# Quantitative Characterization of Complex Systems--An Information Theoretic Approach
## 1 Introduction  
Complex systems are referred to as systems that are composed of many independent system elements playing a key role in the whole system's behavior. Although the independent system elements follow their own logic and behavior, and interact among themselves, such elements define the dynamic behavior observed in a collectively complex system. Based on the principles of complexity science, the complexity of any system can be characterized by understanding the relations, interactions, and behavior of the constituent system elements. Systems can be formalized by two distinctions, disorganized complexity, and organized complexity [1]. Disorganized complexity is a result of many constituent independent system elements. In such a case, the system elements interact among several others, contributing to random system behavior. To understand such a system, statistical and probabilistic methods are usually used. By comparison, organized complexity comprises non-random behavior of the system elements where the number of parts need not be large for the system to be knowledge emergent. Properties of such a system can be understood with the help of simulations and various modeling techniques. It is to be noted that the whole idea of a complex system is still uncertain, based on the fact that it differs in terms of its definition, understanding, and concept between authors according to their perspectives [2]. To provide a few frequently contested definitions [3] of complex systems over the years:  
\"_Complex systems are those with many strongly interdependent variables. This excludes systems with only a few effective variables, the kind we meet in elementary dynamics. Italso excludes systems with many independent variables; we learn how to deal with them in elementary statistical mechanics. Complexity appears where coupling is important but doesn't freeze out most degrees of freedom\" [4]. \"Complex system can be defined as a system with large number of components, often called agents or constituent system elements, that interact, adapt and learn\" [5]. \"Complex systems exhibit several defining characteristics, including feedback, strongly interdependent variables, extreme sensitivity to initial conditions, fractal geometry, and self-organized criticality, multiple metastable states, and a non-gaussian distribution of outputs\" [6]. \"Systems that tend to be robust and at the same time capable of producing large events. They can attain equilibria, both fixed pints and simple patterns, as well as produce long random sequence\" [7]. \"Complex Systems are composed out of interdependent parts and violate the ergodic theorem. They have many degrees of freedom whose time dependence is very slow on a microscopic scale\" [8]. The varying perspectives on complex systems are spearheading the development of several metrics in the literature to measure systems complexity. These include identifying system complexity based on requirements difficulty, stakeholder relationships, and several similar other measures. However, these measures are not exhaustive, considering the difficulty in accurately measuring systems complexity [3]. As an attempt to contribute to the scientific body of knowledge that addresses understanding systems complexity, this paper illustrates a framework to characterize complex systems based on the concept of Information Entropy, using a two-stage examination method. The novelty of the proposed approach lies in establishing a translatable framework across complex system domains to use information entropy. This includes incorporating a well-established and validated statistical foundation that shall be able to assess the relationships between sub-systems/components of a complex system, provide information to understand a system, assess and identify the interaction patterns of components/sub-systems in a system, and be able to differentiate between the input and output information of a system.