Contents lists available at ScienceDirect
Transportation Research Part E
journal homepage: www.elsevier.com/locate/tre
Topological data analysis for aviation applications
Max Z. Li
a,
⁎
, Megan S. Ryerson
b,c
, Hamsa Balakrishnan
a
a
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA
b
Department of City and Regional Planning, University of Pennsylvania, Philadelphia, PA, USA
c
Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA
ARTICLE INFO
Keywords:
Aviation data
Topological data analysis
Persistent homology
Airline networks
ABSTRACT
Aviation data sets are increasingly high-dimensional and sparse. Consequently, the underlying
features and interactions are not easily uncovered by traditional data analysis methods. Recent
advancements in applied mathematics introduce topological methods, offering a new approach to
obtain these features. This paper applies the fundamental notions underlying topological data
analysis and persistent homology (TDA/PH) to aviation data analytics. We review past aviation
research that leverage topological methods, and present a new computational case study ex-
ploringthetopologyofairportsurfaceconnectivity.Ineachcase,weconnectabstracttopological
features with real-world processes in aviation, and highlight potential operational and man-
agerial insights.
1. Introduction
The sustained growth of air traffic worldwide has been accompanied by an influx of aviation data (Li and Ryerson, 2019). The
1300commercialairlinesandtheirnearly32,000in-serviceaircraftoperated41.9millionflightsbetweenmorethan3700airportsin
2017, transporting 4.1 billion passengers across 45,000 routes (Air Transport Action Group, 2018). Each of these components of the
air transportation system is also a source of data. The interconnectedness of the system is reflected in its data: Embedded sensors
onboard individual aircraft not only report data pertaining to that specific aircraft, but also keep track of its interactions with other
assets in the air and on the surface. It is in this context that the “Flight 4.0” era of smart and connected technologies has emerged,
couplingtraditionalaviationparadigmswithconceptssuchas theInternetofThings(IoT)andcyber-physicalsystems(Durak,2018).
The application of these concepts to aviation has already produced promising results (Mott and Bullock, 2018; Chatterjee et al.,
2017).
While the ever-growing quantity of data is, barring computational tractability,generally welcomed, the increasing complexity of
the data poses a challenge (Cook et al., 2015). The complexity of aviation data reflects the interdependencies within the air trans-
portation network. We provide a more in-depth review and discussion of big data in aviation in Section 1.1. Additionally, many
aviation data sets are high-dimensional and sparse; graph-based models that only encode pairwise relationships do not have the
flexibility to capture higher-order relational information. We review elements of complexity within the aviation system, as well as
previous research investigating network-level modeling in aviation in Section 2.
Given the prevalence of such data sets in aviation that resist accurate representation and analysis via classical statistical, graph-
theoretic, and other data analysis methods, we propose switching to a global, topologically-driven perspective to extract pertinent
features. Recent advances in applied mathematics introduce topological methods – most prominently topological data analysis and
https://doi.org/10.1016/j.tre.2019.05.017
Received 30 December 2018; Received in revised form 22 May 2019; Accepted 30 May 2019
⁎
Corresponding author.
E-mail addresses: [email protected] (M.Z. Li), [email protected] (M.S. Ryerson), [email protected] (H. Balakrishnan).
Transportation Research Part E 128 (2019) 149–174
Available online 13 June 2019
1366-5545/ © 2019 Elsevier Ltd. All rights reserved.
T
persistent homology (TDA/PH) – that offer a compelling new direction from which these salient features may be obtained (Carlsson,
2009;Ghrist,2014).TDA/PHusesresultsfromalgebraictopologytoextractinformationsuchasclusters,holes,voids,andcomplexes
from high-dimensional data sets; these topological methods have already found a diverse range of data science applications, a
sampling of which we review in Section 2.
1.1. Aviation and big data
We briefly discuss some key characteristics of aeronautics-specific big data to provide context (Burmester et al., 2018). The first
aspect is volume, characterized by the increasing quantities of interdependent data, rendering traditional data analysis methods
inadequateinsomesettings.Asingledayofair trafficoperationswithintheUSNationalAirspaceSystem(NAS)generates (1)
O (10 )
4
reports pertaining to arrival and departure information, boundary crossing updates, oceanic reports, planed positions, and actual
positions;(2)
O (10 )
5
recordsofflightmanagementinformation,flightplan-specificdata,aircraftspecifications,andflightroutedata;
and (3) more than
O (10 )
6
observations related to flight tracks, airways, centers, fixes, sectors, and waypoints (Comitz et al., 2013).
The interdependencies within and interconnectedness of data elements also present scenarios in which topological methods may be
useful. Furthermore, various inconsistencies, errors, and sparsities that are present in the data result in high levels of variability
(Burmester et al., 2018). Due to the inherent ability of TDA/PH to search for global topological invariants within a data set, the
methodswediscussinSection4are robustagainstnoisewithinaviation datasets.Inasense,theoverallshapeofthedatashouldnot
change in a dramatic way if slightly perturbed, either by introducing noise or by inducing sparsities (Ghrist, 2014).
Fig. 1 illustrates that even in thelimited example of convective impacts on the NAS, the complexity of the system is evident. The
forward layers in Fig. 1 that describe various NAS airspace partitions are highly organized and structured, while the last layer
containingrawmeteorologicaldata and weather radarreturnsis comparatively less structured,butwithhigher data dimensions and
quantities.
TakingthelayersdescribedinFig.1intoaccountwithotherfeatureswithintheaviationnetwork(airportsasnodes,airroutesas
edges, origin-destination delays, ground stops, Ground Delay Programs, Airspace Flow Programs, etc.), a highly complex and high-
dimensional network begins to surface. It is often important to keep track of many tuples of relationships between airports, airlines,
and other aviation stakeholders, not just pairwise relations. A graph-theoretic analysis of this network is limited to dyadic re-
lationships.Bycontrast,toolsbasedonTDA/PHenableananalysisoftheoveralltopologyofNASdatasets.Wewillmotivatethecase
for introducing TDA/PH as a complementary data analysis tool in aviation data analytics in the succeeding sections.
1.2. Manuscript outline
We provide an encompassing literature review of network-level models and complexity science in aviation, as well as successful
applications of TDA/PH, in Section 2. We highlight specific contributions of our work in Section 3, then present a mathematically
rigorousbutconcreteintroductionto TDA/PH anditsalgebraic-topologicalfoundationsinSection4.Tomotivatetheapplicability of
TDA/PH in aviation, we conduct an in-depth review of three aviation research works with a topological flavor in Section 5, before
presenting our novel computational case study where we apply TDA/PH to airport surface operations in Section 6. We holistically
round out our work with a detailed discussion of potential operational and managerial insights derivable from using TDA/PH in
aviation data analytics (Section 7). To encourage the continued exploration of TDA/PH in aviation, we provide an array of future
research directions in Section 8. We summarize the main contributions of our work and our vision for TDA/PH in aviation data
science in Section 9.
2. Literature review
In a comprehensive look at applying complexity science to air transportation, Cook et al. (2015) explores how high-dimensional
Fig. 1. ThearchitecturallayersoftheNASwith respecttoconvectiveweather;eachlayerprovidesitsownsetofdatathatarecloselyrelatedtodata
from another layer.
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
150
and sparse data sets, along with operational, equipment, and weather uncertainty lead to tremendous challenges in modeling,
predicting, and improving the performance of the air transportation system. Examples of complexity factors throughout the air
transportationsystemincludenumberofaircraft,aircraftproximitymeasures,anddensityindicators(Djokicetal.,2010).Theability
to extract insights and patterns from high-dimensional and complex data sets – both in aviation and other industries – has the
potential to inform the creation of better models and predictive analytics (Govindan et al., 2018; Oracle, 2012; Ayasdi, 2015;
FlightGlobal, 2019). Furthermore, the specific adoption of TDA/PH by several data science industries has already yielded novel
insights in areas such as healthcare and retail consumer behavior (Ayasdi, 2015).
Previousresearchfocusesonabroaderviewoftheairtransportationsystem,choosingtoexaminetheentiretyofagivenairroute
networkusing network science (Zanin and Lillo, 2013). Through such a wide scope of analysis, insights such as the varying levels of
dynamic complexity and how that variability plays into delay and fuel inefficiencies have been uncovered (Simić and Babić, 2015;
Rocha, 2017). These insights help address another facet of airspace complexity: the impacts of different stakeholders – airport
operators, airlines, air navigation service providers (Schaar and Sherry, 2010) – on the air transportation network as a whole. Each
stakeholder has a different perspective of the system, resulting in a different valuation of system utility and performance efficiency
(Kotegawa et al., 2014). While these insights are critical quantitative foundations upon which routing decisions and airspace utili-
zationshouldbemadeatthestakeholderlevel,itmayalsobe necessary to infer global features and characteristics regardingvarious
aviation subsystems from local data.
More recently, in line with the increasing ubiquity of big data in aviation, aviation data science research has shifted towards
dealing with high-dimensional and multifaceted data sets. New predictive models in aviation using methods such as deep belief
networks (DBNs) harness such complex data sets to estimate flight delays (Yu et al., 2019); however, such methods cannot identify
nor explicitly discern triadic and higher-order relationships, a concept that naturally arises in TDA/PH. Topological motifs analyzed
inDuetal. (2018) offeraproxyofhigher-orderinteractionsbetweenairportdelays, but ultimatelystillreliesonunderlyingpairwise
Granger causality tests. Other previous works use analogues of hypergraphs via a multi-layered network representations in order to
approximatehigher-orderrelationshipsbetweenairports(Belkouraetal.,2016;Duetal.,2016);however,eachindividualnetworkis
still restricted to node-edge airport pairs, and is not flexible enough to admit multi-airport interactions. TDA/PH utilize so-called
simplicial complex representations, and is not constrained to pairwise relationships, as we will see in Section 4.
TDA/PHleveragetools fromalgebraictopologytoprovidetopologicallyqualitativeandglobalinformationaboutadatasetusing
local information, while also naturally allowing for higher-order relationships. While this subject is slowly transitioning from pure
mathematics to applications in engineering, the usage of topological structures such as networks and graphs in air transportation
research is much more mature. In the scope of air route networks, graph-based approaches have been used to characterize the
complexity of airspace sectors (Hongyong et al., 2015), to quantify the connectivity and robustness of the route network structure
(Weiet al., 2014; Zhou et al., 2019), and to find correlations between stakeholder metrics and the network topology ofthe air route
network (Kotegawa et al., 2014). In addition, Yousefi and Zadeh (2013) applied methods in fluid flow dynamics to characterize en
route airspace corridors, resulting in a quantitative benefit assessment of new Area Naviation (RNAV)-enabled “Q” jet routes.
TDA/PH have already found a diverse range of applications in engineering, particularly in problems involving large, high-di-
mensional data sets with non-trivial degrees of connectivity and loops. These topological methods have opened new avenues in
networkedneuroscience(Giustietal.,2016),aswellasinpath-planningforautonomousrobots(Bhattacharyaetal.,2013,2015)and
modelingcoverageandexplorationbehaviorsforteamsofmultiplerobots(Bhattacharyaetal.,2014).Topologicalmethodshavealso
been used to extract persistent features within fluid flow (Kasten et al., 2011), identify pertinent structures within LiDaR data sets
(Keller et al., 2011), and characterize the complex connections and networks within the airways of the human lung (Szymczak,
2011).
3. Contribution of work
Ourwork in this paper embraces the fact that aviation,like manyother fields, is entering the era of big data. Such an abundance
of high-dimensional trajectory data lends itself well to augmentation routines and topologically-qualitative methods from the
emerging toolbox of TDA/PH. We have identified the need for complementary data analytics tools that offer insights into global,
topological features within complex and high-dimensional aviation data sets in Sections 1.1 and 2. To directly address this metho-
dologicalgap,thefirstmajorcontributionofourworkistoprovideaconsistent,formal,andconcreteintroductiontoTDA/PHwithin
the aviation domain. This is accomplished through Section 4, where we use an aviation-based example to build up the algebraic-
topological notions required for TDA/PH. Our second major contribution is bridging the connection between TDA/PH and potential
actionable operational and managerial insights within the air transportation system. We accomplish this through providing an in-
depthreviewofpreviousresearchwork(Section5),presentinganewcomputationalairportsurfacecasestudy(Section6),discussing
how TDA/PH yields insights directly relatable to physical processes and operations within the aviation domain (Section 7), and
providing a plethora of future research directions in aviation data analytics that leverage TDA/PH (Section 8).
We emphasize that TDA/PH does not replace, but instead complement existing topological and graphical methods employed in
aviationresearch. We surveyed a sampling of aviation research that employed network-based and graph-theoretic models in Section
2; the array of important and impactful results from Section 2 are based off of dyadic relationships. TDA/PH complement these
approaches by capturing triadic and higher-order relationships (Ramanathan et al., 2011; Giusti et al., 2016; Sizemore et al., 2018;
Xia, 2018). This indicates that one challenge regarding applying TDA/PH to aviation data sets is first identifying whether or not
characterizing pairwise relationships are enough. For example, in studies of causal interactions between airport delays, it may be
desirable to only examine pairwise airport interactions via Granger causality (Du et al., 2018); however, if the goal is instead to
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
151
capture hub-and-spoke dynamics with stronger forms of causality such as structural equation models (SEMs), then higher-order
relationships may be necessary. Another challenge is the need for aviation-specific subject matter expertise in mapping topological
results to real-world processes and phenomenon; we address this challenge directly in Section 7.
4. From simplices to holes: fundamental concepts from algebraic topology
TDA/PH are built up from a collection of fundamental ideas, definitions, and concepts borrowed from algebraic topology. We
introduce some fundamentals in this section, now that we are equipped with the contextual understanding of the types of aviation
datasetsthatmaybewell-suitedfor TDA/PH.Ahigh-levelexplanationofalgebraictopologyisthatitisarigorouswayofcomparing
spaces, and keeping track of non-trivial topological features within these spaces. In other words, the goal is to find holes in spaces.
Imagine two spaces where one is a disc, and another is a disc with a hole punctured in the middle of it. In the first space (the whole
disc), if we envision moving a rubber band around in the space, there are no restrictions to how we move this rubber band. In other
words, there is only one configuration for this rubber band. However, in the second space (the punctured disc), there are two
configurations for this rubber band: When the rubber band surrounds the punctured hole, and when the rubber band does not. The
rubber band cannot transition between the two configurations without breaking, due to the hole in the space of the punctured disc.
Algebraic topology provides a framework drawn from abstract algebra and topology for this natural intuition of holes within
spaces. TDA/PH provide the bridge between data and the kinds of topological spaces that can be examined through concepts from
algebraic topology. We build up to a rigorous definition of holes and higher-dimensional holes (Definition 7) in this section, cul-
minatingin an algebraic structure endowed with the ability to keep track of holes between spaces (homology groups; Definition 13)
that formalizes our rubber band example. With an understanding of the various topological and algebraic foundations, the natural
question then is how these abstract notions translate to aviation data science applications (Sections 5 and 6) and specifically how
topological features such as holes map to physical processes and phenomena within the aviation domain (Section 7), translating to
potential operational and managerial insights.
In this section, we present the background required to apply TDA/PH methods. Using an illustrative example, we develop the
algebraic-topologicalconstructknownas a simplicial complex,anddefinevariousalgebraicobjectsonthesimplicialcomplex.Finally,
we define homology groups, whose sizes inform us whether or not there are holes in our dataset. For a deeper discussion of algebraic
topology and TDA/PH, we recommend other references (Hatcher, 2002; Edelsbrunner and Harer, 2009; Ghrist, 2014).
4.1. Simplicial complexes
We begin our exploration of TDA/PH by considering a particular way of structuring datathat has its roots in algebraic topology.
Table 1 shows a sample data set containing a collection of major airports within the US, as well as a list of associated features. The
airports are Chicago O’Hare International Airport (ORD), Chicago Midway International Airport (MDW), New York-LaGuardia Air-
port(LGA),NewarkLibertyInternationalAirport(EWR),SanFranciscoInternationalAirport (SFO), and Miami International Airport
(MIA). The features need not share contextual commonalities: information regarding metroplexes are geographic in nature, hub
characteristicsareairline-dependent,andGroundDelayPrograms(GDPs)aretacticalfeaturesrelatedtoairtrafficflowmanagement
(ATFM).Itwould be difficulttocapture the relationships andstructure in thisdatawith a graphthatmodels only pairwiserelations.
Using the vertex labels for each airport given by the leftmost column in Table 1, we define the set of vertices
v v v v v v{ , , , , , }
0 1 2 3 4 5
along with a superset containing sets of vertex relations:
Table 1
Example data set containing airports along with the associated features. UAL and AAL denote United Airlines and American Airlines; Departure
delays, Miles-In-Trail (MIT) and Minutes-In-Trail (MINIT), Flow Constrained Areas (FCA), routing flow constraints (e.g., wind routes), and Ground
Delay Programs (GDPs) are all traffic management initiative (TMI) features.
Vertex Airport Features
v
0
ORD Chicago UAL hub AAL hub FCA TMI Wind route
metroplex TMI
v
1
MDW Chicago Secondary
metroplex airport
v
2
LGA New York Secondary
metroplex airport
v
3
EWR New York UAL hub Departure MINIT Wind route MIT TMI
metroplex delays TMI TMI
SFO Ongoing UAL hub Departure FCA TMI MIT TMI
GDP delays
MIA Ongoing AAL hub Departure MINIT
GDP delays TMI
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=
v v v v v v
v v v v v v v v v v v v v v v v v v
v v v v v v
{sets of vertices with same features}
{ }, { }, { }, { }, { }, { }
{ , }, { , }, { , }, { , }, { , }, { , }, { , }, { , }, { , }
{ , , }, { , , }
.
0 1 2 3 4 5
0 1 2 3 1 2 4 5 0 5 3 4 3 5 0 4 0 3
0 3 4 3 4 5
(1)
Theelementsof
areinterpretedusing the shared features among the vertices,asshown in Table1.Forexample, an element in
containingonevertexcorrespondstothefeatureofthevertex
v
i
representingacertainairport(e.g.,
v{ }
0
isassociatedwiththe
featureof representingORD),anelementin
containingtwoverticesindicatefeaturessharedbyanairportpair(e.g.,
v v{ , }
0 1
is
associated with the feature that both ORD and MDW are in the Chicago metroplex), and so on. The algebraic structure provided by
{ , }
is known as a simplicial complex. Formally, an (abstract) simplicial complex is defined in the following manner in terms of
and :
Definition 1. An (abstract) simplicial complex, given a set of vertices
and superset containing sets of vertex relations, is an
algebraic structure such that if
and .
Alternate ways of forming simplicial complexes from data include flag complexes and Vietoris-Rips complexes (Sizemore et al.,
2018). We provide an illustrative example in Fig. 2 of additional simplicial complexes that could be formed from different types of
aviation domain data.
We can decompose our simplicial complex into smaller simplices, and begin visualizing it from a geometric perspective. To this
end, we consider a geometrically-inspired definition of a k-simplex (Hatcher, 2002):
Definition 2. A k-simplex, denoted as
k
, is the convex hull of
+k 1
affine (non-collinear) points. The standard k-simplex can be
defined explicitly as:
= … =
+
t t t t i{( , , ) 1, 0, }.
k
k
k
i
i i0
1
(2)
Weprovideanillustrationofthestandard2-simplex
2
inFig.3.Inordertobeginvisualizingthesimplicialcomplexgiven by the
datainTable1,weneedtodiscussthe conventionregardingstandard orientations on thek-simplices.Fig.4providesanillustrationof
k-simplices for
=k 0, 1, 2
, and 3. The edges connecting each vertex in the k-simplices for
k 1
are oriented with respect to the
vertexindices.Forexample, given a 1-simplex
1
with
v v{ , }
i j
and
<i j
,thestandardorientationpointsfrom
v
i
to
v
j
.Thisextends
tothelabelingsonthe2-and3-simplexaswellinFig.4.Wecan formalize the notion of a standard orientation on a k-simplex by the
following definition:
Definition 3. Supposewehaveak-simplex
… k0, ,
withtheindiceswritteninastrictly-ascendingorder,i.e.
< < k0
.Thisorderingis
the standard orientation on a k-simplex, and can be visualized via the decomposition
< < … <k k0 1, 1 2, , 1
consisting of 1-
simplices
…, , ,
k0,1 1,2 1
wheretheorientededgefor
i j,
exitsthe0-simplex
v
i
andentersthe0-simplex
v
j
,giventhat
= +j i 1
and
Fig. 2. Example of multi-layered simplicial complexes corresponding to different types of aviation data.
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
153
… × …i j k k( , ) {0, , } {0, , }
.
Another intuitive relationship between a k- and a
±k( 1)
-simplex for
k 1
is the notion of boundary decompositions. Referring
back to Fig. 4, k-simplices for
k 1
compose the boundary of the
+k( 1)
-simplex, and can be decomposed into its own boundary
formed from
k( 1)
-simplices. This is a notion that we will formalize and exploit in our search for holes in our data.
We map our algebraic simplicial complex
{ , }
to a topological simplicial complex. Let
X
NAS
, referred to as the NAS simplicial
complex,denotethetopological space thatisthesimplicialcomplexcreatedfromthedatainTable1.Weprovideanillustrationof
X
NAS
in Fig. 5. An issue of notation must also be addressed here as well, since we can view k-simplices and simplicial complexes as
algebraicortopologicalobjects.Ak-simplexcanbedenotedas
k
or
… k0, ,
withtheusualordering
< < k0
;theformercarriesa
topological interpretation whereas the latter carries an algebraic one. For the purposes of introducing algebraic topology and
Fig. 3. The standard 2-simplex, denoted as
2
. Note that
x x x( , , )
1 2 3
spans
3
.
Fig. 4. Illustration of a 0-simplex, 1-simplex, 2-simplex, and 3-simplex. The edges are marked according to standard orientation conventions.
Fig. 5. Our simplicial complex
X
NAS
visualized using 0-, 1-, and 2-simplices. It is the topological counterpart of our algebraic structure
{ , }
derived from the NAS data in Table 1.
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Transportation Research Part E 128 (2019) 149–174
154
homology to understand the basics of TDA/PH, we will use both notations depending on the context.
In the case of our simplicial complex
X
NAS
in Fig. 5, each black 0-simplex
i
corresponds to the element
v{ }
i
; each orange 1-
simplex
<i j,
i j,
(recall standard orientation conventions) corresponds to the element
v v{ , }
i j
; and each purple, filled-in 2-
simplex
i j k, ,
with
< <i j k
correspondsto theelement
v v v{ , , }
i j k
.Forexample,thepurpletriangledenotingthe2-simplex
0,3,4
formedby
v v v{ , , }
0 3 4
encodesthesharedfeatureamongstORD, EWR, and SFO ofbeingUALhubs.Furthermore,wecandefine a
substructure on
X
NAS
called a k-skeleton, typically denoted as
X
k
. A k-skeleton is the collection of all k-simplices within a larger
simplicial complex.
4.2. Algebra with simplicial complexes: chains and chain groups
We will now introduce a linear-algebraic abstraction that allows us to perform algebra on
X
NAS
with surprisingly intuitive vi-
sualizations. First, we formalize the idea of chains within simplicial complexes such as
X
NAS
with the definition of chain groups:
Definition 4. Given a simplicial complex
{ , }
, the 1
st
chain group
1
is the group:
= a a X
{
, derived from { , }
}
.
n i j n i j1 , ,
1
(3)
Elements of
1
are known as 1-chains. Note that from the definition of the 1
st
chain group
1
, 1-chains are simply linear
combinationsof1-simplices
i j,
withscalarintegercoefficients
a
n
.Thenotionofchaingroupscanbeextendedtoalldimensions
naturally:
=
… …
a a X
{
, derived from { , }
}
.
k n k n k
k
0, , 0, ,
(4)
Analogously, elements of
k
are known as k-chains. Note that the summation in the definitions for the 1
st
and k
th
chain group is
formally the operator
+ ×:
k k k
k
that sums together two k-simplices. For example, for the 1
st
chain group, we have that
+
i j k l, ,
1
resultsinthenew1-simplex
+
i j k l, , 1
.Weomitthesubscriptonthesummationoperatorwhenthechaingroupwe
are working over is obvious. Furthermore, recalling the standard orientation on a k-simplex, any k-chains with negative coefficients
correspond to the same k-chain but with reversed entry-exit orders.
Whilethedefinitionfor
k
maybereminiscentofavectorspacespannedbyk-simplices,itisactuallyamoregeneralizedalgebraic
structure known as a group.
4.3. Paths and holes
We can now visualize 1-chains in
1
in the context of the simplicial complex
X
NAS
. If we were to pick the 1-chain
+ + +
0,1 1,2 2,3 3,5 4,5 1
,payingattentionto the signsofthecoefficientsofeach1-simplex
i j,
,weseethat this isthe1-chain
consisting of 1-simplices in the standard orientation connecting
v v v v, , ,
0 1 2 3
, and
v
5
plus the 1-simplex connecting
v
5
to
v
4
with its
orientation reversed by the negative coefficient on
4,5
. Besides being a 1-chain, it is also a path consisting of 1-simplices, as the
orientations on each 1-simplex agree consistently. Formally, we can define a path of k-simplices as:
Definition 5. A path consisting of k-simplices allows for a traversal with the same number of entrances and exits of each
k( 1)
-simplex along the k-chain, with the exception of the initiating and terminating
k( 1)
-simplexes. The initiating
k( 1)
-simplex allows for only one exit, and the terminating
k( 1)
-simplex allows for only one entrance.
Intuitively,we note a specialtypeof path that colloquially-speakingdoesnot have an endor a beginning. Formally,aclosed path
is defined as follows:
Definition 6. A closed path consisting of k-simplices allows for a traversal with the same number of entrances and exits of each
k( 1)
-simplex along the k-chain.
This 1-chain path
+ + +
0,1 1,2 2,3 3,5 4,5 1
can be visualized in Fig. 6 (left). While all paths consisting of 1-simplices are
also1-chains,notall1-chainsarepaths.Forexample,picktwonegativeintegers
a
1
and
a
2
,then
+a a
1 0,5 2 2,3
isavalid1-chaininour
NAS simplicial complex (Fig. 6 (right)), but not a valid path as the 1-simplices are disconnected.
The notion of paths can also be extended into higher dimensions using the same rules as in the case of 1-chains. In the NAS
simplicial complex, there are two 2-simplices given by
0,3,4
and
3,4,5
representing UAL hub features shared by ORD, EWR and SFO,
and departure delay features shared by EWR, SFO and MIA, respectively. The 2
nd
chain group
2
for the NAS simplicial complex is
spanned by these two 2-simplices. The orientations do not match on the two 2-simplices – this can be visualized via Fig. 7. Thus,
+
3,4,5 0,3,4
isavalid2-chain,butnota valid path consistingof2-simplicies.Modifyingthesignon,forexample,the2-simplex
0,3,4
,
gives us a different 2-chain that is a valid path (Fig. 8).
We now have the necessary background to formalize the process of finding topological holes within the simplicial complex
representing our NAS data set. In the context of TDA/PH, we use the following definition to distinguish between holes and other
features in our simplicial complex:
Definition 7. A k-dimensional hole (also known as a void or cavity) is a valid closed path composed of k-simplices within a simplicial
complex. Furthermore, this valid closed path cannot be the boundary of some
+k( 1)
-simplex.
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Suppose that we examine the following two paths composed of 1-simplices in our NAS simplicial complex:
+ +
0,1 1,2 2,3 0,3 1
and
0,4 3,4 0,3 1
. Both 1-chains are valid paths; they are indistinguishable until they are visua-
lized on our NAS simplicial complex (Fig. 9).
Thebrown path inFig. 9 depictingthe second1-chain does not qualify as a1-dimensional hole, becauseit is theboundary of the
2-simplex encoding for the UAL hub feature. On the other hand, the cyan path surrounds a 1-dimensional hole. We can verify that
therearetwo1-dimensionalholeswithinthis data set (thecurrentpresentationof
X
NAS
yieldstheansweroftwo 1-dimensionalholes
more intuitively in the equivalent presentation shown in Fig. 10). However, it would be difficult to visually locate all 1-dimensional
holes in a larger or richer data set, and impossible to do so for higher-dimensional holes. Given the definition for a k-dimensional
hole, we now formalize the notion of a boundary and of boundary operators. These notions are needed to understand the basics of
simplicial homology, an algebraic-topological construct that rigorously keeps track of holes across a given topological space.
4.4. Boundaries and boundary operators
From our example in Fig. 9, we note the need to define precisely a notion of boundaries as it relates to simplicial complexes.
Insteadofprovidinga definitionof aboundary,itwillbemorebeneficialtoinstead introducetheconceptofaboundary operator that
maps a k-simplex to its boundary. There are computational and methodological reasons for why we use boundary operators.
Fig. 6. Two 1-chains in the NAS simplicial complex, where the one on the left also forms a path, but the one on the right does not.
Fig. 7. The 2-chain
+
3,4,5 0,3,4 2
does not form a path in our NAS simplicial complex.
Fig. 8. The 2-chain
3,4,5 0,3,4 2
forms a path in our NAS simplicial complex.
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Computationally, boundary operators admit matrix representations (Section 4.5), allowing for computer algebra software manip-
ulations. Methodologically, they are important in the definition of homology groups in Section 4.6. The boundary of a k-simplex and
the boundary operator is defined as follows:
Definition 8. The boundary operator, denoted as
:
k k 1
with
>k 0
, takes a k-simplex
k
and decomposes it into alternating
sumsof
k( 1)
-simplices.This alternating sum of
k( 1)
-simplices is the boundary of the original k-simplex. The boundary operator
maps explicitly by (5).
… … = … …
=
v v v v v v v, , , , ( 1) , , , , .
k k
i
k
i
i k0 0
0
0
(5)
The notation
v
i
or
i
(the v is sometimes omitted when it is clear that we are referring to vertices) indicates that vertex
v
i
has been
deleted.
The boundary operator
:
k k 1
formalizes the decomposition relations in Fig. 4. We illustrate a straightforward appli-
cationoftheboundaryoperatorona3-simplexinFig.11.Aquicknoteregardinglabelingconventions:thevertexlabelsonagivenk-
simplex can always be relabeled to match the canonical
… …v v v, , , ,
i k0
labeling scheme used in the definition of a boundary
operator, as long as orientation is preserved. For example, in our NAS simplicial complex, the vertices of the 2-simplex
0,3,4
can be
relabeled as
0,1,2
, and then the boundary operator can be explicitly applied to the relabeled 2-simplex. For consistency, the vertices
on the resultant boundary of 1-simplices can be “unlabeled” according to the original 2-simplex vertex labels.
Twoimportantpropertiesoftheboundaryoperatoristhat (1) the boundary operator isalinear operator,and(2)thecomposition
k k1
always maps to nullity.Bothpropertiesare surprisingly intuitive – the former indicates thattakingtheboundaryofmultiplek-
simplices should give their overall boundary, and the latter indicates that the boundary of a boundary does not exist. While both
properties can be proven generally by using the explicit definition of boundary operators given in (5), we will demonstrate the
intuitions behind both properties via examples on
X
NAS
. Recall the 2-chain in
X
NAS
from Fig. 8; the expected result of applying the
boundary operator on this 2-chain is the boundary consisting of the vertices
v v v, ,
0 4 5
, and
v
3
. From the linearity of boundary
operators, we have:
=( ) ( ) ( ).
2 3,4,5 0,3,4 2 3,4,5 2 0,3,4
(6)
The boundary operator
k
written with an index of
=k 2
denotes that this is the boundary operator that sends a 2-simplex to its
boundaryconsisting of 1-simplices.Recall that we can relabel vertices inorder to directly use theexplicit definition of the boundary
operator given in (5). Specifically, for
3,4,5
, we relabel
{3, 4, 5} {0, 1, 2}
, and for
0,3,4
, we relabel
{0, 3, 4} {0 , 1 , 2 }
. Con-
tinuing with the boundary calculation, we have:
Fig. 9. Visualizationofthetwovalidpaths
+ +
0,1 1,2 2,3 0,3 1
and
0,4 3,4 0,3 1
.Intuitively,onlythecyanpathontheleftsurrounds
a 1-dimensional hole.
Fig. 10.
X
NAS
re-drawn to emphasize the existence of two 1-dimensional holes.
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= + + + +
= + +
= + +
= +
= +
( ) ( ) (( 1) ( 1) ( 1) ) (( 1) ( 1) ( 1) )
( ) ( )
( ) ( )
.
2 3,4,5 2 0,3,4
0
1,2
1
0,2
2
0,1
relabeled vertex labeling for
0
1 ,2
1
0 ,2
2
0 ,1
relabeled vertex labeling for
1,2 0,2 0,1 1 ,2 0 ,2 0 ,1
4,5 3,5 3,4 3,4 0,4 0,3
un-labeled vertices
4,5 3,5 0,4 0,3
0,4 4,5 3,5 0,3 1
3,4,5 0,3,4
(7)
“”The resultant 1-chain
+
0,4 4,5 3,5 0,3 1
is visualized as the green path in Fig. 12. To demonstrate the second property of
boundary operators, we apply the next boundary operator
1
to the resultant 1-chain
+
0,4 4,5 3,5 0,3 1
:
Fig. 11. An application of the boundary operator on a 3-simplex.
Fig. 12. Linearity of boundary operators demonstrated with the computation of a 2-chain’s boundary.
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= +
= +
=
( ( )) ( )
( ) ( ) ( ) ( )
0.
1 2 3,4,5 0,3,4 1 0,4 4,5 3,5 0,3
4 0 5 4 5 3 3 0
(8)
Thus, compositions of consecutive boundary operators map to nullity, i.e.,
:
k k k1
, formalizing the intuition that the
boundary of a boundary does not exist.
4.5. Matrix representation of boundary operators
While our illustrations of simplicial complexes and boundary operators are useful for purposes of introducing the algebraic-
topologicalfoundationsofTDA,softwareimplementationstypicallystoretheseconstructsasmatrices(Fasyetal.,2014;Baueretal.,
2017; Henselman et al., 2018). Any linear operator
V W:T
can be represented by a matrix with dimensions related to the
dimensions of the domain and codomain of
T
. More specifically, if
M
T
is the matrix representation of the linear operator
T
, then it
will have dimensions
×W Vdim ( ) dim ( )
. We can explicitly write down the matrix representation
D
2
of the boundary operator
:
2 2 1
that we applied to the example in Fig. 12 (colored text is used to annotate the columns and
rows):
The domain of
2
is
2
spanned by the two 2-simplices
0,3,4
and
3,4,5
, so we have that
=dim( ) 2
2
, and hence
D
2
must have two
columns.Thecodomainisthe1
st
chaingroup
1
associatedtotheNASsimplicialsubcomplexinFig.12,spannedbyfive1-simplices.
Thus,
D
2
must have five rows. The
( )
column of
D
2
constructs the 2-simplex
0,3,4
; since the standard orientation on
0,3,4
is
counterclockwise,thecoefficienton
0,3
and
3,4
ispositiveas it follows the orientation,and the coefficient on
0,4
isnegativesince it
is reversed with respect to the orientation. By the same construction, the
( )
column of
D
2
represents the other 2-simplex
3,4,5
.
Recall that the 2-chain we applied the boundary operator to was
3,4,5 0,3,4 2
. We can rewrite this in a vectorized form
:
Note that the row order on matches the column order on
D
2
. This is important because the vector representing the 2-chain
3,4,5 0,3,4
livesin thedomain ofthe boundaryoperator being represented by
D
2
.We cannow applythe boundary operator on our
2-chainasamatrixleft-multiplication,yieldingthe1-chainpathillustratedinFig.12(right).Allsimplicialcomplexes,subcomplexes,
k-simplices, and boundary operators can be stored in a convenient matrix form, thus enabling
computation.
We can now count the number of holes within the NAS simplicial complex, creating the mathematical construct that reflects the
intuitionfoundinFig.10.InSection 4.6,we introduce two new setsthatcanbenaturallydefinedonour NAS simplicial complex via
boundaryoperators.Informally,onesetcontainscandidatesthatcouldbeholes,with somefalsepositives(non-holes)thrownin.The
othersetcontainstheidentityofthefalsepositives.Thenaturalnextstepwouldbeto“divideout”thelatterfromtheformeruntilwe
are only left with the holes, which is the idea behind simplicial homology.
4.6. Simplicial homology
Recall, from Fig. 9, some of the differences between the two paths: One path surrounds a 1-dimensional hole, whereas the other
pathdoesnot.The secondpathfailedtosurroundaholebecause itistheboundaryofahighersimplex.We impose twoconditionsto
keepourexamplessimple:(1)Werestrictourselvestolookingfor1-dimensionalholeswithin
X
NAS
,and(2)weconsiderchain groups
that admit coefficients from the set of integers modulo 2 (that is, from the set
{0, 1}
2
). The second restriction greatly simplifies
thek-chainswewillexamine;insteadof integercoefficientsinourk-chain,alleven integers are mapped to 0 andallodd integersare
mapped to 1. Surprisingly, this choice of coefficients to work over does not affect TDA/PH. Specifically, this is due to the universal
coefficient theorem; while this theorem is integral to algebraic topology, a full understanding isnot needed for TDA/PH, and we refer
readers to any standard algebraic topology textbook (e.g. (Hatcher, 2002)) for a much more in-depth overview.
Under the previous
coefficients, a cancellation would occur if we had
= 0
i j i j, ,
. Now, under
2
coefficients, we have
+ = mod2 0 2
i j i j i j, , ,
.Notethatunder
2
coefficients,orientationsdonotmattersinceanyodd combinations of k-simplices map
to 1, and any even combinations of k-simplices map to 0. This simplification – and more generally the ability to flexibly pick
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coefficients to work over – has additional computational advantages (Fugacci et al., 2014; Bhattacharya et al., 2015).
4.6.1. Cycles and boundaries
Following the aforementioned line of thought regarding constructing a set of all possible hole candidates, we go back to our
redrawnNASsimplicialcomplexin Fig. 10. There arefourpossible candidates
, ,
1 2 3
,and
4
aspresentedinFig. 13. While itmay
betemptingto call these fourcandidates closed paths,notethat the orientationsonthe 1-simplices donot line up, althoughtheyare
certainlyclosedloops.The formalterminologyfortheseobjects
, ,
1 2 3
,and
4
are1-cycles,withk-cycles being higher-dimensional
equivalents:
Definition 9. A k-cycle isanycombinationofk-simplicesthat formavalidclosedpathwitha finitenumberoforientationexchanges.
The next observation is that any other 1-cycles can be found in the form
a
i
i i
with
a
i 2
. For example, to form the 1-cycle
that goes around the entire NAS simplicial complex (Fig. 14), we can examine the linear combination
+ +
1 3 4
:
+ + = + + + + + + + + +
= + + + + + + +
+ + + + +
( ) ( ) ( )
2 2
.
1 3 4 0,1 1,2 2,3 0,3 3,4 4,5 3,5 0,3 3,5 0,5
0,1 1,2 2,3 3,4 4,5 0,5 0,3 3,5
0,1 1,2 2,3 3,4 4,5 0,5
(9)
Notethatsince we areworking under
2
coefficients,the terms
2
0,3
and
2
3,5
vanishbecause the coefficientsare even. Theresultant
1-cycle
+ + + + +
0,1 1,2 2,3 3,4 4,5 0,5
isdepicted in blue on the left-hand sideof Fig. 14. We cannow define the following space
that contains all possible hole candidates:
Definition 10. The space of 1-cycles, denoted
, for our NAS simplicial complex is defined as follows:
Fig. 13. The four 1-cycles that exist in our NAS simplicial complex.
Fig. 14. Creating the 1-cycle
HEX
through linear combinations of
, ,
1 2 3
, and
4
under
2
coefficients.
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(10)
Sincetheboundaryoperator
1
wouldmapallfour1-cyclesthatspan totriviality(theboundaryofaboundarydoesnotexist),
we arrive at the following general definition of
:
Definition 11. The space of 1-cycles, denoted by
, is defined as follows:
(11)
Analogously, the space of k-cycles, denoted by , is defined as follows:
(12)
Recall that the kernel of some linear operator
V W:T
is the subset of V that contains all elements
v V
such that
=v W( ) 0T
. In this case, since
=( ) 0
i1
for all 1-cycles that span , the two sets
ker( )
1
and coincide.
If
contains all possible hole candidates, then we need to find an analogous space containing all false positives that cannot
possiblybe holes.Theappearanceofboundaryoperatorsinthedefinitionof
canhelp usidentifythesenon-holes,byapplyingthe
boundary operator
2
to
X
NAS
. We illustrate such an operation in Fig. 15.
Sincetheboundaryoperator
2
onlyaffects2-chainsin
2
,itsendsthose2-chainstotheirboundariescomposingof1-chains(more
specifically,1-cycles) in
1
.These are theelements that wewant to excludefrom .We can nowgive this space of false positives a
name, and a definition that hinges on the boundary operator
2
:
Definition 12. The space of 1-boundaries, denoted by
1
, is defined as follows:
= im( ).
1 2
(13)
Analogously, the space of k-boundaries, denoted by
k
, is defined as follows:
=
+
im( ).
k k 1
(14)
Thenotation
im( )
2
denotestheimageoftheboundaryoperator,whichisasubsetof1-cyclesin .Notethatsinceall1-
boundaries must be 1-cycles, but not all 1-cycles are 1-boundaries (the ones that are not are exactly the 1-dimensional holes we are
looking for), we must have that
. Analogously, we can always construct the space of k-cycles and the space of k-
boundaries
k
for
k 0
,andtheintuitionbehindtherelation remainsthesame.Furthermore,thespacesofk-cyclesand
k-boundaries are actually groups as well.
4.6.2. Homology groups and Betti numbers
Homology groups ariseastheformal results of “dividingout”theimposters(thek-boundaries)fromthepossiblecandidates(thek-
cycles). Note that since
and
k
are sets (more formally, groups) of all possible k-cycles and k-boundaries formed from linear
combinations of a finite number of cycles (these cycles can be thought of as being analogous to basis for vector spaces in linear
algebra), the notion of quotients carries through in this context by acting on the cycles themselves. In other words, shared cycles in
thebasisforboth
and
k
canbedividedoutiftheappropriatequotientistaken.Since ,onlyonesensiblequotientcan
be taken, and this quotient actually forms another group:
Definition 13. Thek
th
homology group foratopologicalspaceX,denotedas
X( )
k
,isthegroupobtainedfromthefollowingquotient:
(15)
or alternatively,
=
+
X( ) ker( )/im( ).
k k k 1
(16)
The dimension of
X( )
k
is exactly equal to the number of k-dimensional holes in X. This quantity is known as the k
th
Betti number,
denoted by
k
.
Thus, for our NAS simplicial complex, we can compute the 1
st
homology group as follows:
Fig. 15. The result of applying the boundary operator
:
2 2 1
to our NAS simplicial complex.
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(17)
The brackets notation is a shorthand for indicating that the space is spanned by linear combinations of elements within the
brackets with coefficients from
2
. Finally, the number of 1-dimensional holes, or alternatively, the 1
st
Betti number
1
is equal to
=Xdim( ( )) 2
NAS1
. Note that from the definition of what the 0
th
homology group should be, the 0
th
Betti number
0
measures the
number of connected components in the simplicial complex. Thus, for our NAS simplicial complex, we have that
= 1
0
.
4.7. Topological data analysis and persistent homology (TDA/PH)
The notions from algebraic topology discussed above serve as the starting foundations for TDA/PH methods. In general, TDA
focuses on transforming a given data set into complex-like structures such as
X
NAS
. Other more nuanced constructions of complexes
from data can also be used, as we will see in Sections 6 and 7. After transforming the data into a simplicial complex-like structure,
someTDAtoolsfocusonrunningsummarystatisticsonthestructureanditsvarioussubstructures.OtherTDAtoolsexploretheglobal
“shape” of the data trying to find topologically significant features, such as the k-dimensional holes. Another possible approach is
throughpersistenthomology(PH);thekeyideahereis to dynamically generate successionsofsimplicialcomplex-likestructuresand
to look for persistent topological features.
We now illustrate ways to bridge the gap between the abstract results stemming from TDA/PH to actual aviation applications by
reviewing three studies that leverage a topological analysis of aviation data (Section 5). To further emphasize the potential insights
gained from examining aviation data from a topological perspective, we present in Section 6 a computational case study examining
the topology of surface operations at five major US airports using so-called nerve complexes, a sub-type of the simplicial complex we
introduced in Section 4.1. Thereafter, in Section 7 we discuss the results of our case study, as well as the range of operational and
managerial insights derivable from applying TDA/PH to aviation data. Section 7 also explicitly maps topological features such as k-
dimensional holes to physical processes and phenomena in the aviation domain. We conclude our work with numerous potential
avenues of future research utilizing TDA/PH in aviation (Section 8), followed by a summary of our work and contributions (Section
9).
5. Review of applications of TDA/PH to aviation
With an understanding of important algebraic-topological concepts such as simplicial complexes, homology groups, and Betti
numbers,wereviewthreeapplicationsofTDA/PHmethodologiestovariousproblemsettingsinaviation(LiandRyerson,2018;Zhou
et al., 2018; Cho and Yoon, 2018). This is followed by a computational case study investigating the relationship between simplicial
complexesandairportsurface assets in Section6.We provide an in-depthdiscussiononthe operational and managerialimplications
of topological features (e.g. holes) studied in Li and Ryerson (2018), Zhou et al. (2018), and Cho and Yoon (2018) as well as the
computational case study in Section 7.
5.1. Detection of anomalous trajectories
The trajectories of aircraft executing airborne holding patterns and missed approach procedures can be classified as anomalous
trajectories within the terminal arrival airspace. Although rare under nominal conditions, it is important to be able to better detect
andcharacterizethese anomalous trajectories toimproveterminalareaoperations.Priorworkuses PH toexplorethetopologyofthe
filtrations of simplicial complexes that emerge from examining an aircraft’s trajectory as a planar latitude-longitude point cloud (Li
and Ryerson, 2018). Specifically, the topological features of interest are 1-dimensional holes that signify anomalous trajectories due
to airborne holding.
The simplicial complex considered in Li and Ryerson (2018) is an Alpha complex; Alpha complexes are a sub-type of simplicial
complexesthatareformedfroma different geometric perspective(EdelsbrunnerandHarer,2009).Specifically,Alphacomplexesare
closely related to
-shapes, which we define and analyze in-depth for the study reviewed in Section 5.3. The authors construct the
Alpha complexes for each aircraft trajectory and ascertain if the aircraft flew an anomalous trajectory (Fasy et al., 2014; Li and
Ryerson, 2018). The results of examining persistent topological features in the case of an aircraft that experienced airborne holding
priortoInstrument Landing System(ILS) establishment isshown in Fig.16. The 1-dimensionalholeindicative of airborneholding is
shown as a red-highlighted loop in Fig. 16 (right). Correspondingly, the Alpha complex persistence diagram shows the birth and
death of several holes – the most significant hole with the longest survival time is detected and shown in the persistence diagram as
the red triangle farthest in distance from the diagonal. This persistent topological feature indicates cyclic behavior within the tra-
jectory data set, correctly identifying this aircraft as having encountered airborne holding.
5.2. Characterizing network robustness via homology
Analgebraic-topological approachis coupled with a graph-theoretic approach in Zhou etal. (2018)to characterize robustness in
the context of random and targeted node removal-type attacks in highly connected and complex networks. Given a network re-
presentedbyacanonicalgraph
= ( , )
andaprobabilityq ofnoderemovalduringanattack,themoretraditionalgraph-theoretic
measures computed in Zhou et al. (2018) include the network connectivity
( )
, and a fraction
Q( )
that indicates the fraction of
nodeswithin so-called giant components after the removal of
=Q q
nodesdue to an attack.
Q( )
providesa measure of the giant
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component size relative to the whole network, and is used by the authors to compute a robustness R-index. These metrics are
evaluated using (18).
= =
=
R Q( )
2
; ( )
1
( ).
Q
2
1
(18)
The algebraic-topological measure that is used in Zhou et al. (2018) to characterize the topology of the network under node-
removalattacksis a ratio
k
h
ofthe number of k-dimensionalholes to the numberof connected components inthenetwork. This ratio
k
h
leverages homology to succinctly measure the impact of removing a node on the intrinsic topology of the network. The scenario
showninFig.17representsthedelaynetworkexperiencedbyAmericanAirlineshubsduringaperiodofbadweatheracrosstheNAS.
The 1-simplices encode pairwise departure delays, and the 2-simplex encodes Miles-In-Trail restrictions affecting three airports. The
delaytopologyontherighthastwo1-dimensionalholesandoneconnectedcomponent,resultinginaratioof
= 2
1
h
.Supposethatan
“attack” results in the node representing John F. Kennedy International Airport (JFK) being removed; this may be due to an airline-
specifictechnicalissuethatrequiresagroundstop.Thedelaytopologychangesdramatically,ascapturedbythechangeintheratioto
= 1
1
h
, due to the fact that there are now two connected components.
5.3.
-shapes and geofencing for unmanned aerial vehicles (UAVs)
WereviewworkbyChoandYoon(2018)thatpresentsanapplicationoftopologicalmethodstoUAVgeofencingapplications.The
current expansion of UAV technology is exemplified both in the increasing affordability and accessibility of small UAVs for personal
use as well as the growing interest on the part of companies in harnessing the ability of UAVs for commercial and industrial
Fig. 16. Alpha complex persistence diagram (left) and representative loop in latitude-longitude space (right) for the base leg and final approach
trajectory of a flight that experienced in-air holding prior to establishment on the ILS (Li and Ryerson, 2018).
Fig. 17. Hypothetical delay topologies at AAL hubs before and after a node-removal “attack”.
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operations. The natural progression of this technology is the development of low-altitude, high-density urban airspace used by both
guided and autonomous UAVs. While the structure of higher altitude airspace occupied by general and commercial aviation is
relatively rigid with strict separation minima and traffic procedures, such characterizations for the future UAV-centric airspace are
active areas of research (Sunil et al., 2018).
Researchconcerningunmannedaircraftsystemtrafficmanagementcontainsettingsanddatasetsripefortopologicalanalysis;we
reviewoneprominent examplepresentedinChoandYoon (2018).TheauthorsofChoandYoon(2018)investigatethestructureand
capacity of a given urban airspace in the context of UAV usage by using topological keep-in and keep-out geofences constructed via
-shapes. In Fig. 18 we give an illustration of the -shape construction used in Cho and Yoon (2018). For ease of visualization, we
present this illustrative example in
2
, but the construction generalizes to higher dimensions. Suppose we sample
X
2
via
samplingfunction
X S:
,obtaininga set of datapointsS (Fig. 18(a)). Wecan construct a closeddisk
D D
r r
wheretheradius
depends inversely on the
parameter and
D
r
is the boundary.
We first examine the construction of the
-shape (denoted as
S( )
) when
(0, )
. As shown in Fig. 18(b), we construct an
edgeof
S( )
ifitispossibletoconstructacloseddisk
D D
r r
suchthattwomembersofS lieon
D
r
and
=S D
r
.Notethatin
Fig.18(b),onecouldselecttheappropriate
parametersuchthataroughoutlineof
X
isrecoveredby
S( )
.Twodegeneratecasesas
tends to and 0 are illustrated in Fig. 18(c) and (d), respectively. Intuitively, as , the radii of all the possible disks
D D
r r
we could construct vanishes, and we cover the entirety of
S
2
. Hence, the case where we have that
S( )
degeneratestoS. On theotherhand,if
0
wehavethatthe radius ofeachcloseddiskgrowstoinfinity,wherein eachcloseddisk
–losingitscurvatureas
r
–becomesindistinguishablefromaclosedhalf-plane.Intryingtoconstruct
S( )
withtheseunwieldy
closed half-planes, we end up with the convex hull
Sconv( )
as the -shape of S (Fig. 18(d)).
The authors of Cho and Yoon (2018) use the construction of
-shapes given above, albeit in
3
, to construct three-dimensional
keep-in geofences. They found that these keep-in spaces via
-shapes provide an upper-bound on the amount of usable airspace,
whereaskeep-outgeofencesgivemoreconservativeresults.Keep-outgeofenceswerenotmodeledvia
-shapessincetheirgoalwasto
provide a uniform airspace buffer between UAVs and static obstacles such as buildings and other infrastructures. This produces a
naturalprimal-dualrelationshipbetween the two geofences,thetrade-offs of whichareexplored in Cho andYoon (2018) using both
simulated urban environments as well as actual terrain data from a heavily built-up district of Seoul, South Korea. The method of
-shapes for constructing keep-in geofences presented in Cho and Yoon (2018) is a pertinent example of harnessing computational
topologyinaviationresearch.Furthermore,theauthorsofChoandYoon(2018) note in their conclusion thatTDA/PHcouldbeused
inthefutureto assess theconnectivitiesandcontinuitiesofusableurbanairspacedata setsextractedfromtheirkeep-inandkeep-out
partitioning.
6. Case study: nerve topologies of airport configurations
Thesettingofnetworksnaturallyappearswhenexaminingtheairportsurface.Theactiverunwaysandrampareas can beseen as
sinks and sources where aircraft originate and depart from, traveling on paths given by taxiways and inactive runways. Current
research in modeling, optimizing, and controlling airport surface operations focuses on graph-based approaches to represent the
system of runways and taxiways at an airport (Khadilkar and Balakrishnan, 2014; Guépet et al., 2017). However, this graph re-
presentation is constrained to pairwise relationships, unlike the aforementioned ability of simplicial complexes to represent higher-
order interactions. An interesting application of TDA/PH may be to examine topological features within the nerve complex corre-
spondingto the prevailing configuration at anairport. We introduce the notion of covers for topological spaces in Section 6.1, which
will allow us to introduce nerve complexes as a sub-type of simplicial complexes. We then present our computational case study
investigating the nerve complexes of airports within the US NAS.
Fig. 18. (a)Sampledsetofredpoints
S
2
from
X
space;(b) -shapefornonzero,finiteradius
=r 1/
;(c) -shapedegeneratestosetofpointsS
as r vanishes (
); (d) -shape degenerates to convex hull
Sconv( )
as
r
(
0
).
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
164
6.1. Covers and nerves
Depending on prevailing wind conditions and operational demand, airports tend to operate a specific runway configuration that
assigns some subset of available runways as arrival and departure runways. Furthermore, depending on the runway configuration,
there will be specific taxiway flow patterns to guide aircraft to and from the runways and ramp areas. Given a simplified example
airport such as the ones presented in Figs. 19 and 20, we can define the set of ramp areas as
= T{ }
, the set of active taxiways as
= A A{ , }
1 2
, and the set of active runways as
= R R{ , }
1 2
. Given an airport and its current runway-taxiway configuration, we can
construct a simple topological space X that contains
+ +
point elements, each corresponding one-to-one with a specific
ramp area, taxiway, or runway.
Fig. 19. Underlying nerve topology of an airport configuration: An example of an airport with parallel runways (e.g. Seattle-Tacoma International
Airport).
Fig. 20. Underlyingnervetopologyofanairportconfiguration:Anexampleofanairportwithtwoperpendicularintersectingrunwaysisshownhere
(e.g. New York-LaGuardia Airport).
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Inordertoconstructthenervecomplexcorrespondingtotherunway-taxiwayconfigurationataspecificairport,wefirstconstruct
acover forthetopologicalspaceX associatedwiththerunway-taxiwayconfigurationatanairport.Let
= T R R{ , , }
1 2
beanindexset;
we now define a topological cover as follows:
Definition 14. The cover of a topological space X is a collection of subsets
U i,
i
such that their union forms X.
Weassociateeach
U
i
witharampareain orrunwayin .ThepointelementsinX isincludedwithinaparticular
U
i
ifandonlyif
the airport surface structure associated with that point element is directly reachable from the ramp area or runway associated with
theindex
i
forthatparticular
U
i
.IfwelookattheleftmostconfigurationfortheairportshowninFig.19,taxiways
A
1
and
A
2
are directly reachable from ramp area T, so the green-colored subset
U
T
encompasses three point elements in X representing
T A,
1
,
and
A
2
. The other subsets
U
R
1
and
U
R
2
are constructed analogously. We now give the formal definition of the nerve complex of a
topological space X using the language of covers:
Definition 15. The nerve complex of a topological space X, denoted by
Xnerve( )
, corresponding to a valid cover of X given by is
thecollectionofk-simplicesmappedtobynonempty intersections of
U
i
,wherek indicatesthenumberof
U
i
subsetsthatintersect.
The indices of the intersecting subsets map to the vertices of the k-simplices.
We can explicitly use the definition of nerve complexes to construct the nerve complex for airport surface assets. Using the same
leftmost configuration in Fig. 19 as an example, there are two non-empty intersections given by
=U U U A{ }
T R R 1
1 2
and
=U U A{ }
T R 2
2
.Thefirstnonempty intersection of threesubsetsmaptothe3-simplex
T R R, ,
1 2
,andthesecond non-empty intersection
of two subsets map redundantly to the 2-simplex
T R,
2
. Thus,
Xnerve( )
in this example is exactly the 3-simplex
T R R, ,
1 2
. All nerve
complexes at the bottom of Figs. 19 and 20 are constructed in the same manner.
Through examining nerve complexes, we can already begin to connect various topological features to operational realities.
Although the taxiways in use are different between the leftmost and center configurations in Fig. 19, their underlying nerve com-
plexesareidentical.TherightmostconfigurationinFig.19showsanon-trivialchangeinthenervecomplex;adegenerationfroma3-
simplextoa2-chainconsistingof2-simplicesoccurswhentaxiway
A
1
servesonlyrunway
R
1
,andtaxiway
A
2
servicesonlyrunway
R
2
.
In an airport with intersecting runways, such as the one presented in Fig. 20, a similar configuration where taxiway
A
1
servicesonly
runway
R
1
and taxiway
A
2
services only runway
R
2
results in an underlying nerve complex that is a 2-chain with a detectable 1-
dimensional hole. Loosely speaking, these nerve complexes trade local information that a graph-theoretic representation contains in
order to look at the underlying topology associated with a particular airport runway-taxiway configuration.
6.2. Constructing nerve complexes of airports from real operational data
Inordertoconstructandanalyzenervecomplexesforourairportsofinterestin this computationalcasestudy,wefirstcollectthe
necessary runway configuration data and declared capacities from the Aviation System Performance Metrics (ASPM) database
maintainedbytheUS Federal Aviation Administration(Section6.2.1).SinceactivetaxiwaysarenotincludedinASPM, we develop a
frameworkforinferringactive taxiway flow patternsviaairtrafficcontrol audio (Section 6.2.2).Wethen construct and interpret the
corresponding nerve complexes for our case study airports (Sections 6.2.3 and 6.2.4).
6.2.1. Runway configurations and declared capacities
The five case study airports are the five highest-ranked US airports in terms of total enplanement statistics for 2017 (Federal
Aviation Administration, 2016); in order, these airports are Hartsfield-Jackson Atlanta International Airport (ATL), Los Angeles
International Airport (LAX), Chicago O’Hare International Airport (ORD), Dallas/Fort Worth International Airport (DFW), and
Denver International Airport (DEN). We retrieve the runway configurations and declared capacities data for these airports for No-
vember 2018 via ASPM (Federal Aviation Administration, 2018a). For each airport, the maximum total declared capacity – the
AirportArrivalRate(AAR)plustheAirportDepartureRate(ADR)–andtheminimumtotaldeclaredcapacityislocatedandretrieved,
along with the accompanying runway configurations. Since ASPM records hourly data, we will refer to the maximum total declared
capacityanditsrunwayconfigurationasthe“high-capacity scenario”oftheairport,andtheminimumtotaldeclaredcapacityandits
runway configuration as the “low-capacity scenario”.
6.2.2. Inferring active taxiway configurations via air traffic control audio
Since taxiway configurations and active taxiways are not recorded in ASPM, in order to accurately re-create the active taxiways
duringhigh-andlow-capacityscenarios,weretrievearchivedaudiorecordingsofgroundcontrollerfrequenciesatthefivecasestudy
airports (LiveATC, 2018). Since these ground controller audio archives are given in 30-min intervals, for each capacity scenario, the
first30-min segment was used to retrieve the list of active taxiways used by arriving and departing aircraft. arrivals and departures.
Repeating this process for all five case study airports and combining information regarding active taxiways with the runway con-
figurationsprovided viaASPM, weare able to construct airport runway-taxiway configuration diagrams for all five casestudy airports.
An example of these airport runway-taxiway configuration diagrams for ORD during its high- and low-capacity scenarios can be
foundinthe leftpanelsofFigs.21and22, respectively.Terminalandrampareasaregiveninblue,taxiwaysingray,arrival runways
in red, and departure runways in green.
6.2.3. Constructing nerve complexes for our case study airports
Recallthedefinitions of coversand nerve complexes associatedwith a given topologicalspaceprovided in Section6.1.Given the
M.Z. Li, et al.
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runway-taxiway configuration diagram at an airport, we can construct the associated nerve complex. Using ORD during its high-
capacity scenario as the guiding example, we first attach proxy labels
…R R, ,
N1
to each runway, and labels
…A A, ,
M1
to each
taxiway.TherampareaislabeledasT.Fromtheairportrunway-taxiwayconfigurationdiagramforORDinitshigh-capacityscenario
(Fig.21(a)),weconstructtheassociatedintersection graph.Theintersectiongraph(Fig.21(b))containsanodeforeachactivetaxiway
and runway asset, as well as a node representing the ramp area. For illustrative purposes, the nodes in the intersection graph are
colored in accordance to the airport runway-taxiway configuration diagram. Edges are drawn between nodes in the intersection
graphrepresenting a direct connectionbetween the rampand runway nodes toa taxiway node.For example, taxiway
A
1
connects
R
1
and
R
2
; thus, an edge
A R{ , }
1 1
is drawn along with an edge
A R{ , }
1 2
.
The intersection graph allows us to easily write down the various non-empty intersections of covers needed to construct the
Fig. 21. (a) Complete airport runway-taxiwayconfiguration diagram forORD high-capacity scenario,(b)the associated intersectiongraph, and (c)
the associated nerve complex.
Fig. 22. (a) Complete airport runway-taxiway configuration diagram for ORD low-capacity scenario, (b) the associated intersection graph, and (c)
the associated nerve complex.
M.Z. Li, et al.
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correspondingnervecomplex.Thenumberof edges attached to each taxiwaynodeisexactlythe number of covers that intersectina
non-emptymanner,andthenodesthatareconnected tothattaxiwaynodebytheedgesaretheindicesof theintersectingcovers.For
example,the
A
1
taxiwaynodehastwoedgesconnectingittorunways
R
1
and
R
2
.Thus,thisisanon-emptyintersectionoftwocovers,
i.e.
=U U A{ }
R R 1
1 2
.RecallfromSection6.1thatthis correspondstothe1-simplex
R R,
1 2
,whichcan befoundinFig.21(c) asthe
orange-colorededge (1-simplex)connecting nodes(0-simplices)
R
1
and
R
2
.Analogously, note that both
A
5
and
A
6
taxiwaynodes are
connected to
T R,
3
, and
R
4
. Thus, these are both non-empty intersections of two covers:
=U A A, .
i T R R
i
{ , , }
5 6
3 4
(19)
These intersections both correspond to the 2-simplex
T R R, ,
3 4
, found in Fig. 21(c) as the purple-shaded triangle (2-simplex) con-
necting nodes
T R,
3
, and
R
4
. Every non-empty intersection in the intersection graph can be mapped to corresponding k-simplices in
thenervecomplexrepresentation.Acompressionofinformationcanbeseenintranslatingfromthecompleteairportrunway-taxiway
configuration diagram to the nerve complex representation: The complete airport runway-taxiway configuration diagram conveys
geographic and geometric information as well as information regarding connectivities, whereas only topologically non-trivial con-
nectivities are preserved in the nerve complex representation.
As a comparison, we give the complete airport runway-taxiway configuration diagram, intersection graph, and nerve complex
representation for ORD during its low-capacity scenario as well in Fig. 22. Note the drastic changes in the intersection graph as well
as the corresponding nerve complex representation. This particular instance was exacerbated by an ongoing winter storm impacting
ORD, causing many taxiways to be impassable due to snow drifts. In particular, the loss of the 2-simplex within the nerve complex
represents a large loss in airport surface connectivity.
6.2.4. Loss of maximal simplex indicate large changes in airfield topology
For each of the five case study airports, we construct the associated intersection graphs for the full airport runway-taxiway
Table 2
Summary of computational results from the airport surface nerve complex case study.
Airport GMT Date (m/d/
y)
Declared rates Runway configurations 0-simplices 1-simplices 2-simplices 3-simplices
ATL
HIGH
1400Z
11/11/2018
AAR: 132
ADR: 118
Arr: 8L, 9R, 10
Dep: 8R, 9L
T,
8L, 9R, 10,
8R, 9L
{T, 8L}, {8L, 8R},
{T, 8R}, {T, 9L},
{9L, 9R}, {T, 9R},
{9R, 10}
{T, 8L, 8R},
{T, 9L, 9R}
ATL
LOW
1200Z
11/5/2018
AAR: 72
ADR: 80
Arr: 8L, 9R, 10
Dep: 8R, 9L
T,
8L, 9R, 10,
8R, 9L
{T, 8L}, {8L, 8R},
{T, 8R}, {T, 9L},
{9L, 9R}, {9R, 10}
{T, 8L, 8R}
LAX
HIGH
1400Z
11/4/2018
AAR: 74
ADR: 74
Arr: 24R, 25L
Dep: 24L, 25R
T,
24R, 25L, 24L,
25R
{24R, 24L}, {24L,
25R},
{24L, T}, {T, 25R},
{T, 25L}, {25R, 25L}
{T, 25R, 25L}
LAX
LOW
0900Z
11/8/2018
AAR: 29
ADR: 29
Arr: 6R
Dep: 25R
T,
6R, 25R
{T, 6R}, {6R, 25R},
{25R, T}
ORD
HIGH
0500Z
11/1/2018
AAR: 114
ADR: 114
Arr: 27L, 27R, 28C
Dep: 22L, 28R
T,
27L, 27R, 28C,
22L, 28R
{27R, 27L}, {27L, T},
{27L, 28R}, {22L, T},
{22L, 28R}, {28R, T},
{28R, 28C}, {T, 28C}
{T, 28R, 28C}
ORD
LOW
1200Z
11/26/2018
AAR: 16
ADR: 20
Arr: 4R
Dep: 4L
T,
4R, 4L
{T, 4L}, {T, 4R}
DFW
HIGH
1200Z
11/10/2018
AAR: 84
ADR: 100
Arr: 31R, 35R, 36L
Dep: 31L, 35L, 36R
T,
31R, 35R, 36L,
31L, 35L, 36R
{31L, 36L}, {36L, T},
{T, 31L}, {31L, 36R},
{T, 36R}, {36L, 36R},
{35L, 31R}, {31R,
35R},
{35R, 35L}, {35L, T},
{31R, T}, {35R, T}
{T, 31L, 36L},
{T, 31L, 36R},
{31L, 36L, 36R},
{T, 36L, 36R},
{T, 35L, 31R},
{T, 35L, 35R},
{T, 31R, 35R},
{35L, 31R, 35R}
{T,31L,36L,36R},
{T, 35L, 31R, 35R}
DFW
LOW
0200Z
11/23/2018
AAR: 60
ADR: 80
Arr: 17L, 18R
Dep: 17R, 18L
T,
17L, 18R, 17R,
18L
{T, 18R}, {18R, 18L},
{T, 18L}, {T, 17R},
{17R, 17L}, {T, 17L}
{T, 18R, 18L},
{T, 17R, 17L}
DEN
HIGH
2300Z
11/8/2018
AAR: 152
ADR: 114
Arr: 16L, 16R, 35L, 35R
Dep: 8, 25, 34L, 34R
T,
16L, 16R, 35L,
35R, 8, 25,
34L, 34R
{T, 16R/34L}, {T, 25},
{T, 35L}, {T, 35R},
{T, 34R/16L}, {T, 8},
{34R/16L, 8}
{T, 8, 34R/16L}
DEN
LOW
0600Z
11/24/2018
AAR: 32
ADR: 32
Arr: 34R, 35L, 35R
Dep: 34L
T,
34R, 35L, 35R,
34L
{T, 34L}, {T, 34R},
{T, 35L}, {T, 35R}
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Transportation Research Part E 128 (2019) 149–174
168
configuration diagram, as well as the corresponding nerve complex representations. We list the date, declared capacities, runway
configurations,andk-simplexrecords for each ofthe casestudyairportsunderhigh-andlow-capacityscenariosin Table2.Notethat
some nerve complex representations do not have higher-order k-simplices beyond 1-simplices – in these cases, “
” is recorded to
indicate that there were no 3- or 4-fold intersections of covers to map to a 2- or 3-simplex.
From Table 2, we see that for ourfive case study airports, the loss of the maximal simplices can be an indication that there was a
large loss in surface connectivity during low-capacity scenarios. A maximal simplex within a simplicial complex – in this case, the
nerve complex – is the “largest” existing k-simplex, where k is maximized across all existing k-simplices. For example, the maximal
simplexintheORDhigh-capacityscenariowouldbethe2-simplexgivenby
T R C,28 ,28
,whereasthe maximalsimplexintheDFW high-
capacity scenario would be a 3-simplex given by
T L L R,31 ,36 ,36
or
T L R R,35 ,31 ,35
. We see that in the case of every case study airport with
theexception of ATL,the maximal simplex that exists in the high-capacity scenario disappears inthe low-capacityscenario. The loss
ofa2-simplexmeansthelossofataxiwaynodethatfacilitateduptothreeconnectionsbetweenrampandrunwayassets,andtheloss
of a 3-simplex means the loss of a taxiway node that facilitated up to four connections. In Section 7 we discuss the operational and
managerialimplications of this case study, alongwith the other three reviewed studies,all of whichleverage topological methodsto
examine aviation data sets.
7. Potential operational and managerial insights
Thereviewedpaststudiesandourowncomputationalcase studyinSections5and 6,respectively,bridgethegapbetweentheory
(Section4)andaviationapplications.Wenowdiscussin-depththeconnectionsbetweentheresultsfromSections5and6toaviation,
specifically in terms of operational and managerial insights. Furthermore, we explicitly map holes and other abstract topological
features to concrete physical processes occurring within the specific aviation domain that TDA/PH was utilized in. In doing so, we
hopetoemphasizethewidespreadapplicabilityandflexibilityofthesetopologicalmethods,aswellashighlightthefactthatTDA/PH
serve to complement and augment existing data analysis methods, not succeed them. This discussion will lead into a presentation of
future research avenues for TDA/PH within the aviation domain in Section 8.
7.1. Topological features map to trajectory characteristics
Understanding terminal airspace constraints and situations that may result in excessive in-air holding patterns and missed ap-
proachesiscriticalforairtrafficflowmanagersaswellasairspaceplanners.TheapplicationofTDA/PHindetectingindividualcyclic
trajectories indicative of holding patterns and missed approaches reviewed in Section 5.1 could provide decision support insight
regarding terminal airspace conditions that may result in an increased number of these anomalous trajectories. Data resulting from
using TDA/PH to locate anomalous trajectories is of importance to airspace planners and airport managers with more long-term,
strategicgoalsaswell.Forexample,whenairportsplanforcapitalprojectssuchastheextensionorconstructionofanewrunway,the
terminalairspacealsoneedstobereadjustedtoplanfornewarrivalanddepartureroutesgiventhenewsurfaceinfrastructure(Liand
Ryerson, 2017). Any new arrival and departure routes within the terminal airspace should not interfere with sectors of the airspace
generallyreservedfor in-air holdingsand missed approach clearances.The topological approachin Section 5.1usefulfor identifying
such anomalous operations within the terminal airspace complement other methods such as trajectory clustering (Murça and
Hansman,2018)andlearning anomalous trajectories viamachinelearning (Li et al.,2015),butwith advantages suchasadecreased
reliance on parameter tuning and training. Anomalous cyclic trajectories with holes are natural objects identifiable via TDA/PH.
Furthermore, there is an increased tolerance for noise in the data set, as noisy, non-persistent data can be filtered out via PH (Ghrist,
2014, 2017).
7.2. Topological features map to robustness in air transportation networks
Payinghomagetothe flexibility ofTDA/PHandsimplicialcomplexrepresentation,the same algebraic-topologicalfoundationsof
holesandconnectivitiescanalsobeappliedtoadatasetofairportnodeswithintheUSNAS(Section5.2),acompletelydifferentdata
context from a point cloud representation of aircraft trajectories (Section 5.1). The approach leveraged by the reviewed study in
Section 5.2 combines traditional graph-theoretic measures with algebraic-topolgical measures based on the ratio of various Betti
numbers, mapping both to metrics describing network robustness. Here, topological holes indicate dyadic relationships between
airportswithnohigher-orderconnectivities,whereasafilled-insimplex(i.e. noholes)indicatenon-trivial,higher-orderrelationships.
The ability to include higher-order relationships gives a more nuanced and realistic network robustness model than graph-based
approaches (Lordan et al., 2014; Wei et al., 2014). From a tactical, operational standpoint, if it is known a priori that a particular
clique of airports often propagate delays between themselves, then traffic management initiatives and airspace flow programs could
be better tailored to target such specific cliques and prevent a larger cascade of delays and cancellations. On the other hand, from a
managerial and strategic standpoint, identification of such cliques within the air transportation network could lead to revised re-
routingpoliciesthatlocallyincreasenetworkrobustnessagainstperturbationssuchasconvectiveweather.Anexampleofwheresuch
amanagerialamendmentcouldoccurwithintheUSNASisatthelevelof“plays”withinthe“playbook”ofSevereWeatherAvoidance
Plans (SWAPs) (Federal Aviation Administration, 2018b).
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
169
7.3. Topological features map to airspace partitions
Elements of topology and geometric analysis are used to inform urban air mobility (UAM) airspace planning scenarios in the
researchreviewedinSection5.3. The need todesigncorridorsembeddedwithinthelow-altitude,high-densityairspacethatmitigate
UAV-on-infrastructure collision risks is crucial towards the future adoption and expansion of UAM. The underlying data set in this
study arises from urban features such as buildings and other public utilities infrastructures, with topological
-shapes partitioning
and demarcating safe versus unsafe UAV operating airspace regions. The airspace partitions and geofences procured via such a
topological analysis could help inform future city planners and Municipal Planning Organizations (MPOs) regarding how to best
policytheuseofUAVsforcommercial,personal,andgovernmentalpurposeswithinanurbanenvironment.Designingandoptimizing
topologically efficient urban airspace corridors for UAV operations have the potential to lead to new avenues in logistics and traffic
flow management research (Sunil et al., 2018).
7.4. Topological features map to airport surface characteristics
It is well-known that the arrival and departure capacities at an airport is largely a function of the service rate of the runway (de
Neufvilleetal.,2013);thearrivalanddepartureserviceratesofairportrunwaysplayanimportantroleinmanymodelsrangingfrom
simpledeterministic time-based models to complex network queueing models (Badrinathet al.,2019). In Fig. 23, we plot separately
theAARandADRforeachairport,colorized by the drop in the number ofarrivalanddeparturerunways,respectively.FromFig.23,
wecan see thatwhilethe number of runways in servicedoes dictate thecapacity to somedegree, there arevariations that cannotbe
explainedbythenumberofrunwaysalone.Forexample,ATLexperiencesanoticeabledropinbothAARandADR,butthenumberof
arrival and departure runways in use during the high- and low-capacity scenarios remained the same. On the other hand, airports
suchas DEN and ORD not only suffered large drops in capacities, but also the loss of several departure and arrival runways between
high- and low-capacity scenarios.
OurcomputationalcasestudyinSection6depictsyetanotheraviationdatacontextwithinwhichTDA/PHcanbeapplied:Airport
surface taxiway and runway reachability and connectivity. By representing the full airport runway-taxiway configuration as a
simplicial complex, we were able to not only give a sparse representation that preserves the important connectivities, but also glean
insight regarding how airport surface operations map to easier-to-understand topological shapes. In particular, while the AAR and
ADR plot in Fig. 23 confirms the relationship between number of active runways and the airport capacity, the results from the
computation case study (Table 2) suggest that one could simply keep track of the maximal simplex within the simplicial complex
representationofthefullairportrunway-taxiway configurationand get a sense of how well-connectedandreachablevariousairfield
Fig. 23. Change in ADR and AAR at each airport between high- and low-capacity hours during November 2018, with case study airports high-
lighted. Runway asset losses are also noted.
M.Z. Li, et al.
Transportation Research Part E 128 (2019) 149–174
170
assets are. Furthermore, a simplified representation such as the intersection graph or the nerve complex (Figs. 21 and 22) could be
more helpful to airport surface managers and ground controllers making high-level decisions regarding taxiway flow patterns, as
these representations retain only the important reachability indicators between airfield assets.
8. Future research directions
We have only begun to scratch the surface of applying topological methods such as TDA/PH to examine aviation data sets.
Equippedwithanunderstandingofthefundamentalalgebraic-topologicalnotionsthatenableTDA/PHandapreviewofhowvarious
topological features could map to pertinent features and characteristics previously hidden within aviation data, we dedicate this
penultimate section to a sampling of proposals for future research avenues combining TDA/PH and aviation.
8.1. Topological persistence and embedded strategy complexes of airport surface assets
OurcomputationalcasestudyfromSection6providesaframeworkfortranslatingairportsurfacenetworksintonervecomplexes
amenable to various theorems and techniques found in algebraic topology and TDA/PH. For example, PH could be used to track
topologicalfeaturesofanairport’ssurfacethatpersistthroughawide range of capacity scenarios. The identificationoftransientand
persistenttopologicalfeaturesmayaidinformalizingtheprocessoffinding“hotspots”inairportsurfaceoperationsandtrafficflows.
Furthermore,thetopologyofthenerve complexes themselves may beinteresting, particularlyifthenervecomplexesareviewedasa
strategy complex (Erdmann, 2010). Depending on the similarity between our airport surface nerve complexes and certain classes of
strategycomplexes,thereareguaranteesonwhich“strategies”arepossibleandwhicharenot.Intheaviationcontext,astrategyona
strategy complex could map to, for example, a pre-defined class of taxiway flows.
8.2. Homotopic and homological path planning in aviation
AnotherareainaviationdatasciencethatpresentsopportunitiesforapplyingTDA/PHisorigin-destinationtrajectorygeneration.
Given an origin-destination pair
o d( , )
i i
, it can be thought of as points within a target space
S
2
. One trajectory
S: [0, ]
1
travels from
o
i
to
d
i
via time parameter
[0, ]
by sending
= o(0)
i1
and
= d( )
i1
. We can consider another trajectory
S: [0, ]
2
thatalsotraversesfrom
o
i
to
d
i
,butviaadifferenttrek. Imagine if we can continuously deform trajectory
1
to
2
–ifthis
ispossible,then
1
isfixed end point homotopic to
2
,andthereisafixed end point homotopy
× S: [0, ] [0, ]
1,2
suchthatanyother
trajectories from
o
i
to
d
i
created during the smooth deformation process from
1
to
2
is reachable by
u [0, ]
, with
[0, ]
traveling along this path. Explicitly,
1,2
can be written as:
×
= = =
= = =
S
u
u
: 0, 0, :
( , 0) ( ), (0, ) (0) (0)
( , ) ( ), ( , ) ( ) ( )
.
1,2
1,2 1 1,2 1 2
1,2 2 1,2 1 2
(20)
Homotopy theory can become very complex and rich when considering exotic topologies, but for the purposes of aviation, the
workspace is typically just the surface of the Earth, with various discontinuities that represent airspace features such as no-fly zones
and convective weather which will affect the resultant homotopies. Vidosavljevic et al. (2017) presents one possible application of
homotopy theory in aviation trajectory generation.
8.3. Topological trajectory and fuel flow analysis
Aviation trajectory data sets lend itself well to topological methods due to the high-dimensional and sparse nature of the data.
Anotheractive research areain aviationthat may benefit from an exploration usingTDA/PH is aircraft taxi-in and taxi-out fuelflow
analysis, particularly using raw data from on-board flight data recorders (Chati and Balakrishnan, 2013). An aircraft taxiing to and
from the gate at a busy airport may encounter runway and ramp queues, resulting in a repeated pattern of stop-and-go thrust inputs
as the aircraft taxis behind another aircraft within the queue itself (Levine and Gao, 2007). This pattern of thrust inputs and cor-
responding fuel flow rates can be modeled periodically in time, and this periodicity can be mapped to cycles detectable through
topological means. Similar to the previous study reviewed in Section 5.1, the taxiway network can be analyzed from this viewpoint,
potentially identifying time periods when anomalous trajectories of fuel flow patterns appear. Such analysis may lead to the de-
velopment of better optimization models and system control inputs to reduce taxi-in and taxi-out emissions and delays.
8.4. Aviation sensor coverage problems and higher-order network comparisons
Finally, Dłotko et al. (2012) demonstrated the use of TDA/PH in sensor networks and sensor coverage. Applications in aviation
research include characterizing the expansion and coverage of Automatic Dependent Surveillance-Broadcast (ADS-B) and Multi-
lateration (MLAT) capabilities, particularly in regions of the world with poor primary radar coverage. Zooming back out to the
perspectiveoftheentireairtransportationnetwork,TDA/PH–andinparticularhomologicalalgebra–couldbeusedtoclassifyhow
similar one higher-order network is compared to another. A higher-order network in aviation could resemble something similar to
Fig. 2, where multiple layers of simplicial complexes are used to represent different numerical and categorical data related to the
NAS. This application stems from previous work that investigates using PH to establish approximations of network distances within
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higher-ordergraphs(HuangandRibeiro,2017).Giventhehighlyinterconnectednatureoftheglobalaviationsystemcoupledwitha
plethora of data regarding the edges and nodes of the network, it would be interesting to apply some of the results from Huang and
Ribeiro (2017) to differentiate between various multi-layered networks in the aviation domain using topological features and
homological algebra.
9. Summary and conclusion
In addressing the call for new data science methodologies to help tackle the oncoming big data era of aviation research, we first
discussedtheinfluxofaviationdatafromavarietyofsources.Thishasledtoasurgeindata-drivenaviationresearch.However,some
aviation data sets are high-dimensional, highly-connected, and sparse. As classical data analytics may fail to uncover higher-order
structures and relationships in these data sets, new data science tools that provide a more global perspective are needed.
As TDA/PH may be able to unearth global topological features within aviation data sets, we introduce TDA/PH and propose its
use to further aviation data science research. These topological features could be translated into new benchmarks, models, and
applications.To translate TDA/PH methods to air transportation,one must have boththe necessary aviation domain knowledge and
an understanding of how TDA/PH works; only then can one recognize what TDA/PH can and cannot reveal. In order to make the
algebraic-topological foundations of TDA/PH more readily accessible to aviation researchers, we dedicated a large portion of this
work to an introduction of the fundamental notions, definitions, and constructs behind TDA/PH (Section 4). We created an in-
structiveexample drawn from a sample data setcontaining aviationgeography, airspace, and operational features. This sample data
set is used throughout the manuscript, from constructing simplicial complexes out of data to introducing simplicial homology.
EquippedwithanunderstandingofthetheoryunderlyingTDA/PH,wethenbridgethegapbetweentheabstractionsprovidedby
algebraic topology to actual aviation applications with potential managerial and operational insights. In particular, we motivate the
applicability of TDA/PH in aviation data science research by reviewing past studies that incorporated a topological approach to
examineaviationdatasets(Section5).Thecontrastinthetypesofdatasetsusedinthepaststudies,rangingfromaircrafttrajectories
to geography-based data for constructing UAV geofences, showcases the flexibility of TDA/PH. We add to this growing collection of
studies with our own computational case study exploring nerve complex representations of airport surface networks (Section 6),
where we showed that maximal simplices map to large losses in airport surface connectivity. We elaborate on this translation
between topological features and aviation-specific features in Section 7, as well as describe managerial and operational insights
derivable from these topological data science approaches. To close our foray into TDA/PH and aviation, we suggest in Section 8 an
abundanceofpotentialfutureresearchavenuesfortopologicalmethodsinaviationdatascience.Wehopethatthemotivationforand
introduction to TDA/PH we have provided stimulates an interest in examining aviation data sets through these topological lenses.
Acknowledgements
This work was supported in part by NSF CPS1739505. M. Z. Li was also supported by an NSF Graduate Research Fellowship. We
would also like to thank the anonymous reviewers whose comments, feedback, and suggestions greatly improved this manuscript.
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