Complexity

Volume 2019, Article ID 1303241, 12 pages

https://doi.org/10.1155/2019/1303241

## Mechanism for Measuring System Complexity Applying Sensitivity Analysis

^{1}Experimental and Technological Research and Study Group, Federal Institute of Goias, Goiania, GO, Brazil^{2}School of Electrical, Mechanical and Computer Engineering, Federal University of Goias, Goiania, GO, Brazil^{3}Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada

Correspondence should be addressed to Viviane M. Gomes; rb.ude.gfi@semog.enaiviv

Received 27 December 2018; Revised 24 February 2019; Accepted 11 March 2019; Published 8 April 2019

Academic Editor: Sergey Dashkovskiy

Copyright © 2019 Viviane M. Gomes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work proposes a complexity metric which maps internal connections of the system and its relationship with the environment through the application of sensitivity analysis. The proposed methodology presents (i) system complexity metric, (ii) system sensitivity metric, and (iii) two models as case studies. Based on the system dynamics, the complexity metric maps the internal connections through the states of the system and the metric of sensitivity evaluates the contribution of each parameter to the output variability. The models are simulated in order to quantify the complexity and the sensitivity and to analyze the behavior of the systems leading to the assumption that the system complexity is closely linked to the most sensitive parameters. As findings from results, it may be observed that systems may exhibit high performance as a result of optimized configurations given by their natural complexity.

#### 1. Introduction

The scientific and technological advances of the second half of the twentieth century have generated significant changes in the dynamics of human civilization. The creation of electronic systems and their network structure revolutionized communication systems, modifying the social and economic relations in the world. The systems became more integrated and interdependent, consequently, more complex; since the network structure was not restricted to the computational systems, it is embedded in the human relationships.

Bar-Yam [1] argues that this increase in complexity is directly related to the increasing interdependence of the global economic and social systems, as well as political instabilities. According to Bar-Yam [1], the interdependence is characterized by the network control structure, which considers lateral interactions and transfers decision making to teams due to the high complexity of collective behavior.

The network structure assigns a prominent role to the interactions that, in turn, are responsible for the holistic approach in the study of systems [2–4]. For centuries, scientists tried to explain the whole by its parts, making successive divisions in search of the smallest structure that characterized each system and, ultimately, all systems. According to Bak [5], physicists have been reductionist in considering that the world could be understood in terms of the properties of simple building blocks. Although they have been successful in some cases, the complexity of the systems requires a global analysis instead.

Likewise, in engineering studies, researchers have realized that subdividing systems to analyze them may cause significant losses in the internal structure of the original system [6]. Considering the use of computational tools, it is preferable to carry out the study throughout the system, modeling it in terms of inputs and outputs to simulate its behavior. Another relevant aspect is the number of parameters with their respective variabilities, which may constitute a bottleneck in understanding the systems. In order to reduce the number of variables, several studies use sensitivity analysis for fixing nonessential parameters, since they generate low impact on the output of the system [7, 8].

This holistic approach to systems is based on a philosophical assertion that the whole is more than the sum of the parts. According to Simon [9], in complex systems, this statement means that the properties of the whole cannot be easily inferred from the properties of the parts and their interaction laws. For this reason, complexity has arisen as a unifying feature of our world, regardless of the scale and of the kind of system in analysis [1, 10].

According to Holland [11], the term complexity has assumed such importance that now designates a scientific field with many branches. Some authors consider that the science of the 21st Century is the science of complexity [12]. However, there is no consensus on the quantitative definition of complexity. None of the various measures of complexity is universally accepted by scientists, nor they are practical [13].

Lloyd [14] argues that the measures of complexity are developed to respond to questions about the system with respect to (i) difficulty of description, (ii) difficulty of creation, or (iii) degree of organization. In these categories, the complexity has been approached in different ways, such as entropy [15, 16], statistics [17, 18], fractal dimension [19, 20], algorithmic information content [21, 22], dynamic depth [23], tracking performance [24], and connections [25], among many other forms.

Based on the fractal dimension as measure for self-similar objects, Balaban et al. [26] proposes a metric for quantifying emergence and self-organisation extending fractal dimension to a function, since most of the fractal-like objects have multiple scaling rates. Thus the multifractal analysis investigates the statistical scaling laws of complex fragmented geometrical objects as bacteria aggregates. Balaban et al. [26] observe the evolution of the spatial arrangement of* Enterobacter cloacae* aggregates and apply multifractal analysis to calculate dynamics changes in emergence and self-organisation within the bacterial population. As experimental results, the emergence degree decreases as aggregates populate the plate while the self-organisation degree increases.

Given the relevance of geometrical and computational frameworks, Joosten et al. [27] define the space-time diagrams using small Turing machines with a one-way infinite tape as a computational model and translate these diagrams to fractal dimension. The results from this work have shown that there is a strong relation between the fractal dimension of the Turing machine used and its runtime complexity.

Among the complexity metrics, fractal dimension is frequently applied to the analysis of textures, shapes, and network structures [20, 28, 29]. However, when the detailed system dynamics is available, other metrics may be more effective, such as those based on interactions, for instance, the metrics proposed by Koorehdavoudi and Bogdan [2] for quantifying complexity from spatiotemporal interactions, which estimates the free energy landscape of the states and distinguishes between stable and transition states. This framework was applied to three natural groups: swimming bacteria, flying pigeons, and ants. The analysis has shown that the collective group has had lower energy level and higher degree of complexity at stable states compared to transition ones.

Regarding the connections, the complexity may be measured from the evolution of the system over time, considering the active connections in each state. The connections depend mainly on (i) the transition from one state to another due to the occurrence of events and (ii) the change of the input parameters that lead to variation in the system output [25, 30]. Through sensitivity analysis, the effect of a given input is measured on the output, assessing how the uncertainties in the parameters affect the uncertainty in the system response [31].

Sensitivity analysis is relevant to the study of complexity because certain variables may eventually emerge and have a significant impact on the system. Even if the variables are hidden, the relevance of each one may be defined previously by means of its sensitivity and so anticipate strategies if such variables emerge. According to Holland [11], emergence characterizes complex systems and helps distinguish these systems from others; however this characteristic has no sharp demarcation. To define the system as complex is still a subjective effort.

Here, we focus on the degree of system complexity as a measure that comprises mechanisms related to internal and external interactions of the system. Thus considering (i) the increase in complexity, (ii) the holistic approach to systems, (iii) complexity as a unifying variable, and (iv) the absence of a practical and representative quantitative definition of complexity, we propose a complexity metric based on connections, which may be weighted according to the relevance of each one. This metric is applicable to any system that may be modeled and simulated from its input parameters and output variables.

The proposed metric covers a wide range of systems in the physical world. Using this metric, it is possible (i) to say how complex a particular system is or how much more complex one system is than another, (ii) to use the complexity in the objective function of optimization process, in order to minimize it, or as a constraint, in order not to exceed the value defined as a reference, and (iii) to support decision making.

In order to apply the proposed metrics, Section 2 presents complexity metric developed by adapting the Second Law of Thermodynamics. Our proposed methodology is presented in Section 3, where the connection-based complexity and sensitivity metrics are defined and two systems are modelled as case study. The complexity and sensitivity of different models are analyzed in Section 4, leading to the proposal to include the sensitivity index in the systems complexity metric as a factor of relevance of each connection (Section 5). By including this factor, the complexity metric will consider the descriptive and organizational aspects of the systems, verified by the number of connections and the relevance of each connection, respectively.

#### 2. Metric of System Complexity

Several metrics to calculate the complexity have been developed based on the size of the system, entropy, information, cost, hierarchy, organization, and other criteria [14]. The complexity measures are used to compare systems or different configurations of the same system [13, 32]. In some cases, these measures are dimensionless, allowing to compare one value to another one measured in the same system or in different systems, as long as the nature of them allows comparison [32, 33].

Some complexity metrics are proposed based on the information entropy [32, 34–37]. Considering the system connections, Paiva [33] presents the modeling of Shannon [38] adapted by Lemes [35] to measure the system complexity usingwhere is the complexity of the system connections equivalent to the entropy in information exchange, is the set of connections between elements of the system, is the total number of connections, , is the frequency with which the connections between elements and occur, where is given by , in which is the number of connections between the elements and .

#### 3. Methodology

##### 3.1. Proposed Metric of System Complexity

Based on the Paiva’s [33] modeling, the proposed method measures the complexity of real systems using expression (2). This metric considers the connections regardless of information exchange, observing their probabilities according to expression (3).where is the system complexity based on connections, is the probability of occurrence of the connection between two elements, and is the number of active connections at the instant , expressed by (4). The variables , , and correspond to the number of entities, resources, and queues at the instant , respectively. These variables are components of the system according to the discrete-event modeling, which is applied to the systems investigated in the case studies. In this kind of model, the dynamics of the system is known with respect to the interaction between its components, which enables its modeling in terms of connections.where is the number of entities states, is the number of active connections per entity in each state, and is the number of entities in each state.

In (3), may be generalized to the function , seeing that some constraints may rule out the connection between entities and resources or queues. In this function, corresponds to the constraints, which lead to the decrease in the number of relationship possibilities. The function may assume values in the interval .

Expressions (3) and (4) are useful in the context of discrete event systems, which consist of a class of dynamic systems that depend on the occurrence of events to the state change, i.e., new set of values of the attributes at a given instant [30]. The concept of queue is commonly used in discrete-event modeling, since the entities often need to share the system resources. In these cases, the entities have to wait in queue in order to use certain resources, which provide them some service or something they need [39].

The proposed metric maps the active connections related to entities, resources, and queues at any given time , expressed through the relationship matrix . Figure 1 shows a system configuration with the entities to (in orange) and the resources , , , , , , and (in gray).