**2. Theoretical Background**

As previously mentioned, CT represents a multi-disciplinary, modern approach that studies CAS, following its own specific set of laws, behaviors, and characteristics, such as self-organization, and emergence. In principle, CAS can be considered as open systems consisting of several agents locally interacting in a non-linear manner and forming a unique, organized, and dynamic entity; this entity is capable of adapting to, and evolving within, the environment [16]. In other words, CAS have many features in common with living systems; they adapt and evolve through learning.

As mentioned above, a first important characteristic of CAS is the concept of self-organization. The Austrian biologist Von Bertalanffy [17] seminally coins this term in reference to the growth of organisms over time. Self-organizing reflects the ability of CAS to establish an internal organization through adaptation and evolution, without central control.

Relatedly, emergence is a characteristic showed by CAS, where "the behavior of the whole is much more complex than the behavior of its parts" [18] (p. 12). The peculiarity of emergence is that its nature is not necessarily linked to that of the agents [19]. For example, in PM, it has been conjectured that the complex interactions of various parts of a project can generate a specific behavior of the project itself, which can be explained through systemic analysis.

In order to understand how CAS behave, we need to model them, i.e., identify a set of variables that operationally describe these systems. System theory helps with this operationalization [20–22]. In particular, we can define a state variable of CAS as a measurable element of the systems that describes their conditions in a given moment. The state of CAS at a given time is, thus, the set of values held, at that time, by all their state variables [11]. In this regard, there is no formal rule for choosing the appropriate number and type of state variables; however, we can assume that the greater the complexity of CAS (in terms of number of agents and level of interdependence), the greater the variety in type and number of the state variables [13]. Moreover, state variables are represented in an n-dimension space, where *n* = number of state variables. In this space, each point defines a precise state of the systems (such a state is the state space of CAS). Given a set of state variables, the evolution in time of CAS is a trajectory in its state space [14].

Accordingly, another important characteristic of CAS is that their trajectories in the state space can have three main types of behavior [23]:


The most interesting type of trajectory appears to be the third (i.e., complex regime), since CAS in this regime show their most relevant behaviors. When CAS reach the complex regime, the conditions are set for all of its peculiarities, i.e., self-organization and emergent behavior, respectively, to be present. However, despite the tendency of the trajectory to orbit around its strange attractor, the evolution of CAS is generally unpredictable [11].

To date, CAS may be found in different contexts, such as economics (e.g., a market), sociology (e.g., a human group), biology (e.g., a cell), business (e.g., an organization), or EM (e.g., a NPD process). In this regard, approaching these contexts through the lens of complexity can, appropriately help face uncertainty and unpredictability [24,25]. In particular, complexity can help model the real world through describing its main characteristics, especially when the deterministic approach seemingly unveils its limits. To do so, to date there are many methodological tools available in the scientific arena. Agent-Based Modeling (ABM), for example, allows simulating the actions and interactions of simple agents, and capturing the emergent and usually complex behavior of the system to which they belong [26]. ABM could also generate adaptive-learning models, which assume that agents have non-linear behaviors, generally based on very simple agent rules [27]. Another tool is fuzzy modeling, which helps face the ambiguity of complexity contexts by introducing un-precise values for the selected variables [28,29]. Likewise, stochastic models countervail the inability to accurately measure well-defined parameters, assuming that an optimal representation may be indeed found within a probability distribution of such measures [30]. Finally, a contribution to help understanding and modeling of complex systems can also be provided by the system of the systems approach [31] because of its tendency to pool resources and capabilities from single systems into a more complex entity, which performs more than the sum of the systems taken separately.
