Modeling Infrastructural Territory

1. Introduction

This paper describes the development and analysis of a model designed to explore the spatial dynamics of territory and infrastructure. The model’s purpose is to explore the impact of the gradual development of infrastructure on the emergence of bounded, homogenous territories. Somewhat contrary to popular usage, infrastructure is here thought of quite broadly, referring not just to the sewage systems, communication networks, and transportation systems that first come to mind, but also any productive transformation of the environment (Carse 2012, Cronon 1991). Infrastructure, particularly broadly defined, exhibits a powerful ability to shape the emergence of spaces: transportation networks draw distant places temporally closer even as others recede; utility provision networks forge new relations between the fates of once separate areas; communication networks enable linguistic and cultural flows over unprecedentedly wide scales (Harvey 2012, Swyngedouw 2004 & 2007). All this serves to transform humans’ relation with place. Most transformative, however, is the rise of sophisticated skills required to navigate through and survive within these increasingly elaborate infrastructural environments: these skills tie humans to particular culturally-specific infrastructural habitats (Adey et. al. 2012, Löfgren 2008, Watts 2008). Infrastructure, then, defines some aspects of an emerging economic and cultural area bound together by a shared human environment: a territory.

Territory is a concept of considerable interest in the literature of political geography. Delaney (2005, 14) provides a succinct definition of territory: “a bounded social space that inscribes a certain sort of meaning onto defined segments of the material world.” In political geography, as John Agnew famously pointed out, the de facto territory is the nation-state (1994). Since then, despite a widespread recognition of the limits of that conception of territory, leading to more studies of non-national territorial units; however the field has not yet had a deeper reckoning with why the nation-state so consistently is the de-facto unit (Agnew 2010). My contention is that this is in part due to the field’s tendency to adopt top-down understandings of territory, relying on the territoriality of states and their state-making projects to understand the production of territories (Brenner and Elden 2009). While such top-down influences are undeniably influential, I contend that exploring bottom-up processes of territorialization are equally necessary to any understanding of territory.

In order to further explore the dynamics of territorialization from a bottom-up, infrastructurally-driven perspective, I have adopted an agent-based modeling approach. Infrastructure is built due to a complex interaction of humans, environmental constraints, and feedback loops that collectively constitute a complex system. Building a simulation model of the system forces us to explicitly define the elements of the system and their interactions, refining our understanding of the system. An agent-based approach captures the role of individual interactions–choosing where to move, developing local infrastructure–in producing system-level results, i.e. the emergence of homogenous territories. The objective of this research project is to explore the dynamics of human-infrastructure interactions and the conditions under which it leads to the emergence of distinct and bounded spaces. As this is an early iteration of this project, the primary goal is relatively basic: to examine the influence of several key parameters on the final levels of homogeneity in cell and agent distributions. This project achieves that objective by performing a parameter sweep of the chosen variables and performing a statistical analysis of the resulting outcomes.

2. Methods

2.1. Purpose

The model’s purpose is to explore the impact of the development of infrastructure and its associated skilled populations on the emergence of bounded, homogenous territories. The description below follows the ODD protocol laid out in Grimm et. al. 2010.

2.2. Entities, state variables and scales

The model contains two basic types of entities: agents and grid cells. The agents are defined by a unique identity number, a variable describing their location in the grid, and two variables defining their productive capacity (skill A and skill B) with the corresponding type of infrastructure. Agents cannot die or reproduce, fixing the number of agents over the course of the run. The grid cells are defined by their location in the grid matrix, and by two variables defining the cell’s level of productive capacity (infrastructure A and infrastructure B) when utilized with the corresponding skills.

There are a number of model-wide state variables that govern turtle behavior: All agents share a parameter which is used, in combination with their skill variables, to determine their gaze, or sensing range. Another parameter defines a bias against changing cells. Two last parameters define the manner in which infrastructure values increase when the agents produce: one that determines how much increase on one cell benefits neighboring cells, and another that determines how much the opposing infrastructure value is decreased. Each cell can hold a number of agents constrained by a function of its infrastructure value and a maximum agents parameter. The number of cells remained fixed–1024–but the number of agents was a parameter varied across runs. There are no collective variables. At this stage the model is sufficiently abstract to be independent of any real spatial or temporal scales: space is represented by a two-dimensional, toroidally-wrapped lattice, and time by an abstract “time step”.

2.3. Process overview and scheduling

All processes in the model are driven by agent decisions. Each time step abstractly represents an opportunity for agents to move or produce. At each time step, agents first look within their gaze radius and determine the cell with the highest productive capacity. They then either move or produce; if the productivity of the best cell exceeds the productivity of the current cell by a value governed by the move preference parameter, the agent will move–unless the cell is at capacity, in which case it stays and produces. All agents look before any agents move or produce, and order of agents is determined randomly by the modeling program.

2.4. Design Concepts

2.4.1 Basic principles – The basic principles and assumptions that structure this model are as follows. Individual agents are assumed to be workers who a) attempt to maximize production, b) utilize available infrastructure to do work, and c) whose skills are particular to a specific type of infrastructure. Infrastructure is here thought of as a product of labor which is a) also a means of production, b) fixed in space and c) is embedded with a certain set of arbitrary standards. Following the approach to infrastructure outlined above, no distinction between “natural” infrastructure (such as especially fertile farmland or natural resources) and built infrastructure is made in the model.

Both infrastructural and skill are capable of change over time: the more they are used, the more developed they become. Different types of skill-specific infrastructure are assumed to be somewhat mutually exclusive: an increase in one results in the diminishment, to a lesser degree of the other.

Individuals’ use of infrastructure thus relies on the fit between their skill set and those embedded on the infrastructure, producing the most when the skill and infrastructure are closely fitted and most developed. According to the theoretical principles, the mutually reinforcing tendencies of skill and infrastructure are expected to tend towards a level of correspondence between the skills of resident agents and the type of infrastructure present.

2.4.2 Emergence – The model will help in exploring under what conditions the real world observed phenomenon of infrastructure-skill fit manifests as the emergence of bounded territories. Territories are not modeled by any explicit variables or entities within the model: if they appear it will be solely due to the model’s emergent properties. It is expected that over time spatial patterns in the distribution of agents, and the level of skill/infrastructure correspondence will emerge; the research questions concern what type of spatial patterns emerge and whether they resemble their real-world counterparts.

2.4.3 Adaptation – Agents are capable of adaptation via changing values of their skill variables: when using a particular skill, it increases. Additionally, agents are capable of adapting their environment—the cell—through the same process: using a skill while in the cell increases the corresponding infrastructure value.

2.4.4 Objectives – Agents’ objective is to produce as much as possible: production is a function of agent skill and cell infrastructure. The production of homogenous territories is not an agent objective.

2.4.5 Learning – Agents are capable of basic learning: the agents “learn” how to produce more efficiently through practice, and as their skills increase the range of their gaze expands as well. Agents are not capable of adopting novel movement or production strategies.

2.4.6 Prediction – The agents in this model engage in only a very primitive sort of prediction. Because moving occurs only at the cost of foregoing production for the turn, there is a bias built into the move or produce procedure to prevent constant and costly movement. The potential gain must be significant before the agent will move, with significance judges according to a model parameter.

2.4.7 Sensing – Agents sense their surroundings via the look procedure, which evaluates the potential production for neighboring cells. The range of the look procedure is dependent on the agent’s skill, and each skill is calculated separately: an agent with a higher skill in A can “see” further away infrastructural A values. Agents cannot see each other except when not moving into an overcrowded cell.

2.4.8 Interaction – Agents do not directly interact with each other. Agents’ actions modify themselves and their local cells, and the cells then have an impact on the actions of other agents, which creates an indirect interaction.

2.4.9 Stochasticity – This model employs stochasticity to set up initial conditions: agents’ skill values and positions, and cells’ infrastructure values are randomly seeded with values between 0 and .2. Thereafter all agent actions are deterministic, though random agent action order (as determined by the modeling program’s pseudo-random number generators) nonetheless introduces a stochastic element into the model. The separation of the look and move or produce procedures is done in order to minimize the impact of stochasticity.

2.4.10 Collectives – No collective entities are explicitly included within this model.

2.4.11 Observation – The relevant output of this model are the densities of self-similar cells and self-similar agents at the end of the model run (200 time steps). Self-similarity for cells is measured by how many of their neighbors have the higher values in the same infrastructural variable, i.e. if the cell has a higher infrastructure A value, how many of its neighbors also have a higher infrastructure A value. Agent self-similarity is calculated in the same manner but for skill variables of nearby agents: this includes agents on the same cell and on neighboring cells. These outputs will be used to run a statistical analysis of the model’s sensitivity to four different input parameters: number of agents, level of infrastructural diffusion, level of hostility, and maximum agents per cell.

2.5. Initialization

The model will be initialized via random number generation of relevant variables. Because the purpose of this model is early-stage, exploratory analysis of the system, smaller model sizes are preferable: each run is more easily visually analyzed and more runs can be completed. Accordingly, the initial model will be run with a 32 x 32 cell grid. For each run, initial values for agent skill and cell infrastructure variables will be arbitrarily assigned random values between 0 and .2 out of a maximum value of 1. Initial locations of agents will also be randomly assigned.

2.6. Input data

There is little available data on the variables that drive this model. Indeed, part of the purpose of this model is to explore the relations between a set of characteristics that it is nigh impossible to collect empirical data for. While later, more detailed and specific iterations of this model may be capable of incorporating input data, this one is not.

Analysis and Evaluation

The model will be evaluated by means of a sensitivity analysis, to determine how different starting parameters affect the output variables. This will allow me to parameterize the input variables, gaining a valuable understanding of how sensitive the model is to the different inputs. The sensitivity analysis focuses on four parameters of interest: number of agents, level of infrastructural diffusion, level of hostility, and maximum agents per cell. These four parameters each reflect to some degree a real world equivalent characteristic that is of interest. The number of agents reflects landscapes of different population densities, and coupled with the maximum agents per cell,  infrastructural development capable of supporting denser settlement. Denser and more lightly populated landscapes in the real world show a divergent level of infrastructural development and cultural homogeneity, the understanding of which are the objective of this study. Infrastructural diffusion reflects the degree to which local changes affect neighboring areas, and the level of hostility reflects the degree to which different types of infrastructure are mutually exclusive (or compatible). The sensitivity analysis will illustrate how these different variables influence self-similarity levels of the landscape and its inhabitants.

3. Results

An example model run, with parameters of: number of agents 1536, maximum agents per cell 5, infrastructural diffusion of 0.5, hostility of 0.5.

This study found that all the parameters explored had a unidirectional relationship with the output variables; the density of self-similar agents and cells varied directly with three, and inversely with one (number of agents). Cell self-similarity varied between .47 to .73; agent self-similarity exhibited higher values and more variability, ranging between .53 to .92.

As the boxplots above show, by far the most dramatic impact on cell self-similarity came from the infrastructural diffusion parameter. As this parameter directly impacts the infrastructural values of neighboring cells, this is not unsurprising–however, it reflects an increase in homogeneity driven not by emergent patterns but rather by direct, code-level manipulation. The other three parameters all exhibited a degree of influence on the output values, particularly at the upper bound: at higher hostility and maximum agent (and lower number of agent) parameters, the upper limit of the distribution went up while the lower limit remained relatively steady.

The self-similarity levels of agents is a more interesting result in that, unlike cell self-similarity, it is not directly driven by any model parameters. Interestingly, the infrastructural diffusion parameter has the smallest impact on agent self-similarity, while the number of agents, maximum agents and hostility parameter have a much greater influence. More agents, limited to lower numbers per cell, produced the lowest agent self-similarity percentages: unable to move away from transition zones, unlike agents were forced to exist side-by-side.

The agent-level processes that drive global processes of homogenization can be further explored using the narrative modeling approach laid out by Millington et al. (2012). Homogenization is created when multiple agents on nearby cells become more similar, through a combination of infrastructural diffusion reinforcing similar values and hostility reducing opposing types of infrastructure. Examined at an individual scale, single cell clusters of agents can emerge through multiple pathways. The first type of pathway is through conversion. In the example illustrated in Figure 1, the cell of interest (circled) begins the model with an infrastructure dominated by infrastructure A, here shown in blue (relative values are represented by intensity). However, it holds four randomly placed agents, three of whom have higher B skills, shown in orange. Because these agents have an advantage in producing B over A, but no sufficiently superior cell to move to, they stay in place and produce B, raising the cell’s infrastructure B value until the cell becomes B dominant as well at time step 5. Meanwhile, the fourth agent, which has been producing A, i.e. developing its A skill and the cell’s A infrastructure, maintains A dominance for a while longer: only after the B infrastructure significantly outpaces the A does it become the best option for the fourth agent despite its superior A value. Once it switches however, it is only a matter of time until it also becomes B dominant, at timestep 8. Clusters of agents with similar skills exert a strong influence on the cells they inhabit and consequently on the other agents which share them. The second type of clustering results from agent movement. The bottom half of Figure 1 depicts the movement decision of the skill B-dominant agent: randomly placed on a A dominant cell, it decides to move to an adjacent cell with a superior B value rather than produce in place. This also leads to an increased cell-agent clustering, which lays the foundation for the cell and agent self-similarity observed in the model output.

As an exploratory, early-stage model, this model has performed well, providing an enhanced understanding of the model’s characteristics and the relative influence of model parameters. The sensitivity analysis, by teasing apart the relative influences of different parameters, allows me to understand the specific interactions of abstract parameter values, thereby laying the foundation for future parameterization based on empirical data. The narrative exploration of local-level processes has also illuminated important aspects of the model’s function and pointed out opportunities for further refinement. In exploring the movement of individual agents, I found that few agents made more than a small number of moves: they often find the best in-range option is a cell that is full, and stymied, remain in place indefinitely. Re-designing the movement function to allow agents to move to second- or third-best choices would create a more fluid field of movement.

4. Discussion:
This model embodies a complex adaptive system: driven by the decision-making of individual, autonomous agents, different constraints (parameters) on the system produce different emergent outcomes. Certain constraints, such as low infrastructural diffusion, and low numbers of agents tended to produce a patternless landscape; others such as high diffusion and hostility parameters produce bounded, homogenous territories in the form of relatively self-similar clusters.

The development trajectory of any given region in the model is highly path-dependent: minuscule early variations in skill and infrastructure values determine where self-similar clusters emerge (or fail to). These clusters result from positive feedback loops: clusters of agents drive up infrastructural values, attracting more agents, and so on in a virtuous circle. The constraint of the limit on the number of agents per cell acts as a negative feedback, keeping values in check. In contrast to some complex systems, this model exhibits little non-linearity (within the parameters analyzed): most outcomes fall along a recognizably linear spectrum.

5. Conclusion:
This study entailed the development and analysis of a model designed to explore the spatial dynamics of territory and infrastructure. Treating bounded self-similar spaces as a form of territory, it examined how individual-level decisions about maximizing production can work to create emergent patterns in spatial distribution of cell and agent characteristics. The analysis took two approaches: a sensitivity analysis of the influence of four key parameters with some real world relevance, and an individual-level narrative exploration of agent behavioral processes. The study concluded that all four parameters had unidirectional influence on levels of self-similarity, laying the basis for future model parameterization, and the narrative exploration revealed problems with the agent’s decision-making code, pointing towards future improvements. In summary, the study, while an early stage exploration of the problem, nonetheless made significant contributions to a bottom-up understanding of the emergence of territory.

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