Many realworld spatial systems can be conceptualized as networks. In these conceptualizations, nodes and links represent system components and their interactions, respectively. Traditional network analysis applies graph theory measures to static network datasets. However, recent interest lies in the representation and analysis of evolving networks. Existing network automata approaches simulate evolving network structures, but do not consider the representation of evolving networks embedded in geographic space nor integrating actual geospatial data. Therefore, the objective of this study is to integrate network automata with geographic information systems (GIS) to develop a novel modelling framework, Geographic Network Automata (GNA), for representing and analyzing complex dynamic spatial systems as evolving geospatial networks. The GNA framework is implemented and presented for two case studies including a spatial network representation of (1) Conway’s Game of Life model and (2) Schelling’s model of segregation. The simulated evolving spatial network structures are measured using graph theory. Obtained results demonstrate that the integration of concepts from geographic information science, complex systems, and network theory offers new means to represent and analyze complex spatial systems. The presented GNA modelling framework is both general and flexible, useful for modelling a variety of real geospatial phenomena and characterizing and exploring network structure, dynamics, and evolution of real spatial systems. The proposed GNA modelling framework fits within the larger framework of geographic automata systems (GAS) alongside cellular automata and agentbased modelling.
As geospatial data becomes increasingly available, networks are used as a powerful conceptual framework to represent and analyze a wide array of complex spatial systems in social, urban, and ecological contexts [
Network dynamics can be distinguished between dynamics
Network evolution is not well understood since detailed datasets representing real phenomena as networks over geographic space and time have been limited due to the lack of appropriate datasets. Conventional spatial network analysis tends to focus on describing and cataloguing
Models representing phenomena such as predator–prey dynamics [
Therefore, the objectives of this study are to integrate concepts of geographic information science (GIScience) and systems (GIS), complex systems, and network theory to (1) propose a theoretical framework for a novel modelling approach called Geographic Network Automata (GNA) that is used to represent and analyze complex spatial systems as evolving networks, (2) demonstrate the proposed theoretical framework by developing and implementing two GNA models based on Conway’s Game of Life [
This section first presents the general GNA modelling framework for the network representation of realworld spatial phenomena and second introduces the theoretical background for the application of graph theory to analyze the GNA
A geographic network automaton (GNA) is a mathematical representation of a complex system or parts of a system as an evolving spatial network and can be expressed as follows:
The GNA approach can be operationalized to model a variety of phenomena by executing each stage detailed in
The GNA framework requires data acquired from different sources including synthetic geospatial data, actual geospatial data, and data from the literature as an input to initialize node location, parameterize nodes, and to implement the network matrix and any potential geographic barriers. These types of data are also required to parameterize transition rules and connection cost. Datasets independent of model development are required for model testing.
In stage 1, a system or part of a system of interest is conceptualized as an evolving spatial network
As demonstrated in the example above, it is often the case that a spatial network of primary interest
In the less common case where there is no underlying network
Both the evolving spatial network SN generated by the GNA and the underlying network
An underlying network
The sets of nodes
Each node
Links
Transition rules
The spatial and topological organization of the spatial network
In the case of simulating dynamics
The output of a GNA is a sequence of evolving spatial networks of primary interest
Networks that exhibit properties of regular graphs are composed of a set of nodes and links, where each node has the exact same number of links of degree
Unlike regular graphs, properties of nonspatial random graphs differ significantly from random spatial graphs. Networks that exhibit the properties of random graphs that are nonspatial are composed of nodes that are connected to other nodes at random [
There are two main types of random spatial graphs. The first type is referred to as a random geometric graph (RGG), composed of nodes that are randomly located in geographic space. Unlike a nonspatial random graph, nodes in an RGG are not connected randomly, but rather connect to other nodes if the distance between the two nodes falls within a selected distance threshold
Smallworld graphs, both their nonspatial and spatial counterparts, are characterized by a structure that falls between regular graphs that have no randomness at all and random graphs that are entirely random [
Networks, either nonspatial or spatial, with properties characteristic of scalefree graphs typically have a degree distribution
Global graph theory measures can be used to characterize the overall network structures, to give insight into spatial dynamics that take place on those structures, to compare between different systems, and to compare between the same system as it evolves over time. Some important global graph theory measures are presented in
In the following sections, the application of the proposed GNA framework to the spatially explicit network version of the Game of Life GNA_{GOL} and Schelling’s Segregation GNA_{SEG} is presented. Both models are developed using the Java programming language in the Eclipse integrated development environment using the REcursive Porous Agent Simulation Toolkit (Repast) [
The Game of Life is selected as the first case study to present the GNA framework because it is a wellknown model of a theoretical system that is inherently simple and operates in space and time. The original Game of Life is a cellular automaton developed by John Conway in 1970 that was designed to simulate dynamics of reproduction, death, and survival of cells in a lattice. Applying these dynamics to nodes in a network permits the exploration of spatial network evolution, specifically spatial network growth and shrinkage as nodes are added and removed over time. Therefore, this case study facilitates broader learning about spatial network dynamics and evolution.
The GNA_{GOL} model simulates an evolving spatial network
The underlying RGG
The spatial network
Although the influence of the cost matrix on system dynamics is not formally explored in the traditional Game of Life, a barrier is introduced into the GNA_{GOL} to demonstrate the use of the connection cost
For any model run, the underlying network structure
Two scenarios were developed by adjusting the transition rules
The output of the GNA_{GOL} is a series of spatial networks
The obtained simulation results from both scenarios are presented in
Scenario 1: based on the results presented in
Scenario 2: the shrinking network structure
In general, for the evolving spatial networks
The Game of Life is a wellknown model of a theoretical spatial system selected as a case study to demonstrate clearly the GNA modelling framework, however the presented GNA_{GOL} does not incorporate real geospatial datasets. In the case of a geospatial application of the GNA modelling framework to realworld phenomena, the elements of the GNA including the initial network state, the underlying network, the transition rules, the connection cost, and the spatial and temporal resolution would need to be designed to properly reflect a particular realworld system and include geospatial data to permit GNA development, calibration, sensitivity analysis to initial conditions and parameters, and validation. This process would be the same for designing any cellular automaton or for an agentbased model. In the following section, a second model that incorporates real geospatial data into the GNA modelling framework is presented.
Schelling [
In this section, a second GNA is developed. The GNA_{SEG} model is a prototype for representing patterns of segregation in an urban environment. The model presented is more advanced than the GNA_{GOL} model as it incorporates real geospatial data forming an underlying network, accesses several node neighborhood types for which the transition rules are implemented, and explores dynamics between several different node types. In this network, unlike the GOL, the network evolution is not characterized by addition or removal of nodes, but rather the rewiring of a similar number of nodes change location over time.
The GNA_{SEG} model simulates an evolving spatial network
The underlying spatial network
For any model run, the underlying network structure
The spatial network
The spatial network
Each property node in the underlying network
The GNA_{SEG} outputs consist of a series of spatial networks
The obtained simulation results from the GNA_{SEG} are presented in
In general, the number of nodes connected to the spatial network
When comparing the obtained results for the two case studies, the
This study introduces the novel modelling framework of Geographic Network Automata (GNA) that can be used for the representation and analysis of complex spatiotemporal systems as evolving and dynamic networks. The proposed GNA modelling approach presented in this study fits within the larger modelling framework of geographic automata systems (GAS) [
While the proposed GNA modelling approach fits well within the framework of GAS, it maintains a strong departure from the classic cellbased CA and vectorbased ABM. The developed GNA approach is designed to explicitly leverage network representations, networkbased neighborhoods, networkbased transition rules, and network analysis using graph theory for the simulation of complex spatiotemporal phenomena. The GNA modelling framework differs from traditional GAS including CA and ABM because of its uniquely explicit view of the networkbased relationships and interactions between the spatial features that is represented by links and the
In CA and ABMs, transition rules are implemented to simulate relationships, interactions, and flows, but they are not often represented explicitly nor measured discretely. Instead, the way in which the system responds to these interactions is measured. The GNA offers a more flexible modelling framework than traditional CA where nodes may be mobile, may have several defined neighborhood types for which the transition rules are implemented, and nondeterministic systemlevel behavior. Furthermore, the GNA offers explicit representation of interactions as links and thus provides “Xray” vision of the model that can be used for measuring and visualizing large sets of interactions between components of a system in a way that ABMs traditionally do not.
“Networks are everywhere” is a phrase found in many studies that review network research that ultimately speaks to the interdisciplinary nature and usefulness of abstracting real systems into complex spatial dynamic networks. This speaks to the potential for the GNA to be implemented on many other geospatial applications for the representation, characterization, and analysis of a variety of complex systems. Complex spatial networks are a natural fit for representing and analyzing relationships and interactions and as such, the GNA modelling framework is an ideal approach for applications when interaction, relationships, dynamics, and flows between sets of components are of interest. The application potential is vast and includes movement and flows of information, people, resources, money, ecological species, energy, disease, and transportation vehicles over time and across points in geographic space. Naturally, the study of spatial and nonspatial relationships between individuals is also an ideal application for these modelling approaches. In addition, the proposed GNA modelling approach would be ideal to better understand interactions between two or more tightly coupled systems over space and time such as interactions among policy, social, and environmental systems. The proposed GNA modelling framework was not developed with the intention to replace other GAS nor does it claim to be better than, but instead, offers novel means for representation and a new lens for analysis of complex spatiotemporal phenomena.
The presented GNA models successfully represent dynamic spatial phenomena as networks as demonstrated using the two case studies. The quantification of the simulated network structures using network measures can reveal new understanding of the phenomena. In the Game of Life example, the network measures offer a way to quantify common behaviors of spatiotemporal phenomena such as growth and shrinkage. In the segregation example, it is revealed that the network measure assortativity can be very useful for quantifying segregation at the individual level and as a whole. This has been a challenge with traditional segregation indices which tend to summarize segregation within various units of measure i.e., census boundaries and as such are subject to the modifiable areal unit problem.
The Game of Life GNA_{GOL} example presented in this research study, simulates the evolution of network structures composed of nodes that are geographically referenced. The segregation GNA_{SEG} example uses real GIS data for the study area in City of Vancouver to represent the actual location of nodes where each node represents a residential property in the city. There is a qualitative difference between the two case studies where the level of GIS integration is increased from the GNA_{GOL} to the GNA_{SEG.} Integrating GIS and network automata and thus providing the framework of geographic network automata are advantageous in three ways: (1) the network structure and dynamics can be linked to georeferenced data representing real world phenomena, (2) the geovisualization of simulated evolving spatial networks are georeferenced to a study area, and (3) both the spatial analysis using GIS and network analysis of the generated spatial network structures can be leveraged.
Even though the GNA_{SEG} incorporates actual spatial data to form the underlying network
In conclusion, this study presents geographic network automata (GNA), a modelling approach developed for the simulation of spatial systems as evolving complex spatial networks. The novelty of the GNA approach lies in its ability to represent the tight coupling between spatial network structure and network dynamics, resulting in network space–time evolution. The GNA approach acknowledges that for many phenomena network evolution occurs in geographic space, and thus geospatial data and geographic information systems can be leveraged for representing and analyzing real systems. The GNA modelling framework adopts a complex systems approach by simulating local spatial interactions between georeferenced nodes represented by links from which a complex network emerges. Ultimately, this approach is a new class of GAS alongside ABM and CA models. The framework is implemented using a spatial network representation of two GNA models including Conway’s Game of Life and Shelling’s Segregation Model where the implemented transition rules simulate dynamics between nodes at the very local level, altering the structure of the spatial network, which in turn influences the dynamics between nodes. Graph theory is then used to characterize and measure the structure and behavior of simulated networks. The developed GNA approach is both general and flexible so that it can be applied to represent and analyze many real geographical systems including urban, social, and ecological and has the potential to be used in knowledge discovery and decisionmaking processes.
Conceptualization, Taylor Anderson and Suzana Dragićević; Formal analysis, Taylor Anderson; Funding acquisition, Taylor Anderson and Suzana Dragićević; Investigation, Taylor Anderson and Suzana Dragićević; Methodology, Taylor Anderson; Resources, Suzana Dragićević; Supervision, Suzana Dragićević; Visualization, Taylor Anderson; Writing—original draft, Taylor Anderson; Writing—review and editing, Suzana Dragićević. All authors have read and agreed to the published version of the manuscript.
This research was funded by Natural Sciences and Engineering Research Council (NSERC) Canadian Graduate ScholarshipDoctoral (CGS D) and the Discovery Grant awarded to the first and second author, respectively. In addition, partial funding was from the SFUSSHRC small institutional grant awarded to the second author.
The authors are thankful to the Natural Sciences and Engineering Research Council (NSERC) of Canada programs and SFUSSHRC small institutional grant for the support of this research study. The authors appreciate valuable feedback from the three anonymous reviewers.
The authors declare no conflict of interest.
Different complex
Based on (
The calculated number of nodes, number of links, average clustering coefficient, average degree, and average path length for the obtained spatial network
Calculated degree distribution for each evolving spatial network
Based on (
Values obtained for number of nodes, number of links, average clustering coefficient, average degree, average path length, and assortativity for the spatial network
Obtained degree distributions for each evolving spatial network
Operationalizing the GNA approach.
Stage  Description 

1. Conceptualize system of interest as spatial network 

2. Identify important graph theory measures  
3. Determine the neighborhood 

4. Develop transition rules  
5. Develop connection costs  
6. Implement the GNA  
7. Test the GNA  
8. Execute model and scenarios  
9. Apply graph theory to characterize network structures 
Examples of spatial, nonspatial, and network properties for both nodes
Property Type  Description  Examples of Node Properties  Examples of Link Properties 

Spatial  Geometric properties pertaining to the node 
Location 
Length, coordinates of end points, direction 
NonSpatial  Qualitative and quantitative nonspatial attributes pertaining to the node 
Name, ID, color, value, type, state  Name, ID, color, value, type, state 
Network  Measurements derived from network theory pertaining to the node 
Degree, betweenness, weight, clustering coefficient, list of neighbors  Weight, list of end nodes 
Selected graph theory measures to characterize and analyze spatial networks.
Graph Theory Measure  Definition 

Average degree 
The number of connections a node has to other nodes in the network is a localized measure, specific to each node, and is referred to as node degree 
Degree distribution 
The fraction of nodes in the network with degree 
Average clustering coefficient 
Clustering coefficient 
Average shortest path length 
The average number of intermediate nodes and links in the shortest path between all pairs of nodes in the network is referred to as average path length 
Transition rules and their parameterization specific to each scenario for GNA_{GOL}.


Parameter x 

Scenario 1 


Scenario 2 


Scenario 1 


Scenario 2 


Scenario 1 


Scenario 2 


Scenario 1 


Scenario 2 

Correlation matrix presenting the Pearson’s Correlation (








1.00  
0.99  1.00  
−0.97  −0.98  1.00  
0.79  0.79  −0.82  1.00  
0.99  0.99  −0.98  0.8  1.00  


Graph Theory Measure 





1.00  
0.94  1.00  
−0.73  −0.67  1.00  
0.74  0.94  −0.58  1.00  
0.99  0.95  −0.72  0.74  1.00 
Correlation matrix presenting Pearson’s Correlation
Graph Theory Measures  (1)  (2)  (3)  (4)  (5)  (6) 

1.00  
−0.82  1.00  
−0.72  0.97  1.00  
0.42  −0.74  −0.80  1.00  
−0.83  0.99  0.96  −0.74  1.00  
−0.70  0.95  0.99  −0.82  0.95  1.00 