SMALL WORLDS IN CONTEXT: HOW GENERALIZABLE IS INTERORGANIZATIONAL NETWORK STRUCTURE?

Our field’s growing attention to interorganizational network structure frequently builds on the Watts and Strogatz (1998) small world model. Our literature has identified “small worlds” -actual networks which simultaneously obtain relatively high clustering and short path length – in many contexts. Yet as we illustrate, these approaches use incommensurable methods and overgeneralized data, while yielding equivocal findings of structure on performance. To address these issues, we discuss underlying characteristics of primary network data and encourage researchers to detail their data and measures in order to facilitate comparative assessments. Further, we demonstrate a more rigorous approach, the “small world omega” statistic (Telesford et al., 2011), which may serve as a basis for comparing small world networks more systematically. We close with suggestions for ongoing research. Small worlds in context 3 Introduction The long-held organization-theoretic belief that social structures shape the behaviors and outcomes of social actors has gained purchase in a host of strategic management studies of network effects on firm conduct (Baum, Calabrese, & Silverman, 2000; Gulati, Nohria, & Zaheer, 2000; Phelps, Heidl, & Wadhwa, 2012; Rosenkopf, Metiu, & George, 2001). Yet the wide contextual range of these studies, across not only varied industries and timeframes but also varied types of network ties and empirical approaches, yields findings that may not be commensurable or generalizable (Shi, Sorenson, & Waguespack, 2017). As an example, consider how one popular network structure, the “small world,” affects performance. In a study of alliance networks in 11 high-tech industries over 8 years, Schilling and Phelps (2007) demonstrated that the combination of network clustering and reach (the two essential characteristics of small worlds) was associated with higher firm innovative productivity, suggesting a linear relationship between “small-worldliness” and performance. In contrast, in a study of Broadway musical collaboration networks from 1945 to 1989, Uzzi and Spiro (2005) demonstrated that the small-worldliness of the artists’ network has a curvilinear relationship with box-office receipts, which they attribute to the desirability of blending novelty and routines. Another contrast arises from a study of inventor collaboration networks across 337 metropolitan statistical areas over 21 years, where Fleming, King Iii, and Juda (2007) find no clear effect of small-worldliness on regional innovativeness, though they draw an interesting contrast between the network structures of Silicon Valley and Boston. Does this mean that the relationship between small-worldliness and performance differs for different types of networks? Or might it mean that the measurement of small worlds differs across network types, or even across researchers? How can we distinguish these issues? Small worlds in context 4 To address these questions, we contend that further progress on the relation between network structure and performance requires some rethinking of our ideas about modeling, measuring and comparing network structures. We review the theory of small worlds and its application in strategic management, and then offer two divergent approaches that can help sort the variety of extant findings about small worlds. First, we argue that most network analyses in strategic management research are derived from what have been termed “affiliation networks” (Wasserman & Faust, 1994), and we differentiate affiliation networks by considering both the frequency of affiliative events as well as the number of participants in a typical affiliation. With this approach established, we then address the limitations of current small world measures and introduce a new network statistic (“omega”) borrowed from brain science so that we can assess the variability in traditional small world network studies systematically. Extrapolating from these small world findings, we conclude with recommendations for more uniform approaches that would facilitate comparative work across, and generalization within, each network type. The Watts-Strogatz Small World Model While earlier work by social scientists studied the so-called “small world” problem, that is, how long are acquaintance chains between seemingly well-separated others (Milgram, 1967; Pool & Kochen, 1978), it was several decades later before network scientists formalized a model which might comport with the “six degrees of separation” concept. Watts and Strogatz (1998), heretofore “W-S”, formalized the small world model as a regular (ring lattice) network with nominal rewiring to increase disorder, yielding networks characterized by both a short average path length and a high clustering coefficient simultaneously1. This was an important 1 Mathematically, path length is the average distance between all pairs of vertices in the network (assuming a connected network where all nodes are reachable from each other), where distance between two vertices is the length of the shortest path between them. Watts and Strogatz define the clustering coefficient this way: "Suppose that a vertex Small worlds in context 5 contribution to the graph-theoretic literature, as two key reference classes of graphs had each demonstrated one of these characteristics without the other. Specifically, Erdos-Renyi random graphs tended to exhibit short average path length, but did not have the clustering typical of social networks. In contrast, ring lattice graphs exhibited clustering, but had high path length. As a result, the W-S model represents a hybrid of these two types, and has received substantial attention across a wide variety of literatures. As seen in Figure 1, in order to construct network candidates for small worlds, the WattsStrogatz model starts with a lattice (characterized by n actors with k links to nearest neighbors) and then randomly rewires the links between nodes with a certain probability (p). When a link is selected for rewiring, one node of the dyad is held constant, while the other is randomly assigned to a new destination to preserve connectivity among all the nodes. When p=0, the network is a lattice, while for p=1, since all links are randomly rewired, the network generated is an ErdosRenyi random graph2. Only for some moderately low values of p does the small world network emerge. -----------------------------------------Insert Figure 1 about here ------------------------------------------To understand these dynamics more fully, Figure 2 demonstrates the transitions between lattice, small world, and random networks for the example of n=1000 and k=20 by plotting a v has kv neighbors; then at most kv(kv 1)/2 edges can exist between them (this occurs when every neighbor of v is connected to every other neighbor of v). Let Cv denote the fraction of these allowable edges that actually exist. Define C as the average of Cv over all v. " 2 The Erdos-Renyi graph is one particular type of many possible random graphs, where all edges are equally likely to be included, yielding a Poisson degree distribution. It is worth noting that different types of random graphs obtain if the random rewiring follows a different rule such as rewiring with a probability proportional to the degree of the node, which would result in a scale-free network. Small worlds in context 6 normalized3 clustering coefficient and average path length against log(p). When moving from left to right along the x-axis, we observe that there is a region in which the clustering coefficient remains high while the average path length decreases due to the shortcuts generated by the rewiring. As log p gets larger (that is, p approaches 1), the W-S networks approximate ErdosRenyi random networks, and clustering drops significantly. Thus, small world networks occur in the range of p between ring lattices and random networks where the clustering coefficient remains high while the path length has dropped. -----------------------------------------Insert Figure 2 about here ------------------------------------------Ultimately, for a given n and k, varying p yields a family of W-S networks. The specific value of p determines whether this network is best classified as a lattice, a small world, or a random network. When n>>k (a key condition for all small world studies), then small worlds arise at relatively low values of p. In other words, only a small fraction of shortcuts are need to dramatically decrease path length while still preserving clustering. At the same time, it is important to note that while the lattice (p=0) has a uniform degree distribution (all nodes with degree k), as p grows and tends to 1, the rewired network will display a Poisson degree distribution (Barrat & Weigt, 2000). Said differently, at any value of p, the W-S network, by its construction, has little if any skewness in its degree distribution. Indeed, in a follow-up paper focused on social networks, Watts (1999) explicitly notes that a small world network is decentralized, as it would not be surprising for a centralized network to have short path length. 3 Following Watts & Strogatz (1998) the average path length and clustering coefficients are normalized by the corresponding values for an equivalent ring lattice (i.e, for the network with p=0) since both path length and clustering are maximized at this value. Small worlds in context 7 Application of the W-S Model in Management Within the strategy and organization theory fields, the W-S model has captivated many researchers because this network structure seems better suited to efficient search and diffusion than other well-established networks such as a lattice or a randomly generated graph. The imagery of clusters and shortcuts also strikes a chord with our theorizing on social capital formation and exploratory search (Fleming et al., 2007; Fleming & Marx, 2006; Rosenkopf & Padula, 2008; Schilling & Phelps, 2007; Verspagen & Duysters, 2004). As a re