General Theory of Topological Explanations and Explanatory Asymmetry

Abstract

In this paper, I present a general theory of topological explanations, and illustrate its fruitfulness by showing how it accounts for explanatory asymmetry. My argument is developed in three steps. In the first step, I show what it is for some topological property A to explain some physical or dynamical property B . Based on that, I derive three key criteria of successful topological explanations: a criterion concerning the facticity of topological explanations, i.e. what makes it true of a particular system; a criterion for describing counterfactual dependencies in two explanatory modes, i.e. the vertical and the horizontal and, finally, a third perspectival one that tells us when to use the vertical and when to use the horizontal mode. In the second step, I show how this general theory of topological explanations accounts for explanatory asymmetry in both the vertical and horizontal explanatory modes. Finally, in the third step, I argue that this theory is universally applicable across biological sciences, which helps in unifying essential concepts of biological networks.
This article is part of the theme issue ‘Unifying the essential concepts of biological networks: biological insights and philosophical foundations'.

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royalsocietypublishing.org/journal/rstb
Research
Cite this article: Kosti
c D. 2020 General
theory of topological explanations and
explanatory asymmetry. Phil. Trans. R. Soc. B
375: 20190321.
http://dx.doi.org/10.1098/rstb.2019.0321
Accepted: 23 September 2019
One contribution of 11 to a theme issue
‘Unifying the essential concepts of biological
networks: biological insights and philosophical
foundations’.
Subject Areas:
neuroscience, behaviour
Keywords:
philosophical theory of explanation,
explanatory asymmetries, explanatory
perspectivism, counterfactual dependencies,
facticity of explanation, explanatory unification
Author for correspondence:
Daniel Kosti
c
e-mail: daniel.kostic@gmail.com
General theory of topological explanations
and explanatory asymmetry
Daniel Kosti
c
University Bordeaux Montaigne, Department of Philosophy and EA 4574 ‘Sciences, Philosophie, Humanités’
(SPH) at University of Bordeaux, Allée Geoffroy Saint-Hilaire, Bâtiment B2, 33615 Pessac cedex, France
DK, 0000-0001-5729-1476
In this paper, I present a general theory of topological explanations, and
illustrate its fruitfulness by showing how it accounts for explanatory asym-
metry. My argument is developed in three steps. In the first step, I show
what it is for some topological property Ato explain some physical or dyna-
mical property B. Based on that, I derive three key criteria of successful
topological explanations: a criterion concerning the facticity of topological
explanations, i.e. what makes it true of a particular system; a criterion for
describing counterfactual dependencies in two explanatory modes, i.e. the
vertical and the horizontal and, finally, a third perspectival one that tells
us when to use the vertical and when to use the horizontal mode. In the
second step, I show how this general theory of topological explanations
accounts for explanatory asymmetry in both the vertical and horizontal
explanatory modes. Finally, in the third step, I argue that this theory is uni-
versally applicable across biological sciences, which helps in unifying
essential concepts of biological networks.
This article is part of the theme issue ‘Unifying the essential concepts of
biological networks: biological insights and philosophical foundations’.
1. Introduction
Even though network-based explanations (hereafter, topological explanations)
have been used in sciences for several decades, we do not yet have a theory of
topological explanations that can delimit good from bad topological explanations
in a principled way. The most general idea of topological explanation is that it
describes how the mathematical properties of connectivity patterns in complex
networks determine the dynamics of the systems exhibiting those patterns [1].
A network is a collection of nodes and edges that are connected in certain
ways; and a graph is a mathematical description of such a network ([2], p. 683).
Topological properties are mathematical properties obtained by quantifying net-
works by using graph theory and similar approaches. Since many biological
systems can be modelled as networks, i.e. they have many interconnected
elements that can be considered as nodes and edges, this approach clearly has
enormous explanatory potential. This fact is even more significant given the
sheer microphysical diversity of biological systems, e.g. the same topological
explanatory pattern can be used to explain the robustness of the brain, acomputer
network, an ecological community, a protein interaction network and so on.
One of the major demands on such a theory of topological explanations
is to account for what it is for a certain pattern to be explanatory as a matter of
a principle (why the explanation succeeds, or why it does not succeed), as
well as to account for explanatory asymmetries, i.e. in a good explanation, if A
explains Bthen Bshould not explain A; otherwise, the explanation is circular or
too permissive.
In this paper, I present a general theory of topological explanations and
illustrate its fruitfulness by showing how it accounts for explanatory asymmetries.
My argument is developed in the three steps. In the first step, I show what it is for
some topological property Ato explain some physical or dynamical property B
(§2). Based on that, I derive three key criteria of successful topological
© 2020 The Author(s) Published by the Royal Society. All rights reserved.

explanation: a criterion concerning the facticity of topological
explanations, i.e. what makes it true of a particular system; a
criterion for two explanatory modes, i.e. describing what I
dub ‘vertical’and ‘horizontal’counterfactual dependencies
and, finally, a third one that tells us when to use the vertical
and when to use the horizontal mode. In the second step, I
show how this theory of topological explanations accounts
for explanatory asymmetry in several different ways (§§3 and
4). Finally, in the third step, I argue that this theory applies
across all biological networks (§5).
2. A general theory of topological explanations
A general theory of topological explanations should tell us
what it is for a certain network pattern to be explanatory for
any particular system or a phenomenon that we want to
explain using that pattern. To that end, I characterize topologi-
cal explanation in the most general sense, and precisely define
its structure so as to capture different explanatory modes, i.e.
the vertical and the horizontal ones.
Let us start with the most general definition of kind
of things a topological explanation is: a topological explana-
tion supports counterfactuals that describe a counterfactual
1
dependency between a system’s topological properties and
its network dynamics.
There are two different ways in which topological expla-
nations may describe counterfactual dependency relations,
i.e. which I call the ‘vertical’and the ‘horizontal’.By‘vertical’,
I mean an explanation in which a global topological property
determines certain general properties of the real-world
system. On the other hand, by ‘horizontal’I mean an expla-
nation in which a local topological property determines
certain local dynamical properties of the real-world system.
Only whole networks possess global properties; only parts of
networks possess local properties. Examples of the global–
local distinction include the distinctions between within-scale
(horizontal) and between-scales (vertical) and intra-modular
(horizontal) and hierarchical–modular (vertical) and so on.
To that effect, I propose a definition of topological
explanation that is sensitive to both vertical and horizontal
approaches:
Atopologically explains Bif and only if:
1. (Facticity): Aand Bare approximately true; and
2. Either
(a) (Vertical mode): Adescribes a global topology of the net-
work, Bdescribes some general physical property, and
had Ahad not obtained, then Bwould not have
obtained either; or
(b) (Horizontal mode): Adescribes a set of local topological
properties,Bdescribes a set of local physical properties,
and had the values of Abeen different, then the values
of Bwould have been different.
3. (Explanatory perspectivism): Ais an answer to the relevant
explanation-seeking question Q about B, such that
the Q determines whether to use the vertical or horizontal
explanatory mode.
The facticity requirement is a general condition according
to which the explanans
2
and explanandum
3
have to be approxi-
mately true, and it works the same in both vertical and
horizontal explanatory modes.
4
In short, correct topological
explanations cannot rely on gross misrepresentations of a
system’s topological properties or physical properties. This
point can also be framed in terms of conditions of applica-
tion, which are elaborated in great detail in another
contribution [9] in this theme issue.
Turning to the second condition, one can illustrate the
difference between vertical and horizontal modes of topologi-
cal explanation by looking at how they apply to closely
related explananda. The brain is able to process information
extremely efficiently by engaging in a variety of very complex
behaviours, e.g. it performs various cognitive tasks and can
go through many different states. To understand some of
its dynamics, i.e. how it is possible for it to go through differ-
ent states (for example, from a resting state, to a state of
solving a logical puzzle, or from a state of solving a logical
puzzle to a state of retrieving information from memory)
we want to understand how it achieves cognitive control,
i.e. how the brain as a dynamical system achieves efficient
transition from one of its internal states to another. This can
be done by looking into what keeps the energy cost of such
transitions low.
When explainingcognitive control in either mode, the brain
is represented as a network of brain regions connected through
white matter tracts. In the vertical mode, the explanans is
the global topological property such as small-worldliness
and the explanandum is the global physical property such as
global cognitive control (if global cognitive control is theoreti-
cally possible). The most basic way to illustrate small-
worldliness as a global topological property is through the
Watts & Strogatz [10] model. Among the most established
ways to quantify networks are the average path length L(p)
and clustering coefficient C(p). The L(p) measures the average
number of edges that have to be traversed in order to reach one
node from the other. Clustering is understood as a tendency of
a small group of nodes to form connected triangles around one
central node, which indicates that the connected neighbouring
nodes are also neighbours among themselves; hence they form
a cluster or a clique. The clustering coefficient is a measure of
this tendency, which characterizes a value for all nodes in a net-
work ([11], p. 312; [2], p. 683). Networks that have high
clustering coefficients and low path lengths are called small-
world networks [10]. Small-worldliness as a global topological
property indicates that almost any two nodes in the network
will be connected either directly or through a minimal
number of indirect connections, which shortens the distance
between the nodes within a neighbourhood of nodes as well
as between neighbourhoods of nodes, and neighbourhoods
of neighbourhoods, which further ensures that the energy
requirements for changing any of the trajectories will be mini-
mal, and thus explains why the network is globally or in
principle controllable.
If the explanation-seeking question is: why is the brain con-
trollable, then the explanation will have to begiven in the vertical
mode. The relevant ‘vertical counterfactual’then is
Had the brain not been a small-world network it wouldn’thave
been controllable.
On the other hand, in the horizontal approach, Adescribes a set
of local topological properties,Bdescribes a set of local phys-
ical properties, and had the values of Abeen different, then
the values of Bwould have been different. For example,
when applied to a particular case of cognitive controllability,
the explanation-seeking question is: How and why is the brain
able to efficiently transition from one state to the other? The
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2

answer proposed in a series of studies [12–14] is that local topo-
logical properties determine energy requirements for those
movements ([13], p. 1). Specifically, in recent literature in cogni-
tive neuroscience [12–15], the problem of cognitive control is
being treated through the notion of network control.
It should be emphasized from the outset that these two
notions of control are very different. The notion of cognitive
control refers to a brain’s dynamics, which can be understood
as a neural regional activity that can be elicited by neurofeed-
back in fMRI imaging, or by non-invasive brain stimulation
such as TMS (trans-cranial-magnetic stimulation). In this
sense, to control means: how to perturb the system in order
to reach a desired state. By contrast, the notion of network con-
trol is a purely mathematical notion in network control theory
which refers to topological constraints on such perturbations.
5
The dependency that the explanation describes in this case is
between the topological properties and cognitive control—
the idea that the topological properties exert network control
over cognitive control understood as an aspect of the brain’s
dynamics, e.g. certain connectivity patterns enable more
cognitive control, while others enable less cognitive control.
As we shall shortly see, this difference will have to do
with the specific values of local topological properties, e.g.
the higher the network communicability
6
value the lower the
energy requirements for changing the particular trajectories.
In one of the programmatic papers on this approach in
cognitive neuroscience, Gu and colleagues [12] make this
distinction explicit:
Importantly, this notion of control is based on a very detailed
mathematical construct and is therefore necessarily quite distinct
from the cognitive neuroscientist’s common notion of ‘cognitive
control’and the distributed sets of brain regions implicated in
its performance. To minimize obfuscation, we henceforth refer
to these two notions as ‘network control’and ‘cognitive control’,
respectively. ([12], p. 8)
The distinction between cognitive control and the network
control is of great importance here because it helps to clearly
delimit the explanans from explanandum in the horizontal
mode of topological explanation, i.e. we want to explain
how the cognitive control is achieved by using the tools of net-
work control theory to describe counterfactual dependencies
between topological properties and the network dynamics.
The horizontal counterfactual would then be: ‘Had the
network’s communicability been higher, then the energy
required to change a trajectory in a state space would have
been lower.’When we plug this example into the horizontal
explanatory mode condition, we can ascertain that the com-
municability is a local topological property, energy
requirements are local dynamical properties, and the energy
requirements counterfactually depend on the communicability
measure.
This pattern of counterfactual dependency is explanatory
precisely because it tells us how hypothetical changes in the
values of topological properties would affect the system
dynamics.
This example also highlights key differences between the
vertical and horizontal modes of topological explanation. In
the horizontal mode, the facticity requirement is satisfied by
describing the system dynamics as a state space in which tran-
sitions from one state to the next are described as trajectories
in a state space. Understanding why some trajectories are con-
trollable has to do with understanding the counterfactual
dependency between local topological measures of the
network and the particular trajectories in the state space, i.e.
the explanation tells us in what ways exactly the trajectories
in the state space will be affected by the changes of relevant
topological properties. As opposed to cases of vertical counter-
factual dependencies, horizontal counterfactual dependencies
hold between variables that are at the same local level in the
network. Thus, in the horizontal approach, the relationship
between the explanans and explanandum is more direct: just by
describing the topological properties of the trajectories in the
state space we are almost immediately able to understand the
relevant counterfactual dependency relations, without needing
to appeal to any kind of inferential patterns between different
levels. This indicates that in the horizontal approach the topo-
logical explanation has what Kosti
c [1] calls minimal structure,
where the relationship between the explanans and explanandum
is more direct.
The final criterion tells us when to use the vertical and
when to use the horizontal mode. The idea, foreshadowed by
Achinstein [16] and van Fraassen [17] in more general discus-
sions of explanation, is what Kosti
c [18] calls explanatory
perspectivism: explanation-seeking questions determine the
mode (horizontal or vertical) of topological explanations. To
appreciate this point, consider the idea recently proposed by
Hilgetag & Goulas [19]. They question the very notion of
small-world topology as the crucial aspect for understanding
the efficiency of the brain organization in signal processing.
Given the multitudes of ways in which the small-worldiness
can be realized in very diverse arrangements of topological
properties, they argue that, if each of the arrangements of
topological properties that can be used to describe the small-
world topology describes a different pattern of dependencies
between topological structure and the dynamical features in
the brain, then the global notion of small-world topology
does not seem very informative. The truly explanatory depen-
dencies should be the ones that tell us various hypothetical
ways in which the changes in the local values of topological
variables would affect the properties and behaviours of dyna-
mical elements such as trajectories in the state space. However,
this does not imply that the vertical approach is explanatorily
superfluous. Rather, it suggests that explanation-seeking ques-
tions dictate whether to use the vertical or horizontal approach.
For example, if we want to explain some very general property
of the system or of a phenomenon, e.g. if a system is globally
controllable at all or why it is stable at its most global level,
we will use the vertical approach, because it gives us a very
general answer. However, if we want to know why the
system is controllable in particular ways, we will use the hori-
zontal approach. As illustrated above, it is possible to use both
the vertical and horizontal explanatory modes to explain the
same phenomenon, depending on what we want to know
about the phenomenon.
The theory of topological explanations developed in this
section thus provides three criteria: a criterion of what makes
the explanation true of a particular system; a clear distinction
between two different modes of topological explanations, the
vertical and the horizontal modes; and a criterion for when
to use one or the other modes of topological explanations.
3. Background of the asymmetry problem
In this section, I provide some background of the asymmetry
problem. This sets the stage for how the general theory of
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3

topological explanations developed above accounts for the
asymmetry problem, and how it also helps to unify essential
concepts of biological networks in the remainder of the paper.
Beginning in the second part of the twentieth century, one
of the central topics in the philosophy of science was the
scientific explanation. Several very sophisticated philosophi-
cal accounts of scientific explanation have emerged, each of
which proposed a set of universal epistemic norms to
govern any successful scientific explanation.
The first and most influential account was the deductive-
nomological model (DN model hereafter) developed by
Hempel & Oppenheim [20] and Hempel [21]. They argued
that the explanation has an argument structure, in which the
explanandum is the conclusion in the logical argument that is
derived from a set of premises that constitute the explanans.
The set of premises in the explanans are constituted, respect-
ively, by the statements describing the antecedent conditions
and the statements describing the general laws of nature, and
the only constraining conditions are that the statements in
the explanans must be empirically true and that the deductive
argument must be valid. Famously, Hempel & Oppenheim
[20] represented the model as in figure 1.
However, Bromberger [22] and Salmon [23,24] pointed out
several very important shortcomings of this model of expla-
nation, viz. its failure to account for explanatory relevance,
7
its permissive definition of laws of nature,
8
and its failure to
account for explanatory asymmetries. Even though the pro-
blems of explanatory relevance and permissive definition of
laws of nature are unique to the DN model, the problem of
explanatory asymmetries puts any theory of explanation to
the test. To illustrate the importance of explanatory asymme-
tries, consider one of the most famous objections to the DN
model, i.e. the flagpole example, which is adapted from Brom-
berger [22]. Suppose we want to explain why the shadow of a
flagpole is of certain length. According to the DN model, the
explanans will include the antecedent conditions such as the
height of the flagpole and the sun’s elevation at a particular
time, and together with the laws of optics, we would be able
to explain why the flagpole shadow has a certain length, as is
shown in figure 2.
But we can reverse this calculation and deduce the length
of the flagpole or the elevation of the Sun based on the
length of the shadow. The resulting ‘explanation’would not
strike us as particularly good, even though the argument is
formally equivalent to the original one and the premises are
empirically true. The trouble is, we might say, that the
height of the flagpole and the position of the Sun do not
depend on the length of the shadow. This has become
known as the asymmetry problem in theories of explanation.
A good theory of explanation should account for explanatory
asymmetry.
9
One obvious way to solve the asymmetry problem is to
switch focus from inferential patterns to patterns of causation,
or at the very least this is the lesson that many philosophers
took from the flagpole problem. The general idea is that
causes explain their effects and not the other way around,
hence the causal patterns provide directionality to expla-
nations, and thereby prevent explanatory symmetries. In the
above flagpole example, this would imply that the flagpole’s
height causes the shadow’s length, but not vice versa. This
kind of asymmetry will have to be preserved across all
the counterfactuals related to that explanation. This points
toward another lesson one could take as well: it is the role of
counterfactual dependence in explanations. If we take these
lessons, a clearer path emerges when it comes to the solution
of the asymmetry problem in topological explanations,
namely, the counterfactual dependence is a structural feature
of explanations that is available in both causal and non-
causal explanations. Given that many philosophers argue
that topological explanations are non-causal, because their
explanans does not cite any causes, but rather mathematical
properties of network topology [1,26–32], then appealing to
causation or causal facts is not an option for solving the asym-
metry problem in topological explanations. A better way to
approach the problem is to inquire whether the counterfactual
dependence in topological explanations can account for
explanatory asymmetry. In his recent paper, Lange [33]
argues that in several available counterfactual accounts of
non-causal explanations, including the one espoused by
Jansson [25], the counterfactual dependence alone is not
sufficient to account for explanatory asymmetry. Jansson’s con-
tribution to this theme issue [9], takes up Lange’s challenge
directly, by focusing on the conditions of application in non-
causal explanations. In the next section, two more possible
bases of explanatory asymmetry in topological explanations
are discussed; however, responding directly to Lange’s [33]
challenge is beyond the scope of this paper.
logical deduction
statements of antecedent
conditions
general laws
explanans
C1,C2,…,Ck
L1,L2,…,Lr
E
explanandum
description of the
empirical phenomenon
to be explained
Figure 1. Hempel and Oppenheim ([20], p. 138).
10 m
ground
x
q
Figure 2. The flagpole example. (Online version in colour.)
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4

4. The bases of topological explanatory
asymmetries
As we have seen, one of the most influential ways to solve the
asymmetry problem for the DN model of explanation was to
switch focus from inferential patterns to patterns of causa-
tion. The idea is that causes explain their effects, and not
the other way around; and thus, in causal explanations
the explanatory asymmetry is rooted in the directionality of
causation. But it is not initially obvious what could be a
basis of explanatory asymmetry in topological explanations,
for it is unclear if they appeal to causation.
In this section, I provide several possible strategies
to account for explanatory asymmetries in topological explana-
tions, i.e. in terms of property, counterfactual and perspectival
asymmetries, instead of asymmetry of causation. All three of
these bases of topological explanatory asymmetries are tightly
interconnected and rooted in the three conditions of the general
theory of topological explanations.
One could conceive of property asymmetry in the follow-
ing way. Suppose Aexplains Btopologically. Then Ais a
topological property and Bis a physical property. Conse-
quently, the symmetry problem only arises if Btopologically
explains A. However, if Btopologically explains A, then B
must be a topological property as well. Some physical proper-
ties are topological properties thus whenever Bis not a
topological property the asymmetry holds. Recall from the
example above about topological explanation of cognitive con-
trollability. Cognitive controllability is not a topological
property; therefore, the topological explanations of it exhibit
property asymmetry.
The counterfactual asymmetry can be best understood in
the following terms. Suppose Aexplains Btopologically.
Then Bcounterfactually depends on A. If there is an asymme-
try problem, then Aalso counterfactually depends on B.
But there will be cases in which Bcounterfactually depends
on Abut not vice versa. And in these cases, there will be no
asymmetry problem.
The perspectival asymmetry has to do with the fact that
although Ais a relevant answer to an explanation-seeking
question about B, sometimes Bwill not be an answer to the
relevant explanation-seeking question Qabout A, meaning
that in the vertical mode, the fact that the system is stable
will not be an answer to the question why the system is a
small-world network. Equally, in the horizontal mode, the
fact that the cognitive control has certain energy requirements
does not answer why the local communicability measure has
a certain value. Thus, reversing the direction of explanation
would deem it non-explanatory, and the symmetry problem
will be blocked.
Moreover, in the horizontal mode, explanatory perspecti-
vism tends to figure prominently, as many ‘local asymmetries’
figure prominently. Very roughly, explanation-seeking ques-
tions are relevant, in part, because they serve scientists’
interests. Often, the properties that interest scientists have asym-
metries that are highly specific to the phenomena they are
studying. As an illustration, consider cases in which various
local features of brain networks determine certain behaviours
or processes, e.g. asymmetry of information flow [34], or asym-
metry of topological organization of the auditory cortex [35].
The asymmetry of information flow can be thought of in
terms of the efficiency of network communication. Efficiency
in this context is measured by shortest path lengths (the fewer
edges that have to be traversed between two nodes the more
efficient the communication between them), or in terms of
strongest and more reliable connections between the nodes.
For example, in complex systems such as the brain, communi-
cation between the brain region iand the brain region jis
asymmetric if it can be achieved more efficiently from ito j,
than the other way around. The explanation-seeking question
here is: Why are some propagation strategies of signalling
pathways in communication networks better than others? An
answer to the question appeals to the graph-theoretical effi-
ciency of communication between the nodes in a network
([34], p. 2).
Such an asymmetry is determined by using several net-
work communication measures, e.g. navigation efficiency,
diffusion efficiency and search information [34]. If region i
(the source region) and region j(the target region) are not
directly connected, the information flow from ito jmust
use one or more intermediate nodes. This implies that there
could be different signalling pathways in the network, in
virtue of which the communication efficiency may differ for
each set of pathways. And the communication efficiency of
a particular pathway may depend on the direction of the
information flow. Seguin et al. conclude that these results
‘may be primarily driven by specific properties of brain net-
works, rather than by aspects particular to one network
communication measure’([34], p. 10). Furthermore, they
also suggest that ‘complex organisational properties of ner-
vous systems are necessary to shape the directionality of
neural signaling’([34], p. 10). This approach is particularly
useful when applied to undirected topology of brain
networks and in predicting large-scale neural signalling.
The direction of an explanation in this case follows the
efficiency of network communication, which is best captured
through the property of ‘send–receive communication
asymmetry’([34], p. 1). Given that the explanation-seeking
question was: Why are some propagation strategies of signalling
pathways in communication networks better than the other?, The
explanation is: because it is more efficient to send and receive
signal from the region ito region jthan the other way around,
where the efficiency of direct and indirect send–receive com-
munication is measured by the network navigation efficiency,
diffusion efficiency and search information ([34], p. 2). In
other words, the direction of explanation is the direction of
send–receive efficiency—which only has one direction, since
by the network communication efficiency definition, a node
cannot simultaneously be both a sender and a receiver.
So, in this case, scientists are interested in efficient infor-
mation flows. However, this is very much indexed to the
specific question being asked, and hence is a ‘local asymmetry’
that is captured by the perspectival criterion of topological
explanations.
It should be noted that the actual wiring patterns of
course vary with particular complex systems; however, the
explanation for why some signalling patterns are more effi-
cient than other is based on underlying facts about the
counterfactual dependence between the topological proper-
ties (such as network navigation efficiency, diffusion
efficiency and search information) and the network
dynamics. The explanation thus does not appeal to contin-
gent facts about wiring in any particular system, but to
counterfactual dependency between topological properties
that determine network communication, and thus that
holds independently from any particular system. The facticity
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5

criterion, or the fact that the system instantiates a certain top-
ology, only tells us why such an explanation is true of a
particular system, but the explanatory force stems from the
counterfactual dependence between the topological proper-
ties (communication efficiency) and network dynamics
(direction of neural signalling), and that is why the instantia-
tion of topology in the system is not explanatory in itself [31].
A similar idea about the basis of topological directionality
comes from consideration of asymmetric network embedding
in auditory cortices. Structural and functional differences in
auditory cortex affect the performance of a variety of sensory
and cognitive tasks, such as speech and tonal processing
([35], p. 2656). The structural and functional lateralization
of the auditory cortex are explained by appealing to topologi-
cal properties that determine the network embedding in
the auditory cortex. The explanation appeals to the property
of topological centrality of the brain auditory networks,
which is determined by using network communication
measures—such as closeness centrality or nodal efficiency,
which measure direct and indirect communication pathways
between two hemispheres. The hemisphere with higher close-
ness centrality or nodal efficiency is better integrated into the
overall network communication, which allows better global
communication in the network, and most importantly this
topological asymmetry is driven by differences in com-
munication pathways between two hemispheres, i.e. the
hemisphere that has higher nodal efficiency is better
integrated into the global communication network, thus the
difference in topological efficiency drives the functional later-
alization in the cortex ([35], p. 2660). Here the explanation has
the direction of the counterfactual dependence as well as a
local asymmetry, i.e. asymmetric embedding of networks in
auditory cortex determine the communication efficiency
in both direct and indirect connections, and not the other
way around. Equally, in terms of explanatory perspectivism,
asymmetric embedding of networks in auditory cortex is
the answer to the question of why there is a lateralization
in cognitive function and not the other way around.
These are just two examples of the general idea about
how counterfactual dependence and explanation-seeking
questions provide bases of explanatory asymmetries in
topological explanations, without appealing to causation.
Property, counterfactual and perspectival asymmetries
dampen possible objections such as that topological expla-
nations in the vertical mode will require some non-
negligible ontic commitments in order to be considered
asymmetric, e.g. that in some cases it might allow retrodiction
which, in turn, requires causal assumptions.
10
This kind of
objection can be preempted if one looks into the second cri-
terion in the general theory of topological explanation,
which stipulates that Atopologically explains Bin the vertical
mode only if Adescribes a global topological property of the
network, such that, had Ahad not obtained some general
physical property Bwould not have obtained either. Thus,
only the counterfactuals of this form should be considered
(vertical) topological explanations, and the patterns that do
not describe this exact type of counterfactual dependency
should either not be considered cases of vertical topological
explanations or are not explanations at all. As for the retrodic-
tion, topological explanations in the vertical mode are
synchronous, i.e. they rarely explain the dynamics, but
rather some general empirical property of the system. To
that effect, retrodiction in the vertical mode would require a
different explanandum, which is a consequence of the explana-
tory perspectivism condition, i.e. that the kind of explanatory
questions determine whether we use vertical or horizontal
explanatory mode, in which case the explanation may as
well be causal.
Having discussed some ways in which the general theory
of topological explanations accounts for explanatory asym-
metries in both vertical and horizontal modes, in the next
section, I argue how the general theory and its accounting
for the explanatory asymmetries can help to unify essential
concepts of biological networks.
5. Generalizability of the epistemic norms for
topological explanatory asymmetries
I have argued in this paper that determining whether a topo-
logical explanation is successful (or not) and understanding
why it is successful (or not) can only be assessed through a
general theory of topological explanations, which tells us
what it is for Ato topologically explain B. In §2, I laid out
such a theory, according to which Atopologically explains
Bif and only if:
1. (Facticity):Aand Bare approximately true; and
2. Either
(a) (Vertical mode): Adescribes a global topology of the net-
work, Bdescribes some general physical property, and
had Ahad not obtained, then Bwould not have
obtained either; or
(b) (Horizontal mode): Adescribes a set of local topological
properties,Bdescribes a set of local physical properties,
and had the values of Abeen different, then the values
of Bwould have been different.
3. (Explanatory perspectivism):Ais an answer to the relevant
explanation-seeking question Q about B, such that the Q
determines whether to use vertical or horizontal explana-
tory mode.
This theory provides three criteria for evaluating the suc-
cess of any topological explanation:
(a) The criterion concerning the facticity of topological expla-
nations, i.e. what makes it true of a particular system.
(b) The criterion that governs two explanatory modes
of topological explanations, i.e. the vertical and the
horizontal;
(c) Finally, the third criterion tells us when to use the vertical
and when to use horizontal mode. This last criterion
is based on the idea of explanatory perspectivism,
according to which the explanation-seeking questions
determine whether we use vertical or horizontal mode
in describing counterfactual dependencies.
One of the most important features of any theory of expla-
nation is that it can account for explanatory asymmetries. In
§3, I provided a background of the asymmetry problem in
order to demonstrate why it is of foundational importance
that a theory of explanation can account for explanatory
asymmetries. In §4, I argued how the general theory of topo-
logical explanations accounts for explanatory asymmetries
under two explanatory modes, by using examples of cogni-
tive controllability and the asymmetry of information flow
royalsocietypublishing.org/journal/rstb Phil. Trans. R. Soc. B 375: 20190321
6

[34], and the asymmetry of topological organization of the
auditory cortex [35].
The final piece of my argument in this paper is to show
how the general theory of topological explanations helps to
unify essential concepts of biological networks. The best
way to do so is by considering universal applicability of the
three criteria that the theory provides.
The facticity criterion as it was shown works in both ver-
tical and horizontal modes of topological explanations. Given
that topological explanations are non-causal and that coun-
terfactual dependency relations under horizontal and
vertical modes hold independently from the contingent
facts about any particular system, the facticity criterion tells
us how this universally applicable explanation is to be suc-
cessfully used in any particular case, regardless of the area
of science.
The criterion about vertical and horizontal modes of topo-
logical explanations can be understood also in terms of
counterfactual dependencies within-the-scale (horizontal)
and between-the-scales (vertical), intra-modular (horizontal)
and hierarchical-modular (vertical), or in general in terms
of the distinction between local (horizontal) and global (ver-
tical) network levels. The distinctions between different
scales, modules and hierarchies of modules or between
local and global levels are not limited to any particular
class of cases or to any particular area of biology. Given
this, they should be applicable to all networks, regardless
of the discipline, and ipso facto, the criteria about vertical
and horizontal modes of topological explanations are appli-
cable across biological sciences.
Finally, the criterion about explanatory perspectivism also
helps to delimit distinctively topological explanations from
the other types of explanation or ensembles of explanations
[36,37]. For example, sometimes we want to understand
very particular phenomena (such as why certain connections
exist) or perhaps very complex phenomena (such as develop-
ment of certain dynamical constraints, e.g. the particular
wiring costs). In those cases, we will have a combination of
different kinds of explanation, e.g. mechanistic, dynamical,
statistical and topological. This is particularly evident in the
case of homophily, where regions with similar connectivity
properties connect directly among themselves. We might
ask why do they connect among themselves, and the
answer is: owing to various environmental mechanisms in
the evolution of a particular network. For example, an expla-
nation of why particular nodes in a network are connected
among themselves will appeal to local wiring costs and avail-
able energy. An explanation will also appeal to regularity
rather than some teleological principle, e.g. connections that
are established on a more regular basis will have lower
wiring costs than connections that are established only
occasionally. In this case, the explanation will not be distinc-
tively topological, but rather mechanistic, despite involving
some network concepts. More generally, when the expla-
nation-seeking question is about how and why the system
arrived at having certain topological properties, the expla-
nation will not be distinctively topological, it will more
likely be mechanistic. On the other hand, if the explanation-
seeking question is why the system displays certain
behaviours or dynamical properties given the topology that
it instantiates, the explanation will be distinctively topologi-
cal and it will employ either vertical or horizontal mode of
describing counterfactual dependencies.
The examples that I used to illustrate these points come
from computational neuroscience, biology and ecology,
which indicate that these criteria are applicable to network
explanations in all of those areas. More importantly,
because these criteria are not limited to particular classes
of cases or specific areas of biological sciences, they are
in principle universally applicable across any sciences
that use network explanations, thus their universal applica-
bility may help to unify network concepts across biological
sciences.
Data accessibility. This article has no additional data.
Competing interests. I declare I have no competing interests.
Funding. This research is funded by the European Commission, Hor-
izon 2020 Framework Programme, Excellent Science, Marie
Skłodowska-Curie Actions under REA grant agreement no 703662;
project: Philosophical Foundations of Topological Explanations (Pro-
posal acronym: TOPEX) and the project Neuroessentialisme:
explication neuroscientifique et diffusion des neurosciences, le cas
du neurodroit, funded by the Conseil régional Nouvelle-Aquitaine
and Bordeaux Montaigne University.
Acknowledgements. I am grateful to Claus Hilgetag, Kareem Khalifa,
Angela Potochnik, Thomas Polger and Cédric Brun for tremendously
helpful comments and discussions of the ideas and various versions
of this paper.
Endnotes
1
A counterfactual is a statement describing a hypothetically different
situation relative the actual state of affairs. In currently fashionable
terms, it describes what-if-things-have-been-different relative to a
postulated explanation.
2
Technical term denoting the part of an explanation with which we
are explaining (plural is explanantia).
3
Technical term denoting the part of an explanation that describes
what is being explained (plural is explananda).
4
Topological explanations raise ontological questions about math-
ematical objects. Philosophers of mathematics cast this in terms of
the so-called indispensability arguments, according to which we
ought to rationally believe that mathematical entities that are
indispensable in explaining physical facts are real in the same sense
as other unobservable theoretically postulated entities such as
quarks, dark matter or black holes [3–8]. However, this paper shall
remain neutral about this issue, because addressing it exceeds
its scope.
5
The idea of using network control theory in this context comes from
control theory in engineering. Gu et al. explicitly acknowledge it:
‘Network control theory is a branch of traditional control theory in
engineering that addresses the question of how to control a system
whose components are linked in a web of interconnections; here
the term control indicates perturbing a system to reach a desired
state’. ([12], p. 8).
6
The measure of network communicability assesses to what extent
the nodes in a network are connected indirectly. Network communic-
ability is calculated by quantifying weighted sum of walks of all
lengths in the communicability matrix, with a direct implication
that longer lengths require more energy, and the shorter ones require
less energy. Now we see that the topology that is described by com-
municability measures can affect energy requirements for changing
the trajectories in the state space, and therefore how from particular
values of these measures it is possible to find both the exact energy
requirements and also the minimal ones.
7
According to the explanatory relevance objection, the DN fails to
distinguish between relevant and irrelevant antecedent facts, thus
the following case would be considered a successful explanation:
(1) birth control pills prevent pregnancy, (2) Jones (a male) has
taken birth control pills, (3) therefore, Jones did not get pregnant.
Clearly, Jones did not get pregnant because he does not have appro-
priate reproductive organs to become pregnant in the first place, thus
the DN model focuses solely on inferential patterns, without provid-
ing a norm for explanatory relevance [24].
royalsocietypublishing.org/journal/rstb Phil. Trans. R. Soc. B 375: 20190321
7

8
Namely, the definition of the laws of nature does not distinguish
between accidental generalizations and lawful generalizations.
9
However, some phenomena seem to permit symmetric explanations,
e.g. in one system of gas, decrease of volume can account for
increased pressure; in another, increased pressure can account for
decreased volume; or some laws of nature, especially in physics
also seem to provide symmetric explanations [25]. A good theory
of explanation should account for such cases as well. As far as I
can tell this does not apply to topological explanations (in either
vertical or horizontal mode).
10
I thank Referee 1 for raising this point.
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