Scale Invariant Pareto Optimality A Meta–Formalism For Characterizing and Modeling Cooperativity in Evolutionary Systems Mark Fleischer Johns Hopkins University Applied Physics Laboratory Laurel, Maryland Mark.Fleischer@jhuapl.edu ABSTRACT This article describes a mathematical framework for characterizing cooperativity in complex systems subject to evolutionary pressures. This framework uses three foundational components that constitute a meta-formalism that can be utilized in a host of research and de- velopment settings to improve the management, control, and under- standing of large numbers of interacting systems such as in com- munication, computer, and sensor networks. A new concept, Scale Invariant Pareto Optimality, provides a mathematical basis for the efficient tradeoffs of efficiency on many scales and the measurement of cooperativity in complex systems. A mathematically oriented definition of self-organized behavior is also described. Discussion and conjectures are offered. Categories and Subject Descriptors I.2.6 [Artificial Intelligence]: Learning—Parameter learning, Con- nectionism and neural nets and; I.2.11 [Artificial Intelligence]: Distributed Artificial Intelligence—Coherence and coordination General Terms Algorithms, Design, Economics, Experimentation, Management, Measurement, Performance, Theory. Keywords Complex Systems, Self-Organization, Self-Organized Criticality, Multi-objective Optimization, Pareto Optima, Swarm Intelligence 1. INTRODUCTION The ubiquity of computers and other forms of advanced technol- ogy have created a number of difficult problems. Many are of a practical nature relating to the management of large ensembles of interacting systems. To be effective, these systems must work to- gether harmoniously, i.e., cooperatively. Unfortunately, their har- monious operation become increasingly difficult to achieve and Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO’05, June 25–29, 2005, Washington, DC, USA. Copyright 2005 ACM 1-59593-010-8/05/0006 ...$5.00. maintain as their size and numbers increase. Indeed, there is a growing consensus among experts that current approaches for man- aging these large systems will be insufficient to handle their in- creased complexity. For example, today’s communications net- works have become enormously complex systems—for the past three decades, this growth has approximately doubled every 18 months according to Moore’s Law [14, p.32]. These new technolo- gies, configurations, protocols, network and computer architectures and so forth are constantly and relentlessly challenging our abilities to effectively manage them. In addition to these practical issues and difficulties are problems of a more theoretical nature—problems stemming from an incom- plete, or incoherent understanding of fundamental phenomena. The complete understanding of complex systems still lies well beyond our grasp, yet our growing dependence on them impels us to con- tinue to explore ideas and increase our understanding. Many different approaches for studying the behavior of complex systems have been described. One approach is based on Swarm Intelligence (SI) and represents the view that it is possible to control and manage large, complex systems of interacting entities with only “minimal”, or stigmergic communications channels, where only a relatively small amount of information is communicated [9]. In recent years, new insights have been obtained based on obser- vations of social insects [6]. Ant colonies and beehives, e.g., seem to conduct their affairs in a very organized and purposeful way that enhances their collective survival. Needless to say, these insects do not have very large brains and their capability to communicate com- plex information for planning and resource allocation seem very limited. Yet, their collective behavior has often been characterized using the terms “intelligent”, “emergent”, and “self-organized” [6]. Their behaviors are also reminiscent of those observed in other do- mains of inquiry such as cellular automata (CA) [17, 18] which have perplexed scientists for many years. A major problem confronting scientists working in these areas is that no widely agreed upon definition of SI or self-organized behav- ior (SOB) exists. How could or should these terms be mathemati- cally defined or characterized? This difficulty is often reflected by the many descriptions of SI and SOB that are couched in terms of “self-organization”. Such circular definitions obscure what is really going on and how complex systems can be more simply character- ized. Moreover, the absence of precise definitions and therefore, theoretical foundations, itself creates problems caused by many dis- parate concepts. Yet, progress is continually being made (see e.g., [5, 4, 12]). Nevertheless, it seems useful to provide some new ideas in the hopes of contributing to a greater synthesis. This article provides some new perspectives for describing fun-