The winner takes it all-Competitiveness of single nodes in globalized supply networks.

Quantifying the importance and power of individual nodes depending on their position in socio-economic networks constitutes a problem across a variety of applications. Examples include the reach of individuals in (online) social networks, the importance of individual banks or loans in financial networks, the relevance of individual companies in supply networks, and the role of traffic hubs in transport networks. Which features characterize the importance of a node in a trade network during the emergence of a globalized, connected market? Here we analyze a model that maps the evolution of global connectivity in a supply network to a percolation problem. In particular, we focus on the influence of topological features of the node within the underlying transport network. Our results reveal that an advantageous position with respect to different length scales determines the competitiveness of a node at different stages of the percolation process and depending on the speed of the cluster growth.

Introduction

Global connectivity is central to our social, economic and technological development [1-4]. The growth of a global transportation network has dramatically changed world economy and led to increased efficiency and more centralized production [5]. But this global connectivity also bears new, systemic risks—highlighted in particular in the financial sector [6, 7].

Economies of scale are a major driving force in the formation of many of these socio-economic networks. Generally, a well developed economic agent with high connectivity is more attractive or competitive compared to smaller, less developed agents. The larger agents thus naturally attract even more connections [8-10]. In social network theory, this principle is commonly referred to as preferential attachment, driving the formation of scale-free networks [11]. In economic theory, economies of scale have been identified as a key mechanism leading to the emergence of trade networks and globalization [5, 12]. More recently, we have seen the emergence of quasi-monopolies in digital platform economies where economies of scale are particularly strong [13-15]. In this case the winner takes it all. But who wins and how?

Understanding which node in a network is the most competitive one and how it ‘wins’ over the competition as the network evolves toward global connectivity is still largely an open question. In particular, a systematic study of network formation in a heterogeneous geographic environment is a demanding task. Percolation models describing network growth typically involve random processes [16-18], while optimization models of the network structure typically start from a single global objective function [19-23]. However, neither model class fully describes socio-economic networks, whose formation is determined by the individual decisions (optimization, non-random) of interacting agents (multiple different objective functions). Economic equilibrium models and game-theoretic models capture these interactions and the individual decision but quickly become intractable as the number of agents increases [3, 24–28].

In this article, we study a simplified supply network model that explicitly includes nonlinear nonconvex economies of scale and transportation costs while simultaneously enabling a semi-analytical treatment by mapping the evolution of the network to a percolation problem [29]. In the model, agents try to satisfy a given demand at minimum costs, either through domestic production or via imports. Economies of scale favor the centralization of production and the emergence of trade. On the other hand, non-zero transportation costs favor distributed production. Simulating the evolution of the emerging trade network in this model allows us to systematically study how the transition to a globally connected supply network takes place, how the transportation network affects this transition, and last but not least which geographic factors provide an advantage for the competitiveness of the economic agents. In particular, we demonstrate that the way to be successful in the globalization process is to be in an advantageous position on the correct length scale. We show that the length scale characterizing the competitiveness of a node changes depending on the stage of the percolation process and the speed of the cluster growth.

Methods

Economic percolation model

We analyze the influence of topological features on the importance of nodes in a network formation model recently introduced by Schröder et al. [29]. The model describes the formation of global connectivity in networks inspired by the evolution of trade interactions in a fundamental network supply problem [5, 12]. The idea is as follows: Each node (or economic agent) i ∈ {1, 2, …, N} in the network has a fixed demand D (identical for all nodes). A node i can either fill this demand by domestic production or by making purchases from other nodes it is connected to via the underlying transport network. Filling this demand always incurs costs for node i: (I) production costs for production at node k, even for domestic production where k = i, and (II) transport costs for transport from node k to node i if node i makes purchases from other nodes (k ≠ i). This general setup is illustrated in Fig 1.

Fig 1

Network supply problem.

Each node i chooses a supplier k to satisfy its demand D at minimal cost K = min K. These costs include: (I) production costs at node k, where the costs per unit depend on the total amount of production S at that node (left panel), and (II) transport costs that depend on the distance T between the nodes k and i in the underlying transport network (dashed line). All nodes in the network (including k) simultaneously solve their individual optimization problem.

The production costs of goods manufactured at node k and consumed at node i are given by where S denotes the amount of goods produced at node k and consumed at node i. The costs per unit p are decreasing with the total production due to economies of scale at node k. This means production becomes more efficient for larger quantities. Throughout this article we assume a linear relation for the sake of simplicity, where the parameter a ≥ 0 directly quantifies the effective strength of the economies of scale and b is a constant offset different for each node, describing inherent production cost advantages.

The transport costs are proportional to the amount of purchased goods S and the distance T between the nodes in the underlying transport network. The proportionality factor pT controls the importance of transport costs relative to production costs. In real-world settings, it typically decreases over time due to technological advancements in the transport sector and serves as the main control parameter for the network formation model. Together, the total costs for node i read as illustrated in Fig 1. This cost structure captures the fundamental incentives for the agents in this supply network percolation process.

Each node i chooses its purchases S in order to minimize its costs under the constraint that it exactly satisfies its demand, ∑ S = D. In general, this leads to N interacting nonlinear and nonconvex optimization problems as the production costs depend on the purchases of all (other) nodes. Nevertheless, a resulting Nash equilibrium, where no node can further decrease its costs by changing its supplier, can be computed efficiently as shown in [29]: Each node i chooses only a single supplier k (either itself or one other node in the network) that can be found efficiently with an adapted breadth-first-search due to the mapping to a local percolation problem. While multiple Nash equilibria exists for each value of p, this mapping uniquely defines the sequence of Nash equilibria describing the states of the supply network during the slow decrease of p depending on the parameters and initial conditions.

We study the evolution of the supply network starting from the limit of infinite transport costs, pT = ∞, such that all nodes purchase locally and no trade takes place. As the importance of transport costs decreases, some nodes start to make non-local purchases such that the production S of other nodes increases. Eventually, large common markets (clusters) emerge in the network of trades S, each with a single supplier node k. In the end, when transport costs disappear, pT = 0, only one giant cluster remains with a single supplier k* with globally centralized production S = ND. This evolution is illustrated in Fig 2 for a small planar network.

Fig 2

Cluster growth in the percolation model.

(a) Evolution of the size S of four clusters measured by the production S of the clusters supplier i (the number of nodes relative to the size of the whole network). Every node in the network optimizes its costs to satisfy its demand as described in the main text. As the importance of transport costs pT decreases, nodes make external purchases and clusters (common markets) emerge where production is centralized at a single node k. As pT → 0, only a single, global cluster with a central supplier k* = 16 and S16 = 1 remains (blue line). (b-e) Snapshots of the network for different values of pT. The four clusters with centralized production shown in panel (a) are illustrated in their respective colors and the central supplier node is highlighted. Black nodes do not belong to any of these four clusters. Solid colored lines indicate active links in the transport network, dashed lines indicate potential transport links that are not used by the four large markets. Parameters are D = 1/N, b ∈ [0, 1] distributed uniformly at random and a = 10−3. The planar network is created as the Delaunay triangulation from N = 100 points distributed uniformly at random in the unit square (see Methods for more details).

In this article we study two main aspects of the formation of this trade network: First, how does centralization occur? That is, how does the transition from local production at large pT to centralized production at low pT take place? Second, we analyze which node k* becomes the final supplier (the center of the globally connected cluster) as production becomes fully centralized for pT → 0.

Analysis of network structure

The economic percolation model includes heterogeneous geographical conditions explicitly. The matrix T encodes the distances of all pairs of nodes (k, i) which depends on their geographic location and the structure of the underlying transportation network. Hence, the model allows to systematically study the influence of geographical or topological properties on the formation of connectivity and trade and the centralization of production. Are there any geographical or topological features that determine which node becomes the final supplier and which does not?

To study the impact of the transport network topology, we consider four different random network ensembles. We start from an ensemble of geographically embedded networks obtained by distributing N = 1000 nodes uniformly at random on the unit square. Edges are constructed by a Delaunay triangulation with periodic boundary conditions. Each of the resulting M = 3000 links is undirected and assigned a distance equal to the Euclidean distance between the connected nodes. The distance T of two arbitrary nodes i, j in the network is finally obtained as the geodesic or shortest path distance in the network.

The other random network ensembles are obtained from the initial ensemble by a reshuffling of the edges. This procedure keeps the number of connections and the distribution of the individual edge lengths identical and thus leaves the networks comparable to each other. We apply three different reshuffling procedures creating randomizations with different properties: First, we keep the structure of the network the same but choose a random permutation of the distances (random weights). This breaks correlations between the link distances and the node position. Second, we uniformly randomly rewire all links to different nodes under the constraint that the resulting network is connected. The network then has a topology corresponding to a Poisson random network [2]. Comparison of this randomization to the original network allows us to understand the impact of regular versus random network topologies. Third, we create a Barabasi-Albert scale-free network with the same number of links and the same distances for the links [11]. We thus create four different ensembles with identical average degree and edge lengths, but vastly different global structures. For instance, the degree distribution changes from narrow for the geometric and Poisson random networks to heavy-tailed for scale-free networks.

Model parameters

In addition to the structure of the transportation network, several model parameters determine the evolution of the trade network. First, we note that the system evolution is invariant with respect to a rescaling of the costs. In particular, we can set D = 1/N by choosing an appropriate unit system. A rescaling of the distances can be absorbed into the main control parameter pT describing the transport cost per unit. It characterizes the relative importance of transportation costs with respect to production costs.

Two parameters a and b characterize the production costs via the costs per unit p(S) = b − aS [Eq (2)]. Since only the relative ordering of the costs are relevant to compare different suppliers (in the form of K < K), we scale the costs such that all b ∈ [0, 1] with min b = 0 and max b = 1. In particular, we choose the b uniformly at random from the interval [0, 1]. The second parameter a characterizes the economies of scale and has a strong impact on the model behavior. We perform simulations for vastly different values a ∈ {10−5, 10−4, …101} to cover all different regimes. To put this into context, note that total centralization of production leads to a decrease of production costs by exactly NDa = a for D = 1/N. Economies of scale are negligible if a is much smaller than typical differences of the cost parameter b, i.e., for a ≪ 1/N = 10−3. Economies of scale are dominant if a is of the order of the largest difference of the b, i.e. for a ≈ 1. The range a ∈ {10−5, 10−4, …101} covers both regimes.

In summary, we perform simulations for four different transportation network ensembles and several values of a. For each case we consider 1000 different random realizations of the transportation network with 10 different permutations of the b each, resulting in 10.000 measurements per ensemble and value of a. For each realization, we start the simulation in the limit of large transport costs, pT = ∞, without any trade interactions. We gradually lower pT and record the emergence of a trade network, i.e., the emergence of connected components of the network defined by the purchases S, as well as the final supplier for pT = 0.

Results

How does global connectivity emerge?

To understand the emergence of a globally connected network we record the size of the largest clusters as the transport costs decrease from pT = ∞ (no trade) to pT = 0 (single, globally connected cluster). A trade network between nodes emerges as transportation costs decrease. An example of the centralization of production is shown in Fig 2 for a small geographically embedded random network. For pT = 1.0, several nodes have already decided to purchase their goods from other neighboring nodes and multiple clusters have formed where production is centralized to a single node. The clusters grow when pT decreases to pT = 0.5 as further nodes decide to purchase non-locally. Finally, many nodes again change their supplier, joining one large, global cluster with strong economies of scale instead of the smaller local clusters. In the end, as pT → 0, production is fully centralized at a single node. The size of the four largest clusters is shown in Fig 2(a) as a function of the transportation cost parameter pT.

Inspecting this evolution, we are directly led to the question how the transition to global connectivity takes place under different circumstances. Is it very sudden with a single large change in the size of the largest cluster or is the transition slow and the largest cluster grows gradually as pT decreases? Does a single node expand its cluster or do multiple large clusters grow and only later merge to one global cluster? To answer these questions, we measure the largest gap max[ΔS(1)] in the size (total production) of the largest cluster [30] as well as the maximum size of the second largest cluster max[S(2)], the third largest cluster max[S(3)] and so on over the course of the evolution from infinite to zero transport costs (see Fig 3). The maximal size max[S(2)] of the second largest cluster in particular measures how much clusters grow before global centralization occurs. If it is small, only a single large cluster emerges and local competitiveness is relevant to gain an early advantage. If it is large, at least two large clusters expand side by side before one of them becomes globally dominant and production is completely centralized. Here, the central nodes of the clusters have to compete against each other on a larger length scale. The maximal size max[S(2)] of the second largest cluster serves as a proxy for this length scale.

Fig 3

Multiple clusters or sudden growth?

Distribution of the maximum size max[S(n)] of the n-th largest cluster and largest change max[ΔS(1)] in the size of the largest cluster (insets) during the emergence of global connectivity for (a) the random planar network, (b) the network with randomized weights, (c) the network with uniformly randomized links and (d) the network with scale-free randomized links. For small a, multiple large clusters appear and merge slowly in all networks. For large a, a globally connected cluster suddenly forms from the individual nodes in a single large cascade before any other cluster had the chance to grow significantly. Depending on the value of the parameter a, nodes have to be competitive at different length scales to become the final supplier. The maximal size of the second largest cluster max[S(2)](red) can serve as a proxy for this length scale.

If economies of scale are weak (small values of a), multiple large clusters coexist before they finally merge. As a becomes larger, the maximum size of all clusters except the largest one decreases. Finally, for strong economies of scale a, only a single cluster grows. Correspondingly, the transition to global connectivity becomes more and more abrupt with increasing a, measured by the growth of the gap max[ΔS(1)]. We thus obtain the following picture: For weak economies of scale, several clusters grow and finally merge in a gradual process. For strong economies of scale, only local clusters exist until a globally connected cluster emerges in abruptly. After this sudden transition, exactly one globally connected cluster remains.

We observe rather little differences between the four network ensembles under consideration. The transition from gradual to abrupt emergence of global connectivity is qualitatively the same in all networks and also the transition point is remarkably similar. While the transition is gradual (no large gaps) for a = 10−5, it is sudden for a = 10−3 for all networks. Slight differences are observed only for a = 10−4. While the maximum gap is larger than 0.1 for all realization of the random planar network, the transition is still gradual with smaller changes of the largest cluster for most realizations of a scale-free network.

This is rather surprising, as scale free networks are characterized by the existence of hubs, a few nodes with very high degree. At first glance, one might expect that these hubs can exploit economies of scale most easily, making the transition abrupt already for small a. Our results show that this simple reasoning fails. The impact of economies of scale on the transition and on the competitiveness of nodes is more subtle. In fact, different hubs have to compete when the economies of scale are not dominant (small a). Thus, while hubs allow for the easier formation of local clusters, these hubs then have to compete on a larger length scale (measured by the maximum size of the second largest cluster), where the local properties of the central supplier, such as the high degree of the hubs, are less important. Overall, this competition slows down the centralization of production in scale-free networks. This idea is similar to the mechanism preventing or delaying the merger of large clusters in models resulting in explosive and discontinuous percolation transitions [18, 31, 32].

Who becomes the central supplier?

Understanding how global connectivity emerges, we now address the question who wins the competition in this model. That is, which node i becomes the central supplier of the network for pT → 0? Are there any geographic features that determine a node’s competitiveness?

To characterize the geographical location of a node in a network, we consider several different centrality measures that measure different aspects of a node’s position in the network:

cost centrality 1/b

local closeness centrality 1/minT

global closeness centrality 1/∑ T [33, 34]

degree centrality [34]

betweenness centrality [34, 35].

These quantities measure the advantage of the nodes in terms of (i) global production costs, (ii) small transport costs to a local trade partner, (iii) small transport costs to the whole network, (iv) immediate access to different trade partners and (v) position of the node along many trade routes.

We generally expect that all these properties are beneficial for the nodes. For example, a high cost centrality implies that production is cheap—at least until production costs decrease significantly due to economies of scale. The node with the highest cost centrality would be the socially optimal supplier when pT = 0 and minimize the total costs across all nodes. Similarly, a high global closeness centrality implies that transportation is cheap on average, making the node an attractive global supplier when transport costs are not zero. The remaining three centrality measures also point to a favorable position in the network, but their implication is less clear. High degree and local closeness point to an attractive local environment, while high betweenness centrality is a typical measure of importance in social networks and means that many shortest transportation routes cross the respective node.

To understand which of these properties most strongly influences the competitiveness of a node, we rank all nodes according to their centralities and evaluate if the final suppliers typically have a high or low ranking. We record the final supplier and its centrality ranking x for each random realization of the percolation process. The resulting distributions of the ranks of the final supplier are shown in Fig 4 for the four network ensembles under consideration. In addition, we fit a distribution P(x) ∼ exp[−m(N − x)] to the observed centrality rankings to quantify the importance of the respective centrality. A value of m = 0 indicates a flat distribution, i.e., no influence of the centrality rank x on the chance to become the final supplier. The higher the value of |m|, the stronger the correlation, and the more meaningful the respective centrality to predict which node becomes the central supplier.

Fig 4

How to become the central supplier?

Distribution of the ranking of the final supplier in various centrality measures (see main text) in (a) a random planar network, (b) the network with a random permutation of edge distances, (c) a Poisson random network with a random permutation of the edge distances, and (d) a scale-free network with a random permutation of the edge distances. All networks are constructed from a Delaunay triangulation of N = 1000 points uniformly randomly distributed in the unit square, resulting in M = 3000 links with distances equal to the Euclidean distance between the connected nodes (see Methods for details).

The first, expected observation is the influence of the cost centrality 1/b of a node i. For weak economies of scale (small a) the production costs are dominated by the cost parameters b and low production costs are decisive for the competitiveness of a node. For all network ensembles under consideration, cost centrality is the best indicator for competitiveness for small a, whereas its importance decreases for stronger economies of scale.

The second, more striking observation is the importance of the local closeness centrality. In the case of strong economies of scale a = 1, this centrality measure provides the best indicator for the competitiveness of a node. The histogram of the centrality ranking peaks strongly at top ranks. Local closeness is even more important than global closeness, although we evaluate the global competitiveness of the nodes. Again, this finding holds true for all four network ensembles.

A surprising correlation is found for the two remaining centrality measures, degree and betweenness, for the spatially embedded random network. Contrary to our expectation, the final supplier typically has a low degree and betweenness centrality for strong economies of scale a. This effect is lost or even reversed for the other network ensembles and can be attributed to a subtle geometric property of spatially embedded random networks. In this network class, local closeness centrality is anti-correlated with degree and betweenness centrality. As competitive nodes have a high local closeness, they are likely to have a low degree and betweenness centrality. This observation is particularly relevant since real-world transportation networks are typically spatially embedded, with the exception of digital, data exchange networks. Note that similar correlations exist for other network ensembles as well. For example, nodes with a high degree centrality in the reshuffled scale free networks typically also have high local closeness centrality, due to more opportunities for a short link.

Finally, a more subtle implication of the centrality measures is that, depending on the parameter a, the size or length scale of the relevant neighborhood changes. This length scale is defined by the critical size the largest cluster must reach before it becomes the global supplier. The effect is illustrated in Fig 5. For small a, the number of customers does not significantly affect the costs and one new customer allows the supplier to attract customers only in a small additional range [Fig 5(a)]. Consequently, a node must attract a larger number of customers to become globally competitive and the critical size is (almost) equal to the total size of the network. In this regime, global centrality measures like the cost centrality are most relevant. For intermediate a, a single customer allows the supplier to attract nodes in a larger range [Fig 5(b)]. The critical length scale becomes smaller and we need to put more weight to the local structure. In this regime, the global closeness centrality and the degree centrality start to become better predictors, quantifying the centrality of a node in a local neighborhood. Finally, for very large a, the critical size of the largest cluster becomes 2 and one single customer induces a sufficiently large change in production costs for the supplier to become globally competitive immediately [Fig 5(c)]. The centrality of a node in its most local context then becomes the deciding factor. This is best measured by the distance to the nearest neighbor, the local closeness centrality 1/minT.

Fig 5

Impact of a single customer.

Sketch of the effect of a single (new) customer for a node. With the new customer production increases and the production costs per unit decrease by aD (economies of scale). This compensates larger transport costs for nodes further away from the supplier. Consequently, the supplier becomes competitive in a larger range and can potentially attract additional customers. The blue disks indicate the distance that is compensated by the decrease in production costs due to one customer (two customers). (a) For small a, the change in production cost is small and likely has no immediate effect [compare a = 10−4 in Fig 4(a)]. The nodes have to compete at all length scales. (b) For intermediate a, a single customer may reduce the costs sufficiently to cause additional nodes to change their supplier. In this case, nodes have to compete at a local scale until they reach a size sufficiently large to take over the global cluster. (c) For large a, a single customer definitely reduces the costs sufficiently to cause a cascade of purchasing decisions and the first node to attract a customer takes over the whole cluster. Here, only the immediate neighborhood of a node decides about its success [compare a = 1 in Fig 4(a)].

Comparing results across the different network topologies, we find that the network topology becomes more important when the diameter is smaller, i.e., for Poisson and scale-free network structure. Since the total transport costs in these networks are smaller (proportional to the smaller diameter of these networks), the critical size to become the global supplier is also smaller. Thus, local length scales and the (local) network structure become important already for smaller values of a.

Conclusion and discussion

Economies of scale are a decisive factor in the formation of socio-economic networks and the globalization and centralization of economic activities. Eventually, the winner takes it all. Here we have studied core aspects of the question who wins and how in a simplified model of supply network percolation.

The formation of socio-economic networks is a guiding research question across disciplines, including economics [4–6, 12], sociology [3, 27, 36] and statistical physics [2, 11]. Key mechanisms and global properties of network formation through economies of scale have been thoroughly analyzed [5, 11, 27], whereas the microscopic processes in large systems with many heterogeneous actors are much harder to grasp. Most traditional models of network formation do not explicitly capture the behavior of individual actors [11, 17, 37]. Percolation models are based on random processes, while optimization models typically assume a common global objective function. In contrast, game theoretic models describing individual agents [21, 25, 26, 38] are often hard, if not impossible, to solve for large heterogeneous systems. In this article, we have analyzed a supply network model [29] that explicitly includes economies of scale and individual decisions, yet remains simple enough to allow for an efficient simulation of network formation and centralization in large heterogeneous environments. We exploit this fact to reveal the topological properties that determine the importance of a node for the emerging globally connected network.

The model yields the structure of a trade network given an underlying transportation network as a function of two main parameters: the strength of economies of scale a and the transport costs per distance pT. As transport costs decrease, trade links are established and the production is centralized to fewer and fewer nodes. For weak economies of scale, this process is gradual. Nodes compete at all length scales and the merger of two large clusters is inhibited while transport costs are large, similar to mechanisms of explosive percolation [18, 31, 32]. The internal cost parameters are decisive for the competitiveness of a node. Only nodes with low productions costs b have a chance to become the final supplier of the network once production is centralized completely. The geographic location of the nodes in the network, characterized by different centrality measures, plays only a minor role. In contrast, if economies of scale become dominant, this picture changes entirely: Production is centralized in a single, discontinous percolation transition once transportation costs decrease below a critical value. Only a single node attracts a significant number of customers and wins the competition almost instantly. Moreover, the transition becomes abrupt and as such hard to foresee. The chance of a node to become the central supplier is now mostly determined by the location of the node in the network. Interestingly, however, global centrality measures are not the best indicator for competitiveness. Instead, a local measure of the distance to the nearest neighbor, referred to as local closeness, is the best indicator for the success of a node. These results remain qualitatively unchanged for a broad range of cost functions describing economies of scale [29]. While modifications, for example stopping the process at non-zero transportation costs, change the quantitative evolution, the mechanistic insights into which length scales determine the importance of nodes during the emergence of (global) connectivity are generally applicable.

Loosely speaking, our findings are as follows: For weak economies of scale the internal properties of a node or economic agent are decisive. Competition occurs across all length scales in the network and basic efficiency provides the greatest advantage in all stages of the emergence of global connectivity. Only the (globally) most efficient nodes have a chance to take over the network. For strong economies of scale speed becomes the most important factor, rather than efficiency or global location. Competition occurs only locally to gain a first advantage and only the agent with the highest local closeness can rapidly attract the first external customers and then exploit economies of scale to grow its market, skipping over the competition in later stages of process. For the future it would be of eminent interest to study how other factors influencing economic globalization processes confirm or modify these findings and whether they can be confirmed in real world settings.

Information on the realization of network typologies (10 different realizations for each reshuffling method) indexed by r.

Legends can be found in te readme.txt file.

(ZIP)

Click here for additional data file.

Simulation results.

Legends can be found in te readme.txt file.

(ZIP)

Click here for additional data file.

10 Aug 2019

PONE-D-19-16528

The winner takes it all - How to win network globalization

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Reviewer #1: A serious problem with this paper is that it has very little to do with trade & globalization, and is therefore misleading as far as it's title/advertising goes. The formulation of the problem is a very narrow, cost side approach, whereas "globalization," it's structure and who "wins" (the use of this term is questionable too), involve both demand and supply side considerations. More realistic additions to the model, such as the addition of variable costs on the production side (at the moment there are only fixed costs) and fixed costs on the transportation side (at the moment there are only variable costs) would invalidate the results of the model. In this sense, the model presented in the paper is a caricature of trade and globalization.

More specifically:

-The supply problem is divorced from the demand side. Why is this about globalization? The authors provide no interpretation or intuition of this.

-The production side of the economy is assumed to have a fixed cost, and the economies of scale come from this sole feature. There are no variable costs of production in the model. Such a cost structure is restrictive and applies to very few industries in the real world. There is no explanation for this omission.

-Transportation costs on the other hand are modeled as variable costs. Why? There could very well be fixed costs on the transportation side too. A small change such as this would invalidate the results.

The problem in the paper is more related to a cost-minimization engineering problem, that perhaps fits in logistics or operations research, but is a caricature of the economics of trade and globalization. It is a very narrow and misleading characterization of the problem.

Reviewer #2: In this work, the authors assess the importance of node in trade networks using a percolation-type model of economy of scale. They answer two questions: how a globalized market emerges and who wins the competition and takes the all in the globalization. They conclude two different scenarios depending on the strength of economies of scale. For weak economies of scale, internal properties (costs) of nodes is an important factor while for strong economies of scale, the efficiency is an important factor to find out the power of nodes.

Identification of the power of node in terms of the network structure is a central problem in complex network society and physics of complex systems. In this work, the authors successfully provide a method to identify and analyze the importance of nodes. This work is novel and all the results and discussion in the paper are well supported by their analysis. Therefore, I recommend the paper to be accepted in PloS ONE with a minor revision taking into account the following issues.

1. All the analysis in the paper is dependent on a resulting Nash equilibrium state in the model. Is this equilibrium state uniquely defined? If not, is the following analysis reliable?

2. The mechanism of merging processes with weak economies of scale is reminiscent the explosive percolation which have been popular in physics community recently. Can the authors add some discussion about a association between them?

3. The first sentence in the very last paragraph is not complete: Loosely speaking, we our findings are as follows.

**********

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Reviewer #2: No

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17 Oct 2019

Response to Reviewers

We thank both reviewers for taking the time and effort to read the manuscript and comment on our work. Reviewer 2 considers the manuscript to address a ‘central problem in complex network’ science and recommends publication. Reviewer 1 raises some points regarding our choice of (economic) terminology. We understand the confusion with the terminology in the presented context and have adjusted the terminology, specifically in the title ‘The winner takes it all - Competitiveness of single nodes in globalized supply networks’ and abstract in the direction of network science to more accurately reflect the intent of the manuscript: to quantify the (topological) features that determine the importance of a node in a simplified percolation model inspired by the fundamental forces driving the emergence of global connectivity in a supply network. As such, while we do not aim to exactly model the emergence of globalization, we hope that our results may help to better understand this process and other processes sharing similar mechanisms. We have revised the manuscript to more clearly point out this goal.

We believe the revised manuscript now well matches the level of contribution desired by PLOS ONE.

Response to Reviewer 1:

“A serious problem with this paper is that it has very little to do with trade & globalization, and is therefore misleading as far as it's title/advertising goes. The formulation of the problem is a very narrow, cost side approach, whereas "globalization," it's structure and who "wins" (the use of this term is questionable too), involve both demand and supply side considerations. More realistic additions to the model, such as the addition of variable costs on the production side (at the moment there are only fixed costs) and fixed costs on the transportation side (at the moment there are only variable costs) would invalidate the results of the model. In this sense, the model presented in the paper is a caricature of trade and globalization.”

We thank the reviewer for raising this point and apologize for the confusion due to our choice of terminology. We indeed intend the model to be a minimal model (“caricature”) ¬¬of a supply network globalization process capturing only a few key aspects. The model is not intended to exactly represent all aspects of globalization, but focus only on the driving forces in the simplified perspective of a percolation model that remains easy to study even for larger networks. This percolation-type model enables us to study the importance of nodes during the emergence of global connectivity in supply networks from the perspective of network science, a “central problem in complex network” as stated by Reviewer 2. In this context, we understand the reviewer’s confusion with our use of the word ‘globalization’. To avoid confusion and make the aim and scope of the model clearer, we have adjusted the wording in the revised manuscript to use network science terminology rather than economic terms (for example, cluster instead of market). We have also modified the title, now ‘The winner takes it all - Competitiveness of single nodes in globalized supply networks’, and the wording throughout to more clearly convey this intention in the revised manuscript.

In general, the choice of cost functions and constraints assumed in the model are made for two reasons: (1) to simplify the model as much as possible while keeping the fundamental driving forces; (2) to allow the mapping to a (local) percolation problem with an efficient solution. Nonetheless, many extensions are already included and simply require a rescaling of the parameters or a small (quantitative, not qualitative) modification of the cost functions. In this sense, the results presented describe the qualitative behavior for a range of processes. For a more detailed discussion of the specific points mentioned by the reviewer (additions to the cost function), we refer to the replies below.

We hope that the change in terminology and additional explanations and clarifications more clearly convey the intent and scope of the manuscript.

“-The supply problem is divorced from the demand side. Why is this about globalization? The authors provide no interpretation or intuition of this.”

As described above, the model is intended to describe the fundamental mechanisms of the growth of a supply network (given fixed demand) from the point of view of a percolation model. We hope that the change in terminology and additional explanations and clarifications more clearly convey the intent and scope of the manuscript.

“-The production side of the economy is assumed to have a fixed cost, and the economies of scale come from this sole feature. There are no variable costs of production in the model. Such a cost structure is restrictive and applies to very few industries in the real world. There is no explanation for this omission.

-Transportation costs on the other hand are modeled as variable costs. Why? There could very well be fixed costs on the transportation side too. A small change such as this would invalidate the results.”

The reviewer asks about the cost functions in the model and why they were chosen as presented. We thank the reviewer for raising this question. To avoid any potential confusion, we first reiterate the cost structure in the model:

• transportation costs per unit are linearly increasing proportionally to both distance and amount

• production costs per unit are affine linearly decreasing (economies of scale)

The absolute production cost is then given as a quadratic function of the total amount x as c(x) = c1 x – c2 x2. The production costs describe the effective result of both variable costs increasing linearly with the amount as well as a discounting term due to the economies of scale. In fact, we do not assume explicit fixed costs in production. The specific choice of these cost functions as (affine) linear is made for the simplicity of the model. In particular, other choices for the production costs are possible under reasonable constraints (non-increasing cost functions and identical demand to guarantee the mapping to the percolation problem, see Ref. [29]).

In this cost structure, we implicitly assume that either the fixed costs in production are negligible compared to the variable costs (no fixed costs) or that fixed costs are identical at all nodes. In the second case, the fixed costs only affect the absolute costs, not the ordering of which supplier is cheaper. Similarly, other extensions are already implicitly included in the model. For example, fixed costs in transportation (as suggested by the reviewer) would qualitatively correspond to a non-zero value of the control parameter pT. At this point in the evolution of the model, there may not be a single globally connected cluster. However, the properties identified in the manuscript still help to understand the evolution, the number of large clusters, and to identify nodes that are likely the center of large clusters. As the goal of the manuscript is to study the transition to global connectivity, we scale all transportation costs with the control parameter pT -> 0 to guarantee a single, globally connected cluster in the end.

These additions or modifications would naturally modify the evolution of the model (e.g. slow down or speed up the transition to a single globally connected component). Yet, the fundamental mechanism of larger clusters growing faster until the effects of the economies of scale are balanced by the transport costs at larger distances, remains the same. As such, the results remain qualitatively valid for a broad range of conditions.

We have added a brief discussion of the choice of the cost function, extensions, and the generality of the results in the revised manuscript.

“The problem in the paper is more related to a cost-minimization engineering problem, that perhaps fits in logistics or operations research, but is a caricature of the economics of trade and globalization. It is a very narrow and misleading characterization of the problem.”

As discussed above, the manuscript aims to understand the fundamental mechanisms of a supply network globalization process in terms of the importance of the topological features of nodes in the transport network. We concede the point that the presented model is not an exact description of "the economics of trade and globalization". Instead, the model studies the problem in a simplified percolation model including (only) the most relevant driving forces. Transferring the presented results to more complex and realistic economic models, they are, of course, quantitatively varied by the additional influences. However, as explained above for the choice of cost functions, these results are qualitatively relevant to a range of processes. In this sense, we argue that the scope is, in fact, not narrow.

We believe the change in wording and additional explanations in the revised manuscript now more accurately represent the aim and contribution of the work presented in the manuscript.

Response to Reviewer 2:

“In this work, the authors assess the importance of node in trade networks using a percolation-type model of economy of scale. They answer two questions: how a globalized market emerges and who wins the competition and takes the all in the globalization. They conclude two different scenarios depending on the strength of economies of scale. For weak economies of scale, internal properties (costs) of nodes is an important factor while for strong economies of scale, the efficiency is an important factor to find out the power of nodes.

Identification of the power of node in terms of the network structure is a central problem in complex network society and physics of complex systems. In this work, the authors successfully provide a method to identify and analyze the importance of nodes. This work is novel and all the results and discussion in the paper are well supported by their analysis. Therefore, I recommend the paper to be accepted in PLoS ONE with a minor revision taking into account the following issues.”

We thank the reviewer for their favorable judgement and their recommendation to publish the manuscript. We address the individual comments in detail below.

“1. All the analysis in the paper is dependent on a resulting Nash equilibrium state in the model. Is this equilibrium state uniquely defined? If not, is the following analysis reliable?”

We thank the reviewer for this important question. Indeed, multiple Nash equilibria coexist in the process. The simplest example can be seen for sufficiently large economies of scale a and no transportation costs pT = 0: centralized production at any node is a Nash equilibrium.

Importantly, during the evolution of the globalized supply network, the transport costs decrease slowly such that only a single node updates its supplier at first. While this decision may cause a cascade of decisions from other nodes (e.g. when a cluster becomes too small and the cost increase too much to keep all nodes), these decisions are well ordered. For example, an ordering can be defined by the new cost for the node making the decision. Other sensible choices of the ordering (e.g. ordering by the largest cost difference) do not qualitatively change the results. However, these other orderings do not necessarily guarantee that nodes make only ‘local’ changes (i.e., switch to a supplier that supplies one of their neighbors, see ref. [29] for a more detailed discussion).

In this sense (given the ordering of the decisions), while other Nash equilibria exist, the sequence of Nash equilibria that the model goes through is uniquely defined based only on the model parameters and the initial conditions (see ref. [29] for a discussion of the resulting hysteresis effect when reversing the process).

We have added a short explanation on the existence of multiple Nash equilibria to the revised manuscript.

“2. The mechanism of merging processes with weak economies of scale is reminiscent the explosive percolation which have been popular in physics community recently. Can the authors add some discussion about a association between them?”

We thank the reviewer for this question. The reviewer correctly identifies the similarities of the network globalization process analyzed in the manuscript to known explosive percolation transitions. The competition between the clusters/markets (effectively not allowing two large clusters to merge until transportation costs decrease sufficiently) is similar to mechanisms in known models of explosive or discontinuous percolation. Additionally, the restriction of transportation costs results in compact clusters, further promoting explosive or discontinuous transitions. In fact, the transition in the presented model is often genuinely discontinuous. The easiest example is the limiting case of very high economies of scale: a single node buying from a new supplier causes all other nodes to also buy from that supplier and the network switches from disconnected nodes to one connected market discontinuously.

A rigorous study of the transition in the presented model in the thermodynamic limit, however, is difficult. In many network topologies, the different scaling of the network size/cluster sizes and the distances in the network breaks the balance between the production cost (economies of scale, cluster sizes) and the transport costs (control parameter, distances), often resulting in trivially discontinuous transitions at pT = 0 in the limit. A sensible study is possible using appropriate network topologies (such as geometrically embedded networks as in the present manuscript or a complete graph) where the corresponding scaling of distances is automatically correct. In these cases, the model in fact exhibits a transition from a slow, gradual transition to a sudden, discontinuous transition.

We have added a short discussion of this connection to the revised manuscript and refer the interested reader (as well as the interested referee) to ref. [29], where the model was first introduced, for a more detailed discussion of the model in a percolation context.

“3. The first sentence in the very last paragraph is not complete: Loosely speaking, we our findings are as follows.”

We thank the reviewer for pointing out the typo and have corrected it.

Submitted filename: Response_to_Reviewers.pdf

Click here for additional data file.

4 Nov 2019

The winner takes it all - Competitiveness of single nodes in globalized supply networks

PONE-D-19-16528R1

Dear Dr. Han,

We are pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it complies with all outstanding technical requirements.

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Academic Editor

PLOS ONE

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Reviewer #2: All comments have been addressed

**********

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Reviewer #2: Yes

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Reviewer #2: Yes

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Reviewer #2: Yes

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Reviewer #2: No

11 Nov 2019

PONE-D-19-16528R1

The winner takes it all - Competitiveness of single nodes in globalized supply networks

Dear Dr. Han:

I am pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please notify them about your upcoming paper at this point, to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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