## How Complex is the Net?

Craig Calhoun's response to Manuel Castells' trilogy The Information Age, brings up an important topic and one which comes to my mind often, namely that of the human experience of infinity, and a related concept, namely complexity. Similarly to Baudrillard, Calhoun questions whether we can track complexity in, e.g., global power relations or finance markets simply using personal experience. He points out that the tools made available to us by analysis etc. are in contrast to personal experience and are or may be required for a deeper understanding of globalisation.

I use the word complexity in the above, whereas Calhoun just intimates it. The complexity I refer to is counter-intuitive and not accessible to ready reasoning by laypersons. So to most of the planet's inhabitants the discussion is in fact incomprehensible and they react in wonderment at developments around them and in the news about politics etc.

The relationship between these two concepts is mathematically founded. Complexity arises in real world networks when networks along several related dimensions are considered. Trying to understand such interrelatedness by simple brute force enumeration (simply counting individual pairs of relationships) in diverse networks as biological, political, social, etc. generates massive numbers. These numbers, although they can be made to dwarf such physical amounts as the number of atoms presumed to exist in the entire universe simply by increasing the number of networks involved in the analysis, still do not come close to infinity. In fact, one can prove that even as one uses this enumeration to "count up toward infinity" (something one cannot really do - hence the inverted commas), one is still just as far from ones goal as when one started. This is a typical counter-intuitive result that one experiences when dealing formally and logically with infinity, and one which cannot be derived from experience. Friends of mine, who say that they have experienced infinity in cosmic appreciation of some natural feature miss the point. That is that there can be no experience as such of infinity, unless it is mediated by the paradoxes formally demonstrated by mathematics, such as Hilbert's paradox of the Grand Hotel (how to get your room in a full hotel). Also in some of the proofs that demonstrate clearly the difference in types of infinity, one of the most elegant enlightening is Cantor's diagonal argument, fantastic and ably explained here (

http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument). Humans cannot experience infinity, we can only experience mediocre amounts of information. We live in a very thin sandwich of reality which while appearing vast to us is very much finite. We are for instance not aware of the infinity of a continuous wave of a single note, our senses cannot perceive it. It is only through imagination and extrapolation that we can appreciate the very very large and the very very small. For instance, we can couple the sound of the note and the expression of that note as a line on an oscillograph, and a mathematical construct, the sine curve in order to get into the very very small. However, our extrapolations and imagination are still bound to our reality and they cannot possibly fathom experience (in a deep human sense) the paradoxes of infinity.

To come back to the discussion on understanding the complex systems that we have built up today, we can pose the question: are these finite or infinite systems? Any network theoretician will tell you, that a snapshot of the system is finite, no mater how many transactions at how many nodes on the graph, or how many arcs there are connecting these nodes. However, where the system does become infinite is in its dynamic nature over time. The arcs on the graph may namely be weighted with shifting weights at any point in time, and the graph can be seen as "multi-dimensional" as various nodes participate in several separate whose topology varies over time. Thus our analytical tools and experience cannot form a complete picture of the processes going on. We can only map portions of the processes in the Net in terms of snapshots or consider dynamic process in terms of reduced dimensionality and the tools we use are not founded in our experience, they are mathematical. The snapshots are thus immediately inaccurate as the network is constantly changing, also the process models are a priori inaccurate (like weather forecasts) -- we are able thus to postulate a "Net Uncertainty Principle" (a la Heisenberg), which states that the analysis of a single transaction won't tell us about the flows in the Net and an analysis of the flows will necessarily abstract which transactions took place. To make matters worse, there are large amounts of transactions about which it is extremely hard to get information. These transactions relate to "sensitive issues", such as weapons or drug transactions. These transactions only work if they are made in secret. Current regulations and the Net architecture allow this anonymity and secrecy.

In conclusion, we are dependent on formal methods to understand our world and to deepen our experience of it, but to make truly informed decisions and to make truly informed statements that can guide our nations and policies, we need to go beyond mere experience to analysis and then back again. This is one of the beautiful things about Manuel Castells' The Information Age trilogy. It is grounded strongly in his experience, but coupled with reviews of formal analyses and proofs. Further, the giddying high-level perspectives still hold today and strongly inform how the world has developed and also why Barack Obama's advances have been so slow. Castells manages to integrate formal ideals and analysis with experience to build new experience and thus to raise ones level of interaction with the world. Thus, in this instance, we come a step closer to experiencing the infinite within our networked world, but mathematically speaking our experience and analyses still leave us infinitely far from it.

**Posted:** December 25th 2010 04:40

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