"All structure is a manifestation of underlying process."
-- Fritjof Capra
Patterns in nature are complex networks that have a physical architecture (form) and associated dynamic processes (function) that control and connect the pattern components within the system. Understanding these patterns comes from looking at both the architectural and the dynamic relationships within these networks. This web page discusses the architecture of network connectivity within nature's patterns. The page on network dynamics explores the flow of energy and processes within these networks.
The study of network architecture seeks to understand the organization of relationships, or connectivity, between the pattern junctions (nodes) in a dynamic system. The physical or logical organization of a network, its pattern of connectivity, is called it's "topology". The organization of a network is important to study because, like the spider's web, there are many different ways to get from one intersection to another. But only certain pathways are advantageous.
Connectivity in a natural patterns is very important to characterize because it is connectivity that permits processes and energy within a system to flow. For example, the topology of social networks influences the spread of both information and disease. The topology of an electrical power grid affects the reliability of power transmission.
One way to understanding networks is to build models and play with these models. Models help us focus on key factors, setting aside the irrelevant. In this way, the simplification has the potential of clarifying the important features of a system under study.
A network model is a graphical portrayal whose purpose is to study connectivity of junctures within a system. These junctures are called "nodes" (or "vertices"). Nodes are connected by "links" (sometimes called "edges") that represent the relationships, energy transfer agents, or processes between nodes. Both nodes and links can be patterns themselves. Any node can accommodate any number of links.
Network science has created certain metrics (measurements) to help quantify connectivity in a network model. Four such metrics are "degree", "degree distribution", "path length", and "clustering".
The number of links that connect to a given node is called the "degree" of that node. Degree is a very important metric because it quantifies the connectivity of a node to the rest of the network. For example, a node with five links attached has a degree of 5. Another node might have a degree of 9. A node with a degree of 9 has a higher connectivity than a node with a degree of 5.
A very important network characterization is the statistical distribution of node degrees within an entire network. Degree distribution of a network characterizes its overall topological structure by defining the nature of connectivity within a pattern. Degree distribution directly effects the operating dynamics of the network.
Degree distribution is one of the mathematical parameters that can be used for classification of networks. If you have a complex system/phenomenon to study, you can often represent this system as a network, or an ensemble of nodes and connections between them, and analyze the connectivity distribution of this network. If it follows a random distribution, your system or phenomenon can be approximated by random network models where each node contributes equally to the overall connectivity of the network. If the degree distribution of your system approximates a power law, then it can be modeled as a scale-free network where hubs offer much higher connectivity than other nodes in the network.
Path length is the shortest distance between any two nodes in the network. Since actual physical distance between nodes is not important in most network models, the path length between a node and a nearest neighbor is defined as 1.0. If the minimum path between a given node and a distant node is three links, the path length is said to equal 3. This metric is important because one characterizes connectivity by looking for the shortest connection between nodes.
"Clustering" is a measure of a node's connectivity to other immediate nodes. It is an indication of network redundancy. Clusters are immediate groups of nodes much like a group of friends.
Network science has found that a combination of short overall path length and high clustering produces high connectivity. In turn, high connectivity produces sufficient energy flow or process channels to sustain a pattern in nature.
Most successful real world networks have a high connectivity that results in emergent behavior within the system. They are considered to be "complex" because they have a topology that doesn't exist in simple network models. Indeed, their topology is somewhere in between totally regular and totally random. Real world networks have enhanced connectivity because they typically have high clustering and short average path lengths between remote nodes. These patterns in nature are said to employ a "small-world" architecture because any two nodes in the network are likely to be connected through a very short sequence of intermediate steps even though most nodes are not neighbors and may have many intermediate regular links. Some nodes are "clustered" -- having a high connectivity to neighbors.
You are invited to explore more detail by looking at regular networks, random networks, small world networks, and scale-free networks.
This document is a draft of work in progress. Please post your comments, thoughts, and suggestions:
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