## The Network Model

An important source of self-organization is provided by the interactions and relationships between the objects that comprise a complex system. Patterns of such relationships are captured by the network model of complexity.

Networks capture the essence of interactions and relationships, which is a fundamental source of complexity. A graph is defined to be a set of nodes (objects) joined by edges (relationships) and a network is a graph in which the nodes and/or edges have values associated with them. In a food web, for instance, the populations form nodes, and the interactions between them (e.g., predation) form the edges. In a landscape, spatial processes and relationships create many networks. For instance, the nodes might be individual plants and the corresponding edges would be any processes that create relationships between them, such as dispersal or overshading. In an animal social group, the nodes would be individuals and the edges would be relationships such as kinship or dominance.

Nodes that are joined by an edge are called neighbors. The degree of a node is the number of immediate neighbors that it has. A path is a sequence of edges in which the end node of one edge is the start node of the next edge, for example, the sequence of edges A-B, B-C, C-D, D-E forms a path from node A to node E. A cycle is a path that ends where it starts, for example, A-B, B-C, C-A. A network is called connected if, for any pair of nodes, there is always some path joining them (otherwise it is disconnected). The diameter of a network is the maximum separation between any pair of nodes. Clusters are highly connected sets of nodes.

The importance of networks stems from their universal nature. Network structure is present wherever a system can be seen to be composed of objects (nodes) and relationships (edges). Less obvious is that networks are also implicit in the behavior ofsystems. In this respect, the nodes are states of the system (e.g., species composition) and the edges are transitions from one state to another.

Sometimes, network structure plays a more important part in determining the behavior of a system than the nature of the individual components. In dynamic systems, for instance, cycles are associated with feedback loops. In disconnected networks, the nodes form small, isolated components, whereas in connected networks, they are influenced by interactions with their neighbors. Self-organization in a network can occur in two ways: by the addition or removal of nodes or edges, or by changes in the values associated with the nodes and edges.

Several kinds of network patterns are common and convey important properties.

• A random network is a network in which the nodes are connected at random. In a random network of n nodes, the degrees of the nodes approximate a Poisson distribution, and the average length (L) of a path between any two nodes is given by L = log(n)/log(d), where d is the average degree.

• A regular network is a network with a consistent pattern of connections, such as a lattice or cycle.

• Small worlds fall between random networks and regular networks. They are typically highly clustered, but with low diameter. A common scenario is a system dominated by short-range connections, but in which some long-range connections are also present.

• A tree is a connected network that contains no cycles. A hierarchy is a tree that has a defined root node. For instance, the descendents of a particular individual animal (the root of the tree) form a hierarchy determined by birth. Trees and hierarchies are closely associated with the idea of encapsulation.

• A scale-free network is a connected network in which the degrees of the nodes follow an inverse power law. That is, some nodes are highly connected, but most have few (usually just one) connections.

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