Probabilistic Networks

November 1, 2011 § Leave a comment

Everything is linked together and related.

There always have been smart people who nor only knew this, but also considered it as primary against the point, the dot, the spot. Thinking in relations is deeply incompatible with one of the most central elements of modernity, the metaphysical belief of independence. Today, in 2013, in the age of the ubiquitous “network,” everything indeed does seem to be improved, doesn’t it, given the fact that for the last 5 or 6 years the concept with the steepest career is the network.

Certainly, one of the main reason the network became a major concept from everyday life to science is given by the fact that connecting things, establishing links between devices and establishing the potential for population of links became a concrete experience, even for private persons. Before the era of WiFi and its almost perfectly automated process to establish a link, the network has been something very palpable. There have been modems for dial-up, confirming their working by a twittering sound, a lot of cables in the office, and the frequent experience of a failure of such technical infrastructure. In other words, networking became an activity with its own specific corporeality.

So, what does it mean to say that things are connected? What are the consequences, both regarding the empiric side concerning the construction or observation of systems or machines, or regarding the conceptual level? For instance, so far there is no particular “network logics”. Whenever networks meet logic, logic wins, meaning that the network will be reduced to individual steps, nodes, transfers, etc., in other words unrelated atoms.

Intuitively, the concept of networks is closely related to the notion of information. Today, this linkage has been integrated deeply into our Form of Life. Through the internet, the world wide web, and of course through the so-called social media we experience and practice this linkage in a rather intensive manner. And the social media just invoke a further important topic that is related to networks: mediality.

Here we meet a first hint for potential friction. Networks are usually well-defined, people speak about nodes and relations. Think just about the telephone network or a network of streets. Even social networks are explicable. Yet, in social media the strict determination starts to get lost. While social media are still based on a network of cables, something different is going on there, which is drastically different from the cable-layer.

The notion of partial indeterminateness brings us to mediality and its inherent element of contingency and probabilism. Yet, what does “probabilistic element” exactly refer to? Particularly with respect to networks? Is it, after all, not just some formalistic exercise to say that there is a random element, largely superfluous when it comes to real systems and problems? Particularly, as cultural artifacts are planned. Actually, I don’t think so. Quite to the opposite, one even would say that in some sense non-probabilistic networks are not networks at all.

In the remainder of this essay we will have to clarify the issues around these concepts, both regarding physical systems and the conceptual aspects, as well as the aspect of application. We will have to take a closer look to the elements of the network, nodes and links, as well as to to the network as an entirety. There is the question of the telos of the network. What is it that networks as a whole introduce? Is it possible to ask about their particular quality, beyond the trivial fact that things are connected?

Such, we first will deal with networks, their elements and the properties of both in a basic manner.

1. Basics of Networks

When dealing with networks, there is immediately a strong reference to topology, that is the way in which items belonging to the network are linked together. More precisely, what actually matters concerning the topology of networks are the symmetry properties of the connectedness. It does not really come as a surprise that the issue of symmetry relates networks to crystals, (mathematical) groups and knots. Yet topology and its symmetry is not the only important dimension.

1.1. Topology

So, before getting precise, let us start with a simple example for a network. What we see here are 3 nodes linked by 3 edges. The nodes represent items, while the edges represent certain relations between them.

o —— o
\       /
o

Actually, this example is almost too simple. Despite the fact that it contains all the basic elements, there are notably only 2, the node and the relation, many would not regard it as a network. What seems to be missing is a certain multiplicity of possible paths. Such a multiplicity would be introduced by at least one “crossing”, that is, we need at least one node that maintains three relations. In turn this means that we need at least 4 nodes to build an arrangement that could be called a network

o —— o
\       /     \
o —— o

On the other hand, we would consider arrangements like the following also as a network, though there is not multiplicity. It is a perfectly hierarchical structure, albeit there are several possible roots for it.

o —— o        o
\                  /
o —— o —-o
|
o — o — o —o
\
o

Obviously, we may distinguish networks by means of their redundancy. In physical systems, if we are going to connect points from a large set within a given “area” among each other, we usually try to avoid redundancy, since redundancy means increased costs for building and maintaining the network. Just think about a street network, the power grid, the water supply grid or the telephone network, in each case the degree of redundancy is quite low.

Yet, things are not that simple, of course. Some degree of redundancy could be quite beneficial if edges or nodes can fail. In case of the internet, for example, it was much higher at its beginnings. Actually, redundancy has been a design goal for the ArpaNet (next figure) for it ought to survive a nuclear attack to the U.S.

Figure 1: The logical layout of the ArpaNet in 1977.

IMG

scale-free Barabasi

1.2. Symmetry

 

1.3. Differentiation

Besides redundancy Taking the case of a street network as an example, the streets between crossings interpreted as edges or relations, we immediately see that beside the redundancy also the transfer capacity of edges is a further important parameter.

 

Those should be clearly distinguished from logistic networks, whose purpose is given by organizing any kind of physical transfer. Associative networks re-arrange, sort, classify and learn

logistics and growth

Yet, we only are at the beginnings to understand what networks “are.” Since there are a lot of prejudices around, we will first give some examples. The second major section discusses the main concepts and adds a few fresh ones. The third section discusses the consequences of changing a network int a probabilistic one.

mapping of items (objects) to nodes and relations to edges

(under construction)

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