Mobili y s. Con igui y: Spa ially Explici G aph
Neu al Ne wo ks o COVID-19 Fo ecas ing
Fengjiao Li1[0009−0006−9075−0818], Meiliu Wu1∗[0000−0002−5246−4603] and Anahid
Basi i1[0000−0002−2399−1797]
1. School o Geog aphical and Ea h Sciences, Uni e si y o Glasgow, Glasgow, UK
* [email p o ec ed]
Abs ac . Assessing how di e en g aph cons uc ions a ec spa io-
empo al o ecas ing is essen ial o a oiding misleading p edic ions and
ine ec i e in e en ions ac oss domains such as epidemic con ol, ans-
po a ion, and en i onmen al moni o ing. G aph neu al ne wo ks (GNNs)
o e a powe ul amewo k o modelling hese p ocesses, bu he choice
o g aph ep esen a ion, whe he based on geog aphic con igui y, ans-
po ne wo ks, o mobili y lows, emains unde explo ed. This s udy e al-
ua es and compa es i e g aph designs o o ecas ing daily COVID-19 in-
ec ion a es pe 100,000 esiden s in Glasgow and Edinbu gh (2020–2023),
including con igui y, oad dis ance, mobili y, and hyb ids ha in eg a e
mobili y wi h ei he con igui y o oad dis ance. Using a 7-day sliding
inpu window o p edic he subsequen 7 days, and assessing pe o -
mance ia MAE, RMSE, MAPE, and R2, we obse e ci y-speci ic he e o-
genei y: mobili y-in o med hyb ids achie e he bes o ecas s in Glasgow
bu no in Edinbu gh. These indings sugges ha he alue o mobili y
is con ex -dependen , shaped by u ban o m and a el in ensi y. Mo e
b oadly, he s udy poin s o he po en ial impo ance o ailo ing g aph
cons uc ion o local condi ions, o e ing me hodological insigh s o epi-
demic modelling and o he spa io- empo al o ecas ing applica ions.
Keywo ds: Geospa ial A i icial In elligence ·Spa ially Explici G aph
Neu al Ne wo ks ·Mobili y ne wo ks ·Spa io- empo al modelling ·COVID-
19 Fo ecas ing
1 In oduc ion
Spa ially explici g aph neu al ne wo ks (GNNs) can model spa ially s uc u ed
ime se ies whe e in e ac ions among places shape ou comes, enhancing many
downs eam asks, e.g., a ic o ecas ing [16], epidemic sp ead p edic ion [9],
and ai pollu ion o ecas ing [10]. Howe e , he choice o g aph (e.g., con igui y,
anspo a ion connec i i y, o measu ed mobili y) emains unde explo ed and
may be con ex dependen . Spa ial in e ac ions can in luence COVID-19 sp ead,
and g aph-based models p o ide a way o ep esen such dependencies. Howe e ,
hei ad an ages depend on con ex , da a quali y, and modelling assump ions.
DOI: h ps://doi.o g/10.5281/zenodo.17660772
2 F. Li e al.
As such, his s udy o ecas s daily COVID-19 in ec ion a es and e alua es i e
di e en g aph designs based on he backbone o Di usion Con olu ional Re-
cu en Neu al Ne wo k (DCRNN) [17] be ween wo Sco ish ci ies ha di e
in popula ion dis ibu ion and a el in ensi y. We aim o answe wo impo an
ques ions: when does mobili y imp o e he p edic ion o e , o combined wi h,
con igui y (i.e., geog aphic adjacency), and how should g aphs be combined?
S udies ha e begun o explo e he po en ial o Spa ially explici GNNs, pa -
icula ly in epidemic o ecas ing. Kapoo e al. [15] examined spa io- empo al
GNNs o COVID-19 o ecas ing, showing ha hei g aph-based ep esen a ions
ou pe o m pu ely empo al baselines. Wi zke e al. [28] u he demons a ed
ha mobili y da a can imp o e he p edic ion o incidence ends using g aph
neu al ne wo ks. Mo e ecen ly, Dua e e al. [11] highligh ed he ad an ages o
in eg a ing mobili y lows di ec ly in o g aph lea ning amewo ks o epidemic
o ecas ing.
Despi e hese ad ances, mos esea ch s udies only ocused on mobili y-based
g aphs, and ew a emp o in eg a e di e en ac o s [3, 8], such as geog aphic
adjacency [12], mobili y lows [7], and oad ne wo ks [18]. An open ques ion
emains: how should g aphs be cons uc ed and po en ially combined? In he do-
main o a ic p edic ion, hyb id g aph designs (e.g., combining oad ne wo ks
wi h mobili y demand) ha e been mo e widely explo ed [29, 30]. Ye , sys em-
a ic compa isons o hese al e na i es ac oss di e en u ban con ex s seem o
be s ill limi ed. I emains unclea whe he mobili y-based g aphs consis en ly
o e bene i s, o whe he hei e ec i eness a ies depending on ci y-speci ic
cha ac e is ics such as popula ion densi y o a el pa e ns.
To add ess hese limi a ions, his s udy explo es he compa a i e pe o -
mance o i e g aph cons uc ion s a egies wi hin he DCRNN a chi ec u e: (1)
geog aphic adjacency, (2) oad ne wo k dis ances, (3) mobili y lows, (4) hyb id
g aphs combining mobili y wi h oad ne wo ks, and (5) hyb id g aphs combining
mobili y wi h geog aphic adjacency. Using daily COVID-19 in ec ion a es pe
100,000 esiden s a he In e Zone le el om 2020 o 2023 and a 7-day inpu
window o o ecas he subsequen 7 days, we conduc a compa a i e analysis in
Glasgow and Edinbu gh, wo Sco ish ci ies wi h con as ing demog aphic and
mobili y p o iles. The s udy aims o examine how di e en g aph designs may
in luence o ecas ing pe o mance and o p o ide guidance on ailo ing g aph-
based neu al ne wo ks o local u ban con ex s in epidemic modelling
2 Da a and Me hods
2.1 Da a Sou ces
This s udy ocuses on Glasgow and Edinbu gh, wo majo Sco ish ci ies wi h
con as ing demog aphics and mobili y pa e ns. Daily COVID-19 in ec ion da a
a he In e Zone le el om Ma ch 2020 o Feb ua y 2023 we e ob ained om
Public Heal h Sco land [21].1Small coun s (≤2cases pe zone pe day) we e
1h ps://www.openda a.nhs.sco /da ase /co id-19-in-sco land
Spa ially Explici G aphs o COVID-19 Fo ecas ing 3
supp essed o con iden iali y. To add ess his, we applied mul iple impu a ion
using he mice package in R [6]. Following s anda d p ac ice in he missing da a
li e a u e, we gene a ed 20 impu ed da ase s. While Rubin o iginally sugges ed
ha as ew as 3–5 impu a ions migh be su icien , subsequen me hodological
s udies ha e shown ha using a la ge numbe (e.g., 20–40) imp o es he s abili y
and p ecision o pa ame e es ima es [5,13,25]. The 20 comple ed da ase s we e
a e aged o p oduce in ec ion a es pe 100,000 esiden s. Figu e 1 illus a es
he empo al e olu ion o in ec ion a es in bo h ci ies, highligh ing subs an-
ial luc ua ions ac oss pandemic wa es and di e ences in magni ude be ween
Glasgow and Edinbu gh. These desc ip i e plo s p o ide he empi ical mo i a-
ion o applying spa io- empo al models capable o adap ing o he e ogeneous
dynamics.
Popula ion mobili y da a we e accessed om he U ban Big Da a Cen e
(UBDC) [26].2These o igin–des ina ion ma ices p o ide anonymised lows be-
ween In e Zones, exp essed as pe cen ages o esiden mo emen s. Road and
ail ne wo k da a we e downloaded om he Geo ab ik OpenS ee Map se e
o Sco land [20] and p ocessed wi h he dodg package o compu e sho es -
pa h dis ances o mul iple anspo modes. Toge he , hese da ase s enabled
he cons uc ion o di e se g aph s uc u es ha cap u e bo h geog aphic adja-
cency and mobili y in e ac ions.
(a) Glasgow (b) Edinbu gh
Fig. 1: Weekly a e age 7days COVID-19 a e pe 100,000 popula ion in Glasgow
and Edinbu gh (2020–2023).
2.2 G aph Cons uc ion
Fi e g aph s uc u es we e designed o ep esen spa ial and mobili y ela ion-
ships.
(1) Geog aphic adjacency:
wgeo
ij =(1,i zones iand jsha e a common bounda y,
0,o he wise.(1)
2h ps://da a.ubdc.ac.uk/da ase s/6d87635c-96bb-4b7a-9a2e-676bb250032d
4 F. Li e al.
(2) Road ne wo k:
w oad
ij =1
d oad
ij +ϵ, i =j, (2)
whe e d oad
ij deno es he sho es -pa h dis ance be ween zones iand j h ough
he mul imodal oad and ail ne wo k, and ϵis a small smoo hing cons an o
a oid di ision by ze o.
S ep 1: Da a
Daily COVID-19 in ec ion coun s a he In e Zone le el (2020–2023);
O igin–des ina ion mobili y lows om he U ban Big Da a Cen e;
Road and ail ne wo k da a om OpenS ee Map;
Geog aphic bounda y da a o In e Zones
S ep 2: P ep ocessing
Small coun s(≤2) impu ed using he mice package in R wi h 20 impu a ions;
Impu ed coun s a e aged and con e ed o in ec ion a es pe 100,000 esiden s;
Z-sco e no malisa ion applied;
Sho es -pa h dis ances calcula ed om he oad and ail ne wo k;
O igin–des ina ion lows symme ised by a e aging lows in bo h di ec ions
S ep 3: G aph Cons uc ion
(1) Geog aphic adjacency (sha ed bounda ies);
(2) Road ne wo k connec i i y (in e se o sho es -pa h dis ance);
(3) Mobili y lows (symme ised o igin–des ina ion lows);
(4) Mobili y + Road;
(5) Mobili y + Geog aphic adjacency
S ep 4: Modelling
Di usion Con olu ional Recu en Neu al Ne wo k (DCRNN);
Inpu : pas 7 days o in ec ion a es; Ou pu : o ecas o he nex 7 days;
Same hype pa ame e s applied o bo h Glasgow and Edinbu gh;
Da a spli ch onologically in o 70% aining, 15% alida ion, 15% es ing
S ep 5: E alua ion
P edic ions ans o med back o o iginal in ec ion a e uni s;
Pe o mance assessed wi h MAE, RMSE, MAPE, and R2
Fig. 2: S ep-by-s ep wo k low o he s udy: da a collec ion, p ep ocessing, g aph
cons uc ion, modelling, and e alua ion.
Spa ially Explici G aphs o COVID-19 Fo ecas ing 5
(3) Mobili y lows:
wmob
ij =1
2 ij + ji, i =j, (3)
whe e ij is he p opo ion o ips om zone i o zone j. The ma ix is sym-
me ized o ob ain an undi ec ed ep esen a ion, whe e edge weigh s e lec he
a e age o lows in bo h di ec ions. Al hough OD ma ices a e inhe en ly di ec-
ional, we adop a symme ic ep esen a ion o wo easons. Fi s , ansmission
isk is bidi ec ional, since mo emen s in bo h di ec ions can acili a e c oss-zone
exposu e. Simila symme ic assump ions a e common in epidemic modelling o
simpli y ansmission s uc u es [3,23]. Second, OD da a a he In e Zone scale
o en con ain sampling noise and di ec ional imbalances; a e aging ij and ji
p oduces a mo e s able connec i i y s uc u e, which imp o es obus ness when
aining g aph neu al ne wo ks.
(4) Mobili y + Road:
WMR =αW mob + (1 −α)W oad,(4)
(5) Mobili y + Geog aphic adjacency:
WMG =αW mob + (1 −α)Wgeo,(5)
which combines dynamic mobili y lows wi h s a ic geog aphic adjacency.
Each ma ix W= [wij]speci ies he weigh ed adjacency s uc u e used in he
DCRNN model, whe e highe alues o wij ep esen s onge spa ial p oximi y
o g ea e mobili y low in ensi y be ween zones.
The coe icien α anged om 0.5 o 0.9, ensu ing ha mobili y con ibu ed
a leas equally o he hyb id s uc u es. This ange e lec s he assump ion
ha mobili y is a dominan d i e o ansmission while allowing sensi i i y
o spa ial cons ain s. Figu e 3 p esen s isualisa ions o he g aph s uc u es,
o e ing an in ui i e illus a ion o he di e ences be ween geog aphic adjacency,
oad connec i i y, and mobili y lows.
2.3 Model and T aining
The DCRNN a chi ec u e was employed, in eg a ing di usion con olu ion wi h
ecu en uni s o cap u e empo al dynamics alongside spa ial di usion o e
g aphs. Fo ecas ing was o mula ed as a mul is ep ask in which a 7-day slid-
ing inpu window was used o p edic he subsequen 7 days a he In e Zone
le el. The window ad anced by one day a each s ep, gene a ing o e lapping
inpu –ou pu sequences ac oss he en i e ime se ies. This olling-window design
unc ions as a ime-se ies analogue o c oss- alida ion, maximising da a u ilisa-
ion while p ese ing empo al o de ing.
To ensu e compa abili y be ween Glasgow and Edinbu gh, all model and
aining hype pa ame e s we e kep iden ical ac oss ci ies, including he numbe
o ecu en laye s (3), hidden uni s pe laye (128), he maximum di usion s ep
6 F. Li e al.
(a) Geog aphic adjacency.
(b) Road ne wo k.
(c) Mobili y lows.
Fig. 3: Visualisa ions o he g aph s uc u es.
Spa ially Explici G aphs o COVID-19 Fo ecas ing 7
(3), he sequence leng h and o ecas ho izon (bo h se o 7), he use o dual
andom-walk il e s, and cu iculum lea ning wi h a decay o 300 s eps. T aining
hype pa ame e s such as he lea ning a e (0.001 wi h mul i-s ep decay), ba ch
size (32), d opou (0.1), g adien clipping (5), and ea ly-s opping pa ience (30)
we e also ixed. Keeping hese hype pa ame e s cons an p e en s di e ences in
model capaci y o op imisa ion beha iou om in luencing pe o mance, allowing
he analysis o isola e he impac o g aph cons uc ion s a egies.
Da a we e spli ch onologically in o aining (70%), alida ion (15%), and
es ing (15%). The spli poin s we e selec ed no only by p opo ion bu also
wi h e e ence o luc ua ions in weekly in ec ion a es, ensu ing ha each subse
con ained ep esen a i e epidemic dynamics. La e s ages o he pandemic we e
ese ed o ou -o -sample e alua ion.
All in ec ion a es we e s anda dised using Z-sco e no malisa ion, de ined as
x′=x−µ
σ,(6)
whe e xis he aw in ec ion a e, µ he mean, and σ he s anda d de ia ion.
The in e se ans o ma ion was applied be o e compu ing he e alua ion me ics
o e ain in e p e abili y in he o iginal uni s.
2.4 E alua ion Me ics
Pe o mance was assessed using ou s anda d e o me ics widely applied in
o ecas ing s udies [14,19]: mean absolu e e o (MAE), oo mean squa ed e -
o (RMSE), mean absolu e pe cen age e o (MAPE), and he coe icien o
de e mina ion (R2).
Since MAPE is unde ined when yi= 0 and uns able o small denomina o s,
we adop a modi ied e sion MAPEεby adding a small cons an ε= 10−3 o
he denomina o . This ensu es nume ical s abili y o small-a ea incidence a es
while e aining he in e p e abili y o pe cen age e o . The me ics a e de ined
as:
MAE =1
n
n
X
i=1
|yi−ˆyi|,(7)
RMSE =
u
u
1
n
n
X
i=1
(yi−ˆyi)2,(8)
MAPEε= 1
n
n
X
i=1
yi−ˆyi
yi+ε!×100%,(9)
R2= 1 −Pn
i=1(yi−ˆyi)2
Pn
i=1(yi−¯y)2.(10)
8 F. Li e al.
3 Resul s
3.1 Glasgow
Table 1 shows ha hyb id g aphs in eg a ing mobili y lows wi h geog aphic
adjacency clea ly ou pe o m he single sou ce al e na i es in Glasgow. The bes
pe o ming speci ica ion assigns 60% weigh o mobili y and 40% o geog aphic
adjacency, achie ing MAE = 80.01, RMSE = 133.22, MAPE = 48.89%, and R2=
0.90. Compa ed wi h he pu e con igui y g aph, his co esponds o a educ ion
o 1.6% in MAE, 14.83% in RMSE, and 4.97% in MAPE, highligh ing he alue o
inco po a ing mobili y in o ma ion. The second and hi d anked con igu a ions
(90% and 50% mobili y weigh s, espec i ely) yield simila ly compe i i e esul s,
wi h RMSE alues a ound 135–138.
To isualise he mul i-me ic pe o mance di e ences mo e clea ly, Figu e 4
p esen s a ada cha ha summa ises all ou no malised me ics (MAE, RMSE,
MAPE, and R2). The hyb id mobili y–adjacency g aphs o m he la ges en-
closed a ea in he ada plo , indica ing consis en ly s ong pe o mance ac oss
all e alua ion me ics. In con as , single-sou ce g aphs (con igui y, oad dis-
ance, o mobili y alone) show mo e une en p o iles, pe o ming well on some
me ics bu wo se on o he s. This g aphical summa y ein o ces he abula ed
esul s, showing isually ha mobili y-enhanced hyb id g aphs p o ide he mos
balanced and accu a e o ecas s in Glasgow.
Table 1: Fo ecas ing pe o mance o di e en g aph s uc u es in Glasgow.
Models MAE (/100k) RMSE (/100k) MAPE (%) R2
Geog aphic adjacency 81.61 148.05 43.92% 0.88
Road 85.89 155.46 47.50% 0.87
Mobili y 81.47 141.32 47.62% 0.89
Mobili y + Road
50% + 50% 82.62 144.64 48.88% 0.89
60% + 40% 84.20 150.78 46.47% 0.88
70% + 30% 83.15 147.41 46.88% 0.88
80% + 20% 84.09 153.50 46.33% 0.87
90% + 10% 83.91 147.55 48.84% 0.88
Mobili y + Geog aphic adjacency
50% + 50%(3) 82.32 137.81 48.43% 0.90
60% + 40%(1) 80.01 133.22 48.89% 0.90
70% + 30% 80.65 135.89 50.27% 0.90
80% + 20% 82.38 136.45 52.55% 0.90
90% + 10%(2) 80.33 135.45 47.55% 0.90
No e: Supe sc ip s (1),(2), and (3) deno e he op h ee models anked by o e all
pe o mance ac oss MAE, RMSE, MAPE, and R2. MAE and RMSE a e epo ed in
uni s o he in ec ion a e, de ined as daily COVID-19 cases pe 100,000 popula ion
(/100k).
The aining and alida ion loss cu es in Figu e 5(a–c) con i m ha he
h ee op hyb id models con e ge smoo hly, wi h alida ion loss s abilising a a
Spa ially Explici G aphs o COVID-19 Fo ecas ing 9
Fig. 4: Rada cha compa ing he o ecas ing pe o mance o all g aph designs
in Glasgow. Fo MAE, RMSE, MAPE: sco e = 1 −x−xmin
xmax−xmin . Fo R2: sco e =
x−xmin
xmax−xmin .
simila le el ac oss uns. These pa e ns indica e ha inco po a ing mobili y in o
spa ial g aphs enhances he model’s abili y o cap u e c oss-zone ansmission
pa hways in Glasgow.
3.2 Edinbu gh
In con as , Table 2 and Figu e 6 highligh a di e en anking o g aph s uc u es
in Edinbu gh. The bes pe o mance is achie ed by he geog aphic adjacency
g aph, wi h MAE = 78.24, RMSE=146.92, MAPE = 43.81%, and R2= 0.88.
The second bes con igu a ion is he mobili y g aph, ollowed by he hyb id o
60% mobili y and 40% adjacency. This o de ing sugges s ha local con igui y e-
la ionships a e mo e p edic i e o in ec ion pa e ns in Edinbu gh han mobili y-
enhanced g aphs, which con as s wi h he indings o Glasgow. Figu e 5(d– )
shows ha he op h ee models in Edinbu gh also exhibi s able con e gence,
hough alida ion loss emains sligh ly highe han in Glasgow, consis en wi h
he o e all la ge e o s epo ed in he me ics.
A key obse a ion is he he e ogenei y in he ole o mobili y ac oss ci ies. In
Glasgow, mobili y-in o med hyb id g aphs deli e clea imp o emen s o e spa-
ial o oad-based s uc u es, whe eas in Edinbu gh, simple geog aphic adjacency
p o ides he bes o ecas s. These indings imply ha he u ili y o mobili y-
enhanced g aphs is con ex dependen , shaped by ci y-speci ic ac o s such as
a el in ensi y, popula ion dis ibu ion, and adminis a i e zoning. They also
unde sco e he impo ance o ailo ing g aph cons uc ion s a egies o local
condi ions a he han assuming uni e sal supe io i y o mobili y-based ep e-
sen a ions.
