Complexity Theory and Network Centric Warfare

Excerpts (with the occasional note) from Complexity Theory and Network Centric Warfare, by James Moffat, DOD Command and Control Research Program (2003):

Attempts have been made to develop some general understanding, and ultimately a theory, of systems that consist of many interacting components and many hierarchical layers. It is common to call these systems complex because it is impossible to reduce the overall behaviour of the system to a set of properties characterising the individual components. Interaction is able to produce properties at the collective level that are simply not present when the components are considered individually. As an example, one may think of mutuality and collaboration in ecology. The function of any ecosystem depends crucially on mutual benefits between the different species present.

Another important feature of complex systems is their sensitivity to even small perturbations. The same action is found to lead to a very broad range of responses, making it exceedingly difficult to perform prediction or to develop any type of experience of a "typical scenario."

This must necessarily lead to great caution: do not expect what worked last time to work this time. The situation is exacerbated since real systems (ecological or social) undergo adaptation. This implies that the response to a given strategy most likely makes the strategy redundant [antibiotics, e.g.].

Complex systems cannot be studied independently of their surroundings. Understanding the behaviour of a complex system necessitates a simultaneous understanding of the environment of the system. One should bear in mind that the separation into system, drive, noise, surroundings, etc. is rather arbitrary and is far from representing a complete analysis.

From these considerations, we see that it is vitally important to consider warfare as a complex system that is linked and interacts (in a coevolving way) with the surrounding socioeconomical and political context.

Further work will need to examine how coevolution across the entire network of military, socioeconomical, and political interactions leads firstly to emergent effects at higher levels, and of equal importance how such effects lead to coevolution at the higher level. It will also be important to consider the robustness of such networks, and their vulnerability to damage.

The intricate interrelationships of elements within a complex system give rise to multiple chains of dependencies. Change happens in the context of this intricate intertwining at all scales. We become aware of change only when a different pattern becomes discernible. But before change at a macro level can be seen, it is taking place at many micro levels simultaneously. Hence, microcomponent interaction and change leads to macrosystem evolution.

Benard cells also show the complexity of movement. The cells unfold along the horizontal axis, adopting successively righthanded or left-handed rotation. Our very small observer can now locate his position in space by considering the rotation of the cell he occupies and by counting the number of cells he passes through. The emergence of this notion of space is known as symmetry breaking. When is below the critical value, the homogeneity of the fluid in the horizontal direction renders its different parts independent of each other. In contrast, beyond the threshold, it is as if each volume element is watching the behaviour of its neighbours and is taking this into account in order to play its role adequately and to participate in the overall pattern. This suggests the existence of correlations of statistically reproducible rate relations between distant parts of the system. The characteristic space dimension of a Benard cell is in the millimetre range, whereas the characteristic space scale of the intermolecular forces is in the Angstrom range. That large numbers of particles can behave in a coherent fashion at this long range, despite random thermal motion, is one of the principal properties characteristic of such self-organisation and emergent complex behaviour.

This experiment is reproducible; the same convection patterns will appear at the same threshold value and the process is subject to a strict determinism. However, the direction of the rotation of the cells is unpredictable. The form of the particular perturbation that prevails at the moment of the experiment will decide whether a given cell is right- or left-handed. When the constraint is sufficiently strong, several solutions are possible for the same parameter values and chance alone will decide which of these solutions is realised. In this way, the system has been perturbed from a state of equilibrium or near equilibrium to a state of self-organisation, with a number of possible modes of behaviour.

To summarise, nonequilibrium has enabled the system to transform part of the energy communicated from the environment into an ordered behaviour of a new type: the dissipative structure. This regime is characterised by symmetry breaking, multiple modes of behaviour, and correlation. Such a system is called "open" since it is open to the effect of energy or information flowing into and out of the system. It is also called "dissipative" because of such energy flows, and the resultant dissipation of energy.

If we think of a guided missile attempting to manoeuvre towards a target, the measure of loss is the miss distance relative to the aim point. The control parameters are the settings for the missile fins at a given time t. For simple forms of linear guidance ( e.g., early forms of laser-guided bombs), this leads to what is called bang-bang control, where the missile fins "bang" from one extreme setting to another in order to keep the missile on course. Applied to a linear control system, this maximum principle leads to the solution of bang-bang control.

A characteristic feature of many of the systems encountered in nature, however, is that the F's (laws of system controlling rate of change) are complicated nonlinear functions of the X's (instantaneous states of system). The equations of evolution of this type of system should then admit, under certain conditions, several solutions (rather than just the one optimal solution) since a multiplicity of solutions is the most typical feature of a nonlinear equation. Our assumption will be that these solutions represent the various modes of behaviour of the underlying system.

The most useful view of equilibrium is as follows. We represent the evolution of the system in a space spanned by the state variables (phase space). An instantaneous state of the system is thus represented in phase space by a point. As the system evolves over time, a succession of such states is produced, giving rise to a curve in phase space, which is called a phase space trajectory. In a dissipative dynamical system, as time progresses, the phase space trajectory will tend to a limit representative of the regime reached by the system when all transients die out. We call this regime the attractor. The attractor representing an equilibrium position is unique and describes a time-independent situation. This gives a phase space point towards which all possible histories converge monotonically. The state of equilibrium is therefore a universal point attractor. The goal of self-organisation is thus the search for new attractors that arise when a system is driven away from its state of equilibrium.

By allowing the intrinsic nonlinearity to be manifested in the regime of detailed balance, nonequilibrium can also lead to the coexistence of multiple attractors in state space. The state space can then be carved up into a set of basins. Each of these corresponds to the set of states that, if the system were to start from there, would evolve to a particular attractor. These are known as the basins of attraction. The ridges separating these basins of attraction are called separatrices. The coexistence of multiple attractors constitutes the natural mode of systems capable of showing adapted behaviour and of performing regulatory tasks.

Stability or "robustness to change" is essentially determined by the response of the system to perturbations.

Such critical systems are of particular scientific interest. Systems in critical states do not have any characteristic scale and may therefore exhibit the full range of behavioural characteristics within the particular system restraints. This means that systems at the point of criticality are in a position of optimal flexibility in some sense, as we have noted. It could thus be argued that one of the requirements of military command is to so arrange things that the forces collaborate locally and thus self-organise into this optimal state.

The system becomes critical in the sense that all of the members of the entire system influence each other. For the example ecosystem above, the system self-organises itself into the critical state corresponding to this ability of the entire system to be influenced through the propagation of local coevolution influences and the resultant clusters/avalanches of species that coevolution created.

We will see that these relate to ideas of correlation in space or time (in contrast to coincidence in space or time). Correlation in space or time is a signal of local clustering and collaboration spatially (e.g., across a battlespace) or in time ( e.g., across an information grid–reading e-mail creates a correlation in time between individuals, taking a phone call creates a coincidence in time).

A signal will be able to evolve through the system as long as it is able to find a connected path of above-threshold regions. When the system is either driven at random or started from a random initial state, regions that are able to transmit a signal will form some kind of random network. This network is correlated by the interaction of the internal dynamics with the external field. The complicated interrelation between the two driving dynamics means that a complex, finely-balanced system is produced. As the system is driven, after this marginally stable selforganised state has been reached, we will see flashes of activity as external perturbations interact with internal drivers to spark off avalanches ( i.e., clusters) of activity through different routes in the system. Bak's assertion is that the structure of this dynamic network is fractal. If the activated clusters consist of fractals of different sizes, then the duration of the induced processes travelling through these fractals will vary greatly. Different timescales of this type lead to what is termed 1/f noise. 1/f noise is a label used to describe a particular form of time correlation in nature. If a time signal fluctuates in a seemingly erratic way, the question is whether the value of the signal at time has any correlation to the signal measured at time ( ).

For a system that is not poised in a critical state and thus not about to change its mode of behaviour, the reaction of the system is described by a characteristic response time and characteristic length of scale over which the perturbation is felt. However, for a critical system, the same perturbation applied at different positions or the same position at different times can lead to a response of any size. The average is not therefore a useful measure of response. The amount of the system involved is a cluster in the spatial dimensions of the system.

We can use the evolution model just described to gain insight into the effect of a Global Information Grid. Imagine such a grid in two dimensions. At each grid point is positioned an element of our force. Each such force element has a "fitness" value corresponding to its ability to evolve and adapt to local circumstances as a function of the information available on the grid. We assume these fitness values are random at first. At each step of the process, we assume that the force element with the smallest fitness is likely to have to adapt fastest to its local environment. In so doing, it will change the fitness values of the units closest to it on the information grid ( i.e., there is local coevolution). Note that these force elements may be separated by large and varying distances in space. With these assumptions, over time the force elements will form clusters of coevolution of the form predicted by the Bak-Sneppen model. In particular, the statistics of emergent cluster size can be predicted mathematically to converge to a power-law.

In natural systems, we can consider the movement of a boundary through a medium (for example, the boundary of an atomic surface, the boundary of a growing cluster of bacteria, or the front of advance of a fluid "invasion" of a medium such as a crystalline rock).

What happens if we restrict ourselves to looking at the boundary between two different regimes (such as two different nationalities or two opposing armed forces), and how this would move over time depending on the local coevolution of the elements involved?

The most relevant case from our point of view is the front of advance of fluid "invasion" of a medium. As described in [9], we can represent the medium itself as consisting of a lattice of cells, each with either a 1 or 0 in it. A "1" represents the fact that that cell can be wetted. The proportion of cells containing a "1" is defined as p. For large configurations, we can also interpret p as the probability that a particular cell contains a "1." A "0" represents the fact that the cell cannot be wetted–it thus "pins" the advance of the fluid through the medium, at least locally.

It turns out that for this case, when the pinning probability p is greater than a critical value pc, the growth of the interface is halted by a spanning path of pinning cells. Such models of interface or boundary movement exhibit fractal properties of the interface, as discussed in detail in [9]. We shall see similar effects later in our discussion in Chapter 4 of the control of the battlespace using ideas based on preventing the flow of opposing forces and/or third parties through the space. Rather than choosing the next cell to invade at random, as in the DPD model, we can use a model of the process that is more akin to the manoeuverist principle of applying your strength where the opponent is weak–in other words, the cell next to be wetted is the one where the local pinning force of the medium is weakest. Such a model of the boundary movement is the Invasion Percolation model.

In natural systems, the boundary of such an interface that is moving through a medium can be characterised by its "roughness."

For many natural systems, the roughness first goes through a transition period before stabilising at an equilibrium value.

The concept of pinning a fluid locally is similar to the idea of trying to exert local control over a boundary to prevent the flow of other forces or third parties across that boundary. We will see later (in Chapter 4) that the idea of control as the prevention of such flows through an area has important implications for the emergent behaviour of a force (or two competing forces) attempting to exert control over a battlespace.

In the case of control along a boundary, Complexity Theory, in terms of the invasion percolation model, can be used to analyse the effect of two forces (an attack and a defence force) interacting across a boundary, when the boundary moves at the point where the defending (pinning) force is weakest. If the defence pinning force is coevolving locally, then the boundary should form a fractal with a fractal dimension in the range 1.33-1.89 [6, Appendix], as we have seen. In Chapter 3, we show that there is historical evidence in warfare for such an effect, and for values in this range.

Brownian motion is also associated with scaling relationships and fractal behaviour. As a nonfractal object is magnified, no new features are revealed. As a fractal object is magnified, finer details are revealed. The size of the smallest feature of a nonfractal object is called the characteristic scale. A measurement made at finer resolution will include more of these smaller pieces. Thus the value measured of a property will depend upon the resolution used to make the measurement. How a measured property depends on the resolution used to make the measurement is called the scaling relationship. A fractal object has features over a broad range of sizes. Fractal phenomenological characteristics are:

1. SELF-SIMILARITY: behavioural characteristics are "similar" at different resolutions.
2. SCALING: the value measured for a property depends upon the resolution at which it is measured.
3. DIMENSION: the dimension of an object gives a quantitative measure of self-similarity and scaling. It tells us how many new pieces of an object are revealed as it is viewed at higher magnification.
4. NONSTATISTICAL PROPERTIES may be observed. Moments may be zero or nonfinite (e.g., the mean tends towards zero and variance tends towards infinity).

There are two types of self-similarity:
1. GEOMETRICAL: pieces of the object are exact smaller copies of the whole object.
2. STATISTICAL: the value of the statistical property Q(r) measured at resolution r is proportional to the value of Q(ar) measured at a resolution ar such that Q(ar) = kQ(r).


What is of particular interest is that Turner and Weigel's data strongly suggest the occurrence of temporal clustering. Over the 1928-1989 period, 12.5 and 37.5 percent of all extreme positive jumps in the S&P 500 occurred within one and five days respectively of another positive jump in equity prices. Positive jumps in the Dow Jones were similarly clustered with 11.3 percent of the positive jumps taking place within one day, and 36.2 percent transpiring within 5 days of each other.

From our analysis of these open and dissipative systems, it is clear that there are a number of key properties of complexity that are important to our consideration of the nature of future warfare. Such futures, involving the exploitation of loosely coupled command systems such as Network Centric Warfare, will have to take account of these key properties. A list of these is given here, and then discussed further in Chapter 2 in the context of Network Centric Warfare.

1. NONLINEAR INTERACTION: this can give rise to surprising and non-intuitive behaviour, on the basis of simple local coevolution.
2. DECENTRALISED CONTROL: the natural systems we have considered, such as the coevolution of an ecosystem or the movement of a fluid front through a crystalline structure, are not controlled centrally. The emergent behaviour is generated through local coevolution.
3. SELF-ORGANISATION: we have seen how such natural systems can evolve over time to an attractor corresponding to a special state of the system, without the need for guidance from outside the system.
4. NONEQUILIBRIUM ORDER: the order (for example, the space and time correlations) inherent in an open, dissipative system far from equilibrium.
5. ADAPTATION: we have seen how such systems are constantly adapting–clusters or avalanches of local interaction are constantly being created and dissolved across the system. These correspond to correlation effects in space and time, rather a top-down imposition of large-scale coincidences in space and time.
6. COLLECTIVIST DYNAMICS: the ability of elements to locally influence each other, and for these effects to ripple through the system, allows continual feedback between the evolving states of the elements of the system.

Chapter 2

Network Centric Warfare is an emerging theory of war based on the concepts of nonlinearity, complexity, and chaos. It is less deterministic and more emergent; it has less focus on the physical than the behavioural; and it has less focus on things than on relationships.

Combat is, by its nature, a complex activity. Ashby's Law of Requisite Variety...which emerged from the theoretical consideration of general systems as part of Cybernetics, indicates that to properly control such a system, the variety of the controller (the number of accessible states which it can occupy) must match the variety of the combat system itself. The control system itself, in other words, has to be complex. Some previous attempts at representing C2 in combat models have taken the view that this must inevitably lead to extremely complex models. However, recent developments in Complexity Theory...indicate another way forward. The essential idea is that a number of interacting units, behaving under small numbers of simple rules or algorithms, can generate extremely complex behaviour, corresponding to an extremely large number of accessible states, or a high variety configuration, in Cybernetic terms. It follows that, if we choose these simple interactions carefully, the resultant representation of C2 will be sufficient to control, in an acceptable way, the underlying combat model. As part of this careful choice, we need to ensure that the potentially chaotic behaviour generated by the interaction of these simple rules is 'damped' by a top-down C2 structure which remains focused on the overall, high level, campaign objectives.... It follows, from what we have just said, that the representation of the C2 process must reflect two different mechanisms. The first is the lower-level interaction of simple rules or algorithms, which generate the required system variety. The second is the need to damp these by a top-down C2 process focused on campaign objectives. Each of these has to be capable of being represented using the same Generic HQ/Command Agent object architecture. We have chosen to do this by following the general psychological structure of Rasmussen's Ladder, as a schema for the decisionmaking process. At the lower levels of command (below about Corps, and equivalent in other environments), this will consist of a stimulus/response mechanism. In cybernetic terms, this is feedback control. At the higher level, a broader (cognitive-based) review of the options available to change the current campaign plan (if necessary) will be carried out. In cybernetic terms, this is feedforward control since it involves the use of a 'model' ( i.e., a model within our model) to predict the effects of a particular system change.

Modelling and analysis to determine the effect of such phenomena underpin our thinking about such future conflict, the representation of information and command being at their heart. A new approach to capturing these effects has been put forward in this book, and is having a significant influence on the approach to modelling these phenomena. However, capturing the process of intelligent agents in conflict, set within a widely divergent set of possible futures, leads to a rich set of possible trajectories of system evolution for analysis to consider. We thus need to complement this effort with other work to categorise and understand the classes of behaviours which might emerge from such a complex situation. This is the domain of Complexity Theory.


COMPLEXITY CONCEPT INFORMATION AGE FORCE
1. Nonlinear interaction -- Combat forces composed of a large number of nonlinearly interacting parts.
2. Decentralised Control -- There is no master "oracle" dictating the actions of each and every combatant.
3. Self-Organization -- Local action, which often appears "chaotic," induces long-range order.
4. Nonequilibrium Order -- Military conflicts, by their nature, proceed far from equilibrium. Correlation of local effects is key.
5. Adaptation -- Combat forces must continually adapt and coevolve in a changing environment.
6. Collectivist Dynamics -- There is a continual feedback between the behaviour of combatants and the command structure.

Complexity is therefore associated with the intricate intertwining or inter-connectivity of elements within a system and between a system and its environment. In a human system, connectivity means that a decision or action by any individual (group, organisation, institution, or human system) will affect all other related individuals and systems. That effect will not have equal or uniform impact, and will vary with the state of each related individual and system at that time. The state of an individual and system will include its history and its constitution, which in turn will include its organisation and structure. Connectivity applies to the interrelatedness of individuals within a system, as well as to the relatedness between human social systems, which include systems of artifacts such as information systems and intellectual systems of ideas.

The phenomenological definition of a complex system is that it exhibits nonlinear, emergent, adaptive behaviour. Nonlinear behaviour is associated with far-from-equilibrium, open systems, in that cause and effect are no longer linearly connected. This is ultimately due to the type of internal-external system interactions (feedback) affecting our system.

Self-organisation in this context is taken to mean the coming together of a group of individuals to perform a particular task. A feature of these groups is that they are informal and often temporary. Enabling self-organisation can often be a source of innovation. Military commanders who understand the nature of auftragstaktik have always understood this.

In systems where the dynamical evolution is a struggle against various types of thresholds or barriers, the action will predominately occur where the net barrier to change is the smallest.

The species with the lowest fitness coevolves first. Similarly, in considering the movement of a fluid through a medium, the boundary moves where the pinning force is smallest.

Forest Fire: The rate (which is [1-p] if each iteration of the process is counted as a unit of time) of sparks dropping onto the grid is termed the sparking frequency. This sparking frequency sp is a key driver of the dynamics of the forest ecosystem. If sp is small, very large clusters of trees are allowed to form, which span the entire grid. When a spark is then dropped, the forest fire wipes out an entire forest stretching from one side of the grid to the other. In Complexity Theory, this is known as snapping noise. This name comes from looking at the behaviour of the system over time–large spikes of tree extinction (forest fires) are created at isolated points in time. If the sparking frequency sp is very large, then tree clusters do not have the chance to grow. Thus, over time, the system produces a large number of small spikes of activity, which are called popping noise. When sp is in the intermediate regime, the system self-organises to a critical state where the clusters of burnt trees have a distribution represented by a power law, and clusters of all sizes can be created. Over time, the spikes produced by this process ( i.e., the time evolution of forest fires of various sizes) have a similar dynamic to that produced by the acoustic dynamics of crumpling paper, and so this regime is termed crackling noise.

It is possible to relate such self-organised behaviour of a forest fire model to the statistics of the scale and intensity of conflicts. This is the beginning of an explanation as to why casualties in war follow a power law distribution.

A war must begin in a manner similar to the ignition of a forest. One country may invade another country, or a prominent politician may be assassinated. The war will then spread over the contiguous region of metastable countries.

Tuning can be seen as a directive way for the macrosystem to attempt to influence the behaviour of the microsystem. A controlling intelligence is deemed to be necessary in order to guide the system towards a particular goal. Varying the tuning parameter (the sparking frequency) of the forest fire model represents intervention from outside the system in order to ensure that it heads towards a particular goal. This question of tuning makes us consider the boundaries of the systems we are examining, and the flux of energy and/or information across the system boundary.

In such open dissipative systems, there will always be fluxes of information and/or energy across the system boundary.

Chapter 3

LTG Sir Francis Tuker indicated that at a threedimensional spatial level, manoeuvre warfare is determined by three conditions:

1. Flanks shall be tactically open or it shall be possible to create a flank by break-in and breakthrough.
2. The mobile arm shall be predominant.
3. It shall be possible to administer the mobile arm to the point where it will decide the battle and gain decisive victory.

In an historical analysis study [7] of the operational level of combat, it was found by Rowland that the occurrence of breakthrough, defined as the destruction of cohesiveness of the defence, was an important event in the eventual success of an offence. Following breakthrough, 86% of operations were successful, whereas if no breakthrough was achieved only 15% were eventually successful. Once breakthrough has been achieved, it becomes possible for the attack force to conduct a type of operation more in the nature of exploitation than combat. Moreover, variations in the time to breakthrough also led to differences in the nature of campaigns.

Immediate breakthroughs actually had a larger failure rate because of the brittleness that pertains to these very quick breakthrough cases.

This process of irruption has been identified as one of the key emergent effects of manoeuvre warfare [8]. We consider now whether such a process has scaling properties of the type discussed in our general consideration of complexity. The historical data indicates (as we have discussed) that for a given type of breakthrough (immediate, quick, or prolonged– I, Q , or P), and subsequent effect on the campaign (Subsequent Success [SS] or Subsequent Failure[SF]), the mean advance at breakthrough turns out to be a log-normal distribution. Of even more interest to us is the fact that if these distributions are plotted for each of the breakthrough/campaign effect categories, then they have a certain scaling character.

The idea is that an essentially straight line frontage between two tactical-level opponents will buckle into a fractal shape, whose fractal dimension can be calculated as a function of the force ratio of the forces involved (the number of attackers to the number of defenders) as derived from Historical Analysis of infantry battles carried out by the UK Dstl.

Lauren was able to show that the combat front will buckle over time and in the limit will have a fractal dimension D = 1.685. From Chapter 1, if we assume that this process is akin to invasion percolation of one fluid by another in a porous medium, the fractal dimension of the boundary of the resulting interface should lie in the range 1.33-1.89, which is what we find from historical data.

It reflects the asymmetry of the infantry battle in the following sense [12]. The attack force aim is to close on the defence position, and fire is used in a general suppressive mode–actual casualties caused to the defence are only a small part of the process at this point. However, from the defence perspective, the aim is to deter the attack, and casualties to the attack force are very important. Such casualties to the attack force are a direct reflection of the intervisibility of targets to the defence force as discussed above.

There is a power-law relationship between the intensity of war and its frequency.

Chapter 4.

Recent work by Perry [9] has exploited the idea of information entropy to address the second question (with a reduction in entropy across the network corresponding to an increase in knowledge, and this then being equivalent to a reduction in delay in prosecuting an action).

A dynamic process is said to be memoryless or Markovian if at each cycle, the state of the system is influenced only by the state of the system in the previous cycle, and not by the specific history of the system.

We draw on information science to develop a knowledge metric that is a function of the average information present in the set of all possible uncertain events. This quantity is referred to as information entropy and it measures the amount of uncertainty in a probability distribution. The amount of information available from the known occurrence of the event, U = u, i.e. that u enemy units are indeed arrayed against the friendly force, is inversely proportional to the likelihood that the event will occur. An event that is very likely to occur provides little information when it does occur. On the other hand, an unlikely event provides considerable information when it occurs.

Information entropy has properties that make it ideal as a metric for measuring the commander's uncertainty prior to making a decision and for measuring the uncertainty in the entire campaign:

1. MAXIMUM ENTROPY: The entropy function is maximised when the uncertainty in the distribution is greatest. Maximum uncertainty occurs when the friendly commander has no sensor assets to deploy. In this case, any number of units might be arrayed against him with equal probability...the more units available to the enemy commander, the less clear we are about their deployment in the absence of sensor outputs. In general, a probability distribution with a wide variance exhibits high entropy.

This network-enabled approach thus allows us to compute the distribution of the response time of the system as a function of the network assumptions. As we increase the collaboration throughout the network in going from platform-centric to network-centric to futuristic network-centric (to use the RAND categories), so the positive effects of enhanced collaboration have to balance off against the downside effects of information overload and increasing network complexity. Going back to the discussion in Chapter 2 on the Conceptual Framework of Complexity, we can call this overall assessed performance of the network the plecticity of the network, since it characterises the combined positive and negative effects of network complexity and collaboration.

There are two ways to define a cluster of agents. The first, and most usual, is to define neighbouring agents only by those that are north, south, east, or west adjacent to the agent in question, known as nearest neighbour clustering. The second (and although most intuitive, less used) definition is to include all eight neighbours of the central agent as part of a cluster, as shown in Figure 5.2, which is known as next nearest neighbour clustering.