# Neighbor product distinguishing total colorings of corona of subcubic graphs
††thanks: This work was supported by the National Natural Science Foundation
of China (Grant No. NSFC12001332). It was also supported by China Postdoctoral
Science Foundation Funded Project (Grant No.2014M561909); the Nature Science
Foundation of Shandong Province of China (Grant No. ZR2014AM028, ZR2017BA009)

Aijun Dong Wenwen Zhang
School of Data and Computer Science,
Shandong Women’s University, Jinan, 250300, China

###### Abstract

A proper $[k]$-total coloring $c$ of a graph $G$ is a mapping $c$ from
$V(G)\bigcup E(G)$ to $[k]=\\{1,2,\cdots,k\\}$ such that $c(x)\neq c(y)$ for
which $x$, $y\in V(G)\bigcup E(G)$ and $x$ is adjacent to or incident with
$y$. Let $\prod(v)$ denote the product of $c(v)$ and the colors on all the
edges incident with $v$. For each edge $uv\in E(G)$, if
$\prod(u)\neq\prod(v)$, then the coloring $c$ is called a neighbor product
distinguishing total coloring of $G$. By $\chi^{\prime\prime}_{\prod}(G)$, we
denote the minimal value of $k$ in such a coloring of $G$. In 2015, Li et al.
conjectured that $\chi^{\prime\prime}_{\prod}(G)\leq\Delta(G)+3$. In this
paper, for the corona graph of two arbitrary subcubic graphs, we give an
algorithm of a neighbor product distinguishing total coloring and confirm the
conjecture put forward by Li et al.

Keywords: total coloring; corona graph; neighbor product distinguishing
coloring

MSC(2010): 05C15

##  1 Introduction

All graphs considered in this paper are simple and undirected. Let $G=(V,E)$
be a graph. $V(G)$, $E(G)$, $\Delta(G)$ and $\delta(G)$ were used to denote
the vertex set, edge set, maximum degree and minimum degree of $G$,
respectively. Let $N_{G}(u)$ be the set of neighbors of $u$ in the graph $G$.
We use $n_{g}$ and $n_{h}$ to denote the number of vertices in graphs $G$ and
$H$, respectively. The notation and terminology used but undefined here can be
found in [1].

Let $k$ be a positive integer. A total $k$-coloring $c$ is a mapping
$c:E(G)\cup V(G)\rightarrow[k]$, such that for any two elements $x,y\in
E(G)\cup V(G)$, if $xy\in E(G)$ or $x$ is incident with $y$, then $c(x)\neq
c(y)$. For each vertex $u\in V(G)$, let $\prod(u)$ (resp. $S(u)$) denote the
product (resp. set) of colors on $u$ and the edges which are incident with
$u$. If $\prod(v)\neq\prod(u)$ (resp. $S(u)\neq S(v)$) for each edge $uv\in
E(G)$, then it is called $a$ $neighbor$ $product$ $distinguishing$ $total$
$coloring$ (resp. $neighbor$ $vertex$ $distinguishing$ $total$ $coloring$) of
$G$. For convenience, they are abbreviated as $NPDTC$ and $NVDTC$,
respectively. The chromatic number of $NPDTC$ (resp. $NVDTC$)
$\chi^{\prime\prime}_{\prod}(G)$ (resp. $tndi(G)$) is the smallest integer
number $k$ such that $G$ admits a neighbor product (resp. vertex)
distinguishing $[k]$-total coloring. Clearly,
$tndi(G)\leq\chi^{\prime\prime}_{\prod}(G)$.

In 2005, Zhang et al. [25] determined the neighbor set distinguishing total
coloring index for graphs which are cliques, paths, cycles, fans, wheels,
stars, complete graphs etc. and made the following conjecture.

###### Conjecture 1

Let $G$ be a connected graph with at least two vertices, then
$tndi(G)\leq\Delta(G)+3$.

Wang, Chen [3, 19] and Lu et al. [13] relayed to verify the Conjecture 1 for
graphs with $\Delta\leq 4$. The Conjecture 1 was confirmed for some special
graphs such as $1$-tree, some sparse graph, $2$-degenerate graph and line and
splitting graph of some graphs in [20], [21], [14] and [17], respectively.
Outer planar graph, series-parallel graphs were studied in [22, 23]. Chang et
al. [2] confirmed the Conjecture 1 for planar graph with $\Delta\geq 8$. More
related results can be seen in [8, 24, 4, 7].

Recently, Li et al. [12] introduced the notation of NPDTC and proposed the
following conjecture.

###### Conjecture 2

If $G$ is a graph with at least two vertices, then
$\chi^{\prime\prime}_{\prod}(G)\leq\Delta(G)+3$.

Li et al. prove that the conjecture holds for complete graphs, cycles, trees,
bipartite graphs, subcubic graphs and $K_{4}$-minor free graphs. In 2017, Dong
et al. confirmed the Conjecture 2 for sparse graph $G$ with bounded maximum
degree [5]. Recently, the Conjecture 2 was confirmed for $2$-degenerate graph
in [26].

In this paper, we discuss the NPDTC of graph products which was introduced by
Frucht and Harary in 1970 [6]. Given two simple graphs $G$ and $H$, the
$corona$ $product$ of $G$ and $H$ is abbreviated as $G\circ H$ which is
obtained by taking one copy of $G$, $|V(G)|$ copies of $H$ ($H_{1}$,
$H_{2}$,$\ldots$,$H_{n_{g}}$), and making every vertex $v_{j}$ of $G$ adjacent
to every vertex $u^{j}_{i}$ (the copy of $u_{i}$ in $H_{j}$) of $H_{j}$ to get
edges $v_{j}u^{j}_{i}$ ($1\leq j\leq n_{g}$, $1\leq i\leq n_{h}$). The edge
$v_{j}u^{j}_{i}$ is called a $corona$ $edge$. Obviously, there are $n_{g}\cdot
n_{h}$ corona edges in $G\circ H$ in total. For convenience, let $d_{G}(v)$,
$d_{H}(v)$ and $d(v)$ denote the degree of a vertex $v$ in $G$, $H$ and
$G\circ H$, respectively. Obviously, the maximum degree of the corona graph
$G\circ H$ is $\Delta(G\circ H)=\Delta(G)+n_{h}$. A corona graph $K_{3}\circ
K_{4}$ can be seen in Figure 1.

Figure 1

The concept of the corona product has some applications in chemistry for
representing chemical compounds [9]. Other applications of this concept
include navigation of robots in networks [10]. Some theoretical results can be
seen in [11, 15]. In this paper, we focus on the corona graph of two arbitrary
subcubic graphs and get the following result.

###### Theorem 1.1

Let $G$, $H$ be two arbitrary subcubic graphs. Then
$\chi^{\prime\prime}_{\prod}(G\circ H)\leq\Delta(G\circ H)+3$.

Since the neighbor product distinguishing $k$-total coloring is a generation
of the neighbor vertex distinguishing $k$-total coloring, we have the
following corollary.

###### Corollary 1.2

Let $G$,$H$ be two arbitrary subcubic graphs. Then $tndi(G\circ
H)\leq\Delta(G\circ H)+3$.

##  2 Preliminaries

Let us recall some known results that will be useful for the forthcoming
proof.

######  Lemma 2.1 ([16])

If $G$ is a subcubic graph, then
$\chi^{\prime\prime}_{\prod}(G)\leq\Delta(G)+3$.

######  Lemma 2.2 ([18])

Let $G$ be a graph. Then $\Delta(G)\leq\chi^{\prime}(G)\leq\Delta(G)+1$.

##  3 Proof of Theorem 1.1

###### Proof.

Since $\Delta(G)\leq 3$, then $G$ is a subcubic graph. We have
$\chi^{\prime\prime}_{\prod}(G)\leq\Delta(G)+3$ by Lemma 2.1. By Lemma 2.2, we
have $\chi^{\prime}(H)\leq\Delta(H)+1$. For convenience, we use
$c^{\prime\prime}$ and $c^{\prime}$ to denote a neighbor product
distinguishing $(\Delta(G)+3)$-total coloring of $G$ and a
$(\Delta(H)+1)$-edge coloring of $H$, respectively. For each vertex $v\in
V(G)$, we use $\prod_{c^{\prime\prime}}(v)$ to denote the product of colors on
the vertex $v$ and the edges which are incident with $v$ in the coloring
$c^{\prime\prime}$ of $G$, i.e.
$\prod_{c^{\prime\prime}}(v)=c^{\prime\prime}(v)\cdot\prod_{v\in
e}(c^{\prime\prime}(e))$. Clearly, for each edge
$v^{\prime}v^{\prime\prime}\in E(G)$, we have
$\prod_{c^{\prime\prime}}(v^{\prime})\neq\prod_{c^{\prime\prime}}(v^{\prime\prime})$.
For each vertex $u\in V(H)$, by $\prod_{c^{\prime}}(u)$ (resp.
$S_{c^{\prime}}(u)$), we denote the product (resp. set) of colors on the edges
which are incident with $u$ in the coloring $c^{\prime}$ of $H$. Without loss
of generality, we assume that
$\prod_{c^{\prime}}(u_{1})\leq\prod_{c^{\prime}}(u_{2})\leq\ldots\leq\prod_{c^{\prime}}(u_{n_{h}})$.
We assume $H_{j}$ (the $j^{th}$ copy of $H$) has the same coloring
$c^{\prime}$ as $H$ for $1\leq j\leq n_{g}$. Obviously, we also have
$\prod_{c^{\prime}}(u^{j}_{1})\leq\prod_{c^{\prime}}(u^{j}_{2})\leq\ldots\leq\prod_{c^{\prime}}(u^{j}_{n_{h}})$.
In the following, we will give colors to the corona edges $v_{j}u^{j}_{i}$ and
the vertices $u^{j}_{i}$ for $1\leq j\leq n_{g}$, $1\leq i\leq n_{h}$ to get a
neighbor product distinguishing $(\Delta(G\circ H)+3)$-total coloring of
$G\circ H$.

Now, we divide the proof into the following cases.

Case $1.$ $\Delta(G)=1$.

Let $G=v_{1}v_{2}$. Clearly, $\chi^{\prime\prime}_{\prod}(G)=3$. Let
$S=\\{1,2,3,4\\}$. Since $|S_{c^{\prime}}(u_{1})|\leq 3$, then
$|(S-S_{c^{\prime}}(u_{1}))|\geq 1$.

Case $1.1.$ $4\notin(S-S_{c^{\prime}}(u_{1}))$.

Since $|(S-S_{c^{\prime}}(u_{1}))|\geq 1$, there exists at least one color
$\beta\in(S-S_{c^{\prime}}(u_{1}))$ such that $\beta\in\\{1,2,3\\}$. Now, let
$c(u_{1}^{j})=c(v_{1}v_{2})=\beta$, $c(v_{1}u_{1}^{1})=c(v_{2}u_{1}^{2})=5$,
and use the other two colors in $\\{1,2,3\\}$ to color $v_{1}$, $v_{2}$,
respectively. Let $c(u_{i}^{j})=i+3$, $c(v_{j}u_{i}^{j})=i+4$ for each $1\leq
j\leq 2$ and $2\leq i\leq n_{h}$. So far, we get a proper
$(\Delta(G)+n_{h}+3)$-total coloring $c$ of $G\circ H$ such that
$c(e)=c^{\prime}(e)$ for each edge $e\in E(H)$. Clearly,
$\prod_{c}(v_{1})=c(v_{1})\cdot\beta\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$,
$\prod_{c}(v_{2})=c(v_{2})\cdot\beta\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$, and
$\prod_{c}(u_{1}^{j})=\prod_{c^{\prime}}(u^{j}_{1})\cdot\beta\cdot 5$ for
$1\leq j\leq 2$. For each vertex $u^{j}_{i}\in V(H_{j})$ where $1\leq j\leq 2$
and $2\leq i\leq n_{h}$, we have
$\prod_{c}(u^{j}_{i})=\prod_{c^{\prime}}(u^{j}_{i})\cdot(i+3)\cdot(i+4)$.

In the following, we will show the coloring $c$ is a neighbor product
distinguishing total coloring of $G\circ H$.

First, we consider the vertices of $G$. Clearly,
$\frac{\prod_{c}(v_{1})}{\prod_{c}(v_{2})}=\frac{c(v_{1})}{c(v_{2})}\neq 1$.
So we have $\prod_{c}(v_{1})\neq\prod_{c}(v_{2})$.

Second, we will show $\prod_{c}(u^{j}_{i_{1}})\neq\prod_{c}(u^{j}_{i_{2}})$
for any two adjacent vertices $u^{j}_{i_{1}}$ and $u^{j}_{i_{2}}$ of $H_{j}$.
Since $\beta\leq 3$, then
$\prod_{c}(u_{1}^{j})=\prod_{c^{\prime}}(u^{j}_{1})\cdot\beta\cdot
5<\prod_{c^{\prime}}(u^{j}_{i})\cdot(i+3)\cdot(i+4)$ for $2\leq i\leq n_{h}$.
For each $2\leq i_{1}<i_{2}\leq n_{h}$, since
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\leq\prod_{c^{\prime}}(u^{j}_{i_{2}})$, then
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\cdot(i_{1}+3)\cdot(i_{1}+4)<\prod_{c^{\prime}}(u^{j}_{i_{2}})\cdot(i_{2}+3)\cdot(i_{2}+4)$,
i.e. $\prod_{c}(u^{j}_{i_{1}})<\prod_{c}(u^{j}_{i_{2}})$.

At last, we consider two adjacent vertices $u^{j}_{i}$ and $v_{j}$ for $1\leq
j\leq 2$, $1\leq i\leq n_{h}$. For each $1\leq j\leq 2$, we have
$\prod_{c}(v_{j})=\prod_{c^{\prime\prime}}(v_{j})\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)\geq 1\cdot 2\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$. Furthermore,
$\prod_{c}(u^{j}_{1})<\prod_{c}(u^{j}_{2})<\ldots<\prod_{c}(u^{j}_{n_{h}})$,
and
$\prod_{c}(u^{j}_{n_{h}})=\prod_{c^{\prime}}(u^{j}_{n_{h}})\cdot(n_{h}+3)\cdot(n_{h}+4)$.
Since $4\in S_{c^{\prime}}(u_{1})$, then $\chi^{\prime}(H)=4$. Obviously,
$n_{h}>4$. We have $\prod_{c}(u^{j}_{n_{h}})\leq 2\cdot 3\cdot
4\cdot(n_{h}+3)\cdot(n_{h}+4)$. Since $\prod_{c}(v_{j})\geq 1\cdot 2\cdot
5\cdot 6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$, then
$\frac{\prod_{c}(v_{j})}{\prod_{c}(u^{j}_{n_{h}})}\geq\frac{5}{2}$. So we have
$\Sigma_{c}(v_{j})>\Sigma_{c}(v^{j}_{i})$ for each $1\leq j\leq 2$ and $1\leq
i\leq n_{h}$.

Case $1.2.$ $4\in(S-S_{c^{\prime}}(u_{1}))$.

Let $c(u_{i}^{j})=i+3$, $c(v_{j}u_{i}^{j})=i+4$ for each $1\leq j\leq 2$ and
$1\leq i\leq n_{h}$. So far, we get a proper $(\Delta(G)+n_{h}+3)$-total
coloring $c$ of $G\circ H$ such that $c(e)=c^{\prime}(e)$ for each edge $e\in
E(H)$ and $c(x)=c^{\prime\prime}(x)$ for each $x\in V(G)\bigcup E(G)$. For
each vertex $v\in V(G)$, we have
$\prod_{c}(v)=\prod_{c^{\prime\prime}}(v)\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$. For each vertex $u^{j}_{i}\in
V(H_{j})$ where $1\leq j\leq 2$ and $1\leq i\leq n_{h}$, we have
$\prod_{c}(u^{j}_{i})=\prod_{c^{\prime}}(u^{j}_{i})\cdot(i+3)\cdot(i+4)$.

In the following, we will show the coloring $c$ is a neighbor product
distinguishing total coloring of $G\circ H$.

First, we consider the vertices of $G$. Since
$\prod_{c^{\prime\prime}}(v_{1})\neq\prod_{c^{\prime\prime}}(v_{2})$, clearly,
we have $\prod_{c^{\prime\prime}}(v_{1})\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)\neq\prod_{c^{\prime\prime}}(v_{2})\cdot
5\cdot 6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$, i.e.
$\prod_{c}(v_{1})\neq\prod_{c}(v_{2})$.

Second, we will show $\prod_{c}(u^{j}_{i_{1}})\neq\prod_{c}(u^{j}_{i_{2}})$
for any two adjacent vertices $u^{j}_{i_{1}}$ and $u^{j}_{i_{2}}$ of $H_{j}$.
Without loss of generality, we assume $i_{1}<i_{2}$. Since
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\leq\prod_{c^{\prime}}(u^{j}_{i_{2}})$, then
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\cdot(i_{1}+3)\cdot(i_{1}+4)<\prod_{c^{\prime}}(u^{j}_{i_{2}})\cdot(i_{2}+3)\cdot(i_{2}+4)$,
i.e. $\prod_{c}(u^{j}_{i_{1}})<\prod_{c}(u^{j}_{i_{2}})$.

At last, we consider two adjacent vertices $u^{j}_{i}$ and $v_{j}$ for $1\leq
j\leq 2$, $1\leq i\leq n_{h}$. For each $1\leq j\leq 2$, we have
$\prod_{c}(v_{j})=\prod_{c^{\prime\prime}}(v_{j})\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)\geq 1\cdot 2\cdot 5\cdot
6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$. Furthermore,
$\prod_{c}(u^{j}_{1})<\prod_{c}(u^{j}_{2})<\ldots<\prod_{c}(u^{j}_{n_{h}})$,
and
$\prod_{c}(u^{j}_{n_{h}})=\prod_{c^{\prime}}(u^{j}_{n_{h}})\cdot(n_{h}+3)\cdot(n_{h}+4)$.

If $n_{h}=2$, then $\chi^{\prime}(H)=1$. We have $\prod_{c}(u^{j}_{2})=1\cdot
5\cdot 6=30$. Since $\prod_{c}(v_{j})\geq 1\cdot 2\cdot 5\cdot 6=60$, we have
$\Sigma_{c}(v_{j})>\Sigma_{c}(u^{j}_{i})$ for each $1\leq j,i\leq 2$.

If $n_{h}=3$, then $\chi^{\prime}(H)\leq 3$. We have $\prod_{c}(u^{j}_{3})\leq
2\cdot 3\cdot 6\cdot 7=252$. Since $\prod_{c}(v_{j})\geq 1\cdot 2\cdot 5\cdot
6\cdot 7=420$, we have $\Sigma_{c}(v_{j})>\Sigma_{c}(u^{j}_{i})$ for each
$1\leq j\leq 2$, $1\leq i\leq 3$.

Otherwise, i.e. $n_{h}\geq 4$. Clearly, $2\leq\chi^{\prime}(H)\leq 4$. We have
$\prod_{c}(u^{j}_{n_{h}})\leq 2\cdot 3\cdot
4\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$. Since $\prod_{c}(v_{j})\geq 1\cdot
2\cdot 5\cdot 6\cdot\ldots\cdot(n_{h}+3)\cdot(n_{h}+4)$, clearly,
$\frac{\prod_{c}(v_{j})}{\prod_{c}(u^{j}_{n_{h}})}\geq\frac{5}{2}$. So we have
$\Sigma_{c}(v_{j})>\Sigma_{c}(v^{j}_{i})$ for each $1\leq j\leq 2$ and $1\leq
i\leq n_{h}$.

Case $2.$ $2\leq\Delta(G)\leq 3$.

Clearly, $\chi^{\prime\prime}_{\prod}(G)\leq 6$. Let $S=\\{1,2,3,4,5\\}$.
Since $|S_{c^{\prime}}(u_{1})\bigcup c^{\prime\prime}(v)|\leq 4$, then
$|S-(S_{c^{\prime}}(u_{1})\bigcup c^{\prime\prime}(v))|\geq 1$. There exists
at least one color $\alpha\in(S-(S_{c^{\prime}}(u_{1})\bigcup
c^{\prime\prime}(v)))$. Let $c(u_{1}^{j})=\alpha$,
$c(v_{j}u_{1}^{j})=\Delta(G)+4$, and let $c(u_{i}^{j})=\Delta(G)+i+2$,
$c(v_{j}u_{i}^{j})=\Delta(G)+i+3$ for each $1\leq j\leq n_{g}$ and $2\leq
i\leq n_{h}$. So far, we get a proper $(\Delta(G)+n_{h}+3)$-total coloring $c$
of $G\circ H$ such that $c(e)=c^{\prime}(e)$ for each edge $e\in E(H)$ and
$c(x)=c^{\prime\prime}(x)$ for each $x\in V(G)\bigcup E(G)$. For each vertex
$v\in V(G)$, we have
$\prod_{c}(v)=\prod_{c^{\prime\prime}}(v)\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+3)$,
$\prod_{c}(u^{j}_{1})=\prod_{c^{\prime}}(u^{j}_{1})\cdot\alpha\cdot(\Delta(G)+4)$.
For each vertex $u^{j}_{i}\in V(H_{j})$ where $1\leq j\leq n_{g}$ and $2\leq
i\leq n_{h}$, we have
$\prod_{c}(u^{j}_{i})=\prod_{c^{\prime}}(u^{j}_{i})\cdot(\Delta(G)+i+2)\cdot(\Delta(G)+i+3)$.

In the following, we will show the coloring $c$ is a neighbor product
distinguishing total coloring of $G\circ H$.

First, we consider two adjacent vertices $v^{\prime}$ and $v^{\prime\prime}$
in $G$. Since
$\prod_{c^{\prime\prime}}(v^{\prime})\neq\prod_{c^{\prime\prime}}(v^{\prime\prime})$,
it is obvious that
$\prod_{c^{\prime\prime}}(v^{\prime})\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+3)\neq\prod_{c^{\prime\prime}}(v^{\prime\prime})\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+3)$,
i.e. $\prod_{c}(v^{\prime})\neq\prod_{c}(v^{\prime\prime})$.

Second, we will show $\prod_{c}(u^{j}_{i_{1}})\neq\prod_{c}(u^{j}_{i_{2}})$
for any two adjacent vertices $u^{j}_{i_{1}}$ and $u^{j}_{i_{2}}$ of $H_{j}$.
Since $\alpha\leq 5$, then
$\prod_{c}(u_{1}^{j})=\prod_{c^{\prime}}(u^{j}_{1})\cdot\alpha\cdot(\Delta(G)+4)<\prod_{c^{\prime}}(u^{j}_{i})\cdot(\Delta(G)+i+2)\cdot(\Delta(G)+i+3)$
for $2\leq i\leq n_{h}$. For each $2\leq i_{1}<i_{2}\leq n_{h}$, since
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\leq\prod_{c^{\prime}}(u^{j}_{i_{2}})$, it
is clear that
$\prod_{c^{\prime}}(u^{j}_{i_{1}})\cdot(\Delta(G)+i_{1}+2)\cdot(\Delta(G)+i_{1}+3)<\prod_{c^{\prime}}(u^{j}_{i_{2}})\cdot(\Delta(G)+i_{2}+2)\cdot(\Delta(G)+i_{2}+3)$,
i.e. $\prod_{c}(u^{j}_{i_{1}})<\prod_{c}(u^{j}_{i_{2}})$.

At last, we consider two adjacent vertices $u^{j}_{i}$ and $v_{j}$ for $1\leq
j\leq n_{g}$, $1\leq i\leq n_{h}$. For each $1\leq j\leq n_{g}$, we have
$\prod_{c}(v_{j})=\prod_{c^{\prime\prime}}(v_{j})\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+3)\geq
1\cdot
2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+3)$.
Furthermore,
$\prod_{c}(u^{j}_{1})<\prod_{c}(u^{j}_{2})<\ldots<\prod_{c}(u^{j}_{n_{h}})$,
and
$\prod_{c}(u^{j}_{n_{h}})=\prod_{c^{\prime}}(u^{j}_{n_{h}})\cdot(\Delta(G)+n_{h}+2)\cdot(\Delta(G)+n_{h}+3)$.

If $n_{h}=2$, then $\chi^{\prime}(H)=1$. We have
$\prod_{c}(u^{j}_{2})=1\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)$. Since
$\prod_{c}(v_{j})\geq 1\cdot 2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)$, then
$\frac{\prod_{c}(v_{j})}{\prod_{c}(u^{j}_{2})}\geq 2$. Clearly, we have
$\Sigma_{c}(v_{j})>\Sigma_{c}(u^{j}_{i})$ for each $1\leq j\leq n_{g}$, $1\leq
i\leq 2$.

If $n_{h}=3$, then $\Delta(H)\leq 2$, $\chi^{\prime}(H)\leq 3$. We have
$\prod_{c}(u^{j}_{3})\leq 2\cdot 3\cdot(\Delta(G)+5)\cdot(\Delta(G)+6)$. Since
$\prod_{c}(v_{j})\geq 1\cdot
2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)(\Delta(G)+6)$, then
$\frac{\prod_{c}(v_{j})}{\prod_{c}(u^{j}_{3})}\geq\frac{5}{3}$. Clearly, we
have $\Sigma_{c}(v_{j})>\Sigma_{c}(u^{j}_{i})$ for each $1\leq j\leq n_{g}$,
$1\leq i\leq 3$.

Otherwise, i.e. $n_{h}\geq 4$. Clearly, $\chi^{\prime}(H)\leq 4$. We have
$\prod_{c}(u^{j}_{n_{h}})\leq 2\cdot 3\cdot
4\cdot(\Delta(G)+n_{h}+2)\cdot(\Delta(G)+n_{h}+3)$. Since
$\prod_{c}(v_{j})\geq 1\cdot
2\cdot(\Delta(G)+4)\cdot(\Delta(G)+5)\cdot\ldots\cdot(\Delta(G)+n_{h}+2)\cdot(\Delta(G)+n_{h}+3)$,
clearly, $\frac{\prod_{c}(v_{j})}{\prod_{c}(u^{j}_{n_{h}})}\geq\frac{5}{2}$.
So we have $\Sigma_{c}(v_{j})>\Sigma_{c}(v^{j}_{i})$ for each $1\leq j\leq
n_{g}$ and $1\leq i\leq n_{h}$.

∎

## References

  * [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, New York, 1976.
  * [2] Y. Chang, J. Hu, G. Wang, X. Yu, Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 8, Discrete Mathematics, (2020) 343(10): 112014.
  * [3] X. Chen, On the adjacent vertex distinguishing total coloring numbers of graphs with $\Delta=3$, Discrete Math., (2008) 308: 4003–4008.
  * [4] X. H. Cheng, G. H. Wang, J. L. Wu, The adjacent vertex distinguishing total chromatic numbers of planar graphs $\Delta=10$, Journal of Combinatorial Optimization, (2017) 34(2): 383–397.
  * [5] A. J. Dong, T. Li, Neighbor product distinguishing total colorings of graphs with bounded maximum average degree, Ars Combinatoria, (2019) 144(1): 1–19.
  * [6] R. Frucht, F. Harary, On the corona of two graphs, Aequations Math., (1970) 4: 322–325.
  * [7] J. Hu, G. Wang, J. Wu, D. Yang, X. Yu, Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9, Discrete Mathematics, (2019) 342: 1392–1402.
  * [8] D. J. Huang, W. F. Wang, Adjacent vertex distinguishing total coloring of planar graphs with large maximum degree, (in Chinese), Sci. Sin. Math., (2012) 42(2): 151–164.
  * [9] M. A. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharmaceutical Stats, (1993)3: 203–236.
  * [10] S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., (1996), 70(3): 217–229.
  * [11] D. Laïche, I. Bouchemakh, Paching coloring of some undirected and oriented corona graphs,Discussiones Mathematicae Graph Theory, (2017) 37: 665–690.
  * [12] T. Li, C. Q. Qu, G. H. Wang, X. W. Yu, Neighbor product distinguishing total colorings, Journal of Combinatorial Optimization, (2017) 33(1): 237–253.
  * [13] Y. Lu, J. A. Li, R. Luo, Z. K. Miao, Adjacent vertex distinguishing total coloring of graphs with maximum degree $4$, Discrete Mathematics, (2017) 340(2): 119–123.
  * [14] Z. K. Miao, R. Shi, X. L. Hu, R. Luo, Adjacent vertex distinguishing total colorings of $2$-degenerate graph, Discrete Mathematics, (2016) 336: 2446–2449.
  * [15] S. Mohan J. Geetha K. Somasundaram, Total coloring of the corona product of two graphs, Australasian Journal of Combinatorics,(2017) 68(1): 15–22.
  * [16] M. Pilśniak, M. Woźniak, On the Total-Neighbor-Distinguishing Index by Sums, Graphs and Combinatorics, (2015) 31(3): 771–782.
  * [17] K. Thirusangu, R. Ezhilarasi, Adjacent vertex distinguishing total coloring of line and splitting graph of some graphs, Annals of Pure and Applied Mathematics, (2017) 13 (2): 173-183.
  * [18] V. G. Vizing, On an estimate of the chromatic class of a $P$-graph, Metody Diskret. Analiz., (1964) 3: 25–30.
  * [19] H. Y. Wang, On the adjacent vertex-distinguishing total chromatic numbers of the graphs with $\Delta(G)=3$, J. Comb. Optim., (2007) 14: 87–109.
  * [20] H. Y. Wang, The adjacent vertex-distinguishing total chromatic number of $1$-tree, Ars Combin., (2009) 91: 183–192.
  * [21] W. F. Wang, Y. Q. Wang, Adjacent vertex distinguishing total coloring of graphs with lower average degree, Tanwanese J. Math., (2008) 12, (4): 979–990.
  * [22] W. F. Wang, P. Wang, Adjacent vertex distinguishing total coloring of $K_{4}$-minor free graphs, (in Chinese), Sci. Sin. Math., (2009) 39 (12): 1462–1472.
  * [23] Y. Q. Wang, W. F. Wang, Adjacent vertex distinguishing total colorings of outerplanar graphs, J. Comb. Optim., (2010) 19: 123–133.
  * [24] W. F. Wang, D. J. Huang, The adjacent vertex distinguishing total coloring of planar graphs, Journal of Combinatorial Optimization, (2014) 27(2): 379–396.
  * [25] Z. F. Zhang, X. E. Chen, J. W. Li, B. Yao, X. Z. Lu, J. F. Wang, On adjacent vertex distinguishing total coloring of graphs, Science in China Ser. A: Mathematics, (2005) 48 (3): 289–299.
  * [26] E.Zhu, C.Liu, J. Yu, Neighbor product distinguishing total colorings of 2-degenerate graphs, Journal of Combinatorial Optimization,(2020) 39: 72–76.


# On the interplay of direct and indirect $C\\!P$ violation in the charm
sector

M Gersabeck1, M Alexander2, S Borghi2,3, VV Gligorov1 and C Parkes3,1 1
European Organization for Nuclear Research (CERN), Geneva, Switzerland  2
School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom  3 School of Physics and Astronomy, University of Manchester,
Manchester, United Kingdom<EMAIL_ADDRESS>
###### Abstract

Charm mixing and $C\\!P$ violation observables are examined in the light of
the recently reported evidence from LHCb for $C\\!P$ violation in the charm
sector. If the result is confirmed as being due to direct $C\\!P$ violation at
the $1\%$ level, its effect will need to be taken into account in the
interpretation of $C\\!P$ violation observables. The contributions of direct
and indirect $C\\!P$ violation to the decay rate asymmetry difference $\Delta
A_{C\\!P}$ and the ratios of effective lifetimes $A_{\Gamma}$ and $y_{CP}$ are
considered here. Terms relevant to the interpretation of future high precision
measurements which have been neglected in previous literature are identified.

###### pacs:

13.25.Ft

††: J. Phys. G: Nucl. Part. Phys.

Charm, after strange and beauty, is the last system of neutral flavoured
mesons where $C\\!P$ violation remains to be discovered. While neutral
$\mathrm{B}$ mesons are characterised by their mass splitting leading to fast
oscillations and neutral kaons by their width splitting which results in a
short-lived and a long-lived state, neutral charm mesons have a very small
splitting in both mass and width. For charm, compared to the beauty sector,
this leads to rather subtle mixing-related effects in time-dependent as well
as in time-integrated charm measurements, which are examined in detail here.

First evidence for $C\\!P$ violation in the charm sector has recently been
reported by the LHCb collaboration in the study of the difference of the time-
integrated asymmetries of $D^{0}\to K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$
decay rates through the parameter $\Delta A_{C\\!P}$ [1]. This measurement is
primarily sensitive to the difference in direct $C\\!P$ violation between the
two final states as discussed further below. Direct $C\\!P$ violation depends
on the final state and is the asymmetry of the rates of particle and
antiparticle decays. It can be caused by a difference in the magnitude of the
decay rates or by a difference in their phase. Indirect $C\\!P$ violation is
considered universal, i.e. final-state independent, and is an asymmetry in the
mixing rate or in its weak phase.

Indirect $C\\!P$ violation can be measured in time-dependent analyses. To
date, two types of measurements were used to search for indirect $C\\!P$
violation in the charm sector. One uses the asymmetry of the lifetimes,
$A_{\Gamma}$, measured in $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the $C\\!P$ eigenstates
$K^{+}K^{-}$ or $\pi^{+}\pi^{-}$ [2, 3, 4]. The other is a time-dependent
Dalitz plot analysis of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to $K^{0}_{\rm\scriptstyle
S}\pi^{+}\pi^{-}$ or $K^{0}_{\rm\scriptstyle S}K^{+}K^{-}$ [5, 6]. Another
observable studied related to $A_{\Gamma}$ is $y_{CP}$, which is given by the
deviation from one of the ratio of the lifetimes measured in decays to a
Cabibbo-allowed, $C\\!P$ averaged, and a Cabibbo-suppressed, $C\\!P$
eigenstate, final state. Any deviation of a measurement of $y_{CP}$ from that
of the mixing parameter $y$ would signal $C\\!P$ violation.

In the interpretation of $A_{\Gamma}$ and $y_{CP}$, direct $C\\!P$ violation
is commonly neglected [7]. In the light of the new evidence this assumption is
no longer justified. The relevance of direct $C\\!P$ violation to a
measurement of $A_{\Gamma}$ has previously been pointed out in [8]. However, a
closer look at both $A_{\Gamma}$ and $y_{CP}$ is necessary to examine the
contribution of direct and indirect $C\\!P$ violation in these observables as
well as their connection to $\Delta A_{C\\!P}$.

The mass eigenstates of neutral $D$ mesons, $|D_{1,2}\rangle$, with masses
$m_{1,2}$ and widths $\Gamma_{1,2}$ can be written as linear combinations of
the flavour eigenstates $|D_{1,2}\rangle=p|D^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, with complex coefficients
$p$ and $q$ which satisfy $|p|^{2}+|q|^{2}=1$. The average mass and width are
defined as $m\equiv(m_{1}+m_{2})/2$ and
$\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The $D$ mixing parameters are defined
using the mass and width difference as $x\equiv(m_{2}-m_{1})/\Gamma$ and
$y\equiv(\Gamma_{2}-\Gamma_{1})/2\Gamma$. The phase convention of $p$ and $q$
is chosen such that $C\\!P|D^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$.

According to [8], the time dependent decay rates of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to the final state $f$,
which is a $C\\!P$ eigenstate with eigenvalue $\eta_{C\\!P}$, can be expressed
as

$\displaystyle\Gamma(D^{0}(t)\to
f)=\frac{1}{2}\rme^{-\tau}\left|A_{f}\right|^{2}\Big{\\{}$
$\displaystyle\left(1+|\lambda_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda_{f}|^{2}\right)\cos(x\tau)$
$\displaystyle+2\Re(\lambda_{f})\sinh(y\tau)-2\Im(\lambda_{f})\sin(x\tau)\Big{\\}},$
$\displaystyle\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(t)\to
f)=\frac{1}{2}\rme^{-\tau}\left|\bar{A}_{f}\right|^{2}\Big{\\{}$
$\displaystyle\left(1+|\lambda^{-1}_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda^{-1}_{f}|^{2}\right)\cos(x\tau)$
(1)
$\displaystyle+2\Re(\lambda^{-1}_{f})\sinh(y\tau)-2\Im(\lambda^{-1}_{f})\sin(x\tau)\Big{\\}},$

where $\tau\equiv\Gamma t$, $\kern-1.99997pt\stackrel{{\scriptstyle\kern
1.39998pt\textsf{(---)}}}{{A}}_{\kern-2.10002ptf}\kern-3.00003pt$ are the
decay amplitudes and $\lambda_{f}$ is given by

$\lambda_{f}=\frac{q\bar{A}_{f}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|\rme^{i\phi},$
(2)

where $\eta_{C\\!P}$ is the $C\\!P$ eigenvalue of the final state $f$ and
$\phi$ is the $C\\!P$ violating relative phase between $q/p$ and
$\bar{A}_{f}/A_{f}$. Introducing $|q/p|^{\pm 2}\approx 1\pm A_{m}$ and
$|\bar{A}_{f}/A_{f}|^{\pm 2}\approx 1\pm A_{d}$, one can write

$|\lambda_{f}^{\pm 1}|^{2}\approx(1\pm A_{m})(1\pm A_{d}),$ (3)

where $A_{m}$ represents a $C\\!P$ violation contribution from mixing and
$A_{d}$ from direct $C\\!P$ violation and where both $A_{m}$ and $A_{d}$ are
assumed to be small.

Expanding (On the interplay of direct and indirect $C\\!P$ violation in the
charm sector) up to second order in $\tau$ one can write the effective
lifetimes, i.e. those measured as a single exponential, as

$\displaystyle\hat{\Gamma}(\kern-1.00006pt\stackrel{{\scriptstyle\kern
0.70004pt\textsf{(---)}}}{{D}}\kern-3.00003pt(t)\to f)\approx$
$\displaystyle\Gamma\Bigg{\\{}1+\left[1\pm\frac{1}{2}(A_{m}+A_{d})-\frac{1}{8}(A_{m}^{2}-2A_{m}A_{d})\right]\eta_{C\\!P}(y\cos\phi\mp
x\sin\phi)$ (4) $\displaystyle\quad\mp A_{m}(x^{2}+y^{2})\pm
2A_{m}y^{2}\cos^{2}\phi\mp 4xy\cos\phi\sin\phi\Bigg{\\}},$

where terms below order $10^{-5}$ have been ignored. The experimental
constraints [9] give $x$, $y$, and $A_{d}$ for the final states $K^{+}K^{-}$
and $\pi^{+}\pi^{-}$ of order $10^{-2}$, and $A_{m}$ and $\sin\phi$ of order
$10^{-1}$. The sum of measurements of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays leads to the definition of
the observable $y_{CP}$ which is given by

$y_{CP}=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1\approx\eta_{C\\!P}\Bigg{\\{}\left[1-\frac{1}{8}(A_{m}^{2}-2A_{m}A_{d})\right]y\cos\phi-\frac{1}{2}(A_{m}+A_{d})x\sin\phi\Bigg{\\}}.$
(5)

The difference of measurements of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays leads to the parameter
$A_{\Gamma}$ which is defined as

$\displaystyle
A_{\Gamma}=(\hat{\Gamma}-\hat{\bar{\Gamma}})(\hat{\Gamma}+\hat{\bar{\Gamma}})^{-1}\approx$
$\displaystyle\bigg{[}\frac{1}{2}(A_{m}+A_{d})y\cos\phi-\left(1-\frac{1}{8}A_{m}^{2}\right)x\sin\phi-
A_{m}(x^{2}+y^{2})$ (6)
$\displaystyle+2A_{m}y^{2}\cos^{2}\phi-4xy\cos\phi\sin\phi\bigg{]}\frac{\eta_{C\\!P}}{1+y_{CP}}.$

The weak phase $\phi$ has not been assumed to be universal. When averaging
measurements from different channels, a potential decay-dependent weak phase
of the amplitude ratio has to be taken into account [8]. Expanding only up to
order $10^{-4}$ leads to

$y_{CP}\approx\eta_{C\\!P}\left[\left(1-\frac{1}{8}A_{m}^{2}\right)y\cos\phi-\frac{1}{2}(A_{m})x\sin\phi\right],$
(7)

and

$A_{\Gamma}\approx\bigg{[}\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\bigg{]}\frac{\eta_{C\\!P}}{1+y_{CP}}\approx\eta_{C\\!P}\left[\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\right].$
(8)

The difference of $y_{CP}$ evaluated in (7) from the expression used in
literature [7] so far is the term $\eta_{C\\!P}\frac{1}{8}A_{m}^{2}y\cos\phi$,
which can be of similar order as $\frac{1}{2}A_{m}x\sin\phi$ and should
therefore not be ignored. Equation (8) shows that there can be a significant
contribution to $A_{\Gamma}$ from direct $C\\!P$ violation. Assuming $y=1\%$
and $\cos\phi=1$, direct $C\\!P$ violation at the level of $A_{d}/2=1\%$ would
lead to a contribution to $A_{\Gamma}$ of $10^{-4}$. Current measurements
yield a sensitivity of a few $10^{-3}$ [2, 3, 4]. Future measurements at LHCb
and future $\mathrm{B}$ factory experiments are expected to reach
uncertainties of the level of $10^{-4}$, i.e. that of the direct $C\\!P$
violation contribution. More precise measurements may well change the
approximations made in (7) and (8), in particular a measurement of
$A_{m}\lesssim A_{d}$.

In time-integrated measurements the rate asymmetry is measured which is
defined as

$A_{C\\!P}\equiv\frac{\Gamma(D^{0}\to f)-\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to f)}{\Gamma(D^{0}\to
f)+\Gamma(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\to f)}.$ (9)

Introducing

$a_{C\\!P}^{dir}\equiv\frac{|A_{f}|^{2}-|\bar{A}_{f}|^{2}}{|A_{f}|^{2}+|\bar{A}_{f}|^{2}}=\frac{1-\left|\frac{\bar{A}_{f}}{A_{f}}\right|^{2}}{1+\left|\frac{\bar{A}_{f}}{A_{f}}\right|^{2}}=\frac{-A_{d}}{2+A_{d}}\approx-\frac{1}{2}A_{d},$
(10)

and using (On the interplay of direct and indirect $C\\!P$ violation in the
charm sector), (9) becomes

$A_{C\\!P}\approx
a_{C\\!P}^{dir}-A_{\Gamma}(1-(a_{C\\!P}^{dir})^{2})\frac{\langle
t\rangle}{\tau}\approx a_{C\\!P}^{dir}-A_{\Gamma}\frac{\langle
t\rangle}{\tau},$ (11)

where $\langle t\rangle$ denotes the average decay time of the observed
candidates. Terms in $\langle t\rangle^{2}$ are below order $10^{-4}$, given
current experimental constraints, and have been ignored.

A common way to reduce experimental systematic uncertainties is to measure the
difference in time-integrated asymmetries in related final states. For the
two-body final states $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$, this difference is
given by

$\displaystyle\Delta A_{C\\!P}$ $\displaystyle\equiv
A_{C\\!P}(K^{+}K^{-})-A_{C\\!P}(\pi^{+}\pi^{-})$ (12)
$\displaystyle=a_{C\\!P}^{dir}(K^{+}K^{-})-a_{C\\!P}^{dir}(\pi^{+}\pi^{-})$
$\displaystyle\quad-A_{\Gamma}(K^{+}K^{-})\frac{\langle
t(K^{+}K^{-})\rangle}{\tau}+A_{\Gamma}(\pi^{+}\pi^{-})\frac{\langle
t(\pi^{+}\pi^{-})\rangle}{\tau}.$

Assuming the $C\\!P$ violating phase $\phi$ to be universal [10] this can be
rewritten as

$\displaystyle\Delta A_{C\\!P}\approx\Delta
a_{C\\!P}^{dir}\left(1+y\cos\phi\frac{\overline{\langle
t\rangle}}{\tau}\right)+\left(a_{C\\!P}^{ind}+\overline{a_{C\\!P}^{dir}}y\cos\phi\right)\frac{\Delta\langle
t\rangle}{\tau}$ (13)

where $\Delta X\equiv X(K^{+}K^{-})-X(\pi^{+}\pi^{-})$, $\Delta X\equiv
X(K^{+}K^{-})-X(\pi^{+}\pi^{-})$, and
$a_{C\\!P}^{ind}=-(A_{m}/2)y\cos\phi+x\sin\phi$. The ratio $\overline{\langle
t\rangle}/\tau$ is equal to one for the lifetime-unbiased $\mathrm{B}$ factory
measurements [11, 12] and is $2.083\pm 0.001$ for LHCb [1] and $2.53\pm 0.02$
for CDF [13], thus leading to a correction of $\Delta a_{C\\!P}^{dir}$ of the
order of $10^{-2}$. The factor $\Delta\langle t\rangle/\tau$ multiplying the
indirect $C\\!P$ violation is zero for the $\mathrm{B}$ factory measurements
and ranges from $0.098\pm 0.003$ to $0.26\pm 0.01$ for LHCb and CDF,
respectively. Therefore, $\Delta A_{C\\!P}$ is largely a measure of direct
$C\\!P$ violation while an obvious contribution from indirect $C\\!P$
violation exists. The contribution from direct $C\\!P$ violation to
$A_{\Gamma}$ pointed out in (8) leads to a term proportional to $y$. This term
may be of similar size as the term proportional to $\Delta\langle t\rangle$
and should therefore be taken into account.

In summary, the mixing and $C\\!P$ violation parameters $y_{CP}$, $A_{\Gamma}$
and $\Delta A_{C\\!P}$ have been discussed in the light of the recent evidence
for $C\\!P$ violation in the $D^{0}$ sector. The parameter $y_{CP}$ is least
affected by direct $C\\!P$ violation, however, it contains a term which has
been neglected in the literature so far and which can be of the same order as
the constribution proportional to $x$. A measurement of $A_{\Gamma}$ can
exhibit a contribution of direct $C\\!P$ violation at the level of $10^{-4}$,
comparable to the expected future experimental sensitivity. The direct $C\\!P$
violation term in the $\Delta A_{C\\!P}$ measurement contains a contribution
proportional to $y$. The interpretation of future high precision measurements
of these observables will need to take account of these contributions.

The authors would like to thank Tim Gershon, Bostjan Golob, Alex Kagan and
Vincenzo Vagnoni for helpful discussions and comments. Furthermore, the
authors thank the LHCb collaboration whose work inspired this paper. MG and
VVG are supported by a Marie Curie Action: “Cofunding of the CERN Fellowship
Programme (COFUND-CERN)” of the European Community’s Seventh Framework
Programme under contract number (PCOFUND-GA-2008-229600). MA, SB and CP
acknowledge the support of the STFC (United Kingdom).

## References

## References

  * [1] Aaij R et al. [LHCb collaboration].  Evidence for $C\\!P$ violation in time-integrated $D^{0}\to h^{-}h^{+}$ decay rates.  2011\.  Preprint hep-ex 1112.0938.
  * [2] Staric M et al. [Belle collaboration].  Evidence for $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing.  Phys. Rev. Lett., 98:211803, 2007.
  * [3] Aubert B et al. [BaBar collaboration].  Measurement of $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing using the Ratio of Lifetimes for the Decays $D^{0}\to K^{-}\pi^{+}$ and $K^{+}K^{-}$.  Phys. Rev., D80:071103, 2009.
  * [4] Aaij R et al. [LHCb collaboration].  Measurement of mixing and $C\\!P$ violation parameters in two-body charm decays.  2011\.  Preprint hep-ex 1112.4698.
  * [5] Abe K et al. [Belle collaboration].  Measurement of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ Mixing Parameters in $D^{0}\to K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ decays.  Phys. Rev. Lett., 99:131803, 2007.
  * [6] Del Amo Sanchez P et al. [BaBar collaboration].  Measurement of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing parameters using $D^{0}\to K^{0}_{\rm\scriptstyle S}\pi^{+}\pi^{-}$ and $D^{0}\to K^{0}_{\rm\scriptstyle S}K^{+}K^{-}$ decays.  Phys. Rev. Lett., 105:081803, 2010.
  * [7] Bergmann S, Grossman Y, Ligeti Z, Nir Y, and Petrov A A.  Lessons from CLEO and FOCUS measurements of $D^{0}$-$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing parameters.  Phys. Lett., B486:418–425, 2000.
  * [8] Kagan A L and Sokoloff M D.  On Indirect $C\\!P$ Violation and Implications for $D^{0}$\- $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ and $\mathrm{B}_{(s)}$ \- $\kern 1.99997pt\overline{\kern-1.99997pt\mathrm{B}}{}_{(s)}$ mixing.  Phys.Rev., D80:076008, 2009.
  * [9] Asner D et al. [Heavy Flavor Averaging Group].  Averages of b-hadron, c-hadron, and $\tau$-lepton Properties.  2010\.  online update at http://www.slac.stanford.edu/xorg/hfag/index.html.
  * [10] Grossman Y, Kagan A L, and Nir Y.  New physics and $C\\!P$ violation in singly Cabibbo suppressed D decays.  Phys.Rev., D75:036008, 2007.
  * [11] Aubert B et al. [BaBar collaboration].  Search for $C\\!P$ violation in the decays $D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$.  Phys. Rev. Lett., 100:061803, 2008.
  * [12] Staric M et al. [Belle collaboration].  Measurement of $C\\!P$ asymmetry in Cabibbo suppressed $D^{0}$ decays.  Phys. Lett., B670:190–195, 2008.
  * [13] Aaltonen T et al. [CDF collaboration].  Measurement of $C\\!P$–violating asymmetries in $D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to K^{+}K^{-}$ decays at CDF.  Preprint hep-ex 1111.5023.


# Attention-Enhanced Reservoir Computing

Felix Köster<EMAIL_ADDRESS>Kazutaka Kanno  Jun Ohkubo  Atsushi
Uchida<EMAIL_ADDRESS>Department of Information and Computer
Sciences, Saitama University 255 Shimo-Okubo, Sakura-ku, Saitama City,
Saitama, 338–8570, Japan

(August 28, 2024)

###### Abstract

Photonic reservoir computing has been recently utilized in time series
forecasting as the need for hardware implementations to accelerate these
predictions has increased. Forecasting chaotic time series remains a
significant challenge, an area where the conventional reservoir computing
framework encounters limitations of prediction accuracy. We introduce an
attention mechanism to the reservoir computing model in the output stage. This
attention layer is designed to prioritize distinct features and temporal
sequences, thereby substantially enhancing the forecasting accuracy. Our
results show that a photonic reservoir computer enhanced with the attention
mechanism exhibits improved forecasting capabilities for smaller reservoirs.
These advancements highlight the transformative possibilities of reservoir
computing for practical applications where accurate forecasting of chaotic
time series is crucial.

††preprint: APS/123-QED

##  I Introduction

The combination of photonics and machine learning has emerged as a promising
field of information processing, utilizing the development of more rapid and
efficient computational methodologies [1]. Central to this interdisciplinary
area is the concept of photonic reservoir computing, a paradigm that utilizes
optic capabilities to enhance reservoir computations [2, 3, 4, 5, 6, 7, 8, 9].
A semiconductor laser is one example of a photonic reservoir with optical
feedback, which is a prime example of the photonic delay-based reservoir
computer model [4, 5, 6, 7, 8]. Numerous studies have underscored the efficacy
of delay-based reservoir computing in tackling a spectrum of complex tasks,
extending from bit error correction [9] to spatiotemporal chaotic systems
[10].

Chaotic time-series prediction is known to be challenging owing to its
inherent sensitivity to accuracy and its pronounced nonlinearity [11]. A prime
example of this challenge is weather forecasting, which is typically limited
to short-term predictions due to the fundamentally chaotic nature of weather
systems.

The framework of reservoir computing has been widely implemented using a
variety of physical devices, such as electronic circuits [12], spintronic
devices [13], soft robots [14], and nanostructure materials [15]. One of the
advantages of reservoir computing is the easy implementation of the reservoir
without high computational cost because input weights and network weights are
randomly fixed, and only the output weights are trained by a learning
algorithm. Classical reservoir computing is often limited because of the
inflexibility of the fixed output structure when faced with the complexities
inherent in such chaotic time series [16]. However, hardware technology
advancements have opened pathways for accelerated forecasting, giving access
to real-time responses and improved computational efficiency [17]. This
progress has enabled the integration of reservoir computing with other
advanced machine learning methods, one example being deep learning and its
most successful recent development, the attention mechanism [18].

To further enhance the predictive capabilities of reservoir computing, we
propose integrating the attention mechanism within a reservoir computer. This
mechanism has demonstrated superior effects in diverse areas of machine
learning [18]. By incorporating an attention layer as the output layer, the
system acquires the capacity to prioritize specific temporal characteristics,
potentially yielding a more dynamic comprehension of the data [19].

This study introduces the attention mechanism to delay-based photonic
reservoir computing using a semiconductor laser with optical feedback. We
perform chaotic time series prediction by deploying two benchmark tasks. The
first is the unidirectional coupling of two Lorenz systems, in which the
reservoir computer is only exposed to the $x$, $y$, and $z$ variables of the
forced Lorenz system [20], while the driving system remains hidden, thus
rendering the task a non-Markovian deterministic chaotic system with hidden
states. The second is a time series consisting of alternating Lorenz and
Rössler data sets [21], which we use to investigate how well the attention
mechanism adapts to swiftly changing tasks.

##  II Attention Enhanced Reservoir Computing

###  II.1 Reservoir Computing

Reservoir Computing (RC) is a paradigm in the field of recurrent neural
networks that offers an efficient approach to tasks involving temporal
dynamics and sequential patterns [17]. The delay-based RC is a prime example
of RC, distinctive for its utilization of time-multiplexed masking for input
representation [12]. Instead of directly feeding the input data $D(t)$ to the
system, it is multiplied, element-wise, with a periodic mask $M(t)$, which is
essentially a high-dimensional random vector repeated periodically. This mask
expands and distributes the input information across the available input span,
thus enriching the temporal dynamics of the system. The masked input $v(t)$ is
formally represented as $v(t)=D(t)\odot M(t\mod T)$, where $M$ is the masking
vector, $T$ is the period, and $\odot$ represents a $D$-elementwise
multiplication with the full $M$ vector. The readouts $R$ in delay-based RC
are also time-distributed, capturing the reservoir’s response at different
time instants within the period $T$. By sampling the system’s response at
multiple points across this period, a high-dimensional representation is
obtained, allowing for rich and diverse dynamics to be captured for downstream
tasks. In this study, we use a mask $m(t)$ that is constant for $N$ intervals
of length $\theta$, i.e. $T=N\theta$, where the values of $m_{n}$ are drawn
randomly from a uniform distribution on the interval between 0 and 1.

We use a delay-based photonic reservoir computer, which captures the dynamics
of semiconductor lasers with weak optical feedback. The Lang-Kobayashi
equations mathematically represent the systems as follows [4, 5, 22].

$\displaystyle\dot{E}(t)$
$\displaystyle=(1+i\tilde{\alpha})E(t)-|E(t)|^{2}E(t)+\eta
E(t-\tau)\exp(i\phi)$ (1) $\displaystyle\dot{n}(t)$ $\displaystyle=p+\eta
v(t)-(1+|E(t)|^{2})n(t)$ (2)

where, $E(t)$ denotes the complex electric field amplitude, $n(t)$ represents
the carrier density, $\tilde{\alpha}$ is the linewidth enhancement factor,
$\eta$ signifies the feedback strength, $\tau$ is the time delay, $\phi$ is
the phase shift, and $p$ is the normalized injection current.

Within the context of delay-based RC, the input modulates the pump term, thus
affecting the system’s dynamics. The modulated pump term is denoted as $\eta
v(t)$, where $\eta$ is the input strength and $v(t)$ is the input to the
reservoir system. The readouts $R$ are derived from the intensity $|E(t)|^{2}$
of the electric field amplitude. In the RC framework, only the readout weights
$W_{\text{out}}$, which connect the reservoir states to the output, are
trained while the internal weights of the reservoir are left unchanged [16].
The output or readout weights are crucial in mapping the reservoir states to
the desired outputs. Denoted typically as $W_{\text{out}}$, these weights are
applied to the reservoir states $R$. Training the output weights involves
optimizing them to minimize the distance between the approximations
$\tilde{U}=W_{\text{out}}R$ and the true target values $U$. Commonly, a linear
regression approach is employed [16], considering a regularization term to
prevent overfitting. The training process can be mathematically represented
as:

$W_{\text{out}}=\arg\min_{W}\left(||U-WR||^{2}+\lambda||W||^{2}\right)$ (3)

where $R$ denotes the collection of reservoir states, $U$ represents the
target outputs, and $\lambda$ is the regularization coefficient.

The analytic solution to the ridge regression problem can be derived by
setting the gradient of the objective function with respect to $W$ to zero.
The resulting equation can be expressed as:

$W_{\text{out}}=(RR^{T}+\lambda I)^{-1}RU^{T},$ (4)

where $I$ denotes the identity matrix. This solution provides a direct way to
compute the optimal $W_{\text{out}}$ without iterative optimization. The
addition of the regularization term $\lambda I$ ensures that the matrix
$RR^{T}+\lambda I$ is invertible, thus guaranteeing a unique solution and
penalizing overfitting.

Parameter |  Numerical Value
---|---
Mask period $T$ |  $N\theta=5\times 10^{-9}$ s
Linewidth enhancement factor $\tilde{\alpha}$ |  $3.01$
Phase shift $\phi$ |  $0.0$
Feedback strength $\eta$ |  $10^{8}$
Time delay $\tau$ |  $1.01T=5.05\times 10^{-9}$
Normalized injection current $p$ |  $1.11$
Input strength $\eta$ |  $0.08$
Number of nodes $N$ |  $50$
Interval length $\theta$ |  $10^{-10}$ s
Table 1: Summary of parameters and their numerical values.

###  II.2 Attention Mechanism

In deep learning, particularly when dealing with sequential data, the
attention mechanism has emerged as a groundbreaking concept [18]. This
mechanism enables models to selectively focus on relevant portions of the
input sequence during output generation, mirroring how humans selectively
prioritize information during decision-making or data processing. Central to
this mechanism is the calculation of attention weights, which subsequently
guide a weighted combination of input values. These weights effectively
signify the level of "attention" or importance each part of the input should
receive.

To clarify the method of attention, we present an example inspired by Ref.
[23]. Imagine a database containing daily temperature records over several
years, illustrated as tuples such as {(“2024-01-01”, “-2°C”), (“2024-01-02”,
“0°C”), (“2024-01-03”, “3°C”)}, where the dates act as keys ($k$) and
temperatures as values ($v$). Querying this database ($q$) with the date
“2024-01-02”, would fetch the exact temperature “0°C”. When an exact match for
the queried date is absent and approximate matching is allowed, the system
might return approximate temperatures close to neighboring dates.

Expanding on this illustration, we compute a distance function between the
query ($q$) and the keys ($k$) within the entire dataset
$\mathbf{D}=\\{(k_{1},v_{1}),\dots,(k_{m},v_{m})\\}$, providing a weighting
(or attention) to the associated values ($v$). This relationship is captured
as:

$\displaystyle\text{attention}(q,K)=\sum_{i=1}^{m}\alpha(q,k_{i})v_{i},$ (5)

where $\text{attention}(q,K)$ is the weighted sum of all data points for the
given query, $i$ indexes the dataset $\mathbf{D}$, and $\alpha(q,k_{i})$
denotes the scalar attention weights. An interesting aspect is the
determination of the function $\alpha$. Several functions can be employed,
from traditional distances like the Boxcar function, defined as
$\alpha_{\text{Boxcar}}=1$ if $||q-k_{i}||\leq 1$, to a Gaussian similarity,
given by $\alpha_{\text{Gaussian}}=\exp(||q-k_{i}||^{2}_{2})$ [23]. In
contemporary deep learning, the function $\alpha$ is often approximated using
neural networks, allowing gradient descent to tailor the function optimally to
the dataset.

A more recent progression in attention mechanisms introduces the concept of
self-attention [24, 18]. In this framework, the queries ($q$), keys ($k$), and
values ($v$) originate from the same data point. For each individual data
point (e.g, the temperature at a specific time step), corresponding queries
($q$), keys ($k$), and values ($v$) are produced. Typically, these elements
are generated through matrix transformations, where the matrix weights are
refined and optimized using a gradient descent algorithm.

(a) Input to Self-Attention(b) Input to
Reservoirqkv$M_{Q}$$M_{K}$$M_{V}$$W_{att,n}=\alpha(q_{n},K)$$W_{att,n}=\alpha_{n}(R(t))$Reservoir$R(t)$Node$r_{n}(t)$
Figure 1: (a): Self-attention mechanism creating queries ($q$), keys ($k$),
and values ($v$) from an input. (b): Reservoir approach transforming an input
into a high dimensional reservoir state $R(t)$. Both projections are then used
in a neural network $\alpha$ to yield the attention weights $W_{att,n}$.

###  II.3 Incorporation of Attention Layer in Reservoir Computing

Next, we delve into incorporating the attention mechanism within a reservoir
computer. This requires a clear understanding of its implementation and the
definitions of queries ($q$), keys ($k$), and values ($v$) in the context of
reservoir computing. While numerous potential locations for integrating the
attention mechanism exist, we focus on its application to improve the
reservoir computer’s output layer. Therefore we replace the traditional
weights $W_{\text{out}}$ with the scalars $\alpha(q,k)$.

$u(t)$$v(t)$Reservoir state $R(t)$Standardized state $S(t)$4\. Neural network
$W_{\text{out}}$Attention weights $W_{\text{att}}(t)$6\. Linear
combination$\tilde{v}(t+1)$$\tilde{u}(t+1)$1\. Standardization2\.
Processing3\. Standardization5\. Attention7\. Prediction7\.
Destandardization8\. Closed-Loop Figure 2: Schematic representation of the
reservoir computing process augmented with an attention mechanism.

In our attention-enhanced reservoir computer, the reservoir states $r_{n}$
simultaneously serve as the queries ($q$), keys ($k$), and values ($v$). This
is reminiscent of the self-attention scheme where every data point of the
input sequence is projected onto queries ($q$), keys ($k$), and values ($v$),
often via matrices learned through a gradient descent algorithm [24, 18].
Consequently, the attention weight associated with a particular reservoir node
is expressed as:

$\displaystyle\tilde{u}(t)=\sum_{n=1}^{N}W_{\text{att},n}(R(t))r_{n}(t)$
$\displaystyle=\sum_{n=1}^{N}\alpha_{n}(R(t))r_{n}(t),$ (6)

where $W_{\text{att},n}(R(t))$ is the weight applied to the $n$-th reservoir
node, $R(t)=(r_{1}(t),\dots,r_{N}(t))$ denotes the whole reservoir state and
$r_{n}(t)$ is an explicit reservoir state of the $n$-th node at time $t$, and
the summation spans all $N$ reservoir nodes. Hence, the weight assigned to the
reservoir state of the $n$-th node is given by some transformation $\alpha$ of
all $N$ reservoir states. Connecting this to the self-attention mechanism, the
reservoir projects the input into a high-dimensional feature space, similar to
the matrix transformation that projects the input data point to the queries
($q$), keys ($k$), and values ($v$). The side-by-side comparison is shown in
Fig. 1. In the self-attention mechanism, the query of a data point combined
with the keys of all other data points in the sequence generates the attention
weights, which are then weighted with the values. From Eq. (6), the role of
all reservoir states $R=\left(r_{1},\dots r_{N}\right)$ in the function
$\alpha$ is equal to the queries and the keys of the attention in Eq. (5). The
resulting attention weights $W_{\text{att},n}$ are multiplied with the values
($v$), which in our case are the reservoir states $r_{n}$. A sum over all
these weighted reservoir states $\left(W_{\text{att},1}r_{1},\dots
W_{\text{att},N}r_{N}\right)$ then yields the final prediction $\tilde{u}$.
For flexibility and optimization, we allow the function $\alpha$ to be
determined by a neural network. In the scope of this paper, this is reduced to
a linear transformation, i.e, a tensor. Thus, this simple linear
transformation approach can concisely be written as:

$\displaystyle W_{\text{att}}(t)$ $\displaystyle=W_{\text{out}}R(t),$ (7)
$\displaystyle\tilde{u}(t)$ $\displaystyle=W_{\text{att}}(t)R(t),$ (8)

where $\tilde{u}$ represents the approximation for the target $u$ of the given
task, $W_{\text{att}}(t)$ are the time-dependent attention scalar weights, and
$W_{\text{out}}$ are weights that are trained via a gradient descent
algorithm. Thus, to get the attention weight $W_{\text{att},n}(t)$ for the
$n$-th reservoir state $r_{n}$, we compute
$W_{\text{att},mn}(t)=\sum_{l=1}^{N}W_{\text{out},mnl}r_{l}$. Note that
$W_{\text{out}}$ is a third-order tensor with dimensions $\mathbb{R}^{N\times
N\times M}$, where $N$ denotes the number of readout nodes in the reservoir,
and $M$ represents the dimensionality of the target. This configuration can
result in a substantial number of weights for a reservoir of significant size
combined with a large target dimension. One potential solution to mitigate
this problem is to compute the attention weights using only a subset of
accessible nodes, reducing the $N$ term to a smaller $n^{\prime}$ term, where
$n^{\prime}\ll N$. However, we employ the full node set in our current example
with smaller dimensions.

A detailed visualization of our complete scheme is shown in the diagram
depicted in Fig. 2. We further unpack this representation by providing a step-
by-step explanation of the underlying operations.

####  II.3.1 Data Standardization

Before processing, the raw input data $u(t)$ undergoes a standardization
process, ensuring the data is in the optimal state for reservoir processing,
not saturating the reservoir state. Standardization ensures that the mean of
the data is zero and the standard deviation is one. The outcome, $v(t)$,
represents the standardized version of the input data.

####  II.3.2 Reservoir Processing

The standardized data, $v(t)$, is then passed into the reservoir, resulting in
a reservoir state, $R(t)$. This reservoir state is an intermediate, high-
dimensional representation of the input data.

####  II.3.3 State Standardization

Like the initial data, the reservoir state $R(t)$ is standardized to obtain
$S(t)$, the standardized reservoir state. This step ensures that the reservoir
state is normalized and ready for the subsequent neural network processing.

####  II.3.4 Neural Network Processing

The standardized reservoir state, $S(t)$, undergoes processing through a
neural network with weights $W_{\text{out}}$. Regarding the training of the
weights $W_{\text{out}}$, deep learning methods can be applied [25] using a
gradient descent algorithm:

$\displaystyle W_{\text{out}}(n+1)=W_{\text{out}}(n)-\gamma\nabla
F(W_{\text{out}}(n),S),$ (9)

with $\gamma$ being a learning rate, which we set to $0.01$, and $n$ iterates
through the gradient descent epochs, and the optimization objective is:

$\displaystyle W_{\text{out}}=\min_{W}\left(F(W,S,U)\right),$ (10)

where $S$ is the set of all reservoir states and $U$ is the target dataset. We
deploy as $F$ a simple squared distance, i.e.
$F(W,S,U)=\frac{1}{2}||\tilde{U}(S,W)-U||_{2}^{2}$, with $\tilde{U}(S,W)$
being the approximation to the target dataset $U$.

The attention-enhanced reservoir thus loses its advantage of utilizing the
analytically solvable ridge regression as an optimization approach. Owing to
the recent improvement in hardware-accelerated optimization of gradient
descent approaches, we believe that the limiting computation factor of
reservoir computing is not the output layer optimization. Therefore,
augmenting reservoir computing with the attention mechanism could couple the
best of two worlds.

####  II.3.5 Attention Mechanism

The attention weights, $W_{\text{att}}(t)$, emerge from the neural network.
These weights dictate how much importance is placed on each node or neuron in
the reservoir at a specific time. In essence, the attention mechanism
dynamically adjusts, prioritizing certain nodes over others based on their
relevance to the output at that particular time step.

A broad range of possible neural networks can be selected. This study focuses
on the simplest approach by adding a linear attention layer to the delay-based
reservoir computing setup. The attention weights $W_{\text{att}}(t)$ are
computed via a linear transformation of the standardized reservoir state
$S(t)$ via the weights $W_{\text{out}}$, represented as:

$\displaystyle W_{\text{att}}(t)=W_{\text{out}}S(t)$ (11)

####  II.3.6 Linear Combination

The attention weights and the standardized reservoir states are then linearly
combined. This combination effectively amplifies the states that the attention
mechanism deems significant while reducing the others. The corresponding
prediction output $\tilde{v}(t+1)$ is given by:

$\displaystyle\tilde{v}(t+1)=W_{att}(t)S(t)$ (12)

####  II.3.7 Prediction and Destandardization

The linear combination yields a prediction $\tilde{v}(t+1)$, which is
subsequently destandardized to produce the output $\tilde{u}(t+1)$. This
output represents the forecasted value of the time series.

Our integration of the attention layer in reservoir computing enables the
system to prioritize important features in the input data, particularly in
complex time series forecasting scenarios. By dynamically shifting focus to
the most important nodes in the reservoir based on context, the attention-
enhanced reservoir computer offers a more nuanced and accurate representation
of the data.

###  II.4 Prediction Quality Metrics

In the context of reservoir computing for time series forecasting, two primary
configurations are commonly used: the open-loop and closed-loop [26].

####  II.4.1 Normalized Root Mean Square Error

In the open-loop configuration, the model performs a one-step-ahead
predictions. We evaluate the model’s prediction accuracy via the normalized
root mean square error (NRMSE) [27]:

$\displaystyle\text{NRMSE}=\sqrt{\frac{\sum_{l=1}^{L}||u_{l}-\tilde{u}_{l}||^{2}}{N\text{Var}(u)}},$
(13)

where $u_{l}$ is the $l$-th target data point, $\tilde{u}_{i}$ is the
prediction data, $l$ is the index up to the $L$-th data point, and
$\text{Var}(u)$ is the variance over the whole data set.

####  II.4.2 Valid Prediction Time

Fig. 2 illustrates the closed-loop feedback configuration. In this setup,
after the model has been trained, it is switched to a closed-loop mode, where
the model’s own predictions are recursively fed back as input. This method
allows for continuous predictions, extending beyond the one-step-ahead
prediction. The quality of predictions in a closed-loop configuration is
evaluated by a metric called the valid prediction time (VPT) [28], defined as:

$\displaystyle\text{VPT}=\delta_{u}(t)>0.4,\quad\delta_{u}(t)=\frac{|u(t)-\tilde{u}(t)|^{2}}{\langle|u(t)-\langle
u(t)\rangle|^{2}\rangle},$ (14)

where VPT measures the first time at which $\delta_{u}(t)$ surpasses the value
of $0.4$, in which $\delta_{u}(t)$ measures the average normalized distance
over the trajectory between the true system $u(t)$, the approximation
$\tilde{u}$, and $\langle\rangle$ denotes the time average. VPT measures the
duration for which the model’s predictions remain within an acceptable error
threshold, thereby indicating the effectiveness of the model in maintaining
reliable forecasts over time. Unlike open-loop predictions, closed-loop
predictions compound potential errors over time, because of the sensitive
dependence on initial conditions on chaotic dynamic systems.

Figure 3: Example time series of open-loop configuration for the UCTLS. The
upper graph shows the x-variable of the time series, while the lower graph
shows the attention weights in the same time intervall for a 50-node
reservoir. The dashed black lines indicate a shift from the attractor,
pinpointing the dynamical adjustment of the weights to the task.

##  III Numerical Results

This section presents the comparative results of the attention-enhanced
photonic reservoir computing method to the conventional linear regression
technique. The evaluation is performed for a unidirectionally-coupled two-
Lorenz system introduced in Sec. III.1 and a Lorenz-Rössler system introduced
in Sec. III.2.

###  III.1 Unidirectionally-Coupled Two-Lorenz System (UCTLS)

####  III.1.1 Model

As a benchmark task, we utilize a unidirectionally-coupled two-Lorenz system
(UCTLS) [20, 29], defined by the following equations:

$\displaystyle\dot{x}_{1}$
$\displaystyle=(a+\sigma_{\text{force}}x_{2})(y_{1}-x_{1}),$ (15)
$\displaystyle\dot{y}_{1}$
$\displaystyle=x_{1}(b+\sigma_{\text{force}}y_{2}-z_{1})-y_{1},$ (16)
$\displaystyle\dot{z}_{1}$
$\displaystyle=x_{1}y_{1}-(c+\sigma_{\text{force}}z_{2})z_{1},$ (17)
$\displaystyle\dot{x}_{2}$ $\displaystyle=a(y_{2}-x_{2}),$ (18)
$\displaystyle\dot{y}_{2}$ $\displaystyle=x_{2}(b-z_{2})-y_{2},$ (19)
$\displaystyle\dot{z}_{2}$ $\displaystyle=x_{2}y_{2}-cz_{2}.$ (20)

These equations represent two Lorenz systems, with the first one being driven
by the second. The parameters used, namely $a=10$, $b=28$, and
$c=\frac{8}{3}$, are the standard Lorenz parameters [20]. The key distinction
is the introduction of a forcing term, $\sigma_{\text{force}}$, that couples
the two systems unidirectionally.

Figure 4: Closed loop time series of a UCTLS for a 50-node reservoir. The true
trajectory and closed loop prediction are depicted by the solid blue and
dashed black lines, respectively. The color bar shows the magnitude of the
attention weights. The vertical dashed black line indicates the time point for
VPT computation.

For the reservoir computing task, only the $x_{1}$, $y_{1}$, and $z_{1}$
variables are exposed, while $x_{2}$, $y_{2}$, and $z_{2}$ are hidden.
Predicting such a non-markovian deterministic chaotic time series is
challenging [30], especially when it has underlying hidden states that are not
directly observable by the reservoir computer. The system was integrated with
a Runge-Kutta 45-step approach with a step size of $dt=0.01$, while the
sampling time is $dt_{\text{sample}}=0.1$. Regarding the largest Lyapunov
exponent of the system, our simulations indicate that no significant variation
up to a force of $\sigma_{\text{force}}=0.15$ is exhibited relative to the
classic Lorenz value of $\lambda_{\text{Lorenz}}\approx 0.91$. For all
simulations we use 25000 training and 5000 testing data points.

We begin with the UCTLS as defined by Eqs. (15)-(20) subjected to a
perturbation strength of $\sigma_{\text{force}}=0.05$, implemented in a
photonic reservoir utilizing 50 nodes, with other parameters specified in
Table 1.

####  III.1.2 Results of Time Series and Weights

Fig. 3(a) illustrates an open-loop time series. The upper graph displays the
time series of the $x_{1}$-variable from the second Lorenz attractor of the
true system (in blue) alongside the predicted time series (dashed black line),
plotted against the time in Lyapunov times with
$\lambda_{\text{Lorenz}}\approx 0.91$. The similarity between the prediction
and the actual system’s behavior is accurate.

Figure 5: Comparison of VPT and NRMSE for classical linear regression (dashed
orange line) and attention-enhanced reservoir computing (solid blue line) with
varying reservoir size $N$ of UCTLS.

A more interesting insight comes from Fig. 3(b), which shows the computed
attention weights for each node within the reservoir across the same period.
Nodes with higher weight magnitudes are assumed to be more important for
predictions at specific time points. Vertical dashed lines indicate a pattern
change corresponding to moments where the system’s trajectory shifts from one
side of the attractor to the other. A corresponding shift in the attention
weights’ structure implies that the attention-augmented reservoir can adapt
dynamically to the time-dependent characteristics of the task. This feature is
essential, as the UCTLS demands time-dependent transformations depending on
the current position within the attractor space.

The attention mechanism provides a dynamic adjustment unlike the static nature
of weights employed in linear regression, which cannot distinguish this
characteristic. Given that nearly all complex systems evolve with time,
adaptively adjusting the significance of different nodes is not just
beneficial but necessary. This strongly illustrates the inherent limitations
of static approaches when dealing with systems characterized by dynamic
evolution.

This behavior continues even if the closed-loop configuration is applied, as
shown in Fig. 4. The reservoir’s prediction and the true trajectory stay close
to each other until around 5 Lyapunov times, indicated by a vertical dashed
black line pinpointing to the time at which the VPT (see Sec. II.4) is
computed. The weights in the lower graph show the distinctive feature of
adjusting to the dynamical problem itself despite being configured in the
closed-loop scheme.

####  III.1.3 Results of Quality Metrics

To showcase the efficacy of dynamic adjustment, we simulated an ensemble
prediction across ten distinct trajectories for varying reservoir sizes. The
results are shown in Fig. 5. We computed the NRMSE for the open loop depicted
in Fig. 5(a) and the VPT for the closed loop illustrated in Fig. 5(b). These
metrics were calculated for both the classical linear regression model (dashed
orange line) and the attention-augmented model (solid blue line) across a
spectrum of reservoir sizes, from 10 to 150 nodes.

The results indicate that the attention-augmented model yields a lower NRMSE
for smaller-sized reservoirs while simultaneously achieving a higher VPT.
Although this advantage diminishes as the reservoir size increases, it stays
consistent. Notably, the optimal performance of the attention-enhanced
reservoir is observed with as few as 20 nodes. This suggests that the
attention mechanism significantly reduces the required size of the reservoir.
However, this size reduction comes at the costs of additional weights in the
output layer. Nonetheless, this approach is particularly interesting when only
limited reservoir sizes are realistically accessible.

We analyze the average power spectrum of a reservoir with 20 nodes,
reproducing the UCTLS task with $\sigma_{\text{force}}=0.05$. Fig. 6 displays
the resulting spectra: the true system’s spectrum is depicted by the blue
curve, the orange curve represents the closed-loop outcome of the classical
linear regression approach, while the green curve corresponds to the linear
attention approach.

Figure 6: The Fourier Spectrum of a predicted time series in the closed-loop
configuration with a reservoir size of $N=30$. The true system is depicted as
a solid blue line, while the classic linear regression and the attention
enhanced reservoir are shown as dashed orange and green lines respectively

An examination of Fig. 6 reveals that the linear regression approach
significantly deviates in reconstructing the spectrum of the original
attractor, failing to identify the correct main frequency and the secondary
peak at a frequency of $2.9$Hz. In contrast, the linear attention method
successfully captures both frequencies, reproducing very accurately the power
spectrum.

To deepen our analysis of the UCTLS, we evaluate the dependency of the NRMSE
and VPT on the driving force $\sigma_{\text{force}}$. We keep the reservoir
size constant with $N=50$ nodes, while the other parameters are unchanged. The
results are illustrated in Fig. 7, using the same color coding and line style
as in Fig. 5. The behavior of the classic single Lorenz system is represented
on the left side of the graph with a driving force of
$\sigma_{\text{force}}=0$. The linear attention method outperforms the
classical linear regression approach for the VPT and NRMSE metrics in this
regime.

Figure 7: Mean over ten trajectories of the the VPT and NRMSE for classical
linear regression (dashed orange line) and attention enhanced reservoir
computing (solid blue line) with varying coupling strength
$\sigma_{\text{force}}$ of UCTLS. The shaded areas depict the standard
deviation. The photonic reservoir size was set to $N=50$ nodes.

When the forcing strength is increased, the observed improvement for VPT
diminishes, while the NRMSE gradually increases for both methods. However, the
attention-enhanced approach yields lower NRMSE values throughout. These
results underline the improved predictive performance of the attention-
enhanced reservoir relative to the traditional approach.

###  III.2 Alternating Lorenz and Rössler System (ALRS)

####  III.2.1 Model

The second benchmark is a target dataset constructed using two canonical
chaotic systems: the Rössler and Lorenz models [31]. For the Rössler model,
the equations of motion are given by:

$\displaystyle\frac{dx_{R}}{dt}$ $\displaystyle=-y_{R}-z_{R},$ (21)
$\displaystyle\frac{dy_{R}}{dt}$ $\displaystyle=x_{R}+a_{R}y_{R},$ (22)
$\displaystyle\frac{dz_{R}}{dt}$ $\displaystyle=b_{R}+x_{R}z_{R}-c_{R}z_{R}.$
(23)

For the Lorenz model, the dynamics are described by:

$\displaystyle\frac{dx_{L}}{dt}$ $\displaystyle=a_{L}(y_{L}-x_{L}),$ (24)
$\displaystyle\frac{dy_{L}}{dt}$ $\displaystyle=-x_{L}z_{L}+b_{L}x_{L}-y_{L},$
(25) $\displaystyle\frac{dz_{L}}{dt}$ $\displaystyle=x_{L}y_{L}-c_{L}z_{L}.$
(26)

The parameters used in our simulations are $a_{R}=0.2,b_{R}=0.2,c_{R}=-5.7$
for the Rössler system and $a_{L}=10,b_{L}=28,c_{L}=-\frac{8}{3}$ for the
Lorenz system. Both these systems are known for generating deterministic
chaos. For data generation, we exclusively consider the $x$-variable
trajectories ($x_{R}$ for Rössler and $x_{L}$ for Lorenz) if the open loop
configuration is applied, while for the closed loop configuration the $x$,
$y$, $z$ variables are exposed. The Rössler system is integrated with a time
step of $dt=0.05$, while every ten points are downsampled in an effective time
step of $dt_{\text{sample}}=0.5$. Similarly, the Lorenz system was integrated
using a time step of $dt=0.01$ and subsequently downsampled to achieve an
effective time step of $dt_{\text{sample}}=0.1$. For each system, multiple
500-step trajectories are generated. These trajectories are then alternatively
concatenated, yielding a time series that alternates between Rössler and
Lorenz every 500 points for 25000 training data points and 5000 testing data
points [31]. Before concatenation, the time series data from both models are
standardized to ensure zero mean and unit standard deviation. We call this
benchmark the ALRS. For both systems, the largest Lyapunov exponents are
calculated with $\lambda_{\text{Lorenz}}\approx 0.91$ and
$\lambda_{\text{R\"{o}ssler}}\approx 0.071$. For all simulations, we use 25000
training and 5000 test data points.

Figure 8: Example time series of open-loop configuration for ALRS. The upper
graph shows the x-variable of the time series, while the lower graph shows the
attention weights in the same time interval. The dashed black lines indicate
the change in task from Rössler to Lorenz or vice versa, pinpointing to the
dynamical adjustment of the weights to the task.

####  III.2.2 Results

We investigate the ALRS, defined by Eqs. $\eqref{eq:lorenz_roessler_1}$ to
$\eqref{eq:lorenz_roessler_last}$ in the same manner. An exemplary time series
for this system is depicted in Fig. 8. Fig. 8(a) indicates that the prediction
closely approximates the true system, suggesting a high degree of accuracy in
forecasting the next state.

Fig. 8(b) provides the interesting part of the attention-enhanced reservoir.
The vertical dashed lines denote the transitions between the Lorenz and
Rössler dynamics within the task. A shift in the attention weights in response
to these changes can be observed, implying that the attention-enhanced
reservoir can adapt to different dynamical problems presented by the input
signal.

To quantitatively analyze the prediction quality, we computed the NRMSE for
the open loop and the VPT for the closed loop in ALRS across reservoir sizes,
spanning from 10 to 150 nodes, as shown in Fig. 9. Note that in the ALRS
benchmark, two different VPTs are computed by starting the closed loop
configuration either in a window of Rössler inputs or in a window of Lorenz
inputs. Here, the NRMSE for the classical linear regression method is
presented by the dashed orange line, while the solid blue line represents the
attention-based approach. These findings agree with previous observations: the
attention-enhanced reservoir yields smaller NRMSE (higher VPT), with the
effect being particularly pronounced for smaller reservoir sizes.
Interestingly that this dynamic adjustment can tackle a problem that swiftly
changes between two distinct attractors.

Finally, we analyze the NRMSE for predicting more than one step ahead for the
UCTLS and ALRS [8]. The results are shown in Fig. 10, depicting the NRMSE over
$k$-steps for future predictions. The UCTLS system was simulated with a
driving force of $\sigma_{\text{force}}=0.05$. Both reservoirs for the UCTLS
and ALRS tasks utilize 50 nodes. The dashed orange line shows the classical
ridge regression approach while the solid blue line depicts the attention-
enhanced scheme. The improvement for the UCTLS case is clear for all
prediction steps into the future, decreasing the NRMSE by a small amount. More
impressively, the NRMSE decreases for the ALRS approach in the lower graph,
where a significant reduction in the NRMSE is clearly observed.

Figure 9: NRMSE of the open loop task and VPTs of the closed loop
configuration for the ALRS over the reservoir size $N$. The mean of ten
trajectories for the classic linear regression is shown as a dashed orange
line, while the attention-enhanced reservoir approach is depicted as a solid
blue line. The shaded areas are the standard deviation.

##  IV Conclusion

Figure 10: NRMSE of the open loop task for the UCTLS and the ALRS system over
the number of steps $k$ to predict into the future. The mean of ten
trajectories for the classic linear regression is shown as a dashed orange
line, while the attention-enhanced reservoir approach is depicted as a solid
blue line. The shaded areas represent the standard deviation.

We introduced an attention mechanism to reservoir computing to demonstrate a
notable improvement predicting complex dynamical systems. Across our
simulations of the UCTLS and ALRS, the attention-enhanced reservoir
consistently outperformed classical linear regression methods. It has shown
particular advantages in scenarios with smaller reservoir sizes, indicating
its potential to reduce system requirements without sacrificing performance
but at the cost of an attention layer.

This study highlights the importance of dynamically-adjustable weights,
characteristics of the attention mechanism, and their effect in predicting the
complex behaviors of nonlinear dynamic systems. Notably, the attention-
enhanced reservoir exhibits swift adaptability to sudden changes in dynamic
input, a quality that promises significant improvements in real-time
applications. By achieving lower prediction errors and capturing key spectral
features more accurately, the attention-enhanced approach provides an
efficient method for integrating attention within various reservoir computing
schemes.

###### Acknowledgements.

This study was partly supported by JSPS KAKENHI (JP19H00868, JP20K15185,
JP22H05195). We thank Ryugo Iwami, André Röhm, and Dhruvit Patel for
productive discussions.

## References

  * Shen  _et al._ [2017] Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, Deep learning with coherent nanophotonic circuits, Nature Photonics 11, 441 (2017).
  * Larger  _et al._ [2012] L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutiérrez, L. Pesquera, C. R. Mirasso, and I. Fischer, Photonic information processing beyond turing: an optoelectronic implementation of reservoir computing, Optics Express 20, 3241 (2012).
  * Nakajima  _et al._ [2021] M. Nakajima, K. Tanaka, and T. Hashimoto, Scalable reservoir computing on coherent linear photonic processor, Communications Physics 4, 20 (2021).
  * Brunner  _et al._ [2013] D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, Parallel photonic information processing at gigabyte per second data rates using transient states, Nature Communications 4 (2013).
  * Nakayama  _et al._ [2016] J. Nakayama, K. Kanno, and A. Uchida, Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal, Optics Express 24, 8679 (2016).
  * Bueno  _et al._ [2017] J. Bueno, D. Brunner, M. C. Soriano, and I. Fischer, Conditions for reservoir computing performance using semiconductor lasers with delayed optical feedback, Optics Express 25, 2401 (2017).
  * Sugano  _et al._ [2020] C. Sugano, K. Kanno, and A. Uchida, Reservoir computing using multiple lasers with feedback on a photonic integrated circuit, IEEE Journal of Selected Topics in Quantum Electronics 26, 1 (2020).
  * Takano  _et al._ [2018] K. Takano, C. Sugano, M. Inubushi, K. Yoshimura, S. Sunada, K. Kanno, and A. Uchida, Compact reservoir computing with a photonic integrated circuit, Optics Express 26, 29424 (2018).
  * Sackesyn  _et al._ [2021] S. Sackesyn, C. Ma, J. Dambre, and P. Bienstman, Experimental realization of integrated photonic reservoir computing for nonlinear fiber distortion compensation, Optics Express 29, 30991 (2021).
  * Rafayelyan  _et al._ [2020] M. Rafayelyan, J. Dong, Y. Tan, F. Krzakala, and S. Gigan, Large-scale optical reservoir computing for spatiotemporal chaotic systems prediction, Physical Review X 10, 041037 (2020).
  * Strogatz [2000] S. H. Strogatz,  _Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering_ (Westview Press, 2000).
  * Appeltant  _et al._ [2011] L. Appeltant, M. C. Soriano, G. Van der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, Information processing using a single dynamical node as complex system, Nature Communications 2, 468 (2011).
  * Torrejon  _et al._ [2017] J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V. Cros, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier, Neuromorphic computing with nanoscale spintronic oscillators (2017), arXiv:1701.07715 [cs.ET] .
  * Nakajima  _et al._ [2015] K. Nakajima, H. Hauser, T. Li, and R. Pfeifer, Information processing via physical soft body, Scientific Reports 5, 10.1038/srep10487 (2015).
  * Kotooka  _et al._ [2021] T. Kotooka, S. Lilak, A. Stieg, J. Gimzewski, N. Sugiyama, Y. Tanaka, H. Tamukoh, Y. Usami, and H. Tanaka, Ag2se nanowire network as an effective in-materio reservoir computing device (2021), preprint, not peer-reviewed, ResearchSquare:10.21203/rs.3.rs-322405/v1 .
  * Jaeger [2001] H. Jaeger, The “echo state” approach to analysing and training recurrent neural networks-with an erratum note, Bonn, Germany: German National Research Center for Information Technology GMD Technical Report 148, 13 (2001).
  * Tanaka  _et al._ [2019] G. Tanaka, T. Yamane, J. A. He, R. Nakane, S. Kanazawa, H. Takeda, H. Numata, D. Nakano, and A. Fujii, Recent advances in physical reservoir computing, Neural Networks 115, 100 (2019).
  * Vaswani  _et al._ [2017] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, Attention is all you need, in  _Advances in neural information processing systems_ (2017) pp. 5998–6008.
  * Bai  _et al._ [2018] S. Bai, J. Z. Kolter, and V. Koltun, An empirical evaluation of generic convolutional and recurrent networks for sequence modeling, arXiv preprint arXiv:1803.01271 (2018).
  * Lorenz [1963] E. N. Lorenz, Deterministic nonperiodic flow, Journal of Atmospheric Sciences 20, 130 (1963).
  * Igra [2023] E. Igra, Knots and chaos in the rössler system (2023), arXiv:2306.04772 [math.DS] .
  * Uchida [1994] A. Uchida, Optical communication with chaotic lasers: Applications of nonlinear dynamics and synchronization (1994).
  * Zhang  _et al._ [2023] A. Zhang, Z. C. Lipton, M. Li, and A. J. Smola,  _Dive into Deep Learning_ (Cambridge University Press, 2023) https://D2L.ai.
  * Bahdanau  _et al._ [2014] D. Bahdanau, K. Cho, and Y. Bengio, Neural machine translation by jointly learning to align and translate, arXiv preprint arXiv:1409.0473 (2014).
  * Goodfellow  _et al._ [2016] I. Goodfellow, Y. Bengio, and A. Courville,  _Deep learning_ (MIT press, 2016).
  * Cucchi  _et al._ [2022] M. Cucchi, S. Abreu, G. Ciccone, D. Brunner, and H. Kleemann, Hands-on reservoir computing: a tutorial for practical implementation, Neuromorphic Computing and Engineering 2, 032002 (2022).
  * James  _et al._ [2013] G. James, D. Witten, T. Hastie, and R. Tibshirani,  _An introduction to statistical learning: with applications in R_ (Springer, New York, 2013) xiv, 426 pages: illustrations (some color); 24 cm.
  * Pathak  _et al._ [2018] J. Pathak, B. Hunt, M. Girvan, Z. Lu, and E. Ott, Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Physical Review Letters 120, 024102 (2018).
  * Fen [2017] M. O. Fen, Persistence of chaos in coupled lorenz systems, Chaos, Solitons & Fractals 95, 200 (2017).
  * Abarbanel  _et al._ [1993] H. D. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, The analysis of observed chaotic data in physical systems, Reviews of Modern Physics 65, 1331 (1993).
  * Kanno  _et al._ [2020] K. Kanno, M. Naruse, and A. Uchida, Adaptive model selection in photonic reservoir computing by reinforcement learning, Scientific Reports 10 (2020).