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2. Components of Network Theory

In network theory the key concepts are structure and dynamics. A network's structure consists of a configuration of nodes connected together by links into definable clusters. You need to take a snapshot of the network to see it's structure.

Nodes are identifiable parts of a system. They can be physical (e.g. individual computers or humans) or virtual (e.g. particular web pages or beliefs).

Links represent whatever happens between nodes. Links operate on two levels: physical and informational. The physical medium (e.g. roads or telephone cables) makes the transfer of information possible (e.g. volume of traffic or messages between computers). Links are also known as ties, bonds and connections.

Clusters are groups of nodes that are inter-linked into a unit. The great majority of interactions between nodes is contained within the group. Most people's friends are also friends of each other. Clustering breeds redundancy (multiple pathways between nodes) which keeps the network going when nodes or links fail, but it also breeds parochialism -- it keeps local events local and prevents them from escaping into the rest of the network.

Dynamics has two meanings:

"The first meaning is what we might call dynamics of the network. In this sense of the word, dynamics refers to the evolving structure of the network itself, the making and breaking of network ties. ... A dynamical view of networks, claims that existing structure can only be properly understood in terms of the nature of the processes that led to it. The second meaning, is what we might call dynamics on the network. From this perspective, we can imagine the network as a fixed substrate linking a population of individuals, but now the individuals are doing something -- the outcome of which is influenced by what their neighbors are doing and, therefore, the structure of the network. ... In the real world, both kinds of dynamics are going on all the time. ... The structure of the network could change, but so could the pattern of activity on the network." Six Degrees, p. 54-55

Certain configurations of nodes and links have been identified as signatures of complex adaptative networks:

Nodes are either just nodes or they are hubs - exceptionally large nodes in terms of the number of other nodes they are connected to. Hubs {'keystone species' in ecology and 'connectors' in sociology} are not just well connected - they are massively connected. e.g.

"90% of all documents on the web have 10 or fewer links pointing to them, while a few, about 3, are referenced by close to a million other pages." Linked, p. 58

Links can either be weak or strong depending on the amount of interaction between two nodes. Weak links can also be local or long-distance. Local and distant are metaphors. In network theory they do not represent physical distance, they are a measure of interconnectivity. Local links are those within a cluster and long-distance links connect clusters.

That means there are three types of links:

Strong and Local - high interaction between two nodes within a cluster. This is where most of the action takes place, say between good friends.

Weak and Local - low interaction between two nodes within a cluster. These links make up the majority of the network, say between you and your neighbour's or work colleagues.

Weak and Distant - low interaction between nodes in different clusters. These "weak ties" (Mark Granovetter) or 'shortcuts' are the 'bridges' that bind a network together and create small worlds. This means they are also an expression of the relationship between the individual and the group level, even though the individuals at the ends of a weak tie will rarely be aware of the role they are playing, say between you and an acquaintance in another country.

[Note: The fourth category 'Strong and Distant' links, do not exist because in network theory if two nodes are strongly connected they are automatically part of the same cluster and therefore 'local'.]

A measure of the connectedness of a network is the average minimum path length (number of links or hops) between any two points. This is know as the degree of separation. [see footnote]

As you will see below, to grasp the significance of the ideas emerging from network theory you need to grasp the central role played by hubs and weak ties. Hubs will inevitably have all three types of links and therefore they are importance not just because of their size but also because of the central role they play in the architecture of a network. Weak ties are what make a 'small world' small.

It seems that complex adaptive networks are almost always what is known as small-world networks. That is they are highly clustered and highly interconnected at the same time. e.g. the brain has lots of local interconnections, it is discernibly modular, and it has a few long distant connections that link the clusters and make the network an integrated whole. (Gerald Edleman calls the weak links in the brain "re-entrant pathways"). The result is that small-world networks have a surprisingly low degree of separation.

"What distinguishes a small-world network is not only that it has a low number of degrees of separation but also that it remains highly clustered. We might say that the fabric of the network is densely weaved, so that any element remains comfortable and tightly enmeshed within a local web of connections. Consequently, the network overall can be viewed as a collection of clusters, within which the elements are intimately linked, as in a group of friends. A few 'weak' links between clusters serve to keep the whole world small. ... On the other hand, there are drawbacks to too much clustering. ... At its core lies the idea that too much order and familiarity is just as bad as too much disorder and novelty. We instead need to strike some delicate balance between the two." Nexus, p. 199-207

There are two kinds of small-world network, "egalitarian and aristocratic" (Mark Buchanan), i.e. those without hubs and those with hubs.

In egalitarian networks most of the nodes have approximately the same number of links. Examples are: road, rail and air networks; the electrical power grid and other distribution networks; the network of synchronising fireflies; the neural network of the nematode worm and the human brain.

A NEURAL NETWORK

In aristocratic networks a few elements - hubs - have a disproportionally large number of links (see below: power law distribution). Examples are: The Internet and the World Wide Web; ecosystems, food webs and the metabolic network of cells; the networks of individual airline companies; the networks of references in scientific papers and of the co-authors of those papers; networks of directors of major corporations and sexual partners; and the closeness of words used in English sentences.

AN AIRLINE NETWORK

A few caveats to bear in mind:

Whenever considering 'a network' it is worth remembering that a network is a construct {map, model, metaphor} created by someone to make sense of some aspect of the world. A network comes into existence when someone conceives of some bits of the world as a network - and usually draws a map of their conception. There is no network of species or even World Wide Web pages but it currently seems like a useful way to understand the interactions between living things or how the WWW is interconnected.

This stuff is so new (all the popular books on network theory have been published in the last ten years) that there are bound to be major revisions. It really is 'work in progress'.

"Claiming that everything is a small-world network or a scale-free network not only oversimplifies the truth but does so in a way that can mislead one to think that the same set of characteristics is relevant to every problem. If we want to understand the connected age in any more than a superficial manner, we need to recognize that different classes of networked systems require us to explore different sorts of network properties." p. 304 Six Degrees.

[JL And how do we do that? We use bottom-up modelling]

Except where there are physical entities that are physically connected together, all talk of nodes, links, weak and strong ties, hubs and connectors is metaphor. And for every aspect that a metaphor illuminates, it hides something else in shadow.

Because the study of networks is such a new field and because many of the researchers have different scientific backgrounds there are many terms for similar phenomenon. I have tried to reflect this by giving several of the most commonly occurring names. Also, the terminology of networks can be mapped onto the terminology we use in Metaphors in Mind:

COMPARING TERMINOLOGY: LEVELS OF ORGANISATION

NETWORK THEORY


SYMBOLIC MODELLING

Network

<-->

Metaphor Landscape

Cluster

<-->

Perception

Links

<-->

Relationships

Nodes

<-->
Components / Symbols

Footnote

"The 'average pathlength,' formalizes the intuitive idea of degrees of separation. To calculate it, take any pair of nodes and count the number of links in the shortest chain between them; then repeat for all other pairs of nodes, and average the resulting chain length. ... The average amount of overlap in a network is quantified by a second statistic, the 'clustering,' defined as the probability that two nodes linked to a common node will also be linked to each other. ... Average pathlength reflects the global structure; it depends on the way the entire network is connected, and cannot be inferred from any local measurement. Clustering reflects the local structure; it depends only on the interconnectedness of a typical neighborhood, the inbreeding among nodes tied to a common center. Roughly speaking pathlength measures how big the network is, clustering measures how incestuous it is." Sync p. 239-241

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