A conceptual model for automating spatial network analysis

Abstract

Spatial network analysis is a collection of methods for measuring accessibility potentials as well as for analyzing flows over transport networks. Though it has been part of the practice of Geographic Information Systems (GIS) for a long time, designing network analytical workflows still requires a considerable amount of expertise. In principle, Artificial Intelligence (AI) methods for workflow synthesis could be used to automate this task. This would improve the (re)usability of analytic resources. However, though underlying graph algorithms are well understood, we still lack a conceptual model that captures the required methodological know-how. The reason is that in practice, this know-how goes beyond graph theory to a significant extent. In this article, we suggest to interpret spatial networks in terms of quantified relations between spatial objects, where both the objects themselves as well as their relations can be quantified in an extensive or an intensive manner. Using this model, it becomes possible to effectively organize data sources and network functions towards common analytical goals for answering questions. We tested our model based on 12 analytical tasks, and evaluated automatically synthesized work-flows with network experts. Results show that standard data models are insufficient for answering questions, and that our model adds information crucial for understanding spatial network functionality.

Full text

ORIGINAL ARTICLE
Transactions in GIS
A conceptual model for automating
spatial network analysis
Simon Scheider 1| Tom de Jong 2
1Department of Human Geography and
Spatial Planning, Utrecht University, NL
2Department of Logistics, Stellenbosch
University, ZA
Correspondence
Simon Scheider, Department of Human
Geography and Spatial Planning, Utrecht
University, the Netherlands
Email: s.scheider@uu.nl
Funding information
This work was supported by the European
Research Council (ERC) under the European
Union’s Horizon 2020 research and
innovation programme (grant agreement
No 803498).
Spatial network analysis is a collection of methods for mea-
suring accessibility potentials as well as for analyzing flows
over transport networks. Though it has been part of the
practice of Geographic Information Systems (GIS) for a long
time, designing network analytical workflows still requires
a considerable amount of expertise. In principle, Artificial
Intelligence (AI) methods for workflow synthesis could be
used to automate this task. This would improve the (re)usability
of analytic resources. However, though underlying graph
algorithms are well understood, we still lack a conceptual
model that captures the required methodological know-how.
The reason is that in practice, this know-how goes beyond
graph theory to a significant extent. In this article, we sug-
gest to interpret spatial networks in terms of quantified
relations between spatial objects, where both the objects
themselves as well as their relations can be quantified in
an extensive or an intensive manner. Using this model, it
becomes possible to effectively organize data sources and
network functions towards common analytical goals for an-
swering questions. We tested our model based on 12 ana-
lytical tasks, and evaluated automatically synthesized work-
flows with network experts. Results show that standard
data models are insufficient for answering questions, and
that our model adds information crucial for understanding
spatial network functionality.
1

2
KEYWORDS
Geo-analytical workflows;Spatial network analysis;Task
automation;Core concepts of spatial information
1|INTRODUCTION
Computational models of spatial networks for Geographic Information Systems (GIS) have been known for a long time
(Sutton, 1998). They are frequently used in applications such as spatial planning (Geertman et al., 2003), transport
analysis (Thill, 2000), supply infrastructures, and the analysis of flows (Curry, 1972) (cf. Miller et al. (2001) for an
overview). Corresponding functions are nowadays implemented in many GIS software tools, such as ArcGIS Network
analyst1, as well as in Web APIs and geoservices2.
Yet, despite the ubiquity of technical resources, answering questions about spatial networks still requires orga-
nizing analytic functionality into workflows, and the latter presupposes a considerable amount of expertise. Suppose
our task is to assess the accessibility and distribution of transport flows within a road network. Could ArcGIS’s service
area tool3be used for this task, or rather a different one? And is a road network dataset enough, or do we need travel
statistics in addition? It is clear that while such tasks are of relevance for many data scientists, manual identification of
functions and data is a time-consuming process (Scheider and Tomko, 2016), and manual composition of workflows
remains a non-trivial craft.
To address this challenge, program synthesis algorithms were developed in (symbolic) Artificial Intelligence (AI)4
(Naujokat et al., 2011). They provide a way to automate this task, allowing analysts to loosely specify workflows
without knowing the details about available resources (Kasalica and Lamprecht, 2020b). These algorithms have pre-
decessors in GI service composition (Lutz, 2007), but go beyond by searching through the composition space of func-
tions described by an information ontology, in order to satisfy a given task specification (Lamprecht et al., 2010).
To automate spatial network analysis, the main challenge lies in finding the right semantic constraints for both task
specifications and function descriptions (Kruiger et al., 2021).
Yet, Geographic Information Science (GIScience) has struggled to come up with a model that is able to capture
the semantic constraints implied by this practice (cf. Sect. 2). The difficulty seems to lie in a frequent confusion of
networks as concepts used in geographic practice, with networks as data models implemented in particular information
systems (Kuhn and Ballatore, 2015). Network data models are usually understood as embedded graphs (Scheider and
Kuhn, 2008), where vertices are embedded as points in Euclidean space allowing us to assess metric distances. While
sufficient for implementing network procedures, this model seems to disregard important concepts needed to analyse
spatial networks, and in consequence, fails to capture underlying analytical tasks. To illustrate, suppose our goal is to
assess the effect of football games on traffic load on the streets, caused by football fans travelling to their respective
clubs. How could a graph model be used to the specify the task of determining flows of fans from residential areas
to clubs based on the amounts of residents and their distances to clubs? There is no concept in embedded graphs
that would allow us to distinguish amounts from ratios on nodes or flows from distances on edges. Another sub-task
is to assign flows to particular paths on a road network to assess the traffic caused by fans. To handle this problem,
different kinds of weights for different kinds of edges need to be distinguished, yet we currently lack a theory that
makes such distinctions. Sticking to graph-theoretic terms seems to merely transfer the problem to the semantics of
1https://www.esri.com/en-us/arcgis/products/arcgis- network-analyst/overview
2https://developer.here.com/
3https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/service- area-analysis- layer.htm
4SAT solvers, e.g., form a basis for algorithms behind workflow synthesis (Lamprecht et al., 2010) and belong to symbolic AI.

3
graph labels, or of edge and node labels (Kanjilal and Schneider, 2010).
We therefore argue that the concepts underlying spatial network data models need to go beyond embedded
graphs. An explicit model of these concepts would help us better understand not only what kind of information a
spatial network contains, and which questions can therefore be answered with it (Scheider et al., 2021), but also what
kinds of analyses are possible. This leads the way towards automating the analysis process itself. To address this goal,
we argue that spatial networks should be conceived in terms of core concepts of spatial information (Kuhn, 2012), which
implies important restrictions on the applicability of functions. More precisely, we consider networks as quantified
relations between spatial objects 5, where both object and relational qualities can be considered extensive or intensive
(Scheider and Huisjes, 2018). Spatially extensive measures are additive w.r.t. the spatial extent of their controlling
objects, whereas intensive measures are not additive in this sense. Consider, for example, the potential of football
fans living in districts of a city. These potentials add up when merging underlying districts, as opposed to the distance
to the city center. Notice that both potential and distance measures are required for estimating travel flow in our
example above, and more generally, to model spatial network analysis (see Sect. 4.2).
We argue that this new model, by its very simplicity, can go a long way in clearing up a pathway through the
jungle of available functionality and corresponding network tasks. We focus on the following questions:
1. How can spatial network tasks be specified in terms of core concepts and extensivity, to assess the suitability of
resources?
2. To what extent can network functionality be distinguished in terms of concept transformations?
3. What is the quality of automatically synthesized workflows that are based on such concepts?
Note that in this article, spatial network analysis is not a method, but an object of investigation. Correspondingly,
we are not targeting empirical questions about spatial networks, as usually intended by GIS analysts. Instead, our study
is about conceptual modeling (Guarino et al., 2020) of geographic information, and a network analysis scenario serves
merely as our empirical basis. Even though our goal is to distinguish network concepts from other kinds of concepts
relevant in GIS, we are aware the underlying functions always form an integrated whole in practice. Correspondingly,
our model feeds into a more general geographic information ontology (Scheider et al., 2020). Our goal is a lightweight
type system that is able to model the part of this practice needed to compose workflows for answering questions
(Scheider et al., 2021). In the following, we start with a review of spatial network theory and corresponding conceptual
models (Sect. 2), before giving an overview of our methodological approach (Sect. 3). Our own conceptual model is
developed in Sect. 4, and is then used to introduce computational signatures for spatial network functions (Sect. 5),
as well as to specify 12 spatial network tasks in an application scenario (Sect. 6). Finally, we evaluate our model by
automatically synthesizing workflows for each scenario task and by assessing their quality (Sect. 7).
2|RELATED WORK
If we look at current standard textbooks on GIS, spatial networks seem to occupy only a minor role (Burrough et al.,
2015; Chrisman, 2002; Longley et al., 2015; Heywood et al., 2010). Yet, the relevance of spatial networks for geospa-
tial analysis has been known to Geographers since the rise of quantitative methods in the second half of the 20th
century. It is insightful to take a look at the history of spatial network related concepts, which runs in parallel to the
change of research paradigms within Geography and GIScience. Furthermore, we review recent work on geo-spatial
5This idea is rooted in a suggestion made by Werner Kuhn (cf. Kuhn (2012)) in personal conversation, so we can only partially take credit for it.

4
semantics as a basis for modeling spatial network concepts.
2.1 |Spatial network analysis
Peter Haggett’s and Richard Chorley’s book (Haggett and Chorley, 1969) provides an early integrated view on passive
(drainage networks) and active transportation networks (e.g. roads). In this text, graph theory occupies a minor part,
including definitions of trees and circular graphs, as well as shortest path algorithms. Beyond graphs, the authors
focused their discussion on flow networks versus barrier networks; relations of channel order numbers,flow and lengths
in drainage networks; geometric shapes, densities and orientations of networks; the relation between distance,flow and
efficiency/costs of networks, relating to Christaller’s optimal settlement system (Christaller, 1933), as well as network
change over time. Furthermore, optimization methods include not only shortest path algorithms, but also districting
and problems of regionalization (how to divide space into tessellated regions using networks).
In the 70’s and 80’s of the 20th century, when GIS evolved, Human Geographers discovered the powerful concept
of a potential in geographic space (Rich, 1980). This is related to the idea of accessibility, which combines the concepts
distance with utility of activities that can be performed at the destinations in a network (Moseley, 1979; Ingram, 1971).
Accessibility allows us to assess a potential interaction (Masser and Brown, 1977) of amounts ofpeople or goods between
places, in analogy to a gravity model (Curry, 1972; Batty, 1976; Wilson, 1974). These methods have become essential
tools of spatial planning with GIS (Geertman and Ritsema van Eck, 1995; Jong and Ritsema van Eck, 1996). Besides
path algorithms, Ritsema van Eck (1993) identified zoning,districting and origin-destination matrix methods as essential
for spatial network analysis in GIS.
Research on spatial network models and GIS during the 90ies of the last century, in contrast, focused less on
conceptual or methodological issues, and more on network data models that would allow integration of transport
science functionality into GIS databases (Thill, 2000; Sutton, 1998; Miller et al., 2001). These systems were called
GIS-T, and researchers were mainly concerned how data structures and algorithms for transportation research could
best be integrated within a GIS infrastructure. This ”structural” view on networks continues to the present day, though
the focus has shifted from implementation models to formal models that would support efficient design of databases
across software environments (Kanjilal and Schneider, 2010; Qi et al., 2016), as well as efficient querying of network
data, including graph databases (Güting, 1994) and moving objects on networks (Güting et al., 2006). Other authors
have focused on network complexity measures for spatial graphs (Arlinghaus et al., 2002; Jiang and Claramunt, 2004).
The latter approach, however, largely abstracts from the conceptual basis of network analysis in Geography.
2.2 |Networks as core concepts of spatial information
What kind of semantics should be adopted to model spatial networks as concepts? Some researchers have been
investigating transport networks from the viewpoint of environmental cognition, such as wayfinding activities and
affordances (Winter, 2002; Scheider and Kuhn, 2010, 2008). A more general, trans-disciplinary account of networks
was given by Kuhn in terms of the core concepts of spatial information (Kuhn, 2012). On this account, networks are one
out of a range of concepts needed for interpreting the environment and for reasoning with GIS. These concepts consti-
tute conceptual “lenses” through which the environment can be studied independently from technical representations
(Allen et al., 2016; Kuhn and Ballatore, 2015). Besides the base concept location, allowing for metric distance assess-
ments in space, Kuhn distinguished the following content concepts, which we interpret here in a broader research
context:

5
•Fields: Are understood as continuous functions (Galton, 2004) whose domain is time and location, and whose
range may be any kind of measurable quality. Prime examples are temperature fields.
•Objects: Are understood as functions from time to locations and qualities (Galton, 2004). Objects are distinct
from fields and events in the sense that they have an identity and that they are fully localized in each moment of
their existence. We assume that objects include both, bona fide (perceivable) and fiat (conventional) boundaries,
as in the case of administrative units.
•Events: are understood as entities that, besides having identity and having qualities like objects, happen during
some temporal interval. Prime examples are earth quakes, having a time, a location as well as a magnitude.
•Networks: Are quantified relations between objects, i.e., functions from pairs of objects to some quality. In this
way, networks measure a relationship between objects. Kuhn (2012) distinguished link networks which connect
objects in a qualitative way (e.g. friendship, treaty or business relation) from path networks, which can measure
flows or paths between objects. Similar distinctions can be drawn in our model.
We believe that geo-analytical tasks, and network analysis in particular, can only be understood when modeling these
concepts in combination, because they depend on each other. Yet, so far, computational models of core concepts have
not taken networks into focus (Kuhn and Ballatore, 2015). Furthermore, it is an open question how core concepts
combine with other semantic concepts needed for geographic analysis (Scheider et al., 2020). Our model of spatial
networks was designed to reflect precisely this underlying practice.
2.3 |Ontologies for geo-analytic workflow synthesis
Automated workflow composition first appears in the context of GI Web processing services (Yue et al., 2007). How-
ever, its effectiveness mainly depends on the quality of the ontology used to describe the information resources (Hofer
et al., 2017). As recognized early on (Giordano et al., 1994; Albrecht, 1998), this includes the need for generalized
taxonomies of GIS that focus on functionality rather than technicalities. The main difficulty seems to lie in the fact
that analytical concepts are not fully reflected in data types, and thus can occur in various syntactical variations. In
Scheider et al. (2020), we have therefore suggested an OWL6ontology of types of core concepts that can occur in
combination with measurement levels and data types, to serve as a method for reasoning about GIS workflows and
geo-analytical tasks. Based on this work, there have been recent attempts at automating GIS workflow synthesis for
tasks that are not network related (Kruiger et al., 2021). Computationally, this approach is based on loose program-
ming, i.e., the sequencing of functions satisfying task constraints specified over an ontology with some temporal logic
(Lamprecht et al., 2010) (see Sect. 7.1). To handle spatial network analysis tasks in the same manner, network con-
cepts need to be combined with other core concepts. Yet, formal models of the role that networks play in this respect
are lacking. We also do not know of any studies about modeling network functionality with the goal of automating
geo-analytical tasks. This gap is addressed in the current paper.
3|METHODOLOGY AND APPROACH
In this section, we explain the steps taken towards developing and testing a conceptual model of spatial network anal-
ysis. Empirically, our study is based on a network analysis scenario: The analysis of football clubs and their fans in the
Netherlands, as outlined below. For one, this scenario gives us a way to explore core tasks of spatial network analysis
6Web Ontology Language, cf. https://www.w3.org/OWL/.

6
as a basis for developing our model (Grüninger and Fox, 1995). Furthermore, to evaluate our model, we manually gen-
erated expert-level workflows for these tasks, and compared them with workflows automatically synthesized using
our conceptual model.
3.1 |Network analysis scenario and task design
The following scenario was selected based on whether it captures precisely those practices that distinguish spatial
network analysis from other types of spatial analysis. This mainly includes the capabilities of handling spatial inter-
action data, going beyond geometrical GIS models that focus on topological relations and distances. Dejonghe et al.
(2006) published a book on professional football clubs and their fan base in the Netherlands. One of the datasets
they used is the 2003 nationwide complete list of the number of seasonal ticket holders per football club and per
municipality. Football fans in the Netherlands are usually season ticket holders, and as such form regular transport
flows when traveling towards their clubs.
(a) Threshold distance map, shows travel distance of fan
amount reachable from municipalities on a scale from red
(5,000 tickets within a five minute drive time) via orange
(5,000 within 10 minutes) to yellow (5,000 within 15
minutes). Stadiums are shown as points.
(b) Catchment areas are drawn in different colors for each
club and the allocation lines on top indicate the closest
club for each municipality. The analysis generates both
shortest distances to closest club and information about
this club.
FIGURE 1 Fan potential (a) and club accessibility (b).
We assume an analyst plans a follow up GIS study exploring spatial interaction of the fan base at a municipal
scale. Suppose he or she is given municipal data about population numbers, football clubs (within municipalities), a
road network, and some data about fan (ticket) statistics. Using this data, the analyst can answer various network
related questions. In total, we formulated 12 different workflow tasks that cover major forms of analysis (Sect. 6). For
illustration purposes, we explain the first three examples:

7
1. What is the suitability of municipalities (e.g. as a place for a new stadium) in terms of the fan potential reachable
within a certain distance?
2. What is the suitability of municipalities in terms of the minimal travel distance to reach a certain amount of football
fans?
3. What is the accessibility of football clubs for people living in municipalities?
Workflows to answer the first two questions can be found using threshold distance/amount analysis. For example (Task
1), we can assess the minimal distance that needs to be traveled to reach a threshold amount of potential fans, which
generates a map of municipal travel times (Fig.1a). Alternatively (Task 2), one could assess the amount of football fans
reachable within a threshold distance (not shown here). Answers to the third question can be found by generating
a map of catchment areas (Task 3), where each municipality is assigned to its nearest club according to some club
capacity. This results in a map like in Fig. 1b, where the smaller the distance, the more accessible clubs are. The data
can be used to do accessibility statistics, revealing e.g. that over 77% of ticket holders live near at least one club within
15 minutes car driving time.
3.2 |Expert-level workflow design
Once analytical tasks were formulated, we designed workflows manually as a basis for developing and evaluating our
model. We were interested in understanding how experts choose and organize software tools into a workflow graph
which generates valid answer maps. The answers were computed and illustrated using the software Flowmap, which is
specialized on spatial interaction7. Some of this functionality can also be found in other GIS software, such as ArcGIS
Network Analyst8. Example workflows for answering tasks 1,2 and 3 can be seen in Fig. 2.
FIGURE 2 Expert workflow implementing tasks 1, 2 and 3 in Flowmap. Ellipses denote computational steps,
rectangles denote data sets. We have generated such expert solutions for every task (not shown here because of
lack of space), cf. Sect. 7.
To computationally solve these three tasks, we first need to measure the length of road segments (or their travel
impedance) using street data. Then we need to turn the latter into a transport network (graph), by taking segment
ends as intersections. This also includes checking segment topology. Origin and destination locations (municipalities)
together with the transport network then need to be fed into a distance matrix function to compute a matrix of
shortest paths between municipalities (including also the ”last mile” feedlinks from origins and destinations to the
closest network intersection). The distance matrix together with the origin (destination) locations including their
capacity (demand) then feed into either a threshold or a catchment area function, to produce either a suitability or an
accessibility map. Parameters (not shown here) are the use of travel speed for computing distances as travel time, as
7http://flowmap.geo.uu.nl/
8https://www.esri.com/en-us/arcgis/products/arcgis- network-analyst

8
well as the choice of threshold distance or amount.
3.3 |Conceptual modeling and workflow synthesis study
The goal of our investigation is to learn how to produce workflows comparable to the examples above in an automated
manner, given just the task descriptions and the starting data. Since these computational steps are implicit in the task,
they need to be figured out automatically. This is done based on some conceptual model that can be used to describe
the task, the data and the computational functions. We develop such a model in Sect. 4 in the form of an ontology.
We used this ontology to describe typical spatial network functions as transformations of concepts in our model. This
means we described functions in terms of their input/output types, resulting in a type signature in Sect. 5. Furthermore,
we specified the 12 analytical tasks in terms of concept transformations using the same types (Sect. 6).
The conceptual model, together with the task specification and the function type signatures were then fed into
aloose programming algorithm. The latter searches for ontologically consistent sequences of function applications
of increasing complexity that satisfy a given task description (Lamprecht et al., 2010). As explained in Sect. 7, we
evaluated synthesized workflows based on expert assessments.
4|A CONCEPTUAL MODEL OF SPATIAL NETWORKS
The model introduced in this section is less about computation, but more on the level of thinking in GIS. Thinking
happens in parallel to computation by interpreting the computational products in terms of concepts (Guarino et al.,
2020). In a nutshell, we suggest to regard spatial networks as quantified relations between objects embedded in a
metric space, such that both objects and their relations can be quantified in a spatially extensive or intensive manner.
This model is used to formulate analytical tasks and to guide the composition of workflows.
4.1 |Spatial networks as quantified relations
One way to think of core concepts of spatial information (Kuhn, 2012) is in terms of particular kinds of relations in the
sense of relational algebra9(Codd, 1979). For example, information about a spatial field can be regarded as a relation
between locations and some quality (”at this location, the temperature is 15◦C”), and information about objects as a
relation between object identifiers and object qualities (”this building has a height of 10 m”). In the first case, locations
form the primary key, in the second case, object identifiers serve as primary key, while qualities are foreign keys in
all cases. We call such relations unary qualities, because the measured quality is controlled by a single entity. A spatial
network, in contrast, captures the idea of a relation with composite key: the key consists of some pair of instances of
objects or other concepts, and we measure some quality for each pair. For example, a distance matrix between cities
has pairs of objects as a primary key and distance measurements as foreign key. We call such relations quantified
relations, and their qualities binary qualities.
In principle, all core concepts can play a role in determining quantified relations. The measured quality e.g. can be
generated by various kinds of concepts: To analyse a drainage network in a catchment area requires summation of a
hydrological field (rainfall, water content) within the river catchment to determine network flow (Haggett and Chorley,
1969). To study movement or changes in a transport network, traffic or construction events need to be summarized.
9Relations behave similar to tables in a database or toEntity Relationship (ER) models in that they have primary and foreign keys. In the following, such analogy
is used only to illustrate our idea, not to imply that core concepts are actually implemented as tables.

9
Measure Unary
quality
Binary
quality
S
(spatial
geometry)
OS
(object
geometry)
OSO
(path
network)
B
(boolean
quality)
OB
(boolean
object quality)
OBO
(boolean
network)
N
(nominal
quality)
ON
(nominal
object quality)
ONO
(nominal
network)
I
(intensive
quality)
OI
(intensive
object quality)
OIO
(intensive
network)
E
(extensive
quality)
OE
(extensive
object quality)
OEO
(extensive
network)
TA B L E 1 Concepts as types of
relations of objects and measured
qualities. Unary object qualities have a
simple primary key, networks are binary
object qualities (composite primary key).
FIGURE 3 ER diagram of spatial network concepts,
with realization examples (tables with primary/foreign
keys) taken from our football scenario (see Sect. 6). Note
how different tables can be realizations of a given
concept.
Furthermore, the primary key of a quantified relation can be formed by different concepts: Prominent GIS methods
such as visibility analysis and Euclidean distance analysis can be conceived in terms of a Boolean or ratio scaled relation
between locations in space. We might call the latter relational fields, given that they quantify a measure for pairs
of locations, similar to ordinary fields quantifying a single location. Such hybrid models have been proposed earlier,
compare e.g. Cova and Goodchild (2002)’s idea of object fields. However, within the limited scope of this article, we
focus only on object-based primary keys. This interpretation may correspond to a default understanding of spatial
networks.
4.2 |Measuring extensive and intensive network qualities
Unary and binary qualities can be measured on different levels, and in this way determine whether functions are
applicable or not (Scheider and Tomko, 2016). For example, it is well known that different levels of measurement,
including count, ratio, interval, ordinal and nominal, are relevant for understanding analysis in geographic information
systems (Chrisman, 2002). In this article, we will make use of a Boolean quality including the values True and False, as
well as plain nominal qualities, which correspond to qualities that are on a nominal level and not on any other level. We
will also consider the regions of space that an object occupies as a measurable quality of that object.
The most important distinction for network qualities, however, is the one between spatially extensive and intensive
qualities (Scheider and Huisjes, 2018). Extensivity is known to influence the applicability of arithmetic functions, such
as the possibility of forming sums:

10
•Extensive qualities, which are closely related to amounts, are ratio-scaled qualities that are additive with respect to
the spatial extent of non-overlapping control units. An example for an extensive quality would be the population
of administrative units. If we merge two such units into a larger one (assuming the units do not overlap), then their
population counts add up in a corresponding way (Scheider and Huisjes, 2018). And vice versa, the population
count of a region shrunken to zero size becomes zero, making it ratio-scaled (Chrisman, 2002). We consider
extensivity as a class not only of unary qualities, but also of binary qualities or networks. Following this idea,
extensive binary qualities are determined by the extents of the objects that constitute the network relation. Take
the example of commuter flows: when merging a destination region, e.g. a city, with a new destination (a satellite
town), the commuter flow between origin and destination will increase by the sum of flows from the origin to the
new destination10.
•Intensive qualities, in contrast, are ratio-scaled qualities that do not add up when merging units. An example would
be the percentage of elderly people of a municipality, or the distance to the closest sport club. When merging
control units, the first quality needs to be aggregated using weighted averages, not sums. For spatial networks,
we consider intensive binary qualities. An example would be a distance measured between two regions, which
needs to be not summed but minimized when merging one of these regions with others.
These ideas give rise to the relational types listed in Table 1. In Fig. 3, these types are illustrated by Entity Rela-
tionship (ER) diagrams, with primary keys (PK) taken from data examples in our scenario (see Sect. 6). For example,
layers of municipalities and football clubs are modeled as unary qualities with objects as primary key and some geom-
etry as foreign key (OS). Road sizes and population numbers are examples for extensive unary qualities (OE). Distance
networks (between municipalities and clubs), in contrast, correspond to intensive binary object qualities of type OIO,
whereas traffic flows between road intersections correspond to extensive binary object qualities of type OEO. Binary
qualities can also be Boolean, indicating whether paths go through a pair of objects, or consist of geometries that
denote such a path (= path networks, type OSO).
FIGURE 4 Modeling spatial networks in terms of qualities of objects and their relations. Both kinds of qualities
can be extensive or intensive. Spatial network analysis essentially transforms these qualities into each other.
10Extensivity in this case would need to take account of the product of origin and destination regions. Formalization of this kind of unary and binary extensivity
is considered future work.

11
Just like in relational algebra, we leave open how complete a given relation is with respect to its set of tuples and
the domains that make up its key. Binary concepts that consist of an incomplete subset of the cross-product of two
given sets of objects are called networks. Networks might consist of only a single pair of objects as key. Sometimes,
we want to be more exhaustive, then the complete cross-product of two sets of objects makes up the primary key of
that relation, which we call a matrix. We use the star symbol * to refer to relations of that latter sort (e.g. OEO*).
Spatial network analysis, in essence, consists of transformations between such qualities (Fig. 4). For example, a
catchment area analysis, which computes network distances to the closest object in a layer, transforms an intensive
(distance-based) network between spatial objects with extensive amounts into intensive object qualities (distance to
closest object). This corresponds to going from the middle layer to the upper layer in Fig. 4. Gravity models (Batty,
1976), in contrast, allow us to estimate amounts of interactions between objects. In essence, they convert an intensive
(distance-based) quality between spatial objects with extensive amounts (middle layer) into some extensive quality
(lower layer).
4.3 |Representing object and network qualities as data types
The concepts discussed above are interpretations of input or output data of network functions, i.e., they constitute
intermediary types. Which formal type system should be used to add such interpretations to the data? A given core
concept can be represented by various geometry types, and vice versa, a given geometric model might be interpreted
in terms of different concepts (Scheider et al., 2020). A field, for example, may be represented by vector lines or
polygons (think about contours or land cover polygons), as well as by some raster layer. Similarly, networks may be
represented by many kinds of geometries, not only by lines11. And vice versa, a line dataset alone does not yet imply
the existence of a network: To turn a roads file into a network, we first need to build a network topology. We take
account of this representational variety simply by three orthogonal semantic dimensions: the core concept represented
by a given attribute, its measurement level, and the geometry type of its layer. Each dimension forms an independent
subsumption hierarchy, where subsumed classes are interpreted as sub-classes. Classes can be combined arbitrarily
between hierarchies, while leaf classes of one dimension are considered mutually exclusive.
(a) Denotes the core concept
represented by some geodata attribute. (b) Denotes the geometry type of some
geodata attribute.
(c) Denotes the measurement level of
some geodata attribute.
FIGURE 5 Three semantic dimensions of CCD types used in this article, including core concept, geometric types
and measurement levels of attributes. Arrows denote subsumption relations.
11Think e.g. about grid or lattice graphs, which represent a network of raster cells or lattice polygons.

12
Dimensions were encoded by extending the Core Concept Data Types (CCD) ontology12 (Scheider et al., 2020)
with corresponding OWL classes (see Fig.5): The first dimension (a) includes the hierarchy of core concept types.
CoreConceptQ is the upper bound of this hierarchy and subsumes ObjectQ (object quality), NetworkQ and MatrixQ
(network/matrix quality). The latter two are subsumed by RelationalQ (≈binary quality). AmountQ denotes amounts
of objects or other content that is not bound to any object quality. We use this class to denote summary statistics.
In the layer geometry dimension (b), LayerA subsumes LineA (line attribute), VectorTessellationA (polygon tessellation
attribute) and PlainVectorRegionA (attribute of a non-tessellated polygon layer). The third dimension (c) subsumes
measurement levels of an attribute, with NominalA being the upper bound. IRA/ERA are considered subtypes of
RatioA standing for intensive/extensive region attributes. PlainNominalA denotes nominal attributes that are not on a
more specific measurement level. Conjunctions of these classes are used in the following to specify tasks, describe
functions and compute workflows.
5|SPATIAL NETWORK TRANSFORMATIONS
Building on our model, we can distinguish available network functions based on how they transform one concept into
another. This is done based on type signatures using the types from our model. The signatures of functions relevant
for our scenario are given in Table 2, and each one is shortly explained in the following. The table contains software
examples from ArcGIS as well as Flowmap13. Tool annotations in RDF14 areavailable online15 , including also geometry
types for function inputs and outputs which are omitted here. Tasks illustrating their use are mentioned together with
each function, cf. Appendix A.
We start with basic functions that are underlying yet not usually considered to be network analysis. Usually, the
first step in constructing an intensive (distance) network is to measure road lengths using street segment lines. We call
this operation measure size, and it takes object regions (OS;in this case lines) and generates object sizes (OE; in this case
lengths of lines), which are extensive measurements. Object sizes can then be used together with the geometry of their
object regions in order to construct a distance network, based on topological (touch) relations between geometries.
The latter are used to generate new (intersection) object pairs in the network, while the object sizes become intensive
distance qualities of the network (OIO). This step corresponds to ”building a topological network” in GIS. Following our
logic to name functions according to their outputs, we call it distance network here. The distance matrix function takes
an intensive network of distances (OIO), as well as a set of object regions (OS), and generates a matrix of network
distances between all pairs of objects. Commonly this is the shortest path between these objects on the network,
and involves, in case some objects are not in the network, also a metric distance measurement between these objects
and their entry points to the network.
Functional clustering (Brown and Horton, 1970) between two locations in space is the reverse of the amount of
interaction between them. For example, the Intramax Method developed by Brown and Masser clusters (adjacent) lo-
cations based on the amount (Masser and Brown, 1975) or relative amount (Masser and Brown, 1977) of interaction.
It therefore takes an extensive (interaction) matrix, as well as some object regions, and generates a nominal object
quality, where the nominal value indicates the cluster to which a given object belongs. Object regions are needed to
determine whether objects are neighbors. An example is given with Task 6. A catchment area function takes an inten-
sive (distance) matrix, some object regions as origins, as well as some object regions as destinations, and indicates, for
12http://geographicknowledge.de/vocab/CoreConceptData
13In the future, we plan to extend the coverage of software to further network functions, such as the functions available in QGIS.
14http://www.w3.org/TR/rdf11- primer/
15https://figshare.com/s/1f794030aaf8b78175ab

13
Function Software examples Inputs Outputs
ArcGIS Flowmap
Measure size
Calculate
Geometry
Attributes
Calculate
Length/size
Object
regions OS
Object
size OE
Distance network Build Network
Import to
Flowmap
Object
size OE
Object
regions OS
Distance
network OIO
Network distance
matrix
OD Cost
Matrix Solver
Network
Distance Matrix
Distance
network OIO
Object
regions OS
Distance
matrix OIO*
Functional
clustering
-Intramax
Analysis
Flow
matrix OEO*
Object
regions OS
Object
nominal ON
Catchment area
Closest
Facility
analysis
Calculate
Catchment
Areas
Distance
matrix OIO*
Object
regions OS
Object
regions OS
Object
distances OI
Network analysis -
Transport
Network
Analysis
Distance
network OIO
Object
regions OS
Object
distances OI
Boolean
network OBO
Threshold amount
(amount within
distance)
(Service Area
Analysis*)
Proximity
count
Distance
matrix OIO*
Object
amounts OE
Object
amounts OE
Threshold distance
(distance to
amount)
-
Regular
Treshold
Distance
Distance
matrix OIO*
Object
amounts OE
Object
distances OI
Accessibility
analysis
Summary
Statistics
Catchment
Profile
Object
distances OI
Statistics
I
Flow matrix
estimation
(doubly-constrained)
doubly-c.
Gravity Model
Distance
matrix OIO*
Object
amounts OE
Object
amounts OE
Flow
matrix OEO*
Attr./prod.
scores OI
Flow matrix
estimation
(singly-constrained)
Huff model singly-c.
Gravity Model
Distance
matrix OIO*
Object
amounts OE
attr./prod.
factors OI
Flow
matrix OEO*
Object
amounts OE
Flow
summation
-Interaction
Summation
Flow
matrix OEO*
Object
amounts OE
Trip length
analysis
-Trip End
Ranking
Distance
matrix OIO*
Flow
matrix OEO*
Statistics
I
Trade area (Probability
Based Markets*)
Trade Area
Analysis
Distance
matrix OIO*
Flow
matrix OEO*
Object
regions OS
Object
regions OS
Flow
assignment
-
Flow
Assignment
to Network
Flow
matrix OEO*
Distance
network OIO
Object
regions OS
Flow
network OEO
TA B L E 2 Functional signatures of basic spatial network transformations. Software tools with asterisk* are quasi
equivalent.

14
each origin object, its distance to the closest destination object, as illustrated in Task 3. Network analysis does a similar
thing, only based on an intensive distance network and some destination object regions, computing shortest distances
to the closest object for all possible origins given within this network (Task 4). The resulting distance measurements on
objects can be used to compute accessibility statistics. In addition, this function also outputs corresponding shortest
paths given as a boolean network, where true indicates that some path goes through the corresponding pair of objects.
Threshold distance and Threshold amount functions both take an intensive (distance) matrix and some extensive object
quality (amount). The latter generates, for each object, the sum of amounts reachable within some distance, and the
former the minimal distance to a given sum of object-based amounts. In our scenario, an example is given in terms of
fan potential analysis as part of answers to Tasks 1 and 2.
Adoubly-constrained flow matrix function takes some intensive (distance) matrix and two extensive object qualities
(amounts) and generates an extensive (interaction) matrix between these objects, as well as some attractiveness/pro-
ductivity score on objects, which is intensive. For example, a gravity model (Batty, 1976; Wilson, 1974; Huff, 1964)
can be used to estimate interactions between municipalities and football clubs based on both the amount of ticket
holders residing in each municipality and the number of tickets sold by each football club using some distance decay
function. The parameter of the distance decay function is either given or fitted to a measured mean trip length. A
singly-constrained model, in contrast, takes some attractiveness/productivity score on destinations (origins) and some
capacity on origins (destinations) to generate interaction estimations and amounts for destinations (origins). Examples
are the different sorts of gravity models that can be used in Tasks 9, 10, and 11. Flow summation takes an exten-
sive (interaction) matrix and sums up all outgoing flows to corresponding amounts on origin objects, as illustrated in
Task 5. Trip length analysis is a statistical summary of the distribution of interactions over distances between objects,
resulting in some trip statistics (average trip length, Task 7, or average trip end ranking), like the average car travel
time for all trips being approximately 16 minutes. Trade area functions also take a distance and an interaction matrix
as inputs, as well as some object regions, and determine some smallest (minimal distance based) object region that
contains a particular sum of interactions. For example, it allows us to demarcate an area around each football club
that contains a certain percentage of its closest ticket holders (Task 8). Finally, a flow assignment function takes some
interaction matrix and some distance network as well as some object regions, and assigns flows to the network ac-
cording to the shortest paths between flow origin and destination objects (Task 12). Functions are also summarized
in the computational diagram in Fig 6.
Note that only three of the 15 functions in Table 2 require an actual transport network file. Most of the other
functions (namely 10) require a distance table that can be based on transport network distance but also on airline
distances, time schedules, tariff structures or functional distances. This illustrates that spatial network analysis is way
broader than implied by the common focus on transport networks. Furthermore, note that 7 of the 15 functions did
not have an equivalent in the standard software ArcGIS, though this functionality can of course be reprogrammed.
6|SPECIFICATION OF WORKFLOW TASKS IN FOOTBALL SCENARIO
Starting from a simple data source, we went through 12 different analytical tasks16 as an empirical basis for evaluating
our model. We begin with a description of the available data sources. Note that an in-depth study of the data and the
results is beyond scope17.
16Because of limited space, detailed tasks descriptions can be found in Appendix A.
17A more comprehensive documentation of the analysis is available under http://geographicknowledge.de/pdf/Flowmap%20Applications%20Q1-16.pdf.

15
FIGURE 6 Computational diagram of spatial network transformations. Note that some signatures have been
simplified in this diagram.
6.1 |Data source specification
There are five different data sources, which were interpreted in terms of the following types in our model:
•MUNCLUB (ObjectQ,VectorTessellationA,PlainNominalA)Polygon layer, containing 489 municipalities, plus the four-
digit postcode areas of 37 professional football clubs in the Netherlands. Field ”LABEL” contains residential mu-
nicipal name or football club name, and Field ”FC” is 1 in the latter and 2 in the former case. Conceptually, this
corresponds to a collection of objects, including ON and OS. Municipalities form a vector tessellation of the area.
•POP_2003 ∗MUNCLUB (ObjectQ,VectorTessellationA,ERA). Table POP_2003 contains the total population per mu-
nicipality (CBS Statline), corresponding to an extensive object based vector tessellation attribute, and correspond-
ing to OE.
•ROADS08 (ObjectQ,LineA,PlainNominalA). Dutch Road Transport Network ROADS08 (BASNET by Adviesdienst
Verkeer en Vervoer (AVV)), contains names for roads and line geometries, thus including both ON and OS. Note
that roads are conceived as linear objects, not networks.
•STICKET2(MatrixQ,VectorTessellationA,ERA). Spatial interaction table of the royal Dutch football association
(KNVB); contains total number of season tickets per combination of residential municipality and club. In total,
in 2003, there were 349.538 ticket holders distributed over 3.459 different combinations of municipality and
club. The interaction table reports amounts of ticketholders. Thus it corresponds to an extensive matrix quality
OEO*.
•NEWCLUBS (ObjectQ,VectorTessellationA,IRA). Hypothetical attractiveness score for each football club (OI) in a
scenario where the lower professional league is abolished.

16
Though these sources cover only a limited set of types, further types of data are generated as part of the workflows
described in the following.
6.2 |Specification of analytical tasks and expert workflows
Each task was described by a unique question (workflow task, cf. Tab. 3 and Appendix A). The latter was then specified
in terms of our type model (CCD), including input data types,goal types and (optionally) requests for intermediate data
types that should be used in the workflow. Specifications were later used as as basis for automatic workflow synthesis.
Furthermore, we manually generated one expert workflow for each question (examples below). In Appendix A, we
explain in more detail how each task specification reflects the information given in the question, which computational
steps are needed to answer it, and how resulting maps look like.
Distance based analysis
We first considered analytical tasks 1-4 that exploit distances between residential areas and football clubs measured
on a road network, in addition to amounts measured at origins or destinations. Workflow tasks include the assess-
ment of fan potentials and accessibility analysis. Computationally, these tasks require the generation of a distance
matrix between objects, by computing shortest path distances on the road network and including the last mile be-
tween road intersections and these objects (use types). To assess fan potentials, the goal types are extensive/intensive
object qualities. Accessibility analysis requires intensive (distance based) object qualities, represented either as regions
(municipality level) or lines (street level). Workflows for tasks 1-3 were already discussed in Sect. 3.
Interaction based analysis
Here we focus on tasks that analyse spatial interaction or flows between residential areas and clubs, in addition to the
network distance, making use of a (measured or modeled) interactionmatrix (type OEO*). This includes flow summation
(Task 5) to summarize flows of destination/origin amount totals, and which was specified by requesting extensive
object qualities as goal type. Functional distance clustering (Task 6) was specified by requesting nominal values (cluster
identifiers) for objects. Trip length distribution (Task 7) was specified by requesting some intensive measure. Finally,
Trade area analysis (Task 8) was specified by requesting an object based region. Workflow solutions for tasks 7 and 8
are shown in Fig. 7.
FIGURE 7 Expert workflows solving the tasks 7 and 8.
Flow generation
The final type of analysis provides ways of estimating interactions from other kinds of spatial information. Expert
workflows solving these tasks are depicted in Fig. 8.

17
Task
category
Task
subcategory
Workflow task Task specification
Fan potential
1 "What is the potential amount
of fans within a travel distance
for each municipality?"
input: (1) ROADS08, (2) POP_2003
goal types: OE (ObjectQ, RegionA, ERA)
use type: OIO* (MatrixQ, IRA)
Distance
based
analysis
2 "What is the potential minimal
travel distance to reach a certain amount
of fans for each municipality?"
input: (1) ROADS08, (2) POP_2003
goal types: OI (ObjectQ, RegionA, IRA)
use types: OIO* (MatrixQ, RegionA, IRA)
Accessibility
3 "What is the accessibility
of clubs from each municipality?"
input: (1) ROADS08, (2) MUNCLUB
goal types: OI (ObjectQ, RegionA, IRA)
use types: OIO* (MatrixQ, RegionA, IRA)
4 "What is the accessibility
of clubs from each road?"
input: (1) ROADS08, (2) MUNCLUB
goal types: OI (ObjectQ, LineA, IRA)
use types: OIO (NetworkQ, LineA, IRA)
Flow
summation
5 "What is the amount of
fans for each municipality?"
input: (1) STICKET2
goal types: OE (ObjectQ, RegionA, ERA)
Interaction
based
analysis Functional
clustering
6 "To which functional cluster
does a club belong?"
input: (1) STICKET2, (2) MUNCLUB
goal types: OS, ON (ObjectQ, RegionA, PlainNominalA)
Trip length
distribution
7 "What is the average travel
time to/trip rank of a club?"
input: (1) ROADS08, (2) MUNCLUB, (3) STICKET2
goal types: I (IRA)
Trade area
analysis
8 "What is the area enclosing
60% of the amount of fans
closest to each club?"
input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OS, ON (ObjectQ, RegionA, PlainNominalA)
use types: OEO* (MatrixQ, ERA), OIO* (MatrixQ, IRA)
9 "What is the potential amount
of fans in each municipality
for each club assuming distance decay?"
input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OEO* (MatrixQ, RegionA, ERA)
Flow
generation
Gravity
modeling 10 "What is the attractiveness of clubs
for fans?"
input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OI (ObjectQ, RegionA, IRA)
11 "What is the potential amount
of fans for each club when the lower
professional league is closed?"
input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2,
(4) NEWCLUBS
goal types: OE (ObjectQ, RegionA, ERA)
11b (a more challenging version of 11)
input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OE (ObjectQ, RegionA, ERA)
use types: OI (ObjectQ, RegionA, IRA)
Traffic
load
analysis
12 "What is the potential traffic load for
each road assuming fans travel by car
at the same time?"
input: (1) MUNCLUB, (2) STICKET2, (3) ROADS08
goal types: OEO (NetworkQ, LineA, ERA)
TA B L E 3 Collection of spatial network analysis tasks for synthesizing workflows. Tasks were formulated as
questions and specified using CCD types.

18
FIGURE 8 Expert workflows solving the tasks 9, 10 and 11.
The task of estimating the potential amount of season ticket holders (Task 9) was specified by requesting an
extensive matrix. Another task was to estimate relative attractiveness scores (Task 10) for clubs, based on the product
of (club or municipal) amount and their matching ”balancing” factor. This was specified by an intensive object quality.
Finally, suppose the lower professional football league is abolished and their attractiveness becomes zero. What will
happen to the fans and the rest of the clubs? To answer this question, the task (Task 11) was to generate an extensive
object qualities as goal. A more challenging version of the same task (Task 11b) is to start without manually generated
attractiveness scores, but requiring the generation of attractiveness scores in an intermediate step, via use types. The
final flow generation task takes an interaction matrix between municipalities and clubs, as well as a street network
as input, and generates more fine grained flows between road intersections, based on assuming that trips are made
on the shortest paths on this network. This task is called traffic load analysis (Task 12), specified by requesting an
extensive network quality on lines.
7|EVAL UATI ON
When thinking is turned into workflows, concepts need to be translated into concrete tools and data sources. Our
hypothesis is that common geodata models alone, as well as graph theoretic models, are insufficient to perform such
a translation. To test this hypothesis, we follow an approach of workflow synthesis quality assessment that was
developed in Kruiger et al. (2021). An overview of the evaluation process is shown in Fig. 9. We compare the quality of
automatically synthesized workflows that were generated using our conceptual model against two benchmark models.
In the following, we explain the synthesis algorithm, the benchmark models and our workflow quality assessment
approach.
7.1 |Synthesis algorithm and workflow repository
We used a workflow composition algorithm as described in Kasalica and Lamprecht (2020a). Automated Pipeline
Explorer (APE)18 generates sequences of tool applications satisfying logical (type) constraints as used in our task
specification (input types, output types, use types). The latter are expressed in semantic linear-time logic (SLT) using
the classes of our ontology. The three semantic dimensions of the CCD model were used independently as constraints
18https://github.com/sanctuuary/APE

19
Ontology
(3) Task specification(2) Taxonomy preparation
(1) Tool annotation
Taxonomy
(4) APE
Task specifications
(5) Evaluation
Workflow evaluations
Questions
Error types
Too l a nn o ta t io n s
Workflows
FIGURE 9 A summary of our ontology evaluation framework for workflow synthesis. For an ontology, five steps
are performed. All steps are done both for the ontology and the benchmarks to measure improvements (Kruiger
et al., 2021).
for this kind of reasoning, and class combinations were automatically interpreted as class conjunctions. Furthermore,
leaf classes in the ontology were interpreted as mutually exclusive and jointly exhaustive. In APE, workflow models
satisfying these task specifications are generated with increasing size, drawing from a repository of tool signatures (cf.
Table 2) annotated with the same types. Maximum number and size of workflows were given as parameters. In our
test, we generated five workflows up to a length of 10 tool applications for each variant of a task. More workflows
increased only the amount of soft errors (cf. Sect. 7.3). Furthermore, we used the constraints that all given input data
should be used in the workflow, and that at least one of the data instances that are generated as output, per tool, has
to be used. The workflow synthesis repository with all resources is available online19, including task specification files
(ape.configuration and constraints.json) for the 12 tasks as well as resulting workflows, for both CCD and the benchmark
solutions. Workflow outputs are generally encoded as a directed acyclic graph with function applications as vertices.
Examples of automatically synthesized workflows are shown in Appendix B. In APE, workflows can also be exported
in a serialized form, as an executable script. This requires, however, a way to deal with function parameters (cf. the
discussion below).
7.2 |Benchmarking
We compared the synthesized workflows from our model against workflows obtained under the exact same condi-
tions, except that we used some modified type system reflecting the kind of information available in current data
models used to represent spatial networks. We considered two benchmark variants:
19https://figshare.com/s/aea5c00a9858db69e37f

20
1. Geometric benchmark (abbreviated bench). This is a proper subset of CCD where the two conceptual dimensions
(including core concepts and measurement levels) were removed, including only one dimension related to geom-
etry types, namely the distinction between raster and vector attributes, as well as between point, line and region
attributes (cf. Fig. 5 b). The distinction between VectorTessellationA and PlainVectorRegionA was also removed,
since it does not occur in current data structures.
2. Embedded graph benchmark (abbreviated graph). This version retains the idea of a graph embedded into geo-
metric space. We distinguish between nodes and directed edges (≈relations between nodes) based on the core
concept superclasses ObjectQ and RelationalQ (cf. Fig. 5 a) respectively, together forming one dimension. Further-
more, nodes as well as edges can be embedded by either of the geometric types in the geometric benchmark.
This is encoded by taking the geometric benchmark types as a second dimension.
Using these benchmark versions of the ontology, we manually created corresponding tool annotations by substituting
every type with the least upper bound (supremum) concept that is still in the corresponding benchmark ontology. In
the same way, we generated benchmark versions of all task specifications, by substituting input, use and goal types
with their benchmark equivalents, respectively.
7.3 |Evaluation metrics and quality assessment
We treated workflow synthesis like a retrieval process, measuring its quality with respect to an expert judgment
and considering expert workflows produced independently with Flowmap. We decided to measure both precision
(the proportion of retrieved answers that are correct given all retrieved answers) as well as recall (the proportion of
retrieved answers that are correct given all correct answers).
For assessing recall, an expert on spatial network analysis went through the tasks ahead of our study and manually
generated a gold standard of expert workflows, using the set of spatial network functions in Table 2. Afterwards,
when going through the synthesized workflows for each task, the expert simply indicated whether one of them cor-
responded to the expert workflow for this task.
For assessing precision, our expert assessed synthesized workflows individually based on different error types. We
used three error types on two different severity levels, which are summarized and illustrated in Table 4.
Error severity Error type Example workflows
Hard Syntax Fig. 16
Semantic Fig. 15
Soft Redundancy Fig. 17
TA B L E 4 An overview of the different error types, figures in Appendix B.
Hard errors are critical errors which result either in a wrong or non-meaningful answer, or in a workflow that is
non-executable due to wrong data formats. We distinguish two kinds of hard errors: syntax errors, which have a part
of the workflow that can not be executed because a tool is incorrectly applied, and semantic errors, which produce
a meaningless or invalid answer for the given question. Soft errors are non-critical errors where workflows do entail
a correct answer, but which are in some sense of lesser quality. We focused on redundancy errors, where workflows
make use of unnecessary tool applications.

21
8|RESULTS
Evaluation result datasets are available online20. In Table 5, evaluation results for each task variant are shown as a
statistic over the first five workflows generated using each task specification. Num indicates the number of workflows
for each task variant, which can be less than five in case not more options were found. Semantic error denotes the num-
ber of semantic errors in these workflows, Syntactic error denotes the number of syntactic errors in these workflows,
Correct denotes the number of workflows without hard errors, Rdn denotes the number of correct workflows with
redundancy errors. Expert solution denotes the number of correct workflows that correspond to an expert solution.
Expert order denotes the order of occurrence of the first expert solution, in case it occurred within the set of gener-
ated workflows, and ∞otherwise 21. Results are listed for workflows generated with the CCD model (CCD[x]), the
embedded graph model (graph[x]), and with the geometric benchmark model (geom[x]). In the row total, we summed
up all workflow counts and averaged the length and order measurements for each of these three test variants. In total,
we checked the quality of 181 workflows. Our interpretation of these results is summarized as follows:
•Our study shows that the CCD model is capable of reproducing at least one expert workflow for each single task
(cf. Expert solution). In total, 22 expert workflows could be recalled by the CCD model. Removing duplicates,
this amounts to 13 unique expert workflows (including tasks 11 and 11b, cf. Fig. 14), which is a recall of 100
%. This is in stark contrast to the geometric benchmark, which only produced a single expert solution (for task
5) over all 12 tasks (recall 8%), as well as the embedded graph model, which found four expert solutions (recall
33%). Whether the exceptionally high recall value of the CCD model can be sustained for larger sets of expert
workflows or other kinds of tasks remains to be seen. However, it shows that our model indeed is capable of
accounting for a significant amount of such expert knowledge.
•Furthermore, the expert solutions that were found by CCD appear very early in the process (cf. Expert order). Most
often they appeared as the first solution, except for tasks 10 and 11b, where they appeared as number five and
two in the row. In the four cases in which the graph model was able to produce experts solutions, these were
generated on places five,two,three,five. This indicates that despite of the presence of semantically incorrect or re-
dundant workflows, high quality solutions produced by the CCD model may be filtered out simply by constraining
the number of generated workflows.
•CCD solutions are on average much longer than benchmark workflows (4.5 nodes compared to 1.8 in the geo-
metric and 2.5 in the graph model) (cf. Avg length). This indicates that the CCD ontology adds more constraints to
the space of workflow composition, and thus contains more information than both the geometric and the graph
model.
•59 % (36 out of 61) of all CCD workflows were correct solutions of the task (without any semantic or syntactic
errors) (cf. Correct). This is again in stark contrast to the geometric model, with a precision of less than 2 %, and
also to the graph model, with a precision of 6%. This indicates that without deeper semantics, it becomes nearly
impossible to generate high quality solutions, even if using an embedded graph. Furthermore, since errors tend
to occur with larger solutions, the precision of the CCD model dramatically increases to 84 % (11 out of 13) when
selecting the first workflow as a solution for each task. Still, there remain quite a lot of semantic and syntactic
errors in the CCD solutions. The 13 semantic errors were due to missing workflow constraints implicitly contained
in the task (cf. discussion below). The 16 syntax errors were mainly due to the fact that some of the computational
functions in our model, which are treated independently, are actually not implemented in terms of independent
20https://figshare.com/s/7f44d57de058b51c19e3
21Meaning that the workflow may still appear later in the process, but not within the first five workflows.

22
Task Variant Num Avg
length
Semantic
error
Syntax
error
Correct Rdn Expert
solution
Expert
order
1 CCD1 5 4.8 2 1 2 1 1 1
graph1 5 3.0 4 5 0 0 0 ∞
geom1 5 1.8 5 5 0 0 0 ∞
2 CCD2 5 4.6 3 2 1 0 1 1
graph2 5 3.0 5 5 0 0 0 ∞
geom2 5 1.8 5 5 0 0 0 ∞
3 CCD3 5 5.6 2 3 2 1 1 1
graph3 5 3.0 4 5 0 0 0 ∞
geom3 5 1.8 5 5 0 0 0 ∞
4 CCD4 5 4.6 0 3 2 1 1 1
graph4 5 2.8 0 4 1 0 1 5
geom4 5 1.6 4 5 0 0 0 ∞
5 CCD5 1 1.0 0 0 1 0 1 1
graph5 5 1.6 2 4 1 0 1 2
geom5 5 1.0 4 3 1 0 1 1
6 CCD6 5 3.0 0 4 1 0 1 1
graph6 5 1.0 4 4 1 0 1 3
geom6 5 1.0 4 5 0 0 0 ∞
7 CCD7 5 4.8 1 1 4 3 1 1
graph7 5 2.4 4 5 0 0 0 ∞
geom7 5 2.8 0 5 0 0 0 ∞
8 CCD8 5 4.8 0 0 5 4 1 1
graph8 5 2.4 4 5 0 0 0 ∞
geom8 5 2.0 4 5 0 0 0 ∞
9 CCD9 5 5.2 0 1 4 0 4 1
graph9 5 2.6 5 5 0 0 0 ∞
geom9 5 2.0 5 5 0 0 0 ∞
10 CCD10 5 5.0 4 0 1 0 1 5
graph10 5 2.4 5 5 0 0 0 ∞
geom10 5 2.0 5 5 0 0 0 ∞
11 CCD11 5 5.2 0 0 5 1 4 1
CCD11b 5 6.0 1 0 4 0 4 2
graph11 5 3.0 5 5 0 0 0 ∞
geom11 5 2.0 5 5 0 0 0 ∞
12 CCD12 5 3.8 0 1 4 3 1 1
graph12 5 2.8 0 4 1 0 1 5
geom12 5 1.8 3 5 0 0 0 ∞
total CCD 61 4.5 13 16 36 14 22 1.4
graph 60 2.5 42 56 4 0 4 -
geom 60 1.8 49 58 1 0 1 -
TA B L E 5 Results of evaluating the core concept (CCD) model of spatial networks against the benchmark models.
Each task variant included the first 5 workflows generated by APE under the given specification. In total, 181
workflows were evaluated. See text for explanation.

23
components in the Flowmap software22. In consequence, some possible combinations and repetitions of these
tools in our model are actually syntactically impossible in Flowmap. These errors can be easily avoided by forcing
the tools to be used only once or only in conjunction with others. Furthermore, syntactic errors due to repetitions
can be considered redundancy errors. If we count these errors as redundancy errors instead, the hard error rate
of the CCD solutions falls by 10, resulting in a precision of ≈75% (46 out of 61).
•Redundancy errors occur within CCD workflows mainly because CCD imposes increased constraints on the work-
flow composition process, and so the only possibility of generating longer workflows is to repeat function ap-
plications. This is compatible with earlier results (Kruiger et al., 2021). The problem can be handled by further
restricting the number of workflows produced for each task.
Regarding the validity of these results, we would like to add the following considerations. First, one may ask
whether the chosen benchmark for comparison is of sufficient quality. Our argument is that the benchmarks cover
precisely the concepts used and available in current spatial network information systems. These are, on the one hand,
geometric data types, and on the other hand, graph theoretic models. We were rather lenient with the combinability of
graph elements and geometry types to distinguish functions, which in practice is rather more restricted. Second, one
might ask whether our chosen tasks and scenarios are not too limited in range. Our list indeed misses some common
network functions, including more complex routing functions, such as traveling salesman or Chinese postman routing,
or location allocation methods. However, the first two of these can be seen as a special case of the network distance
matrix function. Shortest-path routing deals with a single origin and destination and some path network (OSO) as
output that contains all trips as geometries between origin and destination objects. In the traveling salesman variant,
the only thing added is another object input, namely, the objects to be visited on a tour. Location-allocation func-
tions are methods to place objects in respect of both amounts and distances, and thus should also fit well into our
framework. Third, regarding the complexity of our tasks, we believe they correspond to the level required in practice.
Nevertheless, it should be investigated in the future how longer tasks and larger repositories of functions influence
the quality of workflows. And fourth, in the practice of spatial network analysis, parameter settings and fitting of
parameters values (e.g. the distance decay parameter for gravity models) and manual interventions are essential parts
of a workflow. In this respect, our model still commits to a considerable simplification, leaving completely autom-
atized workflow synthesis beyond current reach. However, this could be addressed in the future by incorporating
abstract parameter semantics. Which kinds of concepts could be used for this purpose, however, is an open question.
Finally, in compliance with previous results (Kruiger et al., 2021), it seems that the amount of semantic errors can
only be further reduced when incorporating information about the type of transformation. As shown in Fig. 15, this
workflow about task 10 fails because the threshold distance function has the same result type as the (required) at-
tractiveness score of the doubly constrained flow matrix function. To prevent this error, we would need to distinguish
between measuring threshold distances vs. measuring attractiveness, which is beyond the current model. However,
the workflow synthesis algorithm would allow incorporating such tool constraints (Lamprecht et al., 2010).
9|DISCUSSION AND CONCLUSION
In this article, we suggested and tested the idea that spatial network analysis, as implemented in Geographic Infor-
mation Systems, and as envisioned by early writers in network related Geography, can be fruitfully understood as a
repertoire of functions that transform between relations of objects and their qualities. Qualities can be unary or bi-
22”Measure size”, e.g. is implemented in Flowmap only as part of the network topology generation.

24
nary, extensive or intensive (depending on whether they are additive w.r.t the spatial extent of the controlling objects),
and on different levels of measurement. To this end, we extended the core concept data (CCD) types ontology with
new classes along three semantic dimensions, including core concept, measurement level and geometry type. We
also included two benchmark models, one of them corresponding to a geometrically embedded graph.
We tested our model against the benchmarks on a scenario with 12 different network analysis tasks. We evaluated
automatically synthesized workflows by expert judgements and by comparing them with independently generated ex-
pert workflows. Despite its simplicity, we demonstrated that the model helps us not only more clearly understand the
underlying functions, but also to automate spatial network analysis to a degree that can support analysts in answering
questions. Our model distinguishes (question 1) 12 network analysis tasks in terms of input/output and intermediary
types, which was sufficient to instruct corresponding workflow synthesis. Only in few cases (e.g. Task 10) the model
was not able to distinguish between tasks that should result in different workflows. Furthermore, the model was suffi-
cient (question 2) to distinguish between all relevant spatial network functions, except for functional differences that
depend on function parameters or type-equivalent transformations, (e.g. threshold distances and attractiveness scores)
which were not distinguished in this study. Furthermore, regarding the quality of synthesized workflows (question 3),
results show not only that the model was capable of regenerating all expert workflows, but also that the semantic
depth added by our model over and above graph theory is crucial for high quality workflows, improving their accuracy
from 6% to 60%, and potentially over 75% under certain adjustments.
To enable fully automatized workflows and executable workflow scripts, there are still several open issues. First,
future work should focus on models for incorporating method parameters (which were not considered here) and for
removing remaining syntax errors. To remove the considerable amount of semantic errors, the model needs to be
extended to types of network transformations. Modeling parameter semantics is closely related to a transformation
model, because function parameters are often functions themselves (e.g. ”averaging” trip lengths versus ”taking the
median” of trip ranks). We are currently working on a transformation algebra that is based on a higher-order type
system for specifying such conceptual transformations. Finally, tool annotations should be extended to encompass
further relevant software for spatial network analysis, including QGIS, ArcGIS and Python libraries, allowing for cross-
software comparisons.
What are the wider implications of these results? We see our work in the context of symbolic AI for GIS (Janowicz
et al., 2019). For purposes of GIS automation, we can learn from this study that the know-how required to deal with
spatial information generally goes beyond knowing the computational procedures or having the data. Thus reducing
know-how to knowledge extraction bears the danger of underestimating this task. This is especially important in an
age where intelligence tends to be reduced to a variant of machine learning. By reducing analysis to the computa-
tional process on data, we disregard the underlying reasoning process that is necessary to arrive at meaningful results.
As our study demonstrates, this reasoning process requires concepts instilled into data, not extracted from data. Cor-
respondingly, while Janowicz et al. (2019) claim that ”GeoAI research will have to make a case for spatially explicit
models”, our study clearly shows that for purposes of automation, explicit spatial models are beyond question, and
that even such models can still be insufficient. While we have made a suggestion for the kind of knowledge lacking,
it remains unknown what we will lose once our network experts are substituted by machines.
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Appendix A
This appendix contains more detailed descriptions of the analytical tasks 1-12 for designing workflows, which were
used to build and evaluate our conceptual model.
|Distance based network analysis
|Fan potential of municipalities
Starting from the number of inhabitants of municipalities, the potential of fans for a club in a municipality can be
assessed by assuming a fixed distance threshold that these potential football fans would be willing to travel:
Workflow Task 1 ”What is the potential amount of football fans with a certain travel distance for each municipality in the
Netherlands?”
Task Specification 1 input: (1) ROADS08, (2) POP_2003
goal types: OE (ObjectQ, RegionA, ERA)
use type: OIO* (MatrixQ, IRA)
For this task, we start with the roads file and the population data on municipalities, and the goal is to assess some object
based extensive measure (the amount of football fans reachable at some travel distance from a given municipality). To
account for the concept of travel distances, we request in addition that some intensive matrix be used in the solution.
Alternatively, we can measure a minimum travel distance to reach a threshold amount of fans that a football club
can attract:
Workflow Task 2 ”What is the potential minimal travel distance to reach a certain amount of football fans for each munic-
ipality in the Netherlands?”
Task Specification 2 input: (1) ROADS08, (2) POP_2003
goal types: OI (ObjectQ, RegionA, IRA)
use types: OIO* (MatrixQ, RegionA, IRA)
Starting again with the roads file and the population data, our goals is here to estimate some object based intensive
measure (minimal travel distance of some amount of fans). For the same reason as above, we require that some
distance matrix be used in the solution.
Both kinds of analysis result in a map that shows a potential for each municipality. Fig. 1a shows the map for Task
2, in which all high ranking (red) municipalities are covered by at least one actual stadium. This supports the validity
of the chosen potential measure.
|Accessibility of football clubs from municipalities
In this task, we are interested in finding out how accessible football clubs are for each municipality:
Workflow Task 3 ”What is the accessibility of football clubs for each municipality in the Netherlands?”
Task Specification 3 input: (1) ROADS08, (2) MUNCLUB
goal types: OI (ObjectQ, RegionA, IRA)
use types: OIO* (MatrixQ, RegionA, IRA)

29
Here we start with plain municipalities (including some nominal attribute) and roads, to assess some object-based
intensive measure (the accessibility of football clubs). Since accessibility implies distance measurements, we likewise
require that some distance matrix be used in the solution.
|Accessibility of football clubs from roads
Roads and intersections are the objects that constitute a road network. Here we determine the distance between
each road and its closest football club.
Workflow Task 4 ”What is the accessibility of football clubs from each road?”
Task Specification 4 input: (1) ROADS08, (2) MUNCLUB
goal types: OI (ObjectQ, LineA, IRA)
use types: OIO (NetworkQ, LineA, IRA)
Using roads and clubs as input, we request some object-based intensive measure on lines, representing road objects.
We require distances measured on some line network to account for the concept of accessibility from roads (Fig. 10).
FIGURE 10 The map shows for each road segment the travel time to the closest football club, from more than
half an hour in yellow, to half an hour in red, to less than five minutes in purple. The black lines indicate the shortest
path from each municipality to its closest football club.
|Interaction based network analysis
|Amounts and flows of football fans
Flows in a matrix are shown in terms of the thickness of connecting lines in Fig. 11a. Starting from an interaction
matrix, a simple transformation is needed in order to assess how many football fans originate in each municipality.

30
(a) Amount of flow of ticket holders, displayed as desire
lines.
(b) Amounts of ticket holders per municipality, displayed
as bar chart.
FIGURE 11 Amounts and flows of football fans in the Netherlands.
Workflow Task 5 ”What is the amount of season ticket holders per municipality in the Netherlands?”
Task Specification 5 input: (1) STICKET2
goal types: OE (ObjectQ, RegionA, ERA)
In this task, we start from the interaction table with ticketholders and request some extensive object-based measure
(amount of ticket holders). To generate the map in Fig.11b, we need to summarize the interaction table over one of
its keys. This shows that season ticket holders seem concentrated around existing football clubs.
|Functional distance clustering of football clubs
Workflow Task 6 ”To which functional cluster does a football club belong?”
Task Specification 6 input: (1) STICKET2, (2) MUNCLUB
goal types: OS, ON (ObjectQ, RegionA, PlainNominalA)
Here, we start again with the interaction table. Together with the plain municipality data, it should be used to derive
some nominal attribute for objects (cluster labels for municipalities).
The dendrogram in Fig. 12a shows the progress of the fusion process as residential municipalities are merged
with football clubs. The map in Fig. 12b shows, in purple, the fusion lines representing a merge between a residential
municipality and a football club. The blue fusion lines indicate the first 24 merges between football clubs resulting in

31
(a) Functional distance clusters. (b) Functional distance clusters in map.
FIGURE 12 Functional distance based clustering.
14 (sub-)regional clusters; After this stage over 86.6% of all ticket holders are internalized in one of the clusters. The
red fusion lines indicate the next 9 steps after which five clusters at the national level remain.
|Football trip length distribution and trip end ranking
Workflow Task 7 ”What is the average travel time to/rank of their football club in case all ticket holders would travel by
car?”
Task Specification 7 input: (1) ROADS08, (2) MUNCLUB, (3) STICKET2,
goal types: I (IRA)
In this task, we use roads, municipalities and some interaction table to assess some intensive measure, namely
the mean travel time. Alternatively, one can also compare a given trip with potentially closer trip alternatives, by
ranking destinations (football clubs) for each given origin (municipality) with respect to the closest destination. When
we weight this rank by the amount of interaction and average it over all flows, we obtain an average rank number.
This is called Trip End Ranking. In this example, it shows that out of a choice of 37 clubs the average season ticket
holder chooses the 2.5 closest club.
|Trade area analysis of football clubs
Workflow Task 8 ”What is the area enclosing 60% of the amount of football ticket holders (trade area) closest to each
football club in the Netherlands?”
Task Specification 8 input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2

32
goal types: OS, ON (ObjectQ, RegionA, PlainNominalA)
use types: OEO* (MatrixQ, ERA), OIO* (MatrixQ, IRA)
In this task, we are requesting some object based region enclosing a given amount of the closest ticket holders. The
term "closest" in this task implies some distance matrix be used, and the "amount" implies some extensive matrix
between clubs and municipalities. We are interested in the size and the extent of overlapof these trade areas (Fig. 13a).
It can be seen that the big three clubs (Ajax Amsterdam, Feyenoord Rotterdam and PSV Eindhoven) fully dominate
their neighbours.
|Flow generation
|Gravity model of football fan interaction
Workflow Task 9 ”What is the potential amount of season tickets to be sold in each municipality for each football club in
the Netherlands, in case some form of distance decay is assumed?”
Task Specification 9 input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OEO* (MatrixQ, RegionA, ERA)
Given some road data, some ticket interaction data and some municipality/club data, we are interested in predicting
an extensive matrix, denoting the amounts of tickets sold for a municipality and some club.
Workflow Task 10 ”What is the attractiveness of football clubs for season ticket holders?”
Task Specification 10 input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OI (ObjectQ, RegionA, IRA)
In this task, our goal is to assess some intensive object-based measure (attractiveness of clubs) using the same input.
Workflow Task 11 ”What is the potential amount of season ticket holders for remaining football clubs, when the same
distance decay effect and the same attractiveness for the remaining clubs are assumed as before closure?”
Task Specification 11 input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2, (4) NEWCLUBS
goal types: OE (ObjectQ, RegionA, ERA)
In this task, we use some hypothetical club attractiveness together with roads and other data to obtain some object-
based amounts (ticket holders for each club).
Task Specification 11b input: (1) MUNCLUB, (2) ROADS08, (3) STICKET2
goal types: OE (ObjectQ, RegionA, ERA)
use types: OI (ObjectQ, RegionA, IRA)
Note that in 11b we leave out the fourth input, but additionally require some intermediate step that generates this
input.

33
(a) Trade areas of football clubs: Convex hulls around
origin municipalities that demarcate the closest 60 % of
interactions with destinations.
(b) Flows of football ticket holders assigned to street
network. Amounts designated by line width and color
hue (red: 3.000 - 49.000 ticket holders).
FIGURE 13 Trade areas of football clubs and flow assignments of football fan trips in the Netherlands.
|Traffic load analysis for football trip flow
Workflow Task 12 ”What would be the traffic load for each road of the Netherlands, if assuming all season ticket holders
would travel by car at the same time?”
Task Specification 12 Traffic load for each road in a network assuming shortest paths:
input: (1) MUNCLUB, (2) STICKET2, (3) ROADS08
goal types: OEO (NetworkQ, LineA, ERA)
In this task, we start with the interaction table, the municipalities and the roads to estimate some extensive (flow)
measure on these roads. Based on flow assignment, we find that the traffic load caused by season ticket holders may
run up to almost 49,000 on a single road segment in the vicinity of the most popular football clubs (Fig 13b).

34
Appendix B
FIGURE 14 Example of an expert solution for task 11b automatically generated with the CCD model in APE.
Here we model potential amounts of ticket holders for each football club in a scenario where some clubs are closing.
We start from a roads file (2), municipalities and clubs given as an object tessellation (1), and some ticket interaction
table between these objects (3). Attractiveness scores for clubs (used to obtain amounts in the last step applying a
singly constrained gravity model) are generated on the fly using some doubly constrained gravity model.

35
FIGURE 15 Example of a semantic error produced by the CCD model for task 10. The task is to generate
attractiveness scores for clubs, based on some municipality/club tessellation (1), some roads file (2) and some
interaction table (3). The problem is that threshold distances are not attractiveness scores, and that the task
specification misses semantic detail to prevent this confusion.

36
FIGURE 16 Example of a syntax error produced by the geometric benchmark model for task 7. The distance
matrix function needs a topological network as data input, but it is given a roads file, resulting in a syntax error.
FIGURE 17 Example of a redundancy error produced by the CCD model for task 7. The workflow produces a
correct result of trip length analysis from roads (1), municipalities (2) and an interaction file (3), but an unnecessary
functional clustering step is added.