How many interacting elements are required for emergent behavior to arise? The answer depends upon the complexity of the elements and their interactions.
At one extreme, vast numbers of simple elements with simple
interactions can form complex behavior. Gravitational attraction between
atoms created galaxies, stars and black holes
(see other examples of emergence).
At the other extreme unexpected behavior can emerge from only two
interacting elements if their interactions are rich and they are sufficiently
adaptable (consider a marriage).
In general, the less complex the interactions the more individuals are required to produce surprises. But unless the interactions are very simple, it doesn't take all that many. Small flocks of birds such as a few pellicans, or schools of a few dozen fish can provide surprising collective behavior. And, as every programmer has discovered, even small software systems all too often demonstrate surprising emergent behavior (bugs) that are difficult to understand and fix.
Multicellular systems, whether in biological or digital realms, have many elements that are themselves complex dynamic systems that interact with each other in quite complex ways.
Neurons are much smaller than grains of sand yet their interactions are far richer. Neural "circuits" of only a few hundred to a few thousand neurons can produce amazing behavior. The entire central nervous system of C. elegans, a roundworm used in biological studies of development, has 302 neurons and about 5000 synapses, yet it demonstrates social behaviors of a primitive sort and something akin to associative learning. The nervous system of Aplysia (a mollusk) contains 18-20 thousand neurons. These animals aren't noted for their intellectual gifts, yet their behavioral repertoire is far greater and more "purposeful" than a hurricane or a sand dune: they seek food, avoid danger, mate, and learn. Aplysia, demonstrates learning in both classical conditioning experiments and operant reward conditioning.
The earliest computers had a few thousand gates and programs with a few thousand machine instructions. While their planned emergent behavior, i.e., what they did when a program was run, was primitive by today's standards, they still could calculate artillery ballistic tables, prime numbers and the like. Unplanned emergent behavior (bugs) popped up even then. In modern computers with upwards of a billions gates and hundreds of millions of lines of code the behavior, both planned and unplanned, is beyond our comprehension.
What, then, of the Internet with more than a billion elements,
each quite adaptable and each interacting in complex ways with
multiple other elements? The simplest answer is,
no one knows, nor can know, nor ever will know in detail.
Emergence in the Internet is exploding at such a pace and with
such subtle and high-speed interactions that it is already literally
incomprehensible. A given element
(say one PC or smart-phone in the net) simultaneously
participates in many emergent multicellular phenomena that operate
at different time scales. For example, a smartphone, wireless Netbook,
or iPad is interacting at millisecond scales with cell towers or WiFi
hubs while perhaps doing some web eCommerce interaction at second
or minute scales, or interacting with FaceBook blogs or Twitter at
minute or hourly time scales, or requesting a Google search of
Internet-scale information that itself is changing second by second
as Google's crawlers and ever-adapting page ranks do their work.
Individual computers participating in the Internet are only the
first metalevel of Internet interaction. The trails of browsing
left by these individual computers become the fodder for huge
advertising networks, e.g. Google Adsense among many others, that
compete with each other to organize consumer marketing campaigns
in other sets of computers, or to guide political messaging campaigns.
Sophisticated and well organized sets of computers create and/or
exploit financial markets that affect the world banking system and
therefore its economy.
Massive intelligence operations track and analyze all emails, blogs,
and phone conversations world-wide in an effort to identify and/or
foil unfriendly efforts by international bad-actors. Malware botnets
form and are exploited by cyber-criminals in the cracks of the
systems. Geotracking and social networks create their own sots of
stigmergy structures that are the territory of the next generation of
We can no more understand the emergent Internet phenomena in which our personal digital devices participate than can a neuron "understand" the brain of which it is a part.