Diversification versus specialization -- lessons from a noise driven linear dynamical system

Specialization and diversification are two major strategies that complex systems might exploit. Given a fixed amount of resources, the question is whether to invest this in elements that respond in a correlated manner to external perturbations, or to build a diversified system with groups of elements that respond in a not necessarily correlated manner. This general dilemma is investigated here using a high dimensional discrete dynamical system subject to an external noise, analyzing the statistical properties of an order parameter that quantifies growth. Our analytical solution suggests that diversification is a good strategy once the system has a fair amount of resources. For systems with small or extremely large supplies, we argue that specialization might be a more successful strategy. We discuss the results also from the perspective of economic and biologic systems. Many physical systems are in a first approximation linear [1]. Well-know examples are electric circuits with simple elements, mechanical strain-stress relations, transport phenomena or kinetics in chemical reactions. These systems often lead to iteratively repeating dynamics. Even in high dimensional cases, the dynamics of linear systems are easy to predict [2]. In case however they are subjected to an external noise, their dynamical response and consecutively their statistical properties might become interesting [3]. The observed processes may be similar to the ones perceived in a variety of other, not necessarily physical systems [4–7]. Here we plan to address the problem in a more general, interdisciplinary context, using a simple, analytically solvable model. Particularly, we are interested in comparing the statistical properties of linear systems composed entirely by positively correlating elements with that of systems that have elements that anti-correlate with each other. We argue that the former one can be viewed as a complex system using a specialization strategy, while the latter one would correspond to a system that exploits the advantage of diversification. We can put now the problem in a more clear interdis-ciplinary context, quoting, for instance, the golden rules of portfolio optimization, " don't pull all your eggs in the same basket ". Investors have been applying the principle of diversification for millennia, the concept being mentioned also in the Bible (Ecclesiastes 11:2). With the emerging of modern portfolio theory different algorithms were developed to assemble a group of diverse assets which satisfies certain predefined criteria with respect to the perceived risks and the expected returns [8– 11]. The topic however is highly debated …