# A Generalized Theory of Interactions - I. The General Problem

Santiago Núñez-Corrales1∗ and Eric Jakobsson2†
1Illinois Informatics and National Center for Supercomputing Applications,
University of Illinois at Urbana-Champaign. Urbana IL, USA.
2Deparment of Physiology, Molecular and Cellular Biology and National Center
for Supercomputing Applications,
University of Illinois at Urbana-Champaign. Urbana IL, USA.
* Corresponding author<EMAIL_ADDRESS>
$\dagger$ In memoriam

###### Abstract

Understanding realistic complex systems requires confronting significant
conceptual, theoretical and experimental limitations rooted in the persistence
of views that originated in the mechanics of simple moving bodies. We define
the category of complex multiscale stochastic systems as a useful device for
capturing the minimally required complexity of many types of phenomena. In
doing so, we provide evidence indicating that determinism, continuity and
reversibility can lead to theoretical inadequacies that manifest as
intractability, inaccuracies or non-representativeness when applied to complex
systems. We take the view that interactions are fundamental and summarize
their portrayal across many disciplines. Despite their centrality,
interactions remain largely neglected as subjects of research interest of
their own. We hypothesize that a generalized theory of interactions may help
organize evidence from multiple scientific domains towards a more unified
realistic view of systems.

##  1 Introduction

The Internet [5], bacterial communities [67], the global economy [10],
ecosystems [66], societies [211], distributed information systems [23],
international diplomatic bodies [247], biomimetic nanomaterials [210] and
organizations in general [75] exemplify systems with scientific and practical
relevance characterized by dynamical complexity in their structure and
function [143]. All of them exhibit intricate dynamics and responses coupled
to their environment that impact our ability to predict their future
behaviour. Elements within these systems appear simple, in stark contrast to
the great variety and number of possible stimuli the system as a whole can
receive [15]. All these systems also exhibit emergent properties and self-
regulation, which seem to provide resilience to small external perturbations.
At a closer look, we find that their architecture depends on intricate and
flexible organization patterns that are not directly dependent on external
driving agents, but which lead to diverse internal microstates that work to
stabilize the system as a whole against destructive influences of external
driving agents and confer unity to the system [14].

Constructing causal explanations and predictions for densely interconnected
systems may or may not admit strong simplifications. Models that omit even a
few elements of the problem sometimes show loss of empirical resemblance to
the instance under study. Some of the most frequent simplifications involve
the supposition that changes are described by linear functions or that for
relevant time periods they behave _almost_ linearly, or that sources of noise
can be safely neglected in the mathematical description. In this proposal, we
provide evidence against such simplifications arguing that they are inadequate
for many instances of densely interconnected systems; correspondingly, we will
also describe in detail properties of these systems in an attempt to identify
particular instances and formulate strategies for their modeling and
simulation. Prior efforts include nonlinear science [215, 187] and complex
systems science [71, 219]. Both have made attempts to explain the structure
and dynamics of these systems, but both face many challenges when used to
either make empirically verified predictions or provide causal accounts of
changes across temporal and spatial dimensions.

Many explorations of nonlinear science begin with viewing systems through the
lens of dynamical systems theory, supposing that sets of coupled differential
equations completely capture trajectories of all relevant events: a manifold
contains the trajectory of variation of system states (i.e. points in their
trajectories) through time, and transforming the manifold by removing time
produces a phase space representation that captures the global geometry of
state changes [140]. In this view, instantaneous transformations derived from
a small set of governing laws specify the changes that occur in the system.
Analytic or numerical integration of these transformations –usually specified
by systems of differential equations- yields values for macroscale observables
that correspond to measurable properties of the real system. We discuss below
several problems that exist with this approach preventing it from being
effective for solving some problems in many systems.

A second view is that of networks, which has contributed in the last three
decades to a vast number of discoveries and applications. A network is,
briefly, a collection of entities (i.e. nodes) related in some manner (i.e.
links) that determines the local and global structure of the collection, its
dynamical properties or in many cases the likelihood of emergence of
properties that cannot be explained by the individual action of the entities
alone. Despite the success of networks in capturing essential aspects of many
systems with high complexity, the picture they portray is incomplete and
requires additional elements to derive causal explanation or prediction from
them.

The research described here concerns itself with a third alternative aimed at
drastically improving our understanding of apparently intractable and
unrelated phenomena across various systems through the lens of existing theory
and practice where random noise, hierarchical modularity and irreversibility
dominate the structural organization and dynamics of many interacting entities
acting in some coherent and coordinated manner: we refer to them from here
onwards as _complex multiscale stochastic systems_ (CMSS). In this
introductory article we provide a definition of CMSS instances

##  2 Complex Multiscale Stochastic Systems

In order to provide a unified account of phenomena and systems as varied as
those mentioned above, we focus on understanding the properties of CMSS. We
start by defining complex, multiscale stochastic systems (CMSS) as the class
of systems –whether natural, artificial or informational- whose structural
organization is best described by nested hierarchies of constituent systems,
comprised of objects and relations subject to non-negligible and intrinsic
random noise, and bearing recognizable collective identity [18]. Some of these
systems have evolved in a biological sense, others have arisen from the
combination of physical laws at different scales and, in many recent cases,
these systems are being engineered by humans. Some CMSS are still in their
infancy and have not reached sufficient critical connectivity for emergent
phenomena to arise.

We observe that these systems exhibit four critical properties:

  1. 1.

Composition. A large variety of objects and object types, driven by processes,
that interact with each other in non-trivial manners.

  2. 2.

Multiscale structure. Their structure and behavior can be described at several
nested, coupled scales of aggregation by different principles and laws, yet
their action remains causally connected when looked at from the outside.

  3. 3.

Stochasticity. Random perturbations can initiate and drive sudden changes in
the internal structure or dynamics of the system, leading to uncertain
measurements that demand statistical treatment.

  4. 4.

Overarching laws. CMSS obey the laws of thermodynamics: local conservation of
energy (First Law) and thermodynamic irreversibility (Second Law).

CMSS dissipate heat [216], hence producing entropy, and also exhibit noise
with a fractal spectrum. Living organisms [214] in general and the brain in
particular [217] organize in hierarchical ways and also exhibit noise with a
generally fractal spectrum.. Action in CMSS follows trajectories determined by
the Maximum Entropy Production (MEP) principle, which appears to create the
context for the emergence of information processing [73]. Information in
dissipative systems (i.e. those with friction and irreversible energy loss)
often exhibit hysteresis and are characterized by one or more sources of noise
for which dampening mechanisms exist at some phenomenological level. We stress
here that in all prior CMSS examples our understanding of their integrated
phenomenology in relation to entropy production is still at its infancy.
Moving from infancy to adolescence depends on finding efficient ways to
exploit general properties of underlying governing laws, and the relations
between objects at different spatio-temporal scales as experimentally
measured.

###  2.1 Complexity

Succinctly, a system is complex if the laws that govern the trajectory of
parts do not permit a straightforward reconstruction of the trajectory of the
complete system [182], and simultaneously some type of organization is
recognizable [223]. Care must be exercised not to confuse the _apparent_
complexity of simple dynamical systems [230] with that studied by systems
captured by complex networks [61]: we wish to study systems such as clocks,
made by a thousand interacting parts, instead of describing the movement of a
single hinged pendulum. The complexity of the systems of interest here is
beyond apparent; it manifests in their structural and dynamical properties and
to some extent is captured by analogues of algorithmic complexity [239].
Moreover, the kind of complexity that captures the interest of this work is
that which is simultaneously hierarchical in its structure, emergent [104] and
adaptive [128] by virtue of its governing laws and composition. CMSS also
manifest decentralized control strategies [222] that sustain the efficiency of
internal processes by exploiting their modular structure [189] through
internal communication mechanisms. While some aspects of the energetics [77,
78, 79, 80, 171, 22] and the dynamics [227, 134] of systems in general have
been explored through statistical physics, we lack a unified relational view
of CMSS where causal relations provide information towards prediction,
analysis and retrodiction based on empirical findings.

###  2.2 Multiscale structure

A complex system is also multiscale when (a) more than one valid account of
same system can be given using phenomenologies that involve objects, relations
and dynamics that differ widely in spatial and/or temporal scales, (b) the
collective effect of governing laws and objects at one level (i.e. microscale)
can be mapped to a single, much larger persistent objects at the next level
(i.e. macroscale) whose properties cannot be additively explained from
objects, relations and dynamics in the former one, (c) changes in a
significant111 _Significant_ depends here on the particular system, its scales
and laws proportion. What we claims is that, if a complete theory of CMSS
exists, it must provide a way to compute the threshold for significance of a
given current proportion. proportion of objects in the microscale lead to
distinct and mutually exclusive macrostates under certain critical conditions
(i.e. phase transitions), (d) scales of structure and action can be ordered
according to the amount and nature of information that can be extracted from
them and (e) the architecture of the system is modular and nearly decomposable
[224], leading to information representations [164] while still determined by
irreducible couplings. CMSS tend to be extremely effective negative entropy
producers [21] that actively preserve internal stability through various
mechanisms. Part of the reason seems to be connected to the numerical
relations derived from their structure as described by either Zipf’s law
[181], power law [22] or Rent’s law [24].

Network science has enabled the development of various methodologies capable
of extracting putative structures of complex multiscale systems from data in
various knowledge domains [209]. Linking many relevant scales and including
dynamics, however, remains limited in practice mainly because of the
combinatorial explosion derived from keeping record of the links between
different objects across scales. Partial solutions to this problem often rely
on pragmatic choices that include neglecting scales, objects or relations,
devising sampling strategies while preserving the general phenomenology or
focusing only on one level of description to focus on a certain regime of
applicability. Still, mounting research on many disciplines including physics
[152], chemistry [153], biology [91, 242] and socio-technical systems [238]
indicates that current multiscale methods face a growing number of challenges.
Two specific issues are how to devise empirically correct mappings between
microscales and macroscales that capture all relevant details without
sacrificing efficiency, and understanding how the underlying laws of motion at
one microscale constrain or determine laws at the corresponding macroscale.

###  2.3 Stochasticity

Stochasticity refers to the presence of non-negligible noise in the dynamics
of a system, often associated with nonlinearity [93]. For a system to be
stochastic, two types of forces must be present: drift forces that bias the
motion towards some preferential direction and diffusion forces that randomize
the overall trajectory of the system [12]. If noise levels are small enough
and fluctuations are not amplified (e.g. through multiplicative terms in the
driving forces), then the system can be treated as deterministic for
simplicity. However deterministic representations suffer from a deep
theoretical flaw. Because they are reversible they violate the Second Law of
Thermodynamics. Essentially they are an extension into dynamics of complex
systems of the concept of the ”frictionless plane” that is often introduced in
introductory physics courses. Just as one would be skeptical of the
performance of a machine engineered based on the assumption of zero friction,
so should one be skeptical of descriptions of complex systems based on
deterministic models. Specifically in CMSS the magnitude and frequency of
fluctuations is often large enough to prompt signal propagation and autonomous
reorganization through stochastic amplification [240, 220, 151, 145]. In such
conditions, CMSS entities and dynamics appear to become organized into
discrete units [253] –i.e. modules- that maximize internal stability in
irreversible ways [29]. It is known that stability increases exponentially in
complex discrete systems whose structure is described through subsystems [38],
even those that contain nonlinear elements driven by time-dependent gain
functions [13]. The combination of these elements produces a rich landscape of
possible decentralized control strategies [221]. At present, the main
challenge stochasticity introduces resides in the mathematical incompatibility
of noise with the usual form of equations of motion and the computational
complexity associated to finding solutions in CMSS.

###  2.4 Summary: the need for a science of CMSS structure and function

Due to multiple scales, noisy and nonlinear dynamics, and hierarchical nature
of CMSS, restricting ourselves to simplified models mostly ends up in sub-
optimal solutions at best, and non-solutions often, to the problem of finding
empirically relevant causal mappings across scales. In this proposal, we focus
on describing various examples of how –and most important why- simplified
theories and models fail to capture essential empirical aspects of CMSS. In
systems with highly coupled positive and negative feedback mechanisms (e.g.
_wicked problems_), the density of true solutions for many problems tends to
vanish compared to the size of the search space [244]. However, many pressing
issues in contemporary social, economic and environmental research and
practice require true solutions to avoid financial, human or environmental
losses.

If the principles that govern CMSS were accessible in terms of causal mappings
between microscales and macroscales, gaining information about phenomena would
only be a matter of computing consequences of the respective theories
confident that the observables will likely be consistent with empirical
measurements, including changes and their occurrence thresholds. However,
finding theories that integrate nicely and universally at extreme regimes of
action (e.g. quantum-classical systems) have not been found. Changes at the
microscale usually involve classes of entities or relations rather than on
single objects, yet many cases exist where small events (i.e. localized,
unique and often very improbable) are sufficient to trigger large changes
across the system [129, 7, 228]; catastrophe theory has remained an evolving
field that attempts to deal with sudden changes in systems, but its
description is at the macroscale and phenomenological in nature [196].
Catastrophe theory often focuses on identifying conditions under which
deterministic descriptions of dynamical systems fail, but the theory itself
fails to provide an alternative description.

Our understanding of the causality of events and relations in CMSS remains
limited and largely disconnected. When a system reaches a certain critical
degree of complexity, it is not easy (or sometimes even possible) to
accurately reconstruct the trajectory of events that lead to the current
state. More importantly, even with such information, there is no guarantee
that the available theories –most of them formulated as systems of
differential equations or assertions about networks- will integrate nicely.
The state of multiscale modeling in most areas is that of constructing a
theoretical _patchwork_ that must be painstakingly mended to avoid breaking
apart as revealed by contemporary multiscale modeling of complex materials as
an example [74]. Changing this situation –the main objective of this proposal-
is critical for the solution of many challenges of intellectual and practical
importance in our century.

##  3 Current problems with CMSS models across scientific theories

The Internet, cells, ecosystems, the global economy and many other CMSS
examples are characterized fundamentally by local exchanges of matter, energy
or information between smaller components at all scales. Each exchange,
whether evolved or designed, is constrained by governing laws of the
interacting entities –or rather, the laws of their substrates which allow some
notion of identity- and at the same time collections of these exchanges
provide the basis for emergent constraints at critical system sizes and
densities (i.e. their thermodynamic limit). We concisely review here how
various interactions are addressed through various scientific fields,
including how their representation and suppositions may be limited in each
case. In essence, we argue that the structure of interactions is either
represented explicitly without a concise explanatory picture of their dynamics
or captured by detailed dynamics without any reference to its structure,
becoming problematic for the purpose of having effective and efficient ways of
understanding phenomena in CMSS.

Methodologically, we explore in detail the existing set of alternatives to
model CMSS across various domains. First, we make explicit various assumptions
about the character of dynamical equations in the context of creating
effective models for CMSS. Second, we unmask a series of biases that
simultaneously pervade and negatively impact the ability to reason in
empirically correct ways about CMSS phenomena. Finally, we discuss the most
relevant representations of interactions across the natural and social
sciences.

###  3.1 On the character of dynamical equations _vis-à-vis_ CMSS

The study of fundamental relations between energy, information, time, matter
and space has led to the discovery of two types of symbolic descriptions:
those that pertain to concrete realizations of physical systems and those that
pertain to _abstract_ classes of physical systems that ubiquitously constrain
all concrete ones [191]. Correspondingly, these relations are expressed in one
case as sets of dynamical equations and as physical laws in the other one. A
set of dynamical equations is expected to be valid only in relation to
specific settings; physical laws are more general, and the laws of
thermodynamics are universal.

The usual route to arrive at specific sets of dynamical equations in
contemporary physical sciences has been to depart from one of at least three
starting points: its Hamiltonian [155] (i.e. a function of potential and
kinetic energy), its Lagrangian [193] (i.e. the functional that assigns a
value to a configuration of the elements of a system depending on its laws of
motion) or its symmetries [127] (i.e. equivalence relations in the manifold
and their consequences for motion). The existence of the three of them depends
on conservation laws that ensure that no amount of motion will remain
unexplained under a closed system, whether locally or globally [55].
Conservation principles have been used used to process trajectory data of
various systems to construct nontrivial useful theories in the form of laws;
generalizations which are mostly true in many instances and can be used to
organize knowledge. Whether by manual or automatic means, the existence of
such laws is assumed and they are sought, generally to apply to one level of
system behavior.

In the context of CMSS however, characterizing the system at its outermost
phenomenological level can be problematic. Choosing a level of description
implies selecting certain aspects of reality, including the collection of
conservation laws that apply at each level. When couplings between various
scales occur, it may be either impractical to compute the consequences of laws
of motion for many entities or unrealistic after simplification. More
importantly, the form of the laws (and correspondingly of the dynamical
equations) is of little help when stochastic fluctuations are dominant at the
scale of interest. When two or more scales are included, different types of
equations and laws are discontinuously patched together through approximations
which lead often to semi-empirical solutions that work only for reduced
regimes of action.

Yet, the existence of critical phenomena and phase transitions across a
multitude of systems and scales (from the cosmological to the quantum) [225]
indicates that despite complex systems being nearly decomposable [224, 24]
they should be treated as emergent, including their governing laws. Our
perception of nature as a collection of nearly decomposable systems appears to
arise from our perceptual biases, the law of large numbers and the exponential
decay of large deviations arising from stochastic fluctuations [186, 233].
Adaptive emergence may trigger aggregation as a way to minimize action
globally across the system as a means to reach regions of dynamical stability
[34]. Hence, the likelihood for any arbitrary set of theories, each of them
applicable only to a certain scale of action in a system –especially those
that have been obtained through experimental observations at only one such
scale- to integrate nicely and describe CMSS adequately (including detailed
mechanisms) appears to be slim and decreases in an inversely proportional
fashion with the number of scales involved.

Many, perhaps most, descriptions of dynamical systems are made through
differential equations, and most often through deterministic ones. Arriving at
the appropriate set of them is the bread and butter of modeling from physics
to social science. In order to understand at greater depth what is involved in
the choice of sets of dynamical equations, we concentrate next on their
representation relative to the view of the universe as emergent from a set of
fundamental laws of nature. The first two elements to be considered are
concerned with how the form of the equations is derived. The third element is
the determination of whether transformational pre- and post-conditions can be
recovered from differential equation models. Finally we will consider physical
laws as dynamical extrema under the law of large numbers.

####  3.1.1 The manifold choice problem

Let us consider the case of the classical $N$-body problem, known to be non-
integrable for $N>3$. At any time the 3D manifold contains $N$ particles, with
resulting embedding of $3N$ degrees of freedom in total. The embedding itself
is described as a vector product $\mathbb{R}^{N}\otimes\mathbb{R}^{3}$ of two
vector spaces, that of the system of coordinates for one particle
$\mathbb{R}^{3}$ and the collection of all particles $\mathbb{R}^{N}$. In
general, any non-trivial motion can be described for one particle by adding
its momentum to the position, leading to $\mathbb{R}^{6N}$ simultaneous
variables to keep track of. Solving this computationally expensive problem is
well known in computational molecular dynamics [30, 194].

The input of the $N$-body problem consists of the initial positions of all the
particles, the function for the pairwise (or $k$-wise) interactions between
particles and the description of the energy potential surface. The more exact
the model, the more computationally expensive it becomes, since the coupling
of all forces (e.g. ion-induced dipole, ion-dipole, electrostatic, and van der
Waals forces in molecular dynamics) in conjunction with the geometry of the
objects can quickly lead to many correlations impinging on one another in
small portions of the volume of interest. When the forces act at a short range
(e.g. van der Waals), the problem is simplified by modeling its action with
cutoff functions. In the presence of long-range forces the cutoff leads to
some error which one seeks to minimize by an approximate accounting for
surroundings, but some residual error is unavoidable..

Let us pause for a moment and consider various systems described by the same
Hamiltonian. If we assume only classical forces are at play, all systems of
interacting particles are captured by it, from very dilute gasses [37],
liquids [125], metallic solids [98] to core-shell nanoparticles [252]. Four
distinctively different classes of systems are captured by the same physical
model, yet the complexity associated with solving their systems of equations
varies drastically on each case. Given that pressing practical questions often
depend on their solution, finding the smallest embedding that preserves all
the features of the original system is advisable.

The problem above may be stated as that of finding an appropriate scaling
relation, or a renormalization group (RG) whose effect is drastically reducing
the degrees of freedom while providing a good approximation [114]. The scaling
relation should provide three advantages: (1) reduction of the number of
particles or degrees of freedom through a mean field theory, (2)
identification of the critical exponents that dominate force ranges and (3)
analysis schemes to reveal the emergence of structure. For the $N$-body
problem with long-range correlations, a formal RG only makes sense with a few
bodies (often $N\leq 4$) [46, 213, 138]. For systems with high thermal motion,
RG reduces to some form of noise across the problem domain. In both cases,
information loss is guaranteed in some form of another.

On the other hand, classical mechanics provides plenty of examples of scaling
relations that, though not thought of immediately as emergent, produce the
same outcomes at the scale of interest and are simple to compute. For
instance, take a metallic solid under translation and rotation forces. Any
transformation applied to the solid is propagated across its structure to each
one of its constituent atoms. Suddenly, the $\mathbb{R}^{6N}$ manifold for its
dynamics reduces to $\mathbb{R}^{6}$, no RG required or evident at first
glance. A similar case occurs for laminar flow with a suitable Reynolds
coefficient, in which the dynamical equations reduce to $\mathbb{R}^{2}$ or
even $\mathbb{R}^{1}$. Those are indeed extreme cases of scaling relations,
the last with an effectiveness of $1/6N$; consider the implications for
laminar flow of one liter of water along a pipe ($N\approx 3.346\times
10^{25}$ water molecules). It should be noted that this simple renormalization
is strictly true only for a metallic solid of infinite size or laminar flow
through a pipe of infinite length.

What fundamental properties of the universe as the archetype of self-organized
criticality at all scales justifies such choice of degrees of freedom
different from detailed descriptions of phenomenology or equational parsimony?
The intuitive answer appears to be that some classes of self organization lead
to collective identity [184], defined as the ability of an entity to retain
its structure and functions for long periods despite small perturbations.
Hence, alternative models and representations of systems that may deviate from
the underpinning reality become physical fictions. Such is the case of
metallic crystals [166]: atoms are assumed to “jiggle in place” in a rigid
lattice of (fictional) harmonic oscillators connected by springs governed by
Hooke’s law. A great deal of progress in crystal physics has been made in this
fashion [251], driving a large share of the present digital revolution. The
springs propagate external perturbations and, at the same time, fix the global
geometry of an object in place by damping propagation to adjacent atoms. The
object acquires identity, and the effects of the rotation and translation of
an object in $\mathbb{R}^{3}$ are nicely translated simultaneously (at least
with respect to measurements performed at the macroscale) to the
$\mathbb{R}^{6N}$ particles.

CMSS appear to lie in the middle of both purely elastic molecular collisions
and metallic solids when viewed as an $N-body$ problem, yet their complexity
is much higher than that in either extreme. Hierarchical modularity increases
the likelihood of finding a scaling relation, but it enters immediately in
tension with randomized interactions between levels of the hierarchy. In
addition, and in contrast to the types of questions posed on systems modeled
through the $N$-body problem, CMSS are interrogated simultaneously at multiple
levels and not only at the bottom or the top of the hierarchy. Since one
macroscale may be explained by a large number of equivalent microstates while
events at the mesoscale remain of importance, keeping track of all entities in
the hierarchy becomes a difficult task.

Concisely, except for various well-known limiting cases, finding scaling
relations that yield an economic and representative manifold in hierarchically
coupled systems is not a systematic process as long as Hamiltonians (or
equivalent formulations) are used. Solutions are achieved by either
constructing relevant abstractions (e.g. ordered lattices) or by losing
information through the reduction of degrees of freedom. At present, such
essential processes for finding the most efficient and realistic manifold for
CMSS instances are unknown.

####  3.1.2 The functional choice problem

Once the the manifold is chosen along with its embeddings, the next step is to
determine –based on empirical observations- the most appropriate mathematical
description of the phenomenon. Again, conservation laws, invariants and
symmetries are the standard tools most theorists have access to for devising
the equational form of the governing laws at present [9]. In the case of
physics –and back to the $N$-body system in particular- the choice of
dynamical functions occurs in a way that (a) accounts for both potential and
kinetic energy present in it, (b) dictates the specific laws of motion for
each body through application of differential operators to the main invariant,
(c) provides coefficients that represent material aspects of the system. By
fixing them, one can parameterize the equations of motion to the particular
instance. Motion, in this sense, is represented as the trace in time of
positions and velocities either through time series (i.e. position or velocity
per every time instant) or phase diagrams (i.e. reachable states as
represented by statistical properties).

As discussed above, interactions appear to be mostly hidden in dynamical
descriptions of CMSS, or may be neglected. In the case of a pendulum, it is
not easy to derive a picture containing interactions, yet many certainly
exist. We can imagine interactions between the pendulum mechanism, the
pendulum and the air (when the experiment is not carried in vacuum) and the
interactions between the atoms in the rod and the bob. However, that
information is unavailable, or rather is _obscured_ by the emphasis on
position and momenta. Fourier analysis has become a standard tool [39] that
uncovers the frequency of events in a system, from which interactions can be
derived to some extent. Detecting the frequency of events is a starting point
to the investigation of their mechanisms and internal dynamics. Both the time
series and the frequency domain representation are naturally expected to be
much richer and harder to unravel for CMSS.

The evident limitations of dynamical models of systems for capturing
interactions requires some historical investigation about their origin. Our
modern depictions of dynamical systems, and of motion in general, can be
traced back to Descartes [72] and his philosophical investigations on the
material conditions required for a science of motion. His analytic geometry
[45] provided a foundation benefiting from both geometry and algebra as means
to understand motion as a sequence of instances, each one of them localized at
a given instant of time. Representing motions as functions was only natural
after this, presupposing that motion at any given future instant was
predictable from motion at previous instance provided the universe contained
some sort of regularity. Note that this particular definition of motion does
not require any reference to interaction (or rather, a model of it) to be
considered complete. In that sense, choosing the functional form to describe
the motion of a dynamical system immediately refers to the selection of the
appropriate number of spatial dimensions such that, at every instant, the
position of an object can be located and its future position for any given
time step can be known using information from the present and the past.
Newton’s Principia [183] adds the concept of interactions with a functional
form and thus represents the pinnacle of this sort of mechanics, providing a
mathematical toolkit capable of answering questions about position, velocity
and momentum at any instant provided the functions were guaranteed to be
continuous.

However, even at the time in which the Principia became popular, important
objections were raised by Leibniz’s [148]. Disregarding interpersonal disputes
and feuds [54], two of the limitations he stated of Newton’s mechanics would
remain valid criticisms centuries later: (a) the mathematics of the Principia
could not easily account for the material constitution of objects –or its
internal motion if any- and, most importantly, (b) it disregarded completely
the relation between entities and objects. The foundations of Leibniz’s
scientific work were placed on the putative existence of relations between
objects as defined by changes in their properties, which could be identified
as motion when these properties were represented in a Cartesian manner.
However, the lack of impact and success of his theory of relations –in
contrast to Newtonian mechanics- appears to have been largely due to his
inability to articulate what interactions _are_ instead of what they _are not_
[173]. In that sense, Leibniz was unable to provided a computable account of
motion starting from relations, and his notion of interacting bodies became
confounded often with his interacting monads, which pertain to formal logic
[50]. As an addendum, Fourier showed that trajectories could be reversibly
converted into frequencies, which when taken at infinitesimally small
intervals, resembled discrete events [116]. The latter would be used
extensively by Dirac [123] through his delta function to give operational
meaning to interactions in quantum mechanics.

With the advances in the mathematical language of calculus and its increasing
use in science and technology, the representation of motion as trajectories
that are constrained by laws and given by differentiable laws of motion became
the rock on which Maxwell’s work on electromagnetism [169] stands. However,
the renaissance of atomic theory through Boltzmann’s equations [42, 43] for
the theory of gases gave way to the possibility of computing thermodynamic
quantities in a straightforward fashion restricted to gas molecules that do
not interact, which only holds for ideal gases. In order to include
interactions, nonlinear (e.g. product) terms need to be introduced in the
equations of motion, and the information that can be obtained locally is how
the particular force involved (for instance, electromagnetism) would be
rendered in space-time as a landscape of potential energy with valleys and
peaks. The latter does not only leave the problem of describing the _internal
mechanics_ of interactions unsolved, but also makes their numerical treatment
intractable for many molecules. At present, the combinatoric explosion that
arises when simulating CMSS with many scales and classes of entities using
Newtonian mechanics is a particular case of the latter problem [203].

Ernst Mach’s [163] objections to the foundations of mechanics between the XIX
and XX centuries rested on the same arguments that Leibniz had raised, only in
a slightly different form. Ignoring interactions, or rather the composition
and properties of the relations between objects, leaves a plethora of
unanswered questions and a complicated mathematical conundrum. For Mach, the
ultimate form of any law of nature should be given in terms of properties of
one system in relation to other systems. Einstein’s special [85, 84, 86, 87]
and general relativity [88, 89, 83] were non-obvious but natural extensions of
Leibniz and Mach insofar as the principle of relativity was applied to frames
of reference, which led to the interpretation of space-time as a dynamical
entity. The objects that interact in general relativity are frames of
reference and the messengers are light particles. By sending photons and
experiencing internal changes, one frame of reference exerts a causal effect
on another frame of reference. Although actual computations are performed by
unfolding the tensor expressions into their full differential form, the most
relevant representation is given by the metric tensors that indicate how
energy and matter constrain space-time. These expression are compact and
appear be universal.

The development of quantum mechanics required inevitably describing
interactions (i.e. measurements), and marked, with relativity, one of the two
most profound revolutions in thinking about the physical world of the
twentieth century [137]. The necessary introduction of probability in the laws
of motion, based both on wave equations and positions/momenta required several
amendments to Cartesian spaces in order to cope with singularities and
discontinuities [118] arising from intrinsic randomness and wavefunction
collapse. Quantum theory showed that posterior mutual state relaxation in the
form of decoherence [255] is a fundamental property of interactions below the
atomic scale. Quantum field theory [136] is one of the most accurate
formulations of reality in existence today, based on a model of interaction
between fields that, when observed by a system, produces particles
corresponding to field excitations. However, the transition functions still
refer to elements of the Cartesian view and their effects are expressed as
changes in the geometry of the fields rather than the changes in the entities
themselves. The situation appears to be no different in modern particle
physics [232], where Feynman diagrams are used to derive gruesome expressions
posing a myriad of challenges when using the Standard Model to obtain
particular solutions that explain experimental data. Although these are the
two most successful theories in the physical sciences, some of their remaining
limitations can still be traced thanks to Leibniz’s objections.

Regardless of the theoretical background, two types of problems remain. First,
the practical choice of parameters to accompany the functional descriptions is
expected to be a matter of experimental verification through fitting.
Nevertheless, CMSS instances pose challenges in this stage for a variety of
reasons. For instance, it may be hard to differentiate parameters pertaining
to two different trajectories that are spatially close at a measurement time,
requiring a large number of measurements for their differentiation. It is not
always possible to obtain good, repeatable experimental measurements. Second,
two deeper theoretical problems arise in connection with the search for
appropriate functional choices. The form of the laws must be guessed at some
level with only circumstantial information from more fundamental scales, but
the apparently connected structure of the universe [224, 225] and the
Correspondence Principle in quantum mechanics [82, 94] suggest that physical
laws are dynamical extrema at the thermodynamic limit. The form of that limit,
however, can only be guessed for systems of simple particles and homogeneous
interactions (if any). An open question is whether the number characterizing
the thermodynamic limit for a given system (e.g. a CMSS) is system independent
or is a function of the types and diversity of interactions. The second
problem is the apparent independence between laws at the microscale and the
organization of systems at the macroscale: if the macroscopic laws are
independent from the physical substrates, then either the scales of
organization cannot be truly connected, or the formulation of the laws are
biased by our selection of relevant entities at the microscale [108]. We are
inclined to support the latter rather than the former, since the possible
formulation of interactions as Barabási networks suggests that laws appear to
be topologically connected through substrates and bridge equations.

Although it is common (and correct) to think of macroscale properties as
emerging from microscale objects and connections, in nature the macroscale
exerts a formative influence on the microscale through adaptive evolution. In
constructed systems desired functional attributes plays the same role as
adaptive evolution in constraining the nature of the microscale subsystems.

In summary, our usual choice of dynamical functions is most frequently
determined by the Cartesian view of motion as trajectory, and the emphasis on
their prediction and retrodiction rather by interrogations of its properties
or structure. In most theories, interactions are either entirely neglected or
indirectly represented by transition functions that refer to changes in fields
or potentials, both contained in generalized versions of Cartesian spaces.
Maintaining this view limits the possibility of understanding systems of
larger complexity, either because of the combinatorial explosion of
calculations or because of the inability of our methods to efficiently
traverse the granular landscape of the resulting potential energy surfaces. A
focus on interactions and relations between objects has brought significant
understanding of the natural world on previous occasions, but interactions
largely remain circumstantial, derived or implicit despite their fundamental
character. Our aim in this project is to provide an alternative view of motion
in CMSS where interactions put at the center or descriptions or nature and are
explicitly described, investigated and operationalized.

####  3.1.3 Pre- and post-conditions from dynamical equations of motion point
to opaque interaction mechanisms

The formalization of CMSS using sets of deterministic differential equations
is thought to express instantaneous rates of change. Let us suppose that a
given CMSS is studied using ODEs for the present argument. Take a simple ODE
such as

$\frac{dx(t)}{dt}=f(x,t)$ (1)

and, for the moment, remove concerns about the specific properties of
$f(x,t)$. At first impression, this may be thought of as a definition only
that encodes two statements: one about equality between two quantities and one
about dynamics. The transformation $f$, dependent on $x$ and $t$, can be
intuitively thought of as either a fixed relation in the $(t,x(t))$ manifold
–more properly a graph of such instantaneous relations– or a unique trajectory
across the manifold. However, when we expand the definition of
$\frac{dx(t)}{dt}$ into Eq. 1

$\lim_{\alpha\to 0}\frac{x(t+\alpha)-x(t)}{\alpha}=f(x,t)$ (2)

it becomes clear that, for the limit to exist, both sides of the limit must
converge to $f(x,t)$ from the left and the right simultaneously. Recalling
that $|\alpha|>0$ and $|\alpha|\to 0$ in order for it to be an infinitesimal,
we may think at any given moment that, for any $\alpha^{-}=-\alpha$ and
$\alpha^{+}=+\alpha$ it must be true that

$\frac{x(t+\alpha^{-})-x(t)}{\alpha^{-}}<f(x,t)<\frac{x(t+\alpha^{+})-x(t)}{\alpha^{+}}.$
(3)

More generally, we may think of $\alpha$ as the limit of a Cauchy sequence
$\alpha=\lim_{n\to\infty}\alpha_{n}$ with $\forall\alpha_{i}.\alpha_{i}\neq
0$. Hence, we may think of the left and right limits more properly as

$\frac{x(t+\alpha^{-}_{n})-x(t)}{\alpha^{-}_{n}}<f(x,t)<\frac{x(t+\alpha^{+}_{n})-x(t)}{\alpha^{+}_{n}}$
(4)

and, after reorganizing terms,

$0<f(x,t)-\frac{x(t+\alpha^{-}_{n})-x(t)}{\alpha^{-}_{n}}<\frac{x(t+\alpha^{+}_{n})-x(t)}{\alpha^{+}_{n}}-\frac{x(t+\alpha^{-}_{n})-x(t)}{\alpha^{-}_{n}}$
(5)

the last term in the inequality can be interpreted as a time equivalent to the
center of mass (i.e. CoD: ”center of dynamics”)

$\mathrm{CoD}(\alpha,x,t)=\frac{\alpha^{-}_{n}[x(t+\alpha^{+}_{n})-x(t)]-\alpha^{+}_{n}[x(t+\alpha^{-}_{n})-x(t)]}{\alpha^{-}_{n}\cdot\alpha^{+}_{n}}$
(6)

which can be given the following interpretation: the CoD is the centroid of
the interval that bounds $f(x,t)$ to the left by an event
$x(t+\alpha^{-}_{n})$ and to the right by another event $x(t+\alpha^{+}_{n})$
for any $a_{n}>0$ in the corresponding Cauchy sequence. To that extent,
independent of how small $\alpha^{-}_{n},\alpha^{+}_{n}$ become as given by
approximations of the sequence, both bounding points are identified and have a
definite value. It is then possible to label $\alpha^{-}_{n}$ as a pre-
condition and $\alpha^{+}_{n}$ as a post-condition. Even with the
instantaneous mode of action expected of Newtonian mechanics, pre-conditions
and post-conditions are inescapable. In practice, this becomes more evident by
the suitable choice of the integration algorithm and the integration step. One
or more past points in the trajectory and estimates of one or more future
points in the trajectory are used to compute the current state of the
system222Pragmatically, however, equations for the current state under
computation are performed for $t+\Delta t$ with $\Delta t$ being the time
step..

What information is then contained in dynamical equations, and more
importantly, what information is excluded? From the discussion above, sets of
coupled deterministic differential equations do contain an implicit
specification of local pre-conditions and post-conditions at every point in
the manifold where the action functional is defined. We must note that, in
terms of causality, trajectories are not contingent on external factors in
Newtonian mechanics –-they are inevitable- but at the same time every point in
the trajectory appears as if it were contingent on the (densely) immediate
past. Also, these equations provide a global geometry, a field that strictly
conditions the set of all possible trajectories for a given set of parameters;
this does not, however, ensure that the trajectories will be sensible always
or that local information suffices to predict behavior at large. Even if the
equations govern the trajectory of one entity instead of the evolution of sets
of entities described through proportions, there is no information whatsoever
about how entities are internally structured, how their properties vary by
being in contact with the medium as captured by the manifold and, in the case
of coupled equations, what types of interactions lead to the coupling itself.

In summary, deterministic coupled differential equations contain a normative
prescription for the action in a system by describing possible trajectories
under fixed parameterizations. However, the equations themselves contain no
information about mechanisms that explain the form of differential functions
(validly interpreted as a transformation of state), the connection of the
medium of action and the entities that explains the metric that arises under
integration, and how the magnitude and frequency of couplings occurs as given
by covarying quantities or proportions in a system. Intuitively, the latter
framework should be considered as extremely limited to describe CMSS
phenomena.

##  4 Interactions: a theoretical inventory

Interactions are necessary to understand all phenomena across the natural and
social sciences. However, their representation is not transparent in terms of
internal structure or processes, and in the majority of cases their presence
is inferred indirectly from dynamical equations that focus on the trajectories
of systems, their components or statistical correlations. Moreover,
interactions in quantum mechanics and field theory are represented by unitary
transformations that dictate how a prior state transforms into a posterior one
without accounting for the mechanics of the change in a direct or detailed
manner. A subset of social theories explicitly model the dynamics of the
interactions locally, but are yet to produce an integrated understanding of
the dynamics and possible macroscale configurations of social systems at
large. Changes in the _modes_ of interaction of simpler parts of a system,
however, have been experimentally shown to have aggregate or collective
effects in the system as a whole. The following is a detailed description of
how scientific disciplines capture interactions according to their theoretical
concepts and tools.

###  4.1 Statistics

In statistics, interactions are modeled by non-additive correlations between
two or more variables either explicitly through their product, or transitively
by composition of other products [229, 103]. Many statistical models follow
the Fisherian (i.e. frequentist) approach that asserts the statistical
significance of events [81, 254] by computing values of various statistics and
their various moments. Its main sources of trouble are the inability to
straightforwardly link statistics to observables in a system, and the need to
assume normally distributed data for many of the interpretations to be correct
while non-normally distributed data dominates empirical observations.
Statistical models (possibly capturing the effects of interactions indirectly)
often require dummy variables with no phenomenological significance for a good
fit [47].

Mixed models containing positive and negative regulatory effects of
interactions may not lead at all to statistically significant conclusions for
complex situations [170]. When the interactions are complex –beyond two-
variable couplings- computational cost grows exponentially [236]. Various
methods and heuristics are used to give structure to the statistical search
space based on parsimony as a desirable property of good models [17, 33].
Nevertheless, the problem of deriving mechanisms from statistical models for a
given system –assuming interactions exist- in a more direct way [162] remains
open in Fisherian statistics. The needs for analysis of ever-increasing
datasets in biology is suggestive of the latter [243]. Bayesian [147] and non-
parametric [65] statistics have arisen and matured as responses to some of
these limitations.

###  4.2 Computing and logic

In computer science and formal logic, interactions have been defined multiple
times in relation to hardware, formal descriptions of computations, events in
software systems, or their various combinations. The geometry of interactions
[112] in logic attempts to capture the causal effect of sequential steps in
proofs, where proofs are actions denoting observable transformations of
statements grounded on an underlying operational semantics. In the classical
theory of computation [69], a Turing machine measures (i.e. interacts) with an
infinite tape with cells that contain symbols: the measurement is used by a
transition function to select the next state through pattern matching and its
posterior selection of the associated action (e.g. read, write, move left,
move right, halt). Consequently, the Turing machine is one example where
interaction mechanisms in imperative programming are entirely visible thanks
to the simplicity of the components and steps required to implement the
robotics (i.e. automatizable manipulation) of symbolic pattern matching. In
object oriented programming, the effect of a program –as formalized in sigma
calculus [1]\- is a result of the interaction between s random-access memory
(RAM), a stack and a set of instructions that specify how the internal state
and the computer (i.e. stack + RAM) are simultaneously modified under
appropriate conditions. In constraint logic programming [241], not only proofs
are sequences of action that capture the interaction between denotational and
operational contents of programs (e.g. the interaction between knowledge bases
with rules of inference [175]) but they are bound by the contours of internal
laws (i.e. constraints) that prevent or allow only certain interactions to
occur. Finally, monads were introduced in lambda calculus and functional
programming languages as a means to express interactions between purely
functional programs and operations that alter the machine state (i.e. the
outside world) through persistent, secondary effects [178].

With the development of distributed computing and information systems, their
interaction became a significant area of research [11]. Milner’s seminal
Turing Award Lecture [174] recognized the need for a theory of interactions in
computer systems beyond C. A. R. Hoare’s foundational work con communicating
sequential processes [126]. Simultaneously, actor-based parallel programming
models for distributed systems were developed based on various empirical
observations that collections of agents can achieve coordination as a result
of the product of their internal (active) state and the messages being
exchanged between them [2]. The expected internal complexity of agents can be
derived through the law of requisite variety [15] as well as through measures
of effective complexity of systems [107]: the information content of the
control of a distributed system is proportional to the information required to
describe its state and dynamics [202].How interaction patterns determine the
dynamics of a distributed system is a well-established research area at
present [235].

Research on distributed systems has also focused on modeling and expressing
interactions through programming languages [3, 197] and their semantics [231,
25], necessary to build and execute computer experiments that probe their
effects controlled ways. Another view on distributed systems has been
developed from organization theory by attempting to understand how
organization patterns emerge based on which interactions are present and what
the responses of the agents are [101], similar to what happens in human
organizations. The latter has been critical to postulate and simplify the
process of reasoning about collective and emergent properties of distributed
systems in a succinct and intuitive manner [110]. Establishing causality in
distributed computing and information systems, despite their simplicity in
comparison with other types of distributed systems in nature, has proven to be
a hard problem. Lattice theory and discrete time [205] have been used to
understand collective problems in distributed information systems, which
appear to strongly depend on whether interactions are synchronous or
asynchronous [237]. An interesting aspect that has been explored is how law-
governed interactions shape coordination and control [176], similar to what
occurs in physical and biological systems with direct applicability to cyber
physical systems [201]. In terms of theoretical advances, tangle machines [52,
53] a model based on the intersection of interaction histories as processing.

###  4.3 Social theory

The literature describing social interactions is vast, spanning microbiology
[154], developmental biology [109], neuroscience [185], primatology [51, 16],
human psychology [150], linguistics [58, 144], social science [204, 172],
organization theory [135, 105, 41], economics [27, 212], policy-making [177],
Internet-based communication [132, 57], complexity theory [120] and more
recently discussions of how humans [48, 44, 95] and robots may coexist, to
name a few. Identifying [115, 70, 40] and measuring [113] social interactions
–and their effects [165]\- is central to converting observations into reliable
data to either postulate new theories or to put existing ones to the test. In
contrast to the situation in the physical sciences, describing and
understanding interactions (e.g. their structure, transformations,
regularities) between two social agents has proven to be somewhat tractable
and useful compared to efforts that have focused on obtaining governing laws.
These laws are generally expected to be emergent and its form depend on the
internal properties of agents and evident from various collective phenomena
[92]. Social coordination is a prime example of collective behavior that
depends heavily on the properties of interactions [248, 59, 188, 234], which
also appear to allow the existence of scaling laws of various types [106, 36,
207]. To that extent, social systems constitute a prototypical CMSS example.

Given that one aim in this project is to clarify the structures of CMSS that
can be interpreted simultaneously through both network and dynamical theories
while remaining agnostic of particular substrates –systems in which
interactions among agents appear to be crucial-, we have chosen three
particular views on social systems as the most relevant descriptions of social
individual and collective organization: Luhmann’s systemic view of society
[159, 160], Berger’s model of the function of communication as a form of
interaction that decreases future uncertainty of encounters, [32, 31] and
Pentland’s conceptualization of routines as recurring patterns of interaction
that simultaneously endow systems with structure and flexibility [96, 97].
Societies are complex systems in which events are coupled at multiple
interconnected levels that are structured by self-organization mechanisms.
These usually harness feed-back loops through coordination tokens (e.g.
messengers) that are exchanged during the interaction and encode complex
transformations. These tokens, to a large extent, summarize the abstract space
of state transformations as given by past and future states of the system in
relation to the message being conveyed. The token therefore also encodes
aspects of the internal changes in each state (e.g. relaxation, new forms of
self-organization) and causal properties of state transformations in other
agents including its knowledge and actions. As ensured again by various
measures of total information [107] and the law of requisite variety [15], we
can known –at least approximately- the amount of degrees of freedom exchanged
between both agents by analyzing the tokens. However, the exchange is often
imperfect thanks to noisy channels (i.e. environmental perturbations) or
problems of interpretation (i.e. encoding mismatches), extending the effects
of uncertainty to all causally connected events in the system at various
degrees. Considering routines as persistent interactions introduces frequency
as a relevant factor, modulated by system size and diversity.

###  4.4 Biology

Interactions are essential in biology at all scales. In ecology, the largest
class of interactions are defined between the ecosystem and evolutionary rules
in a feedback cycle [195]. Species interact through the establishment of
various relations, including predator-prey dynamics [26], parasitism [63] and
symbiosis [76]. These relations exist even at the cellular level [167] and are
complemented by chemically mediated interactions [158]. Within the environment
of the cell, interactions occur in a complex biochemical environment
articulated by molecular expression pathways whose activation is stochastic
and threshold-dependent [218], and whose action is carried out by protein-
protein interaction networks as determined by the affinity between various
binding sites [131]. An estimate of $10^{14}$ atoms in a cell and a similar
number of cells in higher organisms suffice to make numerical simulations
intractable if all forces and interactions are included. However, hierarchical
modularity also guarantees that approximate simulations of these systems or
their components may be both tractable and remain biologically meaningful
[198]. Multiscale modeling often help maintain consistency across all nested
hierarchical scales without sacrificing good performance and repeatability.
This network embedding can also be used to apply analogues of renormalization
group theory to reduce complexity through mean-field approximations [4] that
contain scaled down quantities of entities, interactions or even hierarchical
levels while preserving the main character of events and relations present in
biological phenomena. Life is a prime example of a coupled, multiscale
problem. The longest relevant times scale in biology is evolutionary. Over
time scales ranging up to millions of years, selection pressures applied at
the macroscale in the form of environmental change select variations at the
molecular genetic level that are manifest as changes at the organismic level.

At the most abstract extreme in the research spectrum, Barabási and Albert’s
seminal work on the emergence of scale-free networks [22] has been applied
widely in biology, in particular to organisms as one of the most important
mesoscales in biology. These systems exhibit organized complexity [142]
arising from interactions that produce identity, robustness while remaining
constrained by various types of underlying laws. Contrary to artificial
complex systems, biological entities have emerged through evolution by natural
selection, operating over whole ensembles of live entities. The rules of
interaction follow the contours of laws that appear to emerge at more
fundamental scales and are expected to define the ultimate boundaries of life
itself. Cells are constantly subject a changing and complex environment whose
response depends on stochastic delays driven by drift-diffusion forces,
captured by Langevin equations [102]. At the same time, phylogeny suggests
that selection, repetition and variation thread the fabric of biological
diversity. Understanding the massive complexity and variety observed in living
organisms is one of the most important tasks in biology [249]. It is also one
of imperfect reconstruction of interactions of extremely modular [19] and
plastic [119] systems. At present, one of the few principles that facilitates
this reconstruction is Hennig’s Auxiliary Principle, stating that sequence
changes between genotypes are related to the molecular clocks in organisms
[206], but more principles –and ideally, more evidence to support them- is
required for a more solid scientific ground. We hypothesize that some of those
new principles rest on properties and dynamics of interactions.

###  4.5 Chemistry

Interactions in chemistry at the atomic level are represented by various types
of bonds, emerge from orbital structures [8] which determine atomic valence
and polarization due to electron motion. Bonding is determined by the time-
dependent energy potential surface mediating interactions, described as short-
range forces between atoms. Predicting interactions in chemistry at the
molecular level, however, is extremely expensive. From a generative
perspective, the combinatorial problem of finding suitable reactions from sets
of molecules becomes intractable [20, 246] since the space of possible
molecules and interactions grows exponentially. If the set of molecules of
interest is known as well as some of their interactions, predicting new ones
requires accounting for the quantum character of the laws from which bonding
emerges, introducing a large set of concerns.

For a given molecule, the strict computation of the state of nuclei and
electrons across atomic orbitals depends simultaneously on how forces act on
their position and spin [250]. With each new electron, computing
superpositions of orbital wave functions is exponentially more expensive. Many
quantum chemistry methods start from the Born-Oppenheimer approximation where
(a) nuclei are removed from the computation since their mass and position are
stable relative to the motion of electrons and (b) the individual electron
wave functions $\psi_{i}(\vec{r},\vec{s})$ are approximated as the product of
a position-dependent wave function and a spin-dependent wave function
independent of each other. We thus obtain the Hartree-Fock equations, which
constitute the basis of most self-consistent field theories [100] used in
quantum chemistry. Other methods such as Density Functional Theory [190])
dispense with explicit representation of the electron cloud and rather operate
on electron densities that vary depending on how the statistics of how
electrons are spatially (and spin) correlated, and how they exchange energy
changes in various situations as given by functionals dependent on the
chemical species in the system. Along with many other approximations of the
electronic structure of the atom, most of quantum chemistry is devoted to
understanding the effects of various interactions within electron clouds on
the change of the geometrical conformation of molecules, their polarization
and reaction surfaces, and emitted spectra [64].

Several approximations have been devised to alleviate difficulties in quantum
computational chemistry methods. Stochastic Monte Carlo sampling helps
decrease computational cost in all quantum chemistry methods [121]. Moreover,
semi-empirical methods are used whenever any variables can be considered as
fixed and experimental data supply their value [208]. At a larger yet specific
scale, interactions are more approximately captured by molecular reaction
dynamics [149], which contain rules that seek to reproduce bonding behavior in
simpler, more prescribed ways. Finally, molecular dynamics describes
interactions at large between the position of atoms and potential energy
surfaces [60] when the latter are previously known, without involving quantum
phenomena. More recently, the rising field of supramolecular systems chemistry
attempt to provide a better account of the (non-covalent) chemical
interactions responsible for molecular recognition and self-assembly [168].

###  4.6 Classical mechanics

Interactions in dynamical systems governed by classical mechanics are
implicitly represented as non-linear terms containing two or more variables in
the differential equations representation of the system trajectory.
Interaction events may be often equated with either normal modes [56], non-
linear oscillations [117], field excitations [179], or bifurcation points
[122]. Interactions only arise indirectly as the underlying cause for
observable deviations from additivity and linearity [199, 161, 157] but the
interaction mechanisms themselves causing the deviation in the trajectory
remain hidden at various degrees.

When dynamical systems are simultaneously not analytically computable and
sufficiently complex due to the presence of nonlinear terms and non-trivial
couplings, Monte Carlo methods [156] can be used to approximate a solution by
collecting independent samples with lower computational cost. Adjusting
measurement density permits to approximately reconstruct nonlinearities with
varying degrees of success. More refined well-known techniques remove the
restriction of sample independence by introducing stochastic processes [111,
35, 226] in an attempt to preserve time dependencies arising during successive
interactions. For spatial dependencies, a generalization exists in the form of
sequential Monte Carlo sampling [180]. In all Monte Carlo methods, probability
models can be setup to mimic the outcome of particular interaction mechanisms
at the expense of often impractically expensive computations if the systems
are large and/or complex. However, the interactions themselves remain opaque.

###  4.7 Quantum mechanics

In quantum mechanics, interactions are modeled in general as interference
patterns in the wave function resulting from coupled systems whose states are
quantized [6]. In measurement processes, one system is used to irreversibly
interrogate another system, collapsing the state of one or more degrees of
freedom that are superposed resulting in a particular _observable_ with a
probability proportional to wave function amplitude. Superposition,
entanglement and decoherence provide critical insights into interactions at
the quantum level. Superposition entails the existence of shared global
information across many possible states within a single system prior to being
observed. Superposition also involves correlated histories of states prior to
their measurement, a fact exploited by quantum computing architectures [49].
Quantum entanglement [90, 28, 130, 200] corresponds to a interaction where a
collective system contains a non-separable quantum state capable of being
preserved even when the system components are separated across arbitrary
distances; operators applied to one part of the system impact the state as a
whole immediately. Finally, quantum decoherence refers to the interaction
between a quantum system and the larger environment, resulting in fast decay
of state superpositions [255].

In terms of representation, the evolution of a quantum system with $N$ bodies
is given by a Hamiltonian operator with exponentially more information than
its classical or relativistic counterparts [62, 133]. Approximations used to
compute outcomes usually contain wave functions for interacting pairs of
quantum states in terms of their position, momenta or spin. In the case of
coherent collective quantum phenomena such as Bose-Einstein condensation,
interactions are represented as lattices of quantum harmonic oscillators whose
properties depend on whether the lattice points are fermions or bosons [141].
While the spectral nature of quantum states lends itself naturally to
frequency domain analysis, the main focus remains around system trajectories.

###  4.8 Quantum field theory

In quantum field theory [146], particle interactions arise as interaction
terms for excitation modes of local states of quantum fields: whenever two
particles interact, a field can be inferred. Field theory has been
instrumental in organizing and making sense of the diversity of particles and
their interactions in high energy physics [124]. Feynman diagrams capture
particle interactions as perturbative contributions to transition amplitudes
between causally connected quantum states [192]. Translating these interaction
diagrams into actual computations starts from obtainig rules derived from the
interaction Lagrangian in quantum electrodynamics [99], then finding a well-
defined grounded state of the system, integrating over all possible histories
for the quantum process in the field and using a Wick rotation to obtain an
expression in terms of imaginary time [245]. Even when the interactions are
explicit, the formal treatment of pairs of interacting field excitations
remains constrained by trajectory-based representations.

###  4.9 Summary: interactions in CMSS require better representations

All prior descriptions of interactions share one of two limitations. Either
the description avoids describing what happens during an interaction by
concentrating on trajectories or frequencies, or specific interaction
mechanisms becomes transparent at the expense of predictive or retrodictive
power for system outcomes. We argue that, in order to make progress on
describing complex problems with strongly-coupled scales of action towards
higher realism requires constructing a theory _de novo_ capable of overcoming
the epistemic biases and limitations derived from non-relational or partially-
relational theories that do not promote interactions as first-class citizens.

##  5 Conclusion

Complex stochastic dynamical systems are ubiquitous and their understanding is
necessary to find accurate and efficient solutions to frontier problems across
a wide spectrum of disciplines in the natural and social sciences. Each CMSS
exhibits multiple phenomenologies whose understanding remains largely
incomplete and is often performed through multiple formal levels of
description. These are often constructed assuming independence from one
another in terms of governing laws and involved entities, are expressed
through sets of deterministic, continuous, reversible dynamical equations, and
are expected to integrate nicely _ab initio_.

On the contrary, cosmology, biology and information theory strongly suggest
simultaneously that not only the architecture of the universe is nicely
integrated across all scales, but that the emergence of nearly decomposable
systems appears to be one manifestation of a more general correspondence
principle between a sequence of microstates associated with macrostates
through bridge equations. In this view both objects and laws are emergent and
dependent on the geometry defined by interactions yet in most dynamical
descriptions of systems interactions are described implicitly through
transformations, themselves often buried within the vanishing infinitesimals
that define differential operations.

Such a state of affairs limits the ability to provide causal explanations of
CMSS across multiple levels in general, resulting further in (a) the inability
to predict responses of these systems at various horizons and (b) the
inability to appropriately perform retrodiction by capturing all relevant
contingencies. In practice, these two limitations seem to materialize as
either theoretical incompatibilities between adjacent levels of description,
manifold hardships to ensure appropriate formal conditions for numerical
methods to yield approximately correct answers, and spurious results with
respect to experimental measurements known to be rigorous. To a large extent,
this is just a symptom of a deeper abyss dividing research in the physical
sciences. Roughly speaking, we can identify two communities: those who insist
on the preeminence of calculating [139], and those that insist on the
importance of working on the conceptual foundations [68]. Most contemporary
approaches discussed here, eager to obtain results from readily observable
entities fits in the first category, belongs to the first group. Our work is
rather inspired by the second category.

We hypothesize that a universal solution to all the above mentioned problems
exists in the form of a novel theoretical framework that focuses its attention
on interactions rather than on trajectories, states or equations of motion.
This generalized theory of interactions should be expressed as a differential
geometry that maps manifolds of events in a CMSS as usually expressed into a
3-manifold named the _interaction space_ of the system that contains
information about classes of interactions present in the system in a form
suitable for the law of large numbers. In addition, a translation of
invariants between the two classes of spaces should not only be possible, but
proven to be more economic and effective in interaction space than in usual
differential manifolds.

In consequence with the vast collection of differential methods, we also
hypothesize that it is possible to partially recover the natural and expected
variation produced by interactions currently neglected in existing
differential models by (1) using spectral analysis of data from stochastic
systems to obtain pure probabilistic models that capture relevant interaction
scales, (2) introducing stochasticity into coupled, multiscale differential
equation models (3) developing agent-based models that rigorously model
interactions and (4) providing cyberinfrastructure to facilitate the gradual
adoption of both the theory and associated methods.

### Contributions

S. Núñez-Corrales devised the theoretical formulation. E. Jakobsson edited and
revised the manuscript

### Competing interest

Authors declare no competing interest.

### Funding

This work was supported by Illinois Informatics and the ACM/Intel SIGHPC
Computational and Data Science Fellowship, 2017 cohort.

### Acknowledgments

This work is dedicated to the memory of late Prof. Eric Jakobsson, who passed
away in October 2021.

## References

  * [1] Martín Abadi and Luca Cardelli.  An imperative object calculus.  In Colloquium on Trees in Algebra and Programming, pages 469–485. Springer, 1995.
  * [2] Gul A Agha.  Actors: A model of concurrent computation in distributed systems.  Technical report, Massachusetts Institute of Technology, Cambridge Artificial Intelligence Laboratory, 1985.
  * [3] Gul A Agha.  Abstracting interaction patterns: A programming paradigm for open distributed systems.  In Formal Methods for Open Object-based Distributed Systems, pages 135–153. Springer, 1997.
  * [4] Réka Albert and Albert-László Barabási.  Statistical mechanics of complex networks.  Reviews of modern physics, 74(1):47, 2002.
  * [5] Réka Albert, Hawoong Jeong, and Albert-László Barabási.  Error and attack tolerance of complex networks.  nature, 406(6794):378, 2000.
  * [6] Sergio Albeverio, Friedrich Gesztesy, Raphael Hoegh-Krohn, and Helge Holden.  Solvable models in quantum mechanics.  Springer Science & Business Media, 2012.
  * [7] Sergio Albeverio, Volker Jentsch, and Holger Kantz.  Extreme events in nature and society.  Springer Science & Business Media, 2006.
  * [8] Thomas A Albright, Jeremy K Burdett, and Myung-Hwan Whangbo.  Orbital interactions in chemistry.  John Wiley & Sons, 2013.
  * [9] Stephen C Anco and Abdul H Kara.  Symmetry-invariant conservation laws of partial differential equations.  European Journal of Applied Mathematics, 29(1):78–117, 2018.
  * [10] Philip W Anderson.  The economy as an evolving complex system.  CRC Press, 2018.
  * [11] Gregory R Andrews.  Paradigms for process interaction in distributed programs.  ACM Computing Surveys (CSUR), 23(1):49–90, 1991.
  * [12] Peter L Antonelli and Curtis Strobeck.  The geometry of random drift i. stochastic distance and diffusion.  Advances in Applied Probability, 9(2):238–249, 1977.
  * [13] Mituhiko Araki and Toshio Kato.  Local stability and stability region of discrete-time composite systems.  Transactions of the Society of Instrument and Control Engineers, 17(1):23–28, 1981.
  * [14] W Ross Ashby.  Self-regulation and requisite variety.  Systems thinking, pages 105–124, 1969.
  * [15] W Ross Ashby and Jeffrey Goldstein.  Variety, constraint, and the law of requisite variety.  Emergence: Complexity and Organization, 13(1/2):190, 2011.
  * [16] Filippo Aureli, Stephanie D Preston, and Frans de Waal.  Heart rate responses to social interactions in free-moving rhesus macaques (macaca mulatta): a pilot study.  Journal of comparative psychology, 113(1):59, 1999.
  * [17] Peter C Austin and Jack V Tu.  Bootstrap methods for developing predictive models.  The American Statistician, 58(2):131–137, 2004.
  * [18] Sunny Y Auyang.  Foundations of complex-system theories: in economics, evolutionary biology, and statistical physics.  Cambridge University Press, 1999.
  * [19] M Fayez Aziz, Kelsey Caetano-Anollés, and Gustavo Caetano-Anollés.  The early history and emergence of molecular functions and modular scale-free network behavior.  Scientific reports, 6:25058, 2016.
  * [20] K Balasubramanian.  Applications of combinatorics and graph theory to spectroscopy and quantum chemistry.  Chemical Reviews, 85(6):599–618, 1985.
  * [21] Yaneer Bar-Yam.  Multiscale complexity/entropy.  Advances in Complex Systems, 7(01):47–63, 2004.
  * [22] Albert-László Barabási and Réka Albert.  Emergence of scaling in random networks.  science, 286(5439):509–512, 1999.
  * [23] Jon Barwise and Jerry Seligman.  Information flow: the logic of distributed systems, volume 44.  Cambridge University Press, 1997.
  * [24] Danielle S Bassett, Daniel L Greenfield, Andreas Meyer-Lindenberg, Daniel R Weinberger, Simon W Moore, and Edward T Bullmore.  Efficient physical embedding of topologically complex information processing networks in brains and computer circuits.  PLoS computational biology, 6(4), 2010.
  * [25] Ananda Basu, Philippe Bidinger, Marius Bozga, and Joseph Sifakis.  Distributed semantics and implementation for systems with interaction and priority.  In International Conference on Formal Techniques for Networked and Distributed Systems, pages 116–133. Springer, 2008.
  * [26] David A Beauchamp, D Wahl, and Brett M Johnson.  Predator-prey interactions.  Analysis and interpretation of freshwater fisheries data. American Fisheries Society, Bethesda, Maryland, pages 765–842, 2007.
  * [27] Gary S Becker.  A theory of social interactions.  Journal of political economy, 82(6):1063–1093, 1974.
  * [28] John S Bell.  Einstein-podolsky-rosen experiments.  In John S Bell on the Foundations of Quantum Mechanics, pages 74–83. World Scientific, 2001.
  * [29] Charles H Bennett and Geoffrey Grinstein.  Role of irreversibility in stabilizing complex and nonergodic behavior in locally interacting discrete systems.  Physical review letters, 55(7):657, 1985.
  * [30] Herman JC Berendsen, David van der Spoel, and Rudi van Drunen.  Gromacs: a message-passing parallel molecular dynamics implementation.  Computer Physics Communications, 91(1-3):43–56, 1995.
  * [31] Charles R Berger.  Communicating under uncertainty.  1987\.
  * [32] Charles R Berger and James J Bradac.  Language and social knowledge: Uncertainty in interpersonal relations, volume 2.  Hodder Education, 1982.
  * [33] William D Berry, Matt Golder, and Daniel Milton.  Improving tests of theories positing interaction.  The Journal of Politics, 74(3):653–671, 2012.
  * [34] Lorenzo Bertini, Alberto De Sole, Davide Gabrielli, Giovanni Jona-Lasinio, and Claudio Landim.  Lagrangian phase transitions in nonequilibrium thermodynamic systems.  Journal of Statistical Mechanics: Theory and Experiment, 2010(11):L11001, 2010.
  * [35] Carlo Berzuini, Nicola G Best, Walter R Gilks, and Cristiana Larizza.  Dynamic conditional independence models and markov chain monte carlo methods.  Journal of the American Statistical Association, 92(440):1403–1412, 1997.
  * [36] Luís MA Bettencourt, José Lobo, Dirk Helbing, Christian Kühnert, and Geoffrey B West.  Growth, innovation, scaling, and the pace of life in cities.  Proceedings of the national academy of sciences, 104(17):7301–7306, 2007.
  * [37] DK Bhattacharya and GC Lie.  Molecular-dynamics simulations of nonequilibrium heat and momentum transport in very dilute gases.  Physical review letters, 62(8):897, 1989.
  * [38] G Bitsoris and Ch Burgat.  Stability analysis of complex discrete systems with locally and globally stable subsystems.  International Journal of Control, 25(3):413–424, 1977.
  * [39] Peter Bloomfield.  Fourier analysis of time series: an introduction.  John Wiley & Sons, 2004.
  * [40] Lawrence E Blume, William A Brock, Steven N Durlauf, and Yannis M Ioannides.  Identification of social interactions.  In Handbook of social economics, volume 1, pages 853–964. Elsevier, 2011.
  * [41] Marcel Bogers, Ann-Kristin Zobel, Allan Afuah, Esteve Almirall, Sabine Brunswicker, Linus Dahlander, Lars Frederiksen, Annabelle Gawer, Marc Gruber, Stefan Haefliger, et al.  The open innovation research landscape: Established perspectives and emerging themes across different levels of analysis.  Industry and Innovation, 24(1):8–40, 2017.
  * [42] Ludwig Boltzmann.  Zur integration der diffusionsgleichung bei variabeln diffusionscoefficienten.  Annalen der Physik, 289(13):959–964, 1894.
  * [43] Ludwig Boltzmann.  On certain questions of the theory of gases.  Nature, 51(1322):413, 1895.
  * [44] Prashant Bordia and Nicholas DiFonzo.  Problem solving in social interactions on the internet: Rumor as social cognition.  Social Psychology Quarterly, 67(1):33–49, 2004.
  * [45] Carl B Boyer.  History of analytic geometry.  Courier Corporation, 2012.
  * [46] Eric Braaten and H-W Hammer.  Universality in few-body systems with large scattering length.  Physics Reports, 428(5-6):259–390, 2006.
  * [47] Bear F Braumoeller.  Hypothesis testing and multiplicative interaction terms.  International organization, 58(4):807–820, 2004.
  * [48] Cynthia Breazeal.  Social interactions in hri: the robot view.  IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews), 34(2):181–186, 2004.
  * [49] Časlav Brukner.  Quantum causality.  Nature Physics, 10(4):259, 2014.
  * [50] Howard Burdick.  What was leibniz’s problem about relations?  Synthese, 88(1):1–13, 1991.
  * [51] Richard Byrne and Andrew Whiten.  Machiavellian intelligence: social expertise and the evolution of intellect in monkeys, apes, and humans (oxford science publications).  1989\.
  * [52] Avishy Y Carmi and Daniel Moskovich.  Tangle machines.  arXiv preprint arXiv:1404.2862, 2014.
  * [53] Avishy Y Carmi and Daniel Moskovich.  Tangle machines ii: Invariants.  arXiv preprint arXiv:1404.2863, 2014.
  * [54] Ernst Cassirer.  Newton and leibniz.  The Philosophical Review, 52(4):366–391, 1943.
  * [55] Alan Chalmers.  Making sense of laws of physics.  In Causation and laws of nature, pages 3–16. Springer, 1999.
  * [56] GM Chechin and VP Sakhnenko.  Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. exact results.  Physica D: Nonlinear Phenomena, 117(1-4):43–76, 1998.
  * [57] Yubo Chen, Qi Wang, and Jinhong Xie.  Online social interactions: A natural experiment on word of mouth versus observational learning.  Journal of marketing research, 48(2):238–254, 2011.
  * [58] Grace Cho.  The role of heritage language in social interactions and relationships: Reflections from a language minority group.  Bilingual Research Journal, 24(4):369–384, 2000.
  * [59] Michael Suk-Young Chwe.  Communication and coordination in social networks.  The Review of Economic Studies, 67(1):1–16, 2000.
  * [60] Giovanni Ciccotti, Mauro Ferrario, Christof Schuette, et al.  Molecular dynamics simulation.  Entropy, 16:233, 2014.
  * [61] Paul Cilliers.  Boundaries, hierarchies and networks in complex systems.  International Journal of Innovation Management, 5(02):135–147, 2001.
  * [62] Fritz Coester and Wayne Nicholas Polyzou.  Relativistic quantum mechanics of particles with direct interactions.  Physical Review D, 26(6):1348, 1982.
  * [63] Claude Combes.  Parasitism: the ecology and evolution of intimate interactions.  University of Chicago Press, 2001.
  * [64] David B Cook.  Handbook of computational quantum chemistry.  Courier Corporation, 2005.
  * [65] Gregory W Corder and Dale I Foreman.  Nonparametric statistics: A step-by-step approach.  John Wiley & Sons, 2014.
  * [66] Robert Costanza, Lisa Wainger, Carl Folke, and Karl-Göran Mäler.  Modeling complex ecological economic systems: toward an evolutionary, dynamic understanding of people and nature.  In Ecosystem Management, pages 148–163. Springer, 1993.
  * [67] András Czirók, Eshel Ben-Jacob, Inon Cohen, and Tamás Vicsek.  Formation of complex bacterial colonies via self-generated vortices.  Physical Review E, 54(2):1791, 1996.
  * [68] Olivier Darrigol.  ’shut up and contemplate!’: Lucien hardy’s reasonable axioms for quantum theory.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 52:328–342, 2015.
  * [69] Martin Davis.  Computability & unsolvability.  Courier Corporation, 1982.
  * [70] Giacomo De Giorgi, Michele Pellizzari, and Silvia Redaelli.  Identification of social interactions through partially overlapping peer groups.  American Economic Journal: Applied Economics, 2(2):241–75, 2010.
  * [71] Eric B Dent.  Complexity science: A worldview shift.  Emergence, 1(4):5–19, 1999.
  * [72] René Descartes.  1644 principia philosophiae.  J. Cottingham, R. Stoothoff and D. Murdoch (trans.),“Principles of Philosophy”, The Philosophical Writings of Descartes, 1:177–291, 1975.
  * [73] Roderick Dewar.  Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states.  Journal of Physics A: Mathematical and General, 36(3):631, 2003.
  * [74] Adrian Diaz, David McDowell, and Youping Chen.  The limitations and successes of concurrent dynamic multiscale modeling methods at the mesoscale.  In Generalized Models and Non-classical Approaches in Complex Materials 2, pages 55–77. Springer, 2018.
  * [75] Kevin J Dooley.  A complex adaptive systems model of organization change.  Nonlinear dynamics, psychology, and life sciences, 1(1):69–97, 1997.
  * [76] AE Douglas.  Symbiotic Interactions: Oxford Science Pub-lications.  Oxford University Press, Oxford, 1994.
  * [77] Freeman J Dyson.  Statistical theory of the energy levels of complex systems. i.  Journal of Mathematical Physics, 3(1):140–156, 1962.
  * [78] Freeman J Dyson.  Statistical theory of the energy levels of complex systems. ii.  Journal of Mathematical Physics, 3(1):157–165, 1962.
  * [79] Freeman J Dyson.  Statistical theory of the energy levels of complex systems. iii.  Journal of Mathematical Physics, 3(1):166–175, 1962.
  * [80] Freeman J Dyson and Madan Lal Mehta.  Statistical theory of the energy levels of complex systems. iv.  Journal of Mathematical Physics, 4(5):701–712, 1963.
  * [81] Bradley Efron.  Controversies in the foundations of statistics.  The American Mathematical Monthly, 85(4):231–246, 1978.
  * [82] Paul Ehrenfest.  Bemerkung über die angenäherte gültigkeit der klassischen mechanik innerhalb der quantenmechanik.  Zeitschrift für Physik, 45(7-8):455–457, 1927.
  * [83] A Einstein and M Grossmann.  Entwurf einer verallgemeinerten relativitätstheorie und einer theorie der gravitation, teubner, leipzig.  Reprinted in CPAE, 4, 1913.
  * [84] Albert Einstein.  Ist die trägheit eines körpers von seinem energieinhalt abhängig?  Annalen der Physik, 323(13):639–641, 1905.
  * [85] Albert Einstein.  Zur elektrodynamik bewegter körper.  Annalen der physik, 322(10):891–921, 1905.
  * [86] Albert Einstein.  Das prinzip von der erhaltung der schwerpunktsbewegung und die trägheit der energie.  Annalen der Physik, 325(8):627–633, 1906.
  * [87] Albert Einstein.  Über die vom relativitätsprinzip geforderte trägheit der energie.  Annalen der Physik, 328(7):371–384, 1907.
  * [88] Albert Einstein.  Über den einfluß der schwerkraft auf die ausbreitung des lichtes.  Annalen der Physik, 340(10):898–908, 1911.
  * [89] Albert Einstein.  Zur theorie des statischen gravitationsfeldes.  Annalen der Physik, 343(7):443–458, 1912.
  * [90] Albert Einstein, Boris Podolsky, and Nathan Rosen.  Can quantum-mechanical description of physical reality be considered complete?  Physical review, 47(10):777, 1935.
  * [91] Adam J Engler, Patrick O Humbert, Bernhard Wehrle-Haller, and Valerie M Weaver.  Multiscale modeling of form and function.  Science, 324(5924):208–212, 2009.
  * [92] Joshua M Epstein.  Agent-based computational models and generative social science.  Complexity, 4(5):41–60, 1999.
  * [93] Dominique F Escande.  Stochasticity in classical hamiltonian systems: universal aspects.  Physics Reports, 121(3-4):165–261, 1985.
  * [94] Peter Farindon.  The correspondence principle (1918-1923).  In J.R. Nielsen and L Rosenfeld, editors, Niels Bohr Collected Works, volume 3. North Holland, 1976.
  * [95] Alircza Fathi, Jessica K Hodgins, and James M Rehg.  Social interactions: A first-person perspective.  In Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, pages 1226–1233. IEEE, 2012.
  * [96] Martha S Feldman and Brian T Pentland.  Reconceptualizing organizational routines as a source of flexibility and change.  Administrative science quarterly, 48(1):94–118, 2003.
  * [97] Martha S Feldman, Brian T Pentland, Luciana D’Adderio, and Nathalie Lazaric.  Beyond routines as things: Introduction to the special issue on routine dynamics, 2016.
  * [98] GW Fernando, Guo-Xin Qian, M Weinert, and JW Davenport.  First-principles molecular dynamics for metals.  Physical Review B, 40(11):7985, 1989.
  * [99] RP Feynman and AR Hibbs.  Quantum mechanics and path integration, 1965.
  * [100] Charlotte Froese Fischer.  Hartree–fock method for atoms. a numerical approach.  1977\.
  * [101] Mark S Fox.  An organizational view of distributed systems.  In Readings in Distributed Artificial Intelligence, pages 140–150. Elsevier, 1988.
  * [102] TD Frank, PJ Beek, and R Friedrich.  Fokker-planck perspective on stochastic delay systems: Exact solutions and data analysis of biological systems.  Physical Review E, 68(2):021912, 2003.
  * [103] Robert J Friedrich.  In defense of multiplicative terms in multiple regression equations.  American Journal of Political Science, pages 797–833, 1982.
  * [104] Silvio Funtowicz and Jerome R Ravetz.  Emergent complex systems.  Futures, 26(6):568–582, 1994.
  * [105] Les Gasser, Ingemar Hulthage, Brian Leverich, Jon Lieb, and Ann Majchrzak.  Organizations as complex, dynamic design problems.  In Portuguese Conference on Artificial Intelligence, pages 1–12. Springer, 1993.
  * [106] Domenico Delli Gatti, Corrado Di Guilmi, Edoardo Gaffeo, Gianfranco Giulioni, Mauro Gallegati, and Antonio Palestrini.  A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility.  Journal of Economic behavior & organization, 56(4):489–512, 2005.
  * [107] Murray Gell-Mann and Seth Lloyd.  Information measures, effective complexity, and total information.  Complexity, 2(1):44–52, 1996.
  * [108] Carlos Gershenson and Francis Heylighen.  When can we call a system self-organizing?  In European Conference on Artificial Life, pages 606–614. Springer, 2003.
  * [109] Amin S Ghabrial and Mark A Krasnow.  Social interactions among epithelial cells during tracheal branching morphogenesis.  Nature, 441(7094):746, 2006.
  * [110] Hamada Ghenniwa and Mohamed Kamel.  Interaction devices for coordinating cooperative distributed systems.  Intelligent Automation & Soft Computing, 6(3):173–184, 2000.
  * [111] Walter R Gilks, Sylvia Richardson, and David Spiegelhalter.  Markov chain Monte Carlo in practice.  CRC press, 1995.
  * [112] Jean-Yves Girard.  Towards a geometry of interaction.  Contemporary Mathematics, 92(69-108):6, 1989.
  * [113] Edward Glaeser and José Scheinkman.  Measuring social interactions.  Social dynamics, pages 83–132, 2001.
  * [114] Nigel Goldenfeld.  Lectures on phase transitions and the renormalization group.  CRC Press, 2018.
  * [115] John R Graham and Keith S Decker.  Towards a distributed, environment-centered agent framework.  In International Workshop on Agent Theories, Architectures, and Languages, pages 290–304. Springer, 1999.
  * [116] Ivor Grattan-Guiness and Jerome R Ravetz.  Joseph fourier, 1768-1830: A survey of his life and work.  2003\.
  * [117] John Guckenheimer and Philip Holmes.  Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, volume 42.  Springer Science & Business Media, 2013.
  * [118] Stanley P Gudder.  Quantum probability.  Academic Press, 2014.
  * [119] Roger Guimera and Luis A Nunes Amaral.  Functional cartography of complex metabolic networks.  Nature, 433(7028):895–900, 2005.
  * [120] Vishwesha Guttal and Iain D Couzin.  Social interactions, information use, and the evolution of collective migration.  Proceedings of the national academy of sciences, 107(37):16172–16177, 2010.
  * [121] Brian L Hammond, William A Lester, and Peter James Reynolds.  Monte Carlo methods in ab initio quantum chemistry, volume 1.  World Scientific, 1994.
  * [122] Seung Kee Han, Tae Gyu Yim, DE Postnov, and OV Sosnovtseva.  Interacting coherence resonance oscillators.  Physical Review Letters, 83(9):1771, 1999.
  * [123] Sadri Hassani.  Dirac delta function.  In Mathematical methods, pages 139–170. Springer, 2009.
  * [124] Gavin Hesketh.  The Particle Zoo: The Search for the Fundamental Nature of Reality.  Hachette UK, 2016.
  * [125] Berk Hess.  Determining the shear viscosity of model liquids from molecular dynamics simulations.  The Journal of chemical physics, 116(1):209–217, 2002.
  * [126] Charles Antony Richard Hoare.  Communicating sequential processes.  Communications of the ACM, 21(8):666–677, 1978.
  * [127] Sergio A Hojman.  A new conservation law constructed without using either lagrangians or hamiltonians.  Journal of Physics A: Mathematical and General, 25(7):L291, 1992.
  * [128] John H Holland.  Complex adaptive systems.  Daedalus, pages 17–30, 1992.
  * [129] Crawford S Holling.  Understanding the complexity of economic, ecological, and social systems.  Ecosystems, 4(5):390–405, 2001.
  * [130] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki.  Quantum entanglement.  Reviews of modern physics, 81(2):865, 2009.
  * [131] Nazlin K Howell.  Protein-protein interactions.  In Biochemistry of food proteins, pages 35–74. Springer, 1992.
  * [132] Bernardo A Huberman, Daniel M Romero, and Fang Wu.  Social networks that matter: Twitter under the microscope.  arXiv preprint arXiv:0812.1045, 2008.
  * [133] Walter Hunziker and Israel Michael Sigal.  The quantum n-body problem.  Journal of Mathematical Physics, 41(6):3448–3510, 2000.
  * [134] Arthur S Iberall and Harry Soodak.  A physics for complex systems.  In Self-organizing systems, pages 499–520. Springer, 1987.
  * [135] Toru Ishida, Les Gasser, and Makoto Yokoo.  Organization self-design of distributed production systems.  IEEE Transactions on Knowledge and Data Engineering, 4(2):123–134, 1992.
  * [136] Claude Itzykson and Jean-Bernard Zuber.  Quantum field theory.  Courier Corporation, 2006.
  * [137] Max Jammer.  The conceptual development of quantum mechanics.  Tomash, 1989.
  * [138] ED Jurgenson, P Navrátil, and RJ Furnstahl.  Evolving nuclear many-body forces with the similarity renormalization group.  Physical Review C, 83(3):034301, 2011.
  * [139] David Kaiser.  Shut up and calculate!  Nature, 505(7482):153–155, 2014.
  * [140] Anatole Katok and Boris Hasselblatt.  Introduction to the modern theory of dynamical systems, volume 54.  Cambridge university press, 1997.
  * [141] Panayotis G Kevrekidis, Dimitri J Frantzeskakis, and Ricardo Carretero-González.  Emergent nonlinear phenomena in Bose-Einstein condensates: theory and experiment, volume 45.  Springer Science & Business Media, 2007.
  * [142] Hiroaki Kitano.  Computational systems biology.  Nature, 420(6912):206, 2002.
  * [143] James Ladyman, James Lambert, and Karoline Wiesner.  What is a complex system?  European Journal for Philosophy of Science, 3(1):33–67, 2013.
  * [144] Kiran Lakkaraju and Les Gasser.  Norm emergence in complex ambiguous situations.  In Proceedings of the AAAI workshop on coordination, organizations, institutions and norms AAAI, Chicago, 2008.
  * [145] Yueheng Lan and Garegin A Papoian.  The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade.  The Journal of chemical physics, 125(15):154901, 2006.
  * [146] Tom Lancaster and Stephen J Blundell.  Quantum field theory for the gifted amateur.  OUP Oxford, 2014.
  * [147] Peter M Lee.  Bayesian statistics: an introduction.  John Wiley & Sons, 2012.
  * [148] Gottfried Wilhelm Leibniz.  Specimen dynamicum.  In Philosophical Papers and Letters, pages 435–452. Springer, 1989.
  * [149] Raphael D Levine.  Molecular reaction dynamics.  Cambridge University Press, 2009.
  * [150] Michael Lewis and Candice Feiring.  Direct and indirect interactions in social relationships.  Advances in infancy research, 1981.
  * [151] Hong Li and Xianglin Yang.  The influence of stochastic perturbation on dark soliton-distributed amplification transmission system and its suppression.  Microwave and Optical Technology Letters, 17(1):58–62, 1998.
  * [152] Jinghai Li and Mooson Kwauk.  Multiscale nature of complex fluid- particle systems.  Industrial & Engineering Chemistry Research, 40(20):4227–4237, 2001.
  * [153] Jinghai Li, Jiayuan Zhang, Wei Ge, and Xinhua Liu.  Multi-scale methodology for complex systems.  Chemical Engineering Science, 59(8-9):1687–1700, 2004.
  * [154] Yung-Hua Li and Xiaolin Tian.  Quorum sensing and bacterial social interactions in biofilms.  Sensors, 12(3):2519–2538, 2012.
  * [155] Allan J Lichtenberg and Michael A Lieberman.  Regular and stochastic motion, volume 38.  Springer Science & Business Media, 2013.
  * [156] Jun S Liu.  Monte Carlo strategies in scientific computing.  Springer Science & Business Media, 2008.
  * [157] Alexander E Lobkovsky, Yuri I Wolf, and Eugene V Koonin.  Predictability of evolutionary trajectories in fitness landscapes.  PLoS computational biology, 7(12):e1002302, 2011.
  * [158] WERNER R Loewenstein.  Junctional intercellular communication: the cell-to-cell membrane channel.  Physiological Reviews, 61(4):829–913, 1981.
  * [159] Niklas Luhmann.  Social systems.  Stanford University Press, 1995.
  * [160] Niklas Luhmann.  System as difference.  Organization, 13(1):37–57, 2006.
  * [161] Werner K Lutz, Spyros Vamvakas, Annette Kopp-Schneider, Josef Schlatter, and Helga Stopper.  Deviation from additivity in mixture toxicity: relevance of nonlinear dose-response relationships and cell line differences in genotoxicity assays with combinations of chemical mutagens and gamma-radiation.  Environmental health perspectives, 110(Suppl 6):915, 2002.
  * [162] Ranald R Macdonald.  The incompleteness of probability models and the resultant implications for theories of statistical inference.  Understanding Statistics: Statistical Issues in Psychology, Education, and the Social Sciences, 1(3):167–189, 2002.
  * [163] Ernst Mach.  The science of mechanics: A critical and historical account of its development.  Open court publishing Company, 1907.
  * [164] Andrew Majda, Rafail V Abramov, and Marcus J Grote.  Information theory and stochastics for multiscale nonlinear systems, volume 25.  American Mathematical Soc., 2005.
  * [165] Charles F Manski.  Identification of treatment response with social interactions.  The Econometrics Journal, 16(1):S1–S23, 2013.
  * [166] Alexei A Maradudin, Elliott Waters Montroll, George Herbert Weiss, and IP Ipatova.  Theory of lattice dynamics in the harmonic approximation, volume 3.  Academic press New York, 1963.
  * [167] Francisco A Martín, Salvador C Herrera, and Ginés Morata.  Cell competition, growth and size control in the drosophila wing imaginal disc.  Development, 136(22):3747–3756, 2009.
  * [168] Elio Mattia and Sijbren Otto.  Supramolecular systems chemistry.  Nature nanotechnology, 10(2):111, 2015.
  * [169] James Clerk Maxwell and Joseph John Thompson.  A treatise on electricity and magnetism, volume 2.  Clarendon, 1904.
  * [170] Gary H McClelland and Charles M Judd.  Statistical difficulties of detecting interactions and moderator effects.  Psychological bulletin, 114(2):376, 1993.
  * [171] Madan Lal Mehta and Freeman J Dyson.  Statistical theory of the energy levels of complex systems. v.  Journal of Mathematical Physics, 4(5):713–719, 1963.
  * [172] Wendy Berry Mendes, Jim Blascovich, Brian Lickel, and Sarah Hunter.  Challenge and threat during social interactions with white and black men.  Personality and Social Psychology Bulletin, 28(7):939–952, 2002.
  * [173] Richard B Miller.  Leibniz on the interaction of bodies.  History of Philosophy Quarterly, 5(3):245–255, 1988.
  * [174] Robin Milner.  Elements of interaction: Turing award lecture.  Communications of the ACM, 36(1):78–89, 1993.
  * [175] Jack Minker.  Foundations of deductive databases and logic programming.  Morgan Kaufmann, 2014.
  * [176] Naftaly H Minsky and Victoria Ungureanu.  Law-governed interaction: a coordination and control mechanism for heterogeneous distributed systems.  ACM Transactions on Software Engineering and Methodology (TOSEM), 9(3):273–305, 2000.
  * [177] Robert A Moffitt et al.  Policy interventions, low-level equilibria, and social interactions.  Social dynamics, 4(45-82):6–17, 2001.
  * [178] Eugenio Moggi.  Notions of computation and monads.  Information and computation, 93(1):55–92, 1991.
  * [179] Ali H Nayfeh and Char-Ming Chin.  Nonlinear interactions in a parametrically excited system with widely spaced frequencies.  Nonlinear Dynamics, 7(2):195–216, 1995.
  * [180] Christopher Nemeth, Paul Fearnhead, and Lyudmila Mihaylova.  Sequential monte carlo methods for state and parameter estimation in abruptly changing environments.  arXiv preprint arXiv:1510.02604, 2015.
  * [181] Mark EJ Newman.  Power laws, pareto distributions and zipf’s law.  Contemporary physics, 46(5):323–351, 2005.
  * [182] Mark EJ Newman.  Complex systems: A survey.  arXiv preprint arXiv:1112.1440, 2011.
  * [183] Isaac Newton and Edmund Halley.  Philosophiae naturalis principia mathematica, volume 62.  Jussu Societatis Regiae ac typis Josephi Streater, prostant venales apud Sam. Smith, 1744.
  * [184] Gregoire Nicolis.  Physics of far-from-equilibrium systems and self-organisation.  The new physics, 11:316–347, 1989.
  * [185] Berend Olivier, Jan Mos, Jan Van der Heyden, and Jan Hartog.  Serotonergic modulation of social interactions in isolated male mice.  Psychopharmacology, 97(2):154–156, 1989.
  * [186] Yoshitsugu Oono.  Large deviation and statistical physics.  Progress of Theoretical Physics Supplement, 99:165–205, 1989.
  * [187] Yoshitsugu Oono.  The nonlinear world: conceptual analysis and phenomenology.  Springer Science & Business Media, 2012.
  * [188] Olivier Oullier, Gonzalo C De Guzman, Kelly J Jantzen, Julien Lagarde, and JA Scott Kelso.  Social coordination dynamics: Measuring human bonding.  Social neuroscience, 3(2):178–192, 2008.
  * [189] David Lorge Parnas, Paul C Clements, and David M Weiss.  The modular structure of complex systems.  IEEE Transactions on software Engineering, (3):259–266, 1985.
  * [190] Robert G Parr.  Density functional theory.  Annual Review of Physical Chemistry, 34(1):631–656, 1983.
  * [191] Howard Hunt Pattee.  Evolving self-reference: matter, symbols, and semantic closure.  In Laws, language and life, pages 211–226. Springer, 2012.
  * [192] R Penco and D Mauro.  Perturbation theory via feynman diagrams in classical mechanics.  European journal of physics, 27(5):1241, 2006.
  * [193] Michael E Peskin.  An introduction to quantum field theory.  CRC Press, 2018.
  * [194] James C Phillips, Rosemary Braun, Wei Wang, James Gumbart, Emad Tajkhorshid, Elizabeth Villa, Christophe Chipot, Robert D Skeel, Laxmikant Kale, and Klaus Schulten.  Scalable molecular dynamics with namd.  Journal of computational chemistry, 26(16):1781–1802, 2005.
  * [195] David M Post and Eric P Palkovacs.  Eco-evolutionary feedbacks in community and ecosystem ecology: interactions between the ecological theatre and the evolutionary play.  Philosophical Transactions of the Royal Society of London B: Biological Sciences, 364(1523):1629–1640, 2009.
  * [196] Tim Poston and Ian Stewart.  Catastrophe theory and its applications.  Courier Corporation, 2014.
  * [197] Nat Pryce and Steve Crane.  Component interaction in distributed systems.  In Configurable Distributed Systems, 1998. Proceedings. Fourth International Conference on, pages 71–78. IEEE, 1998.
  * [198] Zhilin Qu, Alan Garfinkel, James N Weiss, and Melissa Nivala.  Multi-scale modeling in biology: how to bridge the gaps between scales?  Progress in biophysics and molecular biology, 107(1):21–31, 2011.
  * [199] David C Queller.  Kin selection and frequency dependence: a game theoretic approach.  Biological Journal of the Linnean Society, 23(2-3):133–143, 1984.
  * [200] Jean-Michel Raimond, M Brune, and Serge Haroche.  Manipulating quantum entanglement with atoms and photons in a cavity.  Reviews of Modern Physics, 73(3):565, 2001.
  * [201] Ragunathan Rajkumar, Insup Lee, Lui Sha, and John Stankovic.  Cyber-physical systems: the next computing revolution.  In Design Automation Conference (DAC), 2010 47th ACM/IEEE, pages 731–736. IEEE, 2010.
  * [202] Anand Ranganathan and Roy H Campbell.  What is the complexity of a distributed computing system?  Complexity, 12(6):37–45, 2007.
  * [203] Dennis C Rapaport and Dennis C Rapaport Rapaport.  The art of molecular dynamics simulation.  Cambridge university press, 2004.
  * [204] Werner Raub and Jeroen Weesie.  Reputation and efficiency in social interactions: An example of network effects.  American Journal of Sociology, 96(3):626–654, 1990.
  * [205] Michel Raynal and Mukesh Singhal.  Logical time: Capturing causality in distributed systems.  Computer, 29(2):49–56, 1996.
  * [206] Olivier Rieppel.  The language of systematics, and the philosophy of ’total evidence’.  Systematics and Biodiversity, 2(1):9–19, 2004.
  * [207] Diego Rybski, Sergey V Buldyrev, Shlomo Havlin, Fredrik Liljeros, and Hernán A Makse.  Scaling laws of human interaction activity.  Proceedings of the National Academy of Sciences, 106(31):12640–12645, 2009.
  * [208] Joanna Sadlej.  Semi-empirical methods of quantum chemistry.  Halsted Press, 1985.
  * [209] Marta Sales-Pardo, Roger Guimera, André A Moreira, and Luís A Nunes Amaral.  Extracting the hierarchical organization of complex systems.  Proceedings of the National Academy of Sciences, 104(39):15224–15229, 2007.
  * [210] Clément Sanchez, Hervé Arribart, and Marie Madeleine Giraud Guille.  Biomimetism and bioinspiration as tools for the design of innovative materials and systems.  Nature materials, 4(4):277, 2005.
  * [211] R Keith Sawyer.  Social emergence: Societies as complex systems.  Cambridge University Press, 2005.
  * [212] Jose A Scheinkman.  Social interactions.  The new palgrave dictionary of economics, 2, 2008.
  * [213] Richard Schmidt and Sergej Moroz.  Renormalization-group study of the four-body problem.  Physical Review A, 81(5):052709, 2010.
  * [214] Erwin Schrödinger.  What is Life?: The Physical Aspect Og the Living Cell and Mind and Matter.  Cambridge University Press, 1974.
  * [215] Alwyn Scott.  Nonlinear science: emergence and dynamics of coherent structures.  Oxford Univ. Press, 2003.
  * [216] Andrew JE Seely and Peter Macklem.  Fractal variability: an emergent property of complex dissipative systems.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(1):013108, 2012.
  * [217] Andrew JE Seely, Kimberley D Newman, and Christophe L Herry.  Fractal structure and entropy production within the central nervous system.  Entropy, 16(8):4497–4520, 2014.
  * [218] Eran Segal, Haidong Wang, and Daphne Koller.  Discovering molecular pathways from protein interaction and gene expression data.  Bioinformatics, 19(suppl_1):i264–i272, 2003.
  * [219] Cosma Rohilla Shalizi.  Methods and techniques of complex systems science: An overview.  In Complex systems science in biomedicine, pages 33–114. Springer, 2006.
  * [220] Tatsuo Shibata and Koichi Fujimoto.  Noisy signal amplification in ultrasensitive signal transduction.  Proceedings of the National Academy of Sciences of the United States of America, 102(2):331–336, 2005.
  * [221] DD Šiljak.  On decentralized control of large scale systems.  IFAC Proceedings Volumes, 11(1):1849–1856, 1978.
  * [222] Dragoslav D Siljak.  Decentralized control of complex systems.  Courier Corporation, 2011.
  * [223] Herbert A Simon.  The organization of complex systems.  In Models of discovery, pages 245–261. Springer, 1977.
  * [224] Herbert A Simon.  The architecture of complexity.  In Facets of systems science, pages 457–476. Springer, 1991.
  * [225] Lee Smolin.  Cosmology as a problem in critical phenomena.  In Complex systems and binary networks, pages 184–223. Springer, 1995.
  * [226] Tom AB Snijders.  Markov chain monte carlo estimation of exponential random graph models.  Journal of Social Structure, 3(2):1–40, 2002.
  * [227] Harry Soodak and Arthur Iberall.  Homeokinetics: A physical science for complex systems.  Science, 201(4356):579–582, 1978.
  * [228] Didier Sornette.  Why stock markets crash: critical events in complex financial systems.  Princeton University Press, 2017.
  * [229] Kenneth E Southwood.  Substantive theory and statistical interaction: Five models.  American Journal of Sociology, 83(5):1154–1203, 1978.
  * [230] Jaroslav Stark.  Observing complexity, seeing simplicity.  Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 358(1765):41–61, 2000.
  * [231] Carolyn Talcott.  Interaction semantics for components of distributed systems.  In Formal Methods for Open Object-based Distributed Systems, pages 154–169. Springer, 1997.
  * [232] Mark Thomson.  Modern particle physics.  Cambridge University Press, 2013.
  * [233] Hugo Touchette.  The large deviation approach to statistical mechanics.  Physics Reports, 478(1-3):1–69, 2009.
  * [234] Elliot Turiel.  Social decisions, social interactions, and the coordination of diverse judgments.  Social life and social knowledge: Toward a process account of development, pages 255–276, 2008.
  * [235] Horn S Tzou and Lawrence A Bergman.  Dynamics and control of distributed systems.  Cambridge University Press, 2007.
  * [236] Fred A Van Eeuwijk.  Multiplicative interaction in generalized linear models.  Biometrics, pages 1017–1032, 1995.
  * [237] Rob Van Glabbeek, Ursula Goltz, and Jens-Wolfhard Schicke.  On synchronous and asynchronous interaction in distributed systems.  In International Symposium on Mathematical Foundations of Computer Science, pages 16–35. Springer, 2008.
  * [238] Alessandro Vespignani.  Predicting the behavior of techno-social systems.  Science, 325(5939):425–428, 2009.
  * [239] Paul MB Vitanyi and Ming Li.  An introduction to Kolmogorov complexity and its applications, volume 34.  Springer Heidelberg, 1997.
  * [240] EI Volkov, E Ullner, AA Zaikin, and J Kurths.  Oscillatory amplification of stochastic resonance in excitable systems.  Physical Review E, 68(2):026214, 2003.
  * [241] Mark Wallace.  Constraint logic programming.  In Computational logic: Logic programming and beyond, pages 512–532. Springer, 2002.
  * [242] Joseph Walpole, Jason A Papin, and Shayn M Peirce.  Multiscale computational models of complex biological systems.  Annual review of biomedical engineering, 15:137–154, 2013.
  * [243] Xuefeng Wang, Robert C Elston, and Xiaofeng Zhu.  Statistical interaction in human genetics: how should we model it if we are looking for biological interaction?  Nature Reviews Genetics, 12(1):74, 2011.
  * [244] Edward P Weber, Denise Lach, and Brent S Steel.  New Strategies for Wicked Problems: Science and Solutions in the 21st Century.  Oregon State University Press, 2017.
  * [245] Gian-Carlo Wick.  Properties of bethe-salpeter wave functions.  Physical Review, 96(4):1124, 1954.
  * [246] Thomas Wieland.  Combinatorics of combinatorial chemistry.  Journal of Mathematical Chemistry, 21(2):141–157, 1997.
  * [247] Peter Willetts.  Transnational actors and international organizations in global politics.  The globalization of world politics, 2, 2001.
  * [248] Terry Winograd.  Categories, disciplines, and social coordination.  Computer Supported Cooperative Work (CSCW), 2(3):191–197, 1993.
  * [249] Carl R Woese.  On the evolution of cells.  Proceedings of the National Academy of Sciences, 99(13):8742–8747, 2002.
  * [250] RG Woolley and BT Sutcliffe.  Molecular structure and the born—oppenheimer approximation.  Chemical Physics Letters, 45(2):393–398, 1977.
  * [251] William Alfred Wooster.  A text-book on crystal physics.  Cambridge University Press, 2016.
  * [252] Zhen Yang, Xiaoning Yang, and Zhijun Xu.  Molecular dynamics simulation of the melting behavior of pt- au nanoparticles with core- shell structure.  The Journal of Physical Chemistry C, 112(13):4937–4947, 2008.
  * [253] Bernard P Zeigler, Herbert Praehofer, and Tag Gon Kim.  Theory of modeling and simulation: integrating discrete event and continuous complex dynamic systems.  Academic press, 2000.
  * [254] Steve Ziliak and Deirdre Nansen McCloskey.  The cult of statistical significance: How the standard error costs us jobs, justice, and lives.  University of Michigan Press, 2008.
  * [255] Wojciech H Zurek.  The environment, decoherence, and the transition from quantum to classical.  In Quantum Gravity And Cosmology-Proceedings Of The Xxii Gift International Seminar On Theoretical Physics, page 117. World Scientific, 1992.