Advances in Systems Science and Applications (2014) Vol.14 No.2 144-157

Disease Universe: Visualisation of Population-Wide
Disease-Wide Associations

Max Moldovan1,2, Ruslan Enikeev3, Shabbir Syed-Abdul4, Phung Anh
Nguyen4, Yo-Cheng Chang4 and Yu-Chuan Li4

1Australian Institute of Health Innovation, University of New South Wales, Level 1
AGSM Building, Sydney NSW 2052, Australia

2School of Population Health, South Australian Health & Medical Research Institute
(SAHMRI), University of South Australia, North Terrace, Adelaide SA 5000, Ausralia

3The APAC Sale Group, 3 Petain Rd., #05-07, Singapore 208108
4Graduate Institute of Medical Informatics, College of Medical Science and Technology,

Taipei Medical University, Taipei, Taiwan

Abstract
Over a lifespan, a human organism is affected by multiple disorders of different
origin and severity. We apply a force-directed spring embedding graph layout
approach to electronic health records in order to visualise population-wide asso-
ciations among human disorders as presented in an individual biological organism.
The introduced visualisation is implemented on the basis of the Google maps plat-
form and can be found at http://disease-map.net. We argue that the suggested
method of visualisation can both validate already known specifics of associations
among disorders and identify novel, never noticed association patterns.
Keywords systems biology; phonemics; electronic health records; visualisation;
graph layout

1 Introduction

It is known that many human disorders are positively associated, accompanying
each other due to various, often unknown, genetic, bio-pathological or common
risk factors [?]. There is also evidence that some disorders tend to be associated
negatively, playing a preventative role against each other, or due to other hy-
pothesised but not properly understood reasons [2-3]. We use population-wide
electronic health records data to visualise how human disorders are positioned
against each other in a population with respect to an individual biological organ-
ism. By doing so, we attempt to execute a systems biology approach in order to
reveal the presence of common functional mechanisms influencing pathogenesis
behind groups of human disorders through biological, epidemiological or environ-
mental factors. It is important to note that, due to the specifics of electronic
health records [?], together with biological and environmental mechanisms, the
method may reflect certain aspects of a healthcare system. For example, close-
ly related but distinct diagnoses are often recorded against the same medical
condition. This would induce a positive association between disorders due to
healthcare administration rather than biological reasons.



Advances in Systems Science and Applications (2014) Vol.14 No.2 145

So far the attempts to characterise interactions between human disorders, as
observed in a population, have been mainly implemented through network ap-
proaches [5-6]. While being no doubt informative, network approaches predomi-
nantly focus on positive association patterns. One of the alternative approaches
to the disease-wide association analysis has been reported by Rzhetsky et al. [?].
Among other things, the authors objectively characterised probabilities of a per-
son to be affected by a certain disorder, say A, given that the same person has
been actually affected by an alternative disorder, say B. This approach allowed
us to identify not only positively associated disorders, but also disorders asso-
ciated negatively – the disorders “competing for the same nucleotide site in the
human genome”, as hypothesised by the authors. The shortcoming of the study
by Rzhetsky et al. [?] is that the authors utilised patient records obtained from
a single hospital, also covering a limited pre-selected number of diseases. In the
following study, we use electronic health records covering the entire population
and much wider range of human disorders.

A central objective of the method and its implementation presented below is
to visualise association patterns, both positive and negative, among human dis-
orders as observed in an entire population. This would bring to the surface not
only already known empirical facts, but also information not previously avail-
able. Given that the method implementation can reflect empirical information
already known, e.g., a strong positive association between hypertension and di-
abetes mellitus, as well as unexpected association patterns never noticed before,
the presented visualisation can serve as a starting point for formulating novel
testable hypotheses in the areas of healthcare and medicine. This can further
lead to a better understanding of the complex unobserved dynamics of human
disorders intersecting in a single biological body. Such understanding can prac-
tically improve the delivery of healthcare and medical treatments.

2 Material and methods

2.1 Force-directed spring embedding graph layout algorithm

Imagine a single pair of nodes, A and B, positioned on a plane and connected
by a spring of a certain natural length, δAB. When the distance between A and
B is exactly dAB = δAB, the spring is in a state of equilibrium, creating neither
attraction nor repulsion forces between the nodes (Fig.1(a)). Moving A and B
further apart from each other would create an attraction force (Fig.1(b)), while
moving A and B closer to each other would create a repulsion force between the
nodes (Fig.1(c)).
Given the values of initial required distances δij between multiple pairs of nodes,
it is rarely possible to locate more than three nodes on a plane such that all
required distances between them are satisfied exactly. In fact, it is not even always



146 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

Fig.1 A single pair of nodes in three possible states. (a) Equilibrium: the nodes
neither attract nor repulse; (b) Attraction force: the nodes are shifted far away
from equilibrium and attempt to attract; (c) Repulsion force: the nodes are closer
than if they were in equilibrium and attempt to repulse.

possible to locate three nodes, keeping the pairwise distances intact, see Fig.2.
When distances between the nodes are not satisfied, springs connecting them are
not in equilibrium, creating a certain force – either attraction or repulsion.

Fig.2 A hypothetical system of three nodes. The initial distances δij , ij ∈
{AB,AC,BC} between the nodes are given by the theoretical lines A′B′, B′C ′

and A′C ′. The joint length of A′B′ and A′C ′ is less than the length of B′C ′, i.e.,
δAB+δAC < δBC . As a result, for the nodes to connect, one or more of the initial
distances between the pairs have to be distorted. The three possible states of
springs are equilibrium (AB), attraction (AC) and repulsion (BC).

Aggregated forces created by out-of-equilibrium springs can be expressed by
a specific function, drawn from the principle of physics (Hooke’s law), leading



Advances in Systems Science and Applications (2014) Vol.14 No.2 147

to system’s potential energy U . While a force can be positive (attraction) or
negative (repulsion), an energy level is always non-negative irrespective of the
sign of the force. The potential energy of a system of M nodes connected by
springs of varying stiffness can be expressed as follows:

U =
1

2

∑
ij∈K

(
(dij − δ̂ij)

2 · κij
)
, K =

(
M

2

)
, i ̸= j (2.1)

where dij =
√

(Xi −Xj)2 + (Yi − Yj)2 is an Euclidian distance between nodes i

and j with coordinates (Xi, Yi) and (Xj , Yj), respectively, δ̂ij is a natural length
of a spring between nodes i and j, κij ≥ 0 is an arbitrary parameter that defines
the stiffness of a spring between i and j, and K is a number of all possible springs
connecting M nodes, with

(·
·
)
being a binomial coefficient.

By varying pairwise Euclidian distances dij , the force-directed spring embed-
ding graph layout algorithm [7-9] performs a search for the configuration of node
locations such that system’s potential energy U is minimised. By finding the
minimum energy U , we attempt to obtain a shape of a system of nodes in which
competing forces largely compensate each other. Minimising function (??) is a
complicated task due to the presence of multiple local minima, and it can rarely
be guaranteed that a true global minimum is reached, see Appendix for details.
However, we observed that most nodes have nearly constant “designated” loca-
tions with respect to other nodes across alternative local minima achieved when
minimising (??).

2.2 Defining natural distances between human disorders

Observing a (sub)-population of size N , suppose that over period T there were
CA individuals with at least one occurrence of disorder A, and CB individuals
with at least one occurrence of disorder B. Further, CAB individuals presented
with both disorders A and B, each disorder observed at least once over the same
period. Then the information can be summarised as shown by Table 1.

Table 1 Occurrence counts of disorders A and B in population of size N .

Disorder A
Disorder B A present A absent Total
B present CAB · CB

B absent · · −
Total CA − N

Table 1 is an example of a 2 × 2 table with fixed margins. Assuming that
individuals are affected independently of each other (which can be violated, e.g.,



148 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

for infectious diseases), X follows a non-central hypergeometric distribution X ∼
Hyper(N,CA, CB) given by [10-11]:

Pr(X = CAB) =

(
CB
CAB

)(
N−CB

CA−CAB

)(
N
CA

) eθABCAB (2.2)

where max(0, CA+CB−N) ≤ CAB ≤ min(CA, CB), θ ∈ (−∞,+∞) is a log-odds
ratio, and e = 2.718 . . . is the base of a natural logarithm. For algorithm imple-
mentation, conditional maximum likelihood estimates of θ were approximated by
unconditional log-odds ratios:

θ̂AB = ln

(
CAB(N − CA + CAB − CB)

(CA − CAB)(−CAB + CB)

)
(2.3)

where ln(·) is a natural logarithm. Switching the risk factor from being B for A to
being A for B does not effect log-odds estimates. Natural (equilibrium) lengths
of springs between nodes i and j were obtained through the following reversed
expit transform [?, p.121]:

δ̂ij =
exp(−θ̂ij)

1 + exp(−θ̂ij)
(2.4)

where δ̂ij ∈ [0, 1] by construction. Note that the sign on log-odds estimate θ̂ was
changed to the opposite (i.e., reversed), making stronger positive associations
correspond to the smaller values of δ̂ij . We do so in order to make δ̂ij resemble
Euclidian distances between the nodes.

Due to estimation, there is uncertainty about δ̂ values obtained from the da-
ta. Such an uncertainty is usually handled by reporting confidence intervals
corresponding to δ̂. In the present version of method implementation, we in-
tentionally avoided using confidence intervals, p-values or other statistical tools
normally involved in hypothesis testing. We did so in order to reflect the empiri-
cal information contained in the data without any subjective interpretation that
could otherwise be introduced through, for example, the choice of a significance
level.

2.3 Potential alternative implementations

It should be noted that the force-directed spring embedding graph layout method
for visualising relationships among multiple objects, human disorders in our case,
is not the only approach available. Together with network algorithms already
mentioned above, multidimensional scaling [?] and biplot [?] methods are two
more approaches for two-dimensional visualisation of relationships between mul-
tiple objects. However, it is important to emphasise that both methods, at least



Advances in Systems Science and Applications (2014) Vol.14 No.2 149

in their traditional form, focus on similarities between objects, which would cor-
respond to positive associations between disorders as per our current settings.
While an application of alternative methods to our empirical data set would
be interesting and potentially informative, multidimensional scaling and biplot
methods are unlikely to address negative associations between disorders in a
proper way.

At the same time, Euclidian distances between disorders, as specified by (??)
in our study, could be easily reversed, i.e., positive and negative associations can
be made corresponding to longer and shorter Euclidian distances between the
nodes in Figure 1, respectively. Such an alternative vision of the problem could
reveal an entirely new set of empirical information, not reflected by the current
method implementation. We retain this research direction for later investigation.

3 Practical implementation

3.1 Empirical information

The presented visualisation has been motivated by the Internet Map implemen-
tation [?]. We use electronic health records obtained from the Taiwanese national
health insurance research database covering the entire population of Taiwan over
the period of three years (2000-2002). The same three-year observation window of
the maximum available length has been used to record the counts corresponding
to Table 1. Disorder records are based on ICD9-CM (International Classifica-
tion of Diseases, Ninth Revision, Clinical Modification), five-digit version. The
dataset has been stratified into male and female groups. Each of these two groups
has been further stratified into ten age sub-groups, i.e., 0-9, 10-19, . . . , 90+. Each
subject within a sub-group was noted by his of her first insurance claim start-
ing from 01 January 2000, assigned to a certain age-gender group and followed
for the rest of the period ending on 31 December 2002. Codes corresponding to
E and V categories of ICD9-CM (External causes of injury and Supplemental
classification) were excluded from consideration.

3.2 Prevalence threshold and spring stiffness

We compute δ̂ij given by (??) for each observed pair of disorders i and j. An

empirical examination of δ̂ij revealed that log-odds estimates θ̂ij that underlie δ̂ij ,
exhibit anomalous behaviour for smaller counts Ci and Cj , i.e., they tend to be
much larger than it would be expected under a random process. Such an anomaly
would bias the attention of an optimisation algorithm applied to (??) towards
diseases of a smaller prevalence. We attributed this anomaly to exceptionally high
positive associations between certain low prevalence pairs of disorders as observed
in the context of the entire population and reflected by odds ratio estimates. In
particular, the expected value of X in the hypergeometric distribution function



150 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

Fig.3 Broad disease categories as per ICD9-CM classification and the correspond-
ing colour codes as displayed at http://disease-map.net.

(??) when θ = 0, i.e., there is no association between disorders, is given by [?]:

RECij =
CiCj

N
(3.5)

where REC stands for Random Expected Co-occurrence. We interpret RECij

as a value reflecting “visibility” of co-occurrences between i and j, with higher
visibility (i.e., greater values of RECij) leading to more reliable empirical out-
comes Cij in Table 1. Keeping this interpretation in mind, we have executed
the following ad hoc solution for dealing with the identified anomaly. Firstly, we
imposed the threshold C =

√
2N on disease occurrence counts. This guarantees

that RECij > 2 for all possible Ci and Cj . The meaning behind this restriction

is to ensure that only theoretically “visible” θ̂ij estimates are used for visual-
isation. The cost is that we dismissed smaller prevalence disorders that never
exceeded RECij = 2 in any of the age-gender groups. Secondly to the imposed
lower limit on the observed occurrence counts, we set the stiffness parameter of
a spring between pairs i and j to κij = ln(RECij). This modification makes
sure that less theoretically “visible” co-occurrences Cij are given less importance
when minimising the energy function (??).

3.3 Visualisation

The Google maps platform (https://developers.google.com/maps/) was used to
visualise the outcomes. The sizes of the nodes are set to be proportional to the



Advances in Systems Science and Applications (2014) Vol.14 No.2 151

observed disease prevalence in the corresponding age-gender stratified sub-groups.
The colour codes of the nodes correspond to the broad disease categories as per
ICD9-CM classification, see Fig.3. All maps are displayed in the same coordinate
system with the same scale so that they can be compared against each other.

4 Some examples of using the maps

4.1 An accidental proximity?

When exploring the maps, some regions and mutual locations can attract atten-
tion due to certain, often subjective, reasons. For example, observing the map
for females age 40-49 (F40+), in the central region towards south-east we find
that peptic ulcer (ICD9-CM 533) is located side by side with neurotic disorders
(ICD9-CM 300). Have these disorders fallen close together by chance? A search
through the medical literature has quickly identified that an abnormal association
between peptic ulcer and neurotic disorders was noticed years ago [15-16]. Look-
ing at a wider category of digestive system disorders (ICD9-CM 520 to 579), this
mutual location pattern remains largely the same over multiple maps (e.g., see
M30+, M40+ and F30+), leading to several testable hypotheses. One example
of such a hypothesis would be: “There is no direct psychosomatic link between
digestive system disorders and neurotic disorders”.

Viewed from a different angle, the literature coverage on pairs of closely located
disorders appears not to be accidental. Syed-Abdul et al. [?] found that the
proximity of disorder pairs is positively correlated with the degree of literature
coverage, the latter being represented by a number of hits for a pair of disorders
returned from an appropriate query to the PubMed search engine.

4.2 Unlikely neighbours: a cancer-schizophrenia association puzzle

The topic of observed evidence of associations between schizophrenia and various
cancers has been widely debated [?]. Evidence tends to point towards the presence
of a negative association between schizophrenia and several cancers, even though
there is no absolute consensus [20-21]. A shared genetic architecture has been
proposed as a reason for observed associations [20-21]. Alternatively, there is
evidence that the chances of schizophrenia patients being timely diagnosed with
certain types of cancer, on average, are lower than for general population non-
schizophrenic patients [?].

Exploring the maps, it can be found that schizophrenic disorders (ICD9-CM
295) consistently fall on the southern border of the maps, sometimes being the
most distant points from the imaginary centre of a “galaxy”, e.g., see M50+.
Interestingly, various types of cancers also regularly fall on the same southern
border even though, consistently with the literature, association estimates for
schizophrenia-cancer pairs regularly cross to the negative side, i.e., δ̂ij given by
(??) exceeds the value of 0.5.



152 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

One potential explanation for such an anomaly would be that schizophrenia
and some cancers have closely related underlying causes revealed through similar
relationships with other disorders. In this way, schizophrenia and cancers are
placed to their southern border locations by the forces generated within the sys-
tem. Often being negatively associated, schizophrenia and cancers are like two
sides of the same coin, “competing for the same nucleotide site in the human
genome” as per Rzhetsky et al. [?] vision, but potentially disassociated due to
deeper, not properly recognised and understood reasons which are still to be iden-
tified and investigated. The visualisation we introduced is a tool for originating
and directing such investigations.

5 Conclusion

Electronic health records have become an integral part of national healthcare
systems worldwide, and it is essential to comprehensively utilise the information
contained in the growing number of databases. The method we introduced is one
of the effective and informative tools for doing so. While the current realisation
of the method has its obvious limitations, the presented maps are the first im-
plementation of this kind and intended to set a reference benchmark for further
developments in the same direction. A formal empirical validation of the intro-
duced visualisation is beyond the scope of this paper, but based on the broad
examination of the resulted maps, we argue that the presented implementation
can both assist with validation of already known phenomena as well as with i-
dentification of novel, previously never noticed, association patterns related to
functional aspects of medicine and healthcare. We suggest that the maps be
used for generating testable hypotheses and invite the reader to explore the vast
amount of information contained in them at http://disease-map.net.

Acknowledgements: We are grateful to Chris Lloyd and Nicholas Nechval who
provided their constructive feedback that helped to improve the manuscript. We
thank Hanna Kalkova and Andrey Stepanov for preparing Figures 1 and 2.

Author Contributions: M.M., R.E., S.S-A. and Y-C.L. invented and developed
the concept (IPN-13-000062, NewSouth Innovations). S.S-A., M.M., Y-C.L. and
Y-C.C. performed selective biological validation. R.E. wrote software, implement-
ed the method and developed the website. M.M. and R.E. wrote the manuscript.
Y-C.L. obtained the data. PA.N. organised and validated the data.

Funding: No formal targeted funding was allocated towards the project.

Competing financial interests: The authors declared no competing financial
interests.



Advances in Systems Science and Applications (2014) Vol.14 No.2 153

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Corresponding authors
MaxMoldovan (max.moldovan@gmail.com) and Yu-Chuan Li (jaak88@gmail.com)

Appendix: Energy minimisation method
Finding a global minimum of (??) is a complicated task due to the presence of
multiple local minima of this function. Different approaches of global minimisa-
tion can be applied, but it can be rarely known when and if the global minimum



Advances in Systems Science and Applications (2014) Vol.14 No.2 155

is reached, unless a minimum energy level is known in advance. Our current
implementation of energy minimisation is to use multiple local searches with the
conjugate gradient algorithm from random starting positions in order to obtain
a master map – the map that includes diseases across the entire spectrum of age
groups and both genders. In each of multiple attempts, the nodes are dropped
on the map with random positions (X,Y ), and the conjugate gradient algorithm
runs searching for the closest local minimum of U in (??). If the new local mini-
mum is less than the best (smallest) minimum recorded over previous attempts,
it becomes the new best minimum. The procedure is repeated until the best min-
imum stops changing even after a reasonably large (4000, in our implementation)
number of random allocation attempts, see Algorithm ??. The computational
complexity of the algorithm is O(n2).

The obtained master map served as a collection of starting points for the age-
gender stratified maps, see Algorithm ??. Minimising (??) from a single set of
starting points leads to a local minimum that can be further improved by apply-
ing the minimisation approach used for obtaining the master map. However, we
still used minimisation from the single set of starting points in order to make the
maps comparable across age groups and genders. Table A1 reports the achieved
minimum energy levels using “partial” minimisation as per Algorithm ?? com-
pared to the “full” minimisation implemented through Algorithm ??.



156 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

Table A1 Minimum achieved energy levels from partial and (attempted) full
minimisation approaches.

Group Subjects followed (N) Disorder numbers Partial Full Per cent improve

F 0-9 1,677,365 565 7,807.96 7,700.69 1.39
F 10-19 1,595,057 743 9,470.09 9,166.34 3.31
F 20-29 1,780,095 1041 22,897.04 22,268.39 2.82
F 30-39 1,765,866 1136 25,914.10 25,387.60 2.07
F 40-49 1,631,968 1243 31,126.22 30,913.37 0.69
F 50-59 930,496 1251 33,451.35 33,334.80 0.35
F 60-69 711,096 1271 36,129.76 36,056.87 0.20
F 70-79 427,821 1177 29,935.15 29,857.36 0.26
F 80-89 141,225 783 10,802.72 10,773.33 0.27
F 90-99 8,532 176 318.26 315.74 0.80

M 0-9 1,827,447 630 10,068.44 9,910.12 1.60
M 10-19 1,678,415 721 9,451.03 9,346.16 1.12
M 20-29 1,767,163 859 12,532.53 12,345.32 1.52
M 30-39 1,737,715 948 14,263.07 14,099.18 1.16
M 40-49 1,577,320 1090 19,485.22 19,454.02 0.16
M 50-59 898,150 1065 20,296.22 20,247.20 0.24
M 60-69 692,061 1163 26,737.58 26,563.97 0.65
M 70-79 532,308 1225 30,740.90 30,622.10 0.39
M 80-89 133,480 781 10,636.67 10,599.66 0.35
M 90-99 4,769 151 240.25 238.70 0.65

Master 21,518,574 2298 − 130,381.91 −



Advances in Systems Science and Applications (2014) Vol.14 No.2 157

Algorithm 1 Energy minimisation for the master map.

Require: γ ← 0.01 /* tolerance for the change in objective function (??)
Require: s← 1 /* initial step size
Require: τ ← 0.9 /* step decrease rate
Require: smin ← 0.000001 /* minimum step tolerance

Require: δ̂ij for K =
(
M
2

)
pairs, i ̸= j /* pairwise natural lengths given by (??)

Require: Ucurrent ← +Inf /* current energy level to be reduced
Require: cc← 0 /* random positions attempts counter
Require: ccmax ← 4000 /* maximum number of attempts with no energy reduction

while (cc < ccmax) do
(X0, Y0)← random() /* drop nodes at random positions
U0 ← fE(X0, Y0) /* value of objective function (??)
G← {−∇ (fE(X0, Y0))} /* define antigradients for the first step
(∆X,∆Y )← fG(G) /* step direction
(X,Y )← (X0, Y0) + (∆X,∆Y ) · s /* current coordinates of nodes
U ← fE(X,Y ) /* current value of objective function (??)
∆U ← (U0 − U) /* change in energy
while (∆U > γ) & (s > smin) do

GC ← {∇C (fE(X0, Y0;X,Y ))} /* evaluate conjugate gradients
(∆X,∆Y )← fCG(GC) /* step direction
(Xtemp, Ytemp)← (X,Y ) + (∆X,∆Y ) · s /* trial coordinates of nodes
U ← fE(Xtemp, Ytemp) /* current value of the objective function
if U < U0 then

∆U ← (U0 − U) /* update change in energy
U0 ← U /* update preceding energy value
(X0, Y0)← (X,Y ) /* update preceding coordinates
(X,Y )← (Xtemp, Ytemp) /* assign the values of current coordinates

else
s← s · τ /* reduce step size

end if
end while
if U0 < Ucurrent then

Ucurrent ← U0 /* update minimum energy value
(Xcurrent, Ycurrent)← (X,Y ) /* update coordinates
cc← 0 /* set attempts count to zero

else
cc← cc+ 1 /* next attempt

end if
end while
(Xmaster, Ymaster)← (Xcurrent, Ycurrent)
return (Xmaster, Ymaster) /* nodes’ coordinates under minimum energy achieved



158 Max Moldovan: Disease Universe: Visualisation of Population-Wide Disease-Wide ...

Algorithm 2 Energy minimisation for age and gender stratified maps.

Require: γ ← 1e-5 /* tolerance for the change in objective function (??)
Require: s← 1 /* initial step size
Require: τ ← 0.9 /* step decrease rate
Require: smin ← 0.000001 /* minimum step tolerance

Require: δ̂ij for K =
(
M
2

)
pairs, i ̸= j /* pairwise natural lengths given by (??)

(X0, Y0)← (Xmaster, Ymaster) /* use coordinates from the master map as starting points
U0 ← fE(X0, Y0) /* the value of objective function (??)
G← {−∇ (fE(X0, Y0))} /* define antigradients for the first step
(∆X,∆Y )← fG(G) /* step direction
(X,Y )← (X0, Y0) + (∆X,∆Y ) · s /* current coordinates of nodes
U ← fE(X,Y ) /* current value of objective function (??)
∆U ← (U0 − U) /* change in energy
while (∆U > γ) & (s > smin) do

GC ← {∇C (fE(X0, Y0;X,Y ))} /* evaluate conjugate gradients
(∆X,∆Y )← fCG(GC) /* step direction
(Xtemp, Ytemp)← (X,Y ) + (∆X,∆Y ) · s /* trial coordinates of nodes
U ← fE(Xtemp, Ytemp) /* current value of the objective function
if U < U0 then

∆U ← (U0 − U) /* update change in energy
U0 ← U /* update preceding energy value
(X0, Y0)← (X,Y ) /* update preceding coordinates
(X,Y )← (Xtemp, Ytemp) /* assign values of current coordinates

else
s← s · τ /* reduce step size

end if
end while
(Xstratif , Ystratif )← (X,Y )
return (Xstratif , Ystratif ) /* nodes’ coordinates under minimum energy achieved