We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Software systems dealing with distributed applications in changing environments normally require human supervision to continue operation in all conditions. These (re-)configuring, troubleshooting, and in general maintenance tasks lead to costly and time-consuming procedures during the operating phase. These problems are primarily due to the open-loop structure often followed in software development. Therefore, there is a high demand for management complexity reduction, management automation, robustness, and achieving all of the desired quality requirements within a reasonable cost and time range during operation. Self-adaptive software is a response to these demands; it is a closed-loop system with a feedback loop aiming to adjust itself to changes during its operation. These changes may stem from the software system's self (internal causes, e.g., failure) or context (external events, e.g., increasing requests from users). Such a system is required to monitor itself and its context, detect significant changes, decide how to react, and act to execute such decisions. These processes depend on adaptation properties (called self-a properties), domain characteristics (context information or models), and preferences of stakeholders. Noting these requirements, it is widely believed that new models and frameworks are needed to design self-adaptive software. This survey article presents a taxonomy, based on concerns of adaptation, that is, how, what, when and where, towards providing a unified view of this emerging area. Moreover, as adaptive systems are encountered in many disciplines, it is imperative to learn from the theories and models developed in these other areas. This survey article presents a landscape of research in self-adaptive software by highlighting relevant disciplines and some prominent research projects. This landscape helps to identify the underlying research gaps and elaborates on the corresponding challenges.

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Self-adaptive software: Landscape and research challenges

This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 1970’s (see [S1], [B], [R2]). Since then much progress has been made on two fronts: there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero Lyapunov exponents ([O], [P1], [Ka], [LY]), and there are detailed analyses of specific kinds of dynamical systems including, for example, billiards, 1-dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY1]). Statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero Lyapunov exponents. The goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than Axiom A. This class will not be defined explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unified way of proving a number of results, some new and some old. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps.

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Statistical properties of dynamical systems with some hyperbolicity

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected

Introduction to the Modern Theory of Dynamical Systems: PRINCIPAL CLASSES OF ASYMPTOTIC TOPOLOGICAL INVARIANTS

We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the form\(S_n = \Sigma _{i = 0}^{n - 1} f(T^i x)\), and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.

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Spectral Theory, Zeta Functions and the Distribution of Periodic Points for Collet-Eckmann Maps

An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors.

Non-uniform hyperbolicity and universal bounds for s-unimodal maps

T I lHE scientific community dealing with earth characteristics, marine and air navigation, and modern concepts of national defense is vitally concerned with longitudinal ties between continents. Expanded interest in modern geodesy has resulted particularly from research programs in developing intercontinental guided missiles and earth-circling satellites. The art of navigation has always been the most practical application of longitude, but in this the art does not require any high degree of precision. Concomitant to the age of exploration and rapid expansion of water-borne traffic was the urgent demand for a means of determining longitude at sea. Beginning with Harrison's chronometer, steady refinements were made until modern times, when continental ties are possible with the aid of telegraph and radio. Aside from navigational considerations, several purely scientific requirements involve the precise determination of longitude with reference to a standard meridian, such as Greenwich. The astronomer requires an accurate position for the time coordination of world-wide astronomical observations; the geophysicist needs longitudes of the highest precision in his studies of the drift of continents; and, finally, the geodesist desires to place all triangulation datums as closely as possible in their proper relation one with the other. Longitudinal studies, no matter how localized in place or how distant in time, have an important bearing on the subject. In celebrating the sesquicentennial of the United States Coast and Geodetic Survey during 1957, it is fitting to recall the Survey's contribution to the establishment of precise longitude in the United States. Early in the history of this bureau the need became apparent for a precise tie between some main-scheme triangulation stations in the United States and the meridians of any and all of the European observatories. The Harvard Observatory at Cambridge was adopted for the point of origin in the United States and in due time was tied to Greenwich by the method discussed in this paper. The second superintendent of the Coast Survey, Alexander Dallas Bache, was well aware of the difficulties other nations had encountered in fixing the

Wild Cantor attractors exist

For unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

Invariant measures exist under a summability condition for unimodal maps.

Let G(n,c/n) and Gr(n) be an n-node sparse random graph and a sparse random r-regular graph, respectively, and let I(n,r) and I(n,c) be the sizes of the largest independent set in G(n,c/n) and Gr(n). The asymptotic value of I(n,c)/n as n → ∞, can be computed using the Karp-Sipser algorithm when c ≤ e. For random cubic graphs, r = 3, it is only known that .432 ≤ lim infn I(n,3)/n ≤ lim supn I(n,3)/n ≤ .4591 with high probability (w.h.p.) as n → ∞, as shown in Frieze and Suen [Random Structures Algorithms 5 (1994), 649–664] and Bollabas [European J Combin 1 (1980), 311–316], respectively. In this paper we assume in addition that the nodes of the graph are equipped with nonnegative weights, independently generated according to some common distribution, and we consider instead the maximum weight of an independent set. Surprisingly, we discover that for certain weight distributions, the limit limn I(n,c)/n can be computed exactly even when c > e, and limn I(n,r)/n can be computed exactly for some r ≥ 1. For example, when the weights are exponentially distributed with parameter 1, limn I(n,2e)/n a .5517, and limn I(n,3)/n a .6077. Our results are established using the recently developed local weak convergence method further reduced to a certain local optimality property exhibited by the models we consider. We extend our results to maximum weight matchings in G(n,c/n) and Gr(n). For the case of exponential distributions, we compute the corresponding limits for every c > 0 and every r ≥ 2. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

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Tomasz Nowicki

Papers

Spectral Theory, Zeta Functions and the Distribution of Periodic Points for Collet-Eckmann Maps

Non-uniform hyperbolicity and universal bounds for s-unimodal maps

Wild Cantor attractors exist

Invariant measures exist under a summability condition for unimodal maps.

Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method