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...Faust 1999). There has been shown that informal networks of relationships (e.g., communication, information, and problem-solving networks)--rather than formal organizational charts--determine to a large extent the patterns of coordination and work processes embedded in the organization. In recent years, networks have also become the foundation for understanding numerous and disparate complex systems outside the field of social sciences (e.g., biology, ecology, engineering, and Internet technology (see Albert and Barabasi 2002, Newman 2003, and Bar-Yam 1997)).
The goal of this paper is to examine, for the first time, the statistical properties of an important class of large-scale information-carrying networks--new product development. We discuss the significance of these statistical properties in providing insight into ways of improving the strategic and operational decision making of the organization. In general, information-carrying networks constitute the infrastructure for exchanging knowledge that is important to the achievement of work by individual agents. We believe that our results will also be relevant to other information-carrying networks.
Distributed product development (PD), which often involves an intricate set of interconnected tasks carried out by hundreds of designers, is fundamental to the creation of complex man-made systems (Alexander 1964). The interdependence between the various tasks makes system development fundamentally iterative (Braha and Maimon 1998). Iterations are driven by the repetition (rework) of tasks due to the availability of new information generated by other tasks, such as changes in input, updates of shared assumptions, components, boundaries, or the discovery of errors. In such a network of interactions, iterations occur when some development tasks must be attempted even though the complete predecessor information is not available or known with certainty (Yassine and Braha 2003). As this missing or uncertain information becomes available, the tasks are repeated to come closer to the design specifications or goals. This iterative process proceeds until convergence occurs (Yassine and Braha 2003, Klein et al. 2006, Yassine et al. 2003).
Design iterations, which are the result of the PD network structure, might slow down the PD convergence or have a destabilizing effect on the system's behavior. This will delay the time required for product development, and thus compromise the effectiveness and efficiency of the PD process. For example, it is estimated that iteration costs about one-third of the whole PD time (Osborne 1993), while lost profits result when new products are delayed in development and shipped late (Clark 1989). Characterizing the real-world structure, and eventually the dynamics of complex PD networks, may lead to the development of guidelines for coping with complexity. It would also suggest ways for improving the decision-making process, and the search for innovative design solutions.
The last few years have witnessed substantial and dramatic new advances in understanding the large-scale structural properties of many real-world complex networks (Strogatz 2001, Albert and Barabasi 2002, Newman 2003). The availability of large-scale empirical data on one hand and the advances in computing power and theoretical understanding have led to a series of discoveries that have uncovered statistical properties that are common to a variety of diverse real-world social, biological, and technological networks including the World-Wide Web (Albert et al. 1999), the Internet (Faloutsos et al. 1999), power grids (Watts and Strogatz 1998), metabolic and protein networks (Jeong et al. 2000, 2001), food webs (Montoya and Sole 2002), scientific collaboration networks (Amaral et al. 2000, Newman 2001), citation networks (de S. Price 1965), electronic circuits (Ferrer et al. 2001), and software architecture (Valverde et al. 2002). These studies have shown that many complex networks are sparse, that is, they have only a small fraction of the possible number of links. Despite being primarily locally connected, such networks exhibit the "small-world" property of short average path lengths between any two nodes. Studies also have found that complex networks are characterized by an inhomogeneous distribution of nodal degrees (the number of nodes a particular node is connected to), with this distribution often following a power law (termed "scale-free" networks in Barabasi and Albert 1999). Scale-free networks have been shown to be robust to random failures of nodes, but vulnerable to failure of the highly connected nodes (Albert et al. 2000). A variety of network growth processes that might occur on real networks, and that lead to scale-free and small-world networks, have been proposed (Albert and Barabasi 2002, Newman 2003). The dynamics of networks can be understood to be due to processes propagating through the network of connections (Bar-Yam and Epstein 2004); the range of dynamical processes include disease spreading and diffusion, search and random walks, synchronization, games, Boolean networks and cellular automata, and rumor propagation. Indeed, the raison d'etre of complex network studies might be said to be the finding that topology provides direct information about the characteristics of network dynamics. In this paper, we study network topologies in the context of large-scale product development and discuss their relationship to the functional utility of the system, as well as to the dynamics of the underlying distributed design problem solving.
Planning techniques and analytical models that view the PD process as a network of interacting components have been proposed before (Braha and Maimon 1998, Yassine and Braha 2003, Klein et al. 2006, Yassine et al. 2003, Eppinger et al. 1994, Steward 1981, Mihm et al. 2003). However, others have not yet addressed the large-scale statistical properties of real-world PD task networks. In the research we report here, we study such networks. We show that task networks have properties (sparseness, small-world, scaling regimes) that are like those of other biological, social, and technological networks. We discover a distinctive asymmetry between the distributions of incoming and outgoing information flows (links) of PD networks, which has implications for their functionality, sensitivity, and robustness (error tolerance) properties.
We further present a model of PD dynamics embodying interactions through the network. Using analysis and simulation, we study its behavior to determine the conditions under which all the PD tasks are completed, and the rate of convergence to the completed state. We show that network topology provides key information about the characteristics of convergence, both whether and how rapidly convergence occurs. We find, quite reasonably, that the PD network dynamics will converge unless the total rate at which a task is affected by its neighboring tasks exceeds the "internal completion rate" of the task. Convergence is impeded by the existence of nodes that have high numbers of both incoming and outgoing information flows, i.e., convergence is controlled by the joint distribution of incoming and outgoing links. A more general result, which is presented in Supplement 4 (provided in the e-companion), (1) shows that the characteristics of convergence depend on the incoming and outgoing information flows among multiple tasks.
This paper is organized as follows: In [section]2, we review the basic structural properties of real-world complex networks. In [section]3, we describe the PD data used in this paper. In [section]4, we present an analysis of the PD task networks, their small-world property, and node connectivity distributions. We demonstrate the distinct roles of incoming and outgoing information flows in distributed PD processes by analyzing the corresponding in-degree and out-degree link distributions. In [section]5, we present a dynamical model of PD processes on complex networks, and show analytically and numerically how the empirical structural properties bear on the PD dynamics. In [section]6, we present simulation results. In [section]7, we present our conclusions.
2. Structural Properties of Complex Networks
Complex networks can be defined formally in terms of a graph G = (V, E), which is a pair of nodes V = {1, 2,..., N}, and a set of lines E = {[e.sub.1], [e.sub.2],..., [e.sub.L]} between pairs of nodes. If the line between two nodes is nondirectional, then the network is called undirected; otherwise, the network is called directed. A network is usually represented by a diagram, where nodes are drawn as points, undirected lines are drawn as edges, and directed lines as arcs connecting the corresponding two nodes. Three properties have been used to characterize real-world complex networks (Albert and Barabasi 2002, Newman 2003). The first characteristic is the average distance (geodesic) between two nodes, where the distance d(i, j) between nodes i and j is defined as the number of edges along the shortest path connecting them. The characteristic path length l is the average distance between any two vertices:
l = [1/[N(N-1)]][summation over (i[not equal to]j)][d.sub.ij]. (1)
The second characteristic measures the tendency of vertices to be locally interconnected or to cluster in dense modules. The clustering coefficient [C.sub.i] of a vertex i is defined as follows: Let vertex i be connected to [k.sub.i] neighbors. The total number of edges between these neighbors is at most [k.sub.i]([k.sub.i] - 1)/2. If the actual number of edges between these [k.sub.i] neighbors is [n.sub.i], then the clustering coefficient [C.sub.i] of the vertex i is the ratio
[C.sub.i] = [2[n.sub.i]]/[[k.sub.i]([k.sub.i] - 1)]. (2)
The clustering coefficient of the graph, which is a measure of the network's potential modularity, is the average over all vertices,
C = [1/N][N.summation over (i=1)][C.sub.i]. (3)
The third characteristic is the distribution of degrees of vertices. The degree of a vertex, denoted by [k.sub.i], is the number of nodes adjacent to it. The mean nodal degree is the average degree of the nodes in the network,
= [[[summation].sub.i=1.sup.N][k.sub.i]]/N. (4)
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.sub.in] (i), is the number of nodes that are adjacent to i. The out-degree of a node, [k.sub.out](i), is the number of nodes adjacent from i.
Regular networks, where all the degrees of all the nodes are equal (such as circles, grids, and fully connected graphs), have been traditionally employed in modeling physical systems of atoms (Strogatz 2001). On the other hand, many real-world social, biological, and technological networks appear more random than regular (Strogatz 2001, Albert and Barabasi 2002, Newman 2003). With the scarcity of large-scale empirical data on one hand, and the lack of computing power on the other hand, scientists have been led to model real-world networks as completely random graphs using the probabilistic graph models of Erdos and Renyi (1959). (2)
In their seminal paper on random graphs, Erdos and Renyi (1959) considered a model where N nodes are randomly connected with probability p. In this model, the average degree of the nodes in the network is [congruent to] pN, and...
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