Dave Garvey discussed Alidade’s recent research into network metrics. He detailed the three most important aspects of complex networks:

• Structure – the definition of links, nodes and connection rules.
• Dynamics – the mechanisms by which networked effects are achieved.
• Evolution – the behavior of the network as it adapts to its environment and to competitors.

A great deal of recent study has identified a growing number of statistics that describe important complex network characteristics. These statistics converge to certain ranges or distribution in the many other domains that this research has examined. Some of these statistical values can be used to determine adaptability, robustness, survivability and many other desirable properties that should be engineered into complex military networks.

These properties and their suggested values are offered as thumb rules for Information Age analysis and experimentation. In addition, since they also indicate something about the structure of a network, they can be used as first order search metrics, if only to help searchers understand the networked context of a search, to help navigate within a network or to understand which parts of a network might not yet be detected. In addition, the defense community has begun development of networks that search; these networks can also be examined using these network metrics. The following metrics are important for understanding networks in this context.

Number of Nodes, N. Although some future concepts contain, for example, references to “network-centric warships,” networked effects depend on the presence of a large number of nodes. In general, significant networked effects are unlikely to be realized in a network of fewer than 50 nodes.

Link to Node Ratio, l/N. Just as important as the number of nodes is the number of links. The lengths of paths in the network, local cohesion, survivability and adaptability also perform well in complex networks with far fewer links per node than N-1, the ratio found in maximally connected networks. As a general rule, complex networks should have about two links per node.

Degree Distribution. The number of links per node is not usually uniformly distributed throughout a network. A node’s degree is the number of links connected to it; most complex networks have a skew degree distribution. A skew distribution means that there are a very small number of highly connected nodes, a moderate number of moderately connected nodes, and a very large number of minimally connected nodes. This property is a direct result of cycles in complex networks. Skew degree distributions also are the source of remarkable property of complex networks: the largest hubs can appear, recede, and then re-appear in a different part of the network by re-wiring only about 5 to 10% of the links.

Size, Connectivity of the Largest Hubs. A skewed degree distribution creates a very small number of very well connected nodes. The largest hub typically contains fewer than 100 links and the network can be engineered for survivability so that the largest hubs are not directly connected.

Characteristic Path Length. Although there can be a very large number of lightly connected nodes and only about two links per node in a complex network, the paths from each node to every other node are nonetheless relatively short. The characteristic path length measures this property, and is defined as the median (middle ranked value) of the mean of the lengths of all shortest paths in the network. This value grows only by the order of the number of nodes in the network. In other words, it takes on average only four links to reach any node from any other node in a network of 104 nodes.

Clustering Coefficient. The mechanism of networked effects in complex networks is cyclic compounding feedback. The overall clustering coefficient of a complex network is between about 0.1 and 0.25, meaning that on average about 10-25 per cent of 3-node collections are 3-cycles. The distribution of clustering coefficients among all nodes can be skewed, creating the condition that not all nodes in a cluster of mutually supporting nodes interact directly with nodes outside the cluster. In other words, many nodes have a high clustering coefficient, a moderate number of nodes have a moderate clustering coefficient, and a low number have a low coefficient. A skew distribution of clustering coefficients therefore defines the structure of adaptive hierarchy in a complex network.

Betweenness. Betweenness is a measure of a node’s importance to dynamic behaviors in a complex network. Betweenness measures the proportion of shortest paths that pass through a node, but a node need not be the most well connected node (the largest hub) in order to have the highest betweenness. Betweenness can be used to identify the highest value nodes in a network, to control cascades of pathological behaviors in a network, or to identify potential bottlenecks.

Path Horizon. Path horizon is a measure of how many nodes, on average, a node must interact with for self-synchronization to occur. Only in very simple environments can each node successfully interact with all other nodes and clearly interacting with no other nodes prevents self-synchronization. As a general rule, good self-synchronizing behavior occurs when the path horizon is approximately the order of the number of nodes in the network. For example, a network with 102 nodes will work best with a path horizon of about 2.

Neutrality Rating. Neutrality is additional structure in a complex network above the minimum for required for connectivity. Subtracting the number of links in a network of size N, N-1, from the number of links, l, in a given network of size N, and then normalizing to network size produces the Neutrality Rating, (l – N + 1)/N, which is a good measure of adaptavity. Complex networks should have a neutrality rating of between 0.8 and 1.2.

Coefficient of Networked Effects (CNE). The coefficient of networked effects measures the amount of cyclic behavior per node and compares the potential for networked effects in networks of different sizes.

Susceptibility. Susceptibility is a measure of the number of links or nodes that can be removed before networked effects begin to break down. This breakdown can be measured by degradation of the previously listed properties.

The presentation has a table that summarizes these thumb rules for analysis and experimentation. Dave stressed that these are first approximations inferred from the study of adaptive networks in other domains. One topic for immediate study should be the specific value of these properties for military networks.