# A Game Approach to Multi-dimensional Opinion Dynamics in Social Networks
with Stubborn Strategist Agents

Hossein B. Jond  Department of Computer Science, VSB-Technical University of
Ostrava, Ostrava-Poruba, 708 00, Czech Republic  Aykut Yıldız  Department of
Electrical and Electronics Engineering, TED University, Ankara, 06420, Türkiye

###### Abstract

In a social network, individuals express their opinions on several
interdependent topics, and therefore the evolution of their opinions on these
topics is also mutually dependent. In this work, we propose a differential
game model for the multi-dimensional opinion formation of a social network
whose population of agents interacts according to a communication graph. Each
individual’s opinion evolves according to an aggregation of disagreements
between the agent’s opinions and its graph neighbors on multiple
interdependent topics exposed to an unknown extraneous disturbance. For a
social network with strategist agents the opinions evolve over time with
respect to the minimization of a quadratic cost function that solely
represents each individual’s motives against the disturbance. We find the
unique Nash/worst-case equilibrium solution for the proposed differential game
model of coupled multi-dimensional opinions under an open-loop information
structure. Moreover, we propose a distributed implementation of the
Nash/worst-case equilibrium solution. We examine the non-distributed and
proposed distributed open-loop Nash/worst-case strategies on a small social
network with strategist agents in a two-dimensional opinion space. Then we
compare the opinions evolved based on the Nash/worst-case strategy with the
opinions corresponding to social optimality actions for non-strategist agents.

Keywords: Game theory; Multi-dimensional opinion dynamics; Nash/worst-case
equilibrium; Social networks

##  1 Introduction

Opinion dynamics in social networks has attracted a lot of attention in recent
years from social sciences and control theory [1, 2]. The extensive attention
is due to the widespread applications in business, finance, marketing,
e-commerce, politics, group decision-making, pandemic spread [3, 4], and so
on. See [5, 6] for surveys of applications of opinion dynamics. Researchers
are looking for ideal mathematical models of interaction dynamics in a network
of relationships to answer the critical question of how opinions form. Several
models in this regard have been proposed to explain certain behaviors in the
formation of opinions. The well-known ones are the French-DeGroot (FD) model
[7], the Friedkin-Johnsen (FJ) model [8], the Hegselmann-Krause (HK) model
[9], and Deffuant-Weisbuch (DW) model [10]. The FD model of opinion dynamics
addresses the emergence of consensus in a connected network topology. The FJ
model includes stubborn agents and results in opinion diversity rather than
consensus. In the HK model, the individuals are influenced by their nearest
neighbors, so opinion clustering emerges rather than polarization or
consensus. In the DW model, two randomly selected individuals take part in the
opinion exchange business according to a threshold, resulting in consensus for
a large selection of the threshold or clustering otherwise.

The network of interactions is usually defined as a graph to capture the
patterns of interactions. In this context, the set of nodes represents the
individuals, and each edge denotes a direct interaction, e.g., opinion
transmission, between two individuals. On a communication graph with
nonnegative weights assigned to its edges, the collaborative effort of all
agents causes the network to reach an opinion consensus [11]. Networks with
antagonistic agent interactions modeled by negative edge weights representing
anticooperative behavior have also been studied [12, 13]. Multidimensional
extensions of classical opinion dynamics models for multiple interdependent
topics have recently been a topic of interest. The discrete-time
multidimensional FJ and DG extensions are analyzed in [14] and their
continuous-time counterparts with stubborn agents are presented in [15]. A
discrete-time multidimensional FJ model where each stubborn agent has a time-
evolving stubbornness level is studied in [16].

Previously, the classical opinion dynamics models considered social network
agents as non-strategist entities, where each agent simply updated its
opinions according to attractive interactions with its neighbors. However,
when the agents of a social network become strategists, they become decision-
makers who are selfish and prioritize their own interests in achieving the
network’s opinion consensus. Strategist agents with self-interests show
noncooperative behavior, and thus opinion dynamics become a noncooperative
multi-agent decision-making problem. In this problem, the decisions that other
strategists are making at the same time impact each agent’s opinion. For
dynamical agents under the game theory framework, each agent, as a player of
the game, determines her strategies in accordance with an individual
optimization problem. Opinion dynamics in social networks has been studied
using game theory in the literature for the last two decades [17, 18, 19, 20,
21]. An opinion formation game model for competition and bargaining among
social network agents was studied in [22]. For selfish agents who take actions
independently, the Nash equilibrium applies, and for agents who coordinate
their behavior by negotiating their influence efforts, the Nash bargaining
solution applies. Using a game setting, the cost of disagreement at best-
response dynamics equilibrium was analyzed in [23] by comparing the cost at
equilibrium with the social optimum. The ratio of the worst equilibrium’s
social cost to the optimal social cost as a measure of the price of anarchy in
networks with directed graphs was studied for its boundedness in [24]. Public
opinion evolution in the presence of conformity and manipulation behaviors via
a game-theoretic approach was investigated in [25]. Fast convergence with
limited information exchange as a natural behavior for selfish agents under a
game-theoretic model for social networks was studied in [26]. A game framework
is used for optimal investment strategies for two competing camps, where the
strategy of each camp comprises how much to invest in each node in a social
network in [27, 28, 29]. The problem of competitive information spreading in
the social network, e.g., see [30], as a zero-sum game that admits a unique
Nash equilibrium in pure strategies is analyzed in [31, 32]. Cost function
learning, as opposed to the common assumption that the game players know each
other’s cost function for decision-making, is analyzed for memorized social
networks in [33].

Recently, the differential game and its extension for large networks of
agents, i.e., the mean-field game [34], were proven to provide models that are
examined in a rigorous manner [35, 36, 37, 38, 39]. Differential games are one
of the decentralized extensions of optimal control theory. Unlike in optimal
control, there are multiple cost functions in a different game to minimize
simultaneously [40]. Each agent tries to minimize her cost function, which
consists of the deviation of her opinion from others (i.e., her graph
neighbors) and her initial opinion as behavior characteristics expected from
selfish agents.

In this paper, we propose a differential game model for multi-dimensional
opinion formation in a social network on a communication graph. The proposed
model is a multi-dimensional extension to differential game models in [36, 37,
38] and additionally, an unknown disturbance is included in the formulation.
Furthermore, the proposed model is a differential game contribution to the
current literature in multi-dimensional coupled opinion dynamics studies [14,
41, 42, 43, 44]. The agents communicate on several unrelated or interdependent
topics on an information graph. The game relies on the assumption that the
players know each other’s cost functions for decision-making. Under the
noncooperative mode of play, we derive an explicit open-loop solution for the
Nash/worst-case equilibrium and its associated opinion formation trajectory.
Additionally, we propose a distributed implementation of the solution based on
the information available to each agent. The proposed game provides an
analytical insight into the opinion dynamics of social networks via Nash
equilibrium. Consequently, the main contributions of this work are summarized
as follows: First, a differential model for multi-dimensional opinion dynamics
on directed graphs is established. The cost functions include a stubbornness
term for each agent. Second, the game strategies for each agent were found in
an explicit expression derived by applying the necessary conditions for
optimality using Pontryagin’s principle. Third, we propose a distributed
implementation of game strategies that each agent can execute only using
information from her graph neighbors.

The rest of the manuscript is organized as follows. In Section 2, some basic
concepts in graph theory and the linear quadratic differential game are
presented. In Section 3, the game of opinion formation is introduced. In
Section 4, the explicit solution of the open-loop Nash/worst-case equilibrium
is derived, and in Section 5, its distributed implementation is discussed.
Opinion formation as social optimality is considered in Section 6. In Section
7, numerical simulation results are demonstrated on a small social network
with five agents. Finally, Section 8 is dedicated to the conclusions and
future works.

##  2 Preliminaries and Notation

The differential game framework presented in this paper for opinion formation
in networks on graphs relies on some basic concepts in graph theory and the
linear quadratic differential game that are discussed in the following.

###  2.1 Graph theory

A directed graph is a pair $\mathcal{G}(\mathcal{V},\mathcal{E})$ where
$\mathcal{V}$ is a finite set of vertices or nodes and
$\mathcal{E}\subseteq\\{(i,j):i,j\in\mathcal{V}\\}$ is a set of edges or arcs.
Each edge $(i,j)\in\mathcal{E}$ represents an information flow from node $i$
to node $j$ and is assigned a positive weight $\omega_{ij}>0$. The pair
$(i,i)$ represents a self-loop. The set of neighbors of vertex $i$ is defined
by $\mathcal{N}_{i}=\\{j\in\mathcal{V}:(i,j)\mbox{ or
}(j,i)\in\mathcal{E},j\neq i\\}$. Graph $\mathcal{G}$ is connected if for
every pair of vertices $(i,j)\in\mathcal{V}\times\mathcal{V}$, from $i$ to $j$
for all $j\in\mathcal{V},j\neq i$, there exists a path of (undirected) edges
from $\mathcal{E}$. Matrix $D\in\mathbb{R}^{\mathcal{\lvert
V\rvert}\times\mathcal{\lvert E\rvert}}$ is the incidence matrix of
$\mathcal{G}$ where $D$’s $uv$th element is 1 if the node $u$ is the head of
the edge $v$, $-1$ if the node $u$ is the tail, and $0$, otherwise.

An undirected graph is a pair $\mathcal{G}(\mathcal{V},\mathcal{E})$ where
$\mathcal{V}$ is a finite set of vertices and
$\mathcal{E}\subseteq\\{\\{i,j\\}:i,j\in\mathcal{V}\\}$ is a set of edges.
Each edge $\\{i,j\\}$ represents bidirectional information flow between node
$i$ and node $j$. The set of neighbors of vertex $i$ is defined by
$\mathcal{N}_{i}=\\{j\in\mathcal{V}:\\{i,j\\}\in\mathcal{E},j\neq i\\}$. For
an undirected graph, when constructing $D$, the direction of the edges is
arbitrary.

The graph Laplacian matrix is defined as

$L=DWD^{\top},$ (1)

where
$W=\mathrm{diag}(\cdots,\omega_{ij},\cdots)\in\mathit{\mathbb{R}}^{\mathcal{\lvert
E\rvert}\times\mathcal{\lvert E\rvert}}$ $\forall(i,j)\in\mathcal{E}$. In
other words, the $ij$th entry of the Laplacian matrix is -$\omega_{ij}$, and
its $i$th diagonal entry is $\sum_{j\in\mathcal{N}_{i}}\omega_{ij}$. The
Laplacian $L$ is symmetric $(L=L^{\top})$, positive semidefinite (PSD) $(L\geq
0)$, and satisfies the sum-of-squares property [45]

$\sum_{\\{i,j\\}\in\mathcal{E}}\omega_{ij}\|y_{i}-y_{j}\|^{2}=y^{\top}Ly,$ (2)

where $y=[y_{1},\cdots,y_{n}]^{\top}$ is a nonzero vector and $\|.\|$ is the
Euclidean norm. The nonzero eigenvalues of Laplacian $L$ have nonnegative real
parts. A more compact notation for the matrix expression on the right hand of
(2) is $\|y\|_{L}^{2}$. For any square matrix $A$, the compact form
$\|y\|_{A}^{2}$ means $\|y\|_{A}^{2}=y^{\top}Ay$.

The Kronecker product extends the dimension of a matrix [46]. The extended
dimensional incidence matrix $\mathcal{D}$ is defined as

$\mathcal{D}=D_{n\times\mathcal{\lvert E\rvert}}\otimes I_{m\times
m}=\begin{bmatrix}d_{11}I_{m\times m}&\cdots&d_{1\mathcal{\lvert
E\rvert}}I_{m\times m}\\\ \vdots&\ddots&\vdots\\\ d_{n1}I_{m\times
m}&\cdots&d_{n\mathcal{\lvert E\rvert}}I_{m\times
m}\end{bmatrix}\in\mathit{\mathbb{R}}^{nm\times\mathcal{\lvert E\rvert}m},$

where $D=[d_{ij}]$, $I_{m\times m}$ is the identity matrix of dimension $m$,
and $\otimes$ demonstrates the Kronecker product operator. According to the
Kronecker product properties, $(D\otimes I_{m\times
m})^{\top}=(D^{\top}\otimes I_{m\times m})=\mathcal{D}^{\top}$.

The Laplacian matrix definition in (1) can be revisited as

$\mathcal{L}=(D\otimes I_{m\times m})\mathcal{W}(D\otimes I_{m\times
m})^{\top}=\mathcal{D}\mathcal{W}\mathcal{D}^{\top},$

where $\mathcal{L}$ is the extended dimensional Laplacian matrix,
$\mathcal{W}=\mathrm{diag}(\cdots,\omega_{ij},\\\
\cdots)~{}\forall\\{i,j\\}\in\mathcal{E}$ is a block diagonal weighting matrix
and $\omega_{ij}\in\mathit{\mathbb{R}}^{m\times m}$.

###  2.2 Linear Quadratic Nash/Worst-Case Equilibria

Consider an $N$-player noncooperative linear-quadratic differential game with
an unknown disturbance described by the following linear differential equation
[47]

$\dot{x}(t)=A(t)x(t)+\sum_{i=1}^{N}B_{i}(t)u_{i}(t)+\sum_{i=1}^{N}B_{\varpi_{i}}(t)\varpi_{i}(t),\quad
x(0)=x_{0}$ (3)

where $x(t)\in\mathit{\mathbb{R}}^{n}$ is the state vector, $u_{i}(t)$ denotes
the input control vector for player $i$, and $\varpi_{i}(t)\in$ is an
extraneous input representing the unknown disturbance. We assume that this
unknown disturbance is finite in the sense that
$\int_{0}^{\infty}\varpi_{i}^{2}(t)~{}\mathrm{dt}$ exists as a finite number
(i.e. $\varpi_{i}(t)$ is square integrable or $\varpi_{i}(t)\in
L_{2}^{\dagger}$.

Each player has a quadratic cost function

$J_{i}=\frac{1}{2}x^{\top}(t_{f})Q_{i}x(t_{f})+\frac{1}{2}\int_{0}^{t_{f}}\big{(}u_{i}^{\top}(t)R_{i}u_{i}(t)-\varpi_{i}^{\top}(t)R_{\varpi_{i}}\varpi_{i}(t)\big{)}~{}\mathrm{dt},$
(4)

where $R_{i}$ and $R_{\varpi_{i}}$ are positive definite for $i=1,\cdots,N$.

Due to the uncooperativeness assumption, a so-called set of Nash equilibrium
actions is expected to be played. By adopting a Nash/worst-case equilibrium,
every player has no incentive to change her policy given her worst-case
expectations of the disturbance and the actions of other players. Under open-
loop information, a player’s policy will depend on information only from the
beginning of the game.

Let $\hat{E}_{Nn}$ denote the block-column matrix of $N$ blocks of the
identity matrix of dimension $n$ and $\bar{E}_{i,j}$ be the block-column
matrix of $i$ blocks of zero matrices of dimension $n$ except for block $j$
which is an identity matrix. $\mathrm{diag}(A)_{N}$ also denotes the $N\times
N$ block diagonal matrix with diagonal blocks $A$.

As mentioned in Corollary 3.2 in [47], the linear-quadratic differential game
(3) and (4) is rewritten as the $2n$-player differential game

$\dot{\bar{x}}(t)=\bar{A}_{N}(t)\bar{x}(t)+\sum_{i=1}^{N}\bar{B}_{i}(t)u_{i}(t)+\sum_{i=1}^{N}\bar{B}_{\varpi_{i}}(t)\varpi_{i}(t),\quad\bar{x}(0)=\bar{x}_{0},$
(5)

where player $i$ tries to minimize with respect to $u_{i}(t)$ and player $N+i$
tries to maximize with respect to $\varpi_{i}(t)$ the cost function

$\bar{J}_{i}=\frac{1}{2}\bar{x}^{\top}(t_{f})\bar{Q}_{i}\bar{x}(t_{f})+\frac{1}{2}\int_{0}^{t_{f}}\big{(}u_{i}^{\top}(t)R_{i}u_{i}(t)-\varpi_{i}^{\top}(t)R_{\varpi_{i}}\varpi_{i}(t)\big{)}~{}\mathrm{dt},$
(6)

where $\bar{x}(t)=\hat{E}_{Nn}x(t)$, $\bar{B}_{i}(t)=\hat{E}_{Nn}{B}_{i}(t)$,
$\bar{B}_{\varpi_{i}}(t)=\bar{E}_{N,i}{B}_{\varpi_{i}}(t)$, and
$\bar{Q}_{i}=\bar{E}_{N,i}Q_{i}\bar{E}_{N,i}^{\top}$.

Nash/worst-case equilibria for the linear-quadratic differential game (5) and
(6) are derived using Pontryagin’s principle in terms of a nonsymmetric
Riccati equation.

Let $G=\mathrm{diag}(R_{1},\cdots,R_{n})$, which is invertible since $R_{i}$
($i=1,\cdots,N$) are, $B(t)=[\bar{B}_{1}(t),\cdots,\bar{B}_{n}(t)]^{\top}$,
$\tilde{B}^{\top}(t)=\mathrm{diag}(B_{1}^{\top}(t),\cdots,B_{n}^{\top}(t))$,
$\tilde{S}(t)=BG^{-1}\tilde{B}^{\top}(t)-\mathrm{diag}(S_{\varpi_{i}})$,
$\tilde{M}(t)=\begin{bmatrix}\bar{A}_{N}(t)&-\tilde{S}(t)\\\
-Q&-\bar{A}_{N}^{\top}(t)\end{bmatrix}$,
$\tilde{S}(t)=\hat{E}_{Nn}[S_{1}(t),\cdots,\\\
S_{N}(t)]-\mathrm{diag}(S_{\varpi_{i}})$, $Q=\mathrm{diag}(Q_{i})$,
$S_{j}(t)=B_{j}^{\top}(t)R_{j}^{-1}B_{j}(t)$, and
$S_{\varpi_{i}}(t)=B_{\varpi_{i}}^{\top}(t)R_{\varpi_{i}}^{-1}B_{\varpi_{i}}(t)$.

In [47], it is shown that if the nonsymmetric Riccati differential equation

$\displaystyle\dot{P}(t)+\bar{A}_{N}^{\top}(t)P(t)+P(t)\bar{A}_{N}(t)-P(t)\tilde{S}(t)P(t)=0,\quad
P(t_{f})=Q,$ (7)

has a solution on $[0,t_{f}]$, the linear-quadratic differential game (3) and
(4) has a unique Nash/worst-case equilibrium for every initial state $x_{0}$.
The worst-case Nash equilibrium actions and corresponding worst-case
disturbances are given by the following

$\displaystyle u(t)=-G^{-1}\tilde{B}^{\top}(t)P(t)\bar{x}(t),$
$\displaystyle\varpi_{i}(t)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}(t)P_{i}(t)\bar{x}(t),$

where $u(t)=[u_{1}^{\top}(t),\cdots,u_{N}^{\top}(t)]^{\top}$,
$P^{\top}(t)=[P_{1}^{\top}(t),\cdots,P_{N}^{\top}(t)]$,
$P_{i}(t)\in\mathit{\mathbb{R}}^{n\times nN}$. Moreover, the state trajectory
satisfies

$\dot{\bar{x}}(t)=A_{cl}(t)\bar{x}(t),\quad\bar{x}(0)=\bar{x}_{0},$

where

$A_{cl}(t)=\bar{A}_{N}(t)-BG^{-1}\tilde{B}^{\top}(t)P(t)\bar{x}(t)+\mathrm{diag}(S_{\varpi_{i}})P(t).$

##  3 Game of Opinion Formation

Consider a social network of $n$ (heterogeneous) agents indexed by $1$ through
$n$ on a communication graph $\mathcal{G}(\mathcal{V},\mathcal{E})$
communicating about $m$ interdependent topics ($m\in\mathit{\mathbb{N}}>1$).
The set of vertices $\mathcal{V}=\\{1,\cdots,n\\}$ corresponds to the set of
agents. Each edge $(i,j)\in\mathcal{E}$ represents an opinion flow from node
$i$ to node $j$ and is assigned an interpersonal influences matrix
$W_{ij}\in\mathit{\mathbb{R}}^{m\times m}$ with nonnegative entries. Without a
loss of generality, assume there are two interdependent topics (i.e., $m=2$),
namely, topic $p$ and topic $q$. The off-diagonal entry $\omega_{ij}^{pq}>0$
in $W_{ij}=\begin{bmatrix}\omega_{ij}^{pp}&\omega_{ij}^{pq}\\\
\omega_{ij}^{qp}&\omega_{ij}^{qq}\end{bmatrix}$ imposes a coupling between
topic $p$ of agent $i$ and topic $q$ of agent $j$. Similarly,
$\omega_{ij}^{qp}>0$ relates to a coupling between topic $q$ of agent $i$ and
topic $p$ of agent $j$. Their magnitude corresponds to the level of trust
accorded one agent in the opinion of the other agent. $W_{ii}$ is called the
stubbornness matrix for agent $i$, where its entries are the stubbornness
coefficients. If agent $i$ is not stubborn about a topic, the corresponding
diagonal entry for that topic becomes zero. Similarly, the off-diagonal
entries for topics that are not evolving in coupling by the opinion dynamics
model are also set to zero.

######  Assumption 1.

The opinion network graph $\mathcal{G}$ for every topic is connected.

The connectivity of $\mathcal{G}$ means there exists at least one globally
reachable node for every topic (a root node of a spanning tree on the graph).
In an opinion network on a connected communication graph, each agent for every
topic has at least one neighbor with whom they interact. Two agents $i$ and
$j$ are said to be graph neighbors if a topic in agent $i$ is coupled to
another topic or the same topic in another agent $j$ through a nonzero matrix
$W_{ij}$.

Let $x_{i}(t)\in\mathit{\mathbb{R}}^{m\times 1}$ denote the opinion vector of
agent $i\in\mathcal{V}$ at time $t>0$ which evolves in continuous time
according to the following dynamics [48]

$\dot{x}_{i}(t)=\sum_{j\in\mathcal{N}_{i}}W_{ij}(x_{j}(t)-x_{i}(t)).$ (8)

The meaning of this model is that each agent’s opinion on multiple
interdependent topics evolves according to the sum of disagreements between
the opinions of each agent and its graph neighbors on those topics.

Traditionally, an agent in a social graph is modeled as a node that simply
aggregates the opinions of other agents to update its own opinion according to
a specified rule without strategically influencing other agents’ opinions. In
this study, we assume that every agent in the social network graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$ strategically and directly influences
the opinions of its neighboring agents on the graph. Therefore, all agent
interactions, restricted by the graph topology, are strategic. The agents
decide on their influence efforts, i.e., find strategies under a
noncooperative game framework, by minimizing a personal cost function that
consists of opinion disagreement costs with their neighborhood and their own
initial opinions (i.e., stubbornness costs).

For optimization purposes for strategizing agents in a social network, let the
vector $u_{i}(t)\in\mathit{\mathbb{R}}^{d\times 1}$ denote the agent’s control
or influence, and the input $\varpi_{i}(t)$ be an extraneous input
representing an unknown disturbance influencing the system ($\varpi_{i}(t)\in
L_{2}$). Then, the dynamics (8) becomes

$\dot{x}_{i}(t)=\sum_{j\in\mathcal{N}_{i}}W_{ij}(x_{j}(t)-x_{i}(t))+b_{i}u_{i}(t)+\xi_{i}\varpi_{i}(t).$
(9)

Here, $b_{i}\in\mathit{\mathbb{R}}^{m\times d}$ and
$\xi_{i}\in\mathit{\mathbb{R}}^{m\times d}$ represent the dependency structure
of multiple-control inputs. $b_{i}$’s entries denote the level of trust of
agent $i\in\mathcal{V}$ in the declared opinion or her actions (influence
efforts). $\xi_{i}$ models the disturbance’s internal structure from the point
of view of agent $i\in\mathcal{V}$.

Let us define the following finite time horizon quadratic cost for agent
$i\in\mathcal{V}$ to minimize in $t\in[0,t_{f}]$ in order to determine her
actions

$\displaystyle J_{i}=\frac{1}{2}\|x_{i}(t_{f})-x_{i}(0)\|_{W_{ii}}^{2}$
$\displaystyle+\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}\|x_{i}(t_{f})-x_{j}(t_{f})\|_{W_{ij}}^{2}+\frac{1}{2}\int_{0}^{t_{f}}\big{(}\|u_{i}(t)\|_{R_{i}}^{2}-\|\varpi_{i}(t)\|_{R_{\varpi_{i}}}^{2}\big{)}~{}\mathrm{dt},$
(10)

where $R_{i}\in\mathit{\mathbb{R}}^{d\times d}$ and
$R_{\varpi_{i}}\in\mathit{\mathbb{R}}^{d\times d}$ are positive definite, and
$t_{f}$ is the terminal time. Matrix $R_{\varpi_{i}}$ models the agent’s
expectation about the severity of the disturbance $\xi_{i}\varpi_{i}(t)$.
Agents who expect insignificant disturbances choose large $R_{\varpi_{i}}$
[47]. A finite terminal time refers to situations where the attitudes of
social groups are analyzed in a limited amount of time, like during a
political or advertising campaign with an end date.

The cost function in (10) has three terms. The first term is a weighted
difference between the terminal opinions vector $x_{i}(t_{f})$ and the initial
opinions vector $x_{i}(0)$ (or the agent’s prejudices). Agents with prejudices
(i.e., $x_{i}(0)\neq 0$) are stubborn, thus this term reflects the agent’s
stubbornness. The second term is a weighted sum of differences between the
terminal opinions vector $x_{i}(t_{f})$ and the terminal opinions vector
$x_{j}(t_{f})$ for each neighboring $j\in\mathcal{N}_{i}$ according to the
communication graph $\mathcal{G}$. The last term is the weighted control or
influence effort made during the entire opinion formation process, considering
his expectations of the disturbance.

The proposed cost function in this paper unifies the graph definition of a
network with a control effort in a multi-dimensional opinion space exposed to
disturbances. By adopting it, each agent is trying to minimize their
disagreement with their initial opinions and with the opinions of their
neighbors at the terminal time with the least influence effort during the
whole process of opinion formation. By moving toward their neighbors’
opinions, players can prevent additional costs from being incurred by their
neighbors [23]. Therefore, opinion formation behavior is characterized by
minimizing the cost function at equilibrium. To that end, we like to find a
control policy that minimizes the cost of disagreement with the least effort
against a worst-case assumption of the disturbance.

By concatenating the states of all agents into a vector
$x(t)=[x_{1}^{\top}(t),\cdots,\\\ x_{n}^{\top}(t)]^{\top}$, the state equation
with a given initial state is given by

$\dot{x}(t)=-\mathcal{L}x(t)+\sum_{i=1}^{n}B_{i}u_{i}(t)+\sum_{i=1}^{n}B_{\varpi_{i}}\varpi_{i}(t),\quad
x(0)=x_{0},$ (11)

where $\mathcal{L}=\mathcal{D}\mathcal{W}\mathcal{D}^{\top}$, and
$\mathcal{W}=\mathrm{diag}(\cdots,W_{ij},\cdots)~{}\forall(i,j)\in\mathcal{E}$,
$B_{i}=[0,\cdots,b_{i}^{\top},\\\ \cdots,0]^{\top}$,
$B_{\varpi_{i}}=[0,\cdots,\xi_{i}^{\top},\cdots,0]^{\top}$.

According to the sum-of-squares property of graph Laplacian, $J_{i}$ can be
expressed as

$\displaystyle J_{i}$
$\displaystyle=\frac{1}{2}\|x(t_{f})-x_{0}\|_{F_{i}}^{2}+\frac{1}{2}\|x(t_{f})\|_{\mathcal{L}_{i}}^{2}+\frac{1}{2}\int_{0}^{t_{f}}\big{(}\|u_{i}(t)\|_{R_{i}}^{2}-\|\varpi_{i}(t)\|_{R_{\varpi_{i}}}^{2}\big{)}~{}\mathrm{dt},$
(12)

where $F_{i}=\mathrm{diag}(0,\cdots,W_{ii},\cdots,0)$,
$\mathcal{L}_{i}=\mathcal{D}\mathcal{W}_{i}\mathcal{D}^{\top}$, and
$\mathcal{W}_{i}=\mathrm{diag}(0,\cdots,W_{ij},\cdots,0)~{}\forall
j\in\mathcal{N}_{i}$.

The cost function (12) and the dynamics (11) pose a differential game problem
[49]. This category of problems is very promising for modeling and analyzing
conflict situations in networked systems of self-interested dynamical agents.
In the context of a differential game, each agent of the network is referred
to as a player and thereby the game of opinion formation in (12) and (11) has
$n$ players. In this context, each player seeks the control $u_{i}(t)$ that
minimizes her cost function $J_{i}$ subject to the game state equation with
the given initial state in (11). In other words, the players in the game seek
to minimize their cost functions in order to find their control or influence
strategies $u_{i}(t)$ while their opinions evolve according to the
differential equation (11).

The behavior of self-interested players in the game of opinion formation is
best reflected via noncooperative game theory. Under the framework of
noncooperative games, the players can not make binding agreements, and
therefore, the solution (i.e., the Nash equilibrium) has to be self-enforcing,
meaning that once it is agreed upon, nobody has the incentive to deviate from
[50]. In the next section, we derive the open-loop Nash/worst-case equilibrium
solution for the game of coupled multi-dimensional opinion formation in terms
of (12) and (11).

##  4 Nash/Worst-case Equilibrium Solution

A Nash equilibrium is a strategy combination of all players in the game with
the property that no one can gain a lower cost by unilaterally deviating from
it. The open-loop Nash equilibrium is defined as a set of admissible actions
($u_{1}^{*},\cdots,u_{n}^{*}$) if for all admissible ($u_{1},\cdots,u_{n}$)
the inequalities
$J_{i}(u_{1}^{*},\cdots,u_{i-1}^{*},u_{i}^{*},u_{i+1}^{*},\cdots,u_{n}^{*})\leq
J_{i}(u_{1}^{*},\cdots,u_{i-1}^{*},u_{i},u_{i+1}^{*},\cdots,u_{n}^{*})$ hold
for $i\in\\{1,\cdots,n\\}$ where $u_{i}\in\Gamma_{i}$ and
$\Gamma_{i}=\\{u_{i}(t,x_{0})|t\in[0,t_{f}]\\}$ is the admissible strategy set
for player $i$. The noncooperative linear quadratic differential game and the
unique Nash equilibrium associated with it are discussed thoroughly in [49].

Assume that for every strategy combination ($u_{1},\cdots,u_{n}$) there exists
a worst-case disturbance $\hat{\varpi}_{i}(u_{1},\cdots,u_{n})$ from the point
of view of player $i$, i.e.,
$J_{i}\big{(}u_{1},\cdots,u_{n},\hat{\varpi}_{i}(u_{1},\cdots,u_{n})\big{)}\geq
J_{i}\big{(}u_{1},\cdots,\\\ u_{n},\varpi\big{)}$ holds for each $\varpi$ and
$i\in\\{1,\cdots,n\\}$. Then, a Nash/worst-case equilibrium is defined as a
set of admissible actions ($u_{1}^{*},\cdots,u_{n}^{*}$) that the inequalities
$J_{i}\big{(}u_{1}^{*},\cdots,u_{n}^{*},\hat{\varpi}_{i}(u_{1}^{*},\cdots,u_{n}^{*})\big{)}\leq
J_{i}\big{(}u_{1}^{*},\cdots,u_{i-1}^{*},u_{i},u_{i+1}^{*},\cdots,u_{n}^{*},\hat{\varpi}_{i}(u_{1}^{*},\cdots,u_{i-1}^{*},u_{i},u_{i+1}^{*},\cdots,u_{n}^{*})\big{)}$
hold for $i\in\\{1,\cdots,n\\}$ where $u_{i}\in\Gamma_{i}$.

As presented in Subsection 2.2, the linear quadratic differential game (11)
and (12) converts to the nonsymmetric Riccati differential equation problem
(7). Since solving (7) analytically is difficult, its approximate solution has
to be obtained by numerical methods. Thereby, the Nash/worst-case equilibrium
actions $u_{i}(t)$ and their corresponding worst-case disturbances
$\varpi_{i}(t)$ in Subsection 2.2 could only be approximated.

In the following, we present the open-loop Nash/worst-case equilibrium
solution for the underlying game of opinion formation. The solution is an
explicit expression derived by applying the necessary conditions for
optimality using Pontryagin’s principle. Before we begin, we present the
following assumptions.

######  Assumption 2.

Given an initial state $x_{i}(0)$ and in the time interval $[0,t_{f}]$, the
state dynamics (9) are controllable for all $i\in\mathcal{V}$, meaning that
there exists a control input so that an arbitrary final state $x_{i}(t_{f})$
can be achieved.

######  Lemma 1.

Let $\Phi(t_{f},t)$ denote the state-transition matrix for $-\mathcal{L}$ (and
since $-\mathcal{L}$ is time-invariant,
$\Phi(t_{f},t)=\mathrm{e}^{-(t_{f}-t)\mathcal{L}}$). Matrix

$H(t_{f})=I-\sum_{i=1}^{n}\big{(}\Psi_{i}(t_{f},0)(F_{i}+\mathcal{L}_{i})\big{)},$
(13)

where

$\Psi_{i}(t_{f},0)=\int_{0}^{t_{f}}\Phi(t_{f},\tau)\big{(}B_{\varpi_{i}}R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}-B_{i}R_{i}^{-1}B_{i}^{\top}\big{)}\Phi^{\top}(t_{f},\tau)~{}\mathrm{d\tau},$
(14)

is nonsingular.

###### Proof.

Firstly, the eigenvalues of $F_{i}+\mathcal{L}_{i}$ have nonnegative real
parts. Secondly, $\Psi_{i}(t_{f},0)$ defined in (14) is obviously PSD and its
eigenvalues are real and nonnegative. According to its spectrum property,
eigenvalues of product $\Psi_{i}(t_{f},0)(F_{i}+\mathcal{L}_{i})$ are the
eigenvalue products between $\Psi_{i}(t_{f},0)$ and $F_{i}+\mathcal{L}_{i}$
and hence have nonnegative real parts. Finally, the eigenvalues of $H(t_{f})$
have positive real parts, and the existence of $H^{-1}(t_{f})$ is concluded. ∎

######  Theorem 1.

The opinion formation as the $n$-player noncooperative differential game in
(11) and (12) admits a unique open-loop Nash/worst-case equilibrium. The
unique Nash/worst-case actions, corresponding worst-case disturbances, as well
as the associated state trajectory are given by

$\displaystyle u_{i}(t$
$\displaystyle)=-R_{i}^{-1}B_{i}^{\top}\Phi^{\top}(t_{f},t)\Big{[}(F_{i}+\mathcal{L}_{i})H^{-1}(t_{f})G(t_{f})-F_{i}\Big{]}x_{0},$
(15) $\displaystyle\varpi_{i}(t$
$\displaystyle)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}\Phi^{\top}(t_{f},t)\Big{[}(F_{i}+\mathcal{L}_{i})H^{-1}(t_{f})G(t_{f})-F_{i}\Big{]}x_{0},$
(16) $\displaystyle x(t)$
$\displaystyle=\Big{(}\Phi(t,0)+\Psi(t,0)\Big{[}(F+P)H^{-1}(t_{f})G(t_{f})-F\Big{]}\Big{)}x_{0},$
(17)

where

$\displaystyle H(t_{f})=I-\Psi(t_{f},0)(F+P),$ (18) $\displaystyle
G(t_{f})=\Phi(t_{f},0)-\Psi(t_{f},0)F,$ (19)
$\displaystyle\Psi(t_{f},0)=[\Psi_{1}(t_{f},0),\cdots,\Psi_{n}(t_{f},0)],$
(20) $\displaystyle F=[F_{1}^{\top},\cdots,F_{n}^{\top}]^{\top},$ (21)
$\displaystyle
P=[\mathcal{L}_{1}^{\top},\cdots,\mathcal{L}_{n}^{\top}]^{\top}.$ (22)

###### Proof.

The Hamiltonian for every $i\in\mathcal{V}$ is defined as

$\displaystyle\mathcal{H}_{i}=\frac{1}{2}\|u_{i}(t)\|_{R_{i}}^{2}$
$\displaystyle-\frac{1}{2}\|\varpi(t)\|_{R_{\varpi_{i}}}^{2}+\lambda_{i}^{\top}(t)\Big{(}-\mathcal{L}x(t)+\sum_{i=1}^{n}B_{i}u_{i}(t)+\sum_{i=1}^{n}B_{\varpi_{i}}\varpi_{i}(t)\Big{)},$

where $\lambda_{i}(t)$ is a co-state vector. According to Pontryagin’s
principle, the necessary conditions for optimality are

$\displaystyle\frac{\partial\mathcal{H}_{i}}{\partial
u_{i}}=0,\quad\frac{\partial\mathcal{H}_{i}}{\partial\varpi_{i}}=0,\quad\dot{\lambda}_{i}(t)=-\frac{\partial\mathcal{H}_{i}}{\partial
x}.$

Equivalently,

$\displaystyle u_{i}(t)=-R_{i}^{-1}B_{i}^{\top}\lambda_{i}(t),$ (23)
$\displaystyle\varpi_{i}(t)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}\lambda_{i}(t),$
(24) $\displaystyle\dot{\lambda}_{i}(t)=\mathcal{L}\lambda_{i}(t),$ (25)

subject to the terminal condition

$\lambda_{i}(t_{f})=(F_{i}+\mathcal{L}_{i})x(t_{f})-F_{i}x_{0}.$ (26)

The solution of (25) is

$\lambda_{i}(t)=\Phi(t_{f},t)\lambda_{i}(t_{f}).$ (27)

Substituting this solution in (23) and (24) yield

$u_{i}(t)=-R_{i}^{-1}B_{i}^{\top}\Phi(t_{f},t)\lambda_{i}(t_{f}),$ (28)
$\varpi_{i}(t)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}\Phi(t_{f},t)\lambda_{i}(t_{f}).$
(29)

Substituting (28) and (29) in the state dynamics equation, we get

$\dot{x}(t)=-\mathcal{L}x(t)+\sum_{i=1}^{n}\big{(}B_{\varpi_{i}}R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}-B_{i}R_{i}^{-1}B_{i}^{\top}\big{)}\Phi(t_{f},t)\lambda_{i}(t_{f}).$
(30)

Equation (30) has a solution at $t_{f}$ as follows

$\displaystyle x(t_{f})=\Phi(t_{f},0)x_{0}+\Psi(t_{f},0)\lambda(t_{f}),$ (31)

with $\Psi(t_{f},0)$ given in (20). Stacking (26) for $i=1,\cdots,n$ yields

$\lambda(t_{f})=(F+P)x(t_{f})-Fx_{0},$ (32)

with $F$ and $L$ defined in (21) and (22), respectively, and
$\lambda(t_{f})=[\lambda_{1}^{\top}(t_{f}),\cdots,\\\
\lambda_{n}^{\top}(t_{f})]^{\top}$. Substituting $\lambda(t_{f})$ from (32)
then into $x(t_{f})$ in (31) yields

$\displaystyle
x(t_{f})=\Phi(t_{f},0)x_{0}+\Psi(t_{f},0)(F+P)x(t_{f})-\Psi(t_{f},0)Fx_{0},$

or alternatively,

$\displaystyle\Big{(}I-\Psi(t_{f},0)(F+P)\Big{)}x(t_{f})=\Big{(}\Phi(t_{f},0)-\Psi(t_{f},0)F\Big{)}x_{0}.$
(33)

Therefore, the game has an open-loop Nash equilibrium for any initial state
$x_{0}$ if and only if (33) is satisfied for any arbitrary final state
$x(t_{f})$. If so, the equilibrium actions are unique and exist for all
$t\in[0,t_{f}]$. Otherwise, the game does not have a unique open-loop Nash
equilibrium for every initial state $x_{0}$.

Using the notation $H(t_{f})$ and $G(t_{f})$ in (18) and (19), respectively,
we can rewrite (33) as

$\displaystyle x(t_{f})=H^{-1}(t_{f})G(t_{f})x_{0}.$ (34)

Finally, by substituting (34) into (26) and re-substituting (26) in (23) and
(24), we obtain (15) and (16), respectively.

The solution of (30) at $t$ is given by

$x(t)=\Phi(t,0)x_{0}+\Psi(t,0)\lambda(t_{f}).$ (35)

Substituting $\lambda(t_{f})$ from (32) and then re-submitting $x(t_{f})$ from
(34), we have

$\displaystyle x(t)$
$\displaystyle=\Phi(t,0)x_{0}+\Psi(t,0)(F+P)x(t_{f})-\Psi(t,0)Fx_{0}$
$\displaystyle=\Phi(t,0)x_{0}+\Psi(t,0)(F+P)H^{-1}(t_{f})G(t_{f})x_{0}-\Psi(t,0)Fx_{0},$

or in its final form (17). This concludes the proof. ∎

From Theorem 1, it is obvious that the unique Nash/worst-case equilibrium
actions and their corresponding worst-case disturbances exist if and only if
$H(t_{f})$ in (13) has an inverse.

Let $M=(F_{i}+\mathcal{L}_{i})H^{-1}(t_{f})G(t_{f})-F_{i}$. So,

$\displaystyle u_{i}(t)=-R_{i}^{-1}B_{i}^{\top}\Phi^{\top}(t_{f},t)Mx_{0},$
$\displaystyle\varpi_{i}(t)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}\Phi^{\top}(t_{f},t)Mx_{0}.$

Additionally, assume that $R_{i}=R_{i}^{\top}$ and
$R_{\varpi_{i}}=R_{\varpi_{i}}^{\top}$. Thus,
$(R_{i}^{-1})^{\top}=(R_{i}^{\top})^{-1}=R_{i}^{-1}$.

The worst-case expected costs by player $i$ are

$\displaystyle
J_{i}=\frac{1}{2}\int_{0}^{t_{f}}\big{(}u_{i}^{\top}(t)R_{i}u_{i}(t)-\varpi_{i}^{\top}(t)R_{\varpi_{i}}\varpi_{i}(t)\big{)}~{}\mathrm{dt}$
$\displaystyle=-\frac{1}{2}x_{0}^{\top}M^{\top}\int_{0}^{t_{f}}\Phi(t_{f},t)\big{(}B_{\varpi_{i}}R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}-B_{i}R_{i}^{-1}B_{i}^{\top}\big{)}\Phi^{\top}(t_{f},t)~{}\mathrm{dt}Mx_{0}$
$\displaystyle=-\frac{1}{2}x_{0}^{\top}M^{\top}\Psi_{i}(t_{f},0)Mx_{0}.$

The terminal opinion formation error for player $i$ is

$\displaystyle E_{i}$
$\displaystyle=\frac{1}{2}\|x_{i}(t_{f})-x_{i}(0)\|_{W_{ii}}^{2}+\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}\|x_{i}(t_{f})-x_{j}(t_{f})\|_{W_{ij}}^{2}$
$\displaystyle=\frac{1}{2}\|x(t_{f})-x_{0}\|_{F_{i}}^{2}+\frac{1}{2}\|x(t_{f})\|_{\mathcal{L}_{i}}^{2}.$
(36)

##  5 Distributed Implementation of Game Strategies

It is obvious from (15), (16), and (17) that the Nash/worst-case actions,
their corresponding worst-case disturbances, and state trajectory require the
vector $x_{0}$ from all players or nodes in the communication graph.
Consequently, control actions (15) and (16) are compatible only with a fully
connected communication graph where each agent has access to global
information. A complete graph resembles a social network in which each agent
knows all other agents and communicates with them. In practice, since each
agent usually does not know all the other agents, the graph is incomplete. In
such a network, each player has access to her immediate neighbors’ information
in the opinion network’s communication graph and does not share her
information with non-neighbor agents. Therefore, the distributed
implementation of control actions (15) and (16) is necessary for incomplete
graphs. Moreover, distributed control policies that could be implemented only
with information from neighboring nodes have several advantages, such as
reliability, scalability, and flexibility [51]. However, (15) and (16) are
incapable of being executed with the distributed information.

In the following, we propose a distributed implementation of (15) and (16),
which each agent can execute only with information from her graph neighbors.
Inspired by the Nash strategy design approach in a distributed manner for UAV
(unmanned aerial vehicle) formation control in [52], for each agent $i$, we
propose the following state estimator

$\displaystyle\hat{\dot{x}}_{i}(t)=$
$\displaystyle\delta_{i}\bigg{\\{}\big{(}e_{i}^{\top}\Phi(t_{f},0)e_{i}-e_{i}^{\top}\Psi_{i}(t_{f},0)e_{i}W_{ii}\big{)}x_{i}(0)-\hat{x}_{i}(t)+e_{i}^{\top}\Psi_{i}(t_{f},0)e_{i}\sum_{j\in\mathcal{N}_{i}}W_{ij}(\hat{x}_{j}(t)-\hat{x}_{i}(t))\bigg{\\}},$
(37)

where $\delta_{i}$ is a positive scalar and $e_{i}=[0_{m\times
m},\cdots,I_{m\times m},\cdots,0_{m\times m}]^{\top}$.

######  Theorem 2.

If each agent $i$ for all $i\in\mathcal{V}$ updates its state estimate
$\hat{x}_{i}(t)$ according to (37), then

$\lim_{t\to\infty}\hat{x}(t)=x(t_{f})$ (38)

where
$\hat{x}(t)=[\hat{x}_{1}^{\top}(t),\cdots,\hat{x}_{n}^{\top}(t)]^{\top}$.

###### Proof.

Stacking (37) from $1$ to $n$ yields

$\displaystyle\hat{\dot{x}}(t)$
$\displaystyle=\Delta\left(\big{(}\Phi(t_{f},0)-\Psi(t_{f},0)F\big{)}x_{0}-\Big{(}I-\sum_{i=1}^{n}\big{(}\Psi_{i}(t_{f},0)(F_{i}+\mathcal{L}_{i})\big{)}\Big{)}\hat{x}(t)\right)$
$\displaystyle=\Delta\left(\big{(}\Phi(t_{f},0)-\Psi(t_{f},0)F\big{)}x_{0}-(I-\Psi(t_{f},0)(F+P))\hat{x}(t)\right)$
$\displaystyle=\Delta\left(\big{(}\Phi(t_{f},0)-\Psi(t_{f},0)F\big{)}x_{0}-H(t_{f})\hat{x}(t)\right)$
(39)

where $\Delta=\mathrm{diag}(g_{1},\cdots,g_{n})\otimes I_{m\times m}$.

Using the notation $H(t_{f})$, (33) can be rewritten as

$\displaystyle
H(t_{f})x(t_{f})=\Big{(}\Phi(t_{f},0)-\Psi(t_{f},0)F\Big{)}x_{0}.$ (40)

Substituting the left side from (40) into (5), we have

$\displaystyle\hat{\dot{x}}(t)=-\Delta
H(t_{f})\left(\hat{x}(t)-x(t_{f})\right)$ (41)

As shown in Lemma 1, all the eigenvalues of matrix $H(t_{f})$ have positive
real parts, and thus matrix $-\Delta H(t_{f})$ is Hurwitz. Therefore, the
linear system in (41) is asymptotically stable, and $\hat{x}(t)$ will converge
to $x(t_{f})$ at $t\to\infty$. ∎

Using the distributed state estimator (37), each agent $i\in\mathcal{V}$ can
adopt the following distributed strategy

$\displaystyle\hat{u}_{i}(t$
$\displaystyle)=-R_{i}^{-1}B_{i}^{\top}\Big{(}W_{ii}(\hat{x}_{i}(t)-x_{i}(0))+\sum_{j\in\mathcal{N}_{i}}W_{ij}(\hat{x}_{j}(t)-\hat{x}_{i}(t))\Big{)},$
(42)

and its corresponding distributed worst-case disturbance is

$\displaystyle\hat{\varpi}_{i}(t$
$\displaystyle)=R_{\varpi_{i}}^{-1}B_{\varpi_{i}}^{\top}\Big{(}W_{ii}(\hat{x}_{i}(t)-x_{i}(0))+\sum_{j\in\mathcal{N}_{i}}W_{ij}(\hat{x}_{j}(t)-\hat{x}_{i}(t))\Big{)}.$
(43)

All agents will eventually converge to the open-loop Nash/worst-case strategy,
its corresponding disturbance, and state trajectory with the arbitrary
convergence speed determined by $\delta_{i}$ if they adopt distributed
implementation (42), (43), and (37). Notice that each agent requires only her
own initial state and state estimates and the state estimates of her graph
neighbors, which can be acquired at the beginning of the game.

The terminal opinion formation error for player $i$ when adopting distributed
strategies is

$\displaystyle\hat{E}_{i}$
$\displaystyle=\frac{1}{2}\|\hat{x}_{i}(t_{f})-x_{i}(0)\|_{W_{ii}}^{2}+\frac{1}{2}\sum_{j\in\mathcal{N}_{i}}\|\hat{x}_{i}(t_{f})-\hat{x}_{j}(t_{f})\|_{W_{ij}}^{2}$
$\displaystyle=\frac{1}{2}\|\hat{x}(t_{f})-x_{0}\|_{F_{i}}^{2}+\frac{1}{2}\|\hat{x}(t_{f})\|_{\mathcal{L}_{i}}^{2}.$
(44)

##  6 Social Optimality

In the game of multi-dimensional opinion formation on the communication graph
$\mathcal{G}(\mathcal{V},\mathcal{E})$, each player by minimizing the
individual (local) cost function attains the locally optimal Nash equilibrium.
However, the game equilibrium in general does not correspond to the global
social optimum, which minimizes the sum of all players’ costs. The global
network dynamics can be rewritten as

$\dot{x}(t)=-\mathcal{L}x(t)+Bu(t)+B_{\varpi}\varpi(t),\quad x(0)=x_{0},$ (45)

where $B=[B_{1},\cdots,B_{n}]$,
$u(t)=[u_{1}^{\top}(t),\cdots,u_{n}^{\top}(t)]^{\top}$,
$B_{\varpi}=[B_{\varpi_{1}},\cdots,B_{\varpi_{n}}]$, and
$\varpi(t)=[\varpi_{1}^{\top}(t),\cdots,\varpi_{n}^{\top}(t)]^{\top}$. Social
optimality utilizes an identical global objective (e.g., a cost function to
minimize) as the opinion formation objective by simply adding all the
individual players’ interests together. Therefore, the social optimality
problem is associated with a social network with non-strategist agents in
which individuals do not wish to execute their personal strategies but rather
prefer to commit to the whole network’s common objective. The global cost
function becomes

$J=\sum_{i=1}^{n}J_{i},$

or equivalently

$\displaystyle J$
$\displaystyle=\frac{1}{2}\|x(t_{f})-x_{0}\|_{\bar{F}}^{2}+\frac{1}{2}\|x(t_{f})\|_{\mathcal{L}}^{2}+\frac{1}{2}\int_{0}^{t_{f}}\big{(}\|u(t)\|_{R}^{2}-\|\varpi(t)\|_{R_{\varpi}}^{2}\big{)}~{}\mathrm{dt},$
(46)

where $\bar{F}=\mathrm{diag}(W_{11},\cdots,W_{nn})$,
$R=\mathrm{diag}(R_{1},\cdots,R_{n})$, and
$R_{\varpi}=\mathrm{diag}(R_{\varpi_{1}},\\\ \cdots,R_{\varpi_{n}})$. The
global cost function in (46) is known to all $i\in\mathcal{V}$ and is
minimized with respect to (45).

######  Theorem 3.

The socially optimal actions and their corresponding worst-case disturbances
for opinion formation as a result of the optimal control problem in (45) and
(46) are

$\displaystyle u$
$\displaystyle(t)=-R^{-1}B^{\top}\Phi^{\top}(t_{f},t)\Big{[}(\bar{F}+\mathcal{L})\bar{H}^{-1}(t_{f})\bar{G}(t_{f})-\bar{F}\Big{]}x_{0},$
(47) $\displaystyle\varpi$
$\displaystyle(t)=R_{\varpi}^{-1}B_{\varpi}^{\top}\Phi^{\top}(t_{f},t)\Big{[}(\bar{F}+\mathcal{L})\bar{H}^{-1}(t_{f})\bar{G}(t_{f})-\bar{F}\Big{]}x_{0},$
(48)

where

$\displaystyle\bar{H}(t_{f})=I-\bar{\Psi}(t_{f},0)(\bar{F}+\mathcal{L}),$
$\displaystyle\bar{G}(t_{f})=\Phi(t_{f},0)-\bar{\Psi}(t_{f},0)\bar{F},$
$\displaystyle\bar{\Psi}(t_{f},0)=\int_{0}^{t_{f}}\Phi(t_{f},\tau)\big{(}B_{\varpi}R_{\varpi}^{-1}B_{\varpi}^{\top}-BR^{-1}B^{\top}\big{)}\Phi^{\top}(t_{f},\tau)~{}\mathrm{d\tau}.$

Furthermore, the associated state trajectory is given by

$x(t)=\Big{(}\Phi(t,0)+\bar{\Psi}(t,0)\Big{[}(\bar{F}+\mathcal{L})\bar{H}^{-1}(t_{f})\bar{G}(t_{f})-\bar{F}\Big{]}\Big{)}x_{0}.$
(49)

###### Proof.

Applying Pontryagin’s principle, the optimal control problem in (45) and (46)
converts to the following boundary value problem

$\displaystyle\dot{x}(t)=-\mathcal{L}x(t)+\big{(}B_{\varpi}R_{\varpi}^{-1}B_{\varpi}^{\top}-BR^{-1}B^{\top}\big{)}\bar{\lambda}(t),\quad
x(0)=x_{0},$
$\displaystyle\dot{\bar{\lambda}}(t)=\mathcal{L}\bar{\lambda}(t),\quad\bar{\lambda}(t_{f})=\big{(}\bar{F}+\mathcal{L}\big{)}x(t_{f})-\bar{F}x_{0}.$

Along with that, we have

$\displaystyle u(t)=-R^{-1}B^{\top}\bar{\lambda}(t),$
$\displaystyle\varpi(t)=R_{\varpi}^{-1}B_{\varpi}^{\top}\bar{\lambda}(t).$

The solution for $\dot{\bar{\lambda}}(t)$ is
$\bar{\lambda}(t)=\Phi(t_{f},t)\bar{\lambda}(t_{f})$. Substituting this
solution in the equation for $\dot{x}(t)$, and solving it for $t_{f}$, we get

$\displaystyle
x(t_{f})=\Phi(t_{f},0)x_{0}+\bar{\Psi}(t_{f},0)\bar{\lambda}(t_{f}).$

Substituting $\bar{\lambda}(t_{f})$ into $x(t_{f})$ above yields

$\displaystyle
x(t_{f})=\Phi(t_{f},0)x_{0}+\bar{\Psi}(t_{f},0)(\bar{F}+\mathcal{L})x(t_{f})-\bar{\Psi}(t_{f},0)\bar{F}x_{0},$

or,

$\displaystyle\Big{(}I-\bar{\Psi}(t_{f},0)(\bar{F}+\mathcal{L})\Big{)}x(t_{f})=\Big{(}\Phi(t_{f},0)-\bar{\Psi}(t_{f},0)\bar{F}\Big{)}x_{0}.$

Using the notation $\bar{H}(t_{f})$ and $\bar{G}(t_{f})$, and by appropriate
substitutions similar to the proof in Theorem 1, we obtain (47) and (48) as
well as (49). ∎

Under socially optimal actions, the total terminal opinion formation error is
given by

$\displaystyle E_{o}$
$\displaystyle=\frac{1}{2}\sum_{i\in\mathcal{V}}\|x_{i}(t_{f})-x_{i}(0)\|_{W_{ii}}^{2}+\frac{1}{2}\sum_{\\{i,j\\}\in\mathcal{E}}\|x_{i}(t_{f})-x_{j}(t_{f})\|_{W_{ij}}^{2}$
$\displaystyle=\frac{1}{2}\|x(t_{f})-x_{0}\|_{\bar{F}}^{2}+\frac{1}{2}\|x(t_{f})\|_{\mathcal{L}}^{2}.$
(50)

##  7 Numerical Analysis

For the numerical analysis of the proposed game optimization model, we
consider a small social network with five agents (i.e., $n=5$) in a two-
dimensional opinion space (i.e., $m=2$). The underlying social graph topology
is adapted from [48]. This example graph was used to analyze opinion formation
with the non-stubborn non-strategist agent model on a time-invariant
undirected social graph. We consider this graph with directed edges.

The social graph topology is given in Fig. 1 and the initial opinions of
agents and the interpersonal influence matrices from [48] are as
$x_{1}(0)=[1,2]^{\top}$, $x_{2}(0)=[2,4]^{\top}$, $x_{3}(0)=[3,1]^{\top}$,
$x_{4}(0)=[4,3]^{\top}$, and $x_{5}(0)=[5,6]^{\top}$,
$W_{12}=W_{13}=W_{23}=W_{34}=W_{45}=I_{2\times 2}$. Note that all the weights
of the above matrices are considered nonnegative; while zero denotes no
coupling, a nonzero weight shows a positive coupling between the associated
topics from two agents. This choice of interpersonal influence matrices means
that the agents communicate on several unrelated topics on the information
graph. As we expected and as seen from Fig. 2(a), a consensus of final
opinions about the average opinion in the network for the non-stubborn non-
strategist agent model (8) was reached at a horizon length of $t_{f}=10$. To
examine the evolution of opinions on interdependent topics, we choose
$W_{12}=W_{13}=W_{23}=W_{34}=W_{45}=\begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$,
so for all neighboring agents, both topics are fully coupled. Fig. 2(b) shows
that only agents 1 and 5 reach a consensus on both topics.

Figure 1: Underlying communication graph topology for numerical analysis.

(a) $W_{ij}=I_{2\times 2},i=1,\cdots,5,j\in\mathcal{N}_{i}$

(b) $W_{ij}=[1~{}1;~{}1~{}1],i=1,\cdots,5,j\in\mathcal{N}_{i}$

Figure 2: Opinion trajectories associated with the non-stubborn non-strategist
agent model (8) prior to optimization. (a) A consensus of final opinions about
the average opinion in the network was reached. (b) Only agents 1 and 5
reached a consensus.

We continue the numerical analysis with the proposed game optimization model
to investigate the impact of differential game optimization on opinion
trajectories when the agents are strategists and non-stubborn (i.e.,
$W_{ii}=0$ for all $i$), and then when they are stubborn (i.e.,
$W_{ii}=I_{2\times 2}$ for all $i$). Note that for a time-varying graph,
$\mathcal{L}(t)$ satisfies the commutative condition, i.e.,
$\mathcal{L}(t)\mathcal{L}(\tau)=\mathcal{L}(\tau)\mathcal{L}(t)$ for all $t$
and $\tau$ in $[0,t_{f}]$. Therefore, the state transition matrix
$\Phi(t_{f},t)$ is found in explicit form by the spectral decomposition given
in (52). Let $b_{i}=\xi_{i}=I_{2\times 2}$, $R_{i}=r_{i}I_{2\times 2}$ and
$R_{\varpi_{i}}=r_{\varpi_{i}}I_{2\times 2}$, ($r_{\varpi_{i}}\neq r_{i}$) for
$i\in\\{1,\cdots,5\\}$, then

$\displaystyle\Psi_{i}(t_{f},0)=(r_{\varpi_{i}}-r_{i})\int_{0}^{t_{f}}\Phi(t_{f},t)\Phi^{\top}(t_{f},t)~{}\mathrm{dt}.$

The outcome of the game optimization problem can be quantified by the results
presented in Theorem 1 and their distributed counterparts in Section 5. Fig. 3
shows the evolution of the opinions associated with the game strategies as
well as their distributed counterparts for the parameter selection of
$r_{i}=2$, $r_{\varpi_{i}}=2$, and $\delta_{i}=1$. Although the distributed
estimated opinion trajectories at the beginning are far from the opinion
trajectories corresponding to the game strategies, they converge near the
middle point of the horizon length.

Let all agents expect significant disturbances and thereby choose a smaller
$r_{\varpi_{i}}$. In Fig. 4, we re-illustrate the results for
$r_{\varpi_{i}}=0.5$. Compared to Fig. 3 where $r_{\varpi_{i}}=2$, we observe
that only for stubborn agents with unrelated topics ($W_{ii}=I_{2\times
2},W_{ij}=I_{2\times 2}$), their opinion trajectories change. Therefore, the
decisions of stubborn agents on unrelated topics are affected the most by the
changes in disturbance expectations.

(a) $W_{ii}=0_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=2$

(b) $W_{ii}=0_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=2$

(c) $W_{ii}=I_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=2$

(d) $W_{ii}=I_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=2$

Figure 3: Opinion trajectories associated with the game strategies and their
distributed counterparts (shown with dashed lines). (a) non-stubborn agents
with unrelated topics; (b) non-stubborn agents with fully coupled topics; (c)
stubborn agents with unrelated topics; (d) stubborn agents with fully coupled
topics.

(a) $W_{ii}=0_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=0.5$

(b) $W_{ii}=0_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=0.5$

(c) $W_{ii}=I_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=0.5$

(d) $W_{ii}=I_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=0.5$

Figure 4: Opinion trajectories associated with the game strategies and their
distributed counterparts with disturbances $r_{\varpi_{i}}=0.5$ for
$i=1,\cdots,5$.

The results of applying the socially optimal actions presented in Theorem 3
are shown in Fig. 5. For non-stubborn agents, their opinion trajectories under
the game strategies and social optimal actions overlap, while for stubborn
agents, they are similar. Agent 5 has the least communication with other
agents, and only its final opinion slightly differs when under game strategies
and socially optimal actions.

(a) $W_{ii}=0_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=0.1$

(b) $W_{ii}=0_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=0.1$

(c) $W_{ii}=I_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=0.1$

(d) $W_{ii}=I_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=0.1$

Figure 5: Opinion trajectories associated with socially optimal actions (shown
with a dash-dotted line) versus opinion trajectories associated with game
strategies.

To see whether the changes in stubbornness of stubborn agents affect their
opinion trajectories associated with game strategies and socially optimal
actions, we show both trajectories for highly stubborn agents with the
stubbornness of $W_{ii}=10I_{2\times 2}$ for $i=1,\cdots,5$ in Figure 6. We
observe that at the beginning and end of the horizon length, both trajectories
are very close, but the rest of the time they get a little far from each
other.

(a) $W_{ii}=10I_{2\times 2},W_{ij}=I_{2\times 2},r_{\varpi_{i}}=0.1$

(b) $W_{ii}=10I_{2\times 2},W_{ij}=[1~{}1;~{}1~{}1],r_{\varpi_{i}}=0.1$

Figure 6: Opinion trajectories associated with game strategies and socially
optimal actions for highly stubborn agents.

Furthermore, to quantify the performance of the game strategies against heir
distributed counterparts and the socially optimal actions, we utilize the
terminal opinion formation errors that correspond to each. The terminal
opinion formation error for agent $i$ under the game strategies and their
distributed counterparts are calculated from (4) and (5), respectively. Under
socially optimal actions, the total terminal opinion formation error is given
by (6). The terminal opinion formation errors under the game strategies
($E_{i}$), their distributed counterparts ($\hat{E}_{i}$), and the socially
optimal actions ($E_{o}$) for $t_{f}=10$ are summarized in Tables 1-4. From
these tables, we see that the errors under the game strategies and the errors
under their distributed counterparts are very close to each other, and they
are less than the errors under the socially optimal actions. In addition, the
errors for stubborn agents are much higher than for non-stubborn agents.
Although the socially optimal action approach has nearly the same performance
as the game strategy approach and its distributed counterpart approach in
terms of terminal opinion formation errors, it has significant drawbacks that
make it undesirable. As it is apparent from (47), each agent has to acquire
the complete information of all other agents and their initial state vector.
On the contrary, the agents can implement the counterparts of game strategies
distributedly. In addition, social optimality does not allow individual agents
to be strategists, who are decision-makers prioritizing their own interests.

Table 1: Terminal opinion formation errors for non-stubborn ($W_{ii}=0_{2\times 2}$) agents on unrelated topics ($W_{ij}=I_{2\times 2}$) $i$ |  $E_{i}$ |  $\hat{E}_{i}$ |  $E_{o}$
---|---|---|---
1 |  $7.4116\times 10^{-6}$ |  $7.4229\times 10^{-6}$ |  $5.5011\times 10^{-5}$
2 |  $7.4116\times 10^{-6}$ |  $7.4206\times 10^{-6}$ |  $5.5011\times 10^{-5}$
3 |  $5.9987\times 10^{-5}$ |  $6.0077\times 10^{-5}$ |  $5.5011\times 10^{-5}$
4 |  $6.5443\times 10^{-5}$ |  $6.5548\times 10^{-5}$ |  $5.5011\times 10^{-5}$
5 |  $2.0279\times 10^{-5}$ |  $2.0314\times 10^{-5}$ |  $5.5011\times 10^{-5}$
Table 2: Terminal opinion formation errors non-stubborn agents ($W_{ii}=0_{2\times 2}$) on fully coupled topics ($W_{ij}=[1~{}1;~{}1~{}1]$) $i$ |  $E_{i}$ |  $\hat{E}_{i}$ |  $E_{o}$
---|---|---|---
1 |  $4.5838\times 10^{-10}$ |  $5.8660\times 10^{-10}$ |  $3.4023\times 10^{-9}$
2 |  $4.5839\times 10^{-10}$ |  $5.5744\times 10^{-10}$ |  $3.4023\times 10^{-9}$
3 |  $3.7100\times 10^{-9}$ |  $4.7495\times 10^{-9}$ |  $3.4023\times 10^{-9}$
4 |  $4.0475\times 10^{-9}$ |  $5.2724\times 10^{-9}$ |  $3.4023\times 10^{-9}$
5 |  $1.2542\times 10^{-9}$ |  $1.6661\times 10^{-9}$ |  $3.4023\times 10^{-9}$
Table 3: Terminal opinion formation errors for stubborn ($W_{ii}=I_{2\times 2}$) agents on unrelated topics ($W_{ij}=I_{2\times 2}$) $i$ |  $E_{i}$ |  $\hat{E}_{i}$ |  $E_{o}$
---|---|---|---
1 |  0.2602 |  0.2602 |  1.4263
2 |  0.1645 |  0.1645 |  1.4263
3 |  0.6474 |  0.6474 |  1.4263
4 |  0.7448 |  0.7448 |  1.4263
5 |  0.6275 |  0.6275 |  1.4263
Table 4: Terminal opinion formation errors for stubborn ($W_{ii}=I_{2\times 2}$) agents on fully coupled topics ($W_{ij}=[1~{}1;~{}1~{}1]$) $i$ |  $E_{i}$ |  $\hat{E}_{i}$ |  $E_{o}$
---|---|---|---
1 |  0.2307 |  0.2307 |  1.1261
2 |  0.0251 |  0.0251 |  1.1261
3 |  0.3882 |  0.3882 |  1.1261
4 |  0.5265 |  0.5265 |  1.1261
5 |  0.6037 |  0.6037 |  1.1261

##  8 Conclusions

In this study, a differential game model of opinion formation in social
networks has been proposed. The evolution of the opinions of self-interested
people in social groups formed through friendships in real life served as the
model’s inspiration. The game problem has been solved for an open-loop
information Nash equilibrium solution. Moreover, a distributed implementation
of the solution has been proposed where the game strategies and the associated
opinion trajectory for each player solely depend on the information from the
neighboring players. The model was used to examine the opinion formation of a
small opinion network with five agents. A natural extension to the proposed
model can be a time-varying graph topology with antagonistic agent
interactions.

## Acknowledgments

This work was supported by SGS, VŠB - Technical University of Ostrava, Czech
Republic, under grant No. SP2023/012 “Parallel processing of Big Data X”. This
work was supported in full by the Science and Research Council of Turkey
(TÜBİTAK) under project EEEAG-121E162.

## Appendix

##  Appendix A Commutative linear time-varying systems

Consider the linear time-varying differential state space systems described by

$\dot{x}(t)=A(t)x(t),\quad x(t_{0})=x_{0},$ (51)

where $x(t)$ is the state vector, $A(t)$ is a time-varying matrix, and $x_{0}$
is an initial condition. The solution of (51) is given by

$x(t)=\Phi(t,t_{0})x_{0},$

where $\Phi(t,t_{0})$ denotes the state-transition matrix of $A(t)$ and
satisfies

$\dot{\Phi}(t,t_{0})=A(t)\Phi(t,t_{0}),\quad\Phi(t_{0},t_{0})=I.$

In general, the state-transition matrix for time-varying systems is not
available in explicit form. For a special class of linear time-varying systems
above, $A(t)$ commute with $A(\tau)$ for all $t$ and $\tau$. For these
systems, the unique state-transition matrix is explicitly available in the
exponential form given below

$\Phi(t,t_{0})=\exp{(\int_{t_{0}}^{t}A(\tau)d\tau)}.$

Under the similarity transformation, the expression for the state-transition
matrix further simplifies as

$\Phi(t,t_{0})=P(t)\exp{(\int_{t_{0}}^{t}\Lambda(\tau)d\tau)}P^{-1}(t),$

where $\Lambda(t)$ is a diagonal matrix and $P(t)$ is a nonsingular modal
matrix $P(t)$ [53].

A more effective way of calculation of $\Phi(t,t_{0})$ is using the concept of
an extended eigenvalue and extended eigenvector introduced in [54]. Let the
scalar $\mu(t)$ be an extended eigenvalue and $e(t)$ be its corresponding
extended eigenvector and the pair $\\{\mu(t),e(t)\\}$ is called the extended
eigenpair of $A(t)$. Then, the spectral decomposition of the state-transition
matrix is explicitly given by

$\Phi(t,t_{0})=\sum_{i}\exp{\Big{(}\int_{t_{0}}^{t}\mu_{i}(\tau)d\tau\Big{)}}e_{i}(t)r_{i}^{\top}(t_{0}),$
(52)

where $r_{i}^{\top}(t)$ is the reciprocal basis of $e(t)$, i.e.,

$r_{i}^{\top}(t)e_{j}(t)=\begin{cases}0&\text{$i\neq j$},\\\
1&\text{$i=j$}.\end{cases}$

For a commutative $A(t)$ its extended eigenvectors are constant [55] and its
extended eigenpair are determined by the auxiliary equations

$\displaystyle\mathrm{det}\big{(}\mu(t)I-A(t)\big{)}=0,\quad\big{(}\mu(t)I-A(t)\big{)}e=0.$
(53)

When $A$ is time-invariant and all of its eigenvalues have negative real
parts, (51) becomes asymptotically stable. For a time-varying $A(t)$, (51) is
said to be asymptotically stable if and only if $x(t)$ starting from any
finite initial state $x_{0}$ is bounded and $x(t)\rightarrow 0$ as
$t\rightarrow\infty$. The asymptotic stability of (51) can be characterized in
terms of its extended eigenpairs as quoted below from Theorem 3 in [55].
Define

$\Omega_{i}=\|\exp{\Big{(}\int_{t_{0}}^{t}\mu_{i}(\tau)d\tau\Big{)}}e_{i}(t)\|,$
(54)

for each extended eigenpair $\\{\mu(t),e(t)\\}$ of $A(t)$. If for every
$\\{\mu_{i}(t),e_{i}(t)\\}$, $\Omega_{i}\leq\Xi$ for all $t\geq t_{0}$ and
$\Xi<\infty$, then $x(t)$ is bounded, and $\Omega_{i}\rightarrow 0$ as
$t\rightarrow\infty$, the time varying system (51) is asymptotically stable.

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# Time analyticity for the heat equation under Bakry-Émery Ricci curvature
condition

Ling Wu  School of Mathematical Sciences and Shanghai Key Laboratory of PMMP,
East China Normal University, Shanghai 200241, China
<EMAIL_ADDRESS>
###### Abstract.

On complete noncompact Riemannian manifolds with Bakry-Émery Ricci curvature
bounded below, we prove the analyticity in time for smooth solutions of the
heat equation with exponential quadratic growth in the space variable. This
growth condition is sharp. Our result generalizes Hongjie Dong and Qi S.
Zhang’s result [14] and can be extended to all gradient Ricci solitons. As an
application, we give a necessary and sufficient condition on the solvability
of the backward heat equation in a class of functions with exponential
quadratic growth. In addition, we also consider the solution in certain
$L^{p}$ spaces with $p\in[2,+\infty)$ and prove its analyticity with respect
to time.

##  1\. Introduction

Let $(M^{n},g)$ be an $n$-dimensional Riemmanian manifold. The Bakry-Émery
Ricci curvature tensor of $M$ ([1]) is defined as

$\operatorname{Ric}_{f}:=\operatorname{Ric}+\operatorname{Hess}f,$ (1.1)

where $f$ is a smooth function on $M$ (called the potential function), and
$\operatorname{Ric}$ and $\operatorname{Hess}f$ denote the Ricci curvature
tensor and the hessian of $f$, respectively. It is clear that when $f$ is a
constant, $\operatorname{Ric}_{f}$ reduces to the Ricci curvature tensor. A
gradient Ricci soliton is a Riemannian manifold $(M^{n},g)$ with constant
Bakry-Émery Ricci curvature, namely,

$\operatorname{Ric}+\operatorname{Hess}f=\lambda g$ (1.2)

for some constant $\lambda$. It is called a shrinking, steady, or expanding
Ricci soliton when $\lambda>0$, $=0$, or $<0$, respectively. Also, manifolds
with lower Bakry-Émery Ricci curvature bound are closely related to the
singularity analysis of the Ricci flow and Ricci limit spaces (see e.g., [4,
7, 9, 10, 5]). Therefore, many efforts have been made in extending the results
under Ricci curvature condition to Bakry-Émery Ricci curvature condition.

In this paper, we study the time analyticity for the heat equation of the
Laplace-Beltrami operator $\Delta$ under Bakry-Émery Ricci curvature
condition. For the research of the heat equation, one of the expected results
is that its solution is spatially and temporally analytic. For generic
solutions, as expected, the spatial analyticity is valid. However, the
analytic property of the solution in time is indeed invalid, since in the
Euclidean space, it is not difficult to construct a non-time-analytic solution
of the heat equation in a finite space-time cylinder. Therefore, it is
meaningful to study the time analyticity of the heat equation. We need to
impose some suitable conditions on the solution to ensure the time analyticity
e.g. [17].

In a recent paper [15], Qi S. Zhang proved that the solutions with exponential
growth in the space variable are time-analytic on a complete noncompact
Riemannian manifold with the Ricci curvature bounded below. Later, Hongjie
Dong and Qi S. Zhang [14] extended Qi S. Zhang’s results to the solutions with
exponential quadratic growth in the space variable. Meanwhile, they provided
an example to show that the growth condition of the solution is sharp. In
addition, they gave a sufficient and necessary condition on the solvability of
the backward heat equation in a class of functions with exponential quadratic
growth in the space variable. In [12], Jiayong Wu obtained a similar result on
the time analyticity of the heat equation on the complete noncompact gradient
shrinking Ricci solitons. For more results, see [3], [6], [13] and references
therein.

In this paper, we find that the analyticity in time for smooth solutions of
the heat equation always holds on a complete noncompact Riemannian manifold
with Bakry-Émery Ricci curvature bounded below and the potential function
being of at most quadratic growth provided that the solutions have exponential
quadratic growth in the space variable. This result generalizes Hongjie Dong
and Qi S. Zhang’s result [14] and can be extended to all gradient Ricci
solitons.

######  Theorem 1.1.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold with
$\operatorname{Ric}_{f}\geq-Kg$ for some constant $K\geq 0$. For a fixed point
$o\in M$, assume that there exist non-negative constants $a$ and $b$ such that

$|f(x)|\leq ad^{2}(x,o)+b\ for\ all\ x\in M,$ (1.3)

where $d(x,o)$ is the distance function from $x$ to $o$. Let $u(x,t)$ be a
smooth solution of the heat equation $(\Delta-\partial_{t})u=0$ on
$M\times[-2,0]$. If $u$ satisfies exponential quadratic growth in the space
variable, i.e.,

$|u(x,t)|\leq A_{1}e^{A_{2}d^{2}(x,o)}\ for\ all\ (x,t)\in M\times[-2,0],$
(1.4)

where $A_{1}$ and $A_{2}$ are some positive constants, then $u(x,t)$ is
analytic in time $t\in[-1,0]$ with radius $\delta>0$ depending only on
$n,K,a,b$ and $A_{2}$. Moreover, we have

$u(x,t)=\sum_{j=0}^{\infty}a_{j}(x)\frac{t^{j}}{j!}$ (1.5)

with $\Delta a_{j}(x)=a_{j+1}(x)$ and

$|a_{j}(x)|\leq A_{1}A_{3}^{j+1}(j+1)^{j}e^{A_{4}d^{2}(x,o)},\ j=0,1,2,...,$
(1.6)

where $A_{3}$ and $A_{4}$ are two positive constants depending on
$K,n,a,b,A_{2}$ and $n,a,A_{2}$, respectively.

######  Remark 1.2.

If the potential function is $0$, i.e., $a=b=0$ in (1.3), after careful
calculation, then we get $A_{4}=2A_{2}$ in (1.6). Theorem 1.1 reduces Honejie
Dong and Qi S. Zhang’s result [14].

######  Remark 1.3.

The growth condition in Theorem 1.1 is necessary. As in [14], let $v(x,t)$ be
the Tychonov’s solution of the heat equation in
$\mathbb{R}^{n}\times(-\infty,+\infty)$ such that $v=0$ if $t\leq 0$ and $v$
is nontrivial for $t>0$. Then $u(x,t):=v(x,t+1)$ is a nontrivial solution of
the heat equation and is not analytic in time. Notice that $|u(x,t)|$ grows
faster than $e^{C_{1}|x|^{2}}$ for any constant $C_{1}>0$, but for any
$\epsilon>0$, $|u(x,t)|$ is bounded by $C_{2}e^{C_{3}|x|^{2+\epsilon}}$ for
some constants $C_{2}$ and $C_{3}$. Therefore the growth condition (1.4) is
sharp.

The conditions in the above theorem are especially satisfied on gradient Ricci
solitons. For gradient Ricci solitons, it is well known that

$S+|\nabla f|^{2}=2\lambda f+C,$ (1.7)

where $S$ is the scalar curvature of $M$, $\nabla f$ is the gradient of $f$
and $C$ is a constant.
For gradient shrinking solitons, it is showed in [2] that $S\geq 0$, then
setting $\tilde{f}=f+\frac{C}{2\lambda}$, (1.7) implies

$|\nabla\tilde{f}|^{2}\leq 2\lambda\tilde{f},$

so

$|\tilde{f}(x)|\leq\lambda d^{2}(x,o)+2|\tilde{f}(o)|.$

For gradient expanding solitons, it is showed in [8, 16] that $S\geq
n\lambda$, then (1.7) implies that

$\bigg{|}\nabla\sqrt{-f-\frac{C-n\lambda}{2\lambda}}\bigg{|}\leq\sqrt{-\frac{\lambda}{2}}.$

Hence

$\sqrt{-f(x)-\frac{C-n\lambda}{2\lambda}}\leq\sqrt{-\frac{\lambda}{2}}d(x,o)+\sqrt{-f(o)-\frac{C-n\lambda}{2\lambda}},$

setting $\tilde{f}=f+\frac{C-n\lambda}{2\lambda}$, then

$|\tilde{f}(x)|\leq-\lambda d^{2}(x,o)+2|\tilde{f}(o)|.$

For gradient steady solitons, we know $S\geq 0$ in [2].
If $C=0$ in (1.7), then $f$ is a constant.
If $C\neq 0$ in (1.7), by scaling the metric $g$, we can get

$S+|\nabla f|^{2}=1,$

which implies

$|f(x)|\leq d(x,o)+|f(o)|.$

To sum up, for gradient Ricci solitons, we can always adjust $f$ or $g$ to
make

$|f(x)|\leq ad^{2}(x,o)+b,$ (1.8)

where $a$ and $b$ are two positive constants depending on $\lambda$ and
$f(o)$, respectively.

Therefore, Theorem 1.1 implies the analyticity in time for smooth solutions of
the heat equation on complete noncompact gradient Ricci solitons.

######  Theorem 1.4.

Let $(M^{n},g)$ be a complete noncompact gradient Ricci soliton satisfying
(1.2). Let $u(x,t)$ be a smooth solution of the heat equation
$(\Delta-\partial_{t})u=0$ on $M\times[-2,0]$. For a fixed point $o\in M$, if
$u$ satisfies exponential quadratic growth in the space variable, i.e.,

$|u(x,t)|\leq A_{1}e^{A_{2}d^{2}(x,o)}\ for\ all\ (x,t)\in M\times[-2,0],$
(1.9)

where $A_{1}$ and $A_{2}$ are some positive constants, and $d(x,o)$ is the
distance function from $x$ to $o$, then $u(x,t)$ is analytic in time
$t\in[-1,0]$ with radius $\delta>0$ depending only on $n,\lambda,f(o)$ and
$A_{2}$. Moreover, we have

$u(x,t)=\sum_{j=0}^{\infty}a_{j}(x)\frac{t^{j}}{j!}$ (1.10)

with $\Delta a_{j}(x)=a_{j+1}(x)$ and

$|a_{j}(x)|\leq A_{1}A_{3}^{j+1}(j+1)^{j}e^{A_{4}d^{2}(x,o)},\ j=0,1,2,...,$
(1.11)

where $A_{3}$ and $A_{4}$ are two positive constants depending on
$n,\lambda,f(o),A_{2}$ and $n,\lambda,A_{2}$, respectively.

######  Remark 1.5.

In [12], Jiayong Wu obtained a similar result on the time analyticity of the
heat equation on the complete noncompact gradient shrinking Ricci solitons.
More precisely, he showed that the bound of $a_{j}(x)$ in (1.10) is

$|a_{j}(x)|\leq
A_{1}e^{-\frac{\mu}{2}}e^{\frac{f(x)}{2}}(f(x)+1)^{\frac{n}{4}}A_{3}^{j+1}j^{j}e^{2A_{2}d^{2}(x,o)},\
j=0,1,2,...,$ (1.12)

where $A_{3}$ is a constant depending on $n$ and $A_{2}$ and $\mu=\mu(g,1)$
denotes Perelman’s entropy functional.

Comparing (1.11) with (1.12), it is not difficult to find that our result does
not depend on $\mu$.

In general, the Cauchy problem to the backward heat equation is not solvable.
However, on a complete noncompact Riemannian manifold with Bakry-Émery Ricci
curvature bounded below, we have a solvable result by a simple application of
Theorem 1.1.

######  Corollary 1.6.

Let $(M^{n},g)$ be as in Theorem 1.1. For a fixed point $o\in M$, the Cauchy
problem for the backward heat equation

$\begin{cases}(\Delta+\partial_{t})u=0,&\\\ u(x,0)=a(x)&\end{cases}$ (1.13)

has a smooth solution with exponential quadratic growth of the space variable
in $M\times(0,\delta)$ for some $\delta>0$ if and only if

$|\Delta^{j}a(x)|\leq A_{3}^{j+1}(j+1)^{j}e^{A_{4}d^{2}(x,o)},\ j=0,1,2,...,$
(1.14)

where $A_{3}$ and $A_{4}$ are some positive constants.

In addition, we also consider the solution of the heat equation in $L^{p}$
spaces with $p\in[2,+\infty)$ and prove its analyticity with respect to time.

######  Theorem 1.7.

Let $(M^{n},g)$ be a complete noncompact Riemannian manifold with
$\operatorname{Ric}_{f}\geq-Kg$ for some constant $K\geq 0$. For a fixed point
$o\in M$, assume that there exist non-negative constants $a$ and $b$ such that

$|f(x)|\leq ad^{2}(x,o)+b\ for\ all\ x\in M,$ (1.15)

where $d(x,o)$ is the distance function from $x$ to $o$. Let $u(x,t)$ be a
smooth solution of the heat equation $(\Delta-\partial_{t})u=0$ on
$M\times[-2,0]$. For any $p\geq 2$, assume that there exists a positive
constant $L$ such that

$\left(\int_{M}|u(x,t)|^{p}dv\right)^{\frac{1}{p}}\leq L\ for\ all\
t\in[-2,0].$ (1.16)

Then $u(x,t)$ is analytic in time $t\in[-1,0]$ with radius $\delta>0$
depending only on $n,K,a,b$ and $p$.
Moreover, we have

$u(x,t)=\sum_{j=0}^{\infty}a_{j}(x)\frac{t^{j}}{j!}$ (1.17)

with $\Delta a_{j}(x)=a_{j+1}(x)$ and

$|a_{j}(x)|\leq
A_{6}^{j+1}(j+1)^{j}e^{A_{7}d^{2}(x,o)}\operatorname{Vol}(B_{o}(1))^{-\frac{1}{p}}L,\
j=0,1,2,...,$ (1.18)

where $A_{6}$ and $A_{7}$ are two positive constants depending on $n,K,a,b,p$
and $n,a,K,p$, respectively.

The rest of this paper is organized as follows. In section 2, we recall the
volume comparison theorem and a parabolic mean value inequality from [11] for
complete Riemannian manifolds with Bakry-Émery Ricci curvature bounded below
and the potential function bounded locally. In section 3, adapting Hongjie
Dong and Qi S. Zhang’s proof strategy [14], we apply the mean value type
inequality of section 2 to prove Theorem 1.1, Corollary 1.6 and Theorem 1.7.

##  2\. Preliminaries

From [11], we have the volume comparison theorem for the Laplace-Beltrami
operator $\Delta$ on complete Riemannian manifolds with Bakry-Émery Ricci
curvature bounded below.

For a fixed point $o\in M$ and $R>0$, we define

$L(R)=\sup\limits_{B_{o}(3R)}|f|,$

where $B_{o}(3R)$ is the geodesic ball centered at $o\in M$ with radius $3R$.

######  Theorem 2.1.

Let $(M^{n},g)$ be a complete Riemannian manifold with
$\operatorname{Ric}_{f}\geq-Kg$ for some constant $K\geq 0$. Then the
following conclusions are true.
(a)(Laplacian comparison) Let $r=d(y,p)$ be the distance from any point $y$ to
some fixed point $p\in B_{o}(R)$ with $0<r<R$. Then for $0<r_{1}<r_{2}<R$, we
have

$\int_{r_{1}}^{r_{2}}(\Delta
r-\frac{n-1}{r})dr\leq\frac{K}{6}(r_{2}^{2}-r_{1}^{2})+6L(R).$ (2.1)

(b)(Volume element comparison)Take any point $p\in B_{o}(R)$ and denote the
volume form in geodesic polar coordinates centered at $p$ with
$J(r,\theta,p)drd\theta$, where $r>0$ and $\theta\in S_{p}(M)$, a unit tangent
vector at $p$. Then for $0<r_{1}<r_{2}<R$, we have

$\frac{J(r_{2},\theta,p)}{J(r_{1},\theta,p)}\leq\left(\frac{r_{2}}{r_{1}}\right)^{n-1}e^{\frac{K}{6}(r_{2}^{2}-r_{1}^{2})+6L(R)}.$
(2.2)

(c)(Volume comparison)For any $p\in B_{o}(R),\ 0<r_{1}<r_{2}<R$, we have

$\frac{\operatorname{Vol}(B_{p}(r_{2}))}{\operatorname{Vol}(B_{p}(r_{1}))}\leq\left(\frac{r_{2}}{r_{1}}\right)^{n}e^{\frac{K}{6}(r_{2}^{2}-r_{1}^{2})+6L(R)},$
(2.3)

where $\operatorname{Vol}(.)$ denotes the volume of a region.

From [11], we have the following parabolic mean value inequality, which is
crucial to prove the analyticity of time.

######  Proposition 2.2.

(Mean value inequality) Let $(M^{n},g)$ be a complete Riemannian manifold with
$\operatorname{Ric}_{f}\geq-Kg$ for some constant $K\geq 0$. For any real
number $s$ and any $0<\tilde{\delta}<\delta^{\prime}\leq 1$, let $u$ be a
smooth non-negative subsolution of the heat equation in the cylinder
$Q=B_{o}(r)\times(s-r^{2},s)$, $0<r<R$.

For $2\leq p<\infty$, there exist constants $\tilde{c}_{1}(n)$ and
$\tilde{c}_{2}(n)$ such that

$\sup\limits_{Q_{\tilde{\delta}}}u^{p}\leq\frac{\tilde{c}_{1}(n)e^{\tilde{c}_{2}(n)(Kr^{2}+L(R))}}{(\delta^{\prime}-\tilde{\delta})^{4n}r^{2}\operatorname{Vol}(B_{o}(r))}\cdot\int_{Q_{\delta^{\prime}}}u^{p}dvdt.$
(2.4)

For $0<p<2$, there exist constants $\tilde{c}_{3}(n,p)$ and $\tilde{c}_{4}(n)$
such that

$\sup\limits_{Q_{\tilde{\delta}}}u^{p}\leq\frac{\tilde{c}_{3}(n,p)e^{\tilde{c}_{4}(n)(Kr^{2}+L(R))}}{(\delta^{\prime}-\tilde{\delta})^{4n}r^{2}\operatorname{Vol}(B_{o}(r))}\cdot\int_{Q_{\delta^{\prime}}}u^{p}dvdt.$
(2.5)

Here
$Q_{\tilde{\delta}}=B_{o}(\tilde{\delta}r)\times(s-\tilde{\delta}r^{2},s)$,
$Q_{\delta^{\prime}}=B_{o}(\delta^{\prime}r)\times(s-\delta^{\prime}r^{2},s)$.

##  3\. proof of the main results

In this section, adapting the arguments of Hongjie Dong and Qi S. Zhang [14],
we apply the volume comparison theorem and the mean value inequality in
section 2 to prove the results of this article. We first prove Theorem 1.1.

Proof of Theorem 1.1. Without loss of generality, we may assume that $A_{1}=1$
since the heat equation is linear. To prove the theorem, it suffices to prove
the result at space-time point $(x,0)$ for any $x\in M$.

Since $u$ is a given smooth solution to the heat equation, $u^{2}$ is a non-
negative subsolution to the heat equation. Given a point $(x_{0},t_{0})\in
M\times[-1,0]$ and a positive integer $k$, by letting $p=1$, $s=t_{0}$,
$r=\frac{1}{\sqrt{k}}$, $\tilde{\delta}=\frac{1}{2}$, $\delta^{\prime}=1$ in
(2.5), we have

$u^{2}(x_{0},t_{0})\leq\frac{c_{1}(n)e^{c_{2}(n)(K\frac{1}{k}+\sup\limits_{B_{x_{0}}(\frac{3}{\sqrt{k}})}|f|)}}{(\frac{1}{2})^{4n}\frac{1}{k}\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}\int_{B_{x_{0}}(\frac{1}{\sqrt{k}})\times\left[t_{0}-\frac{1}{k},t_{0}\right]}u^{2}dvdt.$

We observe that

$\sup\limits_{B_{x_{0}}(\frac{3}{\sqrt{k}})}|f|\leq\sup\limits_{B_{o}(d(x_{0},o)+\frac{3}{\sqrt{k}})}|f|\leq
a\left(d(x_{0},o)+\frac{3}{\sqrt{k}}\right)^{2}+b\leq
2ad^{2}(x_{0},o)+\frac{18a}{k}+b,$

then we have

$u^{2}(x_{0},t_{0})\leq\frac{c_{3}(n)e^{c_{4}(n)(K+ad^{2}(x_{0},o)+a+b)}k}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}\int_{B_{x_{0}}(\frac{1}{\sqrt{k}})\times\left[t_{0}-\frac{1}{k},t_{0}\right]}u^{2}dvdt.$
(3.1)

Since $\partial_{t}^{k-1}u$ is also a solution to the heat equation,
substituting this into (3.1) gives

$(\partial_{t}^{k-1}u)^{2}(x_{0},t_{0})\leq\frac{c_{3}(n)e^{c_{4}(n)(K+ad^{2}(x_{0},o)+a+b)}k}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}\int_{B_{x_{0}}(\frac{1}{\sqrt{k}})\times\left[t_{0}-\frac{1}{k},t_{0}\right]}(\partial_{t}^{k-1}u)^{2}dvdt$
(3.2)

for a positive integer $k$.
Next, we will bound the right-hand side of (3.2).
For integers $j=1,2,...,k$, consider the domains:

$\Omega_{j}^{1}=B_{x_{0}}\left(\frac{j}{\sqrt{k}}\right)\times\left[t_{0}-\frac{j}{k},t_{0}\right],$
$\Omega_{j}^{2}=B_{x_{0}}\left(\frac{j+0.5}{\sqrt{k}}\right)\times\left[t_{0}-\frac{j+0.5}{k},t_{0}\right].$

Then it is clear that
$\Omega_{j}^{1}\subset\Omega_{j}^{2}\subset\Omega_{j+1}^{1}$.
Let $\psi_{j}^{(1)}$ be a standard Lipschitz cut-off function supported in

$B_{x_{0}}\left(\frac{j+0.5}{\sqrt{k}}\right)\times\left(t_{0}-\frac{j+0.5}{k},t_{0}+\frac{j+0.5}{k}\right)$

such that

$\psi_{j}^{(1)}=1\ in\ \Omega_{j}^{1}\ and\
|\nabla\psi_{j}^{(1)}|^{2}+|\partial_{t}\psi_{j}^{(1)}|\leq Ck,.$

where $C$ is a universal constant (this includes the following constants $C$)
that may be changed line by line.

For the above cut-off function $\psi=\psi_{j}^{(1)}$, multiplying
$(u_{t})^{2}=u_{t}(\Delta u)$ by $\psi^{2}$, integrating it over
$\Omega_{j}^{2}$, and using integration by parts, we compute that

$\displaystyle\int_{\Omega_{j}^{2}}(u_{t})^{2}\psi^{2}dvdt$
$\displaystyle=\int_{\Omega_{j}^{2}}u_{t}\Delta u\psi^{2}dvdt$
$\displaystyle=-\int_{\Omega_{j}^{2}}\left<(\nabla u)_{t},\nabla
u\right>\psi^{2}dvdt-\int_{\Omega_{j}^{2}}u_{t}\left<\nabla
u,\nabla\psi^{2}\right>dvdt$
$\displaystyle=-\frac{1}{2}\int_{\Omega_{j}^{2}}(|\nabla
u|^{2})_{t}\psi^{2}dvdt-2\int_{\Omega_{j}^{2}}u_{t}\psi\left<\nabla
u,\nabla\psi\right>dvdt$
$\displaystyle=-\frac{1}{2}\int_{B_{x_{0}}\left(\frac{j+0.5}{\sqrt{k}}\right)}(|\nabla
u|^{2}\psi^{2})(x,t_{0})dv+\frac{1}{2}\int_{\Omega_{j}^{2}}|\nabla
u|^{2}(\psi^{2})_{t}dvdt-2\int_{\Omega_{j}^{2}}u_{t}\psi\left<\nabla
u,\nabla\psi\right>dvdt$
$\displaystyle\leq\frac{1}{2}\int_{\Omega_{j}^{2}}|\nabla
u|^{2}(\psi^{2})_{t}dvdt+\frac{1}{2}\int_{\Omega_{j}^{2}}(u_{t})^{2}\psi^{2}dvdt+2\int_{\Omega_{j}^{2}}|\nabla
u|^{2}|\nabla\psi|^{2}dvdt.$

Using $\Omega_{j}^{1}\subset\Omega_{j}^{2}$ and the property of $\psi$, the
above inequality implies that

$\int_{\Omega_{j}^{1}}(u_{t})^{2}dvdt\leq Ck\int_{\Omega_{j}^{2}}|\nabla
u|^{2}dvdt.$ (3.3)

Let $\psi_{j}^{(2)}$ also be a standard Lipschitz cut-off function supported
in

$B_{x_{0}}\left(\frac{j+1}{\sqrt{k}}\right)\times\left(t_{0}-\frac{j+1}{k},t_{0}+\frac{j+1}{k}\right)$

such that $\psi_{j}^{(2)}=1$ in $\Omega_{j}^{2}$ and
$|\nabla\psi_{j}^{(2)}|^{2}+|\partial_{t}\psi_{j}^{(2)}|\leq Ck$. Then we can
obtain

$\int_{\Omega_{j}^{2}}|\nabla u|^{2}dvdt\leq
Ck\int_{\Omega_{j+1}^{1}}u^{2}dvdt.$ (3.4)

Indeed, for the cut-off function $\varphi=\psi_{j}^{(2)}$, multiplying
$uu_{t}=u\Delta u$ by $\varphi^{2}$, integrating it over $\Omega_{j+1}^{1}$,
and using integration by parts, we have that

$\displaystyle\frac{1}{2}\int_{\Omega_{j+1}^{1}}\partial_{t}(u^{2}\varphi^{2})dvdt-\int_{\Omega_{j+1}^{1}}\varphi\varphi_{t}u^{2}dvdt=\int_{\Omega_{j+1}^{1}}uu_{t}\varphi^{2}dvdt$
$\displaystyle=\int_{\Omega_{j+1}^{1}}u\Delta
u\varphi^{2}dvdt=-\int_{\Omega_{j+1}^{1}}|\nabla
u|^{2}\varphi^{2}dvdt-2\int_{\Omega_{j+1}^{1}}u\varphi\left<\nabla
u,\nabla\varphi\right>dvdt.$

Observing that the first term of left hand side is

$\frac{1}{2}\int_{\Omega_{j+1}^{1}}\partial_{t}(u^{2}\varphi^{2})dvdt=\frac{1}{2}\int_{B_{x_{0}}\left(\frac{j+1}{\sqrt{k}}\right)}(u^{2}\varphi^{2})(x,t_{0})dv\geq
0,$

we conclude that

$\displaystyle\int_{\Omega_{j+1}^{1}}|\nabla u|^{2}\varphi^{2}dvdt$
$\displaystyle\leq\int_{\Omega_{j+1}^{1}}\varphi\varphi_{t}u^{2}dvdt-2\int_{\Omega_{j+1}^{1}}u\varphi\left<\nabla
u,\nabla\varphi\right>dvdt$
$\displaystyle\leq\int_{\Omega_{j+1}^{1}}\varphi\varphi_{t}u^{2}dvdt+\frac{1}{2}\int_{\Omega_{j+1}^{1}}|\nabla
u|^{2}\varphi^{2}dvdt+2\int_{\Omega_{j+1}^{1}}u^{2}|\nabla\varphi|^{2}dvdt.$

Therefore,

$\int_{\Omega_{j+1}^{1}}|\nabla u|^{2}\varphi^{2}dvdt\leq
2\int_{\Omega_{j+1}^{1}}\varphi\varphi_{t}u^{2}dvdt+4\int_{\Omega_{j+1}^{1}}u^{2}|\nabla\varphi|^{2}dvdt.$

Then (3.4) follows by $\Omega_{j}^{2}\subset\Omega_{j+1}^{1}$ and the property
of cut-off function $\varphi$.
Combining (3.3) and (3.4) yields that

$\int_{\Omega_{j}^{1}}(u_{t})^{2}dvdt\leq
Ck^{2}\int_{\Omega_{j+1}^{1}}u^{2}dvdt.$

Since $\partial_{t}^{j}u$ is also a solution of the heat equation, we can
replace $u$ in the above inequality by $\partial_{t}^{j}u$ to deduce, by
induction, that

$\int_{\Omega_{1}^{1}}(\partial_{t}^{k-1}u)^{2}dvdt\leq
C^{k-1}k^{2(k-1)}\int_{\Omega_{k}^{1}}u^{2}dvdt.$

Noticing that
$\Omega_{1}^{1}=B_{x_{0}}\left(\frac{1}{\sqrt{k}}\right)\times\left[t_{0}-\frac{1}{k},t_{0}\right]$
and $\Omega_{k}^{1}=B_{x_{0}}(\sqrt{k})\times[t_{0}-1,t_{0}]$, we substitute
the above inequality into (3.2) to get that

$(\partial_{t}^{k-1}u)^{2}(x_{0},t_{0})\leq\frac{c_{3}(n)e^{c_{4}(n)(K+ad^{2}(x_{0},o)+a+b)}k}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}C^{k-1}k^{2(k-1)}\int_{\Omega_{k}^{1}}u^{2}dvdt.$
(3.5)

Using exponential quadratic growth condition $|u(x,t)|\leq
e^{A_{2}d^{2}(x,o)}$ and the triangle inequality, for some point
$(x,t)\in\Omega_{k}^{1}$ we deduce

$|u(x,t)|^{2}\leq e^{2A_{2}(d(x,x_{0})+d(x_{0},o))^{2}}\leq
e^{2A_{2}(\sqrt{k}+d(x_{0},o))^{2}}\leq e^{4A_{2}k+4A_{2}d^{2}(x_{0},o)}.$
(3.6)

By the volume comparison theorem (2.3), we have

$\frac{\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}\leq
k^{n}e^{\frac{K}{6}\left(k-\frac{1}{k}\right)+6\sup\limits_{B_{x_{0}}(3\sqrt{k})}|f|}\leq
k^{n}e^{\frac{K}{6}k+6(2ad^{2}(x_{0},o)+18ak+b)}.$ (3.7)

Substituting (3.6) and (3.7) into (3.5) gives

$|\partial_{t}^{k-1}u(x_{0},t_{0})|\leq
A_{3}^{k}k^{k-1}e^{A_{4}d^{2}(x_{0},o)}$ (3.8)

for all integers $k\geq 1$. Here $A_{3}$ and $A_{4}$ are two positive
constants depending on $K,n,a,b,A_{2}$ and $n,a,A_{2}$, respectively.

Now fix $R>1$. For any point $x\in B_{o}(R)$, we choose a positive integer $j$
and $t\in[-\delta,0]$ for some small $\delta>0$, By Taylor’s theorem,

$u(x,t)-\sum_{i=0}^{j-1}\partial_{t}^{i}u(x,0)\frac{t^{i}}{i!}=\frac{t^{j}}{j!}\partial_{s}^{j}u(x,s),$
(3.9)

where $s=s(x,t,j)\in[t,0]$. Using (3.8), we know that, for
$\delta<\frac{1}{A_{3}e}$, the right hand side of (3.9) converges to $0$
uniformly for $x\in B_{o}(R)$ as $j\rightarrow\infty$. Hence

$u(x,t)=\sum_{j=0}^{\infty}\partial_{t}^{j}u(x,0)\frac{t^{j}}{j!}.$

Then $u(x,t)$ is analytic in time $t$ with radius $\delta$. Set
$a_{j}=a_{j}(x)=\partial_{t}^{j}u(x,0)$. By (3.8) again, we have that

$\partial_{t}u(x,t)=\sum_{j=0}^{\infty}a_{j+1}(x)\frac{t^{j}}{j!}\ and\ \Delta
u(x,t)=\sum_{j=0}^{\infty}\Delta a_{j}(x)\frac{t^{j}}{j!},$

where both series converge uniformly for $(x,t)\in B_{o}(R)\times[-\delta,0]$
for any fixed $R$. Since $u(x,t)$ is a solution of the heat equation, this
implies that

$\Delta a_{j}(x)=a_{j+1}(x)$

with

$|a_{j}(x)|\leq A_{3}^{j+1}(j+1)^{j}e^{A_{4}d^{2}(x,o)}.$

Here $A_{3}$ and $A_{4}$ are two positive constants depending on
$K,n,a,b,A_{2}$ and $n,a,A_{2}$, respectively.
∎

Next we apply Theorem 1.1 to prove Corollary 1.6.

Proof of Corollary 1.6. Assume that $u(x,t)$ is a smooth solution to the
Cauchy problem of the backward heat equation (1.13) with exponential quadratic
growth of the space variable in $M\times(0,\delta)$. Then $u(x,-t)$ is also a
smooth solution of the heat equation with exponential quadratic growth.
By Theorem 1.1, we have

$u(x,-t)=\sum_{j=0}^{\infty}a_{j}(x)\frac{(-t)^{j}}{j!}.$

Then (1.14) follows since $\Delta^{j}a(x)=a_{j}(x)$ in the theorem.

On the other hand, suppose (1.14) holds. We then claim that

$u(x,t)=\sum_{j=0}^{\infty}\Delta^{j}a(x)\frac{t^{j}}{j!}$

is a smooth solution to the heat equation in $M\times[-\delta,0]$ with
$\delta>0$ sufficiently small. Indeed, the bounds (1.14) guarantee that the
above series and the following two series

$\sum_{j=0}^{\infty}\Delta^{j+1}a(x)\frac{t^{j}}{j!}\ and\
\sum_{j=0}^{\infty}\Delta^{j}a(x)\frac{\partial_{t}t^{j}}{j!}$

all converge absolutely and uniformly in $B_{o}(R)\times[-\delta,0]$ for any
fixed $R>0$.
Hence

$\left(\Delta-\partial_{t}\right)u(x,t)=0.$

Moreover, we observe that

$|u(x,t)|\leq\sum_{j=0}^{\infty}|\Delta^{j}a(x)|\frac{|t|^{j}}{j!}\leq
A_{3}e^{A_{4}d^{2}(x,o)}\sum_{j=0}^{\infty}\frac{(A_{3}(j+1)|t|)^{j}}{j!}\leq
A_{5}e^{A_{4}d^{2}(x,o)}$

provided that $t\in[-\delta,0]$ with some sufficiently small constant
$\delta>0$, where we used the fact that the series
$\sum_{j=0}^{\infty}\frac{(A_{3}(j+1)|t|)^{j}}{j!}$ converges in
$[-\delta,0]$.

Thus $u(x,-t)$ is a smooth solution to the Cauchy problem of the backward heat
equation (1.13) of exponential quadratic growth.
∎

The proof of Theorem 1.7 is similar to Theorem 1.1. We only present the key
steps. Proof of Theorem 1.7. From (3.5), we know, for $(x_{0},t_{0})\in
M\times[-1,0]$ and any positive integer $k$,

$\displaystyle(\partial_{t}^{k-1}u)^{2}(x_{0},t_{0})\leq\frac{\bar{c}_{1}(n)e^{\bar{c}_{2}(n)(K+ad^{2}(x_{0},o)+a+b)}k}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}C^{k-1}k^{2(k-1)}\int_{B_{x_{0}}(\sqrt{k})\times[t_{0}-1,t_{0}]}u^{2}dvdt.$
(3.10)

By mean value theorem, there exists $\xi\in(t_{0}-1,t_{0})$ such that

$\displaystyle\int_{B_{x_{0}}(\sqrt{k})\times[t_{0}-1,t_{0}]}u^{2}dvdt$
$\displaystyle=\int_{B_{x_{0}}(\sqrt{k})}u^{2}(x,\xi)dv$ (3.11)
$\displaystyle\leq\left(\int_{B_{x_{0}}(\sqrt{k})}|u|^{p}(x,\xi)dv\right)^{\frac{2}{p}}\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))^{1-\frac{2}{p}}$
$\displaystyle\leq
L^{2}\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))^{1-\frac{2}{p}}$

for $p\geq 2$, where we used Hölder inequality in the second line and in the
last line we used the assumption (1.16).
By volume comparison theorem (2.3) and $k\geq 1$, we have

$\displaystyle\frac{\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))^{1-\frac{2}{p}}}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}$
$\displaystyle=\frac{\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))\operatorname{Vol}(B_{x_{0}}(\sqrt{k}))^{-\frac{2}{p}}}{\operatorname{Vol}(B_{x_{0}}(\frac{1}{\sqrt{k}}))}$
(3.12)
$\displaystyle\leq(k)^{n}e^{\frac{K}{6}k+12ad^{2}(x_{0},o)+108ak+6b}\operatorname{Vol}(B_{x_{0}}(1))^{-\frac{2}{p}}.$

To get a lower bound of $\operatorname{Vol}(B_{x_{0}}(1))$, we use the volume
comparison theorem (2.3) again, then

$\displaystyle\operatorname{Vol}(B_{o}(1))$
$\displaystyle\leq\operatorname{Vol}(B_{x_{0}}(d(x_{0},o)+1))$ (3.13)
$\displaystyle\leq\operatorname{Vol}(B_{x_{0}}(1))(d(x_{0},o)+1)^{n}e^{\frac{K}{6}[(d(x_{0},o)+1)^{2}-1]+6\sup\limits_{B_{x_{0}}(3d(x_{0},o)+3)}|f|}$
$\displaystyle\leq\operatorname{Vol}(B_{x_{0}}(1))e^{nd(x_{0},o)+\frac{K}{6}(d^{2}(x_{0},o)+2d(x_{0},o))+6\sup\limits_{B_{o}(4d(x_{0},o)+3)}|f|}$
$\displaystyle\leq\operatorname{Vol}(B_{x_{0}}(1))e^{(n+\frac{K}{2})d^{2}(x_{0},o)+\frac{1}{4}n+\frac{K}{12}+6(32ad^{2}(x_{0},o)+18a+b)}.$

Combining (3.13), (3.12), (3.11) with (3.10), we arrive at

$|\partial_{t}^{k-1}u(x_{0},t_{0})|\leq
A_{6}^{k}k^{k-1}e^{A_{7}d^{2}(x_{0},o)}\operatorname{Vol}(B_{o}(1))^{-\frac{1}{p}}L$

for all integers $k\geq 1$. Here $A_{6}$ and $A_{7}$ are two positive
constants depending on $n,K,a,b,p$ and $n,a,K,p$, respectively.
∎

## Acknowledgements

Research is partially supported by NSFC Grant No. 11971168, Shanghai Science
and Technology Innovation Program Basic Research Project STCSM 20JC1412900,
and Science and Technology Commission of Shanghai Municipality (STCSM) No.
22DZ2229014.

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