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 = .818181818181818…
Until the 1970s, mathematicians considered random
structures to be the most complicated
Spatial
randomness
Temporal
randomness
Numeric
randomness
dice roll = 396158294106538
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?
Randomness that can be characterized by statistics is
not the highest in complexity
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.
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.
Salt crystal forms
from evaporating
water. Completely
ordered. Trivial
case.
Self-organization tends to produce fractal structures systems
between total order and total disorder
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