# Exact computation of heat capacities for active particles on a graph

Faezeh Khodabandehlou  Instituut voor Theoretische Fysica, KU Leuven  Simon
Krekels<EMAIL_ADDRESS>Instituut voor Theoretische Fysica, KU
Leuven  Irene Maes  Departement Wiskunde, KU Leuven

###### Abstract

The notion of a nonequilibrium heat capacity is important for bio-energetics
and for calorimetry of active materials more generally. It centers around the
notion of excess heat or excess work dissipated during a quasistatic
relaxation between different nonequilibrium conditions. We give exact results
for active random walks moving in an energy landscape on a graph, based on
calculations employing the matrix-tree and matrix-forest theorems. That
graphical method applies to any Markov jump process under the physical
condition of local detailed balance.

heat capacity, active particles, graphical representation

##  I Introduction

From the conception of heat and thermodynamics, through the birth of quantum
mechanics and all the way to black holes, heat capacities have played a
central and foundational role in chemistry and physics [1, 2, 3]. Biology and
complexity science have encountered fewer successes emerging from the study of
heat capacities. On a fundamental level, such progress requires a theory of
heat exchanges for nonequilibrium systems and for driven random motion in an
energy landscape on complex networks. To that effect, we wish to understand
quantitatively how changes in parameters such as environment, temperature, or
activity modify the heat capacity in such systems. Therefore, exact results
concerning nonequilibrium heat capacities should be looked out for. Such is
the motivation of this paper: to introduce nontrivial and interesting toy
models of active systems on graphs and to give exact results for their heat
capacity.

It is important to specify what is meant by a heat capacity for nonequilibrium
systems. It is not a purely thermodynamic quantity and cannot, for example, be
reduced to the temperature-derivative at fixed volume of the mean energy.
Rather, what is needed is a proper extension of the equilibrium notion of heat
capacity, which makes use of the concept of excess heat. Such an extension was
pioneered in [4, 5, 6], building on earlier ideas by e.g. [7, 8, 9, 10]. In
the next section we briefly sketch the main ingredients that go into the
definition of excess heat and heat capacity. We then apply it to the case of
an active random walker on a graph imbued with an energy landscape.

Central in this paper is a graphical method based on matrix-tree and matrix-
forest theorems, extensions in a way of the well-known Kirchhoff formula for
the stationary distribution of a Markov process. The graphical method relies
on the enumeration of trees and forests, which generally makes the algorithm
complexity increase exponentially with the graph size, quickly becoming
unfeasible (more than $\sim\\!20$ vertices). That method and its algorithmic
implementation are discussed in Section III. It is worth emphasizing that the
methods described below are applicable to a much broader category of systems:
we can also imagine the graph to represent lattices or networks for
transmission of matter or information. Even more generally, the graphs may
represent any Markov jump process with a discrete configuration space [11].
However, we restrict our scope and focus on exact computations pertaining to
active systems. Even though collective phenomena such as phase transitions
remain beyond our scope, small active systems are already interesting for
their behavior of the heat capacity as function of temperature, persistence,
energy and activity parameters. Section IV presents several examples and we
illuminate the relevant features in their heat capacities, which are
calculated exactly for all temperatures. These results may be compared to
those from simulation methods for obtaining the heat capacity of thermal run-
and-tumble particles diffusing in one-dimensional energy landscapes are
reported in [12].

##  II Excess work and heat capacity

###  II.1 General idea

Figure 1: A schematic plot of the work as function of time, for a sudden
temperature change at $t=0$. The shaded area represents the excess work
produced with a change in bath temperature $\delta T$ at time zero. The
$w^{T}$ and $w^{T+\delta T}$ are the stationary work performed before $t=0$
and in the far future.

We refer to [4, 5, 6] for details concerning the theory of nonequilibrium heat
capacities. The main idea is to identify the heat capacity $C(T)$ with the
excess heat $\delta Q^{\mathrm{ex}}$ exchanged with the system for
accommodating a (small) change $\delta T$ at environment temperature $T$.

Indeed, imagine an open system in contact with a thermal bath in equilibrium
of which the temperature $T$ can be controlled. To keep the system in a steady
nonequilibrium state, external forces continuously exert power with average
$w^{T}$ on the system by, for example, pushing particles in a given direction.
Now assume that the system undergoes such a sudden and small change in
temperature, say from $T$ to $T+\delta T$ at time $t=0$. Then, the system
relaxes to the new steady condition with bath temperature $T+\delta T$ and
with the new dissipated power $w^{T+\delta T}$. During the relaxation and in
the interval $[0,t]$, the work done on the system is $W_{[0,t]}$, always
obeying with the First Law

$W_{[0,t]}+Q_{[0,t]}=E(t)-E(0)$ (1)

where $E$ is the energy of the system and $Q_{[0,t]}$ is the heat exchanged
with the system. There is an _excess work_ , entirely due to change in
temperature which equals

$\delta W^{\mathrm{ex}}=\lim_{t\to\infty}\left(W_{[0,t]}-w^{T+\delta
T}t\right)$ (2)

see Fig. 1. In terms of that excess work, the steady heat capacity is

$C(T)=\frac{\partial E}{\partial T}-\frac{\delta W^{\mathrm{ex}}}{\delta T}.$
(3)

Indeed $C(T)\mathrm{d}T=\delta Q^{\mathrm{ex}}$ as can be seen by invoking the
First Law (1). The first term in the right-hand side of (3) takes the form of
the usual equilibrium heat capacity at fixed volume; in equilibrium the second
term vanishes.

It is worth mentioning that the excess heat is not unique, see for example
[13]. Note, however, that we consider an excess heat which is geometric and
reversible. These are nontrivial properties and exactly fit what could be
expected from equilibrium thermodynamics. Another reason for our choice of
excess heat, is that it avoids discussions related to entropy and the second
law. For further discussion on this issue, see [14, 15].

###  II.2 Active random walker on a graph

We now restrict ourselves to systems which we call _active systems_ on graphs.
Consider a finite, connected graph $G$, which has vertices $x,y,\ldots$ and
directed edges $(x,y)$ from $x$ to $y$. The vertices have an energy $E(x)$
associated to them, and each directed edge has a rate $k(x,y)$ associated with
it.

Let $X_{t}$ be a random walker in an energy landscape on the vertices of the
graph $G$. In addition, there is a force acting on $X_{t}$, which depends on
an _internal state_ $\sigma$. Specifically, this internal state is represented
by a dichotomous noise, $\sigma\in\\{+1,-1\\}$, occasionally flipping from
$+1$ to $-1$ according to a (fixed and constant) rate $\alpha$. The value of
$\sigma$ affects the transition rates, which we acknowledge by adding a
subscript: $k_{\sigma}(x,y)$. Crucially, the compound process
$(X_{t},\sigma_{t})$ for the dynamics will be Markovian. We thus speak of an
_active_ random walker. Graphically, we imagine two duplicate graphs, $G_{+}$
and $G_{-}$, with identical structure but different transition rates according
to whether $\sigma=\pm 1$. In particular, an ordinary random walker on
$G_{+}\cup G_{-}$, where $G_{\pm}$ are connected by edges with rate $\alpha$,
represents the active random walker on $G$.

The random walker is in contact with a thermal bath in equilibrium at inverse
temperature $\beta=\frac{1}{T}$, letting $k_{B}=1$. The active force performs
work $W_{\sigma}(x,y)=-W_{\sigma}(y,x)$ on the walker during the transition
$x\to y$, which is instantly dissipated into the thermal bath. Effectively, we
assume that the system obeys _local detailed balance_ [16], requiring

$\log\frac{k_{\sigma}(x,y)}{k_{\sigma}(y,x)}=\beta\,[E(x)-E(y)+W_{\sigma}(x,y)].$
(4)

Section IV provides explicit forms of $k_{\sigma}(x,y)$ in terms of $E(x)$,
$E(y)$ and $W_{\sigma}(x,y)$.

In the transition $x\to y$, the heat dissipated to the reference bath is

$Q_{\sigma}(x,y)=E(x)-E(y)+W_{\sigma}(x,y)$ (5)

which is a differential form of the First Law (1) at fixed $\sigma$. As for
many cases for active systems with dichotomous noise, we let

$W_{\sigma}(x,y)=\sigma\,\mathcal{E}(x,y),\quad\mathcal{E}(x,y)=-\mathcal{E}(y,x)$
(6)

meaning that $W_{+}(x,y)+W_{-}(x,y)=0$, thus eliminating any net global bias.

The expected power exerted on the system when in state $x$ is

$w(x,\sigma)=\sum_{y}k_{\sigma}(x,y)W_{\sigma}(x,y)\,.$ (7)

Now let the walker $X_{t}$ start at position $x$ and in (internal) state
$\sigma$ at $t=0$. At bath temperature $T+\mathrm{d}T$, the expected power at
time $t$ is

$\big{\langle}w(X_{t},\sigma_{t})\,|\,(X_{0},\sigma_{0})=(x,\sigma)\big{\rangle}_{T+\mathrm{d}T}\,.$
(8)

The excess work is

$V^{T+\mathrm{d}T}(x,\sigma)=\int_{0}^{\infty}\left[\big{\langle}w(X_{t},\sigma_{t})\,|\,(X_{0},\sigma_{0})=(x,\sigma)\big{\rangle}_{T+\mathrm{d}T}-w^{T+\mathrm{d}T}\right]\mathrm{d}t$
(9)

where we subtracted the asymptotic stationary value

$\big{\langle}w(X_{t},\sigma_{t})\,|\,(X_{0},\sigma_{0})=(x,\sigma)\big{\rangle}_{T+\mathrm{d}T}\xrightarrow[]{t\uparrow\infty}w^{T+\mathrm{d}T}.$
(10)

The convergence is exponential which makes the integral (9) well-defined. Note
that $V^{T+\mathrm{d}T}$ depends on the final temperature $T+\mathrm{d}T$. On
the other hand $\langle V^{T}\rangle_{T}=0$. Therefore, following (3), the
heat capacity at temperature $T$ is

$\displaystyle C(T)$ $\displaystyle=\frac{1}{\mathrm{d}T}[\langle
E\rangle_{T+\mathrm{d}T}-\langle E\rangle_{T}-\langle
V^{T+\mathrm{d}T}\rangle_{T}\,]$ (11) $\displaystyle=\frac{\mathrm{d}\langle
E\rangle_{T}}{\mathrm{d}T}-\Big{\langle}\frac{\mathrm{d}V^{T}}{\mathrm{d}T}\Big{\rangle}_{T}$

where $\langle E\rangle_{T}$ is the average energy.

##  III Graphical representation

The introduction of the active noise $\sigma$ leads to a _double graph_
structure, where each vertex is mirrored by another, but the edges between
them have rates where $\sigma$ is inverted. The rates of flipping between the
graphs is $\alpha$. Consider the simple example of a two-level system, which
is represented by a graph with two connected vertices. Adding dichotomous
noise will “double” the graph to one with four vertices, connected by edges
with rate $\alpha$.

$k_{+}(x,y)$$k_{-}(x,y)$$\alpha$$\alpha$

The active process $\mathcal{I}_{t}=(X_{t},\sigma_{t})$ is thus a Markov jump
process on a finite simple and connected graph with vertices (states) $i$
labeled as $i=(x,\sigma)$. The transition rates $\lambda$ for the Markov
process $(X_{t},\sigma_{t})$ on the double graph are

$\lambda(i,j)=\begin{cases}k_{\sigma}(x,y)&\text{for}\ i=(x,\sigma),\
j=(y,\sigma)\\\ \alpha&\text{for}\ i=(x,\sigma),\ j=(x,-\sigma)\end{cases}$

and the backward generator has matrix elements

$\displaystyle L_{ij}$ $\displaystyle=\lambda(i,j)$ (12) $\displaystyle
L_{ii}$ $\displaystyle=-\sum_{j\neq i}\lambda(i,j).$

We encapsulate the expected power (7) by considering the double graph

$w(i)=\sum_{j}\lambda(i,j)W(i,j).$ (13)

with

$W(i,j)=\begin{cases}\sigma\mathcal{E}(x,y)&\text{for}\ i=(x,\sigma),\
j=(y,\sigma)\\\ 0&\text{for}\ i=(x,\sigma),\ j=(x,-\sigma).\end{cases}$ (14)

We assume here that tumbling ($\sigma\to-\sigma$) produces no heat in the
thermal environment and, as in (6), $\mathcal{E}(x,y)=-\mathcal{E}(y,x)$.

To calculate the heat capacity from (11), we need the stationary distribution
$\rho^{T}(x,\sigma)$ and the excess work $V^{T}(x,\sigma)$. To this end, we
employ _graphical_ methods, the first of which is commonly called the matrix-
tree theorem. The second method makes use of the matrix-forest theorem.

###  III.1 Stationary distribution

$i$$j$$k$$l$

Figure 2: All spanning trees rooted in $i$ for a graph representing an $n=4$
Markov jump process.

The stationary distribution $\rho(x,\sigma)=\rho(i)$ is the normalized row
vector that solves the stationary master equation

$\rho\,L=0.$ (15)

The matrix-tree theorem states that the solution to (15) may be expressed in
terms of weights of the spanning trees of that graph [17, 18, 19]. The
Kirchhoff formula, making use of the matrix-tree theorem, gives a way to
represent the solution to equation (15) in terms of spanning tree weights

$\rho(i)=\frac{\mathcal{M}(\mathcal{T}_{i})}{\sum_{i}\mathcal{M}(\mathcal{T}_{i})}.$
(16)

By a tree, we mean a connected cycle-less component of a graph, and a spanning
tree is a tree including all vertices of the graph. A tree can be defined by
giving all of its constituent edges.

The weight $m$ of a spanning tree $\mathcal{T}$ is simply the product of the
rates associated to all edges in that tree:

$\displaystyle
m(\mathcal{T})=\prod_{(u,u^{\prime})\in\mathcal{T}}\lambda(u,u^{\prime}).$
(17)

Let the set of all rooted spanning trees rooted in $i$ be denoted
$\mathcal{T}_{i}$. Here, _rooted_ means that all edges flow to the same point,
the root. The weight $\mathcal{M}$ of such a set of trees is the sum of the
weights of the trees:

$\displaystyle\mathcal{M}(\mathcal{T}_{i})$
$\displaystyle=\sum_{\mathcal{T}\in\mathcal{T}_{i}}m(\mathcal{T})$ (18)
$\displaystyle=\sum_{\mathcal{T}\in\mathcal{T}_{i}}\prod_{(u,u^{\prime})\in\mathcal{T}}\lambda(u,u^{\prime}).$

Thus, to find $\rho(i)$, we simply need to find the sets of all spanning trees
rooted in $i$ for all $i$; a simple example is shown in Fig. 2, for more
examples and details see [20]. These trees can be generated algorithmically,
yet the computational complexity increases sharply with the number of states.

###  III.2 Excess work

$i$$j$$k$$l$ Figure 3: The spanning forests $\mathcal{F}^{i\to i}$ for a graph
representing an $n=4$ Markov jump process. The sets $\mathcal{F}^{u\to i}$ for
$u\neq i$ are all subsets of $\mathcal{F}^{i\to i}$.

The excess work (9) can be graphically expressed using the _Matrix-Forest_
theorem [21]. Omitting sub- and superscripts now to indicate temperatures, we
use that

$\langle w(\mathcal{I}_{t})\,|\,\mathcal{I}_{0}=i\rangle=e^{Lt}w(i).$ (19)

The excess work (9) can then be written as

$\displaystyle V(i)$
$\displaystyle=\int_{0}^{\infty}[e^{Lt}w(i)-\left<w\right>]\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}e^{Lt}[w(i)-\left<w\right>]\mathrm{d}t$
$\displaystyle=\int_{0}^{\infty}e^{Lt}f(i)\,\mathrm{d}t$ (20)

where $f(i)=w(i)-\left<w\right>$ is the zero-average expected work.

The last expression (20) is also equal to the limit of the resolvent inverse
acting on $f$ [22]

$V(i)=\int_{0}^{\infty}e^{Lt}f(i)\,\mathrm{d}t=\left(\lim_{b\to\infty}b(I-bL)^{-1}f\right)\\!(i)\,.$
(21)

The matrix-forest theorem [23, 24] gives a graphical representation of
$(I-bL)^{-1}$ in terms of $b$ and weights of so-called _forests_. Using this
representation

$\displaystyle V(i)$
$\displaystyle=\sum_{j}\left(\lim_{b\rightarrow\infty}\dfrac{b}{I-bL}\right)_{ij}f(j)$
$\displaystyle=\sum_{j}\lim_{b\to\infty}\frac{\sum_{k=0}^{n-1}b^{k+1}\mathcal{M}(\mathcal{F}^{i\to
j}_{k})f(j)}{\sum_{k=0}^{n-1}b^{k}\mathcal{M}(\mathcal{F}_{k})}$
$\displaystyle=\sum_{j}\lim_{b\to\infty}\frac{\sum_{k=0}^{n-2}b^{k+1}\mathcal{M}(\mathcal{F}^{i\to
j}_{k})f(j)}{\sum_{k=0}^{n-1}b^{k}\mathcal{M}(\mathcal{F}_{k})}+\sum_{j}\lim_{b\to\infty}\frac{b^{n}\mathcal{M}(\mathcal{F}^{i\to
y}_{n-1})f(j)}{\sum_{k=0}^{n-1}b^{k}\mathcal{M}(\mathcal{F}_{k})}$

$f$ has zero average, so the second sum vanishes. More details on the use of
the matrix-forest theorem can be found in [15]. The result is an expression
for $V$ in terms of the weights of forests :

$V(i)=\frac{\sum_{j}\mathcal{M}(\mathcal{F}^{i\to
j})f(j)}{\sum_{j}\mathcal{M}(\mathcal{T}_{j})}.$ (22)

The forests and weights of forests are defined as follows: a forest is a
cycle-less graph, whose connected components are trees; a _spanning_ forest is
a forest which includes all vertices in the graph. A forest is a _rooted_
forest if all the constituent trees are rooted. The weight $m$ of a forest $F$
is the product of the rates of all the edges included in the forest,

$m(F)=\prod_{(u,u^{\prime})\in F}\lambda(u,u^{\prime}).$ (23)

Let $\mathcal{F}_{k}^{i\to j}$ denote the set of all spanning forests
consisting of $k$ edges such that $j$ is the root of one of the trees and $i$
belongs to the same tree as $j$. When $k=n-2$, every forest in
$\mathcal{F}_{n-2}^{i\to j}$ consists of two trees and we use the notation
$\mathcal{F}^{i\to j}$ instead of $\mathcal{F}_{n-2}^{i\to j}$. When $k=n-1$,
every forest in $\mathcal{F}_{n-1}^{i\to j}$ consists of one tree which is a
spanning tree. The weight $\mathcal{M}$ of a set of forests is the sum of the
weights of the forests constituting the set,

$\displaystyle\mathcal{M}(\mathcal{F}^{i\to j})$
$\displaystyle=\sum_{F\in\mathcal{F}^{i\to j}}m(F)$ (24)
$\displaystyle=\sum_{F\in\mathcal{F}^{i\to j}}\prod_{(u,u^{\prime})\in
F}\lambda(u,u^{\prime}).$

Thus, to calculate the excess work one needs knowledge of the spanning trees
of the graph and also the spanning forests of the graph. As a simple example,
see Fig. 3; note that the lone vertices are to be seen as rooted trees, so
there are always two trees in every forest.

##  IV Examples and results

So far we have not specified the transition rates $k_{\sigma}(x,y)$ except by
the requirement of local detailed balance (4). We choose rates of the general
form

$k_{\sigma}(x,y)=\frac{1}{1+e^{-\beta[E(x)-E(y)+W_{\sigma}(x,y)]}}\,$ (25)

compatible with Arrhenius-type reaction rates [25], and depending on the
energy landscape $E(x)$ and the active work
$W_{\sigma}(x,y)=\sigma\mathcal{E}(x,y)=-\sigma\mathcal{E}(y,x)$. Other
choices of rates are possible, and in contrast with equilibrium, the heat
capacity may very well depend on the exact form of the transition rates even
though the same local detailed balance condition is imposed.

###  IV.1 Active motion on a ring

Consider a run-and-tumble particle on a periodic line (i.e., a ring) with $N$
sites. Such a process can be represented by doubling the ring of $N$-sites to
describe two different directions of driving. These two rings are connected by
edges representing the tumble transitions which occur at a fixed rate
$\alpha$; the connected states are naturally at the same energy. Thus in total
there will be $n=2N$ possible states, corresponding to the particle having
clockwise or counterclockwise active orientation, see Fig. 4.

$\alpha$$\varepsilon$$\varepsilon$ Figure 4: Double ring with $N=5$, making
the total number of vertices $n=10$.

The transition rates are given by (25) for the edges drawn with arrows in Fig.
4, where $W_{\sigma}(x,y)=\pm\sigma\varepsilon$ depending on whether the edge
is directed along or against the drawn arrow respectively. We plot the heat
capacity of the active walker, represented by the double ring with $N=5$ for
different $\alpha$ and $\varepsilon$. The energy function in each ring is

$E(x)=E_{0}\,\sin\left(\frac{2\pi}{N}\,x\right)$ (26)

where $x\in\\{0,1,\ldots,N-1\\}$.

(a) Varying $\alpha$, $\varepsilon=1$.

(b) Varying $\varepsilon$, $\alpha=0.5$.

Figure 5: Heat capacities plotted for a range of temperatures for an active
particle on an $N=5$ ring; $E_{0}=0.3$ in (26). Figure 6: $C(T)$ approaches
zero exponentially as $T\to 0$. For low temperature, $C(T)$ decreases as
$e^{-a/T}$ where $a\approx 0.2$ is a fitted parameter. This plot shows
$-1/\log\,C(T)$ for the $N=5$ multi-level system, which becomes approximately
linear for small $T$.

In Fig. 5 the results for $C(T)$ are shown. When the temperature approaches
zero, the heat capacity also goes to zero exponentially [15], see for example
Fig. 6. At low temperatures, the heat capacity can be negative and always
reaches a minimum. In equilibrium, peaks in the heat capacity are referred to
as Schottky anomalies; see e.g. [26]. The peak is largest in absolute value
for the largest values of $\alpha$, but saturates around $C(T^{*})\approx-0.6$
for $\alpha\gtrsim 2$. The temperature at which the minimum occurs is also
$\alpha$-dependent, saturating around $T\approx 0.08$. At higher temperatures,
all heat capacities again become positive, and are larger for larger values of
$\alpha$.

Varying $\varepsilon$ (Fig. 5(b)) gives widely different qualitative results,
even for small changes in $\varepsilon$. That is because in the rate function
(25), the driving strength $\varepsilon$ is divided by the temperature. Thus,
as $T$ becomes small, small changes in $\varepsilon$ make large differences in
the transition rates.

###  IV.2 Sisyphus random walk

(a) Varying $\alpha$, $\varepsilon=1$.

(b) Varying $\varepsilon$, $\alpha=0.5$.

Figure 7: Heat capacity $C(T)$ of active walker on a five-level ladder
$(n=10)$ as function of temperature $T$.

Consider a multi-level system with a switch. The switch reverses the direction
of the driving. The corresponding graph representation is shown for $N=4$ in
Fig. 7. The situation is similar to the previous section, but now with an
active walker on a line segment, with the energy increasing monotonically with
$x$. If we imagine the (double) line segment to be directed upwards, such as
in Fig. 8, the active walker is reminiscent of Sisyphus pushing his boulder up
the mountain, only to tumble down again at a rate $\alpha$ [27].

$E_{0}$${2}E_{0}$${3}E_{0}$${4}E_{0}$$\alpha$$\varepsilon$$\varepsilon$ Figure
8: Energy-ladder

The rates are again determined by (25) for the edges with arrows in Fig. 8,
where again $W_{\sigma}(x,y)=\pm\sigma\varepsilon$ depending on whether the
walker moves along or against the arrows.

The graph representation of this Markov process is a ladder (Fig. 8), i.e. two
lines, one with an upward bias and one with a downward bias. The two lines are
connected by the “tumbling” transitions, where the bias of the walker is
inverted at a rate $\alpha$. We plot the heat capacity for a five-level-ladder
($n=10$) when the energy spacings are equal and $E_{0}=1$, the result is shown
in Fig. 7.

In Fig. 7(a), we plot the heat capacity for different $\alpha$ and fixed
$\varepsilon=1$. The heat capacity is positive for the different values of
$\alpha$ and it vanishes for temperature going to zero. For every $\alpha$,
there is a maximum appearing in the low-temperature region. Increasing
$\alpha$, we see that the peak moves to lower temperatures. At low
temperatures, the heat capacity is largest for the smallest $\alpha$ and at
high temperature it is the other way round. For high $\alpha$ the heat
capacity has two local maxima, the second being less pronounced.

In Fig. 7(b), we plot the heat capacity for different $\varepsilon$ and fixed
$\alpha=0.5$. Again, the heat capacity always vanishes with temperature going
to zero; again, for every epsilon, the heat capacity has a minimum or maximum
in the low-temperature region. Just like in the case of the ring, the behavior
of $C(T)$ at low $T$ is very sensitive to changes in $\varepsilon$.

For $N=2$, the analytical solution obtained from the graphical method is given
in Appendix A.

##  V Conclusion

Thermal properties of active and driven systems are of great interest for
probing the nature of a steady nonequilibrium condition. Excess work and
excess heat are crucial quantities here and the matrix-forest theorem gives
rise to a graphical representation of them. This graphical method actually
yields an _exact_ calculation and no approximations need be made in finding
$\rho$ or $V$. That is especially helpful for small systems, as they already
generate nontrivial steady states which can become far-from-equilibrium. The
graphical method may then even produce analytical expressions for the heat
capacities of such systems. More generally, it enables to study the effect of
the geometry analytically in different limits. For example, in all steps of
the calculation we can directly see the influence of the rates over each edge
on the final result. This is to be contrasted against the more standard, yet
less transparent method of calculating the spectrum of $L$, and using that to
calculate the excess heat.

The proposed method is general and allows the calculation of heat capacities
for any dynamics on any graph. As an example we calculated the heat capacity
for active particles in different geometries, represented as (double) graphs.
The obtained heat capacities show behavior not typically found in equilibrium
systems, notably the frequent occurrence of minima and maxima, and negative
heat capacities. Those fascinating features are waiting to be confirmed
experimentally as well.

###### Acknowledgements.

We thank Christian Maes for many discussions and suggestions.

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##  Appendix A Analytical solution for the active two-level system

In this section we give analytical results for the two-level system with a
switch (the $N=2$ Sisyphus random walk). We call the tumbling rate $\alpha$,
the driving strength $\varepsilon$ and the energy spacing $\Delta$. The
transition rates between the two levels are given by (25). As indicated in
section III, the calculation of the heat capacity requires knowledge of
$\rho^{T}(i)$ and $V^{T}(i)$. These can be calculated by employing the
graphical methods laid out in section III.

The spanning trees rooted in $i$, $\mathcal{T}_{i}$ for the two-level system
are given by Fig. 2, relabeling the vertices as needed for roots other than
$i$.

The spanning forests $\mathcal{F}^{i\to i}$ are given in Fig. 3. Although the
general cases for $\mathcal{F}^{i\to j}$ are not given explicitly, they are
all subsets of $\mathcal{F}^{i\to i}$, and can be found by eliminating the
forests not connecting $i$ and $j$.

To find an expression for the stationary distribution $\rho^{T}(i)$, we use
(16),

$\displaystyle\rho^{T}(i)$
$\displaystyle=\frac{\mathcal{M}(\mathcal{T}_{i})}{\sum_{i}\mathcal{M}(\mathcal{T}_{i})}$
(27)
$\displaystyle=\frac{1}{2(2\alpha+1)\left(e^{\Delta/T}+e^{\frac{\varepsilon}{2T}}\right)\left(e^{\frac{2\Delta+\varepsilon}{2T}}+1\right)}\begin{pmatrix}e^{\Delta/T}\left(\alpha+(2\alpha+1)e^{\frac{2\Delta+\varepsilon}{2T}}+\alpha
e^{\varepsilon/T}+1\right)\\\
(\alpha+1)e^{\frac{\Delta+\varepsilon}{T}}+\alpha
e^{\Delta/T}+(2\alpha+1)e^{\frac{\varepsilon}{2T}}\\\
e^{\Delta/T}\left(\alpha+(2\alpha+1)e^{\frac{2\Delta+\varepsilon}{2T}}+(\alpha+1)e^{\varepsilon/T}\right)\\\
\alpha
e^{\frac{\Delta+\varepsilon}{T}}+(\alpha+1)e^{\Delta/T}+(2\alpha+1)e^{\frac{\varepsilon}{2T}}\end{pmatrix}$

and for $V^{T}(i)$, we use (22),

$\displaystyle V^{T}(i)$
$\displaystyle=\frac{\sum_{j}\mathcal{M}(\mathcal{F}^{i\to
j})f(j)}{\sum_{j}\mathcal{M}(\mathcal{T}_{j})}$ (28)
$\displaystyle=-\frac{\varepsilon}{(2\alpha+1)^{2}\left(e^{\Delta/T}+e^{\frac{\varepsilon}{2T}}\right)\left(e^{\frac{2\Delta+\varepsilon}{2T}}+1\right)}\begin{pmatrix}\left((\alpha+1)e^{\frac{\Delta+\varepsilon}{T}}+\alpha
e^{\Delta/T}+(2\alpha+1)e^{\frac{\varepsilon}{2T}}\right)\\\
-e^{\Delta/T}\left(\alpha+(2\alpha+1)e^{\frac{2\Delta+\varepsilon}{2T}}+\alpha
e^{\varepsilon/T}+1\right)\\\ -\left(\alpha
e^{\frac{\Delta+\varepsilon}{T}}+(\alpha+1)e^{\Delta/T}+(2\alpha+1)e^{\frac{\varepsilon}{2T}}\right)\\\
e^{\Delta/T}\left(\alpha+(2\alpha+1)e^{\frac{2\Delta+\varepsilon}{2T}}+(\alpha+1)e^{\varepsilon/T}\right)\end{pmatrix}.$

Then to calculate the heat capacity (11),

$\displaystyle C(T)$ $\displaystyle=\frac{\mathrm{d}\langle
E\rangle_{T}}{\mathrm{d}T}-\Big{\langle}\frac{\mathrm{d}V^{T}}{\mathrm{d}T}\Big{\rangle}_{T}$
(29)
$\displaystyle=\sum_{i}E(i)\frac{\mathrm{d}\rho^{T}(i)}{\mathrm{d}T}-\sum_{i}\rho^{T}(i)\frac{\mathrm{d}V^{T}(i)}{\mathrm{d}T}\,.$

All put together, the result is

$C(T)=\frac{2\left(\left(\Delta+2\alpha\Delta\right)^{2}+2\varepsilon^{2}\right)\left(1+\cosh\left(\frac{\Delta}{T}\right)\cosh\left(\frac{\varepsilon}{2T}\right)\right)-\left(9+4\alpha\left(1+\alpha\right)\right)\Delta\varepsilon\sinh\left(\frac{\Delta}{T}\right)\sinh\left(\frac{\varepsilon}{2T}\right)}{4T^{2}\left(1+2\alpha\right)^{2}\left(\cosh\left(\frac{3}{4T}\right)+\cosh\left(\frac{1}{T}\right)\right)^{2}}.$

##  Appendix B Finding Forests

Here we briefly sketch an algorithm which can find all sets of spanning
forests $\mathcal{F}^{j\to i}$ for any given connected graph. The forests are
_grown_ by repeatedly adding edges according to certain rules, an important
fact is that any spanning forest with two trees will contain $n-2$ edges. The
steps are as follows

  1. 1.

Find all trees of size smaller than $n-2$ connecting vertex $j$ to vertex $i$
in the direction $j\to i$.

  2. 2.

Choose a tree from step 1, and choose an additional root, other than $i$.

  3. 3.

_Grow_ the forest from the directed tree from step 1 and the chosen root. The
growing has some sub-steps, which are

    1. (a)

Find all edges which can ‘legally’ be added to the forest, respecting the
chosen roots and the directedness of the trees.

    2. (b)

For each edge which can be added, add the edge and put all the newly found
incomplete forests in a _stack_ , i.e., a container for unfinished forests. If
the forest has $n-2$ edges, do not put it in the stack, but keep it in the set
of completed forests.

    3. (c)

Take a forest from the stack and return to step 3a with that forest. Repeat
this procedure until the stack is empty.

  4. 4.

Repeat step 3 for each combination of directed trees from 1 and additional
roots.

To find all spanning _trees_ rooted in some $i$, $\mathcal{T}_{i}$, some minor
modifications in the algorithm above are in order: step 1 and 2 can be
skipped, and the growth can start immediately from the root, $i$. The growth
phase remains the same, bar that the number of edges in a spanning tree is
$n-1$ instead of $n-2$.

The code used to generate the trees and forests for calculations in this paper
is available in [28].


# Big Flip Graphs and their Automorphism Groups

Assaf Bar-Natan, Advay Goel, Brendan Halstead,
Paul Hamrick, Sumedh Shenoy, and Rishi Verma

###### Abstract.

In this paper, we study the relationship between the mapping class group of an
infinite-type surface and the simultaneous flip graph, a variant of the flip
graph for infinite-type surfaces defined by Fossas and Parlier [fossas-
parlier]. We show that the extended mapping class group is isomorphic to a
proper subgroup of the automorphism group of the flip graph, unlike in the
finite-type case. This shows that Ivanov’s metaconjecture, which states that
any “sufficiently rich” object associated to a finite-type surface has the
extended mapping class group as its automorphism group, does not extend to
simultaneous flip graphs of infinite-type surfaces.

##  1\. Introduction

The mapping class group $\operatorname{Map}(S)$ of a surface $S$ is the group
of orientation-preserving homeomorphisms of $S$ up to isotopy. Expanding
$\operatorname{Map}(S)$ to include orientation-reversing homeomorphisms yields
the _extended mapping class group_ $\operatorname{Map}^{\pm}(S)$. The flip
graph $\mathcal{F}(S)$ of $S$ is the graph whose vertices are triangulations
and whose edges are elementary moves, or _flips_. This graph characterizes
finite-type surfaces up to homeomorphism, and the automorphisms of the flip
graph are precisely those induced by the action of the extended mapping class
group [korkmaz-popadopoulos]. This exemplifies a broader pattern described by
the following “metaconjecture” due to Ivanov.

######  Conjecture 1.1 (Problem 6 of [ivanov]).

Every object naturally associated to a surface and having sufficiently rich
structure has the extended mapping class group as its group of automorphisms.

Here, “sufficiently rich” is intentionally left undefined. Ivanov’s
metaconjecture has been shown to hold for many objects in addition to the flip
graph, such as the curve complex [ivanov-curve-complex], the arc complex
[irmak-mccarthy], and Torelli buildings [farb-ivanov], and in 2019, the
metaconjecture was shown to be true for an even larger class of complexes on
finite-type surfaces [brendle-margalit].

For infinite-type surfaces, i.e., surfaces whose fundamental group is not
finitely-generated, it is known that the automorphism group of the curve graph
is the extended mapping class group [bavard-dowdall-rafi, hernandez-morales-
valdez]. Therefore, in this paper, motivated by Ivanov’s metaconjecture and
the above results, we study the same question in the setting of flip graphs on
infinite-type surfaces. We will consider the simultaneous flip graph, a
variant of the flip graph defined by Fossas and Parlier [fossas-parlier],
which adapts the flip graph for infinite-type surfaces. We show that—unlike in
the finite-type case—the symmetries of this graph are no longer fully captured
by the extended mapping class group. In particular, we prove the following
result.

######  Theorem 1.2.

The natural action of the extended mapping class group on the flip graph is an
isomorphism from $\operatorname{Map}^{\pm}(\Sigma)$ to a proper subgroup of
$\operatorname{Aut}(\mathcal{F}(\Sigma))$.

To prove this, we first show that the isotopy class of a homeomorphism is
determined by its action on $\mathcal{F}(\Sigma)$. For other simplicial
complexes on which $\operatorname{Map}^{\pm}(\Sigma)$ acts, such as the curve
complex and the pants complex, this can be accomplished using the Alexander
Method for infinite-type surfaces developed by Hernandez et al. in [alex-ref-
pls]. To apply this result to ideal arcs, we prove the following lemma, which
may be of interest by its own merit. [Alexander Method for Ideal
Arcs]lemmaalex Let $\Sigma$ be an infinite-type surface, and let $f$ be a
homeomorphism of $\Sigma$. For $e\in\operatorname{Ends}(\Sigma)$, let
$\mathcal{L}_{e}$ denote the collection of ideal arcs that are “loops at $e$”
— those which map both ends of $(0,1)$ to $e$. The following are equivalent.

  1. (i)

There exists $e\in\operatorname{Ends}(\Sigma)$ such that for all
$\lambda\in\mathcal{L}_{e}$, the ideal arc $f(\lambda)$ is isotopic to
$\lambda$.

  2. (ii)

The homeomorphism $f$ is isotopic to the identity homeomorphism
$\mathrm{Id}_{\Sigma}$.

Using the terminology of [alex-ref-pls], this means that for any end $e$, the
set $\mathcal{L}_{e}$ forms an Alexander system for $\Sigma$. This also
establishes that the mapping class group acts faithfully on the arc graphs
discussed in [aramayona-fossas-parlier].

### Acknowledgements

We would like to thank the Canada/USA Mathcamp for providing a welcoming space
and setting to meet, discuss, and develop the ideas in this paper. We would
also like to thank Kasra Rafi, Hugo Parlier, and Alan McLeay for their support
and helpful conversations. Finally, we would like to thank Mark Bell for
telling us about the Python package “Bigger” and talking to us about the
subtleties of triangulations on infinite-type surfaces.

##  2\. Preliminaries

This section is an introduction to the concepts and terminology regarding
infinite-type surfaces that will be used throughout the paper. A _surface_ is
a connected, orientable topological 2-manifold. A surface $\Sigma$ is called
_finite-type_ if $\pi_{1}(\Sigma)$ is finitely-generated. Otherwise, $\Sigma$
is called _infinite-type_.

The _end space_ of an infinite-type surface $\Sigma$, introduced by
Freudenthal [freudenthal], and denoted by $\operatorname{Ends}(\Sigma)$, is
$\varprojlim\pi_{0}(\Sigma\setminus K)$, where $K$ ranges over all compact
subsurfaces of $\Sigma$ and the discrete spaces $\pi_{0}(\Sigma\setminus K)$
are ordered by inclusion. We equip $\operatorname{Ends}(\Sigma)$ with the
inverse limit topology, and note that it is totally disconnected, compact, and
metrizable. The set $\bar{\Sigma}:=\Sigma\sqcup\operatorname{Ends}(\Sigma)$,
equipped with an appropriate topology, is called the Freudenthal
compactification of the surface $\Sigma$ [freudenthal]. Concretely, an end can
be thought of as a descending chain $e=U_{1}\supset U_{2}\supset\cdots$ of
connected open sets such that $\cap_{i}U_{i}=\emptyset$. If there exists $n$
such that $U_{n}$ can be embedded in $\mathbb{R}^{2}$, then $e$ is called
_planar_. Otherwise, it is said to be _non-planar_ or _accumulated by genus_.
We denote by $\operatorname{Ends}_{g}(\Sigma)$ the subspace of non-planar
ends. See [fanoni-ghaswala-mcleay] for a more detailed exposition.

In 1963, Richards showed that orientable infinite-type surfaces are classified
up to homeomorphism by their genus and the pair of spaces
$(\operatorname{Ends}(\Sigma),\operatorname{Ends}_{g}(\Sigma))$. For more
information, see [richards].

A _simple closed curve_ is an embedding of the circle $S^{1}$ into $\Sigma$.
An _ideal arc_ , or simply an _arc_ , on a surface $\Sigma$ is an embedding
$\alpha:(0,1)\to\Sigma$ which extends continuously to a map
$\bar{\alpha}:[0,1]\to\bar{\Sigma}$ such that the ends of $(0,1)$ are mapped
to ends of $\Sigma$. A _multi-arc_ is a collection of disjoint arcs. An arc
$\alpha$ in $\Sigma$ is called _peripheral_ if $\Sigma\setminus\alpha$ has a
connected component that is a disk. An arc is called _essential_ if it is not
peripheral. A multi-arc is a possibly-infinite collection of disjoint arcs,
and an isotopy class of a multi-arc is the collection of isotopy classes of
its corresponding disjoint arcs.

Let $[\mu]$ and $[\nu]$ be isotopy classes of multi-arcs. The _intersection
number_ $i(\mu,\nu)$ is the minimum value of $|\mu\cap\nu|$ among all isotopy
representatives of $\mu$ and $\nu$.

######  Definition 2.1.

An _ideal triangulation_ of a surface, denoted $T$, is a locally-finite
maximal collection pairwise disjoint, pairwise non-homotopic, essential ideal
arcs whose complementary components are open disks. We also require that $T$
have a tubular neighbourhood homotopic to itself.

There are other alternative definitions of ideal triangulations on infinite-
type surfaces in the literature, such as in [fossas-parlier, mcleay-parlier].
We will sometimes consider a triangulation of $\Sigma$ to be a subspace of
$\Sigma$ rather than a collection of maps into $\Sigma$. The closures of
complementary components of a triangulation are called triangles. It follows
from maximality that the boundary of a triangle can consist of either two or
three arcs.

If an arc $\alpha$ in a triangulation bounds only one triangle $\Delta$, then
$\partial\Delta$ must consist of a single additional arc $\beta$ [korkmaz-
popadopoulos]. In this situation we call $\beta$ a _petal arc for $\alpha$_.

In [mcleay-parlier], McLeay and Parlier establish that every infinite-type
surface admits an ideal triangulation. Moreover, they determine when exactly a
multiarc is a subset of a triangulation:

######  Lemma 2.2 (Theorem 1.2 of [mcleay-parlier]).

An ideal multiarc $\mu$ is a subset of an ideal triangulation if and only if
for any simple closed curve $\gamma$, the intersection number
$i(\gamma,\mu)<\infty$.

If a multiarc $\mu$ satisfies the conditions above, then there exists a
triangulation $T$ containing $\mu$. We call such a triangulation a
_completion_ of $\mu$.

######  Definition 2.3.

Consider a triangulation $T$ and a disjoint collection of arcs $\mu\subset T$
such that for any $\alpha\in\mu$ bounding distinct triangles $\Delta_{1}$ and
$\Delta_{2}$, we have $\Delta_{1}\cap\Delta_{2}\cap\mu=\alpha$. A
_simultaneous flip_ sends $T$ to a triangulation $T^{\prime}$ by replacing
each $\alpha\in\mu$ by the other diagonal of its bounding quadrilateral.

Figure 1 depicts an example of a simultaneous flip. We can now define the
_flip graph_ $\mathcal{F}(\Sigma)$ of a surface $\Sigma$, whose vertices are
isotopy classes of ideal triangulations of $\Sigma$. If $\Sigma$ is an
infinite-type surface, we connect two triangulations by an edge if they are
related by a simultaneous flip. If $\Sigma$ is finite-type, we connect
triangulations by an edge only if they are related by a single flip.

Figure 1. An example of a simultaneous flip of a triangulation, with the
dotted blue arcs before the flip and the red arcs afterward.

Although the flip graph is connected in the finite-type case [disarlo-
parlier], the flip graph in the infinite-type case has uncountably many
connected components. In particular, [fossas-parlier] establishes conditions
on when two triangulations are in the same connected component.

######  Lemma 2.4 (Theorem 1.1 of [fossas-parlier]).

Let $\Sigma$ be an infinite-type surface. Let $S$ and $T$ be triangulations of
$\Sigma$. Then, $S$ and $T$ are in the same connected component of
$\mathcal{F}(\Sigma)$ if and only if there exists $K\geq 0$ such that for
every arc $\alpha$ of $S$ and every arc $\beta$ of $T$, the intersection
numbers $i(\alpha,T)$ and $i(\alpha,S)$ are bounded by $K$.

Note that there is a natural action of the mapping class group on the flip
graph sending an ideal triangulation to its image under the homeomorphism of
the surface. Since this action commutes with flips, each mapping class induces
an automorphism of the flip graph. In the next section, we show that there
exists an automorphism of the flip graph which is not induced by any mapping
class.

##  3\. Proof of Theorem 1.2

In this section, we show that the extended mapping class group is a strict
subgroup of the automorphism group of the flip graph for infinite-type
surfaces. Our methods were first inspired by those used by Branman to show an
analogous result for pants graphs [branman].

Since the flip graph is disconnected, the connected components can in some
sense be manipulated independently of one another. However, every mapping
class necessarily affects triangulations in every connected component of the
flip graph. This is the essential reason that the symmetries of $\Sigma$ fail
to fully capture the symmetries of $\mathcal{F}(\Sigma)$. After establishing
some properties of the action on $\mathcal{F}(\Sigma)$, we will prove this by
providing an automorphism of $\mathcal{F}(\Sigma)$ that cannot be induced by a
mapping class.

We begin by proving that mapping classes are determined by their action on
certain arcs.

###### Proof.

Assuming (ii), we see that restricting the isotopy from $\mathrm{Id}_{\Sigma}$
to $f$ yields an isotopy from $\lambda$ to $f(\lambda)$. Hence, (ii) implies
(i). To show the converse, we claim that any mapping class satisfying (i) must
fix every isotopy class of curves on $\Sigma$. Then we can use the Alexander
Method of [alex-ref-pls] to conclude that $f$ must be isotopic to
$\mathrm{Id}_{\Sigma}$.

Figure 2. A diagram showing the main ideas in Lemma 1.2.

Let $c$ be a curve in $\Sigma$, and let $\gamma$ be a path from $e$ to a point
$p$ on $c$ such that $|c\cap\gamma|=1$. Choose a tubular neighborhood $U$ of
$\gamma$ such that $|\partial U\cap c|=2$. Let $\overline{\gamma}$ be a path
in $U$ from a point $\overline{p}$ on $c$ to $e$ such that
$\gamma\cap\overline{\gamma}=\emptyset$ and $|c\cap\overline{\gamma}|=1$.
Define $\lambda\in\mathcal{L}_{e}$ to be the arc that begins at $e$, follows
$\gamma$ from $e$ to $p$, then follows the connected component of
$c\setminus\\{p,\overline{p}\\}$ not contained in $U$, and finally runs back
to $e$ along $\overline{\gamma}$. See Figure 2. Since $c\cup f(c)$ is compact,
we can choose a separating curve $b$ such that one component of
$\Sigma\setminus b$ contains $e$ and the other contains $c\cup f(c)$. We also
require that $b$ intersects $\gamma$, $\overline{\gamma}$, $f(\gamma)$, and
$f(\overline{\gamma})$ at exactly one point each.

We now define an isotopy from $c$ to $f(c)$ in terms of the isotopy $I$ from
$\lambda$ to $f(\lambda)$ given by the hypothesis. First, observe that $c$ is
isotopic to a curve $\widetilde{c}$ contained in $\lambda\cup b$. Likewise,
$f(c)$ is isotopic to a curve $\widetilde{f(c)}$ contained in $f(\lambda)\cup
b$. $I$ extends to an isotopy between $\widetilde{c}$ and $\widetilde{f(c)}$
by translating $\widetilde{c}\cap b$ along $b$. ∎

Figure 3. The standard triangulation of the flute surface.

######  Remark.

Initially, one might think that if a mapping class fixes a particular
triangulation, it must be the identity. This is not always the case, however.
For example, consider the triangulation $T$ of the flute surface depicted in
Figure 3. Then, the mapping class that shifts every point on the plane to the
right one unit fixes $T$ (not as a set, but as a triangulation). Note that
this homeomorphism is not isotopic to the identity. In other words, arbitrary
triangulations do not form Alexander systems. This implies that generally,
$\operatorname{Map}^{\pm}(\Sigma)$ does not act freely on
$\mathcal{F}(\Sigma)$.

The following technical lemma will be used in the proof of Lemma 3.2 to glean
information about the action of $\mathcal{F}(\Sigma)$ on individual arcs from
its action on triangulations.

######  Lemma 3.1.

Let $\alpha$ and $\beta$ be distinct ideal arcs on $\Sigma$, such that
$\alpha$ is not a petal arc for $\beta$. Then, there exist ideal
triangulations $T_{1}$, $T_{2}$ in distinct connected components of the flip
graph that each contain $\alpha$ but not $\beta$.

###### Proof.

We first construct $T_{1}$. If $i(\alpha,\beta)>0$, then any completion of
$\alpha$ can be taken as $T_{1}$. Otherwise, by Lemma 2.2, arcs $\alpha$ and
$\beta$ can be completed into a triangulation $T$. If $\beta$ is flippable, we
flip $\beta$ to get a triangulation $T^{\prime}$ that contains $\alpha$ but
not $\beta$. Otherwise, $\beta$ is unflippable, which implies that the
triangles bordering both sides of the arcs are the same, and that $\beta$ is
contained within a petal formed by an arc $\gamma$. Then, $\gamma$ must border
two distinct triangles, so $\gamma$ is flippable. After flipping $\gamma$,
$\beta$ will thus be flippable, and flipping $\beta$ will result in a
triangulation $T_{1}$ that contains $\alpha$ but not $\beta$.

Next, we construct $T_{2}$. Since
$\Sigma^{\prime}=\Sigma\setminus(\alpha\cup\beta)$ is still an infinite-type
surface, we can choose a countable collection of disjoint simple closed curves
$\delta_{i}$ on $\Sigma^{\prime}$. Note that each $\delta_{i}$, when thought
of as a curve on $\Sigma$, is disjoint from $\alpha$ and $\beta$. We apply the
procedure described in Corollary 4.1 of [fossas-parlier] to $T_{1}$ using the
collection $\\{\delta_{i}\\}$ to obtain a triangulation $T_{2}$ in a distinct
connected component of $\mathcal{F}(\Sigma)$, containing $\alpha$ but not
$\beta$. ∎

######  Remark.

Note that the procedure described above can create uncountably many
triangulations, each in a distinct connected component and containing $\alpha$
but not $\beta$. This procedure is done precisely in [fossas-parlier].
However, it is not true that every connected component contains such a
triangulation. Consider the triangulation $T$ of the flute surface shown in
Figure 3. We claim that no triangulation in the connected component of $T$
contains an arc $\alpha$ with an endpoint at the non-isolated end. To see why
this is true, observe that none of the arcs of $T$ connect to the non-isolated
end, so this end cannot be the vertex of a bounding quadrilateral of an arc.
Hence, a simultaneous flip cannot connect any new arcs to this end. Since the
resulting triangulation will retain this property, we conclude that no
triangulation containing $\alpha$ is reachable from $T$ by a series of
simultaneous flips. This implies that the modular flip graph [disarlo-parlier]
is disconnected for certain infinite-type surfaces, a result which is of
interest of its own right.

The next result shows that mapping classes are determined by their action on
the flip graph. In addition, the weaker hypothesis allows us to determine the
action of a mapping class on one connected component of $\mathcal{F}(\Sigma)$
by its action on the other components. We use this in the proof of Theorem 1.2
to show that a particular automorphism of $\mathcal{F}(\Sigma)$ is not induced
by a mapping class.

######  Lemma 3.2.

If $f\in\operatorname{Map}^{\pm}(\Sigma)$ acts trivially on all except for one
connected component of $\mathcal{F}(\Sigma)$, then $f$ is the identity in
$\operatorname{Map}^{\pm}(\Sigma)$.

###### Proof.

Let $\alpha$ an arc in $\Sigma$. We show that $f(\alpha)$ cannot be $\beta$
for any arc $\beta\neq\alpha$, which implies that $f(\alpha)=\alpha$. Let
$\beta\neq\alpha$ be arbitrary.

First, suppose that $\alpha$ is a petal arc of $\beta$. We observe that
$\beta$ must have two distinct endpoints, while $\alpha$ only has one, so
$f(\alpha)$ cannot be $\beta$. In the case where $\alpha$ is not a petal arc
of $\beta$, Lemma 3.1 guarantees the existence of triangulations $T_{1}$,
$T_{2}$ containing $\alpha$ but not $\beta$ such that $T_{1}$ and $T_{2}$ are
in distinct connected components of $\mathcal{F}(\Sigma)$. Then, $f$ must act
trivially on either $T_{1}$ or $T_{2}$, so $f(\alpha)\neq\beta$. Thus,
$f(\alpha)=\alpha$, and by Lemma 1.2, $f$ is the identity in
$\operatorname{Map}^{\pm}(\Sigma)$. ∎

######  Corollary 3.3.

The action of $\operatorname{Map}^{\pm}(\Sigma)$ on $\mathcal{F}(\Sigma)$ is
faithful. In other words, $\operatorname{Map}^{\pm}(\Sigma)$ embeds into
$\operatorname{Aut}(\mathcal{F}(\Sigma)).$

The preceding result establishes that the flip graph has at least as many
symmetries as the surface. But since so little is known about the graph
structure of $\mathcal{F}(\Sigma)$, it is difficult to conceive of
automorphisms that are not induced by specific mapping classes. The following
lemma improves the situation to some extent, allowing us to construct
automorphisms of individual connected components via compactly-supported
mapping classes.

######  Lemma 3.4.

Let $f$ be a compactly-supported mapping class. Then, $f$ induces an
automorphism on every connected component of $\mathcal{F}(\Sigma)$. In
particular, $f$ fixes the connected components of $\mathcal{F}(\Sigma)$.

###### Proof.

Let $T\in\mathcal{F}(\Sigma)$ be arbitrary. We will show that $T$ and $f(T)$
must lie in the same connected component of $\mathcal{F}(\Sigma)$. Since $f$
has compact support, there exists a finite-type subsurface $\Sigma^{\prime}$
of $\Sigma$ containing every point $x\in T\subset\Sigma$ for which $f(x)\neq
x$. It follows that every transverse intersection of arcs of $T$ and arcs of
$f(T)$ is contained within $\Sigma^{\prime}$.

Only finitely many arcs of $T$ and $f(T)$ can pass through $\Sigma^{\prime}$,
because otherwise, the arcs of $T$ or $f(T)$ would accumulate. Let
$\mathcal{A}\subset\Sigma^{\prime}$ be this set of arcs, and let
$K=\max_{\alpha,\beta\in\mathcal{A}}|\alpha\cap\beta|$. This maximum exists,
because intersection points between arcs in $\mathcal{A}$ are discrete and
contained in $\Sigma^{\prime}$. Note that for any $\alpha\in T$, we have
$i(\alpha,f(T))\leq K|\mathcal{A}|$, and for any $\beta\in f(T)$, we have
$i(\beta,T)\leq K|\mathcal{A}|$. Thus, by Lemma 2.4, $T$ and $f(T)$ lie in the
same connected component of $\mathcal{F}(\Sigma)$.

Since $f$ induces an automorphism on $\mathcal{F}(\Sigma)$, it must restrict
to an automorphism on each connected component of $\mathcal{F}(\Sigma)$. ∎

We are now ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

Let $f$ be a nontrivial mapping class with compact support, and let $\Gamma$
be a connected component of $\mathcal{F}(\Sigma)$ that $f$ acts nontrivially
on. Note that $f$ induces an automorphism on $\Gamma$ by Lemma 3.4, which we
also denote $f$. We define an automorphism $\overline{f}$ of
$\mathcal{F}(\Sigma)$ by

$\overline{f}(T)=\begin{cases}f(T)&T\in\Gamma\\\
T&T\in\mathcal{F}(\Sigma)\setminus\Gamma\end{cases}.$

By Lemma 3.2, a mapping class inducing $\overline{f}$ would necessarily act
trivially on $\mathcal{F}(\Sigma)$, which $\overline{f}$ does not. Thus,
$\overline{f}$ cannot be induced by a mapping class. Together with Corollary
3.3, this establishes that the natural action of
$\operatorname{Map}^{\pm}(\Sigma)$ corresponds to a homomorphism
$\operatorname{Map}^{\pm}(\Sigma)\to\operatorname{Aut}(\mathcal{F}(\Sigma))$
that is injective but not surjective. ∎

##  4\. Discussion

In this paper, we constructed an automorphism of the flip graph not induced by
a mapping class. In other words, there are graph isomorphisms between flip
graphs that are not induced by homeomorphisms of the corresponding surfaces. A
natural follow-up question to ask is whether simultaneous flip graphs even
determine a surface:

######  Question 4.1.

If $\mathcal{F}(\Sigma)$ and $\mathcal{F}(\Sigma^{\prime})$ are isomorphic as
graphs, does this mean that $\Sigma$ and $\Sigma^{\prime}$ are homeomorphic?

The method we used in this paper considers how automorphisms of the flip graph
on distinct connected components may be incompatible with mapping classes.
However, we might try to restrict our attention to a single connected
component of the flip graph. We can ask whether every automorphism of a
connected component is induced by a mapping class:

######  Question 4.2.

For a given connected component $\Gamma$ of the flip graph
$\mathcal{F}(\Sigma)$, is $\operatorname{Aut}(\Gamma)$ isomorphic to
$\operatorname{Map}^{\pm}(\Sigma)$?

## References


# Large-scale Quantitative Evidence of Media Impact on Public Opinion toward
China

Junming Huang  Paul and Marcia Wythes Center on Contemporary China and
Department of Sociology, Princeton University  Gavin Cook  Paul and Marcia
Wythes Center on Contemporary China and Department of Sociology, Princeton
University  Yu Xie  Paul and Marcia Wythes Center on Contemporary China and
Department of Sociology, Princeton University<EMAIL_ADDRESS>
###### Abstract

Do mass media influence people’s opinion of other countries? Using BERT, a
deep neural network-based natural language processing model, this study
analyzes a large corpus of 267,907 China-related articles published by The New
York Times since 1970. The output from The New York Times is then compared to
a longitudinal data set constructed from 101 cross-sectional surveys of the
American public’s views on China, revealing that the reporting of The New York
Times on China in one year explains $54\%$ of the variance in American public
opinion on China in the next. This result confirms hypothesized links between
media and public opinion and helps shed light on how mass media can influence
public opinion of foreign countries.

## Introduction

America and China are the world’s two largest economies, and they are
currently locked in a tense trade war. In a democratic system, public opinion
shapes and constrains political action. How the American public views China
thus affects relations between the two countries. Because few Americans have
personally visited China, most Americans form their opinions of China and
other foreign lands from media depictions. Our paper aims to explain how
Americans form their attitudes on China with a case study of how The New York
Times may shape public opinion. Our analysis is not causal, but it is informed
by a causal understanding of how public opinion may flow from media to
citizen.

There is considerable debate among scholars of public opinion and political
communication over how media affect (media is plural) the American mind
(Bennett & Iyengar, 2008). Many things complicate what might seem like a
straightforward relationship. For example, people generally tend to seek out
news sources with which they agree (Iyengar et al., 2008), and politically-
active individuals do so more aggressively than the average person (Zaller et
al., 1992). Additionally, with the advent of social media platforms such as
Facebook and Twitter, traditional media respond to audience demand more
strongly than ever before (Jacobs & Shapiro, 2011).

Most extant survey data indicate that Americans do not seem to like China very
much. Many Americans are reported to harbor doubts about China’s record on
human rights (Cao & Xu, 2015; Aldrich et al., 2015) and are anxious about
China’s burgeoning economic, military, and strategic power (Gries & Crowson,
2010; Yang & Liu, 2012). They also think that the Chinese political system
fails to serve the needs of the Chinese people (Aldrich et al., 2015). Most
Americans, however, recognize a difference between the Chinese state, the
Chinese people, and Chinese culture, and they view the latter two more
favorably (Gries & Crowson, 2010). In Fiske’s Stereotype Content Model (Fiske
et al., 2002), which expresses common stereotypes as a combination of
“competence” and “warmth,” Asians belong to a set of “high-status, competitive
out-groups” and rank high in competence but low in warmth (Lin et al., 2005).

The New York Times, which calls itself the “Newspaper of Record,” is the most
influential newspaper in the USA and possibly even in the Anglophonic world.
The digital edition boasts 91 million unique monthly visitors in the US alone
(The New York Times Company, n.d.), and while the paper’s reach may be
impressive, it is yet more significant that the readership of The New York
Times represents an elite subset of the American public. Print subscribers to
the New York Times have a median household income of $191,000, three times the
median income of US households writ large (Rothbaum & Edwards, 2019). Despite
the paper’s haughty and sometimes condescending reporting, the paper “has had
and still has immense social, political, and economic influence on American
and the world” (Schwarz, 2012, page 81).

A small body of prior work studies the The New York Times and how The New York
Times reports on China. Blood and Phillips uses autoregression methods on time
series data to predict public opinion (Blood & Phillips, 1995). Wu et al. use
a similar autoregression technique and find that public sentiment regarding
the economy predicts economic performance and that people pay more attention
to economic news during recessions (Wu et al., 2002). Peng finds that coverage
of China in the paper has been consistently negative but increasingly frequent
as China became an economic powerhouse (Peng, 2004). There is very little
other scholarship that applies language processing methods to large corpora of
articles from The New York Times or other leading papers. Atalay et al. is an
exception that uses statistical techniques for parsing natural languages to
analyze a corpus of newspaper articles from The New York Times, the Wall
Street Journal, and other leading papers in order to investigate the
increasing frequency of information technologies in newspaper classifieds
(Atalay et al., 2018).

Our paper aims to advance understanding of how Americans form their attitudes
on China with a case study of how The New York Times may shape public opinion.
We hypothesize that media coverage of foreign nations affects how Americans
view the rest of the world. This model deliberately simplifies the
interactions between audience and media and sidesteps many debates. We confine
our analysis to a corpus of 267,907 articles on China from The New York Times
articles because of the paper’s singular influence and importance. We quantify
media sentiment with BERT, a state-of-the-art natural language processing
model with deep neural networks, and segment sentiment into eight topics to
capture the nuance of American opinion toward China. We then use conventional
statistical methods to link media sentiment to a longitudinal data set
constructed from 101 cross-sectional surveys of the American public’s views on
China. We find strong correlations between how The New York Times reports on
China in one year and the views of the public on China in the next. The
correlations agree with our hypothesis and imply a strong connection between
media sentiment and public opinion.

## Results

We begin with a demonstration of how the reporting of The New York Times on
China changes over time, and we follow this with an analysis of how coverage
of China might influence public opinion toward China.

### Quantifying media sentiment and public opinion

We quantify media sentiment with a natural language model on a large-scale
corpus of 267,907 articles on China from The New York Times published between
1970 and 2019. To explore sentiment from this corpus in greater detail, we map
every article to a sentiment category (positive, negative, or neutral) in
eight topics: ideology, government & administration, democracy, economic
development, marketization, welfare and well-being, globalization, and
culture.

We do this with a three-stage modeling procedure. First, two human coders
annotate 873 randomly selected articles with a total of 18,598 sentences as
expressing either positive, negative, or neutral sentiment in each topic. We
treat irrelevant articles as neutral sentiment. Secondly, we fine-tune a
natural language processing model BERT (Bidirectional Encoder Representations
from Transformers (Devlin et al., 2018)) with the human-coded labels. The
model uses a deep neural network with 12 layers. It accepts sentences (i.e.,
word sequences of no more than 128 words) as input and outputs a probability
for each category. We end up with two binary classifiers for each topic for a
grand total of 16 classifiers: an assignment classifier that determines
whether a sentence expresses sentiment in a given topic domain and a sentiment
classifier that then distinguishes positive and negative sentiment in a
sentence classified as belonging to a given topic domain. Thirdly, we run the
16 trained classifiers on each sentence in our corpus and assign category
probabilities to every sentence. We then use the probabilities of all the
sentences in an article to determine the article’s overall sentiment category
(i.e., positive, negative, or neutral) in every topic.

As demonstrated in Table 1, the two classifiers are accurate at both the
sentence level and the article level. The assignment classifier and the
sentiment classifier reach classification accuracy of 89 - 96% and 73 - 90%
respectively on sentences. The combined outcome of the classifiers, namely
article sentiment, is accurate to 65 - 95% across all eight topics. For
comparison, a random guess would reach an accuracy of 50% on each task (See
Supplementary Information for details).

American public opinion towards China is a composite measure drawn from
national surveys that ask respondents for their opinions on China. We collect
101 cross-sectional surveys from 1974 to 2019 that asked relevant questions
about attitudes toward China and develop a probabilistic model to harmonize
different survey series with different scales (e.g., 4 levels, 10 levels) into
a single time series, capitalizing on "seaming" years in which different
survey series overlapped. For every year, there is a single real value
representing American sentiment on China relative to the level in 1974. Put
another way, we use sentiment in 1974 as a baseline measure to normalize the
rest of the time series. A positive value shows a more favorable attitude than
that in 1974, and a negative value represents a less favorable attitude than
that in 1974. Because of this, the trends in sentiment changes year-over-year
are of interest, but the absolute values of sentiment in a given year are not
(See Supplementary Information for details). As shown in Figure 1, public
opinion towards China has varied greatly from 1974 to 2019. It steadily
climbed from a low of -24% in 1976 to a high of 73% in 1987, and has
fluctuated between 10% to 48% in the intervening 30 years.

### Trend of media sentiment

The New York Times has maintained steady interest in China and China-adjacent
topics over the years in our sample and has published at least 3,000 articles
on China in every year of our corpus. Figure 2 displays the yearly volume of
China-related articles from The New York Times on each of the eight topics
since 1970. Articles on China increase sharply after 2000 and eventually reach
a peak around 2010 at almost double their volume from the 1970s. As the number
of articles on China increases, the amount of of attention paid to each of the
eight topics diverges. Articles on government, democracy, globalization, and
culture are consistently common while articles on ideology are consistently
rare. In contrast, articles on China’s economy, marketization, and welfare
were rare before 1990 but become increasingly common after 2000. The timing of
this uptick coincides neatly with worldwide recognition of China’s precipitous
economic ascent and specifically the beginnings of China’s talks to join the
World Trade Organization.

While the proportion of articles in each given topic change over time, the
sentiment of articles in each topic is remarkably consistent. Ignoring neutral
articles, Figure 3 illustrates the yearly fractions of positive and negative
articles about each of the eight topics. We find four topics (economics,
globalization, culture, and marketization) are almost always covered
positively while reporting on the other four topics (ideology, government &
administration, democracy, and welfare & well-being) is overwhelmingly
negative.

The NYT views China’s globalization in a very positive light. Almost 100% of
the articles mentioning this topic are positive for all of the years in our
sample. This implies that New York Times welcomes China’s openness to the
world and, more broadly, may be particularly partial to globalization in
general.

Similarly, economics, marketization, and culture are covered most commonly in
positive tones that have only grown more glowing over time. Positive articles
on these topics begin in 1970s when China and the US began Ping-Pong
diplomacy, and eventually comprise $1/4$ to $1/2$ of articles in these three
topics, the remainder of which are mostly neutral articles. This agrees with
the intuition that most Americans like Chinese culture. The New York Times has
been deeply enamored with Chinese cultural products ranging from Chinese art
to Chinese food since the very beginning of our sample. Following China’s
economic reforms, the number of positive articles and the proportion of
positive articles relative to negative articles increases for both economics
and marketization.

In contrast, welfare & well-being is covered in an almost exclusively negative
light. About $1/4$ of the articles on this topic are negative, and almost no
articles on this topic are positive. Topics regarding or adjacent to politics
are covered very negatively. Negative articles on ideology, government &
administration, and democracy outnumber positive articles on these topics for
all of the years in our sample. Though small fluctuations that coincided with
ebbs in US-China relations are observed for those three topics, coverage has
only grown more negative over time. Government & administration is the only
negatively-covered topic that does feature some positive articles. This
reflects the qualitative understanding that The New York Times thinks that the
Chinese state is evil but capable.

Despite the remarkable diversity of sentiment toward China across the eight
topics, sentiment within each of the topics is startlingly consistent over
time. This consistency attests to the incredible stability of American
stereotypes towards China. If there is any trend to be found here, it is that
the main direction of sentiment in each topic, positive or negative, has grown
more prevalent since the 1970s. This is to say that reporting on China has
become more polarized, which is reflective of broader trends of media
polarization (Jacobs & Shapiro, 2011; Mullainathan & Shleifer, 2005).

### Media sentiment affects public opinion

To reveal the connection between media sentiment and public opinion, we run a
linear regression model (Equ 1) to fit public opinion with media sentiment
from current and preceding years.

$\mu_{t}=\sum_{1\leq k\leq
8}\sum_{j\in[t,t-1,t-2,\cdots]}\sum_{s\in\\{positive,negative\\}}\beta_{kjs}F_{kjs},$
(1)

where $\mu_{t}$ denotes public opinion in year $t$ with possible values
ranging from $-1$ to $1$. $F_{kjs}$ is the fraction of positive ($s=positive$)
or negative ($s=negative$) articles on topic $k$ in year $j$. A non-negative
coefficient $\beta_{kjs}$ quantifies the importance of $F_{kjs}$ in predicting
$\mu_{t}$.

There is inertia to public opinion. A broadly-held opinion is hard to change
in the short term, and it may require a while for media sentiment to affect
how the public views a given issue. For this reason, $j$ is allowed to take
$[t,t-1,t-2,\cdots]$ anywhere from zero to a couple of years ahead of $t$. In
other words, we inspect lagged values of media sentiment as candidate
predictors for public attitudes towards China.

We seek an optimal solution of media sentiment predictors to explain the
largest fraction of variance ($r^{2}$) of public opinion. To reduce the risk
of overfitting, we first constrain the coefficients to be non-negative after
reverse-coding negative sentiment variables, which means we assume that
positive articles have either no impact or positive impact and that negative
articles have either zero or negative impact on public opinion. Secondly, we
require that the solution be sparse and contain no more than one non-zero
coefficient in each topic:

$\displaystyle\underset{\beta}{\text{maximize}}$ $\displaystyle
r^{2}(\mu,\beta,F)$ subject to $\displaystyle\beta_{kjs}\geq 0,\forall k,j,s$
$\displaystyle\|\beta_{k,\cdot,\cdot}\|_{0}\leq 1,\forall k$

where $r^{2}(\mu,\beta,F)$ is the explained variance of $\mu$ fitted with
$(\beta,F)$. The $l_{0}$-norm $\|\beta_{k,\cdot,\cdot}\|_{0}$ gives the number
of non-zero coefficients of topic $k$ predictors.

The solution varies with the number of topics we let the model use for
fitting. As shown in Table 2, if we allow fitting with only one topic, we find
that sentiment on Chinese culture has the most explanatory power, accounting
for $31.2\%$ of the variance in public opinion. We run a greedy strategy to
add additional topics that yield the greatest increase in explanatory power,
resulting in eight nested models (Table 2). The explanatory power of our
models increases monotonically with the number of allowed topics but reaches a
saturation point at which the marginal increase in variance explained per
topics decreases after only two topics are introduced (See Table 2). To strike
a balance between simplicity and explanatory power, we use the top two
predictors, which are the positive sentiment of culture and the negative
sentiment of democracy in the previous year, to build a linear predictor of
public opinion that can be written as:

$\mu_{t}=-0.791+3.112F_{culture,t-1,positive}+1.452F_{democracy,t-1,negative},$
(2)

where $F_{culture,t-1,positive}$ is the yearly fraction of positive articles
on Chinese culture in year $t-1$ and $F_{democracy,t-1,negative}$ is the
yearly fraction of negative articles on Chinese democracy in year $t-1$. This
formula explains $53.9\%$ of the variance of public opinion in the time
series. For example, in 1993 $53.9\%$ of the articles on culture had positive
sentiment, and $44.2\%$ of the articles on democracy had negative sentiment
($F_{culture,1993,positive}=0.539,F_{democracy,1993,negative}=-0.442$).
Substituting those numbers into Equ 2 predicts public opinion in the next year
(1994) to be $0.236$, very close to the actual level of public opinion
($0.218$) (Figure 4).

## Discussion

By analyzing a corpus 267,907 articles from The New York Times with BERT, a
state-of-the-art natural language processing model, we identify major shifts
in media sentiment towards China across eight topic domains over 50 years and
find that media sentiment leads public opinion. Our results show that the
reporting of The New York Times on culture and democracy in one year explains
53.9% of the variation in public opinion on China in the next. Our analysis is
neither conclusive nor causal, but it is suggestive. The story that we draw
from our results is that elite media sentiment on China predicts public
opinion on China. There are a number of potential factors that may complicate
this story. We do not consider intermediary processes through which opinion
from elite media percolates to the masses below. Our results are best
interpreted as a “reduced-form” description of the overall relationship
between media sentiment and public opinion towards China. Though The New York
Times might be biased, and it might have a particular bias to how it covers
China, the paper’s ideological slant does not affect our work explaining the
trends in public opinion of China as long as the relevant biases are
consistent over the time period covered by our analyses.

In addition to those specified above, a number of possible extensions of our
work remain ripe targets for further research. Though a fully causal model of
our text analysis pipeline may prove elusive (Egami et al., 2018), future work
may use randomized vignettes to further our understanding of the causal
effects of the exposure to media on attitudes towards China. Secondly, our
modeling framework is deliberately simplified. The state affects news coverage
before news ever makes its way to the citizenry. It is plausible that multiple
state-level actors may bypass the media and alter public opinion directly and
to different ends. For example, the actions and opinions of individual US
high-profiled politicians may attenuate or exaggerate the impact of state-
level tension on public sentiment toward China. There are presumably a whole
host of intermediary processes through which opinion from elite media on high
percolates to the masses below. Thirdly, the relationship between the
sentiment of The New York Times and public opinion may be very different for
hot-button social issues of first-line importance in the American culture
wars. In our corpus, The New York Times has covered globalization almost
entirely positively, but the 2016 election of President Donald J. Trump
suggests that many Americans do not share the zeal of The Times for
international commerce. We also plan to extend our measure of media sentiment
to include text from other newspapers. The Guardian, a similarly elite,
Anglophonic, and left-leaning paper, will make for a useful comparison case.
Finally, our analysis was launched in the midst of heightened tensions between
the US and China and concluded right before the outbreak of a global pandemic.
Many things have changed since COVID-19. Returning to our analysis with an
additional year or two of data will almost certainly provide new results of
additional interest.

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## Data availability

All data analyzed during the current study are publicly available. The New
York Times data were accessed using official online APIs
(https://developer.nytimes.com/). We use query API to search for 267,907
articles from The New York Times that mention China, Chinese, Beijing, Peking,
or Shanghai. We download the full text and date of each article for natural
language processing. The survey data were obtained from three large public
archives, namely Roper Center for Public Opinion Research (ROPER), the
National Opinion Research Center (NORC), and Inter-university Consortium for
Political and Social Research (ICPSR). The source codes and pretrained
parameters of the natural language processing model (BERT) were publicly
released by Google Inc. (https://github.com/google-research/bert).

## Figures

Figure 1: Public opinion of Americans toward China. This time series is
aggregated from 101 cross-sectional surveys from 1974 to 2019 that asked
relevant questions about attitudes toward China, ranging from -100% to 100%
with the year of 1974 as baseline. Years with attitudes above zero show a more
favorable attitude than that in 1974, with a peak of 73% in 1987. Years with
attitudes below zero show a less favorable attitude than that in 1974, with a
lowest level of -24% in 1976. The time series is shown with 95% confidence
interval.  Figure 2: Topic-specific yearly volume of The New York Times
articles with sentiment on China from 1970 to 2019. In each year we report in
each topic the number of positive and negative articles, while ignoring
neutral/irrelevant articles. The media have consistently high attention on
reporting China government & administration, democracy, globalization, and
culture. There are emerging interests on China’s economics, marketization, and
welfare & well-being since 1990s. Note that the sum of the stacks does not
equal to the total volume of articles about China, because each article may
express sentiment in none or multiple topics. Figure 3: Sentiments on The New
York Times on China in eight topics from 1970 to 2019. Each panel reports the
trend of yearly media attitude toward China in each of eight domains since
1970. The media attitude is measured as the percentages of positive articles
and negative articles respectively. US-China relation milestones are marked as
gray dots. New York Times express diverging but consistent attitudes in the
eight domains, with negative articles consistently common in ideology,
government, democracy, and welfare, and positive sentiments common in
economic, globalization and culture. Standard errors are too small to be
visible (below 1.55% in all topics all years). Figure 4: Regressing public
opinion of Americans toward China on The New York Times sentiments. The public
opinion (solid), as a time series, is well fitted by the media sentiments on
two selected topics, namely Culture and Democracy, in the previous year. The
dash line shows a linear prediction based on the fractions of positive
articles on Culture and negative articles on Democracy in the previous year.
The public opinion is shown with 95% confidence interval, and the fitted line
is shown with one standard error.

## Tables

Table 1: Accurately quantifying media sentiment. Media article sentiment accuracy is measured by the percentage of correctly labeled articles (positive, negative, neutral/irrelevant) in each topic. Sentence assignment accuracy is measured as AUC (Area under ROC curve) on each sentence by a binary classifier (positive/negative vs neutral/irrelevant). Sentence sentiment accuracy is measured AUC on each sentence by a binary classifier (positive vs negative). | Article | Sentence | Sentence
---|---|---|---
| sentiment | assignment | sentiment
Ideology | $88.7\pm 3.9\%$ | $94.1\pm 1.9\%$ | $90.4\pm 7.4\%$
Government | $62.7\pm 6.6\%$ | $95.6\pm 0.8\%$ | $78.1\pm 4.3\%$
Administration |  |  |
Democracy | $87.7\pm 5.2\%$ | $95.7\pm 1.1\%$ | $77.5\pm 5.4\%$
Economic development | $81.2\pm 6.3\%$ | $94.9\pm 1.1\%$ | $79.9\pm 5.0\%$
Marketization | $91.3\pm 4.3\%$ | $96.9\pm 1.7\%$ | $74.9\pm 14.5\%$
Welfare | $82.0\pm 4.4\%$ | $94.5\pm 2.7\%$ | $73.2\pm 9.8\%$
well-being |  |  |
Globalization | $77.4\pm 5.8\%$ | $93.3\pm 1.5\%$ | $80.7\pm 5.6\%$
Culture | $79.8\pm 4.8\%$ | $90.0\pm 1.9\%$ | $73.5\pm 5.4\%$

Table 2: Nested models fitting public opinion with media sentiment. We regress public opinion on various numbers of media sentiment predictors, requiring each topic with (a) no more than one predictor, and (b) non-negative coefficients. The best two topic predictors (yearly fraction of positive articles on Chinese culture in the previous year, and yearly fraction of negative articles on Chinese democracy in the previous year) explain $53.9\%$ of the variation in public opinion. | 1 topic | 2 topics | 3 topics | 4 topics | 5 topics | 6 topics | 7 topics | 8 topics
---|---|---|---|---|---|---|---|---
$F_{culture,t-1,positive}$ | 2.479 | 3.112 | 2.233 | 2.044 | 1.826 | 1.551 | 1.082 | 1.070
$F_{democracy,t-1,negative}$ |  | 1.452 | 1.498 | 1.290 | 1.196 | 1.115 | 1.188 | 1.067
$F_{globalization,t-1,positive}$ |  |  | 1.945 | 1.951 | 2.047 | 1.926 | 1.963 | 1.818
$F_{government\&administration,t-4,negative}$ |  |  |  | 0.762 | 1.087 | 1.197 | 1.246 | 1.249
$F_{marketization,t-4,negative}$ |  |  |  |  | 2.869 | 3.816 | 2.791 | 3.186
$F_{welfare\&well-being,t-5,positive}$ |  |  |  |  |  | 5.412 | 8.245 | 8.297
$F_{economicdevelopment,t-2,negative}$ |  |  |  |  |  |  | 1.607 | 1.794
$F_{ideology,t-2,negative}$ |  |  |  |  |  |  |  | 0.324
Intercept | -1.077 | -0.791 | -2.136 | -1.638 | -1.401 | -1.146 | -0.846 | -0.692
Explained variance | 0.312 | 0.539 | 0.577 | 0.606 | 0.622 | 0.639 | 0.656 | 0.659

## Competing interests

The authors declare no competing interests.

## Acknowledgments

The Acknowledgments are hidden in the review phase to satisfy the requirement
of author anonymisation.