††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Prediction and mitigation of nonlocal cascading failures using graph neural
networks

Bukyoung Jhun  CCSS and CTP, Seoul National University, Seoul 08826, Korea
Hoyun Choi  CCSS and CTP, Seoul National University, Seoul 08826, Korea
Yongsun Lee  CCSS and CTP, Seoul National University, Seoul 08826, Korea
Center for Complex Systems and KI for Grid Modernization, Korea Institute of
Energy Technology, Naju, Jeonnam 58217, Korea  Jongshin Lee  CCSS and CTP,
Seoul National University, Seoul 08826, Korea  Center for Complex Systems and
KI for Grid Modernization, Korea Institute of Energy Technology, Naju, Jeonnam
58217, Korea  Cook Hyun Kim  CCSS and CTP, Seoul National University, Seoul
08826, Korea  Center for Complex Systems and KI for Grid Modernization, Korea
Institute of Energy Technology, Naju, Jeonnam 58217, Korea  B. Kahng
<EMAIL_ADDRESS>Center for Complex Systems and KI for Grid Modernization,
Korea Institute of Energy Technology, Naju, Jeonnam 58217, Korea

###### Abstract

Cascading failures (CFs) in electrical power grids propagate nonlocally; After
a local disturbance, the second failure may be distant. To study the avalanche
dynamics and mitigation strategy of nonlocal CFs, numerical simulation is
necessary; however, computational complexity is high. Here, we first propose
an avalanche centrality (AC) of each node, a measure related to avalanche
size, based on the Motter and Lai model. Second, we train a graph neural
network (GNN) with the AC in small networks. Next, the trained GNN predicts
the AC ranking in much larger networks and real-world electrical grids. This
result can be used effectively for avalanche mitigation. The framework we
develop can be implemented in other complex processes that are computationally
costly to simulate in large networks.

> A small, localized disturbance can result in catastrophic global failure of
> an entire network. Therefore, cascading failure (CF) in infrastructure
> networks such as power grids and Internet, has many practical implications.
> Recently, this subject has become more critical as power-grid systems are
> decentralized by various energy sources such as wind and solar power. The
> electric power from these renewable energy sources may not fit those from
> traditional energy sources, which causes CFs in the electric power grid.
> This CF propagates nonlocally, which differs from what occurs in local
> epidemic disease spread. Here, we study the avalanche dynamics and avalanche
> mitigation strategy of nonlocal CFs. Using a graph neural network (GNN)
> trained in small networks, we can successfully predict the avalanche
> centrality ranking of each node in much larger networks and real-world
> electrical grids. This result enables effective avalanche mitigation.

##  I Introduction

Small, local disturbances in a complex networks can trigger consecutive
failures of other nodes in the network. Most failures remain local and do not
last for an extended period Ewart (1978); Jie Chen, Thorp, and Parashar
(2001); Carreras  _et al._ (2004a). However, there is a small probability that
they will spread throughout the network, resulting in catastrophic global
failure. This type of failure in infrastructure networks such as the Internet
and electrical grids can result in tremendous financial damage and even the
loss of human lives. For example, in the Northeast blackout of 2003, an
initial disturbance in Ohio triggered the largest blackout in North American
history, which affected more than 50 million people and lasted for up to 15
hours in the US and Canada.

The prediction and control of cascading failures in complex networks is a
central topic of research in network science. Conventional epidemic models
Pastor-Satorras  _et al._ (2015) based on local contact process, propagating
in branching process, fail to capture the unique features of CFs that
propagate nonlocally Pagnier and Jacquod (2019); Hines, Dobson, and Rezaei
(2016). A pioneering model associated with the nonlocal CFs de Arcangelis,
Redner, and Herrmann (1985) was proposed to understand the CF dynamics in
random resistor networks, and then this subject has been extensively studied
Gilabert  _et al._ (1987); Kahng  _et al._ (1988).

Most studies of CFs have used numerical simulations and enumeration. In this
study, computational complexity is an important issue. The computational
complexity of the local CF dynamics is generally somewhat lower than that of
the nonlocal dynamics. For instance, for $k$-core percolation Lee, Jo, and
Kahng (2016), CF occurs at nodes with a degree of less than $k$. Thus, CFs
propagate locally. The calculation of a single cascade step has a time
complexity of $O(N)$ in a network with $N$ nodes and can be simulated even in
large-scale networks. By contrast, the random resistor network model de
Arcangelis, Redner, and Herrmann (1985), in which CFs propagate nonlocally,
has a time complexity of $O(N^{2})$. To overcome this problem, Batrouni et al.
Batrouni, Hansen, and Nelkin (1986) proposed the Fourier acceleration
algorithm, which decreased the computation complexity dramatically, to
$O(N\log N)$. In the Motter–Lai (ML) model Motter and Lai (2002), CF
propagates nonlocally, and the time complexity is higher, $O(N^{2}\log N)$,
because the shortest paths must be identified for a single cascade step.
Therefore, the ML model has been simulated only for small networks.

The ML model Motter and Lai (2002) was proposed to study data packet transport
along the shortest paths between two routers on the Internet. When heavily
loaded nodes break down, network traffic is rerouted, causing load
redistribution; consequently, CF can occur at nodes distant from the failed
nodes. This simple dynamics of the ML model makes it possible to understand
the propagation of CF in detail; thus, the model has been widely implemented
Motter (2004); Hayashi and Miyazaki (2005); Zhao, Park, and Lai (2004); Zhao
_et al._ (2005); Wu, Gao, and Sun (2006); Schäfer, Scholz, and Greiner (2006);
Holme and Kim (2002); Lee  _et al._ (2005); Lai, Motter, and Nishikawa (2004);
Kornbluth  _et al._ (2021); Wang and Kim (2007); Fang, Pedroni, and Zio
(2015).

In this study, we addressed the problem of avalanche mitigation. We define the
avalanche centrality (AC) of each node and show that this quantity can be used
to effectively suppress avalanches. To reduce the time complexity of the AC
calculation, which exceeds $O(N^{3}\log N)$, we used a graph neural network
(GNN) approach. The GNN is a deep learning model for analyzing graph data; it
was introduced by Scarselli et al. Scarselli  _et al._ (2008). In particular,
we constructed a scalable GNN structure that is independent of network size
using an inductive learning scheme that is applicable to various type of
networks. The GNN structure was designed to be effective for modeling a large
network, even if it is trained using the simulation results for a small
network. An avalanche mitigation strategy that uses this methodology and is
applicable to large-scale networks, which are impractical to simulate, is
proposed.

Figure 1: Computation time of ML model as a function of network size. The
Schultz–Heitzig–Kurths (SHK) random power grid model was used. The computation
time is proportional to $N^{3.3}$; thus, if the network size doubles, the
computation time increases by approximately tenfold. More than a year of
computation is required to simulate the process in a network of size $N=25000$
using the CPU of model i7-10700.

This paper is organized as follows. First, we introduce the ML model and its
underlying network structure in Sec. II.1. In Sec. II.2, the problem of
avalanche mitigation and an effective strategy for modeling the avalanche
dynamics are discussed. In Sec. II.3, the training dataset and performance
measure are introduced, along with the scalable GNN structure. Avalanche
mitigation using the GNN is presented for both a large network and a real
power grid network.

Figure 2: Avalanche size distribution of SHK network with various numbers of
nodes for the ML model. The control parameter $a$ is taken as 0.25, at which
the avalanche sizes have a scale-free distribution with exponent $\tau\approx
1.6$.

##  II Methods

###  II.1 Avalanche dynamics

Many models of cascading failure dynamics have been proposed, for example,
random fuse networks de Arcangelis, Redner, and Herrmann (1985), $k$-core
percolation Goltsev, Dorogovtsev, and Mendes (2006); Dorogovtsev, Goltsev, and
Mendes (2006); Baxter  _et al._ (2015); Lee, Jo, and Kahng (2016), dynamically
induced cascades Schäfer  _et al._ (2018), and others Crucitti, Latora, and
Marchiori (2004); Kinney  _et al._ (2005); Buldyrev  _et al._ (2010); Cellai
_et al._ (2013); Baxter  _et al._ (2012); Reis  _et al._ (2014); Lee  _et al._
(2016). Among them, the ML model Motter and Lai (2002) is one of the simplest
models capable of capturing the avalanche dynamics of propagation through
nonlocal nodes. It has been applied to power grids Kornbluth  _et al._ (2021).
This simple, shortest-path-based model can be used to understand the avalanche
process. Consequently, it has been extensively applied in studies of
suppression strategies Motter (2004); Hayashi and Miyazaki (2005), the
analytic calculation of the robustness criterion Zhao, Park, and Lai (2004);
Zhao  _et al._ (2005), the effect of network topology Wu, Gao, and Sun (2006),
and other topics Schäfer, Scholz, and Greiner (2006); Holme and Kim (2002);
Lee  _et al._ (2005); Lai, Motter, and Nishikawa (2004); Kornbluth  _et al._
(2021); Wang and Kim (2007); Fang, Pedroni, and Zio (2015).

In the ML model Motter and Lai (2002), the capacity $C_{i}$ of each node $i$
is proportional to the betweenness centrality (BC), $B_{i}^{(0)}$, of node
$i$, as follows:

$C_{i}=(1+a)B_{i}^{(0)}\,,$ (1)

where $a>0$ is a model parameter. The superscript (0) denotes that the BC is
calculated in a network without any failure. The BC is the number of shortest
paths that pass through the node; it is written as

$B_{i}^{(0)}=\sum_{j\neq k\neq i}\frac{m_{jk}(i)}{m_{jk}}\,,$ (2)

where $m_{jk}$ is the number of shortest paths between nodes $j$ and $k$, and
$m_{jk}(i)$ is the number of those paths that pass through node $i$.

We initiate the avalanche dynamics by causing node $i$ on the network to fail.
The failed node is effectively removed from the network, as all the connected
links are disconnected. Then, all the shortest paths that had passed through
node $i$ are rerouted to detours. Consequently, the excess BC is redistributed
over the network. The BC of each node $j$ is then updated to $B_{j}^{(1)}$. If
the updated $B_{j}^{(1)}$ exceeds the capacity of the preassigned node,
$C_{j}$, node $j$ is overloaded and fails. Other nodes with BCs exceeding
their capacities fail simultaneously. Then the BCs of the remaining active
nodes are again updated to $B_{k}^{(2)}$. The process is repeated until there
are no overloaded nodes in the network: $B_{k}^{(t)}<C_{k}$ for all remaining
nodes $k$.

The avalanche size $S_{i}$ of each node ($i=1,\cdots,N$) has a distribution
$P_{a}(S)$. On scale-free networks, the distribution exhibits power-law
behavior as $P_{a}(S)\sim S^{-\tau}$, where $\tau\approx 2.1$ when a
particular value of $a$ is chosen, which is denoted as $a_{c}\approx 0.15$ Lee
_et al._ (2005). This result may indicate that $P_{a}(s)$ exhibits critical
behavior at $a_{c}$. The exponent $\tau$ is insensitive to the degree exponent
$\lambda$ of scale-free networks and seems to be closely related to the
exponent of the diameter-change distribution when nodes are deleted one by one
Kim  _et al._ (2003). It has also been found that the average size $\langle
s(k)\rangle$ of avalanches triggered by removing each node with degree $k$
depends on $k$ as $\langle s(k)\rangle\sim k^{(\lambda-1)/(\tau-1)}$. Note
that the avalanche size distribution observed in cascading failures in real-
world electric power grids has a heavy-tailed distribution Carreras, Newman,
and Dobson (2016); Hines, Balasubramaniam, and Sanchez (2009); Dobson  _et
al._ (2007); Carreras  _et al._ (2004b, a).

In this study, we simulated the ML model on a network model proposed by
Schultz, Heitzig, and Kurths (SHK), which was designed according to the
essential features of real-world electric power grids Schultz, Heitzig, and
Kurths (2014). This model has many control parameters. Using an appropriate
set of parameter values used in Ref. Nitzbon  _et al._ (2017), we obtained
power-law behavior of the avalanche size distribution at $a_{c}\approx 0.25$.
The exponent $\tau\approx 1.6$ was found as shown in Fig. 2, which is close to
$\tau=1.5$, the value of the sandpile model in random networks Goh  _et al._
(2003).

The ML model was modified for application to electric power grids by Carreras
et al. Carreras  _et al._ (2002). In this modified ML model, each node $i$ is
characterized by an input power $P_{i}$, which has positive and negative
values for power generators and consumers, respectively. To account for the
avalanche failures of links rather than nodes, the capacity of each link is
defined as $C_{ij}=(1-a)F_{ij}^{\rm max}$, where $F_{ij}^{\rm max}$ is the
maximum power flow through link $(ij)$ between nodes $i$ and $j$. At each time
step, the direct-current circuit equation is solved, and the power flow
$F_{ij}$ through link $(ij)$ is obtained. If $F_{ij}>C_{ij}$, link $(ij)$ is
disconnected, and the power flow of each link is recalculated. This link
disconnection process is repeated until further link failure occurs. The
avalanche size is obtained by counting the number of failed links.

###  II.2 Avalanche mitigation strategy

Figure 3: Paradoxical effect of reinforcement in electric power grid of
France. Red square (indicated by A) represents the initial failure, red dots
represent secondary failures, and blue triangle (indicated by B) represents a
reinforced node. (a) and (b) If no node is reinforced, the avalanche ends at
$t=2$ with avalanche size $S=5$. (c)–(f) If node B ($\blacktriangle$) is
reinforced, the avalanche lasts until $t=4$, and the avalanche size increases
to 33. Because the reinforced node does not fail, heavy traffic through route
via node B (blue arrow) is maintained, causing other nodes on the route to
fail. Reinforcement thus may result in a significantly larger global cascade.

Various methods of reducing the avalanche size in the ML model have been
implemented. For instance, after an initial failure but before the failure
propagates to other nodes, selected nodes are removed to reduce the avalanche
size Motter (2004); Hayashi and Miyazaki (2005). However, the initial failure
often propagates so rapidly that it is impossible to remove nodes involved in
significant propagation of the failure. As another strategy, a parameter $a$
is assigned to each node under the constraint $\sum_{i}a_{i}=\mathrm{const}$
Wang and Kim (2007). However, there is no way to adjust the capacity of every
node to an appropriate value to minimize the avalanche size.

In this study, we compare the performance of various strategies that minimize
the avalanche size when nodes are reinforced. To simplify the problem, we
suppose that reinforced nodes never fail unless they are intentionally chosen
as the initial failure of the cascade; however, the effectiveness of the
strategies should be maintained even when reinforcement is finite.

The problem with reinforcement is analogous to the problem of vaccination
against epidemic contagion, where epidemic spreading is minimized by
vaccination; however, there is a crucial difference. Specifically, during
epidemics, a node is eliminated after vaccination, and the centrality measures
of the network are recalculated to determine the next node to be vaccinated
Wang  _et al._ (2017); Clusella  _et al._ (2016); Altarelli  _et al._ (2014);
Cohen, Havlin, and Ben-Avraham (2003); Matamalas, Arenas, and Gómez (2018).
This process is repeated until the feasible number of vaccinations is reached.
In the avalanche mitigation problem, however, reinforced nodes can remain in
the power flow; they simply do not fail even if the power flow increases.

Avalanche mitigation poses a novel type of vaccination problem in which the
effects of reinforced nodes on the avalanche are complex. A unique feature of
the avalanche mitigation problem is the paradoxical effect of reinforcement.
In some cases, reinforcement of a node can increase the avalanche size. As
shown in Fig. 3, even if one correctly predicts that node B is at risk and
reinforces it, reinforcement results in a significant increase in avalanche
size. Therefore, reinforcement of nodes without a systematic strategy can be
counterproductive.

To characterize the role of each node in the avalanche dynamics, we define a
binary variable $x^{i}_{j}$, which is 1 if node $j$ fails in an avalanche
triggered by node $i$ and is 0 otherwise ($x^{i}_{i}=1$). Then, the avalanche
fraction of node $i$ is given by $s_{i}=\sum_{j}x^{i}_{j}/N$. This quantity
represents the effect of node $i$ in cascading failures. The failure fraction
$f_{j}$ of node $j$ is defined as $f_{j}=\sum_{i}x^{i}_{j}/N$ and represents
the probability that node $j$ fails in isolation or because of triggering by
another node. These two quantities characterize the role of each node in
cascading failures. For instance, if $s_{i}\gg 0$ but $f_{i}\ll 1$, then node
$i$ triggers a large avalanche, but its effect on the avalanche is rather
limited.

We define the AC of node $i$ as

$\displaystyle A_{i}=s_{i}\left(f_{i}-\frac{1}{N}\right)\,,$ (3)

where the factor $1/N$ accounts for the case where node $i$ is selected as the
initial trigger of the avalanche. Fig. 4 shows that the avalanche size can be
reduced by reinforcing nodes with high ACs.

We define the performance measure $R_{m}$ of each avalanche mitigation
strategy as

$R_{m}=\int_{0}^{1}\varphi(r)dr\simeq\sum_{i}\frac{\varphi(r_{i})+\varphi(r_{i+1})}{2}\Delta
r\,,$ (4)

where

$\varphi(r)=\frac{\bar{s}(r)-1/N}{\bar{s}(0)-1/N}\,.$ (5)

Here $r$ denotes the fraction of reinforced nodes, and $\bar{s}(r)$ is the
mean avalanche fraction; that is, $\sum_{i}s_{i}(r)/N$ when a fraction $r$ of
nodes are reinforced according to the avalanche mitigation strategy. The term
$1/N$ is needed to exclude initial failure from $s$. Consequently,
$\varphi(0)=1$, $\varphi(1)=0$, and $0\leq R_{m}\leq 1$. As $R_{m}$ decreases,
the avalanche mitigation strategy becomes more efficient. To calculate
$R_{m}$, we use $\Delta r=0.01$ for small networks ($N\leq 1000$) and $\Delta
r=0.1$ for large networks ($N>1000$) to avoid excessive computational cost.

Fig. 4 shows the $\varphi(r)$ values of various avalanche mitigation
strategies in the SHK network. The area under each curve represents the
performance measure $R_{m}$. Reinforcing nodes with high network centralities,
such as degree centrality, eigenvector centrality, and BC, blocks the
avalanche dynamics more effectively than random reinforcement. The strategy
based on an avalanche fraction $s$ is effective in a small $r$ range. The
strategy based on the failure fraction $f$ is even less effective than random
reinforcement for small $r$. However, for sufficiently large $r$, strategies
based on $s$ and $f$ both become more effective than other strategies except
that based on the AC, and the cascading failure is reduced to zero at a
certain value $r_{c}$. This value corresponds to the threshold of herd
immunity in epidemics. The strategy based on the AC that takes into account
both the avalanche and failure fractions becomes the most effective strategy
among all the strategies in the broad range $r<r_{c}$.

Table 1 lists the performance measure $R_{m}$ for various avalanche mitigation
strategies in the SHK networks of various sizes and real-world electrical
power grids. The AC-based strategy exhibits fairly good performance compared
with all other methods. We remark that the BC-based strategy is also highly
effective. However, it is not scalable because of the logarithmic correction
of the computational complexity $O(N^{2}\log N)$. Thus, strategies using
scalable centrality measures such as degree centrality and eigenvector
centrality can be employed in the GNN.

Figure 4: Performance of avalanche mitigation strategies in the SHK network of size $N=1000$. Degree, eigenvector centrality (EC), betweenness centrality (BC), avalanche fraction ($s$), failure fraction ($f$), avalanche centrality (AC), and GNN estimation of avalanche centrality (AC-GNN) are employed for avalanche mitigation strategies. Nodes are reinforced in descending order of the corresponding centrality measure. The area under each curve is the performance measure $R_{m}$. Table 1:  Performance measure $R_{m}$ of mitigation strategies: random reinforcement, reinforcement in descending order of degree, eigenvector centrality (EC), betweenness centrality (BC), avalanche size, failure fraction, avalanche centrality (AC), and GNN prediction of AC. Smaller values indicate better mitigation. The most efficient mitigation strategy is marked by bold. Network | Random | Degree | EC | BC | Avalanche fraction | Failure fraction | AC | AC (GNN-pred)
---|---|---|---|---|---|---|---|---
SHK ($N=1000$) | 0.4593 | 0.1923 | 0.2767 | 0.1763 | 0.1691 | 0.2697 | 0.1338 | 0.1467
SHK ($N=2000$) | 0.4547 | 0.1733 | 0.2582 | 0.1624 | 0.1560 | 0.3102 | 0.1273 | 0.1377
SHK ($N=4000$) | 0.4490 | 0.1508 | 0.2423 | 0.1415 | 0.1360 | 0.3569 | 0.1126 | 0.1234
SHK ($N=8000$) | 0.4458 | 0.1322 | 0.2199 | 0.1248 | 0.1202 | 0.4431 | 0.1009 | 0.1107
Spain | 0.4887 | 0.3347 | 0.3604 | 0.2648 | 0.2815 | 0.2410 | 0.2320 | 0.2347
France | 0.4855 | 0.3727 | 0.4593 | 0.2477 | 0.2633 | 0.2171 | 0.2177 | 0.2306
UK | 0.4619 | 0.4220 | 0.2720 | 0.3607 | 0.3203 | 0.2715 | 0.2787 | 0.4227

###  II.3 Graph neural network (GNN)

The AC can be used for effective avalanche suppression; however, it has high
computational complexity as high as $O(N^{3}\log N)$. Thus, it cannot be
calculated directly in large networks. To overcome this problem, we apply the
GNN, a deep learning algorithm that can be used for graph-structured data
Scarselli  _et al._ (2008). We constructed a GNN structure applicable to
networks of different sizes. This GNN was trained using the ACs obtained by
simulations of small networks. Then, the AC of each node in large networks,
where simulations using the ML model are not feasible, can be predicted.

We used the SHK network, a synthetic network mimicking a power grid, to
simulate avalanche failures. The network size was selected at random from 100
to 999 in uniform increments. The dataset consists of $10^{4}$ data.

The GNN is a type of space-based convolutional GNNWu  _et al._ (2020) composed
of graph isomorphism network (GIN) layers Xu  _et al._ (2019), as shown in
Fig. 5. In the GIN layers, the hidden feature $y_{i}$ of node $i$ is updated
according to its own value and those of the nearest–neighbor nodes of $i$,
that is, $\\{y_{j}\\}$, where $j\in$ n.n. of $i$ as

$y_{i}^{\prime}=h_{\Theta}\left((1+\epsilon)y_{i}+\sum_{j~{}\in~{}{\rm{n.n.~{}of~{}}}i}y_{j}\right),$
(6)

where $\epsilon$ is a constant and $h_{\Theta}$ is taken to be a two-layer
perceptron for the trainable parameter $\Theta$. The GNN also has a batch
normalization layer Ioffe and Szegedy (2015) and a rectified linear unit
(ReLU) activation function between subsequently GIN layers.

Figure 5: Structure of GNN. Only the adjacency matrix of the network is input,
and the features of all nodes are initially constant. Batch normalization and
ReLU activation are applied between each pair of adjacent GIN layers, but they
are omitted from the diagram for simplicity. A short skip connection is
applied between each pair of adjacent GIN layers, and an elementwise average
is obtained before the regressor. The regressor is implemented as a
single–layer perceptron followed by a sigmoid function for normalization.

The ML model is based on the BC, which contains a global connection
information of network. Therefore, the GNN must consider not only local
information around a given node, but also the global information of the entire
network. As shown in Eq (6), the GIN layer is used for one-hop calculation.
$y_{i}$ is updated locally. To address the global features at longer hopping
distances, many layers should be stacked. This deep structure may cause the
so-called vanishing gradient problem He  _et al._ (2016). To mitigate this
problem and to take into account the effect of nodes at various hopping
distances Abu-El-Haija  _et al._ (2019); Veit, Wilber, and Belongie (2016),
the short skip connection algorithm Simonyan and Zisserman (2015) is applied
for every pair of adjacent layers. Table. 2 lists the hyperparameters used to
train the GNN.

Table 2:  Hyper parameters used to train GNN. name | node embedding dimension | batch normalization momentum | optimizer | learning rate | loss | L2 regularization
---|---|---|---|---|---|---
value | 128 | 0.1 | RMSprop | $10^{-3}$ | MAE | $10^{-5}$

To perform avalanche mitigation in the GNN framework, it is necessary to rank
the ACs of the nodes. This information makes it possible to handle all
reinforcement states because one can choose a node set depending on the
available reinforcement resources. Because the AC distribution is highly
skewed, simple node feature regression is not an appropriate approach.
Instead, we applied the quantile transformation to the ACs and then the min-
max scale to the obtained values for normalization. The validation and test
datasets were scaled using the scaler fitted to the training dataset.

Let $z_{i}$ is an observable, for example, the AC of node $i$. Rank($i$) is
defined according to the relative size of $z_{i}$ among others. $\sigma(i)$ is
the estimated rank of $z_{i}$ using the GNN. A pair of observables
$\left(z_{i},z_{j}\right)$ is concordant if the order of their ranks is
correctly estimated: for the $\mathrm{Rank}(i)<\mathrm{Rank}(j)$,
$\sigma(i)<\sigma(j)$. The pair is discordant, otherwise.

A well-known metric for ordinal association is the Kendall rank correlation or
Kendall’s tau Kendall (1938), which is defined as

$\tau=\frac{n_{c}-n_{d}}{n_{c}+n_{d}},$ (7)

where $n_{c}$ and $n_{d}$ denote the numbers of concordant and discordant
pairs, respectively. This Kendall’s tau is expressed explicitly as

$\tau=\frac{2}{N(N-1)}\sum_{i<j}\mathrm{sign}\left(z_{i}-z_{j}\right)\mathrm{sign}\left(\sigma(j)-\sigma(i)\right)\,.$
(8)

In the calculation of Kendall’s tau, the incorrect ranking of two values that
differ greatly receives the same penalty as the incorrect ranking of two
similar values. However, the former error is likely to be disproportionately
detrimental (in avalanche mitigation, for example). This problem may be more
important for highly heterogeneous data.

Figure 6: The cumulative fraction $C_{\sigma}(n)$ of AC as a function of $n$
on an SHK network of size $N=1000$. The mean cumulative fraction $\langle
C_{\sigma}\rangle$ is the area under the cumulative fraction $C_{\sigma}(n)$
curve. A large $\langle C_{\sigma}\rangle$ means that the ranking estimation
performs well. Red solid curve is obtained by taking the estimated rank
$\sigma$ as the graph degree, i.e., the number of connections, of each node.
The estimated curve lies between those of the descending order (best case) and
ascending order (worst case).

Accordingly, we propose a new quantity, the cumulative fraction
$C_{\sigma}(n)\equiv\sum_{i=1}^{n}z_{\sigma(i)}/\sum_{j=1}^{N}z_{j}$. This
represents the contribution of $n$ the most highly estimated $z_{i}$s in the
sum of all values in the data. The average of $C_{\sigma}(n)$ over $n$
(denoted as $\langle C_{\sigma}\rangle$) represents the overall performance of
the $\sigma$ estimation.

$\displaystyle\langle C_{\sigma}\rangle$
$\displaystyle=\sum_{n=1}^{N}\frac{1}{N}\sum_{i=1}^{n}\frac{z_{\sigma(i)}}{\sum_{j=1}^{N}z_{j}}$
$\displaystyle=\sum_{i=1}^{N}\frac{N-i+1}{N}\frac{z_{\sigma(i)}}{\sum_{j=1}^{N}z_{j}}\,.$
(9)

Fig. 6 shows the behavior of $C_{\sigma}(n)$ as a function of $n$.
$C_{\sigma}(n)$ is always between the two extremes: the descending and the
ascending orders of Rank($i$) $(i=1,\cdots,N)$. Accordingly, we define the
normalized $\langle C_{\sigma}\rangle$ (denoted as $\langle
C_{\sigma,N}\rangle$) as

$\langle C_{\sigma,N}\rangle\equiv\frac{\langle C_{\sigma,N}\rangle-\langle
C_{\mathrm{ascending}}\rangle}{\langle C_{\mathrm{descending}}\rangle-\langle
C_{\mathrm{ascending}}\rangle}.$ (10)

Then, $0\leq\langle C_{\sigma,N}\rangle\leq 1$.

We also use $R^{2}$ score to evaluate the prediction of $z_{i}$ value, which
is defined as follows Glantz and Slinker (1990):

$R^{2}\equiv
1-\frac{\sum_{i=1}^{N}(z_{i}-\hat{z}_{i})^{2}}{\sum_{i=1}^{N}(z_{i}-\bar{z})^{2}},$
(11)

where $\hat{z}_{i}$ is the estimated value of $z_{i}$ and $\bar{z}$ is the
average of all $z_{i}$.

Fig. 7 shows the performances of the $R^{2}$ score, $\langle
C_{\sigma,N}\rangle$, and Kendall’s tau for the prediction of $z_{i}$ obtained
by the GNN. Although the $R^{2}$ score decreases rapidly as the network size
increases ($N=1000-8000$), the $\langle C_{\sigma,N}\rangle$ and Kendall’s tau
values of the GNN do not change significantly as $N$ increases. This result
suggests that $R^{2}$ score fails to predict the AC values of networks larger
than those in the training dataset. As shown in Fig. 4 and Table. 1, node
reinforcement according to the AC predicted by the GNN is also effective for
suppressing avalanches in the SHK model as well as in several real-world
electric power grids of Spain, France, and UK. We remark that the GNN was
trained on the SHK model, however, the real-world electrical power grids were
not trained.

Figure 7: Performance of the GNN prediction in large networks in the range
$N=1000-8000$, which is trained in various network sizes from 100 to 999. All
the results are averaged over 4 different GNNs with different random seed. (a)
$R^{2}$ scores for different system sizes: the $R^{2}$ score is not
appropriate to predict the performance because the AC values are distributed
very skewed. (b) and (c) $\langle C_{\sigma,N}\rangle$ and Kendall’s tau for
different sizes, respectively. (Blue, dark bars) the GNN predicts ranks of ACs
using a quantile scaled dataset. (Orange, light bars) the GNN predicts values
of ACs using a raw dataset. Rank prediction using the quantile scaling marked
in blue is better and more consistent than value prediction marked in orange.
The representation by $\langle C_{\sigma,N}\rangle$ in (b) is more appropriate
compared with that by the Kendall’s Tau in (c).

##  III Conclusion

In summary, we introduced the concept of the avalanche centrality (AC) of each
node in networks and showed that reinforcing nodes in descending order of AC
is effective for suppressing the nonlocal avalanche propagation in electrical
power grids. However, the calculation of the AC has a high computational cost.
Therefore, we employed a GNN to address this problem. We trained the GNN with
the ACs of small networks and showed that it can be used for larger networks
where the direct calculation of the ACs is not feasible. The GNN predicts the
descending order of AC in large networks, allowing the effective suppression
of avalanche failures in electrical power grids. Conventionally, the Kendall’s
tau was used as a performance measure of ranking estimation. However, this
measure contains intrinsic drawback: the incorrect estimation of the order of
two values that differ greatly is penalized similarly to that of the ranking
of two slightly different values. To overcome such a limitation, here, we
introduced a new performance measure denoted as $\langle C_{\sigma}\rangle$
for ranking prediction. This new performance measure successfully reflects the
importance of the ordering of two values that differ greatly as shown in Fig
6.

In network epidemiology, the microscopic Markov chain approximation (MMCA)
Gómez  _et al._ (2010, 2011); Matamalas, Gómez, and Arenas (2020) was
introduced to bypass the computationally cumbersome Monte Carlo simulation.
The result of MMCA was then used to calculate epidemic prevalence or to
formulate vaccination strategies Matamalas, Arenas, and Gómez (2018); Jhun
(2021a, b). This method, however, cannot be used to approximate avalanche
dynamics that propagates nonlocally, for example, in the ML model. Therefore,
we think the GNN introduced here may replace the role of the MMCA in the
analysis of nonlocal avalanche dynamics and the development of mitigation
strategy, for example, for the blackout in electrical power grid.

## Data and code availability

The data and code that support the findings of this study are openly available
at https://github.com/CNRC-NAJU/Prediction-and-mitigation-of-nonlocal-
cascading-failures-using-graph-neural-networks

###### Acknowledgements.

This research was supported by the NRF, Grant No. NRF-2014R1A3A2069005 and
KENTECH Research Grant No. KRG2021-01-007 (BK).

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# Conditional optimal sets and the quantization coefficients for some uniform
distributions

1Evans Nyanney  ,  2Megha Pandey  and  3Mrinal Kanti Roychowdhury
{1evans.nyanney01<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Bucklew and Wise (1982) showed that the quantization dimension of an
absolutely continuous probability measure on a given Euclidean space is
constant and equals the Euclidean dimension of the space, and the quantization
coefficient exists as a finite positive number. By giving different examples,
in this paper, we have shown that the quantization coefficients for absolutely
continuous probability measures defined on the same Euclidean space can be
different. We have taken uniform distribution as a prototype of an absolutely
continuous probability measure. In addition, we have also calculated the
conditional optimal sets of $n$-points and the $n$th conditional quantization
errors for the uniform distributions in constrained and unconstrained
scenarios.

###### Key words and phrases:

Probability measure, conditional quantization, optimal sets of $n$-points,
quantization coefficient

###### 2010 Mathematics Subject Classification:

60Exx, 94A34.

##  1\. Introduction

The process of transformation of a continuous-valued signal into a discrete-
valued one is called ‘quantization’. It has broad applications in engineering
and technology. We refer to [GG, GN, Z2] for surveys on the subject and
comprehensive lists of references to the literature; see also [AW, GKL, GL1,
Z1].

######  Definition 1.1.

Let $P$ be a Borel probability measure on $\mathbb{R}^{k}$ equipped with a
metric $d$ induced by a norm $\|\cdot\|$ on $\mathbb{R}^{k}$. Let $S$ be a
nonempty closed subset of $\mathbb{R}^{k}$. Let $\beta\subset\mathbb{R}^{k}$
be given with $\text{card}(\beta)=\ell$ for some $\ell\in\mathbb{N}$. Then,
for $n\in\mathbb{N}$ with $n\geq\ell$, the $n$th conditional constrained
quantization error for $P$ with respect to the constraint $S$ and the
conditional set $\beta$, is defined by

$V_{n}:=V_{n}(P)=\inf_{\alpha}\Big{\\{}\int\mathop{\min}\limits_{a\in\alpha\cup\beta}d(x,a)^{2}dP(x):\alpha\subseteq
S,~{}0\leq\text{card}(\alpha)\leq n-\ell\Big{\\}},$ (1)

where $\text{card}(A)$ represents the cardinality of the set $A$.

######  Definition 1.2.

A set $\alpha\cup\beta$, where $\alpha\subseteq S$ and
$P(M(b|\alpha\cup\beta))>0$ for $b\in\beta$, for which the infimum in (1)
exists and contains no less than $\ell$ elements, and no more than $n$
elements is called a conditional constrained optimal set of $n$-points for $P$
with respect to the constraint $S$ and the conditional set $\beta$.

We assume that $\int d(x,0)^{2}dP(x)<\infty$ to make sure that the infimum in
(1) exists (see [PR1]). For a finite set $\gamma\subset\mathbb{R}^{2}$ and
$a\in\gamma$, by $M(a|\gamma)$ we denote the set of all elements in
$\mathbb{R}^{2}$ which are nearest to $a$ among all the elements in $\gamma$,
i.e.,
$M(a|\gamma)=\\{x\in\mathbb{R}^{2}:d(x,a)=\mathop{\min}\limits_{b\in\gamma}d(x,b)\\}.$
$M(a|\gamma)$ is called the Voronoi region in $\mathbb{R}^{2}$ generated by
$a\in\gamma$.

Write $V_{\infty}(P):=\mathop{\lim}\limits_{n\to\infty}V_{n}(P)$. The numbers

$\underline{D}(P):=\liminf_{n\to\infty}\frac{2\log
n}{-\log(V_{n}(P)-V_{\infty}(P))}\text{ and
}\overline{D}(P):=\limsup_{n\to\infty}\frac{2\log
n}{-\log(V_{n}(P)-V_{\infty}(P))},$ (2)

are called the conditional lower and the conditional upper constrained
quantization dimensions of the probability measure $P$, respectively. If
$\underline{D}(P)=\overline{D}(P)$, the common value is called the conditional
constrained quantization dimension of $P$ and is denoted by $D(P)$. For any
$\kappa>0$, the two numbers
$\liminf_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$ and
$\limsup_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$ are, respectively,
called the $\kappa$-dimensional conditional lower and conditional upper
constrained quantization coefficients for $P$. If both of them are equal, then
it is called the $\kappa$-dimensional conditional constrained quantization
coefficient for $P$, and is denoted by
$\lim_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$.

If there is no conditional set, then by the $n$th constrained quantization
error for $P$ with respect to the constraint $S\subseteq\mathbb{R}^{k}$, it is
meant that

$V_{n}:=V_{n}(P)=\inf\Big{\\{}\int\mathop{\min}\limits_{a\in\alpha}d(x,a)^{2}dP(x):\alpha\subseteq
S\text{ and }1\leq\text{card}(\alpha)\leq n\Big{\\}},$ (3)

and then the numbers $D(P)$ and
$\lim_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$, if they exist, are
called the constrained quantization dimension and the $\kappa$-dimensional
constrained quantization coefficient for $P$, respectively. A set
$\alpha\subseteq S$ for which the infimum in (3) exists is called a
constrained optimal set of $n$-points for $P$.

If there is no constraint, i.e., if $S=\mathbb{R}^{k}$, then by the $n$th
conditional unconstrained quantization error with respect to the conditional
set $\beta$, it is meant that

$V_{n}:=V_{n}(P)=\inf_{\alpha}\Big{\\{}\int\mathop{\min}\limits_{a\in\alpha\cup\beta}d(x,a)^{2}dP(x):\alpha\subseteq\mathbb{R}^{k},~{}0\leq\text{card}(\alpha)\leq
n-\ell\Big{\\}},$ (4)

and then the numbers $D(P)$ and
$\lim_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$, if they exist, are
called the conditional unconstrained quantization dimension and the
$\kappa$-dimensional conditional unconstrained quantization coefficient for
$P$, respectively. A set $\alpha\subseteq S$ for which the infimum in (3)
exists is called a conditional unconstrained optimal set of $n$-points for
$P$.

If there is no constraint and no conditional set, then by the $n$th
unconditional quantization error it is meant that

$V_{n}:=V_{n}(P)=\inf\Big{\\{}\int\mathop{\min}\limits_{a\in\alpha}d(x,a)^{2}dP(x):\alpha\subseteq\mathbb{R}^{k},~{}1\leq\text{card}(\alpha)\leq
n\Big{\\}},$ (5)

and then the numbers $D(P)$ and
$\lim_{n}n^{\frac{2}{\kappa}}(V_{n}(P)-V_{\infty}(P))$, if they exist, are
called the unconstrained quantization dimension and the $\kappa$-dimensional
unconstrained quantization coefficient for $P$, respectively. A set
$\alpha\subseteq S$ for which the infimum in (3) exists is called an optimal
set of $n$-means for $P$. It is known that if the support of $P$ contains
infinitely many elements than an optimal set of $n$-means contains exactly $n$
elements, and $V_{\infty}=\lim_{n\to\infty}V_{n}=0$.

Constrained quantization and conditional quantization have recently been
introduced by Pandey and Roychowdhury (see [PR1, PR2, PR4]). After the
introduction of constrained quantization, the quantization theory has now two
classifications: constrained quantization and unconstrained quantization.
Unconstrained quantization is traditionally known as quantization. Thus, the
$n$th unconditional quantization error, given by (5), will traditionally be
refereed to as $n$th quantization error. Likewise, unconstrained quantization
dimension and the $\kappa$-dimensional unconstrained quantization coefficient
for $P$ will be refereed to as quantization dimension and the
$\kappa$-dimensional quantization coefficient for $P$, respectively. For some
other papers in the direction of constrained quantization and conditional
quantization one can see [HNPR, BCDR, BCDRV, OR, PR3, PR5]. For unconstrained
quantization, one can see [DFG, DR, GG, GL, GL1, GL2, GL3, GN, KNZ, P, P1, R1,
R2, R3, Z1, Z2] and the references therein.

Bucklew and Wise (1982) showed that the quantization dimension of an
absolutely continuous probability measure on a given Euclidean space is
constant and equals the Euclidean dimension of the space where the support of
the probability measure is defined, and the quantization coefficient exists as
a finite positive number. By giving different examples, in this paper, we have
shown that the quantization coefficients for absolutely continuous probability
measures defined in the same Euclidean space can be different. We have taken
uniform distribution as a prototype of an absolutely continuous probability
measure. To determine the quantization coefficient for a Borel probability
measure $P$, we sometimes have calculated the conditional quantization
coefficient for $P$ as the quantization coefficient does not depend on the
conditional set. In addition, we have also calculated the conditional optimal
sets of $n$-points and the $n$th conditional quantization errors with respect
to some given conditional sets in constrained and unconstrained scenarios.

###  1.3. Delineation

In Section 2 we give the basic preliminaries. In Section 3, for a uniform
distribution on a line segment taking different conditional sets we have
calculated the conditional optimal sets of $n$-points and the $n$th
conditional quantization errors. Then, for each conditional set we have
calculated the quantization coefficient, and see that the quantization
coefficient does not depend on the conditional set, but depends on the length
of the line segment. In Section 4, we have calculated the conditional optimal
sets of $n$-points, the $n$th conditional quantization errors, the conditional
quantization dimension, and the conditional quantization coefficient, in
constrained scenario, for a uniform distribution defined on a circle of radius
$r$ with respect to a given conditional set and a constraint. In addition, for
the same probability distribution we have investigated the optimal sets of
$n$-means and the $n$th quantization errors, and the quantization coefficient
in unconstrained scenario. From the work in this section, we see that the
quantization coefficient for a uniform distribution defined on a circle
depends on the radius of the circle. In Section 5, we have calculated the
conditional optimal sets of $n$-points, the $n$th conditional quantization
errors, and the conditional quantization coefficient for a uniform
distribution defined on the boundary of a regular polygon which is inscribed
in a circle of radius $r$ with respect to a given conditional set. From the
work in this section, we see that the quantization coefficient for a uniform
distribution defined on the boundary of a regular $m$-sided polygon depends on
both the number of sides of the polygon and the length of the sides.

##  2\. Preliminaries

For any two elements $(a,b)$ and $(c,d)$ in $\mathbb{R}^{2}$, we write

$\rho((a,b),(c,d)):=(a-c)^{2}+(b-d)^{2},$

which gives the squared Euclidean distance between the two elements $(a,b)$
and $(c,d)$. Let $p$ and $q$ be two elements that belong to an optimal set of
$n$-points for some positive integer $n$, and let $e$ be an element on the
boundary of the Voronoi regions of the elements $p$ and $q$. Since the
boundary of the Voronoi regions of any two elements is the perpendicular
bisector of the line segment joining the two elements, we have

$\rho(p,e)-\rho(q,e)=0.$

We call such an equation a canonical equation.

Let $P$ be a Borel probability measure on $\mathbb{R}$ which is uniform on its
support the closed interval $[a,b]$. Then, the probability density function
$f$ for $P$ is given by

$\displaystyle f(x)=\left\\{\begin{array}[]{cc}\frac{1}{b-a}&\text{ if }a\leq
x\leq b,\\\ 0&\text{ otherwise}.\end{array}\right.$ (8)

Hence, we have $dP(x)=P(dx)=f(x)dx$ for any $x\in\mathbb{R}$.

Let us now state the following proposition. For the details of the proof see
[BCDR].

######  Proposition 2.1.

(see [BCDR]) Let $P$ be a uniform distribution on the closed interval $[a,b]$
and $c,d\in[a,b]$ be such that $a<c<d<b$. For $n\in\mathbb{N}$ with $n\geq 2$,
let $\alpha_{n}$ be a conditional unconstrained optimal set of $n$-points for
$P$ with respect to the conditional set $\beta=\\{c,d\\}$ such that
$\alpha_{n}$ contains $k$ elements from the closed interval $[a,c]$, $\ell$
elements from the closed interval $[c,d],$ and $m$ elements from the closed
interval $[d,b]$ for some $k,\ell,m\in\mathbb{N}$ with $k,m\geq 1$ and
$\ell\geq 2$. Then, $k+\ell+m=n+2$,

$\displaystyle\alpha_{n}\cap[a,c]$
$\displaystyle=\Big{\\{}a+\frac{(2j-1)(c-a)}{2k-1}:1\leq j\leq k\Big{\\}},$
$\displaystyle\alpha_{n}\cap[c,d]$
$\displaystyle=\Big{\\{}c+\frac{j-1}{\ell-1}(d-c):1\leq
j\leq\ell\Big{\\}},\text{ and }$ $\displaystyle\alpha_{n}\cap[d,b]$
$\displaystyle=\Big{\\{}d+\frac{2(j-1)(b-d)}{2m-1}:1\leq j\leq m\Big{\\}}$

with the conditional unconstrained quantization error

$V_{n}:=V_{k,\ell,m}(P)=\frac{1}{3(b-a)}\Big{(}\frac{(c-a)^{3}}{(2k-1)^{2}}+\frac{1}{4}\frac{(d-c)^{3}}{(\ell-1)^{2}}+\frac{(b-d)^{3}}{(2m-1)^{2}}\Big{)}.$

######  Remark 2.2.

If nothing is specified, by conditional optimal sets of $n$-points and the
conditional quantization errors, it is meant the conditional unconstrained
optimal sets of $n$-points and the conditional unconstrained quantization
errors, respectively.

The following theorem is known.

######  Theorem 2.3.

(see [PR4]) In both constrained and unconstrained quantization, the lower and
upper quantization dimensions and the lower and upper quantization
coefficients for a Borel probability measure do not depend on the conditional
set.

######  Remark 2.4.

Given that the underlying spaces for all considered probability measures $P$
in this work are one dimensional, their quantization dimensions are given by
$D(P)=1$ (see [BW]). Hence, in the sequel we are mostly interested to
calculate the quantization coefficients for different uniform distributions,
though in some cases we have also calculated the conditional optimal sets of
$n$-points and the $n$th conditional quantization errors in constrained and
unconstrained scenarios.

In the following sections we give the main results of the paper.

##  3\. Conditional optimal sets of $n$-points and the quantization
coefficients for uniform distributions on line segments

Without any loss of generality we can assume the line segment as a closed
interval $[a,b]$, where $0<a<b<+\infty$. Let $P$ be the uniform distribution
defined on the closed interval $[a,b]$. Then, the probability density function
$f$ for $P$ is given by (8). Let us now give the following theorem.

######  Theorem 3.1.

Let $P$ be the uniform distribution on the line segment joining $a$ and $b$,
where $a,b\in\mathbb{R}$ with $a<b$. Then we have:

$(i)$ the conditional optimal set of $n$-points with respect to the
conditional set $\beta:=\\{a,b\\}$ is

$\Big{\\{}a+\frac{j-1}{\ell-1}(b-a):1\leq j\leq n\Big{\\}}\text{ with
conditional quantization error }V_{n}=\frac{(b-a)^{2}}{12(n-1)^{2}};$

$(ii)$ the conditional optimal set of $n$-points with respect to the
conditional set $\beta:=\\{a\\}$ is

$\Big{\\{}a+\frac{2(j-1)(b-a)}{2n-1}:1\leq j\leq n\Big{\\}}\text{ with
conditional quantization error }V_{n}=\frac{(b-a)^{2}}{3(2n-1)^{2}};$

$(iii)$ the conditional optimal set of $n$-points with respect to the
conditional set $\beta:=\\{b\\}$ is

$\Big{\\{}a+\frac{(2j-1)(b-a)}{2n-1}:1\leq j\leq n\Big{\\}}\text{ with
conditional quantization error }V_{n}=\frac{(b-a)^{2}}{3(2n-1)^{2}}.$

###### Proof.

The proof of the theorem follows by Proposition 2.1 upon substitutions $c=a$
and $d=b$, $c=d=a$, and $c=d=b$, respectively. ∎

######  Theorem 3.2.

Let $P$ be the uniform distribution on the line segment joining $a$ and $b$,
where $a,b\in\mathbb{R}$ with $a<b$. Then, the conditional quantization
coefficients for $P$ with respect to the conditional sets $\\{a,b\\}$,
$\\{a\\}$, and $\\{b\\}$ exist as finite positive numbers and each equals
$\frac{(a-b)^{2}}{12}$.

###### Proof.

By Theorem 3.1$(i)$, we obtain the $n$th conditional quantization error for
the uniform distribution $P$ with respect to the conditional set
$\beta:=\\{a,b\\}$ as

$V_{n}=\frac{(b-a)^{2}}{12(n-1)^{2}}.$

Then,

$V_{\infty}=\mathop{\lim}\limits_{n\to\infty}V_{n}=0\text{ yielding
}\lim\limits_{n\to\infty}n^{2}(V_{n}-V_{\infty})=\frac{(a-b)^{2}}{12}.$

Similarly, if the conditional set is $\beta:=\\{a\\}$ or $\beta:=\\{b\\}$, by
$(ii)$ and $(iii$) in Theorem 3.1, we obtain
$\lim\limits_{n\to\infty}n^{2}(V_{n}-V_{\infty})=\frac{(a-b)^{2}}{12}$. Thus,
the proof of the theorem is complete. ∎

######  Remark 3.3.

By Theorem 2.3 and Theorem 3.2, we see that the quantization coefficient for
the uniform distribution $P$ on a line segment depends on the length of the
line segment, and does not depend on the conditional sets.

##  4\. Optimal sets of $n$-points and the quantization coefficients for the
uniform distributions on the circles

In this section, we have two subsections. In the first subsection, we
calculate the conditional constrained optimal sets of $n$-points and the $n$th
conditional constrained quantization errors, the conditional constrained
quantization dimension, and the conditional constrained quantization
coefficient for a uniform distribution $P$ defined on a circle of radius $r$
with respect to a given conditional set and a constraint. In the second
subsection, for the same probability distribution, we investigate the optimal
sets of $n$-means and the $n$th quantization errors, and the quantization
coefficient in unconstrained scenario.

Let $L$ be the circle of radius $r$. Without any loss of generality, we can
take the equation of the circle as $x_{1}^{2}+x_{2}^{2}=r^{2}$, i.e., the
parametric equations of the circle is given by
$L:=\\{(x_{1},x_{2}):x_{1}=r\cos\theta,\,x_{2}=r\sin\theta\text{ for
}0\leq\theta\leq 2\pi\\}.$ Notice that any point on the circle can be given by
$(r\cos\theta,r\sin\theta)$, which will be identified as $\theta$, where
$0\leq\theta\leq 2\pi$. Let the positive direction of the $x_{1}$-axis cut the
circle at the point $A$, i.e., $A$ is represented by the parametric value
$\theta=0$. Let $s$ be the distance of a point on $L$ along the arc starting
from the point $A$ in the counterclockwise direction. Then,

$ds=\sqrt{\Big{(}\frac{dx_{1}}{d\theta}\Big{)}^{2}+\Big{(}\frac{dx_{2}}{d\theta}\Big{)}^{2}}\,d\theta=rd\theta.$

Then, the probability density function (pdf) $f(x_{1},x_{2})$ for $P$ is given
by

$f(x_{1},x_{2})=\left\\{\begin{array}[]{ccc}\frac{1}{2\pi r}&\text{ if
}(x_{1},x_{2})\in L,\\\ 0&\text{ otherwise}.\end{array}\right.$

Thus, we have $dP(s)=P(ds)=f(x_{1},x_{2})ds=\frac{1}{2\pi}d\theta$. Moreover,
we know that if $\hat{\theta}$ radians is the central angle subtended by an
arc of length $S$ of the circle, then $S=r\hat{\theta}$, and

$P(S)=\int_{S}dP(s)=\frac{1}{2\pi}\int_{S}d\theta=\frac{\hat{\theta}}{2\pi}.$

###  4.1. Conditional quantization in constrained scenario

In this subsection to investigate the conditional quantization in constrained
scenario for the uniform distribution $P$ on the circle $L$, we take the
circle $L$ as the constraint and the set $\\{(r,0)\\}$ as the conditional set.
Let us define a function

$T:L\to[0,2\pi r]\text{ such that
}T(\theta):=T((r\cos\theta,r\sin\theta))=r\theta,$

where $0\leq\theta\leq 2\pi$. Then, notice that
$T:L\setminus\\{(r,0)\\}\to(0,2\pi r)$ is a bijective function. Let $Q$ be the
image measure of $P$ under the function $T$, i.e., $Q=TP$ such that for any
Borel subset $A\subseteq[0,2\pi r]$, we have

$Q(A)=TP(A)=P(T^{-1}(A)).$

######  Lemma 4.1.1.

The image measure $Q$ is a uniform distribution on $[0,2\pi r]$.

###### Proof.

Since $P$ is a uniform distribution on $L$, we can assume that $P$ is also a
uniform distribution on $L\setminus\\{(r,0)\\}$, as the deletion, or addition,
of a finite number of points from, or with, the support of a continuous
probability measure does not change the distribution. Take any
$[c,d]\subseteq(0,2\pi r)$, where $0<c<d<2\pi r$. Since $T$ is a bijection,
there exist $\theta_{1}$, $\theta_{2}$, where $0<\theta_{1}<\theta_{2}<2\pi$,
such that $T(\theta_{1})=r\theta_{1}=c$ and $T(\theta_{2})=r\theta_{2}=d$.
Then,

$Q([c,d])=P(T^{-1}([c,d]))=P(\\{(r\cos\theta,r\sin\theta):\theta_{1}\leq\theta\leq\theta_{2}\\})=\frac{\theta_{2}-\theta_{1}}{2\pi}=\frac{r\theta_{2}-r\theta_{1}}{2\pi
r}=\frac{d-c}{2\pi r}.$

Notice $Q([c,d])=\lambda([c,d])$, where $\lambda$ is the normalized Lebesgue
measure on $(0,2\pi r)$. Hence, we can conclude that $Q$ is a uniform
distribution on $(0,2\pi r$), i.e., $Q$ is a uniform distribution on $[0,2\pi
r]$. Thus, the lemma is obtained. ∎

######  Notation 4.1.2.

For any two elements $c,d\in[0,2\pi r]$, by $\ell(\\{c,d\\})$ it is meant the
distance between the two elements $c,d$, i.e., $\ell(\\{c,d\\})=d-c$.
Similarly, for any two elements $\theta_{1},\theta_{2}\in L$ with
$\theta_{1}<\theta_{2}$, by $\ell(\\{\theta_{1},\theta_{2}\\})$ it is meant
the arc distance between the two elements $\theta_{1}$ and $\theta_{2}$, i.e.,
the length of the arc on $L$ subtended by the angle $\theta_{2}-\theta_{1}$,
i.e., $\ell(\\{\theta_{1},\theta_{2}\\})=r(\theta_{2}-\theta_{2}).$

######  Lemma 4.1.3.

The function $T:L\setminus\\{(r,0)\\}\to(0,2\pi r)$ preserves the distance.

###### Proof.

Take any $c,d\in(0,2\pi r)$ such that $0<c<d<2\pi r$. The lemma will be proved
if we can prove that $\ell(\\{c,d\\})=\ell(T^{-1}(\\{c,d\\}))$. Since
$T:L\setminus\\{(r,0)\\}\to(0,2\pi r)$ is a bijection, there exist
$\theta_{1},\theta_{2}\in L\setminus\\{(r,0)\\}$ such that
$T(\theta_{1})=r\theta_{1}=c$ and $T(\theta_{2})=r\theta_{2}=d$. Then,

$\ell(T^{-1}(\\{c,d\\}))=\ell(\\{\theta_{1},\theta_{2}\\})=r(\theta_{2}-\theta_{1})=d-c=\ell(\\{c,d\\}).$

Thus, the lemma is yielded. ∎

######  Lemma 4.1.4.

The conditional unconstrained optimal set for the uniform distribution $Q$
with respect to the conditional set $\\{0,2\pi r\\}$ is given by

$\Big{\\{}\frac{(j-1)2\pi r}{n-1}:1\leq j\leq n\Big{\\}}\text{ with
conditional unconstrained quantization error
}V_{n}(Q)=\frac{\pi^{2}r^{2}}{3(n-1)^{2}}.$

###### Proof.

The lemma is followed by Proposition 2.1 upon substitution $a=c=0$ and
$b=d=2\pi r$. ∎

######  Remark 4.1.5.

Let $\\{a_{1},a_{2},\cdots,a_{n}\\}$ be a conditional unconstrained optimal
set of $n$-points for $Q$ with respect to the conditional set $\\{0,2\pi
r\\}$, where $a_{1}=0$ and $a_{n}=2\pi r$, Since both $P$ and $Q$ are uniform
distributions, and $Q$ is the image measure of $P$ under the function $T$, and
$T:L\setminus\\{(r,0)\\}\to(0,2\pi r)$ preserves the distance, we can say that
the set $T^{-1}(\\{a_{1},a_{2},\cdots,a_{n}\\})$, i.e., the set
$\\{T^{-1}(a_{j}):1\leq j\leq n-1\\}$ forms a conditional constrained optimal
set of $(n-1)$-points for $P$ with respect to the conditional set
$\\{(r,0)\\}$ and the constraint $L$, as $T^{-1}(0)=T^{-1}(2\pi r)=(r,0)$.

Let us now prove the following theorems, which give the main results in this
subsection.

######  Theorem 4.1.6.

Let $P$ be the uniform distribution on the circle of radius $r$ with center
$(0,0)$. Then, the set
$\\{(r\cos\frac{(j-1)2\pi}{n},r\sin\frac{(j-1)2\pi}{n}):1\leq j\leq n\\}$
forms a conditional constrained optimal set of $n$-points with respect to the
conditional set $\\{(r,0)\\}$ and the constraint $L$ with conditional
constrianed quantization error

$V_{n}=2r^{2}(1-\frac{n}{\pi}\sin\frac{\pi}{n}).$

###### Proof.

By Lemma 4.1.4, we know that the set $\\{\frac{(j-1)2\pi r}{n-1}:1\leq j\leq
n\\}$ forms a conditional unconstraied optimal set of $n$-points for $Q$ with
respect to the conditional set $\\{0,2\pi r\\}$, where $n\geq 2$. Hence, by
Remark 5.3, the set $\\{T^{-1}(\frac{(j-1)2\pi r}{n-1}):1\leq j\leq n-1\\}$
forms a conditional constrained optimal set of $(n-1)$-points for $P$ with
respect to the conditional set $\\{(0,\pi)\\}$ and the constraint $L$, where
$(n-1)\geq 1$. Now, notice that

$\displaystyle\\{T^{-1}(\frac{(j-1)2\pi r}{n-1}):1\leq j\leq
n-1\\}=\\{\frac{(j-1)2\pi}{n-1}:1\leq j\leq n-1\\}$
$\displaystyle=\\{(\cos\frac{(j-1)2\pi}{n-1},\sin\frac{(j-1)2\pi}{n-1}):1\leq
j\leq n-1\\}.$

Hence, replacing $n$ by $n+1$, we deduce that the set
$\\{(r\cos\frac{(j-1)2\pi}{n},r\sin\frac{(j-1)2\pi}{n}):1\leq j\leq n\\}$
forms a conditional constrained optimal set of $n$-points with respect to the
conditional set $\\{(0,\pi)\\}$ and the constraint $L$. Due to rotational
symmetry, we obtain the conditional constrained quantization error as

$\displaystyle V_{n}$ $\displaystyle=n(\text{distortion error contributed by
the element }(r,0))$
$\displaystyle=\frac{n}{2\pi}\int_{-\frac{\pi}{n}}^{\frac{\pi}{n}}\rho((r\cos\theta,r\sin\theta),(r,0))\,d\theta$
$\displaystyle=2r^{2}(1-\frac{n}{\pi}\sin\frac{\pi}{n}).$

Thus, the proof of the theorem is complete. ∎

######  Theorem 4.1.7.

Let $P$ be the uniform distribution on the circle of radius $r$ with center
$(0,0)$. Then, with respect to the conditional set $\\{(r,0)\\}$ and the
constraint $L$, the conditional constrained quantization dimension $D(P)$
exists and equals one, and the conditional constrained quantization
coefficient for $P$ exists as a finite positive number and equals
$\frac{\pi^{2}r^{2}}{3}$, i.e.,
$\lim\limits_{n\to\infty}n^{2}(V_{n}-V_{\infty})=\frac{\pi^{2}r^{2}}{3}.$

###### Proof.

By Theorem 4.1.6, we obtain the $n$th conditional constrained quantization
error for the uniform distribution $P$ with respect to the conditional set
$\\{(r,0)\\}$ and the constraint $L$ as
$V_{n}=2r^{2}(1-\frac{n}{\pi}\sin\frac{\pi}{n}).$ Then,
$V_{\infty}=\mathop{\lim}\limits_{n\to\infty}V_{n}=0$. Hence,

$D(P)=\lim_{n\to\infty}\frac{2\log n}{-\log(V_{n}-V_{\infty})}=1\text{ and
}\lim\limits_{n\to\infty}n^{2}(V_{n}-V_{\infty})=\frac{\pi^{2}r^{2}}{3}.$

Thus, the proof of the theorem is complete. ∎

######  Remark 4.1.8.

By Theorem 2.3, we know that quantization dimension and the quantization
coefficient in constrained and unconstrained cases do not depend on the
conditional set. Thus, by Theorem 4.1.7, we see that constrained quantization
dimension of the uniform distribution $P$ with respect to the constraint $L$
equals one, which is the dimension of the underlying space where the support
of the probability measure is defined, and does not depend on the radius $r$
of the circle. This fact is not true, in general, in constrained quantization,
for example, one can see [PR1, PR3]. However, we see that the constrained
quantization coefficient for the uniform distribution $P$ with respect to the
constraint $L$ depends on the radius $r$ of the circle.

###  4.2. Quantization in unconstrained scenario

In this subsection, we investigate the optimal sets of $n$-means, $n$th
quantization errors, and the quantization coefficient for the uniform
distribution $P$ when there is no constraint and no conditional set. The
following lemma is a generalized version of a similar theorem that appears in
[RR].

######  Theorem 4.2.1.

Let $\alpha_{n}$ be an optimal set of $n$-means for the uniform distribution
$P$ on the circle $x_{1}^{2}+x_{2}^{2}=r^{2}$ for $n\in\mathbb{N}$. Then,

$\alpha_{n}:=\Big{\\{}\frac{nr}{\pi}\Big{(}\sin(\frac{\pi}{n})\cdot\cos((2j-1){\frac{\pi}{n}}),\
\sin(\frac{\pi}{n})\cdot\sin((2j-1){\frac{\pi}{n}})\Big{)}:j=1,2,\cdots,n\Big{\\}}$

forms an optimal set of $n$-means, and the corresponding quantization error is
given by $r^{2}(1-\frac{n^{2}}{\pi^{2}}\sin^{2}\frac{\pi}{n}).$

###### Proof.

Let $\alpha_{n}:=\\{a_{1},a_{2},\cdots,a_{n}\\}$ be an optimal set of
$n$-means for $P$. Let the boundaries of the Voronoi regions of $a_{k}$
intersect $L$ at the points given by the parameters $\theta_{k-1}$ and
$\theta_{k}$ such that $\theta_{k-1}<\theta_{k}$ for $1\leq k\leq n$. Without
any loss of generality, we can assume that $\theta_{0}=0$ and
$\theta_{n}=2\pi$. Since, $P$ is a uniform distribution and the circle $L$ is
rotationally symmetric, without going into the much details of calculations,
we see that

$\theta_{1}-\theta_{0}=\theta_{2}-\theta_{1}=\theta_{3}-\theta_{2}=\cdots=\theta_{n}-\theta_{n-1}=\frac{\theta_{n}-\theta_{0}}{n}=\frac{2\pi}{n}.$

implying $\theta_{k}=\frac{2\pi k}{n}$ for $0\leq k\leq n$. It is well-known
that in unconstrained quantization, the elements in an optimal set are the
conditional expectations in their own Voronoi regions. Hence, for $1\leq k\leq
n$, we have

$\displaystyle a_{k}$
$\displaystyle=\frac{2\pi}{\theta_{k}-\theta_{k-1}}\int_{\theta_{k-1}}^{\theta_{k}}\frac{1}{2\pi}(r\cos\theta,r\sin\theta)d\theta=\frac{r}{\theta_{k}-\theta_{k-1}}\left(\sin\theta_{k}-\sin\theta_{k-1},\,\cos\theta_{k-1}-\cos\theta_{k}\right),$

yielding

$a_{k}=\frac{nr}{\pi}\Big{(}\sin(\frac{\pi}{n})\cdot\cos((2k-1){\frac{\pi}{n}}),\
\sin(\frac{\pi}{n})\cdot\sin((2k-1){\frac{\pi}{n}})\Big{)}.$

Let $V_{n}$ be the $n$th quantization error. Due to symmetry, the distortion
errors contributed by $a_{k}$ in their own Voronoi regions are equal for all
$1\leq k\leq n$. Again, notice that $\theta_{0}=0$,
$\theta_{1}=\frac{2\pi}{n}-\theta_{0}=\frac{2\pi}{n}$, and
$a_{1}=\frac{r}{\theta_{1}}(\sin\theta_{1},1-\cos\theta_{1})$. Hence,

$\displaystyle V_{n}$
$\displaystyle=\frac{n}{2\pi}\int_{\theta_{0}}^{\theta_{1}}\rho((r\cos\theta,r\sin\theta),a_{1})\,d\theta=r^{2}(1-\frac{n^{2}}{\pi^{2}}\sin^{2}\frac{\pi}{n}).$

Thus, the proof of the theorem is complete. ∎

######  Theorem 4.2.2.

Let $P$ be the uniform distribution on the circle of radius $r$ with center
$(0,0)$. Then, the quantization coefficient for $P$ exists as a finite
positive number and equals $\frac{\pi^{2}r^{2}}{3}$.

###### Proof.

By Theorem 4.2.1, we have
$V_{n}=r^{2}(1-\frac{n^{2}}{\pi^{2}}\sin^{2}\frac{\pi}{n})$, and hence
$\mathop{\lim}\limits_{n\to\infty}n^{2}V_{n}=\frac{\pi^{2}r^{2}}{3},$ which is
the theorem. ∎

######  Remark 4.2.3.

Theorem 4.2.2 implies that the quantization coefficient for a uniform
distribution on a circle of radius $r$, though exists as a finite positive
number, depends on the radius $r$ of the circle.

##  5\. Conditional optimal sets and the quantization coefficients for the
uniform distributions on the boundaries of the regular polygons

In this section, for a uniform distribution defined on the boundary of a
regular $m$-sided polygon, we calculate the conditional optimal sets of
$n$-points and the $n$th conditional quantization errors, and the conditional
quantization coefficient taking the conditional set as the set of all vertices
of the polygon.

Let $P$ be the uniform distribution defined on the boundary $L$ of a regular
$m$-sided polygon given by $A_{1}A_{2}\cdots A_{m}$ for some $m\geq 3$.
Without any loss of generality we can assume that the polygon is inscribed in
the circle $x^{2}+y^{2}=r^{2}$ which has center $O(0,0)$ and radius $r$ with
the Cartesian coordinates of the vertex $A_{1}$ as $(r,0)$. Let $\theta$ be
the central angle subtended by each side of the polygon, and let $\theta_{j}$
be the polar angles of the vertices $A_{j}$. Then, we have
$\theta=\frac{2\pi}{m}$ and $\theta_{j}=(j-1)\frac{2\pi}{m}$. Then, the polar
coordinates of the vertices $A_{j}$ are given by
$(r\cos\theta_{j},r\sin\theta_{j})$. Hence, if $\ell$ is the length of each of
the sides $A_{j}A_{j+1}$ for $1\leq j\leq m$, where the vertex $A_{m+1}$ is
identified as the vertex $A_{1}$, then we have

$\ell=\text{length of
}A_{j}A_{j+1}=\sqrt{\rho((r\cos\theta_{j},r\sin\theta_{j}),(r\cos\theta_{j+1},r\sin\theta_{j+1}))}=2r\sin\frac{\pi}{m}.$

The probability density function (pdf) $f$ for the uniform distribution $P$ is
given by $f(x,y)=\frac{1}{m\ell}$ for all $(x,y)\in A_{1}A_{2}\cdots A_{m}$,
and zero otherwise. Moreover, we can write

$L=\bigcup_{j=1}^{m}L_{j},\text{ where }L_{j}\text{ represents the side
}A_{j}A_{j+1}\text{ for }1\leq j\leq m.$

Notice that

$A_{j}A_{j+1}=\\{(1-t)(r\cos\theta_{j},r\sin\theta_{j})+t(r\cos\theta_{j+1},r\sin\theta_{j+1}):0\leq
t\leq 1\\}.$

for $1\leq j\leq m$. Write

$\displaystyle a_{j}:$
$\displaystyle=-\sec\frac{\pi}{m}\Big{(}j\sin\frac{2\pi(j-1)}{m}-(j-1)\sin\frac{2\pi
j}{m}\Big{)},$ (9) $\displaystyle b_{j}:$
$\displaystyle=-2\sin\frac{\pi}{m}\csc\frac{2\pi}{m}\Big{(}(j-1)\cos\frac{2\pi
j}{m}-j\cos\frac{2\pi(j-1)}{m}\Big{)},$ (10) $\displaystyle c_{j}:$
$\displaystyle=2(j-1)r\sin\frac{\pi}{m}.$ (11)

Let us consider the affine transformation

$T:L\to[0,2mr\sin\frac{\pi}{m}]\text{ such that }T(x,y)=a_{j}x+b_{j}y\text{ if
}(x,y)\in A_{j}A_{j+1},$

where $a_{j}$ and $b_{j}$ are given by (9) and (10) for all $1\leq j\leq m$.
Then, notice that $T:L\setminus\\{(r,0)\\}\to(0,2mr\sin\frac{\pi}{m})$ is a
bijective function. Let $Q$ be the image measure of $P$ under the function
$T$, i.e., $Q=TP$ such that for any Borel subset
$A\subseteq[0,2mr\sin\frac{\pi}{m}]$, we have

$Q(A)=TP(A)=P(T^{-1}(A)).$

By the distance between any two elements in $[0,2mr\sin\frac{\pi}{m}]$, it is
meant the Euclidean distance between the two elements. On the other hand, by
the distance between any two elements on $L$, it is meant the Euclidean
distance between the two elements along the polygonal arc $L$ in the
counterclockwise direction. Let $T_{j}$ be the restriction of the mapping $T$
to the set $A_{j}A_{j+1}$, i.e., $T_{j}=T|_{A_{j}A_{j+1}}$ for $1\leq j\leq
m$. Notice that each $T_{j}$ is a bijective function.

######  Lemma 5.1.

The function $T:L\setminus\\{(r,0)\\}\to(0,2mr\sin\frac{\pi}{m})$ preserves
the distance.

###### Proof.

Notice that

$\displaystyle[0,2mr\sin\frac{\pi}{m}]=\bigcup_{j=1}^{m}[c_{j},c_{j+1}]\text{
and }T(A_{j}A_{j+1})=T_{j}(A_{j}A_{j+1})=[c_{j},c_{j+1}],$

where $c_{j}$ are given by (11) for all $1\leq j\leq m$. Since the length of
$A_{j}A_{j+1}$ equals the length of the closed interval $[c_{j},c_{j+1}]$, and
$T_{j}$ is a bijection, we can say that
$T:L\setminus\\{(r,0)\\}\to(0,2mr\sin\frac{\pi}{m})$ preserves the distance.
Thus, the lemma is yielded. ∎

The following lemma which is similar to Lemma 4.1.1 is also true here.

######  Lemma 5.2.

The image measure $Q$ is a uniform distribution on $[0,2mr\sin\frac{\pi}{m}]$.

######  Remark 5.3.

Let $\\{a_{1},a_{2},\cdots,a_{n}\\}$ be a conditional unconstrained optimal
set of $n$-points for $Q$ with respect to the conditional set $\\{c_{j}:1\leq
j\leq m+1\\}$ such that $a_{1}<a_{2}<\cdots<a_{n}$. Then, by the definition of
conditional set, we have $n\geq m+1$. Moreover, notice that $a_{1}=c_{1}=0$
and $a_{n}=c_{m+1}=2mr\sin\frac{\pi}{m}$. Since both $P$ and $Q$ are uniform
distributions, and $Q$ is the image measure of $P$ under the function $T$, and
$T:L\setminus\\{(r,0)\\}\to(0,2mr\sin\frac{\pi}{m})$ preserves the distance,
we can say that the set $T^{-1}(\\{a_{1},a_{2},\cdots,a_{n}\\})$, i.e., the
set $\\{T^{-1}(a_{j}):1\leq j\leq n-1\\}$ forms a conditional unconstrained
optimal set of $(n-1)$-points for $P$ with respect to the conditional set
$\\{T^{-1}(c_{j}):1\leq j\leq m\\}$, i.e., with respect to the conditional set
$\\{A_{j}:1\leq j\leq m\\}$.

######  Lemma 5.4.

Let $\gamma_{n}$ be a conditional optimal set of $n$-points for the uniform
distribution $Q$ with respect to the conditional set $\\{c_{j}:1\leq j\leq
m+1\\}$ for any $n\geq m+1$. Let
$n_{j}=\text{card}(\gamma_{n}\cap[c_{j},c_{j+1}])$ for $1\leq j\leq m$. Then,
$n_{j}\geq 2$ and $n_{1}+n_{2}+\cdots+n_{m}=n+m-1$, and $|n_{i}-n_{j}|=0\text{
or }1$ for all $1\leq i\neq j\leq m$.

###### Proof.

Let $\gamma_{n}$ be a conditional optimal set of $n$-points and $n_{j}$ be the
positive integers as defined in the hypothesis. Notice that each of the sets
$\gamma_{n}\cap[c_{j},c_{j+1}]$ always contains the end elements
$c_{j},\,c_{j+1}$ for $1\leq j\leq m$, where $n\geq m+1$. Moreover, except the
two elements $c_{1}$ and $c_{m+1}$ all the end elements $c_{j}$ are counted
two times. Hence, $n_{j}\geq 2$ and
$n_{1}+n_{2}+\cdots+n_{m}=n+(m+1-2)=n+m-1$. Since $Q$ is a unform distribution
and the lengths of the intervals $[c_{j},c_{j+1}]$ for $1\leq j\leq m$ are all
equal, the proof of $|n_{i}-n_{j}|=0\text{ or }1$ for all $1\leq i\neq j\leq
m$ is routine. Thus, the lemma is yielded. ∎

Let us now give the following proposition.

######  Proposition 5.5.

Let $\gamma_{n}$ be a conditional optimal set of $n$-points and $V_{n}(Q)$ be
the $n$th conditional quantization error for the uniform distribution $Q$ with
respect to the conditional set $\\{c_{j}:1\leq j\leq m+1\\}$ for any $n\geq
m+1$. Let $n=mk+1+q$, where $0\leq q<m$. Then, if $q=0$, we have

$\displaystyle\gamma_{n}$
$\displaystyle=\bigcup_{j=1}^{m}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}\text{ with }V_{n}(Q)=\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3k^{2}}.$

On the other hand, if $0<q<m$, then there are ${}^{m}C_{q}$ possible sets
$\gamma_{n}$, one such set is given by

$\gamma_{n}=\Big{(}\bigcup_{j=1}^{q}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k+1}:1\leq
i\leq
k+2\\}\Big{)}\bigcup\Big{(}\bigcup_{j=q+1}^{m}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}\Big{)}$

with
$V_{n}(Q)=\frac{r^{2}q\sin^{2}\frac{\pi}{m}}{3m(k+1)^{2}}+\frac{r^{2}(m-q)\sin^{2}\frac{\pi}{m}}{3mk^{2}}$.

###### Proof.

First assume that $q=0$, then we have $n=mk+1$, i.e., $\gamma_{n}$ contains
$k-1$ elements from each of the intervals $[c_{j},c_{j+1}]$ except the
boundary elements $c_{j}$ and $c_{j+1}$. Hence, if
$n_{j}=\text{card}(\gamma_{n}\cap[c_{j},c_{j+1}])$, we have $n_{j}=k-1+2=k+1$.
Hence, by $(i)$ of Theorem 3.1, we have

$\displaystyle\gamma_{n}$
$\displaystyle=\bigcup_{j=1}^{m}\\{c_{j}+\frac{i-1}{k}(c_{j+1}-c_{j}):1\leq
i\leq
k+1\\}=\bigcup_{j=1}^{m}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}$ $\text{ with
}V_{n}=\sum_{j=1}^{m}\frac{(c_{j+1}-c_{j})^{2}}{12mk^{2}}=\sum_{j=1}^{m}\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3mk^{2}}=\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3k^{2}}.$

On the other hand, if $0<q<m$, then due to Lemma 5.4, we can assume that
$\gamma_{n}$ contains $k$ elements from each of the first $q$ intervals except
the boundary elements, and $\gamma_{n}$ contains $(k-1)$ elements from each of
the remaining $m-q$ intervals except the boundary elements implying
$n_{1}=n_{2}=\cdots=n_{q}=k+2$ and $n_{q+1}=n_{q+2}=\cdots=n_{m}=k+1$. Hence,
the expressions for $\gamma_{n}$ and the corresponding $n$th conditional
quantization errors are obtained by $(i)$ of Theorem 3.1 as

$\gamma_{n}=\Big{(}\bigcup_{j=1}^{q}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k+1}:1\leq
i\leq
k+2\\}\Big{)}\bigcup\Big{(}\bigcup_{j=q+1}^{m}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}\Big{)}$

with

$V_{n}=\sum_{j=1}^{q}\frac{(c_{j+1}-c_{j})^{2}}{12m(k+1)^{2}}+\sum_{j=q+1}^{m}\frac{(c_{j+1}-c_{j})^{2}}{12mk^{2}}=\sum_{j=1}^{q}\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3m(k+1)^{2}}+\sum_{j=q+1}^{m}\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3mk^{2}}$

yielding

$V_{n}=\frac{r^{2}q\sin^{2}\frac{\pi}{m}}{3m(k+1)^{2}}+\frac{r^{2}(m-q)\sin^{2}\frac{\pi}{m}}{3mk^{2}}.$

Notice that if $n=mk+q$, the optimal set $\gamma_{n}$ can be constructed in
${}^{m}C_{q}$ ways. Thus, the proof of the proposition is complete. ∎

The following two theorems give the main results in this section.

######  Theorem 5.6.

Let $\alpha_{n}$ be a conditional optimal set of $n$-points and $V_{n}(P)$ be
the $n$th conditional quantization error for the uniform distribution $P$ with
respect to the conditional set $\\{A_{j}:1\leq j\leq m\\}$ for any $n\geq m$.
Let $n=mk+q$, where $0\leq q<m$. Then, if $q=0$, we have

$\displaystyle\alpha_{n}$
$\displaystyle=\bigcup_{j=1}^{m}T_{j}^{-1}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}\text{ with }V_{n}(P)=\frac{r^{2}\sin^{2}\frac{\pi}{m}}{3k^{2}}.$

On the other hand, if $0<q<m$, then there are ${}^{m}C_{q}$ possible sets
$\alpha_{n}$, one such set is given by

$\displaystyle\alpha_{n}$
$\displaystyle=\Big{(}\bigcup_{j=1}^{q}T_{j}^{-1}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k+1}:1\leq
i\leq k+2\\}\Big{)}$
$\displaystyle\qquad\bigcup\Big{(}\bigcup_{j=q+1}^{m}T_{j}^{-1}\\{c_{j}+\frac{2(i-1)r\sin\frac{\pi}{m}}{k}:1\leq
i\leq k+1\\}\Big{)}$

with
$V_{n}(P)=\frac{r^{2}q\sin^{2}\frac{\pi}{m}}{3m(k+1)^{2}}+\frac{r^{2}(m-q)\sin^{2}\frac{\pi}{m}}{3mk^{2}}$.

###### Proof.

For $n=mk+1+q$, let $\gamma_{n}$ be a conditional optimal set of $n$-points
for $Q$ as given by Proposition 5.5. By Remark 5.3 and Proposition 5.5, it
follows that the set $\alpha_{n}$ given by the statement of the theorem forms
a conditional optimal set of $n$-points for $P$ for $n=mk+q$. Notice that the
$(mk+q)$th-conditional quantization error with respect to the uniform
distribution $P$ remains same as the $(mk+1+q)$th-conditional quantization
error with respect to the uniform distribution $Q$. This happens due to the
fact that the bijective functions $T_{j}$ preserve the distance as well as the
collinearity of the elements in each interval $[c_{j},c_{j+1}]$. ∎

######  Theorem 5.7.

Let $P$ be the uniform distribution defined on the boundary of a regular
$m$-sided polygon inscribed in a circle of radius $r$ with center $(0,0)$.
Then, the conditional quantization coefficient for $P$ exists as a finite
positive number and equals $\frac{1}{3}m^{2}r^{2}\sin^{2}(\frac{\pi}{m})$,
i.e.,
$\lim\limits_{n\to\infty}n^{2}(V_{n}(P)-V_{\infty}(P))=\frac{1}{3}m^{2}r^{2}\sin^{2}\frac{\pi}{m}.$

###### Proof.

Let $n\in\mathbb{N}$ be such that $n\geq m$. Then, there exists a unique
positive integer $k$ such that $n=mk+q$ for some $0\leq q<m$. Then, by Theorem
5.6, we have

$V_{n}(P)=\frac{r^{2}q\sin^{2}\frac{\pi}{m}}{3m(k+1)^{2}}+\frac{r^{2}(m-q)\sin^{2}\frac{\pi}{m}}{3mk^{2}}=\frac{m^{2}r^{2}\sin^{2}\left(\frac{\pi}{m}\right)\left(m^{2}+2mn-3mq+n^{2}-4nq+3q^{2}\right)}{3(n-q)^{2}(m+n-q)^{2}}.$

Then, we see that $V_{\infty}(P)=\lim_{n\to\infty}V_{n}(P)=0$. In fact, we
have

$\lim\limits_{n\to\infty}n^{2}(V_{n}(P)-V_{\infty}(P))=\frac{1}{3}m^{2}r^{2}\sin^{2}\frac{\pi}{m}.$

Thus, the proof of the theorem is complete. ∎

######  Remark 5.8.

By Theorem 2.3 and Theorem 5.7, we can say that the quantization coefficient
for the uniform distribution $P$ defined on the boundary of a regular
$m$-sided polygon is $\frac{1}{3}m^{2}r^{2}\sin^{2}\frac{\pi}{m}$, which is a
finite positive number, but it is not a constant as it depends on both $m$ and
$r$, where $m$ is the number of sides of the polygon and $r$ is the radius of
the circle in which the polygon is inscribed, and this lead us to conclude
that the quantization coefficient for a uniform distribution defined on the
boundary of a regular $m$-sided polygon depends on both the number of sides of
the polygon and the length of the sides.

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