arXiv:1802.05856v3 [q-bio.MN] 18 Mar 2018

Algorithmic Complexity and Reprogrammability of
Chemical Structure Networks∗
Hector Zenil1,2,3,4 , Narsis A. Kiani1,2,3,4 , Ming-Mei Shang 2,3
and Jesper Tegnér2,3,5
1
Algorithmic Dynamics Lab, Centre for Molecular Medicine,
Karolinska Institute, Stockholm, Sweden
2
Unit of Computational Medicine, Department of Medicine,
Karolinska Institute, Stockholm, Sweden
3
Science for Life Laboratory, SciLifeLab, Stockholm, Sweden
4
Algorithmic Nature Group, LABORES for the Natural and
Digital Sciences, Paris, France
5
Biological and Environmental Sciences and Engineering Division,
Computer, Electrical and Mathematical Sciences and Engineering
Division, King Abdullah University of Science and
Technology (KAUST), Kingdom of Saudi Arabia
{hector.zenil, jesper.tegner}@ki.se

Abstract
Here we address the challenge of profiling causal properties and
tracking the transformation of chemical compounds from an algorithmic perspective. We explore the potential of applying a computational
interventional calculus based on the principles of algorithmic probability to chemical structure networks. We profile the sensitivity of the
elements and covalent bonds in a chemical structure network algorithmically, asking whether reprogrammability affords information about
thermodynamic and chemical processes involved in the transformation
of different compound classes. We arrive at numerical results suggesting a correspondence between some physical, structural and functional
properties. Our methods are capable of separating chemical classes
that reflect functional and natural differences without considering any
information about atomic and molecular properties. We conclude that
these methods, with their links to chemoinformatics via algorithmic,
probability hold promise for future research.
Keywords: molecular complexity; algorithmic probability; KolmogorovChaitin complexity; causality; causal path; information signature; chemical compound complexity; algorithmic information theory; Shannon
entropy
∗

An online implementation to estimations of graph complexity is available online at
http://www.complexitycalculator.com

1

1

Background and Preliminaries

One of the major challenges in modern physics is to provide proper and
suitable representations of network systems for use in fields ranging from
physics [3] to chemistry [7]. A common problem is the description of order
parameters with which to characterize the ‘complexity of a network ’. Graph
complexity has traditionally been characterized using graph-theoretic measures such as degree distribution, clustering coefficient, edge density, and
community or modular structure.
A previous algorithmic information-theoretic view of systems toxicity
applicable both to network analysis and pharmacokinetic analysis has been
proposed [13], with the overarching aim of not only describing but also
seeking out causal mechanisms. The suggestion was that since the problem of designing new compounds, aiming to develop drugs, for new targets
is challenging, whereas the prediction problem is easier from an inference
point-of-view compared to elucidating the mechanisms driving toxicity, complementary approaches are warranted.
For example, instead of engineering a drug to target a unique pathway
or mutation of a tiny subset of diseases, drug repositioning involves starting
with approved drugs to find combinations that can be used to treat diseases other than the ones they were designed for, with the advantage that
approved drugs can bypass much regulation if we correctly control for the
effects they can have. Thus prediction and simulation are key. This means
that the whole field has to move towards causal modelling and functional
inference rather than employing traditional statistical and purely geometric
approaches (e.g. distances between compounds or grid-based docking).
Here we are interested in combining techniques originating in fundamental mathematics and theoretical computer science to take a fresh look
at long-standing challenges in molecular complexity from an algorithmic
information perspective as applied to networks [33, 28].
Algorithmic information indices may facilitate the characterization of
some properties of chemical compounds. Statins, for example, are associated with the heart and cholesterol, while morphine, codeine and heroin
share structural properties and effects. Algorithmic information-theoretic
approaches like the one showcased here are concerned with predictive causal
models, going beyond statistical/descriptive approaches (such as structural
alignments). This is important because, for statins, for example, block the
cholesterol synthesis pathway by inhibiting the HMG-CoA reductase because similarity to HMG-CoA structure, which is the rationale used to treat
cardiovascular diseases in patients. The algorithmic approach deployed here
2

is thus equipped to find model-based mechanistic candidates for this kind
of causal processes involved in the transformation and interactions of compounds. Here we will focus on one kind of representation namely chemical
structure networks, first independent of physical properties that can then be
tested and connected back to find what is a consequence of the compound
causal structure/topology and what is a consequence of other intrinsic features such as atomic charge and thermodynamic constraints.
For this approach we will use the techniques and methods developed in
[33, 31]. The basic idea is to estimate the likelihood of similarity between
compounds based on an induced partition of possible common underlying
mechanisms (models are found by an exhaustive algorithm that explains
small pieces of the data). This is, in general, a hard if not impossible
task (uncomputable), but approximations have been shown to be useful
and new numerical methods have been advanced that are complementary
to previous approaches such as the use of lossless compression algorithms to
approximate algorithmic complexity, which are very limited at accounting
for causation [30, 32]. Moreover, even when the method is based on the idea
of finding minimal programs, the more practical aim is to find any, or a set
of programs, explaining the data rather than the smallest one, and so the
problem becomes computationally feasible [9, 21, 27, 30].

1.1

Chemical notation

There are two main notations for chemical substances. The simplified molecularinput line-entry system or SMILES is an ASCII string specification describing the structure of a chemical. The string is obtained by printing the symbol
nodes encountered in a depth-first tree traversal of the chemical graph. The
chemical graph is first trimmed to remove hydrogen atoms, and cycles are
broken to turn it into a spanning tree. Where cycles have been broken,
numerical suffix labels are included to indicate the connected nodes, and
parentheses are used to indicate points of branching on the tree. For example, nicotine is written as CN1CCC[C@H]1c2cccnc2. A SMILES string thus
encodes and contains information about a molecule and is an upper bound
of its information content.
A more standardized notation in chemistry is the IUPAC International
Chemical Identifier or InChI, another textual identifier for chemical substances, its chief advantage over SMILES being that the InChI algorithm
converts the structural information of the chemical substance in a 3-step
process that can be tweaked to a desired level of structural chemical detail, except for an unchanged substring representing the substance (called
3

the main layer). The algorithm then removes redundant information, keeps
as much of the information of the structure as desired, and encodes it in
a string. In some sense, InChI is thus a tighter bound of the algorithmic
complexity of the chemical structure captured by ASCII strings and an improvement over SMILES.

1.2

Molecular complexity

The concept of molecular complexity has been shown to be relevant in the
design of syntheses through minimizing the sum of molecular complexities of
the synthetic intermediates [10]. A number of proposals have been made in
the literature for defining molecular complexity. For example, enumerations
of graph invariants for comparing chemical structures have been used for at
least 3 decades, and QSAR regression models [16] for even longer. Different
approximations emphasize different aspects of the molecule and are heavily
observer dependent, because the observer has to make a pre-selection of
features of interest (e.g. clustering coefficient, some eigenvalues in graph
spectra, degree distributions, etc.). This is what the chemistry community
has been doing with what they call “chemical fingerprints”, and prior to
that in molecular formulae comparing degree distributions.
In 1981, Bertz [5] introduced a measure of molecular complexity by
applying Shannon’s entropy to the distribution of subgraphs in molecular
graphs. That was the starting point of a systematic search in chemical theory for relevant measures of molecular complexity. Ever since, graph and
molecular complexity measures have focused on statistics of the topological properties of graphs, such as the size of the non-repeating subgraph
set, among similar approaches. For recent results and a survey see [4, 15],
including a proposal for using lossless compression as a graph complexity
index [17]. An up-to-date review of mainstream techniques in the area of
molecular networks can be found in [8].
Molecular complexity is not easy to define or to quantify, and all previous approaches have focused on combinatorial or statistical properties of
the molecular graphs, either as a function of bond connectivities, specificity
of structures or diversity of elements. Researchers agree that the complexity of a molecule increases with increasing size, increasing branching, and
increasing cyclicity for acyclic and cyclic structures [19], but no computable
measure can cover all possible enumerable computable features like these
(both currently defined and undefined) at the same time. Indeed, a more
universal and robust measure of molecular complexity should take into account all these features of interest at the same time, without having to
4

enumerate them explicitly or to define an ad-hoc measure for each of them.
A measure that only focuses on some of these properties in the expectation
that it could later be generalized to be able to deal with other properties is
out of the question.
Small Molecules or compounds are commonly represented by their skeletal molecular graphs (see Fig. 1A). That is, the union of a set of points,
symbolizing atoms other than hydrogen, and a set of lines, symbolizing
molecular bonds. A typical similarity formula (see Fig. 1E-F) is given by
the total number of elements in the bin 0 divided by the square of the total
count of atoms of the largest molecule. The closer to 0, the more dissimilar. The formula can be relaxed by taking near 0 bin elements, but this
only works for the most simple cases of structural similarity. This kind of
approach is descriptive rather than predictive. For example, regardless of
different biological mechanisms of action, aspirin and statins have shown
similar beneficial effects on cardiovascular diseases (CVDs) at population
level, combination usage of the two drugs has additive effects, but it is not
yet clear in what precise ways aspirin and statins differ, while the general
agreement is that they possess similarities (e.g. accumulating evidence from
basic and observational research demonstrate the anti-inflammatory effects
of both drugs contribute to CVDs treatment, and combined usage of two
drugs has additive and synergistic effects over the use of only one[1]). It is
not difficult to see how in some cases compounds or compound substructures may share information from complementary regions, e.g. between
drugs and targets, because the structure of the docking entity is energetically and structurally the complement of the docking region of the other
entity, and thus the compounds may display (partially or entirely) similar
classical and algorithmic information properties and estimations.
Molecular complexity is not easy to define or to quantify, and all previous approaches have focused on combinatorial or statistical properties of
the molecular graphs, either as a function of bond connectivities, specificity
of structures or diversity of elements. Researchers agree that the complexity of a molecule increases with increasing size, increasing branching, and
increasing cyclicity for acyclic and cyclic structures [19], but no computable
measure can cover all possible enumerable computable features like these
(both currently defined and undefined) at the same time. Indeed, a more
universal and robust measure of molecular complexity should take into account all these features of interest at the same time, without having to
enumerate them explicitly or to define an ad-hoc measure for each of them.
The number of possible statistical and algorithmic properties in all possible networks is countably infinite, but no effective (computable) measure
5

can account for all of them [14]. This only leave us with uncomputable measures that can serve as general universal measures of complexity equipped to
find any effective (statistical or algorithmic) regularity. Our approach may
find some applications. For example, one may find that low algorithmic
complexity molecules are easier to synthesize or to assemble into larger new
molecules and drugs because high algorithmic complexity molecules would
share fewer physical properties.

1.3

Causality and Algorithmic Probability

The concept of algorithmic complexity [11, 6] is at the core of the challenge of complexity in discrete dynamic systems, as it involves finding the
most statistically likely generating mechanism (computer program) that produces some given data. Formally, the algorithmic complexity (also known
as Kolmogorov-Chaitin complexity) is the length of the shortest computer
program that reproduces the data from its compressed form when running
on a universal Turing machine.
We follow the so-called Coding Theorem (CTM) and Block Decomposition Methods (BDM) as introduced in [9, 21, 27, 30], based on the seminal
concept of Algorithmic Probability [22, 12], which in turn is strongly related to algorithmic complexity [11, 6]. The only parameters used for the
decomposition of BDM as suggested in [30] was the maximum 12 for strings
and 4 for arrays given the current best CTM approximation [21] based on
an empirical distribution based on all Turing machines with up to 5 states,
and no string/array overlapping decomposition for maximum efficiency (as it
runs in linear time) and for which the error (due to boundary conditions) is
bounded [30]. However, the algorithm introduced here is independent of the
method used to approximate algorithmic complexity, such as BDM. BDM
assigns an index associated with the size of the most likely generating mechanism producing the data according to Algorithmic Probability [22]. BDM
is capable of capturing features in data beyond statistical properties [30, 29],
and thus represents an improvement over classical information theory. Because finding the program that reproduces a large object is computationally
very expensive even to approximate, BDM finds short candidate programs
(which are generative models) using a method introduced in [9, 21] that finds
and reproduces fragments of the original object and then puts them together
as a candidate algorithmic model of the whole object [30, 27]. These short
computer programs are effectively candidate mechanistic models explaining
each fragment, with the long finite sequence of short models being itself a
generating mechanism.
6

In this sense, a causal path is a path where the changes between one
state and another is merely the product of an underlying dynamical system
following its normal course, sans external intervention [31].
An important concept is that of the information signature of an object [31]. An information signature quantifies the algorithmic resilience of
an object to transformations, that is how much its most likely mechanistic
model may change after modifying the object. In the case of networks, perturbations can be applied to nodes or edges, that is, in the context of chemical structure networks to atoms and molecular bonds which means that one
can have both node and edge information signatures (see Fig. 1G,H). Comparing information signatures is therefore a way to perform an algorithmic
alignment among different objects such as chemical compounds.

2
2.1

Numerical Experiments and Results
Algorithmic structural complexity

We perturb the structure of a chemical compound network and see the effect
on the set of candidate generating models by performing interventions and
ranking them by the disruptiveness and causal contribution to the networks’
original algorithmic information content and therefore to the networks’ original hypothesize generative models (as found by the CTM/BDM method).
We may attach the rubric in silico alchemy to the digital computer simulation of the types of changes that a molecule can be subject to regardless
of the thermodynamic aspects of said molecule or the processes involved
(later, we will compare it to the known processes and physical properties
associated with the old and new compounds). Central to the ideas exploited
here is the notion of the information signature as the result of an in silico
simulation measuring the sensitivity of causal generative model of a compound to perturbations. The information signature depicted in Fig. 1G
illustrates a set of such interventions/perturbations simulating the kinds of
transformations that a compound such as aspirin can undergo, measuring
the structural sensitivity to single-bond changes and the susceptibility of
aspirin to being reprogrammed (artificially converted) into a different or
similar (causal) structure in what would be a causal path.
This is an alchemy of sorts, because some of these transformations may
be thermodynamically unlikely and the simulation takes no account of any
physical properties (one can easily transform any element into gold under
such conditions). But all likely causally topological paths are studied as
equally possible in order to determine if there are thermodynamic effects that
7

A

B

C

E

D

F

G

H

Figure 1: A: Canonical structural diagram of aspirin. B: The undirected
chemical structure network of aspirin, where shape, particular elements and
bond types are not retained. C and D: Statins have similar contact maps.
Here we feature lovastatin and simvastatin. Contact maps of a compound
are calculated from the distance among its constituent atoms; the further
away, the darker. E: Alignment histograms between lovastatin and simvastatin. The greater the number of atoms in or around the zero bin, the
more similar. The x-axis represents the average distance among atoms. F:
Weaker alignment between lovastatin and aspirin than among statins. G:
Algorithmic (mis)alignment from the node information signatures of aspirin
versus statins (normalized by aspirin size) with lovastatin topping all others. H: The edge information signature of aspirin is all positive i.e. all
single-bond perturbations to aspirin make its generative mechanistic model
even simpler. In comparison, statins (average signature normalized by aspirin size) have very similar signatures but differ in the number of molecular
bonds that when removed send the compound networks towards algorithmic
randomness.
8

can be explained by algorithmic causality, rather than being external effects
following particular laws but being intrinsic properties of the compound in
question.

2.2

Chemical network perturbation analysis

All values in the node (Fig. 1G) and edge (Fig. 1H) information signatures
of aspirin are positive, that is, no intervention targeting any element pushes
aspirin to become a more complex compound. Their algorithmic alignment
is similar to the one produced by classical geometric alignments yet there
are less arbitrary cutoff values (atom to atom distance threshold). The
result is consistent with the literature characterizing aspirin as a simple
structure. More importantly, the signature of aspirin has two clearly identifiable regions. In the disruptive regime at about 25 bits—measuring the
difference between the mutated/disrupted compound and the original aspirin
structure—are the elements of the carbon ring that is identified as the most
stable structure in aspirin. The reduction in algorithmic complexity comes
from the fact that breaking the carbon cycle produces a simple tree graph
with a long path graph, a graph that is of even lower algorithmic complexity because there is an even shorter program that can produce a tree with a
long path than a structure with the ring/cycle. In contrast, removing all hydrogen and oxygen elements makes an algorithmically neutral contribution,
meaning that their removal is less disruptive to the core of the structure of
aspirin (even though it may be more deeply implicated in its functioning, a
limitation of this type of analysis if, e.g., valency values or electric charges
are not incorporated in the network description (e.g. as weights)— which
it is possible to do though we do not cover it in this paper). Yet the signature analysis indicates that such elements may more easily be found in more
causal paths than atoms from the carbon ring. In other words, it is algorithmically less random to find a carbon ring in the middle of a structure than to
add some elements to the molecule by, e.g., methylation or phosphorylation.
Another observation from Fig. 1H is that fluvastatin has the most negative
node information signature among the statins and it is also the statin with
the lowest number of interactions compared to most other statins and lovastatin is the most positive and also similar to aspirin in its information
changes remaining algorithmic simple, with all node and edge perturbations
positive. Lovastatin has similar interactions to atorvastatin and simvastatin
which are also close to Lovastatin in the signature information landscape.
Aspirin is, however, in the middle of the statins pack suggesting a greater
similarity than what classical alignment methods suggest (Fig. 1F).
9

2.3

Algorithmic causal transformations

Chemical compounds that may be in the same causal path will have similar
signatures [31] from single bond or atomic knock-out interventions. One such
example may be that of acids and alcohols compared to compounds that are
structurally—and causal structurally more removed—such as compounds
one may expect to form between organic versus inorganic substances (as
tested in Fig. 2D). This can be seen by analyzing the tails of the signature
distributions (Fig. 2C-D) that provide a network with the means of moving
towards or away from its original algorithmic model.
It is also of interest to find a correspondence between the algorithmic
difficulty of transforming a compound such as an acid to an alcohol. What
is suggested in the preliminary experiments is that it is (slightly) more difficult because it implies a reduction of algorithmic randomness (Fig. 2C),
as compared to transforming an alcohol to an acid, which is consistent with
the literature showing that oxidants able to perform this operation in, e.g,.
complex organic molecules require substantial selectivity, therefore making
it less likely to happen by the chance imposition of a thermodynamic direction, something also suggested by the algorithmic causal calculus. This
is, however, less dramatic than transforming inorganic into organic compounds according to the simulation (Fig. 2D), where inorganic compounds
seem to require a larger increase of algorithmic information content to reach
the complexity of organic compounds, suggesting that inorganic compounds
are algorithmically simpler as they are algorithmically more probable, and
therefore may occur naturally with much greater ease. Counterintuitively,
the results pertaining to organic versus inorganic substances may suggest
that organic compounds are much more stable (less reprogrammable) than
inorganic compounds in general, and structurally this may be the case, given
that the main difference separating the two classes is the stability provided
by carbon atoms that are the building blocks of organic matter.
Figure. 2(A) illustrates the finding that the class of heavy metals is
the least complex according to their algorithmic complexity estimation by
BDM. The reason is that most of them tend to be very simple and small
while possessing properties that endow them with stronger covalent bonds.
In contrast, pyrimidines, for example, which comprise the basis of DNA
and RNA nucleotides, are the most complex (together with purines they
form the other nucleotides among the highest complexity heterocylic aromatic organic compounds (see Fig. 2(A))). The results show that taking the
highest algorithmically complex compounds would profile all pyrimidines
(558) with high accuracy, of all the other compounds considered in this
10

A

B

C

D

Figure 2: A and B: Classification of compound classes according to their estimations of algorithmic probability/complexity by BDM. C and D: In silico
alchemy by intervening in chemical structure networks and analyzing their
algorithmic properties. The algorithmic sensitivity of acids and alcohols is
very similar, corresponding to known mechanistic processes that can transform one into the other. A minor asymmetry can be found similar to the
difficulty of converting one compound into another. In contrast, organic vs
inorganic compounds are among the most dissimilar.

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Class
Acids
Alcohols
Aldehydes
Amides
Amines
Amino Acid Derivatives
Carboxylic Acids
Esters
Ethers
Heavy Molecules
Hydrocarbons

Count
537
1190
723
531
1916
799
1145
1409
619
1060
1499

Class
Iodinated
Ketones
Monomers
Nitriles
Pyridines
Pyrimidines
Aromatic
Biomolecules
Brominated
Chiral
Chlorinated

Count
774
865
994
1100
1435
558
18567
6159
2620
5915
5680

Class
Drugs
Fluorinated
Halogenated
Heterocyclic
Inorganic
Liquids
Organic
Organometallic
Salts
Solids

Count
4630
3470
11223
10964
3605
10474
27969
2315
4718
22936

Table 1: Number of elements per compound class used in the experiments.
A total of 158 399 extracted from ChemicalData[] in the Wolfram Language,
the sources relied upon being provided in the documentation.
database (44 089), and likewise the lowest complexity retrieves all inorganic
compounds followed by heavy and iodinated molecules. The total number
of compounds per class can be found in Table 1.
It is also of interest to note that classes of compounds whose algorithmic
complexity estimation median values are close to each other are causally related. For example, acids and alcohols appear to have very similar algorithmic information content, notwithstanding the fact that alcohol structures
are much larger in size than acids. Their structure and causal origin can
be regarded as similar as they can be derived from each other with their
main structure unchanged. However, esters are difficult to reduce to ethers,
as they decompose to yield alcohols via decomposition of the intermediate
hemiacetals even when esters can be reduced to ethers [25]. It is therefore
interesting to emulate the evolution of these networks through all possible
chemical trajectories and see how algorithmically easy or difficult it is for
them to become other compounds favouring certain properties, and how
(un)stable they may be in the face of perturbations, independently of thermodynamics, while being ultimately related (and we will devise some tests
in this regard). We use a measure of sophistication based on logical depth
and a measure of reprogrammability gauging the susceptibility of a chemical compound to being converted into some other, more random or simpler,
chemical compound.
Statistical overlapping of complexity estimations is to be expected, given
that classes are not distinct. Aspirin is a drug, but it is also classified as a
biomolecule, organic, solid and aromatic. Many classes also share elements

12

Figure 3: Graph branching is one of the most important measures proposed
in molecular complexity. Here it is shown how algorithmic-based measures
such as lossless compression (Compress and Bzip2) are correlated with graph
spectra, which in turn capture graph branching [18]. The correlation is
stronger than that obtained when using only Shannon entropy. All data
come from the Wolfram Language database retrieved by the ChemicalData[]
function
with chirality properties. However, a significant divergence between contrasting classes such as those not closely causally related is to be expected,
as measured by the algorithmic probability of a compound being in the
causal path of another class of compounds (i.e. there being no simple chemical/thermodynamic process—natural or artificial—to convert most elements
from one into another class) such as those of an organic nature (including,
for example, elements under organic, biomolecules, heterocyclic, pyridines,
pyrimidines and monomers) into those of a more inorganic nature (including, for example, the category inorganic itself and heavy molecules). Other
features driving the complexity estimation are properties such as are found
in heterocyclic compounds, compounds containing atoms from at least two
different elements as members of its rings, thus being structurally richer on
average. Nucleic acids and most drugs are also high in algorithmic complexity and have large variance values, indicating that designed compounds tend
to be wide-ranging in nature while inclining towards complexity. Another
contrasting/disjoint pair of classes is liquids versus solids, with statistically
different complexity values. It is interesting to note that chemical structure
networks of liquid compounds are significantly less complex than those of
solids, and that inorganic compounds are the least complex, while drugs are

13

the most complex and feature a high incidence of human-made synthetic
compounds.

Figure 4: SMILES and InChI molecular strings correlate with complexity
even when normalized by size, as would be expected since both notations
capture very important properties of the compounds. InChI has a greater
correlation statistic, as would also be expected from the fact that the sequence notation was designed to capture more information about the molecular compounds it represents.

2.4

Molecular complexity versus compound properties

The simplified molecular-input line-entry standards SMILES and InChI are
specifications in the form of a line notation for describing the structure of
chemical species using short ASCII strings. SMILES is a string obtained
by printing the symbol nodes encountered in a depth-first tree traversal of
a chemical graph. The chemical graph is first trimmed to remove hydrogen
atoms, and cycles are broken to turn it into a spanning tree. Hence one
should find some correlations with properties of the represented molecular
graph. InChI contains all the information contained in a SMILES description
and more atomic information such as bond connectivity, tautomeric information, isotope information, stereochemistry, and electronic charge. Figure. 4
reports the correlation found between these notations and complexity by
several complexity indexes and Fig. 3 describing the correlation between
branching (a common index of order parameter in molecular complexity)
and measure of statistical and algorithmic complexity.
The results reported in Fig. 5 (Appendix) are interesting because one can
think of algorithmic complexity as a method sorting by size of the underly14

Figure 5: Random sample of chemical structures sorted by algorithmic complexity: BDM sorts molecular graphs by their adjacency matrix, not by their
size but their shape. As shown in Fig. 7 this makes for biases reflected in
the class memberships of the molecular compounds, mainly between organic
and inorganic, and between solids, liquids and solvents.
ing generating mechanism causing each structure, and it would be expected
to find compounds that behave or produce similar states to be generated
by similar mechanisms. The results reported together with the class enrichment and depletion analysis shown in Fig. 7(Appendix) are interesting
because, as has been suggested, in the case of organic molecules, the lower
the information content the fewer the possibilities for different interactions
with other molecular compounds. Fig. 6 illustrates how the algorithmic
complexity approximated by BDM correlates with some physical properties
of the molecular compounds. This is interesting because it suggests– if the
correlation is actually correct– that some information about properties that
are global properties, such as temperature, is in the local structure of the
molecule, which is not surprising if one recalls that how rigid or charged
a particle is may have an impact on its dynamic interactions with other
molecular compounds.

15

3

Conclusions

We have identified and illustrated interesting research avenues that a causal
interventional calculus based on algorithmic probability/complexity (as the
study of a system’s changes in algorithmic information) can bring to the
discussion of molecular complexity and chemoinformatics, in particular to
chemical structure networks. Indeed, usually a drug has a backbone from
high throughput screening leading hits and intensive modification of the
chemical structure is done to make it drug like, providing improved stability,
solubility etc.
We have found that this algorithmic approach suggests similarities for
graphs/networks that may be explained by common generative mechanisms,
suggesting an algorithmic likelihood of causal transformations from candidate models found by algorithmic probability. We found that the method
separates distinct classes of chemical compounds, both by estimations of
the algorithmic complexity of chemical structures and for causal sensitivity based on a measure of agnostic (no physical properties being involved)
reprogramability. The experiments with statins whose similar effects to
aspirin are an open question, suggests a measure of similarity and of interactions/toxicity that should be further tested. We have also shown that
the measures show various degrees of correlation with some chemical compound’s physical properties.
Our approach effectively introduces a new dimension in the study of
information-theoretic properties and algorithmic transformations of compounds, and further explorations and generalizations to more general compound networks (where several compounds are connected), binding and reaction networks should be investigated.

Acknowledgements
H.Z. was supported by the Swedish Research Council (Vetenskapsrådet)
grant No. 2015-05299.

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Appendix

Figure 6: Four physical properties found to be slightly correlated to
the graph algorithmic complexity of a large set of molecular networks
(built from contact maps extracted from atomic data in ChemicalData[]
in the Wolfram Language coming from public chemical data banks) according to their BDM values with statistics (Pearson): -0.16, -0.20, 0.34 and -0.3 and p values all < 0.05 except for combustion heat at
p = 0.07. Fitting lines (red) were found by Least square: x2 (0.0133001◦ C) +
x(−3.85595◦ C) + 460.328◦ C, x2 (0.450617kJ/mol)
+ x (−57.8591kJ/mol)
+


2
3
3
5521.25kJ/mol, x 0.0595914kg/m + x −22.2082kg/m + 3306.75kg/m3
and x2 (0.0144063◦ C) + x(−5.23141◦ C) + 546.651◦ C. While the correlation
values are weak, in all these cases no element with high values for each
property was found to also have high BDM. In other words, all elements
with high values for each property also had very low algorithmic probability
estimations and therefore high algorithmic complexity, thereby pinpointing
elements with high values for these physical properties.

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Figure 7: Top enriched and depleted classes separated by estimations of
graph algorithmic complexity (approximated by CTM/BDM v. randomized
membership) out of a random sample of 100 complex chemicals sorted by
class. The probability on the y-axis is given by the Spearman correlation betweeh complexity values and class membership. Liquids, solids, organics and
solvents are the best separated when enriched by algorithmic information
content, i.e. with generating mechanisms with similar computer program
lengths when the underlying networks are explained causally (as generated
by 2-dimensional computer programs). Data source: Wolfram Language
database from the ChemicalData[] function.

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