In any society, people link together and self-organize into groups. Volunteer civic societies form as people find like-minded men to coordinate their activities. There are limits to network growth. Geography, communication, and transportation technology set the maximum distance a network can cover.

The Statistical mechanics of complex networks by Reka Albert, Albert-Laszlo Barabasi provides an overview of network theory and its applications to real life.

There are several common features for networks. A system has many individual components that interact to form complex and emergent results. Because of this, relatively simple parts and small imputs can have major consequences because they enter a reinforcing loop that amplifies their effect. We can use network theory to study epidemiology, crime, warfare, and markets.

Complex social networks are “small worlds networks” and “scale-free networks.” Small Worlds networks means most individual nodes may connect with any other node on the graph with just a few steps. The common example is “Six degrees of Separation”

A Scale-Free Network is even more important. It follows a Power Law. It’s probability distribution is a long-tail graph, like the Pareto distribution. P(k) ~ k^−γ. In English, this means that some connectors grow more influential than others. Some nodes have many connectors; many nodes have few connectors.

Networks have a few basic organizations. Individual nodes have connectors to other nodes. They tend to cluster around “hubs” – critical nodes with many connectors. Matrix organizations are possible as well.

Here is a simple example of a hub: The Network Leader (A) connects with 5 subordinates (Bs). The Subordinates each command 2 other members (Cs). This is a very simple power law graph: A node has 5 connectors; B nodes have 3 connectors; C nodes have 1 connector. This is a small worlds network as everone is at most 4 steps apart.

Adaptation and preferential attachment allows for even greater complexity. For example – take authors. A famous author (one node) can write a book read by millions of consumer nodes. On the other end of the spectrum, there are millions of letter writers, with each letter only read by a single person. In between, there are less popular authors, magazine writers, and the like.

Adaptation and preferences allow for evolution in network structure and operations. This makes it follow non-linear dynamic patterns. Networks are highly robust as a result. There is a high level of redundancy and ability to adapt to changing circumstances. They can overcome internal errors and retain high levels of complexity (this is especially important to understanding biological complexity).

There is a repeated finding that networks are highly vulnerable to targetted attacks. Attacks against random nodes is no more effective than random errors. A targetted attack against critical nodes can fragment and disrupt the operations of a network. We can identify the nodes (k) with the greatest number of connectors (k max). Elimination of critical nodes and hubs reduces the complexity of the network. Individual cells become fragmented and scattered once the hubs and clusters are removed. They may, of course, recombine and form new networks.

This is an image of an emergent network by Vladis Krebs.