Mechanisms describing the propagation of fluctuations in social and economic systems are not well characterized. By analyzing the number of venture capital firms registered in 509 cities of the United States of America between 1981 and 2003, we identified patterns in the spatial-temporal distribution of fluctuations in the number of registered venture capital firms (population fluctuations) in 9 regions. Despite large differences in geographical size, city topology and venture capital firm density, we found that fluctuation dynamics were consistent with spatial contagion. In all regions, fluctuations were more likely to occur in cities that were in close spatial proximity to cities that displayed fluctuations during the preceding year.

We developed a simple model of contagion that was consistent with the empirical data. Simulations suggested that population fluctuations caused fluctuations in nearby cities with a strength that decayed exponentially with distance. The influence of cities was additive: the more surrounding cities that demonstrated fluctuations in the preceding year, the more likely a city would experience a population fluctuation in the following year. Furthermore, the transmission of fluctuations took place on a minimally connected city network that contained a largest connected component. This study has identified and quantified higher-order patterns of economic agent mobility in regions of high venture capital activity. Our results and methods are relevant to understanding the propagation of fluctuations in a broad range of spatially-embedded systems.

### 4.1 Introduction

Spatial patterns characterizing economic activities (Fujihita et al. 2001) and technological innovations (Bettencourt et al. 2007, 2010) exhibit marked inhomogeneities, which can be explained by transport or other infrastructure costs and spill-over effects. For instance, manufacturing industries across US states have been shown to exhibit significant levels of geographic concentration (Krugman 1991), and high-tech regions in particular are characterized by strong spatial clustering (Breschi & Malerba 2005; Saxenian 1994). Key factors believed to drive the co-location of firms in high-tech industries include access to highly skilled labor markets and access to private equity finance in the form of venture capital firms (VCFs) (Stuart & Sorenson 2003a,b; Ferrary & Granovetter 2009).

Proximity to potential target firms makes it easier for a VCF to monitor its investments, and the widespread practice of syndicated investing—to share knowledge and spread risk—generates direct interactions between different VCFs that can also impact location decisions. Typically, more than one VCF will invest in a given target firm, especially when there are multiple funding rounds. The resulting syndication network of VCFs serves as a conduit for information about current investments and 84 future deals (Bygrave 1987, 1988; Sorenson & Stuart 2001; Hochberg et al. 2007), may restrict entry into local venture capital markets (Hochberg et al. 2010), and often builds on repeated interactions between trusted partners (Kogut et al. 2007).

Hence, geographic concentrations of VCFs and the target firms in which they invest can be expected to follow related patterns, where both spatial proximity and network effects play a role. From the perspective of spatial dynamics and network growth, this would suggest that a significant effect is agglomeration and increasing spatial concentration of VCF activity over time (Fleming et al. 2007; Powell et al. 2005).

In many social and economic systems, it is possible to observe the effects of contagion and spatial diffusion processes (Strang & Soule 1998; Dodds & Watts 2004). Prominent examples include the diffusion of innovations (Griliches 1957; Coleman et al. 1957; Rogers 2003), the spatial diffusion of trade union movements (Hedstr¨om 1994), the outbreak of strikes (Biggs 2005), and the spread of obesity (Christakis & Fowler 2007). Recent findings also suggest that many human activities, like the writing and sending of messages in online communities, exhibit large fluctuations that appear to follow simple scaling laws (Rybski et al. 2009).

Such behavior applies to speculative bubbles in financial markets (Shiller 2000), the popularity of online content (Szabo & Hubermann 2010), and the spread of innovations in online environments (Onnela & Reed-Tsochas 2010). For systems that are spatially embedded, this raises the question of whether temporal fluctuations in social and economic activities exhibit spatial contagion, and to what extent the spread of fluctuations differs from the spread of average activity patterns.

We addressed this question in a specific context, by tracking the number of VCFs in 509 US cities between 1981 and 2003. Of the 50 US states (976 cities had at least 1 registered VCF), 8 states had sufficient numbers of cities and VCFs for analysis—the other states were unsuitable because the majority of their cities contained only 1 VCF throughout the 23-year period. The suitable states were divided into 9 geographical regions for comparison: California North, California South, Connecticut, Massachusetts, New Jersey, New York, Pennsylvania, Texas, and Virginia.

We studied the spatial-temporal dynamics of fluctuations in the number of registered VCFs (population fluctuations) between cities in each region. By considering fluctuations, we attempted to identify patterns of contagion that may be masked by heavy-tailed distributions of VCFs that can arise from densitydriven agglomeration and spatial concentration effects. Such an approach also mitigates the influence of exogenous factors—such as economic cycles, and changes in legislation and borrowing rates—enabling higher-order effects to be emphasized. We found that fluctuation dynamics in all regions were consistent with a simple model of spatial contagion.

### 4.2 Model and methods

We analyzed the effect of spatial proximity on fluctuations in cities separated by a range of distances. Our method allowed patterns of fluctuations to be distinguished independently of spatial layout, and is applicable to any system of non-uniformly distributed objects capable of displaying fluctuations. From registered VCF data, we obtained a binary matrix indicating which cities displayed population fluctuations during each year. A fluctuation was defined as having registered numbers of VCFs 1 standard deviation above 86 the 5-year moving average for that city; this threshold was chosen because it represents a significant deviation from the long-term average whilst still retaining sufficient information to conduct quantitative analysis.

The 5-year moving average approximates a typical business cycle (Kitchin 1923). For each fluctuation, we counted the number of fluctuations during the preceding year at cities within an inter-city influence range, δ km, from the focal city. We obtained an average value for the number of preceding fluctuations for all cities and over the entire time period: n. The variation of n with inter-city influence range, nδ, characterized the spatial-temporal dynamics of fluctuations within the system.

To test for patterns in fluctuation dynamics in the empirical data, we compared nδ with samples generated by two null models. Null model 1 assumed no spatial preference regarding which cities displayed fluctuations: each city had an equal probability of producing one of the observed fluctuations in a each year. Null model 2 tested for temporal ordering in the identity of cities displaying fluctuations: the probability of a city producing one of the observed fluctuations in a year was proportional to its empirically observed rate, independent of other cities. It accounted for the observation that some cities produced more fluctuations than others, but assumed no temporal ordering as to when fluctuations occurred. For both null models, the number of fluctuations at each year was fixed to the empirically observed value. We quantified differences between the empirical data and null models using a z-score measure.

Empirical nδ was compared to the two null models using a z-score measure: zδ = (nδ − hn ′ δ i)/σn ′ δ , where hn ′ δ iis the average from a null-model ensemble and σn ′ δ is the standard deviation of the same quantity. zδ for the spatial contagion model was obtained by replacing nδ with the average value from model realizations.

We devised a spatial contagion model of fluctuations to account for the empirical data. Fluctuations in a city are assumed to be produced by a modified Poisson process with firing rate λ:

where k is the number of fluctuations, the probability of which is given by the above function. The firing rate for an individual city, i, can be decomposed into three terms:

where λresting is the unconditional rate of a fluctuation occurring, λself is the rate of a fluctuation occurring given a fluctuation in the focal city during the preceding year, and λexcitation is the rate of a fluctuation occurring given fluctuations in spatially proximate cities during the preceding year.

In practice, the model determines which cities produce fluctuations based on their proximity to cities displaying fluctuations during the preceding year. The probability of a city, i, producing fluctuations during year t + 1 is

where the summation runs over all cities. Hi [y] is the Heaviside function and equals 1 if city i displayed a fluctuation during year t, and equals 0 otherwise; xij is the distance between cities i and j and the summation runs over all cities that displayed fluctuations during year t. The constant term ensures that there is a finite probability of a city producing a fluctuation (λresting). Parameter C represents the propensity for cities to produce runs of fluctuations (λself ). Parameter A represents the strength of inter-city influence and ρ is the characteristic distance of influence; the exponential term determines the increased probability of fluctuations arising from proximity effects (λexcitation).

The probability is additive: the more cities in close proximity that displayed fluctuations during the preceding year, the more likely a city is to produce a fluctuation during the subsequent year. The model explicitly incorporates contagion: fluctuations can appear from multiple sources and can be transmitted between cities, with increased probability to those in close proximity. As with the null models, the number of fluctuations at each year was fixed to the empirically observed value.

We recorded parameter values (C ∗ , A∗ , ρ∗ ) giving greatest similarity to empirical nδ. Similarity was defined as the Euclidean distance between model and empirical nδ’s. The Euclidean distance is drs = [(lr − ls) · (lr − ls)]1/2 , where lr and ls are vectors representing two nδ’s. Lower values for the Euclidean distance indicated greater similarity between vectors. Using these best-fit parameters, we calculated a value for the critical inter-city influence distance, δ ∗ , which represents the effective range of contagion. This is de- fined as the distance where the model inter-city influence term, P j(t) Ae−xij/ρ in Equation (4.3), is equal to 0.1: δ ∗ = ρ ∗ ln A∗ 0.1 . The value of 0.1 is small relative to the other terms, and setting it to be a constant aids comparison between states. Model parameters for the 9 regions are given in Table 4.1. The model reduces to null model 1 with C = 0 and for A, ρ → 0, ∞.

### 4.3 Results

The VCF dataset comprised data on the annual number of registered VCFs in 509 cities in 8 US states for the period 1981 to 2003; and the latitude and longitude coordinates of each city. We required a US state to have at least 5 cities with an average of 1 VCF registered in the city over the 23-year period to be included in the dataset. Cities must have had at least 1 VCF registered during the 23-year period to be included in the dataset. The number of VCFs increased from 318 to 1491 over the 23-year period. The distribution of VCF density was heavy-tailed (Figure 4.1). That is, there were many cities with very few VCFs and some cities with a very large number of VCFs.

Cities with greater numbers of registered VCFs were more likely to display population fluctuations (Figure 4.2). Over the 23-year period, we found

significant positive correlation between the number of fluctuations and the number of registered VCFs in a city (R = 0.79, d.f. = 508, p < 0.0001). This finding can be interpreted as a density-driven process: VCFs locate in cities with high VCF concentration; it follows that more VCF-dense cities are then more likely to experience population fluctuations. However, it remains to be shown whether fluctuations appeared independently or whether they may have been transmitted between cities.

We analyzed the pattern of fluctuations separately for the 9 regions. Despite large variations in geographical size, city topology and VCF density (Table 4.1), the VCF data showed significant deviation from both null models (Figure 4.3; figures for the other 7 regions are provided in the Appendix). For all regions, we observed an increased probability of runs of fluctuations within cities (significant z-score at δ = 0). For δ > 0, the deviation was most pronounced at small δ and gradually decreased with increasing δ. Deviation from null model 1 indicated that fluctuations were localized to distinct spatial scales. Deviation from null model 2 indicated that fluctuations were temporally coupled between cities in close proximity.

The spatial contagion model provided a good fit to the empirical data. The central assumption of the model is an inter-city influence function that decays exponentially with distance and acts additively. This simple assumption accounted for almost all deviation of empirical data from the null models (measured using the z-score, Figure 4.3). Values for the critical inter-city in- fluence distance, δ ∗ , may be explained by the spatial topology of cities in each region. Cities are linked in a hypothetical influence network if they are separated by less than distance δ.

We tracked the proportion of cities and links involved in the network as δ is varied (Figure 4.4). The model-derived value of δ ∗ represents a topology of minimally-connected cities with a largest connected component (Figure 4.5). Thus, fluctuations were typically transmitted between neighboring cities, and there was an indirect propagation route between almost all cities in the network.

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