Does World City Network Research Need Eigenvectors?

Abstract

In a 2011 paper, I introduced the conceptual distinction between positions of centrality and power in world city networks and offered a new measure—now called alter-based centrality—designed to quantify the network positions of cities. Here, I respond to some conceptual and mathematical critiques of those ideas raised by Boyd et al. (2013) in their Comment on Neal (2011). On the conceptual side, I clarify the definitions of centrality and power, the relevance of exchange power in world city networks and the appropriate depth of network measures in this context. On the mathematical side, I clarify the relationship between alter-based centrality and other measures, and explore the limitations of two alternatives: eigenvector and beta centrality. This Reply concludes by noting that each measure has its own strengths and weaknesses, but that researchers should aim to use measures that are no more complex than necessary.

In a 2011 paper, “Differentiating centrality and power in the world city network”, I introduced a conceptual distinction between structural positions of centrality and power into a substantive literature where such a distinction had not typically been made. I also proposed some new measures of centrality and power, comparing them with existing alternatives, but noted that “a detailed discussion of their computation and properties [lies] beyond the scope of this paper” (Neal, 2011, p. 2740). Boyd, Mahutga and Smith’s thoughtful Comment (Boyd et al., 2013) challenges some of the claims made in that original paper, primarily on mathematical grounds, and presents a welcome opportunity to explore the more technical dimensions of centrality and power measurement in the world city network context. It also presents an opportunity to consider the broader issue of methodological complexity by asking what is gained and what is lost when we turn to increasingly complicated (and thus easily misunderstood or misused) measures. That is, does world city network research need eigenvectors?

I have organised my responses to Boyd, Mahutga and Smith’s critiques in three sections. I begin by briefly clarifying several minor issues including the names and formulas of the proposed measures and what Boyd et al. described as ‘discrepancies’. I then turn to conceptual issues concerning what it means for a node to be central, or to be powerful, and the appropriateness of applying these concepts in world city networks. In the final section, I consider the mathematical properties and limitations of beta centrality (Bonacich, 1987) and by extension eigenvector centrality (Bonacich, 1972), which they contend offer superior measurement of centrality and power in world city networks.

Clarifications

Boyd et al. argue that the two measures proposed in the original 2011 paper—recursive centrality and recursive power—are inappropriately labelled because they “are not recursive”. Although they are not recursive in a formal mathematical sense, they are recursive in a more conceptual sense because they view each node’s centrality (or power) as a function of other nodes’ centrality. Indeed, a key strength of the proposed measures is that they achieve a conceptual recursion without the complexity of computational recursion. Nonetheless, to avoid any confusion, since 2011 these measures have been more descriptively named ‘alter-based centrality’ in the Stata programme that computes them (CENTPOW) and in subsequent research by others that have used them (Cihak et al., 2011; Dronen and Lv, 2011; Worrell and Lowe, 2011). I use this updated terminology in what follows.

These measures were originally defined using cellwise computational formulas, as has been conventional in the world city network literature (for example, Taylor, 2001). However, Boyd et al. (2013) present some formulas in matrix form (for example, their equation (1a)). Thus, for the sake of completeness and clarity, I present the proposed measures in both cellwise and matrix form. When a node’s alters’ degrees are positively weighted, alter-based centrality is a measure of centrality ( $A C^{+}$ ) and is defined cellwise as

A C_{i}^{+} = \sum_{j} R_{i j} D C_{j}

(1a)

where; $R_{ij}$ is the weight of the link between i and j; and $D C_{j}$ is j’s degree centrality, or in matrix form as

A C^{+} = R (R 1)

(1b)

where, R is an adjacency matrix and 1 is a column vector of 1s. When a node’s alters’ degrees are inversely weighted, alter-based centrality is a measure of power (AC^–) and is defined cellwise as

{AC}_{i}^{-} = \sum_{j} \frac{R_{ij}}{D C_{j}}

(2a)

or in matrix form as

A C^{-} = R (\frac{1}{R 1})

(2b)

where the division is performed elementwise.

Among their equations, Boyd et al. also describe a normalisation (see equation (1b) in their paper). I agree that normalising these measures can be useful, both to facilitate comparisons across different measures and networks, and to keep the values within a reasonable range. The current version of the CENTPOW programme now offers the option to normalise degree centrality, alter-based centrality and beta centrality using the method described by Bonacich (1987). Under this normalisation, a value of 1

means (approximately) that position i does not have an unusually large or small degree centrality, irrespective of the number of positions in the network (Bonacich, 1987, p. 1173).

As a final point of clarification, Boyd et al. also observed some discrepancies between the figure showing a hypothetical world city network (Figure 2 in Neal, 2011) and the corresponding table of cities’ centrality scores (Table 1 in Neal, 2011). The figure was intended to mirror the hypothetical world city network shown on page 74 of Friedmann’s (1986) seminal paper. However, it incorrectly showed a Rio/São Paulo–Caracas link and did not show links between Rio/São Paulo and New York, Mexico City and Tokyo. The error was a purely visual one, while the underlying data, analyses and results were unaffected.

Conceptual Issues

Discriminant Validity

Boyd et al. begin their conceptual critique by observing that the hypothetical networks shown in the original paper do “not effectively demonstrate that powerful actors will not also be central”. This claim challenges the discriminant validity of alter-based centrality, and has both conceptual and measurement dimensions.

To consider the issue of conceptual discriminant validity, it is necessary to clarify what it means for an actor to be central and how this differs from what it means for an actor to be powerful. The concept of centrality can refer to many different things, but here centrality refers to a structural position that is advantageous for diffusion in a positively connected network. In positively connected networks, where resources like information can diffuse via multiple paths simultaneously, the most advantaged actors are those connected to well-connected others. Such actors can diffuse resources like information widely, and can receive resources from a wide range of sources. Of course, whether such a position is beneficial or detrimental ultimately depends on the nature of the specific resource being diffused: it is generally good to be central in an information diffusion network, but not in a disease diffusion network. The concept of power can likewise refer to many different things, but here power refers to a structural position that is advantageous for exchange in a negatively connected network. In negatively connected networks, where the exchange of a resource with one partner reduces the opportunity to exchange with another partner, the most advantaged actors are those connected to poorly connected others. These actors are able to exercise power over (i.e. control) their exchange partners because poorly connected exchange partners have no alternatives and thus are dependent. In one of the earliest discussions of global networks, McKenzie (1927) offered colonialism as a clear example of this form of advantage in the context of cities and states: the power of imperial centres derives from the fact that their colonies are dependent upon them as sources for supplies and consumers of raw materials.

To consider the issue of measurement discriminant validity, I turn to Figure 1 shown here, which is similar to Figure 1 in the original paper, but offers a clearer illustration by comparing positions of centrality and power in a single network. In this simple network, positions A and B are indistinguishable in terms of degree centrality; they each have three direct connections. However, alter-based centrality clearly demonstrates that these two positions are quite different from one another by taking a wider view, considering the two-step neighbourhood around A and B to assess their positional status. When used to measure centrality (i.e. AC⁺), position A has a much higher score than position B, indicating that position A is more advantageous for diffusion; it can send/receive resources to/from nine different nodes in two steps. In contrast, when used to measure power (i.e. AC^–), position B has a much higher score than position A, indicating that position B is more advantageous for exchange; it can dominate the nodes in position C. Thus, while A and B may initially appear similar, alter-based centrality quantifies the extent to which position A is central but not powerful, while position B is powerful but not central. Although this example illustrates alter-based centrality’s discriminant validity in a very simple network, my original analyses of Friedmann’s world city network and the Internet backbone network also illustrated its discriminant validity in more complex and real-world networks, showing in both cases that a centrality-based ranking of cities differs from a power-based ranking.

Figure 1.

Comparing degree and alter-based centrality.

Relevance

A second element of Boyd et al.’s conceptual critique challenges the relevance of exchange power for city networks, noting that such a conception of power is “odd in the empirical case of many world city networks”. I agree: although exchange power makes sense in the context of colonialism described by McKenzie (1927), other empirical contexts may exist where exchange power is not appropriate. They also note that

we are sceptical that bargaining power in exchange networks is the only way in which power can be conceptualised for world city networks.

Again, I agree: exchange power in negatively connected networks is only one of many ways that power might be conceptualised in the world city network context. However, to critique exchange power on the grounds that it is neither universally applicable nor the only possible conception of power is a strawman. They do discuss one specific type of world city network—one defined by air passenger flows—in which they argue that exchange power is not meaningful, but even here their critique is misdirected. To understand why requires considering how air traffic data can be used to define a world city network.

Mahutga et al. (2010) have previously used air traffic data to define what I have elsewhere called a route network (Neal, 2013), in which intercity linkages reflect passengers departing from one city and landing in the other. Such a network reflects the physical movement of passengers’ bodies and thus can be very useful for exploring practical operational issues like scheduling and air traffic control. Because route networks do not capture the intercity exchange of anything for which bargaining or dominance is relevant, exchange power is not likely to be relevant in world city networks defined in this way. More broadly, many have argued that the route networks that Mahutga et al. examine are generally not useful for studying cities at all (for example, Fleming and Hayuth, 1994; Derudder and Witlox, 2005).

However, air traffic data can also be used to define an origin–destination network (Neal, 2013), in which intercity linkages reflect passengers initially originating in one city and terminally destined for the other, regardless of any intermediate connections or layovers they may have en route. Because passengers’ physical movements often have limited social or economic implications—for example, a passenger’s layover in Atlanta has virtually no implications for Atlanta, except perhaps the local employment of baggage handlers—this approach focuses on passengers’ socially and economically significant movements. That is, for example, it focuses on movement between an origin city where a passenger lives and a destination city where she plans to leave the airport and conduct business, visit family or otherwise engage with the local area. Because passengers bring their spending power with them, such networks indirectly reflect intercity capital flows (Neal, 2010). Thus, a world city network defined by air passenger flows can capture the intercity exchange of resources for which bargaining or dominance is relevant and thus can be an appropriate context for exchange power.

Depth

A third conceptual critique is articulated in methodological terms, but nonetheless has important theoretical implications because it rests on Boyd et al.’s curious but hidden assertion that each city is affected by all other cities, no matter how many network connections separate them. It is often useful to distinguish network characteristics that are measured locally from those that are measured globally. Despite Boyd et al.’s claim that I am “motivated by a desire for a global measure of centrality”, alter-based centrality is not intended to be a global measure. It is an explicitly local measure that assesses the centrality (or power) of a node by considering only its two-step neighbourhood (i.e. the focal node’s alters and their alters). In contrast, eigenvector centrality, which Boyd et al. believe is more appropriate in the context of world city networks, is a global measure that “takes account of all possible long-distance walks.” With this distinction in mind, they contend that local measures like alter-based centrality “will have limited interest in network studies on world cities”. That is, they contend that only global measures like eigenvector centrality are suitable for world city network research.

What such a claim implies conceptually can be illustrated with reference to Friedmann’s world city network. A local measure that assesses centrality within a two-step radius, such as alter-based centrality, suggests for example that Johannesburg is affected by London, to which it is directly connected, and by New York, Rio/São Paulo, Madrid and Paris to which it is indirectly connected via London. In contrast, a global measure looks much deeper, instead implying that all other cities in the network impact Johannesburg. Thus, for a global measure of centrality, Johannesburg is also affected by Singapore, via its indirect connection first to London, then New York, then Chicago, then Los Angeles. This represents the shortest network path between Singapore and Johannesburg and, for some types of durable resources that do not decay as they move from place to place, this long chain of indirect links may be meaningful. However, eigenvector centrality focuses not only on shortest paths, but on all possible walks.¹ Thus, it also suggests that Johannesburg is affected by Singapore via an indirect connection first to London, then to New York, then back to London, then to Rio, then to Mexico City, then back to Rio, then to Tokyo and so on, until eventually reaching Singapore. While possible, it is far from obvious that this is a realistic way to think of cities as affecting one another. This is not to say that global measures are inappropriate for world city network research; indeed, it is likely that they are useful in specific contexts. Yet, it does challenge Boyd et al.’s claim that only global measures are appropriate. Local measures like alter-based centrality offer researchers an alternative way to conceptualise and measure centrality and power in those contexts where the ‘all possible walks’ assumption of global measures simply probes unreasonably deep into the network—for example, in airline traffic networks where passengers generally make at most two connections.

Mathematical Issues

Boyd et al. focus the bulk of their comments on mathematical aspects of centrality computations. These issues have little to do with world city networks per se, but instead apply to the analysis of all types of networks. Nonetheless, I welcome the opportunity to explore the more technical side of these measures here.

Alternatives

It is necessary first to clarify the set of measures for which alter-based centrality is intended to serve as a potential alternative. Borgatti and Everett (2006) classify centrality measures by the properties (volume or length) and positions (radial or medial) of walks they are intended to assess. Using this classification, alter-based centrality is a ‘radial volume’ measure because it evaluates the number (hence, volume) of walks emanating (hence, radial) from a node. This places alter-based centrality in the same category as, and thus a potential alterative to, three other measures I originally discussed: degree centrality, eigenvector centrality and beta centrality.

Boyd et al. dismiss degree centrality as a relevant comparison measure on the grounds that it is not widely used in world city network research. Although research in this area uses many different measures, degree centrality remains, as I originally noted, “the most widely employed measure” (Neal, 2011, p. 2737), and thus is worth taking seriously. The confusion perhaps stems from the fact that when degree centrality is measured in world city networks, it often masquerades under other names, including nodal connection (Taylor, 2001), gross connectivity (Taylor et al., 2002) and global network connectivity (Derudder et al., 2010). Despite its many names, assessing the centrality of cities by counting up their number of direct connections remains the most common approach. Alter-based centrality, eigenvector centrality and beta centrality are all extensions of degree centrality that aim to offer a more nuanced measurement.

Turning to these more sophisticated measures, Boyd et al.’s claim that eigenvector centrality and beta centrality “are both conceptually and computationally unique” is odd because beta centrality was explicitly developed as a conceptual and computational extension of eigenvector centrality (Bonacich, 1987) and because these two measures share many properties (Bonacich, 2007). Although they focus on comparing alter-based centrality with eigenvector centrality, a comparison with beta centrality is more appropriate for several reasons. First, eigenvector centrality is simply a special case of the more general beta centrality (i.e. when $β = 1 / λ_{1}$ ). Secondly, eigenvector centrality is exclusively a measure of centrality, while beta centrality can function as a measure of either centrality or power. Thirdly, whether used to measure centrality or power, beta centrality is conceptually (although not computationally) parallel to alter-based centrality. I thus focus the remainder of this paper on beta centrality, which is defined cellwise as

B C_{i} (β) = \sum_{j} β \times B C_{j} \times R_{ij}

(3a)

or in matrix form as

BC (β) = {(I - β R)}^{- 1} R 1

(3b)

where R is an adjacency matrix, 1 is a column vector of 1s, I is an identity matrix, and β is an attenuation parameter set by the researcher. The measure’s flexibility lies in the fact that beta centrality’s interpretation depends on the value of β. When β >0, beta centrality is a centrality measure that views nodes as central when they are connected to central others. When β<0, beta centrality is a power measure that views nodes as powerful when they are connected to non-central others.² In both cases, larger absolute values of β give greater weight to more distant nodes, widening the measure’s scope. Finally, when β = 0, beta centrality is equal to degree centrality.

Before turning to the limitations of beta centrality, it is also useful to consider two additional centrality measures for which alter-based centrality is not intended to serve as an alternative: closeness and betweenness. First, closeness is a ‘radial length’ measure and thus is not comparable with alter-based centrality because it is not trying to measure the same thing. Additionally, unlike alter-based centrality, closeness cannot be computed on valued networks, thus restricting its application to a subset of (world city) networks. Secondly, betweenness is a ‘medial volume’ measure and thus likewise is not comparable with alter-based centrality because it does not attempt to measure the same thing. Again, unlike alter-based centrality, betweenness cannot be computed on valued networks, restricting its application to a subset of (world city) networks. Boyd et al. correctly point out that a generalised form of betweenness—flow betweenness (Freeman et al., 1991)—can be computed in valued networks. However, lack of awareness of such measures is a practical barrier to their use in world city network research. For example, Alderson and Beckfield (2004) transformed their valued world city network into a binary one, thus discarding the majority of their data, believing it necessary to compute betweenness centrality. Indeed, Mahutga et al. have themselves made a similar mistake, arguing that “eigenvector centrality … is only valid in the context of symmetrical data” (Mahutga et al., 2010, p. 1932), when in fact a generalised form of eigenvector centrality that is valid for asymmetric networks had been available for nearly a decade (Bonacich and Lloyd, 2001).

Selecting Beta

I originally argued that eigenvector-based measures like beta centrality are often not useful in world city networks because such networks’ largest eigenvalues are often too large. Boyd et al. are right to point out that the magnitude of a network’s first eigenvalue ( $λ_{1}$ ) is not the problem. Instead, it is the role that this value plays in the selection of an appropriate value of β.³ Because the interpretation of beta centrality hinges on the value of β, selecting an appropriate value is absolutely critical. Two considerations are important when the researcher makes this selection. The first involves whether the researcher desires a measure of centrality, in which case β should be positive, or a measure of power, in which case β should be negative. The second involves whether the researcher desires a local measure, in which case β should be small, or a global measure, in which case β should be large. This second consideration is complicated by the fact that the magnitude of β cannot exceed the reciprocal of the network’s largest eigenvalue ( $1 / λ_{1}$ ).

It is here that researchers often make mistakes. In the Internet backbone network examined by Choi et al. (2006), $1 / λ_{1} = 0.0000056$ , which sets a very low ceiling on the allowable values of β. On its own, this is not a problem. However, on seeing this low ceiling, the researchers set β = 0, inadvertently transforming beta centrality into a complicated way to compute degree centrality. In other cases, researchers ignore or do not consider the magnitude of the network’s largest eigenvalue when selecting a value for β and thus in an attempt to obtain a global measure with a wide scope, select a value that is larger than $1 / λ_{1}$ . For example, an analysis of economic location theories set β = 0.9 (Irwin and Hughes 1992), while a widely used on-line tutorial of network analysis methods (Hanneman and Riddle, 2005) and an analysis of transport policy networks (Weir et al., 2009) both set β = 0.5. In each of these cases, the computation of beta centrality fails to converge on a unique set of values and the reported centrality (or power) scores are meaningless. These examples do not highlight a technical flaw with beta centrality, but rather suggest that it is easily misspecified. Alter-based centrality is a useful alternative because it measures centrality and power in conceptually similar ways, but its computation does not require researchers to select an attenuation parameter, or indeed to be concerned with eigenvalues at all.

Spectral Gap

While the magnitude of a network’s first eigenvalue can introduce misspecification errors in practice, the magnitude of the first eigenvalue relative to the second, or spectral gap, directly constrains the applicability of eigenvector and beta centrality. Borgatti et al., who have written more about centrality than perhaps anyone, explain that “the ratio of the largest eigenvalue to the next largest should be at least 1.5 and preferably 2.0 or more for the centrality measure to be robust” (Borgatti et al., 2002, unpaginated). Similarly, Hanneman and Riddle recommend that the first eigenvector should capture at least 70 per cent of the total variation, noting that if this assumption is not met “great caution should be exercised in interpreting further results” (Hanneman and Riddle, 2005, unpaginated). Indeed, even Mahutga et al. have previously acknowledged that “the first principal eigenvalue has to explain the highest amount of variation relative to lower-rank eigenvalues for the eigenvector to behave as we describe” (Mahutga et al., 2010, p. 1942). Thus, Boyd et al.’s claim that “there is still no need to throw out [the eigenvector centrality] results” when the data violate this requirement contradicts not only the existing network literature, but also their own past claims.

When the first eigenvalue is not relatively large, they are right to recommend “that the second eigenvector should be examined as well”. However, this is not a recommendation they have followed themselves, reporting in one analysis that the largest eigenvector explained less than 25 per cent of the networks’ structure, but reserving consideration of the other eigenvectors for “future analysis” (Mahutga et al., 2010, p. 1943). Indeed, the examination of anything beyond the largest eigenvector is exceptionally rare, not only in the world city network literature, but in the network analysis literature generally. Okada’s (2008) analysis offers a notable exception and may serve as a model for those wishing to use eigenvector or beta centrality in networks without a distinctively large eigenvalue, but this complicates the measurement of centrality and power still further. In contrast, as a local measure that does not attempt to take account of a network’s global structure, alter-based centrality is not impacted by the size of the spectral gap.

Multiple Components

Just before concluding, Boyd et al. draw attention to an additional limitation of eigenvector and beta centrality that I did not raise in my original discussion: these measures cannot be applied to networks that contain multiple components (Bonacich, 2007). They do offer a solution, recommending that “if there is more than one component (i.e. two networks instead of one), one should consider measuring centrality separately for each”. However, this solution is unsatisfactory because it conflates networks and components. Despite their parenthetical claim to the contrary, there is a big difference between an analysis of two networks and an analysis of one network with multiple components. One might wish to compare the distribution of centralities in the Internet backbone network and a dolphin social network, in which case separate analysis of each network is appropriate because they are two entirely different networks; a router could not, in principle, have been linked to a dolphin. However, one might also wish to examine the distribution of centralities in a network of intercity travel, which during the observation period coincidentally happens to contain two components corresponding to northern and southern hemisphere cities (i.e. people travelled only within their own hemisphere). These are not separate networks, but rather a single network that during the observation period contains multiple components, but could just as easily have been structured as a single component; a northern hemisphere resident, in principle, could have travelled to the southern hemisphere. Eigenvector and beta centrality’s requirement that components be examined separately implicitly requires the researcher to believe that each component could not, in principle, have been linked to the others. In contrast, alter-based centrality imposes no such requirement; it performs equally well in single and multiple component networks.

Conclusion

Aside from clarifying some minor terminological and graphical issues, this discussion addresses several issues raised by Boyd et al.. First, although positions of centrality and positions of power are not necessarily different, when they are, alter-based centrality can effectively distinguish between the two. Secondly, although an exchange-based notion of power may not be relevant in all world city network contexts, it is certainly relevant in some. Thirdly, although some world city phenomena may involve infinitely long and redundant network movements, others may be more localised. Finally, although they have been used in some research on world city networks, eigenvector and beta centrality involve computations that impose significant practical and empirical limitations on their use. This discussion should not be viewed as suggesting that alter-based centrality is always, or even usually, preferable to other measures of centrality. Each measure has its own set of strengths and weaknesses and, whether they are applied in a world city network or some other context, the researcher must select the most conceptually and methodologically appropriate one.

Eigenvector and beta centrality are both excellent choices for those seeking a global measure of centrality in which all possible walks between nodes are important. Beta centrality is likewise an option for those seeking a global measure of exchange power (when β<0) and for those wishing to distinguish between positions of global centrality and global power. However, the computation of these measures is complex and users must be sure they understand the potential limitations and how to interpret the results. For beta centrality, this includes examination of the largest eigenvalue to aid in selecting an appropriate value for β. For both measures, one must examine the spectral gap and, if it is small, undertake still further analysis of subsequent eigenvectors. Additionally, one must determine whether the network contains multiple components and, if so, examine each network component separately, considering the theoretical implications of such a partitioning.

In contrast, alter-based centrality is a strong candidate for those seeking a local measure of centrality or exchange power in which processes are seen as unfolding within the two-step neighbourhood surrounding each node. Compared with eigenvector and beta centrality, the computation and interpretation of this measure is quite simple. Its use does not require the selection of an attenuation parameter, the consideration of eigenvalues or spectral gaps, or the separate analysis of multiple network components.

Yet, perhaps the strongest response to Boyd, Mahutga and Smith’s claim that world city network researchers should be “more confident in the utility of eigenvector centrality” than alter-based centrality can be found in the results of their own comparative analysis. They report that, in Friedmann’s hypothetical world city network, alter-based centrality and eigenvector centrality are very highly correlated (r = 0.97). This raises an obvious question: if these two measures yield roughly the same results, why bother with the more complex and easily misspecified eigenvector centrality? Answering this question goes beyond the scope of this brief discussion, but highlights the danger of allowing our methodological sophistication to outpace our theoretical sophistication. Our analytical tools should be sufficiently complex to make sense of the phenomena under investigation, but no more.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Notes

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