# Path-dependent connectivity, not modularity, consistently predicts
controllability of structural brain networks

Shubhankar P. Patankar  Department of Bioengineering, School of Engineering
and Applied Science, University of Pennsylvania, Philadelphia, PA 19104  Jason
Z. Kim  Department of Bioengineering, School of Engineering and Applied
Science, University of Pennsylvania, Philadelphia, PA 19104  Fabio Pasqualetti
Department of Mechanical Engineering, University of California, Riverside, CA
92521, USA  Danielle S. Bassett  Department of Bioengineering, School of
Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA
19104  Department of Neuroscience, Perelman School of Medicine, University of
Pennsylvania, Philadelphia, PA 19104  Department of Electrical and Systems
Engineering, School of Engineering and Applied Science, University of
Pennsylvania, Philadelphia, PA 19104  Department of Neurology, Perelman School
of Medicine, University of Pennsylvania, Philadelphia, PA 19104  Department of
Physics and Astronomy, College of Arts and Sciences, University of
Pennsylvania, Philadelphia, PA 19104  Department of Psychiatry, Perelman
School of Medicine, University of Pennsylvania, Philadelphia, PA 19104  Santa
Fe Institute, Santa Fe, NM 87501

###### Abstract

The human brain displays rich communication dynamics that are thought to be
particularly well-reflected in its marked community structure. Yet, the
precise relationship between community structure in structural brain networks
and the communication dynamics that can emerge therefrom is not well-
understood. In addition to offering insight into the structure-function
relationship of networked systems, such an understanding is a critical step
towards the ability to manipulate the brain’s large-scale dynamical activity
in a targeted manner. We investigate the role of community structure in the
controllability of structural brain networks. At the region level, we find
that certain network measures of community structure are sometimes
statistically correlated with measures of linear controllability. However, we
then demonstrate that this relationship depends on the distribution of network
edge weights. We highlight the complexity of the relationship between
community structure and controllability by performing numerical simulations
using canonical graph models with varying mesoscale architectures and edge
weight distributions. Finally, we demonstrate that _weighted subgraph
centrality_ , a measure rooted in the graph spectrum, and which captures
higher-order graph architecture, is a stronger and more consistent predictor
of controllability. Our study contributes to an understanding of how the
brain’s diverse mesoscale structure supports transient communication dynamics.

##  1 Summary

A central question in network neuroscience is how the structure of the brain
constrains the patterns of communication dynamics that underlie function. At
the mesoscale of network organization, this question has been examined through
the lens of modularity. Recent work has demonstrated a diversity in the
mesoscale architecture of the human connectome. Further diversity in the
characterization of structural brain networks is introduced by the fact that
the distribution of edge weights in a network depends on the precise empirical
measurement whose value is assigned to an edge. This paper explores network
controllability in light of the variety of community interaction motifs and
edge weight distributions that may be used to characterize structural brain
networks.

##  2 Introduction

The brain is a complex system of interconnected components that can be studied
at a variety of spatial and temporal scales [1]. Signals between communicating
neuronal populations propagate along the white matter structure of the brain
and give rise to the complex repertoire of functional dynamics that underlie
cognition [2, 3, 4, 5]. A key goal of network neuroscience is to elucidate the
relationship between brain network structure and function [6, 7, 8, 9]. At any
scale of interest, the patterns of inter-connectivity between components
constrain the functional dynamics that may evolve on the underlying network
topology [10], and thus the patterns of communication between neural units.
Indeed, structural brain networks display striking features such as small-
worldness [11], hierarchical organization [12], spatial and topological
scaling relationships [13], and modularity [14]. Modularity, in particular, is
a commonly studied feature of interest at the mesoscale of brain network
organization that impacts potential patterns of communication.

Figure 1: Structural brain networks exhibit a diversity of mesoscale
architectures. (a) Assortative communities are internally densely and
externally sparsely connected, whereas (b) disassortative communities are
internally sparsely but externally densely connected. (c) Core-periphery
organization is characterized by a dense core of well-connected nodes, and a
periphery of sparsely connected nodes. (d) Structural brain networks have been
observed to possess a mixed meso-scale architecture that combines assortative,
disassortative, and core-periphery organization [Figure reproduced with
permission from [15]].

The term “mesoscale” refers to the topological level higher than that of a
single node, but lower than that of the entire network. Community detection
techniques have been applied extensively to both structural and functional
brain networks in order to group together nodes that share common features;
each group is commonly referred to as a community or module. The predominant
view is that the brain is composed of assortative modules, in which nodes
connect densely to other nodes within their own community and sparsely to
nodes outside of their community. Assortative modules are observed across
species ranging from humans [16, 17] and non-human primates such as macaques
[18], to the nematode C. elegans [19], and are thought to enable information
integration and segregation in support of flexible cognition and behavior
[20]. However, the field’s focus on assortative modules could in part be an
artifact of our methodologies; popular community detection algorithms
expressly seek internally dense and externally sparse sub-networks and are
agnostic to other forms of mesoscale structure [21, 22, 23]. Recent work has
suggested that while most brain communities are indeed assortative, others
form disassortative and core-periphery structures [24, 15, 25, 26] (Figure 1).
The existence of such a diverse mesoscale architecture could explain the
diversity of the brain’s functional repertoire [27, 15].

Figure 2: Schematic of methods and approach. (a) A variety of empirical
measurements are used to estimate and study brain network structure. This data
is then compiled into a weighted network adjacency matrix $A$ whose entries
$A_{ij}$ describe the connection strength of region $i$ and region $j$, thus
characterizing the brain’s structural network. (b) While brain dynamics are
non-linear, linearization is a convenient modeling approach that has been
demonstrated to yield biologically meaningful insights, and one that allows us
to systematically investigate relationships between model parameters and model
behavior. Linear systems theory provides a natural language in which to
characterize state transitions in the brain. (c) The level of activity in each
brain region is combined into a state vector $\bm{x}$ and modeled using a
linear dynamical system. Linear control theory can be used to assess the
effect of exogenous inputs on the brain’s functional dynamics. Controllability
may be quantified using metrics such as average and modal controllability, and
the minimum energy required to cause a state transition [Figure reproduced
with permission from [28]].

Yet, precisely how the community structure of brain networks constrains,
supports, and explicates the communication dynamics that we observe in
empirical measurements is not well understood. Whole-brain models of neural
dynamics provide an avenue to bridge this knowledge gap by stipulating how
neural activity propagates along the underlying structural network [29, 28].
Further insight into how transient dynamics evolve on networks can be obtained
by perturbing the dynamical model with exogenous inputs. Linear systems theory
and its associated network control framework can be used to probe the
relationship between the structure of networks and the transient dynamics that
they support [30, 31] (Figure 2b). The approach requires that the brain be
represented as a network of regions connected by edges, which are commonly
derived from empirical estimates reflecting the strength, volume, or integrity
of white matter tracts [32, 33] (Figure 2a). Control inputs, which are
representative of changing levels of activity, can then be added to network
nodes to study the evolution of activity dynamics [34, 35] (Figure 2c). From a
biophysical perspective, these inputs may represent an endogenous shift in
neural activity from one cognitive state to another [34, 36], or even direct
exogenous inputs such as during electrical stimulation [37, 38].

We hypothesize that brain regions have different controllability statistics
depending on the extent to which they participate in interactions with nodes
from other communities. We reason that a diversity in connections ought to
lead to greater ability for a node to control the rest of the network. To test
this hypothesis, we partition brain regions into communities by applying the
weighted stochastic block model (WSBM) to structural connectivity matrices
extracted from non-invasive magnetic resonance imaging (MRI) measurements in
humans. Block modeling is a flexible community detection technique that is
able to uncover diverse mesoscale motifs beyond the commonly studied
assortative type [39, 40]. The connectivity matrices we study encode networks
whose nodes represent brain regions. Edges can represent diverse estimates of
inter-node connections, such as white matter streamline counts between
regions, mean quantitative anisotropy (QA) values along the streamlines, and
generalized fractional anisotropy values (GFA) [41, 42, 43, 44].
Unfortunately, there is no consensus in the field yet regarding whether one
type of edge weight has more utility than another type of edge weight, and
therefore the literature contains studies that use a variety. The distribution
of edge weights in the network depends on the precise quantity that the edge
represents, and this fact hampers formal comparison of results across studies.
For example, structural brain networks with QA values [45, 37] and those with
streamline counts have differing edge weight distributions. Both have been
previously used for network control theoretic studies [37, 45, 46, 34, 36, 47,
48, 49], but direct comparisons between the two have not been performed. Here
we seek to obtain a more comprehensive understanding of the relations between
community structure and controllability that is independent of the choice of
edge weight, and the associated differences in edge weight distribution. Thus,
we use multiple data sets containing networks with distinct edge definitions.

We further hypothesize that disrupting the amount of a particular mesoscale
motif such as assortativity, disassortativity, or core-peripheriness in a
network ought to result in a motif-specific controllability profile. We
perform numerical simulations to gradually alter the mesoscale structure of
networks along specific continuums of interest while preserving their binary
density and the distribution from which network edge weights are drawn. At
each stage, we examine their controllability. In one set of simulations we
alter the binary topology on an axis ranging from disassortative to
assortative. In another set of simulations, network topology ranges from
disassortative to core-periphery. We perform both sets of simulations on
networks where edge weights are drawn from the normal distribution as well as
the geometric distribution. The latter distribution is an example of a fat-
tailed distribution, which resembles the weighted degree distributions of many
biologically observed networks [50]. If binary topology of networks is the key
driver of controllability, we expect to observe that regardless of the choice
of distribution used to assign edge weights; similar alterations to network
topology along a structural continuum ought to similarly affect patterns of
network controllability.

##  3 Mathematical Framework

While brain network dynamics are known to be nonlinear (Figure 2b) [51], the
simplification to a linearized network model is often a useful approximation
[52, 53]. We offer a discussion of the utility of the linear framework in the
‘Discussion’ section; for a more comprehensive discussion we point the reader
to the Supplement.

A linear model may be created by linearizing the non-linear system of interest
about a fixed point. System dynamics are then characterized in terms of
deviations about this fixed point. Linear modeling provides a tractable
simplification for the analysis of non-linear dynamical systems, allowing the
use of well-developed theoretical tools from linear systems and control theory
to investigate network dynamics in response to exogenous control inputs [30].
In the context of brain networks, the linear model allows one to study how
signals can propagate along structural links connecting brain regions.

Suppose we have a node set $\mathcal{V}=\\{1,\cdots,n\\}$ with undirected
weighted edges $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$, compiled in
a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ and represented by a symmetric
weighted adjacency matrix $A\in\mathbb{R}^{n\times n}$. Elements of
$\mathcal{V}$ denote brain regions and elements of $\mathcal{E}$ represent the
strengths of the connection between them. The dynamics of a discrete-time
linear time-invariant LTI system are written as

$\bm{{x}}(t+1)=A\bm{x}(t)+B\bm{u}(t),$ (1)

where $A$ is the $n\times n$ symmetric and weighted network adjacency matrix,
which acts as the system matrix in the LTI framework, and $B$ is an $n\times
k$ matrix, where $k$ is the number of independent control inputs. A full
control set implies that all $n$ network nodes receive input, for instance in
the case when $B=I_{n}$, the identity matrix of dimension $n$. The terms
$\bm{x}(t)$ and $\bm{u}(t)$ represent the state of the system and the
exogenous input at time $t$, respectively (see ‘Discussion’ for biophysical
interpretations of $\bm{x}(t)$ and $\bm{u}(t)$).

A particularly useful element of the linear control framework is the matrix
defined as,

${W_{C}}(T)=\sum_{t=0}^{T-1}A^{t}BB^{\top}(A^{\top})^{t}$ (2)

called the finite time controllability Gramian, where $T$ refers to the time
horizon of control [30]. The Gramian plays a vital role in determining the
unique control input of minimum energy that transitions the network state from
some initial state $\bm{x}_{0}$ at $t=0$ to a final state $\bm{x}_{f}$ at a
later time $t=T$ [37, 46]. We create target state vectors by placing a 1 in
$\bm{x}_{f}$ corresponding to the location of each brain region $i$ in turn,
and 0s elsewhere. These one-hot vectors may be thought to represent the
activation of a single brain region with a full control set. With
$\bm{x}_{0}=\mathbf{0}$, the minimum energy of the input required to attain a
state $\bm{x}_{f}$ at time $T$ is written as,

$E_{i}={\bm{x}_{f}}^{\top}{W_{C}}^{-1}(T){\bm{x}_{f}}.$ (3)

We demonstrate in the Supplement that the energies thus computed, by
performing $N$ state transitions to $N$ one-hot vectors, form an upper bound
on the energy required to perform arbitrary non-negative state transitions.

In addition to the useful energy-related interpretation, other controllability
metrics are often defined using the Gramian [54]. Average controllability,
which is the average energy input over all possible target states [55, 56], is
one such metric. It has been used in previous studies examining the
controllability of structural brain networks [48, 57, 58, 47, 49]. Average
controllability is proportional to the trace of the inverse of the
controllability Gramian, $\operatorname{Tr}(W_{C}^{-1})$. In practice however,
this quantity is replaced by the trace of the controllability Gramian,
$\operatorname{Tr}(W_{C})$, since computing the inverse of $W_{C}$ is
typically ill-conditioned, and the two quantities satisfy a bounded relation
of inverse proportionality [54, 59]. We compute average controllability for an
individual node by setting $B=b_{i}$, where $b_{i}$ is a one-hot vector with a
$1$ in the location corresponding to a node. Smaller values of average
controllability for a node may be thought of as implying that the network is
less controllable on average from that node.

Another controllability measure that is often used in the context of
structural brain networks is modal controllability [54, 37, 34, 46, 38, 49].
Modal controllability quantifies the extent to which a network’s eigenmodes,
weighted by the rate of their decay, are influenced by input into a brain
region. For a node $i$, modal controllability is defined as:
$\phi_{i}=\sum_{j=1}^{N}\left(1-\lambda_{j}^{2}(A)\right)v_{ij}^{2}$ [46]. We
note that this functional form of modal controllability is defined
specifically for symmetric matrices. Here, $\lambda_{j}$ represents an
eigenvalue of the weighted adjacency matrix and $v_{ij}$ represents the $i$-th
component of the $j$-th eigenvector of $A$. Since the weighted adjacency
matrix is symmetric, all of its eigenvalues are real. The eigenvectors of $A$
represent independent directions in the state-space along which system
dynamics evolve according to the rate specified by the corresponding
eigenvalues. A quickly decaying mode is harder to control since, intuitively,
it requires more input energy to sustain its activity. As a result, this
metric has been previously described as being a measure of the controllability
to the ‘hard-to-reach’ states of a system [34, 60, 36].

In order to ensure comparability of time scales across networks, we scale the
network adjacency matrices by their largest eigenvalues. In this study we set
$T=4$ for average controllability and minimum energy computations. However, we
demonstrate that our results remain robust to a broad range of choices of $T$
in the Supplement. We also note that whereas average/modal controllability
consider control from a single node, minimum control energy considers
controllability from a larger node set. All minimum control energy results
presented in this paper are computed using a full control set, $B=I$.

##  4 Results

###  4.1 Relationship between network controllability and community structure
for edge weights drawn from a normal distribution

Results presented in this section are obtained from analyses performed on Data
Set 1 (see subsection ‘Data’ in the ‘Methods’ for details), which is comprised
of structural brain networks where edges represent estimates of mean
quantitative anisotropy (QA) values. An element $[A_{ij}]$ of the weighted
adjacency matrix for these networks represents the mean QA weighting across
streamlines connecting two regions $i$ and $j$. Note that edge weights with QA
values approximate a normal distribution.

####  4.1.1 Measures of controllability are not consistently correlated with
measures of modularity for structural brain networks with normally distributed
edge weights

Prior work has reported a statistical correlation between some controllability
metrics and modularity, a summary measure of assortative community structure
[60]; yet, importantly in that study results held even after regressing out
the effects of modularity. Here we began our investigation by assessing
whether controllability of structural brain networks is statistically related
to community structure in a different data set than the one used by Tang _et
al._ , and when using a larger set of measures of a network’s community
structure. Specifically, we compute three metrics of network control for each
brain region: minimum control energy to activate the region, average
controllability, and modal controllability. We then study the relationships
between these measures, and the weighted variant of the participation
coefficient and the intra-module strength $Z$-score. Participation coefficient
measures the diversity of the distribution of a node’s strength amongst
network modules. A value of $0$ for a node implies that all its connection
strength is associated with other nodes in its own module, whereas a value of
$1$ implies that connection strength is distributed uniformly among all
modules. Intra-module strength $Z$-score measures the connectivity strength of
a node to other nodes in its own module [61, 62]. We compute participation
coefficient for brain regions and the intra-module strength $Z$-score after
partitioning the networks into communities using the weighted stochastic block
model (WSBM). We use the normal distribution as the choice of prior for the
edge weight distribution when applying the WSBM, since edge weights in QA
weighted networks are approximately normally distributed.

We begin by testing the relationships between participation coefficient and
the intra-module strength $Z$-score, and the three measures of network
controllability. We observe that participation coefficient relates negatively
with minimum control energy ($\rho=-0.807$, $p\approx 0$) and with modal
controllability ($\rho=-0.810$, $p\approx 0$), whereas it relates positively
with average controllability ($\rho=0.815$, $p\approx 0$). Similarly, intra-
module strength $Z$-score relates negatively with both minimum control energy
($\rho=-0.338$, $p\approx 0$) and modal controllability ($\rho=-0.323$,
$p\approx 0$), and relates positively with average controllability
($\rho=0.244$, $p\approx 0$). These observations suggest the presence of a
statistical relationship between community structure and controllability.

However, it is possible for community structure and controllability to be
related due the influence of a third variable. We hypothesize that node
strength could be such a shared driver since prior work has reported a
correlation between network controllability and node strength [34, 63, 47,
48]. In this dataset, node strength relates negatively with minimum control
energy ($\rho=-0.998$, $p\approx 0$) and with modal controllability
($\rho=-0.998$, $p\approx 0$), whereas it relates positively with average
controllability ($\rho=0.986$, $p\approx 0$). Further, we find that node
strength is also positively related to both participation coefficient
($\rho=0.807$, $p\approx 0$) and intra-module strength $Z$-score
($\rho=0.333$, $p\approx 0$). As a result, node strength may be the potential
driver of any relationship between community structure and controllability.

Figure 3: Relationships between metrics of regional controllability and
metrics of community structure for edge weights approximating a normal
distribution. (a, b, c) Participation coefficient does not relate in a
statistically significant manner with minimum control energy ($\rho=-0.052$,
$p=0.426$) when accounting for node strength. On the other hand, correlations
between participation coefficient with average ($\rho=0.192$, $p=0.003$) and
modal controllability ($\rho=-0.132$, $p=0.044$) survive corrections for node
strength. (d, e, f) Intra-module strength $Z$-score follows a similar pattern;
it does not relate with minimum control energy ($\rho=-0.089$, $p=0.174$), but
relates significantly with average ($\rho=-0.530$, $p\approx 0$) and modal
controllability ($\rho=0.165$, $p=0.011$). Each dot in the scatter plots
represents the mean value of a controllability and modularity measure across
24 (8 subjects in triplicate) network instantiations for a single brain region
resulting in 234 data points.

Therefore, we run partial Spearman correlations between metrics of community
structure and controllability, correcting for node strength (Figure 3). We
find that when node strength is accounted for, participation coefficient no
longer relates to minimum control energy ($\rho=-0.052$, $p=0.426$) (Figure
3a). It continues to relate significantly with average controllability
($\rho=0.192$, $p=0.003$) and modal controllability ($\rho=-0.132$, $p=0.044$)
(Figure 3b, c). Intra-module strength $Z$-score follows a similar trend; it
does not relate significantly with minimum control energy ($\rho=-0.089$,
$p=0.174$), but continues to relate with average controllability
($\rho=-0.530$, $p\approx 0$) and modal controllability ($\rho=0.165$,
$p=0.011$) even when controlling for node strength (Figure 3d, e, f).

From the findings in this section, we conclude that for the examined
structural brain networks where edge weights are approximately normally
distributed, region-level measures of modularity such as participation
coefficient and intra-module strength $Z$-score correlate in a statistically
significant manner with average and modal controllability, but not with
minimum control energy.

####  4.1.2 Numerical simulations using edges drawn from a normal distribution

Next, we seek to better understand the relationship between controllability
and community structure by parsing community structure into distinct motifs,
such as assortativity, or core-peripheriness. We generate synthetic networks
with a specifically determined community structure and examine their
controllability. In silico experiments where network topologies are precisely
enforced and edge weights are drawn from distributions with precisely known
parameters are useful benchmarks in understanding the relationship between
mesoscale organization and controllability. We begin by generating networks
with a $2\times 2$ block structure in their adjacency matrices, and with
normally distributed edge weights (see subsection ‘Numerical Simulations’ in
the ‘Methods’ for details).

Recall that when the diagonal blocks of a network are denser relative to the
off-diagonal blocks, networks possess an assortative block structure (Figure
1a). By contrast, when the off-diagonal blocks are denser relative to the
diagonal blocks, network communities interact disassortatively (Figure 1b).
Another form of mesoscale topology is the core-periphery structure (Figure
1c). Nodes in the core are connected more densely to each other than they are
to the rest of the network. Nodes in the periphery predominantly connect with
nodes in the core but not with each other. We quantify the notion of
modularity in the form of the modularity quality index ($Q$), which is a
network-level measure of how well a given community partition segregates nodes
into modules. It quantifies the extent of modularity by relating the observed
strength of within-module connections in a network to the strength of within-
module connections expected under a null model [21]. The quantity $Q$ can be
positive or negative, with positive values implying the presence of an
assortative community structure [22]. We characterize the relationship between
$Q$ and the fraction of network edges inside of modules (or the core) in the
Supplement.

Figure 4: Controllability for normally weighted networks as a function of
changing mesoscale topology. (a, b) As network topology changes from
disassortative to assortative, mean network control energy and average
controllability first decrease, and then increase tracing out U-shaped curves.
Their values are the lowest when $Q\approx 0$, which corresponds to the point
of randomness. Networks with a balance between disassortativity and coreness
occur when $Q\approx-0.28$. (d) Minimum control energy increases as networks
become less disassortative and more core-like. (e) Average controllability
first decreases and then rapidly increases past $Q\approx-0.28$. (c, f) Modal
controllability, on the other hand, exhibits no discernible trends with
changing network topology. Each point in the scatter plots represents a
$Z$-scored mean network controllability value computed across $100$ network
instantiations at each $Q$-value. Error bars correspond to the standard
deviation of the mean controllability value for networks in a given ensemble.

In the first set of simulations, we generate networks on a range from
disassortative to assortative (see subsection ‘Numerical Simulations’ in the
‘Methods’ for details). At each point along the structural continuum, we
generate an ensemble of $100$ different sparse weighted networks with a known
value of the modularity quality index $Q$. First, for each network in the
ensemble we compute the mean of the 234 obtained values of minimum control
energy, average controllability, and modal controllability. Minimum control
energy and average controllability values are computed using $T=4$ as the
choice of time horizon for consistency. We then compute the mean of the three
network-level controllability metrics across the $100$ network instantiations
in the ensemble. We observe that as network topology becomes more assortative
from disassortative, minimum control energy and average controllability first
decrease, and then increase with a minimum value at $Q\approx 0$ (Figure 4a,
b). The trough corresponds to $Q\approx 0$ where the network topology is
random. Modal controllability has no discernible trend with changing network
topology along the disassortative-assortative continuum (Figure 4c).

In the second set of simulations, we generate networks on a range from
disassortative to core (see subsection ‘Numerical Simulations’ in the
‘Methods’ for details). Along this structural continuum, when the fraction of
edges in the core ($[1,1]$-block) is closer to $0$, a network is
disassortative, whereas when the fraction is closer to $1$, it has a dense
core reminiscent of a core-periphery network. Networks are nearly random when
the fraction is $1/3$ for the $2\times 2$ block adjacency matrix with a single
on-diagonal block ($[2,2]$-block) having zero density. In terms of the
modularity quality index $Q$, the extremes correspond to values of $-0.5$
(disassortative) and $0$ (core), respectively. The extent of disassortativity
and coreness is in balance when $Q\approx-0.28$. Similar to the first set of
simulations, we generate $100$ network instantiations as the topology
gradually changes from disassortative to more core-like. We observe that as
networks become more core-like, mean minimum control energy increases (Figure
4d). There is little change in the mean control energy value in the
disassortative regime; however, this is followed by a sharp rise past
$Q\approx-0.20$. Average controllability, in contrast, first decreases
gradually to $Q\approx-0.28$, followed by a sharp increase (Figure 4e).
Similar to the disassortative-assortative structural continuum, modal
controllability does not exhibit a significant trend along the disassortative-
core continuum (Figure 4f).

In summary, disruptions to particular mesoscale motifs in networks where edges
are drawn from a normal distribution result in motif-specific profiles of
network controllability.

###  4.2 Relationship between network controllability and community structure
for edge weights drawn from a fat-tailed distribution

In the context of structural brain networks, multiple empirical estimates may
be used to quantify the strength of connections between two regions, such as
white matter streamline counts between regions, mean quantitative anisotropy
(QA) values along the streamlines, and generalized fractional anisotropy (GFA)
values. These measures reflect the strength, volume, or integrity of white
matter tracts connecting one region of the brain to another. This diversity in
the characterization of structural networks introduces further complexity in
the modeling of large-scale communication dynamics in the brain. The
distribution of edge weights in a structural brain network is contingent on
the choice of edge definition, which has the potential to cause conflict in
results that relate network topology to controllability.

In order to examine the relationship between the edge weight distribution that
underlies a mesoscale topology and network controllability, we next turn to
brain networks with an edge weight distribution distinct from the already
examined normal distribution from Data Set 1. Results presented in this
section are obtained from analyses performed on Data Set 2 (see subsection
‘Data’ in the ‘Methods’ for details), which is comprised of structural brain
networks where edges represent estimates of streamline counts between regions.
An element $[A_{ij}]$ of an adjacency matrix for these networks represents the
number of streamlines connecting two brain regions $i$ and $j$. Edge weights
with streamline counts approximate a fat-tailed distribution. Recent work has
indicated that real-world networks with fat-tailed distributions can often be
approximated using the log-normal distribution [50]. As a result, we use the
log-normal distribution as the choice of edge weight distribution prior when
inferring communities using the weighted stochastic block model (WSBM). We
demonstrate the robustness of our results to the choice of the edge weight
distribution prior in the Supplement.

####  4.2.1 Measures of controllability are not consistently correlated with
measures of modularity for structural brain networks with a fat-tailed
distribution of edge weights

Similar to our observations in structural brain networks with normally
distributed edge weights (Data Set 1), here we find that the participation
coefficient relates negatively with minimum control energy ($\rho=-0.433$,
$p\approx 0$) and with modal controllability ($\rho=-0.435$, $p\approx 0$),
and positively with average controllability ($\rho=0.450$, $p\approx 0$) for
networks with a fat-tailed edge weight distribution (Data Set 2). Intra-module
strength $Z$-score relates negatively with both minimum control energy
($\rho=-0.638$, $p\approx 0$) and modal controllability ($\rho=-0.630$,
$p\approx 0$), and relates positively with average controllability
($\rho=0.565$, $p\approx 0$). These observations, yet again, suggest the
presence of a statistical relationship between community structure and
controllability.

Figure 5: Relationships between metrics of regional controllability and
metrics of community structure for edge weights approximating a fat-tailed
distribution. (a, b, c) Participation coefficient does not relate in a
statistically significant manner with minimum control energy ($\rho=0.038$,
$p=0.563$), average controllability ($\rho=0.103$, $p=0.117$), or modal
controllability ($\rho=0.023$, $p=0.728$). (d, e) Intra-module strength
$Z$-score relates significantly with minimum control energy ($\rho=-0.190$,
$p=0.004$) and average controllability ($\rho=-0.366$, $p\approx 0$). (f) It
does not relate with modal controllability ($\rho=-0.110$, $p=0.095$). Each
point in the scatter plots represents the mean value of a controllability and
modularity measure across 24 (8 subjects in triplicate) network instantiations
for a single brain region resulting in 234 data points.

Similar to Data Set 1, however, it is possible for these statistical relations
between controllability and community structure to be driven by a third
variable such as node strength. Indeed in Data Set 2, we also observe that
node strength is related to measures of network controllability. Node strength
relates negatively with minimum control energy ($\rho=-0.993$, $p\approx 0$)
and modal controllability ($\rho=-0.993$, $p\approx 0$), and relates
positively with average controllability ($\rho=0.984$, $p\approx 0$). Node
strength is also a predictor of the participation coefficient ($\rho=0.440$,
$p\approx 0$) and the intra-module strength $Z$-score ($\rho=0.625$, $p\approx
0$). Similar to earlier analyses, we run partial Spearman correlations in
order to account for the effects of node strength when characterizing the
relationship between measures of controllability and those of community
structure. We find that participation coefficient no longer significantly
relates to minimum control energy ($\rho=0.038$, $p=0.563$) (Figure 5a),
average controllability ($\rho=0.103$, $p=0.117$) (Figure 5b), or modal
controllability ($\rho=0.023$, $p=0.728$) (Figure 5c). Intra-module strength
$Z$-score continues to relate in a statistically significant manner with
minimum control energy ($\rho=-0.190$, $p=0.004$) (Figure 5d) and average
controllability ($\rho=-0.366$, $p\approx 0$) (Figure 5e), but not with modal
controllability ($\rho=-0.110$, $p=0.095$) (Figure 5f) when accounting for the
effect of node strength.

From the findings in this section, we conclude that for structural brain
networks with a fat-tailed edge weight distribution, region-level minimum
control energy and average controllability are related in a statistically
significant manner with intra-module strength $Z$-score. However, unlike Data
Set 1 no measure of controllability relates with participation coefficient in
a statistically significant manner. Therefore, the hypothesized relationship
between a node’s participation in the community structure, and its associated
controllability metrics, is not general and is also strongly contingent on the
distribution from which network edges are drawn.

####  4.2.2 Numerical simulations using edges drawn from a geometric
distribution

In parallel to the previous set of numerical simulations on networks with
normally distributed edge weights, we next sought to describe the relationship
between mesoscale architecture and network controllability for networks with a
fat-tailed edge weight distribution. We use the geometric distribution as a
representative fat-tailed distribution when drawing network edge weights.

Figure 6: Controllability for weighted networks with a geometric distribution
of edge weights as a function of changing mesoscale topology. (a, c) As
network topology changes from disassortative to assortative, the mean network
control energy and modal controllability first increase and then decrease on
either side of $Q\approx 0$, which marks the point of randomness. (b) By
contrast, average controllability exhibits the opposite trend; first
decreasing and then increasing as networks become more assortative from
disassortative. (d, f) Along the continuum from disassortativity to coreness,
minimum control energy and modal controllability decrease, whereas (e) average
controllability increases. Each point in the scatter plots represents a
$Z$-scored mean network controllability value computed across $100$ network
instantiations. Error bars correspond to the standard deviation of the mean
controllability value for networks in a given ensemble.

In the first set of simulations, we generate networks on a range from
disassortative to assortative. At each value of the modularity quality index
$Q$, we generate an ensemble of $100$ sparse weighted networks with edge
weights drawn from the geometric distribution (see subsection ‘Numerical
Simulations’ in the ‘Methods’ for details). We begin by computing the mean of
the nodal values of minimum control energy, average controllability, and modal
controllability. We then compute the mean of the three controllability
measures across the $100$ instantiations in an ensemble, and repeat this
process at every $Q$ value.

We observe that as the network topology becomes more assortative from
disassortative, minimum control energy and modal controllability first
increase, and then decrease with a peak at $Q\approx 0$, which corresponds to
the point of randomness (Figure 6a, c). Average controllability, on the other
hand, follows the opposite trend, and is the highest at points of greatest
disassortativity and assortativity, with a low at $Q\approx 0$ (Figure 6b).
Importantly, the trends in network controllability observed for networks with
a fat-tailed distribution (Figure 6) of edge weights are not similar to those
observed for networks with a normal distribution of edge weights (Figure 4).

In the second set of simulations, we generate networks on a range from
disassortative to core-like (see subsection ‘Numerical Simulations’ in the
‘Methods’ for details). Along this structural continuum, when the modularity
quality $Q$ index is closer to $-0.5$, a network is disassortative, whereas
when the index is closer to $0$, it has a dense core reminiscent of a core-
periphery network. Networks are nearly random when the index is $-0.28$. We
find that networks with increasingly dense cores have lower mean minimum
control energy and mean modal controllability (Figure 6d, f). Average
controllability, in contrast, increases with an increasingly dense core
(Figure 6e). Trends in the mean network controllability values along the
disassortative-core continuum appear to form traces of U-shaped curves.

For networks where edge weights are drawn from the geometric distribution,
disruptions to particular mesoscale motifs results in motif-specific profiles
of network controllability. However, these profiles are distinct from those
observed for networks with normally distributed edge weights. Had binary
topology been a unique predictor of network controllability, the trends in the
curves in Figures 4 and 6 would have been similar for similarly altered
networks along the continuums.

###  4.3 Weighted subgraph centrality as a predictor of network
controllability

Based on the results thus far, and contrary to the initial hypothesis, the
extent of a node’s participation in the network’s community structure is not a
consistent predictor of its metrics of controllability. In addition, at the
network-level, binary topology does not uniquely determine controllability. It
is apparent that the distribution of edge weights is as important to network
controllability as the binary distribution of edges themselves. Since
modularity and controllability do not uniquely explain one another, perhaps a
different but complementary feature of network organization relates the two.
Since eigenvalues and eigenvectors fully and uniquely describe a matrix, the
spectrum of the weighted network adjacency matrix, which acts as the system
matrix $A$ for our discrete-time LTI system, encodes all features of the
network including those that consistently predict controllability. Therefore,
we hypothesize that a node-level metric that is rooted in the graph spectrum
ought to relate to controllability statistics regardless of the distribution
of edge weights, or the binary distribution of edges.

With a full control set $B=I_{n}$, the controllability Gramian can be written
as,

${W_{C}}(T)=\sum_{t=0}^{T-1}A^{t}BB^{\top}(A^{\top})^{t}=\sum_{t=0}^{T-1}A^{2t}=I+A^{2}+A^{4}+\cdots.$
(4)

Further, in a weighted adjacency matrix $A$, the entry in the $i$-th row and
$j$-th column of $A^{n}$ represents the strength of closed walks from node $j$
to node $i$ along paths of length $n$. _Subgraph centrality_ (SC) is a measure
of centrality defined for unweighted networks that incorporates higher-order
path lengths through a factorial discounted sum of the powers of the adjacency
matrix [64]. We extend the definition of subgraph centrality to a weighted
adjacency matrix $A$ in order to compute the _weighted subgraph centrality_ as
follows:

$WSC(i)=\sum_{k=0}^{\infty}\frac{(A^{k})_{ii}}{k!}=1+(A)_{ii}+\frac{(A^{2})_{ii}}{2!}+\frac{(A^{3})_{ii}}{3!}+\frac{(A^{4})_{ii}}{4!}+\cdots.$
(5)

We note that Equation 5 can also be written in terms of the eigenvalues and
eigenvectors of $A$ [64].

$WSC(i)=\sum_{k=0}^{\infty}\frac{(A^{k})_{ii}}{k!}=\sum_{k=0}^{\infty}\left(\sum_{j=1}^{N}\frac{\lambda_{j}^{k}\left(v_{j}^{i}\right)^{2}}{k!}\right),$
(6)

where $N$ is the number of network nodes, and $\lambda_{j}$ and $v_{j}$ are an
eigenvalue and associated eigenvector, respectively. Practically, we compute
weighted subgraph centrality by noting that the above definition is equivalent
to selecting the diagonal entries of the matrix exponential of $A$,
$WSC(i)=[\mathrm{expm}(A)]_{ii}$. Since minimum control energy and average
controllability are explicitly defined in terms of the controllability
Gramian, and since modal controllability is defined explicitly in terms of the
network spectrum, Equations 4, 5, and 6 suggest that the weighted variant of
subgraph centrality is a promising node level predictor of measures of network
controllability. Hence, in the results that follow, we compute weighted
subgraph centrality on the weighted adjacency matrix $A$.

Figure 7: Relationships between metrics of regional controllability and
weighted subgraph centrality for networks approximating normal and fat-tailed
distributions of edge weights. (a, b, c) Weighted subgraph centrality is
related in a statistically significant manner to controllability when
controlling for node strength in networks with normally distributed edge
weights. (a, c) It relates negatively with minimum control energy
($\rho=-0.461$, $p\approx 0$) and modal controllability ($\rho=-0.795$,
$p\approx 0$), and (b) positively with average controllability ($\rho=0.707$,
$p\approx 0$). (d, e, f) Weighted subgraph centrality is also related in a
statistically significant manner to controllability when controlling for node
strength in networks with a fat-tailed distribution of edge weights. The
relationships follow similar trends as networks with normally distributed edge
weights; (d) negative with minimum control energy ($\rho=-0.898$, $p\approx
0$) and (f) modal controllability ($\rho=-0.954$, $p\approx 0$), and positive
with (f) average controllability ($\rho=0.806$, $p\approx 0$). Each point in
the scatter plots represents the mean value of a controllability measure and
weighted subgraph centrality across 24 (8 subjects in triplicate) network
instantiations for a single brain region resulting in 234 data points.

We test weighted subgraph centrality to examine whether it is an accurate
predictor of controllability that generalizes across structural brain network
data sets with distinct edge weight distributions. Initially we note that
weighted subgraph centrality is related negatively with minimum control energy
($\rho=-0.998$, $p\approx 0$) and modal controllability ($\rho=-0.999$,
$p\approx 0$), and positively with average controllability ($\rho=0.992$,
$p\approx 0$) for Data Set 1, in which the edge weight distribution
approximates a normal distribution. However, it is also related to node
strength ($\rho=0.998$, $p\approx 0$). In order to account for the effects of
node strength, we perform partial Spearman rank correlations, and find that
weighted subgraph centrality continues to relate negatively with minimum
control energy ($\rho=-0.461$, $p\approx 0$) (Figure 7a) and modal
controllability ($\rho=-0.795$, $p\approx 0$) (Figure 7c), and positively with
average controllability ($\rho=0.707$, $p\approx 0$) (Figure 7b).

We then repeat the analyses performed above on Data Set 2, where the
distribution of edge weights approximates a fat-tailed distribution. We find
that weighted subgraph centrality relates negatively with minimum control
energy ($\rho=-0.999$, $p\approx 0$) and modal controllability ($\rho=-0.999$,
$p\approx 0$), and positively with average controllability ($\rho=0.994$,
$p\approx 0$). Since it also relates to node strength ($\rho=0.993$, $p\approx
0$), we examine partial Spearman correlations between weighted subgraph
centrality and measures of network controllability. Similar to results with
Data Set 1, we find that weighted subgraph centrality continues to predict
measures of network controllability in a statistically significant manner for
Data Set 2. It relates negatively with minimum control energy ($\rho=-0.898$,
$p\approx 0$) (Figure 7d) and modal controllability ($\rho=-0.954$, $p\approx
0$) (Figure 7f), and positively with average controllability ($\rho=0.806$,
$p\approx 0$) (Figure 7e). Additionally, we examine the robustness of weighted
subgraph centrality in predicting controllability of potentially directed
structural brain networks in the Supplement. We also examine performance in an
independent high resolution data set (Data Set 3) to verify generalizability
of the weighted subgraph centrality - controllability relationship.

In summary, unlike participation coefficient and intra-module strength
$Z$-score, weighted subgraph centrality reliably and significantly explains
measures of network controllability regardless of the distribution of network
edge weights.

##  5 Discussion

The topology of structural brain networks shapes and constrains the patterns
of signalling between distant neuronal populations [65, 66]. These patterns,
in turn, give rise to the diverse and complex large-scale functional dynamics
of the brain that underlie cognition [67, 68]. In this study, we sought to
probe the relationship between brain network structure and the transient
communication dynamics that the topology can support at the mesoscale of
network organization.

While the structure-function relationship for brain networks is of interest at
all scales of network organization, recent advances in community detection
techniques have made the mesoscale particularly relevant [15, 25]. Distinct
motifs of mesoscale structure serve different roles in the context of
communication dynamics; assortative (or modular) interactions allow for
information integration and segregation [20, 69], core-periphery motifs with
rich-club hubs [70] allow for information broadcast and receipt [71, 72],
while disassortative motifs support information transmission. Controllability,
by contrast, influences state transitions [73], and has been related to the
notion of cognitive control, where the brain shifts from one cognitive state
to another [36]. Through our numerical simulations, we demonstrate that
distinct features of community structure are likely to be implicated in
distinct aspects of neural computation.

A mesoscale feature is any topological feature that cannot be explained by the
local neighborhood of a node, and is better explained by larger neighborhoods
around the node, than it is by the total global architecture [74, 75]. Much of
the literature has focused on modularity and core-periphery structure as the
canonical forms of mesoscale structure [76, 21]. But our results suggest that
another distinct form of mesoscale structure must be considered, and that is
the feature that drives controllability statistics [45]. Here we demonstrate
that _weighted subgraph centrality_ , can potentially assess this distinct
dimension of mesoscale architecture in future studies.

Recent work has sought to define measures of network topology, such as
disassortativity and core-peripheriness, both at the scale of nodes and at the
scale of communities [77, 78, 79, 80]. A natural direction to extend this work
is to examine the distribution of eigenvalues as the network topology
gradually alters to become more assortative or core-periphery from
disassortative. Moments of the eigenvalue distribution such as the mean,
variance, skewness, and kurtosis may hold valuable insights into the behavior
of network control metrics as functions of mesoscale architecture and edge-
weight distribution. More theoretical work is needed in order to relate the
spectra of weighted graphs to properties of network controllability. Recent
work has attempted to create closed-form characterizations of spectral
properties for both assortative [81] and core-periphery networks. In addition,
since structural brain networks simultaneously possess a variety of community
interaction motifs [15], future work might involve characterizing the effects
of mixed interactions in numerical simulations similar to those performed in
this work.

Controllability statistics cannot be explained simply by node strength, nor
can they be explained by mesoscale structure. Through our results, we verify
that node strength is a significant predictor of network controllability in
the classes of graphs we study. However, it does not uniquely explain
controllability. In all our analyses, after verifying the dependence of
controllability on node strength, we proceed to regress out its effects when
examining any dependence on other metrics of interest. We demonstrate in the
Supplement that weighted subgraph centrality correlates more strongly, as well
as linearly, with measures of network controllability than node strength does
across a range of values of the time horizon of control. Additionally, whereas
weighted subgraph centrality survives corrections for node strength, and
continues to significantly predict controllability, modularity often does not.
This distinction indicates that weighted subgraph centrality explains parts of
network controllability that neither node strength nor any modularity metric
we evaluated are able to.

Our results indicate that higher-order path-dependent network structure, as
captured by weighted subgraph centrality, is strongly related to transient
communication dynamics. Indeed, it explains controllability better than
descriptive statistics such as node strength and measures of modularity. At
the network-level communicability is able to separate patients of stroke from
healthy controls [82]. Communicability metrics have been shown to be sensitive
indicators of lesions in patients with relapsing-remitting multiple sclerosis
[83]. It has also been shown that communicability is disrupted in patients of
Alzheimer’s disease [84]. Weighted subgraph centrality is the weighted
extension of the notion of self-communicability. The consistently strong
relationship between weighted subgraph centrality and measures of network
controllability, suggests that statistics derived from linear control theory
(such as average and modal controllability, and minimum energy) are also
likely useful tools in investigating the disruptions to brain network dynamics
in disease.

The distinction between modularity and controllability impacts our
interpretation of previous reports that provide evidence that these two
features change appreciably over normative neurodevelopment. A naive
hypothesis could be that the change in modularity drives a change in
controllability, or _vice versa_. However, Tang _et al._ show that their
network controllability results hold after regressing out modularity [60].
Moreover, we find more generally using multiple data sets and systematic
variation of network modularity in simulations, that the two variables cannot
be explained by one another. In the context of development, our results
suggest that the process of brain development may reflect a more complex
optimization function that coordinates a change in modularity alongside a
change in controllability. What that function is, and what the mechanism of
coordination is, remains to be clearly specified, but would be an important
area for future work. The distinction between modularity and controllability
also calls for care when interpreting reports of either of these features
changing as a function of aging [85], training [86], treatment [87, 88],
injury [89], or disease [90].

###  5.1 Biophysical interpretation of model parameters

In the discrete-time LTI framework, the variable $\bm{x}(t)$ is a real
$N$-dimensional vector, whose $i$-th element corresponds to the level of
activity of brain region $i$. The level of activity of each brain region can
be defined in multiple ways, such as the average blood oxygen level dependent
(BOLD) signal from functional magnetic resonance imaging (fMRI) [91, 92], or
the average electrical activity from electrophysiological recordings [37, 38].
As for the inputs, the variable $\bm{u}(t)$ represents independent control
inputs whose influence can be linearly separated from the activity along white
matter tracts. For instance, these influences may be endogenous
neurotransmitter activity [92], task-based internal modulation of the brain
state [93, 91], or exogenous inputs such as pharmacological agents [92],
direct electrical stimulation or transcranial magnetic stimulation [37, 38].

Hence, while the most immediate and straightforward interpretation of
$\bm{u}(t)$ is as an external electrical or pharmacological perturbation, we
do not discount the possibility of other internal neural mechanisms (e.g.,
local dynamics of gray-matter neurons) that are independent of and take
advantage of these white-matter tracts to influence global dynamics. Keeping
both possibilities in mind, we refer to $\bm{u}(t)$ as the “exogenous input”
for conceptual tractability. In addition, if it is easier for an exogenous
input to globally influence the system by changing the activity of a node
(less energetic cost, more spread of activity), then it is similarly easier
for the endogenous activity of that node to globally influence the system. If
the endogenous nodal activity is generated by a process that is independent of
the white-matter tracts, it can be modeled as a separate input $\bm{u}(t)$ to
the linear dynamical system without making additional assumptions beyond an
interpretation of exogenous inputs.

In the context of structural brain networks and computations of control energy
for state transitions, more work is needed to neurobiologically motivate the
choices for initial and target states. Prior work has made imaging-based
choices for states to model cognitive states of the brain, such as band-
limited power [37] or beta weights from a general linear model of BOLD
activation from functional magnetic resonance imaging [92]. Alternatively,
binary activation of regions corresponding to functional modules has also been
examined [94]. However, since the focus of this paper is to examine network
controllability from the perspective of network community structure, a
thorough investigation of state-pair choices is beyond the current scope. Our
specific choice here is motivated by prior work probing the generic control
properties of a system by formulating an influence maximization problem [95].
We compute minimum control energies by performing $N$ state transitions to $N$
one-hot vectors for each brain region $i$, such that the energies $E_{i}$ form
an upper bound on the energy required to perform arbitrary non-negative state
transitions $\bm{x}^{*}$ (see Supplement for more discussion).

###  5.2 Methodological considerations

The choice of the weighted stochastic block model (WSBM) to uncover network
communities is motivated by the desire to uncover community interaction motifs
extending beyond the traditionally examined assortative type. We hypothesized
that disruptions to specific motifs ought to result in motif-specific profiles
of network controllability. In the context of empirical brain data, the WSBM
uncovers a diverse community structure reflecting the diversity of the
functional dynamics supported. The WSBM is an incredibly flexible community
detection technique. However, this flexibility comes at the price of having to
choose a number of parameters a priori, including the number of communities
that are anticipated to exist in the network, and a prior regarding the nature
of the edge weight distribution. We fix the number of communities by sweeping
over a range of values and choosing the value that maximizes the likelihood of
observing the given network data. Additionally, we verify salient analyses
performed in the paper in the Supplement with a different choice of edge
weight distribution prior.

In our network-level numerical simulations, we adopt the geometric
distribution as a representative fat-tailed distribution from which to draw
edge weights. The geometric distribution is the discrete counterpart to the
exponential distribution. Another fat-tailed distribution that is commonly
explored in network neuroscience is the scale-free distribution characterized
by a power-law [96, 97]. However, recent work has demonstrated that scale-free
networks are not as ubiquitous as previously thought, and that the exponential
distribution is often a suitable alternative [50]. Our motivation in
considering the normal and geometric distributions was to examine
controllability of networks with two different edge weight distributions.
Future work could characterize controllability performance explicitly for
networks with a scale-free distribution of edge weights, instead of relying on
a stand-in fat-tailed distribution [96].

While a linear model of network dynamics lends itself well to control-
theoretic studies of communication dynamics, empirical results have shown that
brain activity is non-linear [51]. However, recent work has demonstrated that
a linear approximation is often useful [8, 52, 63]. In addition, the linear
framework can be adapted to incorporate more complex features of neural
dynamics [98, 99, 100]. Similar to the WSBM, applying linear network control
theory to empirical data involves setting a variety of hyper-parameters, such
as the time horizon over which control is exerted, the target state vector in
computations of minimum control energy, or the normalization scheme employed.
Our hyper-parameter choices are motivated by the desire to investigate and
compare network topology across data sets with very distinct edge weight
distributions. As a result, we choose a non-zero short time horizon after
scaling down the network adjacency matrices by their largest eigenvalues. This
step ensures that the fastest evolving modes across systems stay consistent.
However, we note the need for further work to motivate parameter choices from
a neurophysiological perspective.

Our results demonstrate that the choice of empirical measurement that is used
to characterize structural edges in brain networks is crucial to
investigations of network control. For instance, whereas results derived from
quantitative anisotropy (QA) weighted networks may lead us to conclude that
modularity as measured by the participation coefficient and average
controllability are related (Figure 3), streamline count weighted networks
present contrary results (Figure 5). It is unclear if one type of empirical
estimate for network edges in structural brain networks is better than
another. It is possible that some measures better assess signal speed, others
better assess bundle volume, and yet others better assess micro-structure
integrity [101]. Perhaps the choice of edge weight definition also has
implications for community detection. For instance, are network partitions
likely to be different depending on the distribution of edge weights? More
work is needed to contextualize the impact of edge weights on our
interpretations of modularity, core-periphery structure, and network
controllability, and their relationships to communication, computation, and
dynamics. The WSBM continues to remain a promising tool in this endeavor since
it is comprised of a generative model with a prior over the edge weight
distribution built into its framework.

##  6 Conclusion

We began with the hypothesis that the extent of a node’s participation in the
network community structure ought to be related to its controllability. We
find that modularity as measured by the participation coefficient and intra-
module strength Z-score is a significant predictor of average and modal
controllability for structural brain networks where the distribution of edge
weights approximates a normal distribution. For these networks, neither
participation coefficient nor intra-module strength Z-score are related with
minimum control energy. For networks where edge weights approximate a fat-
tailed distribution, we find that participation coefficient is not related in
a statistically significant manner with any measure of network
controllability. Intra-module strength Z-score is related with minimum control
energy and average controllability, but not with modal controllability.

By contrast, _weighted subgraph centrality_ is a statistically robust
predictor of network controllability, regardless of the distribution of
network edge weights. The relationships between weighted subgraph centrality
and measures of network controllability, indicate that higher-order path-
dependent network structure predicts transient communication dynamics. At the
network level, through numerical simulations, we demonstrate that binary
topology alone is not a predictor of mean network controllability. Along a
structural continuum from disassortative to assortative, or from
disassortative to core, mean controllability profiles are heavily dependent on
the distribution of network edge weights. Our study contributes to an
understanding of how the diverse mesoscale structural architecture of the
brain, characterized by a variety of community interaction motifs and edge
weight distributions, supports transient dynamics in the brain.

##  7 Methods

###  7.1 Data

Structural brain networks used in the analyses are constructed from diffusion
spectrum imaging (DSI) data acquired in triplicate from eight subjects (mean
age $27\pm 5$ years, two female, two left handed) along with T1-weighted
anatomical scans at each scanning session. DSI scans sampled $257$ directions
using a Q5 half-shell acquisition scheme with a maximum $b$-value of $5000$
and an isotropic voxel size of $2.4$ mm. Axial acquisition with the following
parameters was employed: repetition time (TR) = $11.4$ s, echo time (TE) =
$138$ ms, $51$ slices, field of view (FoV) ($231$, $231$, $123$ mm). All
participants volunteered with informed consent in accordance with the
Institutional Review Board/Human Subjects Committee, University of California,
Santa Barbara. Data acquisition and network construction methods are described
elsewhere in further detail [34].

The data contain brain networks where edges represent diverse estimates of
inter-node connections, including white matter streamline counts between
regions, mean quantitative anistropy (QA) values along the streamlines, and
generalized fractional anisotropy (GFA) values. The choice of edge definition
has implications for the distribution of edge weights in the networks.
Streamline counts have a fat-tailed edge weight distribution, whereas QA
values are normally distributed. In the present study, we investigate the
implications of edge weight distribution on network controllability by using
networks with streamline counts as well as QA values. We refer to networks
with QA values as Data Set 1, and to networks with streamline counts as Data
Set 2.

Additionally, we repeat salient analyses in the Supplement on a higher
resolution data set, henceforth termed Data Set 3. This data set is acquired
from ten healthy human subjects as part of an ongoing data collection effort
at the University of Pennsylvania; the subjects provided informed consent in
writing, in accordance with the Institutional Review Board of the University
of Pennsylvania. Similar to Data Set 2, Data Set 3 is comprised of structural
brain networks where edges reflect streamlines counts between regions.

For Data Set 3, all scans are acquired on a Siemens Magnetom Prisma 3 Tesla
scanner with a $64$-channel head/neck array at the University of Pennsylvania.
All participants volunteered with informed consent in accordance with the
Institutional Review Board/Human Subjects Committee, University of
Pennsylvania. Each data acquisition session includes both a diffusion spectrum
imaging (DSI) scan as well as a high-resolution T1-weighted anatomical scan.
The diffusion scan is $730$-directional with a maximum $b$-value of
$5010$s/mm2 and TE/TR = $102$/$4300$ ms, which includes $21$ $b=0$ images.
Matrix size is $144\times 144$ with a slice number of $87$. Field of view is
$260\times 260$ mm2 and slice thickness is $1.80$ mm. Acquisition time per DTI
scan is $53:24$ min, using a multiband acceleration factor of $3$. The
anatomical scan is a high-resolution three-dimensional T1-weighted sagittal
whole-brain image using a magnetization prepared rapid acquisition gradient-
echo (MPRAGE) sequence. It is acquired with TR = $2500$ ms; TE = $2.18$ ms;
flip angle $=7$ degrees; $208$ slices; $0.9$ mm thickness. More detail on data
acquisition and processing is available elsewhere [45].

###  7.2 Weighted Stochastic Block Model

In our effort to probe the relationship between network controllability and
the mesoscale architecture of structural brain networks, the first step is to
partition the networks into communities. We apply block modeling to infer
network partitions from data. Block models uncover diverse mesoscale
architectures [39, 40], which may have implications for network
controllability. The model assumes that connections between nodes are made
independently of one another, and that the probability of a connection between
two nodes depends only on the communities to which the nodes are assigned.
Fitting the model involves estimating the parameters that maximize the
likelihood of observing a given network.

The Stochastic Block Model (SBM) seeks to partition the nodes of a network
into $K$ communities. Let $z_{i}\in\\{1,\cdots,K\\}$ indicate the community
label of node $i$. Under the block model, the probability
$P_{ij}=\theta_{z_{i},z_{j}}$ that any two nodes $i$ and $j$ are connected
depends only on their community labels, $z_{i}$ and $z_{j}$, where
${z_{i},z_{j}}\in\\{1,\cdots,K\\}$. To fit the block model to the observed
data in $A$, we estimate $\theta_{rs}$ for all pairs of communities
$\\{r,s\\}\in\\{1,\cdots,K\\}$ and the community labels $z_{i}$. Assuming that
the placement of edges is independent of one another, the likelihood of the
SBM having generated a network is

$P(A\mid\\{z_{i}\\},\\{\theta_{rs}\\})=\prod_{i,j}({\theta_{z_{i}z_{j}}})^{A_{ij}}({1-\theta_{z_{i}z_{j}}})^{1-A_{ij}}.$
(7)

Fitting the SBM involves determining the parameters $\\{z_{i}\\}$ and
$\\{\theta_{rs}\\}$. However, the SBM is limited to binary networks. By
contrast, the weighted stochastic block model (WBSM) [39, 102, 40]
incorporates edge weights into its framework making weighted graphs such as
brain networks accessible to block models for community detection [24, 15, 25,
26].

In the weighted variant (WSBM) of the block model, the likelihood function in
Eq. (7) is modified to

$P(A\mid\\{z_{i}\\},\\{\theta_{rs}\\})\propto
exp\bigg{(}\sum_{i,j}T(A_{ij})\;.\;\eta(\theta_{z_{i}z_{j}})\bigg{)}.$ (8)

In the binary case (SBM), $T$ and $\eta$ correspond to the vector-valued
function of sufficient statistics and the vector-valued function of natural
parameters for the Bernoulli distribution, respectively. Different choices of
$T$ and $\eta$ can allow for the edge weights to be drawn from different
distributions of the exponential family. The WSBM, just like its classical
variant, is parameterized by the set of community assignments, $\\{z_{i}\\}$,
and the parameters $\\{\theta_{rs}\\}$. The difference is that each
$\theta_{z_{i}z_{j}}$ now specifies the parameters governing the weight
distribution of the edge $z_{i}z_{j}$, and not the probability of edge
existence. For the normal distribution, the vector-valued function of
sufficient statistics is $T=[x,x^{2},1]$, while the vector-valued function of
natural parameters is
$\eta=[\mu/\sigma^{2},-1/2\sigma^{2},\mu^{2}/(2\sigma)^{2}]$. Edges are now
parameterized by a mean and variance,
$\theta_{z_{i}z_{j}}=(\mu_{z_{i}z_{j}},{\sigma^{2}}_{z_{i}z_{j}})$. As a
result, the likelihood function in Eq. (7) can be modified to read

$P\big{(}A\mid\\{z_{i}\\},\\{\mu_{rs}\\},\\{{\sigma^{2}}_{rs}\\}\big{)}=\prod_{i,j}exp\bigg{(}A_{ij}\cdot\frac{\mu_{z_{i},z_{j}}}{\sigma^{2}_{z_{i}z_{j}}}-A_{ij}^{2}\cdot\frac{1}{2\sigma^{2}_{z_{i}z_{j}}}-1\cdot\frac{\mu^{2}_{z_{i},z_{j}}}{\sigma^{2}_{z_{i}z_{j}}}\bigg{)}$
(9)

for edge weights drawn from the normal distribution.

An additional challenge in fitting block models to data is the handling of
sparse networks [40]. This is particularly important for brain networks since
the neural connectome is sparse and most entries in the adjacency matrix $A$
are zero. This sparsity is handled by modeling edge weights as described
above, and separately modeling edge presence with a Bernoulli distribution. If
$T_{e}$ and $\eta_{e}$ represent the edge existence distribution, and $T_{w}$
and $\eta_{w}$ the edge weight distribution, the likelihood function for $A$,
can be written as:

$logP(A\mid\\{z_{i}\\},\\{\theta_{rs}\\})=\alpha\sum_{i,j\in
E}T_{e}(A_{ij})\;.\;\eta_{e}(\theta_{z_{i}z_{j}})+(1-\alpha)\sum_{i,j\in
W}T_{w}(A_{ij})\;.\;\eta_{w}(\theta_{z_{i}z_{j}}).$ (10)

In Eq. (10), $E$ is the set of all edges and $W$ is a subset of $E$
representing the weighted edges. A variational Bayes algorithm is then used to
estimate the model parameters from data, as outlined in [102] and [40].

However, this pipeline is still incomplete as fitting the weighted stochastic
block model (WSBM) to a network requires that the number of blocks $K$ in the
community structure be chosen a priori. A data-driven approach can help
determine the suitable number of blocks present. Since the WSBM is a
generative model, we can estimate the likelihood (see (10)) of observing a
connectivity matrix $A$ for different values of $K$. The $K$ that maximizes
the likelihood of observing the data is chosen as the parameter value when
inferring network partitions downstream. For Data Set 1 and Data Set 2, we run
the WSBM on all structural connectivity matrices derived from the eight
subjects ($8$ subjects $\times\;3=24$ matrices) while sweeping over a range of
$K$ values from $K=6$ to $K=15$. Since the WSBM is not deterministic, we run
$10$ iterations for each subject for each trial at each choice of $K$. We find
that data likelihood is maximized when $K=12$ for networks with normally
distributed edge weights (Data Set 1) with a Gaussian edge weight prior, and
when $K=14$ for networks with a fat-tailed edge weight distribution (Data Set
2) with a log-normal edge weight prior. A by-product of the process of
selecting $K$ is the partitions of the networks into communities that we seek.
At the $K$ that maximizes data likelihood, each network already has $10$
instantiations of partitions. The network partition chosen for the analyses is
the one that is the most central out of all, as defined by variation of
information [25]. For Data Set 3, we run $25$ iterations of the WSBM for each
$K$ and find that the likelihood is maximized when $K=10$ with a log-normal
edge weight distribution prior.

Code to infer community structure from networks using the WSBM is freely
available at http://tuvalu.santafe.edu/~aaronc/wsbm/ [102, 40].

###  7.3 Network Statistics

Recall that our hypotheses depend on the quantification of the extent to which
nodes participate in interactions with nodes from other communities. We
compute the participation coefficient [61], and intra-module strength
$Z$-score [61] to quantify this extent based on the WSBM-generated partitions
of brain networks.

The participation coefficient for a node $i$ is defined as

$PC_{i}=1-\sum_{z=1}^{K}\big{(}\frac{\kappa_{iz}}{\kappa_{i}}\big{)}^{2},$
(11)

where $\kappa_{iz}$ is the strength of connection of node $i$ to nodes in
community $z$, and $\kappa_{i}$ is the total strength of node $i$. The term
$K$ is the number of communities in the partition. Intra-module strength
$Z$-score ($Z$) for node $i$ is defined as

$Z_{i}=\frac{\kappa_{iz_{i}}-\bar{\kappa_{z_{i}}}}{\sigma_{\kappa_{z_{i}}}},$
(12)

where $\kappa_{iz_{i}}$ is the strength of connection of node $i$ to other
nodes in its own community $z_{i}$, $\bar{\kappa_{z_{i}}}$ is the average
strength of connection of all nodes in module $z_{i}$ to other nodes in
$z_{i}$, and $\sigma_{\kappa_{z_{i}}}$ is the standard deviation of
$\kappa_{iz_{i}}$. We compute these metrics using freely available code from
the Brain Connectivity Toolbox (https://sites.google.com/site/bctnet/) [103].

At the network level, the modularity quality index $Q$ measures how well a
given partition of a network compartmentalizes its nodes into modules [21,
22]. We use this measure in conjunction with numerical simulations to quantify
the extent of modularity at the network level. $Q$ is defined as:

$Q=\sum_{ij}\left[A_{ij}-N_{ij}\right]\delta\left(z_{i},z_{j}\right),$ (13)

where $N_{ij}$ is the expected strength of connections between nodes $i$ and
$j$ under the Newman-Girvan null model, which is designed to quantify
assortativity [22]. The Kronecker delta function equals $1$ when the two nodes
belong to the same community, and equals zero otherwise.

###  7.4 Numerical Simulations

In order to generate networks with specific edge weight distributions and
binary topologies, we make use of a $2\times 2$ block structure, and specify
the binary density of each block separately. When the fraction of total edges
inside of the on-diagonal blocks exceeds the fraction in the off-diagonal
blocks, the network has an assortative community structure. By contrast, when
the fraction of total edges in the off-diagonal blocks exceeds the fraction
inside of the diagonal blocks, the network has a disassortative community
structure. If the fraction of edges inside of the block in the $[1,1]$
position is higher than the fractions for the three remaining blocks, the
network has a core-periphery architecture. Upon fixing the value of the
fraction of total edges inside of a block of interest, the remaining edges are
distributed across the network such that the network’s binary density remains
$0.1485$, which is the mean density of structural brain networks from Data Set
1.

For each edge, a corresponding weight value is drawn from a pre-specified
distribution, either a normal distribution or a family of geometric
distributions (see below). Edges drawn from the normal distribution are
parameterized by $\mu=0.5$ and $\sigma=0.12$ [96]. The geometric distribution
was chosen as a representative of the family of fat-tailed distributions that
are ubiquitous in biological systems [96, 97, 50]. Geometric distributions are
parameterized by a single number $p$, which represents the probability of
success of a Bernoulli trial. Weights are then assigned to edges by
incrementing the value of an edge until the first failure of a Bernoulli
trial. Therefore, when $p$ is closer to $0$ edge weights tend to remain small,
and when $p$ is closer to $1$ edge weights tend to take on large values.

During the course of numerical simulations along a structural continuum from
disassortative to assortative, or from disassortative to core-periphery, new
networks are created at each stage with new binary densities for the four
blocks. In the case of the continuum from disassortative to assortative
networks, the fraction of total edges in the on-diagonal blocks is gradually
altered. When this fraction is $0$, all network edges lie in the off-diagonal
blocks giving the network a disassortative architecture. By contrast, when the
fraction is $1$ and all edges lie inside of the on-diagonal blocks, the
network is perfectly modular and possesses an assortative mesoscale structure.
In the case of the continuum from disassortative to core-periphery networks,
the fraction inside of the $[1,1]$-block is gradually altered, and the
$[2,2]$-block is left empty. When the fraction of total edges inside of the
$[1,1]$-block is $0$, the network is disassortative, whereas when the fraction
is $1$, the network only has a single densely connected core. Alternatively,
this process may be thought of as moving edges from the off-diagonal blocks to
either the on-diagonal blocks, or the $[1,1]$-block, depending on the
structural continuum under consideration.

At each stage along the continuum, $100$ networks are created using the set of
parameters that define the network topology of the ensemble. The process of
creating ensembles is intended to ensure roughly similar degree distributions
for networks across a structural continuum. In case of simulations for
networks with geometrically distributed edge weights, a further constraint is
enforced. In order to align network topology to the network geometry, when
drawing edge weights for the numerical simulations, we use multiple geometric
distributions. For each block in the $2\times 2$ block adjacency matrix, $p$
is chosen to be the desired binary density (fraction of total edges)
corresponding to the block [96]. We summarize the extent of modularity in each
network in an ensemble along the continuum using the modularity quality index
$Q$. Since networks are generated with partitions that are known a priori, we
do not perform a re-partitioning of the networks in order to determine $Q$. We
characterize the relationship between $Q$, and the fraction of edges inside of
modules (as well as inside the core) in the Supplement.

##  8 Citation Diversity Statement

Recent work in neuroscience and other fields has identified a bias in citation
practices such that papers from women and other minorities are under-cited
relative to the number of such papers in the field [104, 105, 106, 107, 108].
Here we sought to proactively consider choosing references that reflect the
diversity of the field in thought, form of contribution, gender, race,
geography, and other factors. We used automatic classification of gender based
on the first names of the first and last authors [104], with code freely
available at https://github.com/dalejn/cleanBib. Possible combinations for the
first and senior authors include male/male, male/female, female/male, and
female/female. After excluding self-citations to the first and senior authors
of our current paper, the references in this work contain 58.6% male/male, 8%
male/female, 18.4% female/male, 3.4% female/female, and 11.5% unknown citation
categorizations. We look forward to future work that could help us better
understand how to support equitable practices in science.

##  9 Acknowledgments

The authors gratefully acknowledge helpful discussions with Jennifer Stiso,
Dr. Eli J. Cornblath, Dr. Xiasong He, and Dr. Ann Sizemore-Blevins. DSB would
like to acknowledge support from the John D. and Catherine T. MacArthur
Foundation, the Alfred P. Sloan Foundation, the ISI Foundation, the Paul Allen
Foundation, the Army Research Laboratory (W911NF-10-2-0022), the Army Research
Office (Bassett-W911NF-14-1-0679, Grafton-W911NF-16-1-0474, DCIST-
W911NF-17-2-0181), the Office of Naval Research, the National Institute of
Mental Health (2-R01-DC-009209-11, R01 MH112847, R01-MH107235, R21-M
MH-106799), the National Institute of Child Health and Human Development
(1R01HD086888-01), National Institute of Neurological Disorders and Stroke
(R01 NS099348), and the National Science Foundation (BCS-1441502, BCS-1430087,
NSF PHY-1554488 and BCS-1631550).

##  10 Author Contributions

S.P.P. performed the simulations, analyzed the data, made the figures, and
wrote the paper. J.Z.K. contributed analytical solutions. J.Z.K., F.P., and
D.S.B. participated in discussions and edited the paper.

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