The Complexity of Design Networks: Structure and Dynamics

Abstract

Why was the $6 billion FAA air traffic control project scrapped? How could the 1977 New York City blackout occur? Why do large-scale engineering systems or technology projects fail? How do engineering changes and errors propagate, and how is that related to epidemics and earthquakes? In this chapter, we demonstrate how the emerging science of complex networks provides answers to these intriguing questions.

This chapter is based on keynote lectures delivered on August 6, 2013, at the 39th Design Automation Conference in Portland, Oregon, and on October 12, 2009, at the 11th International DSM Conference in Greenville, South Carolina.

Appendix: Measuring Complex Networks

Complex networks can be defined formally in terms of a graph $G = (V,E)$, which is a set of nodes $V = \{ 1, 2, \ldots , N\}$ and a set of lines $E = \{ e_{1} ,e_{2} , \ldots ,e_{L} \}$ between pairs of nodes. If the line between two nodes is non-directional, then the network is called undirected; otherwise, the network is called directed. A network is usually represented by a diagram, where the nodes are drawn as points, undirected lines are drawn as edges, and directed lines are drawn as arcs connecting the corresponding two nodes. Several properties have been used to characterize ‘real-world’ complex networks:

Density: The density $D$ of a network is defined as the ratio between the number of edges (arcs) $L$ to the number of possible edges (arcs) in the network:

$$D = \frac{2L}{N(N - 1)}\,\left( {\text{undirected networks}} \right)\,D = \frac{L}{N(N - 1)}\,\left( {\text{directed networks}} \right)$$

(8.4)

Characteristic Path Length: The average distance (geodesic) $d(i,j)$ between two nodes $i$ and $j$ is defined as the number of edges along the shortest path connecting them. The characteristic path length $d$ is the average distance between any two vertices:

$$d = \frac{1}{N(N - 1)}\mathop \sum \limits_{i \ne j} d(i,j)$$

(8.5)

Clustering Coefficient: The clustering coefficient measures the tendency of nodes to be locally interconnected or to cluster in dense modules. Let node $i$ be connected to $k_{i}$ neighbours. The total number of edges between these neighbours is at most $k_{i} (k_{i} - 1)/2$. If the actual number of edges between these $k_{i}$ neighbours is $n_{i}$, then the clustering coefficient $C_{i}$ of a node $i$ is the ratio:

$$C_{i} = \frac{{2n_{i} }}{{k_{i} (k_{i} - 1)}}$$

(8.6)

The clustering coefficient of the graph, which is a measure of the network’s potential modularity, is the average over all nodes:

$$C = \frac{{\mathop \sum \nolimits_{i = 1}^{N} C_{i} }}{N}$$

(8.7)

Degree Centrality: The degree of a vertex, denoted by $k_{i}$, is the number of nodes adjacent to it. The mean node degree (the first moment of the degree distribution) is the average degree of the nodes in the network:

$$\langle k\rangle = \frac{{\mathop \sum \nolimits_{i = 1}^{N} k_{i} }}{N} = \frac{2L}{N}$$

(8.8)

If the network is directed, a distinction is made between the in-degree of a node and its out-degree. The in-degree of a node, $k_{\text{in}} (i)$, is the number of nodes that are adjacent to $i$. The out-degree of a node, $k_{\text{out}} (i)$, is the number of nodes adjacent from $i$. For directed networks, $\langle k_{\text{in}}\rangle = \langle k_{\text{out}}\rangle = \langle k\rangle .$ Other node centrality indices were established, including closeness centrality, betweenness centrality, and eigenvector centrality (Braha and Bar-Yam 2004a).

Degree Distribution: The node degree distribution $p(k)$ is the probability that a node has $k$ edges. The corresponding degree distributions for directed networks are $p_{\text{in}} (k)$ and $p_{\text{out}} (k)$.

Connected Components: A weakly (strongly) connected component is a set of nodes in which there exists an undirected (directed) path from any node to any other. The single connected component that contains most of the nodes in the network (and thus many cycles) is referred to as the giant component. For a certain class of networks in which degrees of nearest neighbour nodes are not correlated, the critical threshold for the giant component is found by the following criteria:

$$\frac{{\langle k^{2}\rangle}}{\langle k\rangle } = 2\,\left( {\text{undirected networks}} \right)\,\frac{\langle {k_{\text{in}} k_{\text{out}}\rangle }}{\langle k\rangle } = 1\,\left( {\text{directed networks}} \right)$$

(8.9)

where $\langle k^{2}\rangle$ and $\langle k_{\text{in}} k_{\text{out}}\rangle$ are the second moment and joint moment of the in- and out-degree distributions, respectively. We notice that, for undirected networks, higher variability of the degree distribution leads to a giant component. For directed networks, higher correlation between the in-degree and out-degree of nodes leads to a giant component, and this could lead to significant number of network cycles and further degradation and instability of the system as shown in Fig. 8.8.