# Analysis of Contractions in System Graphs:
Application to State Estimation

###### Abstract

Observability and estimation are closely tied to the system structure, which
can be visualized as a system graph–a graph that captures the inter-
dependencies within the state variables. For example, in social system graphs
such inter-dependencies represent the social interactions of different
individuals. It was recently shown that contractions, a key concept from graph
theory, in the system graph are critical to system observability, as (at
least) one state measurement in every contraction is necessary for
observability. Thus, the size and number of contractions are critical in
recovering for loss of observability. In this paper, the correlation between
the average-size/number of contractions and the global clustering coefficient
(GCC) of the system graph is studied. Our empirical results show that
estimating systems with high GCC requires fewer measurements, and in case of
measurement failure, there are fewer possible options to find substitute
measurement that recovers the system’s observability. This is significant as
by tuning the GCC, we can improve the observability properties of large-scale
engineered networks, such as social networks and smart grid.

Index Terms—  Contraction, clustering coefficient, structural observability,
estimation, system graph

##  1 Introduction

Large-scale networked systems have seen a surge of interest in recent control
and signal processing literature with applications in IoT and CPS [1, 2, 3]. A
key challenge in such networks is state estimation [4, 2, 5] via a distributed
network of measurements. From this perspective of distributed estimation, an
effective tool is the system graph in which nodes represent state variables
and edges between two nodes show coupling among the two state variables [6, 7,
8], motivating structural control and graph signal processing. In this sense,
structural observability is related to certain system graph properties relying
only on the system structure, and not on the exact system parameter values [6,
4, 7, 9].

An important graph-theoretic property to study system observability is the
notion of contraction in the system graph, which is the dual of dilation in
controllability [7]. In a contraction, multiple nodes are contracted
(connected) to a fewer group of nodes. It is known that measuring one state
node in every contraction is essential for network observability [10, 11]. All
states in a contraction are thus observationally equivalent which is
significant for observability recovery, for example, in sensor/measurement
failure [4, 11]. The size of contractions is also a key property, representing
the number of possible options for estimation/observability recovery. A large
contraction presents more choices of equivalent state measurements to replace
the failed/faulty observation, or, for example, to minimize cost [12, 13, 14].
Also, the number of contractions represents the number of necessary
measurement (or sensors) for estimation. The size and distribution of
contractions in a system graph depend on certain graph properties. This work
particularly studies how the global clustering coefficient affects the
distribution of contractions. This paper is a nonlinear model extension of our
previous works [10, 1] on local clustering coefficient and degree
heterogeneity.

This paper models the nonlinear system as a random Scale-Free (SF) graph. The
reason is that the structure of most real-world systems resemble the structure
of SF graphs [15]. To study the effect of the GCC, as our main contribution,
the distribution of size/number of contractions in SF graphs and clustered SF
(CSF) graphs are compared. Due to specific formation in CSF graphs (known as
triad formation) they have higher GCC, while their other properties
(particularly power-law degree distribution) are similar to SF graphs. The
significance of this contribution is that by tuning the network GCC, e.g.,
adopting the results of [16, 17], one can improve/impair system observability
properties. Our results can be used in the design of large-scale man-made
networks to improve their estimation properties in terms of reducing necessary
observer nodes (sensor locations) for cost-optimal estimation. An example of
such re-design of power grid is given in Section 5 as another contribution of
this paper. Another possible application is in changing the structure of
social networks to hinder the possibility of distributed estimation [18, 19]
and, therefore, improve information privacy and reduce the vulnerability
towards information leakage [20].

The rest of this paper is as follows. Section 2 describes graph-theory notions
to define the contractions. Section 3 states the specific application to
system estimation and observability. In Section 4, the distribution of
contractions in SF and CSF graphs are compared, and an illustrative example
application in power grid monitoring is given in Section 5. Conclusions and
future research are presented in Section 6.

##  2 Contractions in Graphs

We consider the complex system, for example a social system, as an undirected
graph or a strongly-connected directed graph (SC digraph) denoted by
$\mathcal{G}=(\mathcal{V},\mathcal{E})$, with the node set $\mathcal{V}$
(representing the $n$ states) and edge/link set
$\mathcal{E}=\\{(v_{i},v_{j})\\}$. The associated bipartite system graph
$\Gamma=(\mathcal{V}^{+},\mathcal{V}^{-},\mathcal{E}_{\Gamma})$ is defined
with two disjoint left/right node sets $\mathcal{V}^{+}$ and
$\mathcal{V}^{-}$, and edges
$\mathcal{E}_{\Gamma}=\\{(v_{j}^{-},v_{i}^{+})|(v_{j},v_{i})\in\mathcal{E}\\}$.
In $\mathcal{G}$, the edges with no common end node are called a matching
$\underline{\mathcal{M}}$, which equivalently in $\Gamma$ represent the subset
of edges not incident on the same node in $\mathcal{V}^{+}$. In other words,
$\underline{\mathcal{M}}$ is a set of pairwise disjoint edges (with no loop).
A matching with maximum size is called maximum (cardinality) matching
$\mathcal{M}$, which is not a subset of any other matching. Note that there
are many possible choices of $\mathcal{M}$ in general. The nodes respectively
in $\mathcal{V}^{+}$ and $\mathcal{V}^{-}$ incident to the chosen
$\mathcal{M}$ are denoted by $\partial\mathcal{M}^{+}$ and
$\partial\mathcal{M}^{-}$, and the nodes in
${\delta\mathcal{M}=\mathcal{V}^{+}\backslash\partial\mathcal{M}^{+}}$ are
unmatched in $\mathcal{V}^{+}$. Given $\mathcal{M}$, let
$\Gamma^{\mathcal{M}}$ be the auxiliary graph made by reversing all edges in
$\mathcal{M}$, and holding all the other edges
$\mathcal{E}_{\Gamma}\backslash\mathcal{M}$ in $\Gamma$. In
$\Gamma^{\mathcal{M}}$, an alternating path associated to ${\mathcal{M}}$
(also called ${\mathcal{M}}$-alternating path), denoted
by$\mathcal{Q}_{\mathcal{M}}$, is a path starting from a node in
$\delta\mathcal{M}$ with its edges alternately in $\mathcal{M}$ and not in
$\mathcal{M}$. An augmenting path $\mathcal{P}_{\mathcal{M}}$ associated to
${\mathcal{M}}$ (also called ${\mathcal{M}}$-augmenting path) is an
${\mathcal{M}}$-alternating path in $\Gamma^{\mathcal{M}}$ that starts from
and ends in $\delta\mathcal{M}$. For a matching $\underline{\mathcal{M}}$ and
associated $\mathcal{P}_{\underline{\mathcal{M}}}$,
$\underline{\mathcal{M}}\oplus\mathcal{P}_{\underline{\mathcal{M}}}$
represents a new matching with one more edge than $\underline{\mathcal{M}}$,
where $\oplus$ is the XOR operator. In $\Gamma^{\mathcal{M}}$, a contraction
$\mathcal{C}_{j}$, associated to an unmatched node
$v_{j}\in\delta\mathcal{M}$, is defined as the set of all state nodes in
$\mathcal{V}^{+}$ reachable by ${\mathcal{M}}$-alternating paths starting from
$v_{j}$. Intuitively speaking, contraction represents subset of nodes linking
to smaller subset of nodes [21, 10]. Algorithm 1 [21, 10] presents the pseudo-
code for finding graph contractions with polynomial order complexity
$\mathcal{O}(n^{2.5})$. Polynomial complexity facilitates applications in
large-scale as in social networks or power grids.

Given: System graph $\mathcal{G}$

Find $\Gamma$;

Find $\underline{\mathcal{M}}$ ;

Find $\Gamma^{\underline{\mathcal{M}}}$ ;

while  _ $\mathcal{P}_{\underline{\mathcal{M}}}$ exist_ do

for  _nodes in $\delta\underline{\mathcal{M}}$_ do

Find $\mathcal{P}_{\underline{\mathcal{M}}}$ ;

$\underline{\mathcal{M}}=\underline{\mathcal{M}}\oplus\mathcal{P}_{\underline{\mathcal{M}}}$
;

Find $\Gamma^{\mathcal{M}}$ ;

for  _state nodes in $\delta\mathcal{M}$_ do

Find $\mathcal{Q}_{\mathcal{M}}$ in $\Gamma^{\mathcal{M}}$ ;

Put nodes in $\mathcal{V}^{+}$ reachable by $\mathcal{Q}_{\mathcal{M}}$ in
$\mathcal{C}_{i}$;

Return $\mathcal{C}_{i},i=\\{1,...,l\\}$;

Algorithm 1 Finding contractions in a graph [10, 21].

In this paper, as our main contribution, we aim to understand possible
correlation between the GCC and prevalence of contractions, and interpret the
implication of this relation through a system estimation perspective.

##  3 Application to state estimation

In this work, a nonlinear autonomous dynamic system (in contrast to the linear
model in [10]) is considered as,

$\displaystyle\dot{\mathbf{x}}$ $\displaystyle=$ $\displaystyle
f(\mathbf{x}(t))+\mathbf{v},$ (1)

where the state variable
$\mathbf{x}=[x^{1},\ldots,x^{n}]^{\top}\in\mathbb{R}^{n}$ is to be estimated
via the measurements,

$\displaystyle\mathbf{y}(t)$ $\displaystyle=$ $\displaystyle
g(\mathbf{x}(t))+\mathbf{r},$ (2)

where $\mathbf{y}=[y^{1},\ldots,y^{m}]\in\mathbb{R}^{m}$ is the measurement,
and $\mathbf{v}$ and $\mathbf{r}$ are Gaussian noise. The system model (1)-(2)
can be represented as a Linear-Structure-Invariant (LSI) model as,

$\displaystyle\dot{\mathbf{x}}$ $\displaystyle=$ $\displaystyle
A(t)\mathbf{x}(t)+\mathbf{v},$ (3) $\displaystyle\mathbf{y}(t)$
$\displaystyle=$ $\displaystyle C(t)\mathbf{x}(t)+\mathbf{r},$ (4)

where $A(t)$ and $C(t)$ are time-dependent system and measurement matrices
representing the linearization of the system and measurement functions
$f(\cdot)$ and $g(\cdot)$ over time. Recall that, from Kalman filtering
theory, the underlying system can be estimated if it is observable via the
given measurements. System observability implies that the global vector,
$\mathbf{x}$, can be uniquely determined by the measurements, $\mathbf{y}$. As
shown in [7], the observability of the nonlinear model (1)-(2) is equivalent
with the observability of the linearized model (3)-(4) over all operating
points. The structure (the zero-nonzero pattern) of the associated linearized
matrices $A(t)$ and $C(t)$ are time-invariant while the numerical values of
their nonzero entries may vary at different operating points, implying the
name structure-invariant. This motivates the concept of structural
observability (or generic observability) based on structured systems theory
[6, 7, 9], which provides a graph-theoretic method to check for system
observability. In structural analysis the system is modeled as a _system
graph_ , where a node $v_{i}$ models a state $x^{i}$ and a link
$v_{j}\rightarrow v_{i}$ models the dependency of the two state variables
$x^{i}$ and $x^{j}$. In other words, if $f^{i}$ is a function of $x^{j}$ then
the entry $\frac{\partial f^{i}}{\partial x^{j}}$ in linearized matrix $A$ is
nonzero while its exact value depends on the operating point and may change
over time [19, 9]. Denote the system graph by
$\mathcal{G}_{A}=(\mathcal{V},\mathcal{E}_{A})$, with state nodes
$\mathcal{V}$ and $\mathcal{E}_{A}=\\{(v_{j},v_{i})~{}|~{}\frac{\partial
f^{i}}{\partial x^{j}}\neq 0\\}$ including the edges $v_{j}\rightarrow v_{i}$.
Using the definitions in Section 2, the next theorem states necessary
conditions for observability of the system graph $\mathcal{G}_{A}$.

###### Theorem 1

Let $\delta\mathcal{M}$ denote the set of unmatched nodes of system graph
$\mathcal{G}_{A}$ associated with an autonomous LSI system. To ensure
observability, it is necessary to measure every unmatched state in
$\delta\mathcal{M}$.

We refer to [7] for the proof (in the dual case of controllability). Based on
the definition, for a given maximum matching $\mathcal{M}$, every node
$v_{j}\in\delta\mathcal{M}$ belongs to a contraction $\mathcal{C}_{i}$, while
the nodes $\mathcal{C}_{i}\backslash v_{j}$ are all matched. This leads to the
following observational equivalence property in contractions:

###### Theorem 2

Consider an LSI system abstracted as a graph (undirected or SC) with
contractions $\mathcal{C}=\\{\mathcal{C}_{1},...,\mathcal{C}_{m}\\}$. The
necessary condition for observability is to measure (at least) one state node
in every contraction.

###### Corollary 1

Nodes in a contraction are equivalent for observability recovery.

See the proofs in [10, 11]. This corollary implies that when a critical
measurement fails, causing loss of observability, measurement of any other
state node in the same contraction recovers the system observability [4].
Recall that (structural) rank deficiency of the system matrix $A$ defines the
cardinality of $\delta\mathcal{M}$ and $\mathcal{C}$ in its associated system
graph $\mathcal{G}_{A}$ [21, 22].

##  4 Contraction Prevalence in SF Graphs

In this section, the distribution of contractions in SF networks, as random
graph-representations of real-world systems, is studied. Such random models
simplify the understanding of different processes, e.g., spreading processes
and cascading failures [15, 23, 24, 25]. The main feature of SF graphs is
their power-law degree distribution [15], which implies that there are few
hubs (nodes with high degree) and large number of low-degree nodes in the SF
network. To construct such networks, Ref. [15] provides an iterative algorithm
initializing with a small seed graph and recursively adding a new node with
$m$ new edges. The main feature of this iterative procedure is that the
linking probability between the new node and an old node is proportional to
its degree. In other words, the new node prefers connecting to old hubs, hence
it is named preferential attachment. Such SF graphs are known to have low
GCC111GCC is defined as the ratio of the triangles $tr$ to the total number of
connected open triplets $trp$ in the graph, i.e.,
$\text{GCC}=\frac{3.tr}{trp}$ [15]., while real-world networks, for example
social networks, show high clustering. Therefore, a modified model with high
GCC is proposed [23, 24, 25], named Clustered Scale Free (CSF). The building
blocks of this model are similar to the preferential attachment, where the
difference is the triad formation step. In this model, the newly added node
directly links to $m_{r}$ nodes, while also making $m_{s}$ preferential
linking to some neighbors of $m_{r}$ preferentially attached nodes to create
triads. This significantly increases the GCC in CSF networks with the same
average node degree as SF networks.

###  4.1 Empirical results and simulation

To study the effect of GCC, we compare the number/average-size of contractions
in CSF/SF graphs. We perform Monte-Carlo simulations over $50$ realizations of
sample CSF and SF graphs with ${m=m_{r}+m_{s}=2}$ and $n=100$ to $n=1000$
nodes. Having $m=m_{r}+m_{s}$ ensures equal number of new edges via
preferential attachment in both CSF and SF networks, implying the same average
node degrees for both types. This is essential for comparison as all features
of both CSF and SF graphs must be similar while only their GCC differs [23].
Fig. 1 shows the Monte-Carlo simulation results.

Fig. 1: Number and size of contractions averaged over $50$ realizations of
growing SF and CSF networks.

As shown in Fig. 1, in SF graphs there are more contractions which are in
average larger (in size) as compared to CSF graphs. Table 1 summarizes the
results shown in Fig. 1.

Table 1: Comparison of CSF and SF graph samples. Graph Type |  SF |  CSF
---|---|---
Average size of Contraction |  $15.8$ |  $6.6$
Average number of Contractions |  $86$ |  $59$
Average GCC |  $0.017$ |  $0.181$

###  4.2 Discussions on the results

From Fig. 1 and Table 1, we see the average size of contractions in SF
networks is significantly larger than CSF networks. Recall that both graph
types are constructed via the preferential attachment method and, therefore,
both share similarity of most graph properties, e.g., (i) logarithmic growth
in mean shortest-path and (ii) power-law degree distribution [23]. Their main
difference stems from the GCC, which is lower in SF graphs. This is the key
feature contributing to the decrease in both size and number of contractions
in CSF graphs. Again we emphasize that the other graph properties of both
types are similar. Thus, one can conclude that, in graphs with power-law
degree distribution, increase in GCC causes fewer contractions with smaller
average-size.

In terms of system observability/estimation, the implication is that for
graphs with higher GCC: (i) fewer state measurements are needed to ensure
observability; and (ii) fewer observationally equivalent states are available
to recover observability loss in case of measurement failure. The first result
deduced from number of contractions while the latter stems from average size
of contractions. This implies that the observability (and consequently
estimation properties) can be improved/deteriorated by tuning the GCC of
(synthetic) system networks via [1, 16, 17]. A such example is given next.

##  5 Illustrative Example: Application to Power Networks

The power grid, as an engineered infrastructural network, can be conceived as
a large-scale dynamical system [26], where the sparcity of its (linearized)
system matrix follows the structure of the power distribution network [27]. To
ensure reliable power delivery, the electrical phasor state (voltage, current,
etc.) is typically measured via advanced sensors, known as phasor measurement
units (PMUs), distributed over the electricity grid. The PMU placement is such
that to ensure observability of the entire power grid for monitoring puposes
[28]. Following the results of this paper, one can reduce the number of
allocated PMUs for observability by changing the structure of the power
network. Consider the European power grid [29] shown in Fig. 2(top) with
$1494$ nodes (buses) and $2156$ edges (transmission lines). The unmatched
nodes represent the possible location of PMU placement in the grid. Following
the results in Section 4, we increase the GCC by adding $40$ edges between
certain bus-nodes in the network as shown in Fig. 2(bottom). The network grid
properties before and after link addition are compared in Table 2. Note that
the change in average node degree is negligible while the GCC is increased
about $19\%$. As expected, contractions and unmatched nodes are reduced by
$47\%$ by adding only $40$ edges which are about $1.8\%$ of the total edges in
the network.

Fig. 2: (top) The European power grid with $151$ unmatched bus-nodes (shown in red) whose states need to be measured via PMUs. An example contraction of $46$ nodes is also shown in black. (bottom) By adding $40$ edges (shown in black) to increase the GCC of power network, the number of unmatched bus-nodes (and minimum required PMUs) is reduced to $80$.  Table 2: Network characteristics before/after link addition. links | average degree | GCC | contractions
---|---|---|---
$2156$ | $2.886$ | $0.094$ | $151$
$2196$ | $2.939$ | $0.112$ | $80$

##  6 Conclusions and Future Research

In this work, distribution of contractions in SC digraphs and its relation
with graph GCC is studied. Note that for general non-SC digraphs, another
component known as root SCC or parent SCC also affects system graph
observability [30, 31]. As future research, we intend to study the correlation
between prevalence/size of both parent SCCs and contractions with other graph
features, such as assortativity/disassortativity and community/hierarchical
structure [15].

## References

  * [1] M. Doostmohammadian and H. R. Rabiee,  “On the observability and controllability of large-scale iot networks: Reducing number of unmatched nodes via link addition,”  IEEE Control Systems Letters, vol. 5, no. 5, pp. 1747–1752, 2020\.
  * [2] M. Rana, L. Li, and S. W. Su,  “Distributed state estimation over unreliable communication networks with an application to smart grids,”  IEEE Transactions on Green Communications and Networking, vol. 1, no. 1, pp. 89–96, 2017.
  * [3] H. Arasteh, V. Hosseinnezhad, V. Loia, A. Tommasetti, O. Troisi, M. Shafie-khah, and P. Siano,  “Iot-based smart cities: A survey,”  in IEEE 16th International Conference on Environment and Electrical Engineering. IEEE, 2016, pp. 1–6.
  * [4] M. Doostmohammadian, H. R. Rabiee, H. Zarrabi, and U. A. Khan,  “Distributed estimation recovery under sensor failure,”  IEEE Signal Processing Letters, vol. 24, no. 10, pp. 1532–1536, 2017.
  * [5] M. Kabiri, N. Amjady, M. Shafie-khah, and J. P. S. Catalao,  “Enhancing power system state estimation by incorporating equality constraints of voltage dependent loads and zero injections,”  International Journal of Electrical Power & Energy Systems, vol. 99, pp. 659–671, 2018.
  * [6] J. M. Dion, C. Commault, and J. van der Woude,  “Generic properties and control of linear structured systems: A survey,”  Automatica, vol. 39, pp. 1125–1144, Mar. 2003.
  * [7] Y. Y. Liu, J. J. Slotine, and A. L. Barabási,  “Controllability of complex networks,”  Nature, vol. 473, no. 7346, pp. 167–173, May 2011.
  * [8] A. Ortega, P. Frossard, J. Kovačević, J. M. F. Moura, and P. Vandergheynst,  “Graph signal processing: Overview, challenges, and applications,”  Proceedings of the IEEE, vol. 106, no. 5, pp. 808–828, 2018.
  * [9] J. F. Carvalho, S. Pequito, A. P. Aguiar, S. Kar, and K. H. Johansson,  “Composability and controllability of structural linear time-invariant systems: Distributed verification,”  Automatica, vol. 78, pp. 123–134, 2017.
  * [10] M. Doostmohammadian, H. R. Rabiee, H. Zarrabi, and U. Khan,  “Observational equivalence in system estimation: Contractions in complex networks,”  IEEE Transactions on Network Science and Engineering, vol. 5, no. 3, pp. 212–224, 2017.
  * [11] C. Commault, J. Dion, D. H. Trinh, and T. H. Do,  “Sensor classification for the fault detection and isolation, a structural approach,”  International Journal of Adaptive Control and Signal Processing, vol. 25, no. 1, pp. 1–17, 2011.
  * [12] B. Guo, O. Karaca, T. H. Summers, and M. Kamgarpour,  “Actuator placement under structural controllability using forward and reverse greedy algorithms,”  IEEE Transactions on Automatic Control, 2020.
  * [13] S. Moothedath, P. Chaporkar, and M. N. Belur,  “Minimum cost feedback selection for arbitrary pole placement in structured systems,”  IEEE Transactions on Automatic Control, vol. 63, no. 11, pp. 3881–3888, 2018.
  * [14] M. Doostmohammadian, H. R. Rabiee, and U. A. Khan,  “Structural cost-optimal design of sensor networks for distributed estimation,”  IEEE Signal Processing Letters, vol. 25, no. 6, pp. 793–797, 2018\.
  * [15] M. Newman, A. Barabási, and D. J. Watts,  The structure and dynamics of networks.,  Princeton university press, 2006.
  * [16] N. Islam,  “Towards a secure and energy efficient wireless sensor network using blockchain and a novel clustering approach,”  M.S. thesis, Dalhousie University, 2018.
  * [17] J. M. Moore, M. Small, and G. Yan,  “Inclusivity enhances robustness and efficiency of social networks,”  Physica A: Statistical Mechanics and its Applications, vol. 563, pp. 125490, 2021.
  * [18] E. Montijano, G. Oliva, and A. Gasparri,  “Distributed estimation and control of node centrality in undirected asymmetric networks,”  IEEE Transactions on Automatic Control, 2020.
  * [19] M. Doostmohammadian, H. R. Rabiee, and U. A. Khan,  “Cyber-social systems: modeling, inference, and optimal design,”  IEEE Systems Journal, vol. 14, no. 1, pp. 73–83, 2019.
  * [20] J. Lovato, A. Allard, R. Harp, and L. Hébert-Dufresne,  “Distributed consent and its impact on privacy and observability in social networks,”  arXiv preprint arXiv:2006.16140, 2020.
  * [21] K. Murota,  Matrices and matroids for systems analysis,  Springer, 2000.
  * [22] M. Doostmohammadian and U. A. Khan,  “On the distributed estimation of rank-deficient dynamical systems: A generic approach,”  in IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE, 2013, pp. 4618–4622.
  * [23] P. Holme and B. J. Kim,  “Growing scale-free networks with tunable clustering,”  Physical review E, vol. 65, no. 2, pp. 026107, 2002.
  * [24] I. Türker,  “Generating clustered scale-free networks using poisson based localization of edges,”  Physica A: Statistical Mechanics and its Applications, vol. 497, pp. 72–85, 2018.
  * [25] C. P. Herrero,  “Ising model in clustered scale-free networks,”  Physical Review E, vol. 91, no. 5, pp. 052812, 2015.
  * [26] J. Machowski, Z. Lubosny, J. W. Bialek, and J. R. Bumby,  Power system dynamics: stability and control,  John Wiley & Sons, 2020.
  * [27] U. A. Khan and M. Doostmohammadian,  “A sensor placement and network design paradigm for future smart grids,”  in 4th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, Puerto Rico, 2011, pp. 137–140.
  * [28] R. Babu and B. Bhattacharyya,  “Optimal allocation of phasor measurement unit for full observability of the connected power network,”  International Journal of Electrical Power & Energy Systems, vol. 79, pp. 89–97, 2016.
  * [29] “The RE-Europe data set,” https://zenodo.org/record/35177.
  * [30] S. Pequito, V. M. Preciado, A. Barabási, and G. J. Pappas,  “Trade-offs between driving nodes and time-to-control in complex networks,”  Scientific reports, vol. 7, no. 1, pp. 1–14, 2017.
  * [31] M. Doostmohammadian and U. A. Khan,  “On the characterization of distributed observability from first principles,”  in 2nd IEEE Global Conference on Signal and Information Processing, 2014, pp. 914–917.