As a matter of definition, systems are composed of a set of entities and their relationships. By now, the reader should have come to expect that these relationships can have two characters: they can be functional relationships or formal relationships.

Functional relationships are relationships between entities that do something; they involve operations, transfers, or exchanges of something between the entities. We sometimes call ­functional relationships interactions to emphasize their dynamic nature. During interactions, operands are exchanged by the entities or acted on jointly. Thus a heart exchanges blood with a lung, and a team member shares results with a colleague. A more thorough presentation of functional interactions can be found in Chapter 5.

Formal relationships are relationships among the entities that exist or could exist stably for some period of time. The adjective “formal,” derived from “form,” is not just a way to describe a dinner party! Formal relations often include a connection or a geometric relationship. For example, when a lung is connected to the heart, or when a person joins a team, a formal relationship exists. We sometimes call formal relationships structure to emphasize their static nature. We ­discuss formal relationships in more detail in Chapter 4.

In general, a functional relationship usually requires a formal relationship. The formal relationship is the instrument of the functional relationship. The heart cannot exchange blood with the lung without a connection. Team members cannot share results without either being in proximity or having an information link.

These formal and functional relationships can be represented in either a relationship diagram or an N-Squared table. Figure 2.10 shows a relationship diagram of a system with two entities within the system and one outside in the context. The formal interactions are represented by the thin double-headed arrows, and the functional interactions are represented by broader arrows, which can be single- or double-headed, depending on the nature of the interactions. Some relationships are within the system, and some cross the system boundary, which is labeled as such and indicated by the dashed line. Best practice for such diagrams is to label the entities with their form and function, and also to label the relationships, but this often produces a very cluttered diagram.

We can now use our running examples to illustrate Task 3 of System Thinking, as expressed in Box 2.7. The relationship diagram for the amplifier circuit is shown in Figure 2.11. The structural relations indicate that the input circuit voltage is connected to Resistor 1, which is connected at its other port to Resistor 2 and the Op Amp. The output of Resistor 2 is connected to the Op Amp at a different port, and to the output circuit. This link provides a type of feedback around the Op Amp. For a circuit, these structural connections or formal relationships are electrical connections. The arrows indicate the functional interactions, which for a circuit are flows of current. In contrast, the standard electrical diagram for this circuit (Figure 2.4) would have only one relationship shown between elements, which would imply to an electrical engineer both a connection and a flow of current.

An alternative representation of relationships uses two N-Squared tables, as shown in Table 2.5. Each N-Squared table lists the N entities on each side. The first shows the formal relationships, the second the functional relationships. The entities within the system are above and to the left of the dashed lines. Off diagonal terms indicate an internal connection or interface. There is value in both the relationship diagram, which is easier to visualize, and the N-Squared table, which has more detail and does not become visually congested as nodes and connections proliferate.

Formal relationships tend to be more concrete, and they are a good starting point for thinking about relationships in general. As you examine each formal relationship, however, try to understand the functional relationship that it enables. The formal relationships are ­important primarily because they are instruments of the functional relationships. Because emergence occurs in the functional domain, it is really the functional interactions that are of primary importance, as will be discussed further in Section 2.6.

Formal and functional relationships can exist across the boundary of the system, between entities in the system and its context. These are called the external interfaces of the system. In the N-Squared table, Table 2.5, these external interfaces are indicated as any relationship outside the part of the table reserved for the system (such as “connected at input”). In the relationship diagram, Figure 2.11, these external interfaces show up as arrows crossing the system boundary and are marked. Similar external interfaces can be found between Team X members and supporters, and between the circulatory system and the air. In fact, it is nearly impossible to define a system that is not connected somehow to entities beyond its boundary by external interfaces.

In summary:

• Fundamental to the definition of a system is the existence of relationships among the entities, which can be formal (they exist or could exist—structure) or functional (they do something—interactions).

• In general, some of the entities of a system will also have both formal and functional relationships with context entities outside the system; such relationships occur across external interfaces.

• Formal and functional interactions can be effectively represented by either relationship diagrams or N-Squared tables.