters. In summary, the main contribution of this paper consists in: (i) defining a simulation-based framework for modeling the decisions of many heterogeneous economic agents involved in spatial interactions; (ii) capturing the decision process using general utility-based equations; (iii) providing a sensitivity analysis of initial parameter values. An example of possible application investigating strategies of various categories of agents (institutions, landowners, homeowners, etc.) facing an intense residential development is provided. The framework consists of abstracting institutional characteristics of actual local Urbanism Plan procedures in France. This model constitutes a first step to be further specified. Copyright S (This abstract was borrowed from another version of this item.)

A multi-agent system is presented here. Agent decisions are based on economic partial utilities. It's a new computational tool using cellular and agent-based models. Decision processes are represented through computational agents. Land-use and land cover changes are represented through a cellular model. It belongs to kind of models called Multi-Agent Systems model of Land-Use Cover Change. Decisions concern the estate development in the neighborhood of the agents. Space is discretized in heterogeneous cells where heterogeneous agents interact together (directly or via space). Based on partial utility probabilities, several stochastic simulations are achieved. These simulations are then summarized, replicated and a parameter sensitivity analysis provided. The main parameters influencing the decisions of an agent depend on his type (ecologist, etc.) and on the expected economic activities in his neighborhood. Little information is assumed to be known about the mechanisms determining the parameter values guiding the decisions of the agents, except their type. The whole system constitutes a general virtual model, mostly independent from initial conditions and real data. Sensitivity analysis are used to explore behavioral paths and influencing parameters. In summary, the main contribution of this paper consists in: (i) defining a simulation-based framework for modeling the decisions of many heterogeneous economic agents involved in spatial interactions; (ii) capturing the decision process using general utility-based equations; (iii) providing a sensitivity analysis of initial parameter values. An example of possible application investigating strategies of various categories of agents (institutions, landowners, homeowners, etc.) facing an intense residential development is provided. The framework consists of abstracting institutional characteristics of actual local Urbanism Plan procedures in France. This model constitutes a first step to be further specified.

We consider the iterated function system (IFS) f q → ( z → ) = z → + q → β , q → ∈ { ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) } . \begin{equation*} f_{\vec {q}}(\vec {z})=\frac {\vec {z}+\vec {q}}{\beta },\,\,\vec {q}\in \{(0,0),(1,0),(0,1)\}. \end{equation*} As is well known, for β = 2 \beta = 2 the attractor, S β S_\beta , is a fractal called the Sierpiński gasket (or sieve) and for β > 2 \beta >2 it is also a fractal. Our goal is to study random β \beta -transformations on the attractor for this IFS with 1 > β ≤ 3 / 2 1>\beta \leq 3/2 . In this case, S β S_\beta is a triangle. We show that all β \beta -expansions of a point z → \vec {z} in S β S_\beta can be generated by a random map K β K_\beta defined on { 0 , 1 } N × { 0 , 1 , 2 } N × S β \{0,1\}^\mathbb {N}\times \{0,1,2\}^\mathbb {N}\times S_\beta and K β K_\beta has a unique invariant measure of maximal entropy. Furthermore, we show the existence of a K β K_\beta -invariant probability measure of the form m 1 ⊗ m 2 ⊗ μ β m_1\otimes m_2 \otimes \mu _{\beta } , where m 1 , m 2 m_1, m_2 are product measures on { 0 , 1 } N , { 0 , 1 , 2 } N \{0,1\}^\mathbb {N},\{0,1,2\}^\mathbb {N} , respectively, and μ β \mu _{\beta } is absolutely continuous with respect to the two-dimensional Lebesgue measure λ 2 \lambda _2 .

Control of complex systems can be achieved via hierarchical multilayered agentbased structures benefiting from their inherent properties such as modularity, scalability, adaptability, flexibility and robustness. The agent-based structures consist of a number of simpler subsystems (or agents), each of which addresses in a coordinated manner a specific sub-objective or sub-task so as to attain the overall design objectives. The complexity of the behavior of such systems arises as a result of interactions between multiple agents and the environment in which they operate. More specifically, multi-agent control systems are fundamental parts of a wide range of safety-critical engineering systems, and are commonly found in aerospace, traffic control, chemical processes, power generation and distribution, flexible manufacturing, robotic system design and self-assembly structures. A multi-agent system can be considered as a loosely coupled network of problem-solver entities that work together to find answers to problems that are beyond the individual capabilities or knowledge of each entity, where local control law has to satisfy decentralized information structure constraints (see, e.g., [20]), and where no global system control (or supervision) is desired.

We give a systematic account of iterated function systems (IFS) of weak contractions of different types (Browder, Rakotch, topological). We show that the existence of attractors and asymptotically stable invariant measures, and the validity of the random iteration algorithm (‘chaos game’), can be obtained rather easily for weakly contractive systems. We show that the class of attractors of weakly contractive IFSs is essentially wider than the class of classical IFSs' fractals. On the other hand, we show that, in reasonable spaces, a typical compact set is not an attractor of any weakly contractive IFS. We explore the possibilities and restrictions to break the contractivity barrier by employing several tools from fixed point theory: geometry of balls, average contractions, remetrization technique, ordered sets, and measures of noncompactness. From these considerations it follows that while the existence of invariant sets and invariant measures can be assured rather easily for general iterated function systems under mild conditions, to establish the existence of attractors and unique invariant measures is a substantially more difficult problem. This explains the central role of contractive systems in the theory of IFSs.

Iterated Function Systems with place-dependent probabilities are considered. Fascinating geometrical invariants, that apply even when there is no unique invariant measure, are presented. Furthermore, it is shown that the invariant measure of a stationary stochastic process, when it contains no atoms and is fully supported, can sometimes be associated with an IFS with probabilities, and with an associated dynamical system. This leads to the idea of an ergodic transform of a string of symbols, with respect to a given string; this is introduced and shown to be useful; it provides a unifying geometrical approach to the description of data compression algorithms such as Huffman, arithmetic, and the Burrows-Wheeler transform.KeywordsInvariant MeasureIterate Function SystemCompression FunctionStationary Stochastic ProcessIntuitive PictureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A computational model for metric spaces

Given a random dynamical system of affine mappings of ℝd whose linear part is hyperbolic, we prove that it has a unique invariant measure. The proof uses multiplicative ergodic theory. The result generalizes and sheds new light on results of Barnsley and Elton on Iterated Function Systems.KeywordsLyapunov ExponentInvariant MeasureTime ReversibilityIterate Function SystemMarkov OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The ultimate navigation efficiency of mobile robots in human environments will depend on how we will appraise them: merely as impersonal machines or as human-like agents. In the latter case, an agent may take advantage of the cooperative collision avoidance, given that it possesses recursive cognition, i.e., the agent's decisions depend on the decisions made by humans that in turn depend on the agent's decisions. To deal with this high-level cognitive skill, we propose a neural network architecture implementing Prediction-for-CompAction paradigm. The network predicts possible human-agent collisions and compacts the time dimension by projecting a given dynamic situation into a static map. Thereby emerging compact cognitive map can be readily used as a "dynamic GPS" for planning actions or mental evaluation of the convenience of cooperation in a given context. We provide numerical evidence that cooperation yields additional room for more efficient navigation in cluttered pedestrian flows, and the agent can choose path to the target significantly shorter than a robot treated by humans as a functional machine. Moreover, the navigation safety, i.e., the chances to avoid accidental collisions, increases under cooperation. Remarkably, these benefits yield no additional load to the mean society effort. Thus, the proposed strategy is socially compliant, and the humanoid agent can behave as "one of us."