As discussed above, I–O systems can be represented by networks in which the nodes (also referred to as *vertices*) are economic sectors or industries, and the links connecting them (also conceptialized as *ties*) represent the flows among those industries. More precisely, I–O systems are *very dense*, *valued*, *directed* networks. In a network, density refers to the proportion of links that actually exist as a share of all possible links. I–O networks are *very dense* because–especially at high levels of aggregation–most, if not all, nodes (industries) will be connected to almost all other nodes.That is, network density (*d*) approaches 1 (in symbols, \(d\rightarrow \>\)1).^{Footnote 2} They are *valued* networks because the links do not only represent the presence of a connection, but such a connection has a specific magnitude. Finally, they are also *directed* networks because I–O systems represent bi-directional flows between economic sectors. That is, each pair of nodes is connected by *two* links, one for each of the directions in which transactions may take place, typically with differing values. At greater levels of sectoral disaggregation, a more granular classification captures narrower and narrower definitions of industries and commodities. This results in more differentiation in the flows between sectors, with one direction potentially overshadowing the other by orders of magnitude (Lovász 2009; Miller and Blair 2009). Ties with similar values in both directions are extremely rare in I–O systems.

If the objective is to determine how important a sector is in the economic network, one may consider using vertex centrality measures such as those introduced by Freeman (Freeman 1977, 1978; Freeman et al. 1991). Common vertex centrality measures are of ambiguous applicability in the case of I–O networks. The difficulties associated with applying such vertex centrality measures to I–O systems become apparent when considering some common features that describe such networks. Of particular relevance are the values of the ties between vertices, loops representing recursive trade within an industry, and the overall density of I–O networks.^{Footnote 3}

The simplest of the measures Freeman defined, *degree*, is calculated as the number of ties that are incident upon a node.^{Footnote 4} As such, the *degree* of a node describes how often each industry participates in the production function of others, and which sectors are part of its own production function. Given the high density found in I–O networks, however, the number of links that are incident upon any given node is not likely to vary greatly throughout the network, making *degree* a relatively poor measure of a given industry’s relative prominence in a local economy.^{Footnote 5} In addition, because *degree* measures only direct access to others, it fails to capture the larger systemic effects that are distributed throughout the wider network.

Path-based measures were introduced to take into account a node’s place within the larger network. Two measures that were introduced to take the entire network into account were *closeness* and *betweenness* Freeman (1978). *Closeness* provides a measure of the inverse distance between a node and all other nodes reachable from it. More specifically, *closeness* centrality for a given node *i* is calculated as

$$\begin{aligned} C_i^{CLO}=\left[ {{\sum _{j=1}^n}d_{(i,j)}}\right] ^{-1} \end{aligned}$$

(1)

where \(d_{ij}\) is the distance of the shortest path (i.e., geodesic distance, or, simply, geodesic) between node *i* and any other node *j*. In this manner, closeness provides a measure of a node’s strategic positioning within an network in terms of the speed or efficiency with which the flows within a network will pass through a particular node. Larger measures of closeness may, for example, indicate a node that will be able to access information or materials more frequently or quickly than others with lower values.

Another means of conceptualizing prominence of a node in terms of flows through a network is *betweenness*, which measures a node’s potential for being able to capture, enable, or impede the passage of informaiton or materials in a network. As such, it is calculated as

$$\begin{aligned} C_i^{BET}=\frac{g_{jk}(i)}{g_{jk}} \end{aligned}$$

(2)

where \(g_{jk}\) is the number of shortest paths (i.e., geodesics) between node *j* and node *k*, and \(g_{jk}(i)\) is the number of those paths that include *i*. In this manner, *betweenness* measures the degree to which a particular node *i* commands strategic junctures within the network.

A major weakness of both *betweenness* and *closeness*, as they are commonly used, is that neither measure was conceived to take the value of the ties into account. For I–O networks, this is a major shortcoming, as the actual magnitudes of intersectoral flows are a critical consideration. A solution has been proposed by Dijkstra (1959), Brandes (2001) and Newman (2001) and further modified by Opsahl et al. (2010) to seek the path with the least cumulative impedance, as opposed to the one with the fewest steps. The idea that drives this modification is that ties with lower values transmit less, and may be considered to impede flow more than ties with greater tie values. The resulting implementation is a weighted distance measure given by

$$\begin{aligned} d^{w\alpha }_{(i,j)}=min\left( \frac{1}{(w_{ih})^\alpha }+\cdots +\frac{1}{(w_{hj})^\alpha }\right) \end{aligned}$$

(3)

where the inverse of the tie values (i.e., weights) \(w_{ij}\) is summed for each of the paths between node *i* and node *j*, and the path of least resistance (i.e., the one with the lowest value) is selected. The \(\alpha\) coefficient functions as a tuning parameter that is used to either emphasize or deemphasize whether the number of steps should be taken into account. Setting \(\alpha =0\) produces the same measure as if the ties were of binary values; setting \(\alpha =1\) sums the inverse tie values; setting \(0<\alpha <1\) favors fewer steps; and setting \(\alpha >1\) favors stronger tie weights in calculating shortest path distances.

Opsahl et al. (2010) employed the weighted geodesic measure shown in (3) in both *closeness* (4) and *betweenness* (5) in a manner that is fairly straightforward.

$$\begin{aligned} C_i^{WCLO}= & {} \left[ {{\sum _{j=1}^n}d^{w\alpha }_{(i,j)}}\right] ^{-1} \end{aligned}$$

(4)

$$\begin{aligned} C_i^{WBET}= & {} \frac{g_{jk}^{w\alpha }(i)}{g^{w\alpha }_{jk}} \end{aligned}$$

(5)

In each case, the weighted geodesic (i.e., shortest path) distance measure has been substituted for the binary form. In the case of *weighted closeness*, depending on the \(\alpha\) setting, the relative distance in terms of summed inverse tie values is substituted for a count of the number of steps in each path when selecting the lowest value. For, *weighted betweenness*, on the other hand, \({g^{w\alpha }_{jk}}\) is a count of the number of geodesics occurring between node *j* and node *k* (Opsahl 2015).

Weighted path-based centrality measures hold the potential to reveal the relative prominence of nodes in a valued network. Each is well suited for use with valued networks, though there are some shortcomings for each that should be noted in regard to their potential for application in I–O networks. The characteristics of I–O networks that make them a challenge for both standard and weighted network metrics include the values of ties in I–O networks, the recursive loops present in aggregated networks, and their density merit consideration in modeling the flow of resources.

If one considers the weighted distance equation given in Eq. (3) to modify closeness (4) and betweenness (5), it should quickly become apparent that such a weighting metric will function in a manner similar to the measures designed for dichotomous ties (Eqs. (1) and (2)) only when the tie values are limited to a relatively narrow range of integer values. Given that I–O networks are expected to take a theoretically unlimited range of positive continuous values, it becomes increasingly likely that the measure will produce one unique shortest path for any given node pair. Such a solution would emphasize the prominence of nodes that are situated in some of the most proiminent production sectors in the region being evaluated. Although prominence within key sectors will produce useful information, the lack of alternate shortest paths between nodes holds the potential to mask the relative importance of other nodes that may hold secondary importance–something that is also important from a planning and disaster mitigation standpoint. The ability to tune the measure using the \(\alpha\) setting helps to reduce this tendency, but also requires a more standardized approach to tuning that has not yet been evaluated in I–O networks at this point.

An additional challenge to the anlaysis of I–O networks occurs when considering the recursive loops that are used to best model flows within a system that has been aggregated to create a set of nodes that would normally trade amongst themselves into a metanode that represents an entire industrial sector. Depending on the level of aggregation, the presence of loops within I–O networks can be substantial. However, both the classic binary and the weighted assessments of shortest paths through a network will logically always ignore such loops, as they would not normally constitute a “shorter” path through the network. This is not a realistic representation of the actual behavior of flows within an I–O network.

The final consideration that should be important to those modeling I–O networks is the high density of ties within the network. Using path-based measures that treat ties as binary tends to make it look as though all nodes are relatively “close” to one another since, on average, the paths throughout exceptionally dense networks will be of roughly the same length. This tendency is somewhat reduced by using weighted ties. When employing weighted measures, both closeness and betweenness will produce measures of prominence that are relative to the total network. They do not provide any special consideration for the more immediate neighborhood (e.g., manufacturing sector) around each node.

One way of avoiding most of those limitations is adopting measures of centrality based on random walks. Newman (2005) explored and further developed the notion of a measure of centrality based on random walks to overcome the need for a pre-determined, known flow from each source (*s*) to each target (*t*), and stated that “the random-walk betweenness of a vertex *i* is equal to the number of times that a random walk starting at *s* and ending at *t* passes through *i* along the way, averaged over all *s* and *t*.”

Experts in I–O modeling, not yet familiar with network metrics, perceive the notion of random walk as one that does not correctly represent the relationships in the economic system. Their complaints are not without merit because, after all, industrial sectors do not consume a random collection of inputs in the hopes of producing a very specific output. Those input are indeed very well defined by a sector’s production function. In the same vein, one could conceptualize random walks as representing a way of exhausting all possible combinations of inputs that produce all outputs in the economy. Because the computation of any random walk-based measure involves the “value” of the flow between two given sectors, only the ties that are present will influence the magnitude of the metric. Therefore, all random walked-based measures of an I–O system will reflect the true underlying production functions that link those sectors.

Network metrics that use random walks could identify those sectors that participate in all other sectors’ production functions more often and with a greater insidence. This is one of the most significant differences between simple measures of centrality (which would identify the “presence” of the tie, in an on–off fashion) and random walk-based measures of centrality, which also include an apt representation of the intensity of the tie.