Spatial Networks: Modeling the Impact of Bridge Nodes in Trade Networks

Introduction

Municipal economies across the world benefit from international commerce, especially port cities where the import and export of goods is often the main source of economic activity. This commerce can be represented using a trade network in which nodes represent coastal cities, land links represent active naval trade routes between said ports. For my final project I wanted to model the impact of disrupting an established sea trade network through the introduction of a bridge node. I decided to model this phenomenon in the fictional world of Westeros from Game of Thrones, as the continent’s shape would make for an interesting port network structure. In Westeros, the endless sea ice found in the landmass’s north forces all cross-continental naval trade networks to snake along the southern coastline. The addition of a canal in the continent's north would allow for significantly shorter routes for ships traveling between northeastern and northwestern ports, and overall facilitate more commercial interaction between the eastern and western coastal cities.

 

To determine the significance of this canal’s introduction I compared two versions of Westeros, a baseline with the original city layout and a version with the canal’s addition. I decided to use Cadaei’s shapefile of Westeros as the unbridged baseline for this simulation as their version included all the notable cities and a detailed coastline of the continent. The bridged version of the model plots a canal city (by the name of Bogharbor) in the narrow Neck of Westeros, which serves to connect the eastern cities of the North to the prosperous Riverland city of Seagard. The addition of Bogharbor transforms Westeros from a peninsula effectively into a circumnavigable island due to the bridge node’s position in the far north of the network.

 

Research Question

To what extent does the addition of a bridge node (Bogharbor) impact the average path length and efficiency of Westeros' naval trade network?

 

Methods

I created a simulation using the BehaviorSpace application of Netlogo, in which each city can create a trade link with itsneighbors which then transfer resources between the ports. To make the network dynamic, I decided to make the cities from each of the 9 kingdoms of Westeros create a unique primary good (furs from The North, timber from The Stormlands); if cities receive a primary good which is foreign to their Kingdom, they will refine it into a secondary good (grapes to wine, fish to sushi). This refinement creates additional value, and to model the increase in municipal economic activity I had this process increase the city’s “development” metric. Cities with higher development values are more appealing to trade with, so the shape of the trade network will evolve. This mechanic also places an emphasis on international trade, as cities near the borders of kingdoms will have direct access to foreign goods and therefore are able to develop quicker than interior cities. As Bogharbor is a city at the trisection of three kingdoms, the bridged version of the model’s increased connection between realms may grow to have a higher average development metric.​​​​​​​​
 

At each time interval (tick), the ports create connections with all available neighbors to exchange primary goods. This link allocation method works by first asking cities to make a list of all ports within 30 patches and then sorts them on their “appeal” value. As Netlogo is an agent-based modeling program, the appeal value is what port agents consider when deciding whether or not to establish a trade route with a city. Appeal is a composite metric comprised of adding together three components: the city’s influence, the potential route’s length, and the city’s development.

 

Influence is determined by a city’s fixed size variable, to model the impact of larger and smaller participants in the trade network. Route length information is collected by calculating the distance in patches between the exporting node and the potential importer. The final component, development, is a total of all the primary goods the city has refined. This represents the increase in economic activity brought about by said refinement and creation of wealth. Coastal cities connect with available ports which have the highest appeal value first, until either they exhaust the list of potential connections, or the city runs out of links to allocate (a limit determined by their fixed city-size variable). At each tick every city trades away the primary goods they produce within the city to other settlements in the trade network. Once cities receive foreign primary goods, they refine it into a secondary good each tick. This refinement limit prevents cities on the borders of the kingdoms from consuming all of the resources and allows for the flow of goods along the network. These secondary goods then remain in the city they are made, and their total represents the city’s development score.

 

To quantify how the network and the cities within it evolved throughout the simulation I

decided to track three metrics: average path length, average link length, and the average

development of the two network versions. I chose average link length to track whether the model evolved to prefer trading with neighbors or with further (but larger) cities. I studied the change in average path length to gain insight as to whether the model’s evolution increased network connectivity, or if it promoted the creation of a more decentralized network. Finally, I compared the rate at which development increased between the bridged and unbridged model versions. Analyzing this metric will provide insight as to whether cities are developing differently due to the greater flow of goods between both coastlines of the continent.

 

Results

Average link length increased significantly throughout both models, rising from ~12.5 patches upon setup, to about 16 patches by tick 100. This increase was inversely correlated with the average path length statistic which decreased significantly over the same time span.

 

 

 

 

 

 

 

 

 

 

 

 

 

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Map 2. The Westeros trade model version with Bogharbor, at the beginning and end of the simulation period

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While the overall trends in average path length and average link length were similar, the extent of the change between model versions differed significantly. The most notable difference is in average path length, the unbridged version of the model begins with a value of 8.5, while the bridged simulation starts with an average path length of just over 6.5. The average path length of the unbridged simulation decreased at a faster rate than the bridged version, as the metric decreased by 2.5 connections by tick 100 while the simulation with the canal only shortened by 1.5 lengths, however both seemed to plateau around tick 70. To test the significance of the results, I performed a two-tail, two-sample, equal variance t-test in excel, using the BehaviorSpace data from both models. This analysis returned a p-value of 3.313E-29, signifying a statistically significant difference between bridged and unbridged simulation types.

 

 

 

 

 

 

 

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Figure 1. Average link length over the simulation period compared between models with the inclusion of Bogharbor, and those without

 

 

 

 

 

 

 

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Figure 2. Average path length over the simulation period compared between models with the inclusion of Bogharbor, and those without

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I applied the same BehaviorSpace analysis to compare how average development evolved throughout the two models and found a notably similar pattern of growth. Average development grew from 0 to roughly 90 after 100 ticks across simulation versions. After performing a t-test of the 4,000 data points from both models, I calculated a p-value of 0.688 – not significant. The unbridged version of the network had a slightly larger average development of 91.144 by the model’s conclusion, while the bridged version’s average development only rose to 90.121. This discrepancy lead me to perform a t-test comparing only the 20 datapoints of each model version at their final tick, so I could determine if there was a significant difference by the simulation’s end. This analysis surprisingly returned an even higher p-score of 0.939.

 

Discussion

When the model starts, route distance and city influence are the dominating factors in choosing how cities allocate their links. Most connections are short and direct, signifying the significance of the route distance metric. Route length is likely the dominant factor at the start of the model as the maximum degree at the start of both models hovers around 8, while it rises to over 15 as the simulation progresses. The reason why the degree is so low at the simulation’s initialization is that cities are likely only taking distance into account, so larger cities which should be appealing are passed up in favor of smaller, closer cities. Development is not taken into consideration when cities make their first links, as the metric is dependent on the consequences of previous trade routes. However, the metric only continues to grow in relevance as the model progresses, and eventually development becomes the dominant factor in link allocation determination, allowing for the creation of highly developed hub cities.

 

As cities gradually prioritize high development cities, the centrality of the network (inversely related to average path length) increases massively because those ports become hubs for the network. These hubs are linked with nearly any city within a 30-patch radius, theoretically allowing a route to travel 60 patches just by connecting to and from the hub. To travel a similar distance at an earlier point in the model would certainly require additional connections because nodes initially highly prioritize shorter link lengths. These outcomes imply that the evolution of the network to prefer cities with high development values resulted in a more centralized, and thus more efficient network.

 

The introduction of the Bogharbor canal node seems to have had no discernible impact on the progression of development growth in the network. The similarity of the average development results disproves an assumption that the increased flow of goods between the North, the Vale, and the Riverlands would boost development in the network. One hypothesis as to why this was not the case is that Bogharbor may have used up a significant amount of the resources passing through it for its own development, reducing the distribution of goods throughout the network.

 

Conclusion

The port trade network of Westeros project demonstrated the significant influence bridge nodes hold when determining the centrality of the system. As but a handful of nodes were directly located on the eastern or western coastlines the network became dominated by those two communities, only interlinked by the few ports along the southern coast. In these highly cliquish networks, the introduction of a bridge node significantly influence on average path length, and leading to a more efficient transfer of information or goods in a network.