Measuring Complexity Ron Eglash, RPI

Intuitively, we know systems are complex if they have “nested” or “nearly decomposable” hierarchies of scale Note that hierarchy of scale is not a hierarchy of control

But 1. The “cut-off points” are too subjective to use 2. This misses many forms of complexity—for example within one scale, such as organism, we can see clear differences in the complexity of spatial structure A sparsely branching cactus vs a rich, densely branching cactus

We can all agree on what is “order” Spatial Order Temporal Order Numeric Order 9/11 = …

Until the 1970s, mathematicians considered random structures to be the most complicated Spatial randomness Temporal randomness Numeric randomness dice roll =

Modern era (1960s) view: complexity as randomness (Kolmogorov-Chaitin measure)

Post-Modern era (post-1970s) view: complexity as between random and ordered (Crutchfield-Smale measure) white noise fractal noise

Which is more complicated? Which is more complex? A gas made of 15 million molecules randomly crashing about? OR A school made of 15 fish gracefully swirling though water?

Temperature: an emergent property of swarms of molecules. But temperature is based on the average velocity of molecules (E=3kT/2). Linear relation, you can use statistics. Flocking, on the other hand, is an emergent property of swarms of birds, and similar collective behavior occurs for ants, fish, etc. Flocks are not well characterized by averages or statistics, their reactions are nonlinear. They are adaptive, anticipative, have memory. Randomness that can be characterized by statistics is not the highest in complexity

From disorder to ordered Toss a handful of particles in the air: “self-organized” but without order. Trival case Sand waves from wind action: a quasi-ordered emergent pattern. Significant case. Self-organization tends to produce fractal structures systems between total order and total disorder Salt crystal forms from evaporating water. Completely ordered. Trivial case.

We now see the most complex structures as those from self-organizing processes: mix of order and randomness lungs fern algae

We now think of the most complex structures as those from self-organizing processes: a mix of order and randomness Lungs fern algae

Fractal structures are common not just in nature but in artifical and social systems as well Tree representation of threaded online conversation

Network Theory example: Measuring the fractal dimension of conversation trees Visually intuitive: Sparse trees = low fractal dimension, lush trees = high fractal dimension. The dimension number can take over when our intuition fails. There is a powerful connection to complexity theory…

Fractals in Complexity Theory Complex adaptive behavior ranges from orderly, equilibrium behavior to random, disequilibrium behavior. Low fractal dimension (eg 1.01) is associated with the order end, high (eg 1.99) with disorder end. Complexity is “inverted U” function: highest between integers (eg 1.5). Kaufman’s “edge of chaos,” Bak’s “self-organized criticality,” etc.

Tree variation as variation in fractal dimension Df = 1.05 Sparse conversation Df = 1.50 Conversation at the "edge of chaos" Df = 1.70 Dense conversation

How to counter the reductive tendancy in network modeling? Reductive example: Linked: The New Science of Networks by Albert-László Barabási. The Internet’s “highly connected hubs” (due to fractal structure) greatly increase vulnerability to planned attack. Used data showing that networks of human sexual contact have a fractal structure. Concluded that HIV infection rates could be greatly reduced by targeting the same “highly connected hubs” – sexually promiscuous individuals.

Targeting “highly connected hubs” in sexual networks can increase HIV rates In Africa, for example, people connected with AIDS risk have been subject to harassment, violence, and even murder. As a result, communication about HIV is very poor Lack of communication greatly increases transmission rates. Fixations on sexual promiscuity in Africa have been closely linked to right-wing religious opposition to condom use.

Conclusion Thus reflexive or recursive engagements are not merely negative barriers to truth claims. They can be a positive force in generating representations that are both diverse and objective. Complexity theory can open up representations to the “trielectic” between natural, social, and technological worlds