Kurt is an Engineer, Physicist and Publisher. As an engineer he has built data-driven web-based applications and has designed microchips for many companies including DIRECTV, Panasonic, Thuraya, SES, Lockheed Martin, SLAC, General Dynamics, and NASA. He was also a Senior Systems Analyst for NASA on their Gamma-Ray Large Area Space Telescope (GLAST, now Fermi). As a physicist he is an active researcher in fundamental complexity theory with a particular interest in the nature of boundaries, and the relationship between form and function in complex dynamical networks. As a publisher he owns and runs Emergent Publications which specializes in publishing academic works concerning Complex Systems Thinking. It's flagship publication is the international journal Emergence: Complexity & Organization.

A common assumption in the ‘modern’ era is that ‘being connected’ can only be a good thing for individuals and for businesses, and even nation states and continents. This short article aims to explore this assumption with the use of Boolean networks. Although the research presented here is in its early stages, it already demonstrates that there is a balance to be met between connectivity and performance, and that being well-connected does not necessarily lead to desirable network performance attributes.

In the ‘information age’ there is a general belief that having access to as much information as possible is a good thing, leading to more effective decision making and thus allowing us to rationally design the world in our image. Of course, any sophisticated view of the information age has to acknowledge that having information for the sake of information is not much more useful than having no information. What is key is having access to the right information at the right time, where knowing both what is the ‘right’ information for the ‘right’ time are highly problematic assessments. We are slowly beginning to acknowledge that in a complex world, ‘states of affairs’ much be considered from multiple directions whilst maintaining a (constructively) critical disposition at the same time. This comes about because the external concrete reality we perceive is not quite as solid as our more recent scientific heritage would suggest. The relationship between our universe and our understanding of it is, dare I say it, very complex.

Along side this popular belief that information should be abundant and freely available, is that being connected is also the preferred state of existence. Like all ideas, ‘being connected’ has limits to its usefulness. Is there a preferred degree of connectedness? An individual who was not connected at all could achieve nothing. Indeed, without some degree of connectedness, an individual could not even be recognized as such - we are, to a large extent, defined by our connections. At the other end of the spectrum, if an individual is too connected then it becomes also most impossible to achieve anything coherent. If too much is going on at once, and no stable patterns emerge (even if only temporarily) then again the notion of the individual and his/her identity is impossible to discern - the whole system becomes the only useful unit of analysis.

This short discussion article represents an early report into some research I have been performing that explores the relationship between connectivity and system behavior. The area I will be focusing upon herein is how a system’s behavior develops as more and more inter-connections are added between individual system components. As in other articles I have written for

I do not plan to go into great detail about Boolean networks and how they are constructed and simulated. The interested reader might read Lucas (2007), or Richardson (2005b). In this section I want to introduce a number of parameters that can be used to characterize the phase space of complex systems (Boolean networks in particular). I hope that after 8½ volumes of

Number of attractors, 1. _{att}

The weight of the largest attractor as a per2. centage of the total size of phase space, _{max}

The maximum number of pre-images for a 3. particular network configuration, _{max}

The longest transient for a particular net4. work, _{max}

The number of Garden-of-Eden states as a 5. percentage of the total size of phase space,

The average robustness for a particular net6. work,

The number of phase space attractors is probably the most single important characteristic of a nonlinear dynamical system. In Boolean networks, all trajectories end up (after some transitory period) on periodic attractors. Given that Boolean networks are discrete networks one would never observe the type of chaotic behavior as seen in certain continuous systems. However, attractors with periods much larger than the size of the network are often referred to as quasi-chaotic or quasi-random. Indeed, it is a relatively trivial task to create Boolean networks that contain attractors with periods so large that for all intents and purposes (and by all standard tests of randomness) they are random. The primary interest in the number of attractors displayed by a particular network, is that this number corresponds directly with the number of qualitatively different behavioral modes that particular network can operate in. So, if a network only exhibits a single attractor then it only has one

The attractors themselves generally only contain a small percentage of the number of states comprising the network’s complete phase space (2^{N}

Although, for Boolean networks, the application of the rule table results in only one unique state (the networks are

The maximum transient length, _{max}

As phase space is finite, if settling times are long then fibrosity is relatively low. At one extreme we have short fibrous attractor branches that ensure that the central attractor is reached quickly from many possible starting points. At the other extreme we have long sparse attractor branches that can result in significant delays between action and desired response (although the system will respond immediately by following a trajectory toward the characteristic attractor).

Figure 1 also indicates a state labeled ‘GoE’ which is shorthand for ‘garden-of-eden’. Mathematically, when rolling back time to construct these attractor basins a point is eventually reached for which there is no solution. These states cannot be reached from any other states — they are, in a sense, the beginning of time for the particular trajectory they ‘create’ — hence, the reference to the Garden of Eden (see Wuensche & Lesser, 1992; Wuensche, 1999). From a control perspective, internal mechanisms do not have any capacity to access GoEs — for all intents and purposes they do not exist, from an

I have already suggested that the number of phase space attractors is one of the more important attributes of a dynamic system. However, although two states may be next to each other on an attractor basin (i.e., in

In one sense the total number of attractors is equivalent to the number of contexts, or

However, dynamic robustness also has an effect on a network’s ‘sight’. Dynamic robustness can be high if the states associated with each basin are distributed in such a way as to maximize R for a particular number of basins (this is more likely when no one basin is significantly heavier than the others). However, networks that are characterized by many basins may also exhibit high dynamic robustness if one of those basins is considerably larger (heavier) than the others. In this latter situation, although the network has the potential to ‘see’ and respond to many environmental archetypes, it overly privileges one particular archetype and so effectively blinds itself to others.

The heart of this article is concerned with how the various parameters presented above vary with increasing network connectivity, but before moving on to consider these relationships a brief presentation of the computer experiment performed will be given.

The computer experiment performed for this study starts with a network of ^{N}

The number of input combinations that a single node can respond is 2^{k}

Input B
Input A
State of node at next time step
0
0
0
0
1
1
1
0
1
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0

The rule for a node with this response configuration can be expressed simply as 0110. Now let us consider that the node has now acquired another input with node C. There are now 23 = 8 possible input configurations. For the algorithm used herein, when the state of node C is 0 (or off), the response to A and B is kept the same as it was before the connection from node C was added. The response to the additional four configurations that are formed when the state of node C is 1 (or on) are selected randomly. So,

C
B
A
State of node at next time step
0
0
0
0
0
0
1
1
0
1
0
1
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1
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1

Simply expressed the rule changes from 0110 to 1100

1 10

2 01

3 11000

4 01100110

5 11010001101011000

Etc.

When the network is fully connected (

Figure 2 shows the relationship between the number of phase space attractors, _{A}, and the percentage of GoEs with the number of connections, _{A} is due to the fact that as each new connection is made _{A} is halved. This is not always the case, but the important characteristic of this ‘ordered’ region is that the qualitative structure of phase space can easily be determined

The second region, which is labeled ‘complex’ sees a continued decrease (although at a much slower rate than in the ordered region) in the number of attractors, down to a minimum of around four attractors. The proportion of GoEs is also constantly high throughout this region, indicating that the vast

The next region, termed ‘chaotic’, is characterized by a steady and relatively rapid drop in the proportion of GoEs, from around 97% to 40%, and a slow rise in the number of attractors from approximately four to six. It is difficult to say much about this region without calling upon other data (which we will do shortly), other than there is a significant increase in states that are not only accessible from outsidethe system. The same is true for the last region — ‘random’ — for which the number of attractors and the proportion of GoEs remain relatively steady at six and ~40%, respectively.

Figure 3 shows the relationship between network connectivity, _{max}

The ‘complex’ region is where it all happens within this particular dataset. The weight of the heaviest basin increases rapidly, and the dynamical robustness increases to 0.85 (at _{max} can be interpreted as an increase in

There is little to distinguish the ‘chaotic’ and ‘random’ regions in this particular dataset. Both _{max} are relatively stable at 0.70 and 0.75, respectively, suggesting that the phase spaces of networks existing in these regions are dominated by a single heavy attractor basin.

Figure 4 shows the relationship between network connectivity, _{max} (or, fibrosity) and _{max} (transient, or settling time). If we consider the ordered region first then we observe that over this region fibrosity increases rapidly to a maximum of around 110, whilst at the same time, the maximum settling time remains very low. This indicates that whatever state a network in this region is initiated from, the trajectory will reach an attractor very quickly (bearing in mind that these attractors will be rather uninteresting as they are analytically trivial).

In the ‘complex’ region we see a rapid decrease in fibrosity along with a modest increase in maximum settling time^{1}. The phase space of the average network in this region,

The ‘chaotic’ and ‘random’ regions, which contains networks whose phase space is dominated by a single heavy attractor, are distinguished by the quite different structure of the branches attached to the states on the central attractor. ‘Chaotic’ branches tend to be quite long and quite fibrous, whereas ‘random’ branches tend to be very long and sparse.

Now that we have considered each of the datasets separately, we can move on to constructing the ‘bigger picture’ by considering all the datasets together and developing a fuller appreciation of each network type. Before doing that, however, I want to highlight that the regional denotations presented herein relate to the characteristic networks’ dynamics and not necessarily their structural topology (although connectivity is an important structural parameter). Network theorists distinguish between ordered, small-world, scale-free and random networks. These

Table 1 attempts to sum-up the observations made in the previous sections, by showing how each parameter changes qualitatively (e.g., ↑↑↑ = rapid increase) over each region. Figure 5 brings presents the data visually (and arguably more effectively) by comparing the network structure, the largest phase space attractor basin, and the state space configuration (the online PDF version of this article is recommended so that full color can be observed) for a ‘typical’ network selected from each region. Figure 5 vividly illustrates the differences in attractor basin structure and state space configuration and allows us to see directly the impact that parameters such as fibrosity and robustness have on these two ‘spaces’.

Qualitative change in principle parameters across each dynamic region

Network Type | N_{att} | PI_{max} | T_{max} | GoE | R | W_{max} |
---|---|---|---|---|---|---|

Ordered | ↓↓↓ | ↑↑↑ | ↑ | ↑↑↑ | ↑↑↑ | ↑ |

Complex | ↓↓ | ↓↓↓ | ↑↑ | ↔ | ↓ | ↑↑↑ |

Chaotic | ↓ | ↓ | ↑↑↑ | ↓↓ | ↓ | ↔ |

Random | ↔ | ↔ | ↔ | ↔ | ↔ | ↔ |

Network Type | N_{att} | PI_{max} | T_{max} | GoE | R | W_{max} |
---|---|---|---|---|---|---|

Ordered | H | M | L | M | M | L |

Complex | M | M | L | H | H | M |

Chaotic | L | L | M | M | H | H |

Random | L | L | H | L | H | H |

Network Type | N_{att} | W_{max} | PI_{max} | T_{max} | GoE | R | C |
---|---|---|---|---|---|---|---|

Ordered | 1 | 0.008 | 48 | 3 | 0.97 | 0.53 | 12 |

Complex | 14 | 0.400 | 84 | 33 | 0.89 | 0.71 | 40 |

Chaotic | 75 | 0.537 | 20 | 120 | 0.58 | 0.46 | 90 |

Random | 75 | 0.870 | 7 | 335 | 0.37 | 0.78 | 210 |

_{max} refers to the period of the heaviest (_{max}) attractor basin for a particular network. This additional data would be useful as the case of _{max} » _{max} varies with _{max} increases with _{max}, and the other parameters discussed thus far, for the four networks depicted in Figure 5. This very small sample confirms that the period of the heaviest basins (_{max}) for typical networks in the ‘chaotic’ and ‘random’ regions are indeed much larger (») than the network size (

We can now attempt to make a ‘standard’ statement about each of the four dynamic regimes: ordered, complex, chaotic and random. It should be noted, however, that the statements that follow are of the ‘on average’ variety, in that there will exist networks in each region that look (in a dynamic sense) very much like networks in other regions. A more obvious limitation of such a categorization process is that the boundaries between each region are fuzzy and certainly not discrete. The statements are, therefore, no more than guidelines. They certainly do not allow us to make statements of the sort “All Boolean networks with

A

A

A

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The emboldened terms are relative assessments that only gain meaning through comparison with the other types. We go even further with this process of data reduction. Consider the following slogans:

The more ‘distance’ however between the raw data and our summations, the more subjective dimensions play in our choice of words. For example, if a different slant was taken we might rewrite the slogan for chaotic as:

My point is simply that the process of categorizing complex datasets is problematic and requires a great deal of care.

The motivation for this article was to explore the merits of a modern assumption that ‘being connected’ is unquestionably a good thing. The research presented here is still in its early phases, but it is clear that different levels of connectivity can be associated with different types of dynamics. One interpretation would suggest that being too connected is actually overly restrictive and that there is a balance to be maintained. Much of the complexity literature argues that “maintaining balance” is central to a sustainable approach for existing within a complex (eco-) system. Of course, we must take care in importing the lessons from abstract models such as Boolean networks into the realm of human organization. That being said, what is rather surprising to me is how narrow the desired ‘balanced’ (complex) region is relative to the (seemingly less desirable) chaotic and random regions. Furthermore, if we consider the width of the complex region, _{width}, in relation to the whole of ‘connectivity space’ we can construct a measure of relative weight, _{complex}, equal to the width of the complex region divided by the total number of a connections in a fully connected network. In this particular study _{complex} ≈ _{complex}^{-1}, and so as

[1] The fact that a large decrease in _{max} is associated with only a modest increase in _{max} is simply that there are more states required to increase _{max} than _{max}. If _{max} increases rapidly then state space quickly runs out of available states before a phase space trajectory can get too far from the central attractor.