Complexity Assessment in Projects Using Small-World Networks for Risk Factor Reduction

Abstract

Despite following standard practices of well-known project management methodologies, some projects fail to achieve expected results, incurring unexplained cost overruns or delays. These problems occur regardless of the type of project, the environment, or the project manager’s experience and are characteristic of complex projects. Such projects require special control using a multidimensional network approach that includes contractual aspects, supply and resource considerations, and information exchange between stakeholders. By modelling project elements as nodes and their interrelations as links within a network, we can analyze how components evolve and influence each other, a phenomenon known as coevolution. This network analysis allows us to observe not only the evolution of individual nodes but also the impact of their interrelations on the overall dynamics of the project. Two metrics are proposed to address the inherent complexity of these projects: one to assess Structural Complexity (SC) and the other to measure Dynamic Complexity (DC). These metrics are based on Boonstra and Reezigt’s studies on the dimensions and domains of complex projects. These two metrics have been combined to create a Global Complexity Index (GCI) for measuring project complexity under uncertainty using fuzzy logic. These concepts are applied to a case of study, the construction of a wastewater treatment plant, a complex project due to the intense interrelations, the integration of new technologies that require R&D, and its location next to a natural park. The application of the GCI allows constant monitoring of dynamic complexity, thus providing a tool for risk anticipation and decision support. Also, the integration of fuzzy logic in the model facilitates the incorporation of imprecise or partially defined information. It makes it possible to deal efficiently with the dynamic variation of complexity parameters in the project, adapting to the inherent uncertainties of the environment.

1. Introduction

Modern projects face increasing levels of complexity, which directly influences their performance results [1]. This complexity comes not only from external factors such as general uncertainty, but also from internal aspects such as cost and duration, team size and dynamics, flexibility and urgency of deliverables, complexity in information technologies, volatility in requirements, number of stakeholders, organizational level, and associated risks, constraints and dependencies [2].

Traditional project management, based on predictability and control through methods such as the waterfall approach, is effective when the objectives and resources remain stable [3]. However, the nature of projects has evolved, especially in the technology sector and volatile markets, presenting a high uncertainty that challenges conventional methods [4]. Therefore, it is imperative to establish stronger relationships with stakeholders and emphasize risk management, implementing new tools, life cycles, and approaches that allow greater adaptability in both project management and final deliverable [5].

In highly dynamic environments, statically treating projects is inefficient [6]. Traditional approaches need to be reconsidered, as conventional or predictive methods can be insufficient and lead to additional costs and planning deviations due to unrealistic initial approaches [7]. The uncertainty inherent in complex projects makes it difficult to predict all risks from the outset, making methodologies that effectively manage such complexity necessary [8].

To address projects with high uncertainty, there are approaches such as iterative, incremental, agile, and hybrid life cycles, which facilitate adaptation to changes and risk mitigation [9]. Determining the degree of complexity of the project is crucial to select the most appropriate approach, whether predictive, adaptive or hybrid [10].

Adaptation and resilience become essential aspects. Resilience refers to the team’s ability to absorb impacts and move toward objectives despite adversity [11], while adaptation involves adjusting processes, resources, and goals as the environment evolves [12].

In complex contexts, project success is not only measured by meeting initial deadlines and budgets but by achieving key objectives, even if these are redefined during development [13]. This calls for a redefinition of the concept of success, focusing on value creation and meeting changing stakeholder needs [14].

Complexity analysis is fundamental; understanding and managing the multiple dimensions of complexity is key to improving project performance and increasing the probability of success [15]. Early identification of the factors that contribute to complexity allows designing more effective strategies [16]. Tools such as complexity ranking, uncertainty matrices and risk mapping are indispensable [17].

Network analysis offers a valuable approach by focusing on the interrelationships and dependencies between components and stakeholders, revealing hidden patterns that can influence project progress [18]. By modeling project elements as nodes and their interactions as links in a network, it is possible to analyze how they evolve and affect each other, a phenomenon known as coevolution.

Although this analysis requires deep understanding and significant resources, which can complicate its implementation [19], its value in high-uncertainty environments is undeniable. Effective communication and collaboration between teams and stakeholders are critical, demanding multidisciplinary coordination and a collaborative approach that promotes transparency and joint problem solving [20]. Digital platforms and collaboration tools facilitate this exchange [21].

Technology plays an increasingly prominent role. Advanced simulation and data analysis tools support decision-making and anticipate problems [22]. Their adoption requires a well-defined strategy and adequate training of the team [23].

The management of complex projects demands the integration of knowledge from various disciplines. Interdisciplinarity is key to address complexity in a comprehensive manner [24].

In summary, complex projects pose challenges that require an adaptive approach and continuous analysis. The adoption of flexible methodologies, the use of advanced tools, and the promotion of a collaborative and agile culture are fundamental to achieve success in uncertain environments [25].

The objective of this research is to develop a versatile model to measure the complexity of a project, applicable both before and during its execution, thus contributing to a more effective and adaptive management. The rest of the article is structured as follows. In Section 2 an introduction about complex systems and their analysis by means of networks is performed, highlighting the importance of observing the interrelationships between components in systems that evolve jointly.

The method section is structured in two parts. First, the methodologies and tools used to analyze networks in the context of complex project management are presented and secondly the proposed method for measuring the complexity of a project is described.

In the Results section, the developed model is applied to a case study, the construction of the “Marismas del Odiel” WWTP. The contractual, supply and stakeholder information networks are detailed, and the structural and dynamic complexity metrics are analyzed, thus evaluating the complexity of the project. Finally, Section 5 shows the conclusions of the research.

2. Complex Systems and Networks

A system is defined as a combination of resources (human, material, information, etc.) that are integrated to perform a specific function and satisfy a defined need [26]. To better understand how a system works, it can be divided into basic units. By analyzing these units and their characteristics, it is possible to understand the overall functionality of the system to which they belong.

This approach is known as systems analysis, a technique that makes it possible to study both the individual elements that make up the system and the interrelationships between them. By identifying and evaluating these interactions, a clearer picture can be obtained of how the various components work together to achieve the overall system objective. Systems analysis is critical to understanding the underlying complexity and ensuring the smooth functioning of the whole, especially in large or multifaceted systems [27].

A complex system contains more information than the sum of its parts, since unexpected interactions between its components generate new information dynamically, both internally and externally. This characteristic implies that analytical precision applied to conventional systems is not sufficient to fully understand a complex system. Instead of a purely macroscopic analytical view, it is necessary to adopt a probabilistic perspective that links the microscopic and macroscopic behaviors of the system. Thus, classical mechanics is replaced by stochastic mechanics [28].

Complex systems are often controlled, but almost always out of equilibrium, which makes their analytical treatment difficult [29]. This non-collapse disequilibrium is explained by the concept of self-organized criticality, which allows us to understand essential elements of statistical mechanics, such as the presence of power laws [30].

A project is defined as a combination of resources-human, material, information, etc.-that are integrated to perform a specific function and satisfy a defined need, and its analytical analysis is possible [31]. Complex projects can be conceived as complex systems, as they present an intricate and dynamic structure of components and interactions that are constantly evolving and constrained by a specific time frame. Because of these characteristics, managing complex projects involves understanding and managing both the structure and emergent behavior of these systems. Therefore, in the remainder of this article, the terms ‘complex system’ and ‘complex project’ will be used interchangeably, emphasizing their interdependent nature [32].

A complex project may not be complex from the outset but become complex due to the presence of unaccounted for or newly emerging activities with uncertainty, ambiguity, volatility and complexity, resulting in new and difficulty control processes. This may be a consequence of a complex development environment or a result that is complex [33].

2.1. Complex Systems Such as Multi-Networks

Understanding how the various components of a system interact is essential for quantitative and predictive analysis. In complex systems, these interactions are rarely uniform, isotropic or consistent, as they can vary significantly between elements. Networks provide an effective tool for mapping who interacts with whom, the intensity of these interactions, when they occur, and in what manner they unfold [34]. In addition, networks are particularly useful for representing structures, flows, and data, since any type of information stored in a relational database can be visualized as a network [35].

It is interesting to be able to further analyze the various interrelationships that may occur between elements, and for this purpose, multilayer networks allow different types of relationships to be represented in separate layers, which facilitates the analysis of how their interactions affect the behavior of the system [36].

Likewise, multilayer networks make it possible to observe the emergence of new complex properties that would not be easy to observe if the system were represented as a simple network [37]. For example: the resilience of a certain biological property of bacteria to increase oxygen in an aerobic reactor in a sewage treatment plant, the way in which information is propagated when there are outages in communication networks, or crisis management in a social conflict.

In all these examples, new properties can emerge that can be better understood through the interaction of multiple layers. In addition, multilayer networks are more flexible and can adapt to the dynamic nature of complex systems, where interactions and states can change over time. This allows modeling the evolution of systems more accurately [38].

Thurner, Hanel and Klimek thus formalize that complex systems are multilayer networks in coevolution, where the states change as a function of the interaction network and, simultaneously, the interactions change as a function of the states [39].

Figure 1 shows two schematic representations of the same multilayer network. The representation on the left shows an acyclic directed network with different types of nodes, each one representing a specific property. On the right we can see the same network seen in perspective where we observe that the network has three layers with interrelationships between them.

Among all the properties that can be defined about a multilayer network, it is interesting to highlight two: the strength of the connection between certain nodes and small-world networks. These concepts are closely related to centrality and strongly connected components, since the centrality of a node reflects its influence on the network, while strongly connected components identify subgroups within the network that maintain significant internal cohesion. Both concepts will be key later in explaining the proposed method, as they allow characterizing the structure and resilience of the network, as well as the communication and influence pathways within the system.

A directed network is considered strongly connected when there is a path in both directions between each pair of nodes. Larger sub-networks that maintain this strong connection are known as strongly connected components (SCC). Nodes that have at least one connection to any node within the SCC are called internal components. On the other hand, nodes that are external to the SCC but can be reached via a path that starts within the SCC are called outbound components.

This type of network is commonly used in studies related to the study of cascading failures of critical systems, project scheduling and risk management, such as the “Bow-Tie” model of risk analysis. The maximum strongly connected sub-networks are called Strongly Connected Components (SCC). Nodes with connection to the SCC are called internal components, while nodes external to the SCC, but reachable from it are called output components, as illustrated in Figure 2 [40].

Small-world networks, introduced by Watts and Strogatz in 1998 [41]. represent a type of network structure in which there is a high degree of clustering (nodes tend to form cliques or groups) and a short average distance between nodes. This means that, although elements are highly connected within small subgroups, a relatively low number of random connections allows these groups to communicate efficiently with each other. This characteristic makes small-world networks especially useful for modeling systems with local and global interaction, a common structure in a variety of applications, from neural networks to social networks, an example of which can be seen in Figure 3.

In project management, small-world networks can have an interesting parallel. Although projects may appear to be small networks due to the limited number of resources, tasks and teams involved, these networks also exhibit small-world characteristics. For example, project team members are often highly connected to each other within their respective roles and disciplines (local clustering), while project leaders or coordinators may act as key nodes connecting different subgroups (long-distance connections) [42].

Another relevant feature of small-world networks is their resilience. Despite the removal of some nodes or connections, the network continues to function effectively. This can be applied to project management when trying to adapt to unexpected changes or loss of resources, which is common in complex and uncertain environments. Projects that are organized as small-world networks are, in theory, more resilient to disruptions, as key connections maintain the cohesion of the entire system [43].

Small-world networks in complex systems enable high local connectivity and efficient global transmission, facilitating adaptability and resilience in the interaction of their components. This optimizes information flow in highly interconnected systems.

2.2. Coevolution in Complex Systems

Evolution in complex systems is a dynamic process that involves the change in the composition of elements or entities over time. These elements can be biological, such as species, or technological, such as projects. Evolution is algorithmic and not based on equilibrium, as these systems are constantly driven by internal and external interactions, resulting in the creation and destruction of new entities. Evolutionary events are often disruptive, generating variability in species or elements and causing changes in the diversity of the system. Periods of stability are often interrupted by “peaks” of change, which generate sufficient potential energy to realize the next evolution [44].

Coevolution in complex systems refers to the process in which the elements of a system not only evolve individually, but also influence each other’s evolution. In this context, both the states of the entities (species, products, ideas, etc.) and their interactions evolve simultaneously. That is to say, the interactions between the elements change according to the states of these, and vice versa. This process is dynamic and open, which implies that it is not in equilibrium and that the interactions between entities can generate new combinations or emerging behaviors [45].

The evolution of the state of the system through the interaction between entities can be formally defined by the following equation:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{dt}}}{\mathsf{\sigma }}_{\mathrm{i}}\left(\mathrm{t}\right)\text{~}\mathrm{F}\left({\mathrm{M}}_{\mathrm{ijk}..}^{\mathsf{\alpha }}\left(\mathrm{t}\right),{\mathsf{\sigma }}_{\mathrm{j}}\left(\mathrm{t}\right)\right)$$

(1)

It indicates that the change in the state of an entity σ_i in the next time step depends on the states of other nodes σ_j in the current time step and on the interaction matrix M_ijk between the analyzed element and the elements interacting with it. Interactions can be of any type depending on the nature of the entity: physical; chemical; economic; power interactions, etc. α expresses the intensity of interaction between the elements involved. The function F, the engine of change, can be deterministic or stochastic.

The second equation of the coevolutionary dynamics specifies how the interactions evolve, through its matrix M in time by means of function G, the engine of that change, which can also be deterministic or stochastic:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{dt}}}{\mathrm{M}}_{\mathrm{ij}}^{\mathsf{\alpha }}\left(\mathrm{t}\right)\text{~}\mathrm{G}\left({\mathrm{M}}_{\mathrm{ijk}..}^{\mathsf{\alpha }}\left(\mathrm{t}\right),{\mathsf{\sigma }}_{\mathrm{j}}\left(\mathrm{t}\right)\right)$$

(2)

These equations show that both states and interactions are mutually updated. Since it is not possible to analyze them analytically, the use of algorithms is required for their study [39].

Applied to complex projects, these equations illustrate how activities and their interdependence evolve simultaneously, affecting the overall behavior of the project. Coevolution in complex systems highlights the need for adaptive and flexible approaches to project management, since changes in one part of the system can have significant effects on other parts.

3. Method

3.1. Metrics on Project Complexity

Pryke stated that Social Network Analysis (SNA) tools can be adapted to study governance in projects, focusing on two key properties: density and centrality. These properties allow for the analysis of important aspects of projects, such as the role of stakeholders and the types of contracts that are established in human activities [46].

Unlike other small-world networks, studies applying SNA to projects, conducted by various authors such as Carrington et al. [47] and Kenis and Oerlemans [48], agree that networks within projects are temporary, non-repetitive and have objectives clearly defined by stakeholders through requirements. Nowadays, projects are not executed by a single company but involve coordination between several companies of different types. These interrelationships can be of a formal nature (through contracts) or informal (verbal agreements or positionings), strong (relationships in a specific phase of the project) or weak (for example, the partnership facing a work affecting a pipe in a street), frequent or infrequent, and can be based on emotional aspects (projects with a social component) or pragmatic (a work without social relevance). Thus, projects generate networks of relationships among their participants, which can present both opportunities and threats.

Borgatti y Halgin [49] mention two network research models to address governance from a structural perspective: the flow model and the coordination model.

The flow model considers how the structure of the network influences a given flow, which allows studying the positions of the actors within the network. This model evaluates three key structural properties:

• Degree centrality, which measures how many connections a node has with other nodes, highlighting those nodes that have high visibility and control over other actors in the periphery. An interesting extension study of this concept is in the strategy-structure term that suggests that performance will be higher when the strategy and structure of the firm are consistent with the strengths of the project [50].
• Closeness centrality, which indicates which nodes are the most efficient for receiving and transmitting information quickly. On this aspect there is a study that delves into the problems of stopping the closeness centrality since many networks are directed and weighted. It is proposed as a solution to use the effective distance versus geometric distance through Dijkstra’s algorithm [51].
• Intermediation centrality evaluates the importance of a node in a network based on how many times this node acts as a bridge in the shortest paths between other nodes. The main problem with this centrality is its calculation since it requires complicated algorithms [52].

Figure 4 shows the three types of centralities defined.

• Density, which is the ratio of existing connections to the maximum possible number of links and reflects the level of connectivity of the network. For large projects, such as international development projects, United Nations peacekeeping missions, etc., calculating density and associating it with centralities is complicated. There are algorithms that allow us to calculate it by successive approximations based on centroid calculations [53].

The complexity of a project is intrinsically related to the centralization and density of its network. Provan and Milward argue that networks with many actors and interdependencies require high levels of coordination, which increases their complexity [54]. Kim, Yan and Dooley point out that larger network size, combined with high levels of centrality and density, is associated with higher complexity [55]. Finally, authors such as Choi and Hong point out that high centralization in the information network can generate weak interactions between the center and the periphery, which could lead to the formation of consortia between peripheral actors. This tends to slow down operations and decision making, which increases costs and the risk of failure, and consequently, the complexity of the project [56].

The coordination model, on the other hand, employs Beta or Bonacich centrality, which measures the influence of a node in a network not only as a function of the degree of connection but also as a function of connections with influential nodes. This type of centrality adjusts the centrality weight according to whether the node is connected to nodes of high or low centrality, which allows a more focused view on the dynamics of influence in networks [57].

Adami and Verschoose propose a structural framework for analyzing complex projects using a multi-network that operates at three levels [58]:

• Supply network, which controls the coordination and supervision of goods and services, in addition to distributing power and authority among the actors. A high level of centralization in this network implies greater control and operational burden for suppliers, while centrality at the actor level helps to identify key buyers and suppliers. Companies with high centrality usually assume the role of integrators, organizing goods and resources for the outcome of the project.
• Contractual network, which manages the formal involvement of firms through contracts, and regulates the formal relationships between them, including supplier changes. A high degree of centralization in this network can generate weak interactions between central and peripheral companies, and disconnection between different levels of the supply chain. A high degree of connections in this network is usually associated with greater complexity.
• Stakeholders’ information network, which deals with informal power related to information and contributes to effective governance. Centralization in this network can restrict the flow of information between central and peripheral nodes, leading to delays in problem solving and the formation of resistant nodes that negatively affect project operations.

The analysis of complex networks in projects requires the use of software tools, such as UCINET 6.790 [59], which facilitates the graphical and analytical analysis of network structures. UCINET offers a comprehensive analysis of social networks, allowing the study of parameters such as centrality, identification of subgroups and analysis of roles within the network. These metrics help to measure the complexity of a project, analyzing both its static structure and its co-evolutionary dynamics, allowing a better understanding and management of complex construction projects.

Determining whether a system is complex is challenging due to its constantly changing nature [60]. Even so, it is essential to have metrics to measure such complexity to better understand its structure. The first step is to define what type of network will represent the complex projects under study, as this will influence the identification of the elements that make up the network. For example, random networks could self-generate, completely changing their structure and the interrelationships between nodes.

Within these, small-world networks are particularly relevant. These networks combine a low average characteristic length (like random networks) with a high level of clustering (like regular networks), making them a common model for phenomena as diverse as social networks, electrical networks, technical collaborations, and business partnerships [61,62].

Regarding the structural analysis of networks representing complex systems, several interesting metrics are proposed. For example, a key metric is information flow, which is often one of the main complexity factors in projects [33]. Although these metrics may seem intuitive, they require specific calculations and in-depth knowledge of the system and its representation in the network.

Summers and Shah propose a toolkit that analyses the structure of systems from three dimensions: size, coupling and solvability. Size encompasses the objectives, problems and processes that connect the two. Coupling defines the level of interrelationship within the network, while solubility measures how aligned the system variables are to achieve the stated objectives. These metrics provide a holistic view of the system structure, although they require an in-depth knowledge of networks [63].

There are other metrics to measure the complexity of the structure in a system that can be found in [64], but the application of these tools in early stages can be complex and difficult to manage. Therefore, there is a need to develop faster and more efficient metrics that do not require an exhaustive analysis of the network or multi-networks of a project created.

A metric that is easy to apply is the one proposed by Lassen and Aalst in their study [65]. In this work, they suggest that the structural complexity of systems can be decomposed into three parameters, like those mentioned by Summers and Shah: size, connections, and solvability. These are inspired by McCabe’s cyclomatic complexity described in [66] but focus only on the number of strongly connected components (SCC). They call this variation Extended Cyclomatic Complexity (ECC), which allows a more detailed analysis (Equation (3)) of the structure of complex systems.

$$\mathrm{ECC}\left(\mathrm{PN}\right)=\left|\mathrm{E}\right|-\left|\mathrm{N}\right|+\mathrm{p}$$

(3)

where PN is a Petri net [67] with three fundamental parameters: P is the number of nodes; T is the number of states or transitions, and F is the number of network interrelationships. E is the number of links, N is the number of nodes, and p is the number of SCC. The following example is proposed, see Figure 5, where the cyclomatic complexity of a supply network consisting of 10 nodes with 12 links and 4 strongly connected components, in orange, can be established.

Applying the cyclomatic complexity formula, its value is obtained:

$$\mathrm{ECC}\left(\mathrm{PN}\right)=\left|\mathrm{E}\right|-\left|\mathrm{N}\right|+\mathrm{p}=12-10+4=6$$

(4)

According to McCabe, this value is considered to correspond to a simple system. If this formula were applied to a system with more links, for example, 30 links, in a network with the same number of nodes: 10 and the same number of connected components: 4, the cyclomatic complexity defined in Equation (3) would be:

$$\mathrm{ECC}\left(\mathrm{PN}\right)=\left|\mathrm{E}\right|-\left|\mathrm{N}\right|+\mathrm{p}=30-10+4=24$$

(5)

Although cyclomatic complexity is effective for measuring complexity in small-world networks, it does so by considering that state transitions-key elements for understanding movement within the network-are subject to numerous initial constraints, such as the elimination of mutually beneficial links due to a lack of suitable conditions. If any of these constraints are removed, the complexity increases significantly. However, a major problem with this metric is that it does not capture changes in interactions caused by coevolution. For this reason, structural metrics must be complemented with other metrics that allow these evolutionary dynamics to be incorporated into the complexity analysis.

When in a complex network both state variables and interactions have a dynamic behaviour, it is crucial to analyse how both elements evolve over time. In this context, Íñiguez and Barrio propose a simple metric to evaluate the coevolution before going into its detailed evolution, since performing a previous analysis could be complicated [67]. To approach this analysis, they establish two temporal parameters: the micro dynamics and the macro dynamics of the system. The micro dynamics describes the rapid changes in the network interactions, limited to a specific set of nodal variables, and is denoted as “dt”. On the other hand, macro dynamics refers to the evolution of the whole system, which requires much more time, symbolized as “dT”.

Coevolution, then, is measured through the parameter g, which is defined as the ratio between the times of macro dynamics and micro dynamics:

$$\mathrm{g}={\displaystyle \frac{\mathrm{dT}}{\mathrm{dt}}}$$

(6)

Depending on the value of g, three different possible scenarios are identified. When g → 0, no coevolution is observed, and the complexity of the system is explained solely by its structure. If g → ∞, the network ceases to exist in topological terms and only a constant evolutionary function remains. Between these two extremes, the interaction between structure and coevolution generates similar emergent properties over time, resulting in a heterogeneous network with the emergence of communities, of small-world networks.

To simplify the calculation of g without the need for a complex network analysis, Íñiguez and Barrio [68] propose the following formula:

$$\mathrm{g}={\displaystyle \frac{〈\mathrm{k}〉}{\left[\mathrm{N}〈\mathrm{C}〉\right]}}$$

(7)

where:

• <k> is the average degree of the network. In small-world networks it can be considered as the highest order degree of the network.
• N is the number of nodes, and
• <C> represents the average clustering coefficient. In small-world networks, with a minimum clustering value around 0.75, it is observed that as N grows, g slowly tends to 0, as shown in Figure 6.

This behavior suggests that, in networks with few nodes, coevolution can lead to faster and possibly much more polarized opinion dynamics, influenced by time dt and limited connectivity between nodes.

The plot shows noticeable fluctuations in the degree of coevolution g when the number of nodes is low, indicating great variability in the structure and interactions of the network in its early stages of growth. These fluctuations reflect the inherent complexity of the system, since, in small networks, changes in the interrelationships between nodes have a greater impact on the overall dynamics of the network. As the number of nodes increases, the coevolution stabilizes, suggesting that the network becomes more structured and less sensitive to small changes, reducing the dynamic complexity of the system. The strong initial fluctuations are therefore associated with the emerging complexity in the early stages of the network.

3.2. GCI: A Model to Assess the Complexity of Projects

The proposed model to assess the complexity of projects is presented in Figure 7. The first step is the study and analysis of the project. After project awareness, the second step consists in the development of the networks of the project, according to the supply data, the contractual data and the stakeholder and their relationships. The built networks and their characteristics allow to calculate two metrics: the Structural Complexity (SC) and the Dynamic Complexity (DC).

The SC and DC metrics are defined by the following definitions and scales:

• Structural Complexity (SC) through the measurement of Extended Cyclomatic Complexity (ECC). This complexity is always present, since it depends on the structure of the network represented by the project. Thus, the following reference values will be considered [66] and their assignments to the complexity level:

  ○

  ECC ≤ 10. Simple project. Structural complexity will have a value of 0.

  ○

  ECC > 10 and ≤20. Complicated project. Structural complexity will have a value of 3.

  ○

  ECC > 20. Complex project. Structural complexity shall have a value of 5.

• Dynamic Complexity (DC) by applying the coevolution parameter g, which will depend on the number of nodes. Thus, the following reference values [67] and assignments to the complexity level will be considered:

  ○

  Number of nodes (N) ≤ 25. High coevolution. The dynamic complexity will have a value of 5.

  ○

  Number of nodes (N) > 25 and ≤100. Medium coevolution. The dynamic complexity will have a value of 3.

  ○

  Number of nodes (N) > 100. Low coevolution. The dynamic complexity will have a value of 0.

Taking into account the two metrics described and following different models reported in the scientific literature [68,69,70], such as the model proposed by Stacey [71] in which he generates a classification of projects according to the degree of agreement in the project team and the complexity of the project interactions, or the model of Williams and Hillson [72] which, for its part, generates a model by means of the complexity of the project structure and its interactions, a model for evaluating the complexity of projects under uncertainty conditions is proposed for the classification of projects according to a Global Complexity Index (GCI).

The modeling of the uncertainty in the assessment of the SC and DC metrics, and the final assessment of the project complexity (GCI) was performed using fuzzy logic [73]. The model consists of two input variables (SC and DC) and one output variable (GCI), with a total of eleven membership functions and nine decision rules. The model uses linear membership functions to transform the input variable into a scale ranging from 1 to 0. A value of 0 indicates that the value in question does not belong to the given set, a value of 1 signifies that the value is undoubtedly a member of the given set, and any value between 0 and 1 represents a degree of potential membership (a higher value implies a greater likelihood of membership).

Table 1. Gamma, Trapezoidal and L membership function definitions.

Gamma Function	Trapezoidal Function	L Function
$L\left(x\right)=\left\{\begin{array}{c}1ifx\le a\\ {\displaystyle \frac{a-x}{b-a}}ifa<x\le b\\ 0ifx>b\end{array}\right.$	$T\left(x\right)\left\{\begin{array}{c}0ifx\le a\\ {\displaystyle \frac{x-a}{b-a}}ifa<x<b\\ 1ifb<x\le c\\ {\displaystyle \frac{d-x}{d-c}}ifc<x\le d\\ 0ifx>b\end{array}\right.$	$\mathsf{\Gamma }\left(x\right)=\left\{\begin{array}{c}0ifx\le a\\ {\displaystyle \frac{x-a}{b-a}}ifa<x\le b\\ 1ifx>b\end{array}\right.$

provides a detailed description of the membership functions utilized in this study. The parameters of each membership function have been modelled using data from previous projects to introduce uncertainty into the values of each model variable. The SC and DC metrics were modelled using gamma, L and trapezoidal functions (Table 1) associated with the three valuation levels defined above for each of the two variables: (

Table 2. Membership functions of the fuzzy model for SC and DC variables.

Variable	Function Name	Function Type	Parameters
SC	SCfm1	Gamma (a; b)	(10; 11.25)
	SCfm2	Trapezoidal (a; b; c; d)	(10; 11.15; 20; 22.5)
	SCfm3	L (a; b)	(22.25; 30)
DC	DCfm4	Gamma (a; b)	(37.5; 56.25)
	DCfm5	Trapezoidal (a; b; c; d)	(37.5; 56.25; 93.75; 112.5)
	DCfm6	L (a; b)	(93.75;170)

). The index (GCI) was modelled using one gamma function, one L function and three trapezoidal functions (

Table 3. Fuzzy model membership functions for output GCI variable.

Variable	Function Name	Function Type	Parameters
GCI	GCIfm1	Gamma (a; b)	(1.57; 1.86)
	GCIfm2	Trapezoidal (a; b; c; d)	(1.57; 1.86; 2.43; 2.21)
	GCIfm3	Trapezoidal (a; b; c; d)	(2.43; 2.71; 3.29;3.57)
	GCIfm4	Trapezoidal (a; b; c; d)	(3.29; 3.57; 4.14; 4.43)
	GCIfm5	L (a; b)	(4.43; 5)

). Fuzzification was performed using the max-min method, and defuzzification using the center of gravity method [74].

The value of the model output variable (GCI), according to the values of the input variables (SC and DC), is defined according to the following decision rules:

• If SC = fm1 & DC = fm1 → GCI = fm4
• If SC = fm1 & DC = fm2 → GCI = fm2
• If SC = fm1 & DC = fm3 → GCI = fm1
• If SC = fm2 & DC = fm1 → GCI = fm4
• If SC = fm2 & DC = fm2 → GCI = fm3
• If SC = fm2 & DC = fm3 → GCI = fm2
• If SC = fm3 & DC = fm1 → GCI = fm5
• If SC = fm3 & DC = fm2 → GCI = fm4
• If SC = fm3 & DC = fm3 → GCI = fm3

The defuzzification process produces values between 0 and 5 for the output variable GCI, which ranks the complexity of the project as described in

Table 4. Complexity of the project according to output GCI variable value of the Fuzzy GCI model: Final assessment.

Variable	Value	Complexity Classification	Remarks
GCI	$0\le$ GCI < 2	Simple	An industrial project involves clear, repetitive tasks.
	$2\le$ GCI < 4	Complicated	Requires specialized knowledge, detailed planning, and precise coordination.
	$4\le$ GCI < 5	Complex	It is characterized by unpredictable interactions, emergent outcomes, and adaptive processes
	$\mathrm{CGI}=$ 5	Anarchy	Extreme uncertainty, lack of structure, and chaotic conditions requiring improvisation

.

The proposed methodology is applicable to larger projects, regardless of the number of nodes and interactions. This is because the networks are managed using the UCINET program, which allows the analysis of large-scale network structures without significant restrictions. However, it is important to note that the complexity of a project is not intrinsically related to its size, but to the nature of its interactions, the density of connections and the level of uncertainty inherent in these interactions. The methodology addresses these factors through an adaptive approach based on fuzzy logic, which may ensure its applicability to a wide variety of projects, including those with larger and more dynamic networks.

4. Results: Application of the Model to the WWTP “Marismas del Odiel” Project

4.1. Description of the Project

The case study for applying the proposed evaluation model is the “Marismas del Odiel” Wastewater Treatment Plant (WWTP) (Figure 8) project located in the municipality of Punta Umbría (Huelva, Spain). Since this project is subject to confidentiality, the data corresponding to the main contractors and subcontractors have been anonymized.

It was designed to treat the wastewater of 142,000 equivalent inhabitants in the horizon year of 2036, belonging to the towns of Punta Umbria, Aljaraque and El Rompido, population centres located within a maximum radius of 12 km from the location of the WWTP. Its construction was carried out between 2008 and 2012 and the technology used was quite innovative at the time, since it reduced using biological treatment (A2O technology, Anaerobic, Anoxic, Oxic) both carbon pollution and nutrients (nitrogen and phosphorus) below the limits required by the administration.

The main design requirements and issues faced by the project management team were:

• Location in the vicinity of a natural park with a high richness of animal species.
• High seasonality of winter/summer discharges due to its location in a summer beach resort area.
• Difficulty of plant distribution due to being located between the fence of the natural park and the access road to the town of Punta Umbria.
• Need for an advanced purification process to eliminate nutrients, since their emission affected the balance of the park’s aquifers.
• Need for a distributed control that would allow changes in the process in real time.
• A broad group of stakeholders made up of the central administration, regional administration, local administration, operation manager, neighborhoods’ groups, environmental groups, universities promoting the study of the process (Basque Country, Valladolid, Seville), and contractors.

4.2. Application of the Model

After the study and analysis of the project, the project network is developed according the three levels proposed by Adami and Verschoose [58]: contractual, supplies and resources, and stakeholders’ information. The analysis was based on a review of the available project documentation.

• Contractual network:

In this network, represented in Figure 9, the General Manager (GM) of the joint venture has a contractual relationship with the administration through the figure of the Project Supervisor (PS) who hires his/her team to control the development of the project. The GM, in turn, has a contractual relationship with the Administration Manager (AM), with the Construction Manager (CM), and with the Electromechanical Equipment and Commissioning Manager (EECM), and in turn, each manager below the manager will hire his/her work team. However, the CM must supervise (not manage) some of the contracted elements since, although they do not report hierarchically of him/her, they will be collaborating with him/her during the work. The network is non-directed, except in the cases of supervision mentioned above, since the contract unites the two actors in the fulfilment of the contract. Hiring is not carried out by the managers, but normally, proposals for hiring or assignment of personnel are made according to the criteria of these managers.

• Supply and resource network:

In this network, represented in Figure 10, the supplies of the main resources and goods purchased during the implementation of the project have been presented. Unlike the previous one, this network has been designed as a directed network, since it is the suppliers who send the necessary equipment or services to the project. It has been considered that the CM is the maximum responsible for the contracts, except for four of the contracts (piling works, blowers, electrical system, and control system) in which the GM participates, since these contracts were of vital importance for the work, and he had to be included. Subsequently, they will be referred to the different subordinates responsible. The figure of the CM is included because the resources must be approved by him/her or by whoever depends on him/her for approval. Suppliers have been marked in red and project members in blue.

• Stakeholders’ information network:

It was said earlier that information is the main element of complexity in a construction project. This study focuses on the stakeholders’ information network. Not all represented stakeholders have equal prominence throughout the project life cycle. Therefore, only a snapshot of the key stakeholders at the peak of the project execution has been included in

Table 5. Stakeholders network WWTP “Marismas del Odiel” (Source: authors).

STAKEHOLDERS WWTP
Id	Description	Functions
CM	Construction Manager	Directs planning, coordination, and execution for project success
GM	General Manager	Oversees operations, strategy, and resources to achieve organizational goals
I1	Council	City government
I2	Construction company 1	Managed the construction of the pretreatment area and the biological reactor
I3	Installation company 1	Handles the installation and integration of mechanical and electrical systems.
I4	Control company 1	Installed the instrumentation systems.
I5	Control Company 2	Installed the PLC control systems and SCADA connections.
I6	Environmental Administration	Owns the contract and oversees construction to ensure compliance.
I7	Construction company 2	Managed the construction of the sedimentation and sludge treatment facilities.
PS	Project Supervisor	Oversees the construction on behalf of the administration to ensure project compliance.

.

First, the centrality parameter is determined, which corresponds to the identification of the most important actors in the network, and this can be dealt with under three concepts: degree centrality, proximity centrality and intermediation centrality. These parameters have been calculated with the UCINET program.

(a)

Degree centrality. It is based on locating those nodes that have more links. That is, they are more interconnected. As can be seen in Figure 10, the CM is the actor that is most interconnected with the rest of the stakeholders, outside and inside the construction site. This representation fits the scenario, since this actor is one of the most important in a construction project. Below him/her is the GM, as the contractor’s legal representative on site. Outside the contractor, there is a stakeholder named I1 who is one of the most active and in this case corresponds to the representative of the town council of the town where the WWTP was placed and who made numerous demands during the project.

(b)

Closeness centrality. It indicates, through which node the information goes faster and more efficiently from one node to another. As can be seen in Figure 11, node I5, which corresponds to the control company 2 that performed the control link with the water company in the area, has a greater preponderance over the network than the rest. It is only related to node I3, which corresponds to the installation company 1 and the CM. This fact corresponds to the continuous changes in the prescriptions to be made in the control and SCADA (Supervisory Control and Data Acquisition) and their repercussion on the need of the installation of the equipment to be controlled. Apparently, one might think that I4, Control Company 1, the company that installed the plant control equipment or the CM might be the most important nodes, but here we are measuring the speed of change, and this is achieved when weakly connected.

(c)

Intermediation centrality. In this case, the frequency in which a node acts as a bridge on the shortest route between two nodes, whatever it may be, is quantified. The intermediate node is the one that has the capacity to control or regulate information flows. Figure 12 shows how the intermediary node is the CM. All the information passes through the site manager, who oversees transmitting it to the rest of the network.

Although it can be achieved by traditional counting methods, the UCINET program can acquire the three parameters needed to obtain the structural complexity. In

Table 6. Node and link parameters obtained from the network through the UCINET program.

Net	Parameter	Value
Stakeholder	N. nodes	10
	N. ties	43
	Avg. Degree	4.3

data of the case of the Marismas WWTP stakeholder information network are presented.

In this network, there are 10 nodes and 43 links. In this case, the stakeholder network is represented as a directed network, therefore, the number of links is the one shown in Table 5. It could happen that the stakeholder network is represented as an undirected network, in this case, the real number of links will be half of those obtained with the UCINET program.

The average degree of each node is 4.3. In addition, the number of strongly connected components (SCC) can be obtained, which is the third variable in the structural complexity calculation. This parameter indicates those actors that are closer and more connected to each other than the rest. This parameter is fundamental in small-world networks, which are the ones we are dealing with in greater depth.

In the case analyzed, there will be 2 SCCs or cliques.

Table 7. Number and network of SCC or cliques obtained from the network through the UCINET program.

Net	Number Cliques	Clique
Stakeholder	1	I3 - GM - I7
	2	I4 - GM - CM

shows the data obtained from UCINET and shows two SCCs, the first one is formed by the relationship between nodes I3, GM, and node I7, and the second one, is by the relationship between nodes I4, GM, and CM.

Analyzing the stakeholders’ information network, the following parameters are obtained: E = 43, N = 10 and SCC = 2. Applying Equation (3) results are presented in (8).

$$\mathrm{ECC}\left(\mathrm{PN}\right)=\left|\mathrm{E}\right|-\left|\mathrm{N}\right|+\mathrm{p}=43-10+2=35$$

(8)

The ECC or for the model, SC, reaches a value of 35. According to the scale, being a value greater than 20, the “Marismas del Odiel” WWTP project has a SC value equal to 5.

For the Dynamic Complexity, it is observed that the number of nodes in the network is 10, according to the scale for DC of the proposed model, the “Marismas del Odiel” WWTP project has a DC value equal to 4.

Introducing the values obtained for the variables SC and DC in the fuzzy model (Table 2 and Table 3 and its decision rules), a value of 4.27 is obtained for the output variable of the GCI model. Entering this value in Table 4, the complexity of the “Marismas del Odiel WWTP project is evaluated as COMPLEX.

The location close to a natural park required rigorous coordination to comply with strict environmental regulations, integrating advanced technology such as A2O treatment to remove nutrients under conditions of high seasonal variability. Furthermore, the participation of numerous stakeholders, including regional and local administrations, contractors, subcontractors, and universities, intensified the need for constant communication and collaboration. These characteristics, together with the need for real-time distributed control and the logistical difficulty of the site, underline the project’s correspondence with the classification of a complex according to the model evaluated.

5. Conclusions

This research has developed a model to assess complexity in technical projects considering both structural and dynamic complexity. This model provides a holistic view of complexity by relying on relationships and interdependencies among stakeholders, thus addressing the emergent and adaptive nature of complex projects. In contrast to traditional metrics that tend to focus on static technical attributes or structural models applied to specific IT projects, the proposed approach allows for capturing the elements of communication and influence flow within the stakeholder network, aspects that largely determine the evolution and success of a technical project. Measurement of these relationships in small-world networks with strongly connected components introduces a quantifiable and practical dimension of complexity, thus overcoming the limitations of conventional approaches that tend to fragment the analysis into partial aspects.

In addition, this model is particularly effective in complex projects due to its ability to reflect how the network structure and stakeholder interactions can adapt and reconfigure, allowing better adaption to the risks that may appear during the life of the project. In environments with high variability and multiple stakeholders, where interactions and communication are critical factors, this approach allows constant monitoring of dynamic complexity, thus providing a tool for risk anticipation and decision support. The graphical representation of the networks and the possibility of visually observing the intensity of connections and the role of critical nodes offer an accessible way to manage the project, which contributes to both initial planning and agile response to changes and new challenges. The DC metric is evaluated statically (at a certain moment of the project), but it can be used iteratively in the different phases of the project, facilitating the decision-making process throughout the life of the project, and incremental and continuous development in the characterization of the project.

Integration of fuzzy logic in the proposed model facilitates the incorporation of imprecise or partially defined information. It makes it possible to deal efficiently with the dynamic variation of complexity parameters in the project, adapting to the inherent uncertainties of the environment.

However, this model has certain limitations in terms of its dependence on the experience of the project manager in interpreting the results in a real context. Although the metrics provide quantitative value, the manager must contextualize the data to adequately reflect the operational reality of the project, suggesting the need for complementary tools to enrich the interpretation. In addition, the analysis has focused mainly on the stakeholder network, without addressing the complexity that a multi-network can provide by incorporating additional layers of relationships. Including a multi-network structure would allow capturing interactions between different types of networks, providing a more holistic view of complexity that could improve the accuracy of analyses in large-scale projects.

Finally, this paper has primarily addressed stakeholder network analysis, providing an initial perspective on complexity in technical projects. To gain a deeper understanding, future work should extend the analysis to contractual networks and resource networks, which are central to project execution and can provide a comprehensive view of structural and operational interdependencies.