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# Fixation probability in evolutionary dynamics on switching temporal networks

Jnanajyoti Bhaumik1 Naoki Masuda1,2

(1Department of Mathematics, State University of New York at Buffalo, NY
14260-2900, USA
2Computational and Data-Enabled Science and Engineering Program, State
University of New York at Buffalo, Buffalo, NY 14260-5030, USA
)

###### Abstract

Population structure has been known to substantially affect evolutionary
dynamics. Networks that promote the spreading of fitter mutants are called
amplifiers of selection, and those that suppress the spreading of fitter
mutants are called suppressors of selection. Research in the past two decades
has found various families of amplifiers while suppressors still remain
somewhat elusive. It has also been discovered that most networks are
amplifiers of selection under the birth-death updating combined with uniform
initialization, which is a standard condition assumed widely in the
literature. In the present study, we extend the birth-death processes to
temporal (i.e., time-varying) networks. For the sake of tractability, we
restrict ourselves to switching temporal networks, in which the network
structure deterministically alternates between two static networks at constant
time intervals or stochastically in a Markovian manner. We show that, in a
majority of cases, switching networks are less amplifying than both of the two
static networks constituting the switching networks. Furthermore, most small
switching networks, i.e., networks on six nodes or less, are suppressors,
which contrasts to the case of static networks.

##  1 Introduction

Evolutionary dynamics models enable us to study how populations change over
time under natural selection and neutral random drift among other factors.
Over the past two decades, the population structure, particularly those
represented by networks (i.e., graphs), has been shown to significantly alter
the spread of mutant types [1, 2, 3, 4, 5]. Mutants may have a fitness that is
different from the fitness of a resident type, which makes the mutants either
more or less likely to produce offsprings. The fitness of each type may vary
depending on the type of the neighboring individuals’ types as in the case of
evolutionary games on networks. On the other hand, the simplest assumption on
the fitness is to assume that the fitness of each type is constant over time.
This latter case, which we refer to as constant selection, has also been
studied as biased voter models, modeling stochastic opinion formation in
networks (and well-mixed populations)[6, 7, 8, 9].

Networks on which real-world dynamical processes approximated by evolutionary
dynamics occur may be time-varying. Temporal (i.e., time-varying) networks and
dynamical processes on them have been extensively studied [10, 11, 12, 13, 14,
15, 16]. Evolutionary game dynamics on time-varying networks are no exception.
It has been shown that temporal networks enhance the evolution of cooperation
as compared to static networks [17, 18, 19, 20, 21]. It has also been known
for a longer time that coevolutionary dynamics of a social dilemma game and
network structure, in which the dynamics of the network structure depend on
the state of the nodes (e.g., cooperator or defector), enhance overall
cooperation if players tend to avoid creating or maintaining edges connecting
to defectors [22, 23, 24, 5, 25].

In this study, we investigate constant-selection evolutionary dynamics on
temporal networks to clarify how the time dependence of the network structure
impacts evolutionary processes. In particular, a key question in studies of
constant-selection evolutionary dynamics on networks is the fixation
probability, defined as the probability that a single mutant type introduced
to a node in the network eventually fixates, i.e., occupies all the nodes of
the network. The fixation probability depends on the fitness of the mutant
type relative to the fitness of the resident type, denoted by $r$. A network
is called an amplifier of selection if it has a higher fixation probability
than the complete graph, which corresponds to the Moran process, when $r>1$
and a lower fixation probability when $r<1$; conversely, a network is called a
suppressor of selection if the fixation probability is smaller than for the
Moran process on $r>1$ and larger for $r<1$ [1, 26]. In Fig. 1, we show
hypothetical examples of the fixation probability as a function of $r$ for
three networks: the complete graph (i.e., Moran process), an amplifier, and a
suppressor. Under the so-called birth-death updating rule and uniform
initialization, most static networks are amplifiers of selection [27, 28]. In
fact, there is only one suppressing static network with six nodes among the
112 connected six-node networks [29].

Figure 1: Concept of amplifier and suppressor of selection. The fitness of the
resident and mutant type is equal to $1$ and $r$, respectively. A value of
$r>1$ indicates that a mutant node is more likely to be selected for
reproduction than a resident node. A value of $r<1$ indicates vice versa. The
fixation probability of a single mutant type for an amplifier is smaller than
that for the Moran process when $r<1$ and larger than that for the Moran
process when $r>1$. Conversely, the fixation probability for a suppressor is
larger than that for the Moran process when $r<1$ and smaller than that for
the Moran process when $r>1$. The Moran process, amplifier, and suppressor
have the same fixation probability at $r=1$, which is equal to $1/N$. In the
figure, the fixation probabilities for the Moran process are given by Eq. (24)
with $N=5$, and those for the amplifier and suppressor are hypothetical ones
for expository purposes.

Furthermore, various families of amplifiers of selection have been found [30,
31, 32, 33, 34], whereas suppressors of selection still remain elusive [29,
35]. On these grounds, we ask the following two main questions in the present
study. First, as in the case of static networks, are many temporal networks
comprised of sequences of unweighted networks amplifiers of selection under
the same condition (i.e., birth-death updating rule and uniform
initialization)? Second, if we combine amplifying static networks, $G_{1}$ and
$G_{2}$, into a temporal network, can the obtained temporal network be a
suppressor of selection or a less amplifying temporal network than both
$G_{1}$ and $G_{2}$?

##  2 Model

Let $G$ be a static weighted network with $N$ nodes. We assume undirected
networks for simplicity although extending the following evolutionary dynamics
to the case of directed networks is straightforward. We assume that each node
takes either the resident or mutant type at any discrete time. The resident
and mutant have fitness $1$ and $r$, respectively. The fitness represents the
propensity with which each type is selected for reproduction in each time
step. The mutant type initially occupies just one node, which is selected
uniformly at random among the $N$ nodes. The other $N-1$ nodes are occupied by
the resident type. We then run the birth-death process, which is a
generalization of the Moran process to networks [1, 36, 37, 3, 4, 5].
Specifically, in every discrete time step, we select a node $v$ to reproduce
with the probability proportional to its fitness value. Next, we select a
neighbor of $v$, denoted by $v^{\prime}$, with the probability proportional to
the weight of the undirected edge ($v$, $v^{\prime}$). Then, the type at $v$
(i.e., either resident or mutant) replaces that at $v^{\prime}$. We repeat
this process until the entire population is of a single type, either resident
or mutant, which we call the fixation.

In this study, we extend this birth-death process to temporal networks in
which two static networks $G_{1}$ and $G_{2}$, both having $N$ nodes,
alternate with constant intervals $\tau$. We call this temporal network model
the switching network and denote it by $(G_{1},G_{2},\tau)$. Switching
networks have been used for studying various dynamics on temporal networks
including synchronization [38, 39, 37, 40, 41, 42, 43], random walk [44, 45,
46], epidemic processing [48, 49, 47, 50], network control [51], and reaction-
diffusion systems [52]. Specifically, we first run the birth-death process on
$G_{1}$ for $\tau$ time steps. Then, we switch to $G_{2}$ and run the same
birth-death process on $G_{2}$ for $\tau$ time steps. Then, we switch back to
$G_{1}.$ We keep flipping between $G_{1}$ and $G_{2}$ every $\tau$ time steps
until the fixation of either type occurs.

##  3 Computation and theoretical properties of the fixation probability in
switching networks

In this section, we describe the methods for calculating the fixation
probability of a single mutant, i.e., the probability that the mutant type of
fitness $r$ fixates when there is initially just one node of the mutant type
that is selected uniformly at random. We extend the methods for static
networks [53] to our model. We also state some mathematical properties of the
fixation probability in switching networks.

###  3.1 Fixation probability in static networks

We first explain the known procedure for calculating the fixation probability
of the mutant type, which we simply refer to as the fixation probability in
the following text, in any static weighted network using Markov chains [1,
53]. We describe the state of the evolutionary dynamics by an $N$-dimensional
binary vector $\bm{s}=(s_{1},\ldots,s_{N})$, where $s_{i}\in\\{0,1\\},\forall
i\in\\{1,\ldots,N\\}$. For each $i$, let $s_{i}=0$ or $s_{i}=1$ indicate that
node $i$ is occupied by a resident or a mutant, respectively. Let $S$ be the
set of all states. Note that $S$ has cardinality $2^{N}$, that is, there are
$2^{N}$ states and that there are $\binom{N}{m}$ states with $m$ mutants. We
label the states by a bijective map, denoted by $f$, from $S$ to
$\\{1,\dots,2^{N}\\}$. The transition probability matrix of the Markov chain,
denoted by $T=(T_{ij})$, is a $2^{N}\times 2^{N}$ matrix. Its entry
$T_{f(\bm{s}),f(\bm{s^{\prime}})}$ represents the probability that the state
changes from $\bm{s}$ to $\bm{s^{\prime}}$ in one time step. It should be
noted that $T_{f(\bm{s}),f(\bm{s^{\prime}})}$ can be non-zero if and only if
vectors $\bm{s}$ and $\bm{s^{\prime}}$ differ in at most one entry. Therefore,
each row of $T$ has at most $N+1$ non-zero entries.

Let $\bm{s}$ be a state with $m$ mutants, $s_{i}=1$ for
$i\in\\{g(1),\ldots,g(m)\\}$, and $s_{i}=0$ for
$i\in\\{g(m+1),\ldots,g(N)\\}$, where $g$ is a permutation on
$\\{1,\ldots,N\\}$. Let $\bm{s^{\prime}}$ be the state with $m+1$ mutants in
which $s^{\prime}_{i}=1$ for $i\in\\{g(1),\ldots,g(m),g(m+1)\\}$ and
$s^{\prime}_{i}=0$ for $i\in\\{g(m+2),\ldots,g(N)\\}$. Note that $\bm{s}$ and
$\bm{s^{\prime}}$ differ only at the $g(m+1)$th node, where $\bm{s}$ has a
resident and $\bm{s^{\prime}}$ has a mutant. We obtain

$T_{f(\bm{s}),f(\bm{s^{\prime}})}=\frac{r}{rm+N-m}\sum_{m^{\prime}=1}^{m}\frac{A_{g(m^{\prime}),g(m+1)}}{w(g(m^{\prime}))},$
(1)

where $A$ denotes the weighted adjacency matrix of the network, i.e., $A_{ij}$
is the weight of edge $(i,j)$, and $w(i)\equiv\sum_{j=1}^{N}A_{ij}$ represents
the weighted degree of the $i$th node, also called the strength of the node.
Next, consider a state $\bm{s^{\prime\prime}}$ with $m-1$ mutants such that
$s_{i}^{\prime\prime}=1$ for
$i\in\\{g(1),\ldots,g(\tilde{m}-1),g(\tilde{m}+1),\ldots,g(m)\\}$ and
$s^{\prime\prime}_{i}=0$ for
$i\in\\{g(\tilde{m}),g(m+1),g(m+2),\ldots,g(N)\\}$,  where
$\tilde{m}\in\\{1,\ldots,m\\}$. We obtain

$T_{f(\bm{s}),f(\bm{s^{\prime\prime}})}=\frac{1}{rm+N-m}\sum_{m^{\prime}=m+1}^{N}\frac{A_{g(m^{\prime}),g(\tilde{m})}}{w(g(m^{\prime}))}.$
(2)

The probability that the state does not change after one time step is given by

$T_{f(\bm{s}),f(\bm{s})}=1-\frac{r}{rm+N-m}\sum_{\ell=m+1}^{N}\sum_{m^{\prime}=1}^{m}\frac{A_{g(m^{\prime}),g({\ell)}}}{w(g(m^{\prime}))}-\frac{1}{rm+N-m}\sum_{\tilde{m}=1}^{m}\sum_{m^{\prime}={m+1}}^{N}\frac{A_{g(m^{\prime}),g(\tilde{m})}}{w(g(m^{\prime}))}.$
(3)

Let $x_{f(\bm{s})}$ denote the probability that the mutant fixates when the
evolutionary dynamics start from state $\bm{s}$. Because

$x_{f(\bm{s})}=\sum_{\bm{s^{\prime}}\in
S}T_{f(\bm{s}),f(\bm{s^{\prime}})}x_{f(\bm{s^{\prime}})},$ (4)

we obtain $T\bm{x}=\bm{x}$, where
$\bm{x}=\left(x_{1},\dots,x_{2^{N}}\right)^{\top}$, and ⊤ represents the
transposition. Because $x_{f(\left(0,\ldots,0\right))}=0$ and
$x_{f(\left(1,\ldots,1\right))}=1$, we need to solve the set of $2^{N}-2$
linear equations to obtain the fixation probabilities starting from an
arbitrary initial state.

###  3.2 Fixation probability in switching networks

We now consider the same birth-death process on switching network
$(G_{1},G_{2},\tau)$. To calculate the fixation probability in
$(G_{1},G_{2},\tau)$, we denote by $T^{(1)}$ and $T^{(2)}$ the transition
probability matrices for the birth-death process on static network $G_{1}$ and
$G_{2}$, respectively. Let $x_{i}(t)$ be the fixation probability when the
evolutionary dynamics start from the $i$th state (with
$i\in\\{1,\ldots,2^{N}\\}$) at time $t$. We obtain

$\displaystyle\bm{x}(t)=\begin{cases}T^{(1)}\bm{x}(t+1)&\text{ if }2n\tau\leq
t<\left(2n+1\right)\tau,\\\ T^{(2)}\bm{x}(t+1)&\text{ if }(2n+1)\tau\leq
t<\left(2n+2\right)\tau,\end{cases}$ (5)

where $\bm{x}(t)=\left(x_{1}(t),\ldots,x_{2^{N}}(t)\right)^{\top}$ and
$n\in\\{0,1,\ldots\\}$. We recursively use Eq. (5) to obtain

$\displaystyle\bm{x}\left(0\right)=$ $\displaystyle
T^{(1)}\bm{x}\left(1\right)=\cdots=\left(T^{(1)}\right)^{\tau}\bm{x}\left(\tau\right)=\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)\bm{x}\left(\tau+1\right)=\cdots$
$\displaystyle=$
$\displaystyle\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)^{\tau}\bm{x}\left(2\tau\right).$
(6)

Because of the periodicity of the switching network, we obtain
$\bm{x}\left(0\right)=\bm{x}\left(2\tau\right)$. Therefore, the fixation
probability is given as the solution of

$\bm{x^{*}}=\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)^{\tau}\bm{x^{*}}.$
(7)

Let $\tilde{S}^{(1)}$ be the set of the $N$ states with just one mutant. Then,
the fixation probability when there is initially a single mutant located on a
node that is selected uniformly at random is given by

$\rho\equiv\frac{1}{N}\sum_{\bm{s}\in\tilde{S}^{(1)}}x^{*}_{f(\bm{s})}.$ (8)

Note that $\rho$ is a function of $r$ and depends on the network structure.
Because $\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)^{\tau}$ is a
stochastic matrix with two absorbing states, it has a unique solution [54,
55].

The birth-death process on switching networks has the following property.

######  Theorem 1.

(Neutral drift) If $r=1$, then $\rho=\frac{1}{N}$ for arbitrary $G_{1}$,
$G_{2}$, and $\tau\in\mathbb{N}$.

###### Proof.

We imitate the proof given in [56]. Assume a switching network
$(G_{1},G_{2},\tau)$ on $N$ nodes and that each node is initially occupied by
a mutant of distinct type, i.e., node $i$ is occupied by a mutant of type
$A_{i}$. We also assume that each mutant has fitness $1$. We denote the
probability that mutant $A_{i}$ fixates by $q_{i}$. Note that
$\sum_{i=1}^{N}q_{i}=1$. Now we reconsider our original evolutionary dynamics
with $r=1$, in which there are only equally strong two types, i.e., resident
type and mutant type, with the initial condition in which the mutant type
occupies the $i$th node and the resident type occupies all the other $N-1$
nodes. Then, the fixation probability of the mutant is equal to $q_{i}$
because this model is equivalent to the previous model if we identify $A_{i}$
with the mutant type and the other $N-1$ types with the resident type.
Therefore, the fixation probability for the original model with $r=1$ and the
uniform initialization is given by $\sum_{i=1}^{N}q_{i}/N=1/N$. ∎

######  Remark 1.

We acknowledge that a recent study proved a more general version of this
theorem and provided extensive discussion on neutral drift [21].

######  Remark 2.

The theorem holds true even if we switch among more than two static networks
or if the switching intervals, $\tau$, deterministically change from one
switching interval to another. The proof remains unchanged.

###  3.3 Initialization at random time

In this section, we discuss the case in which the initial mutant arises in the
switching network $\left(G_{1},G_{2},\tau\right)$ at a time selected uniformly
at random. Without loss of generality, we assume that the initial mutant
arises at time $t_{0}$, where $t_{0}\in\\{0,1,\dots,2\tau-1\\}$. If the
initial mutant appears at time $t\geq 2N$, then we can set $t_{0}=t\mod
2\tau$. Similar to Eq. (5), we obtain

$\displaystyle{\color[rgb]{0,0,0}\bm{x}(t_{0})=\begin{cases}T^{(1)}\bm{x}(t_{0}+1)&\text{
if }0\leq t_{0}<\tau,\\\ T^{(2)}\bm{x}(t_{0}+1)&\text{ if }\tau\leq
t_{0}<2\tau.\end{cases}}$ (9)

We use Eq. (9) to obtain

$\displaystyle{\color[rgb]{0,0,0}\bm{x}\left(t_{0}\right)=}$
$\displaystyle{\color[rgb]{0,0,0}T^{(1)}\bm{x}\left(t_{0}+1\right)=\cdots}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-
t_{0}}\bm{x}\left(\tau\right)}$ $\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-
t_{0}}\left(T^{(2)}\right)\bm{x}\left(\tau+1\right)=\cdots}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-
t_{0}}\left(T^{(2)}\right)^{\tau}\bm{x}\left(2\tau\right)}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-
t_{0}}\left(T^{(2)}\right)^{\tau}\left(T^{(1)}\right)^{t_{0}}\bm{x}\left(2\tau+t_{0}\right)}$
(10)

when $0\leq t_{0}<\tau$ and

$\displaystyle{\color[rgb]{0,0,0}\bm{x}\left(t_{0}\right)=}$
$\displaystyle{\color[rgb]{0,0,0}T^{(2)}\bm{x}\left(t_{0}+1\right)=\cdots}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(2)}\right)^{2\tau-
t_{0}}\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)^{t_{0}-\tau}\bm{x}\left(2\tau+t_{0}\right)}$
(11)

when $\tau\leq t_{0}<2\tau$. Because of the periodicity of the switching
network, we obtain $\bm{x}\left(t_{0}\right)=\bm{x}\left(2\tau+t_{0}\right)$.
Therefore, the fixation probability, which depends on $t_{0}$ in the present
case, is given as the solution of

$\displaystyle\bm{x^{*}}(t_{0})=\begin{cases}\left(T^{(1)}\right)^{\tau-
t_{0}}\left(T^{(2)}\right)^{\tau}\left(T^{(1)}\right)^{t_{0}}\bm{x^{*}}(t_{0})&\text{
if }0\leq t_{0}<\tau,\\\ \left(T^{(2)}\right)^{2\tau-
t_{0}}\left(T^{(1)}\right)^{\tau}\left(T^{(2)}\right)^{t_{0}-\tau}\bm{x^{*}}(t_{0})&\text{
if }\tau\leq t_{0}<2\tau.\end{cases}$ (12)

Equation (12) yields

$\displaystyle{\color[rgb]{0,0,0}\bm{y^{*}}}\equiv$
$\displaystyle{\color[rgb]{0,0,0}T^{\left(1\right)}\bm{x^{*}}\left(t_{0}\right)}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-
t_{0}+1}\left(T^{(2)}\right)^{\tau}\left(T^{(1)}\right)^{t_{0}-1}T^{(1)}\bm{x^{*}}\left(t_{0}\right)}$
$\displaystyle{\color[rgb]{0,0,0}=}$
$\displaystyle{\color[rgb]{0,0,0}\left(T^{(1)}\right)^{\tau-(t_{0}-1)}\left(T^{(2)}\right)^{\tau}\left(T^{(1)}\right)^{t_{0}-1}\bm{y^{*}}.}$

Therefore, we obtain $\bm{y^{*}}=\bm{x^{*}}\left(t_{0}-1\right)$ when $1\leq
t_{0}<\tau$. Using this relationship recursively, we obtain

$\displaystyle{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\bm{x^{*}}\left(\tau-1-k\right)=}\left(T^{\left(1\right)}\right)^{k}\bm{x^{*}}\left(\tau-1\right)}$

for $0\leq k<\tau-1$. Similarly, we obtain

$\displaystyle{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}\bm{x^{*}}\left(2\tau-1-k\right)=}\left(T^{\left(2\right)}\right)^{k}\bm{x^{*}}\left(2\tau-1\right)}$

for $0\leq k<\tau-1$. By combining Eqs. (3.3) and (3.3), we obtain

$\bm{x^{*}}=\frac{1}{2\tau}\left\\{\left[\sum_{k=0}^{\tau-1}\left(T^{\left(1\right)}\right)^{k}\right]\bm{x^{*}}\left(\tau-1\right)+\left[\sum_{k=0}^{\tau-1}\left(T^{\left(2\right)}\right)^{k}\right]\bm{x^{*}}\left(2\tau-1\right)\right\\},$
(16)

where $\bm{x^{*}}$ is the fixation probability vector when the initial mutant
appears at a uniformly randomly drawn time.

As in Eq. (8), let $\tilde{S}^{(1)}$ be the set of the $N$ states with just
one mutant. Then, the fixation probability for a single mutant when the
initial mutant appears at a uniformly randomly drawn time is given by

$\rho=\frac{1}{N}\sum_{\bm{s}\in\tilde{S}^{(1)}}x^{*}_{f(\bm{s})}.$ (17)

###  3.4 Stochastic switching

In this section, we formulate the fixation probability for stochastic
switching networks. We adapt the methods proposed for epidemic spreading [57,
58] and evolutionary games [21] to the case of constant-selection dynamics. We
assume that the network switches with probability $p$ at every time step. In
other words, if the network is $G_{1}$ at time $t$, then it switches to
$G_{2}$ at time $t+1$ with probability $p$ and remains $G_{1}$ with
probability $1-p$. Likewise, if the network is $G_{2}$ at time $t$, then it
switches to $G_{1}$ at time $t+1$ with probability $p$ and remains $G_{2}$
with probability $1-p$. The duration of $G_{1}$ and that of $G_{2}$ before
switching to the other network, $\tau$, obeys the geometric distribution with
${\displaystyle\Pr(\tau)=(1-p)^{\tau-1}p}$, where $\Pr$ denotes the
probability.

We can write the state of the dynamics at any time $t$ as
$\left(\bm{s},G_{i}\right)$, where $\bm{s}$ is one of the $2^{N}$ states
(i.e., $\bm{s}\in S$) as in the deterministic switching case, and
$i\in\\{1,2\\}$. Let $x_{\left(\bm{s},G_{i}\right)}\left(t\right)$ denote the
probability that the dynamics attains fixation when starting in state $\bm{s}$
at time $t$. We obtain

${\color[rgb]{0,0,0}x_{\left(\bm{s},G_{i}\right)}\left(t\right)=\sum_{\bm{s^{\prime}}\in
S}T_{\bm{s}\rightarrow\bm{s^{\prime}}}^{(i)}\left[p\cdot
x_{\left(\bm{s^{\prime}},G_{i^{\prime}}\right)}\left(t+1\right)+\left(1-p\right)\cdot
x_{\left(\bm{s^{\prime}},G_{i}\right)}\left(t+1\right)\right],}$ (18)

where $i^{\prime}=2$ if $i=1$ and $i^{\prime}=1$ if $i=2$. Let $\tilde{T}$ be
the $\left(2^{N}\times 2\right)\times\left(2^{N}\times 2\right)$ transition
probability matrix defined by
$\tilde{T}_{\left(\left(\bm{s},G_{i}\right),\left(\bm{s^{\prime}},G_{i^{\prime}}\right)\right)}=pT_{\bm{s}\rightarrow\bm{s^{\prime}}}^{(i)}$
and
$\tilde{T}_{\left(\left(\bm{s},G_{i}\right),\left(\bm{s^{\prime}},G_{i}\right)\right)}=\left(1-p\right)T_{\bm{s}\rightarrow\bm{s^{\prime}}}^{(i)}$
for $i\in\\{1,2\\}$. Matrix $\tilde{T}$ is the following block matrix:

$\tilde{T}=\begin{bmatrix}pT^{(1)}&\left(1-p\right)T^{(1)}\\\
pT^{(2)}&\left(1-p\right)T^{(2)}\end{bmatrix}.$ (19)

Using $\tilde{T}$, we rewrite Eq. (18) as

${\color[rgb]{0,0,0}\bm{x}\left(t\right)=\tilde{T}\bm{x}\left(t+1\right),}$
(20)

where

$\bm{x}\left(t\right)\equiv\begin{bmatrix}x_{\left(\left(0,\ldots,0\right),G_{1}\right)}\left(t\right)\\\
\vdots\\\ x_{\left(\left(1,\ldots,1\right),G_{1}\right)}\left(t\right)\\\
x_{\left(\left(0,\ldots,0\right),G_{2}\right)}\left(t\right)\\\ \vdots\\\
x_{\left(\left(1,\ldots,1\right),G_{2}\right)}\left(t\right)\end{bmatrix}.$
(21)

In fact, $\bm{x}\left(t\right)$ does not depend on $t$. Therefore, to find the
fixation probability, we need to solve

${\color[rgb]{0,0,0}\bm{x^{*}}=\tilde{T}\bm{x^{*}}.}$ (22)

Similar to the derivation of Eq. (8), we find that the fixation probability
when there is initially just one mutant on a node selected uniformly at random
and the initial network is selected uniformly at random is given by

${\color[rgb]{0,0,0}\rho=\frac{1}{2N}\sum_{i=1}^{2}\sum_{\bm{s}\in\tilde{S}^{(1)}}x_{\left(\bm{s},G_{i}\right)}.}$
(23)

###  3.5 Identifying amplifiers and suppressors of selection

We operationally define amplifiers and suppressors of selection as follows;
similar definitions were used in the literature [1, 59]. For a given switching
or static network, we computed the fixation probability for several values of
$r$. We say that the network is amplifier of selection if the fixation
probability is larger than for that for the complete graph with the same
number of nodes, or equivalently, the Moran process, at six values of $r>1$,
i.e., $r\in\\{1.1,1.2,1.3,1.4,1.6,1.8\\}$ and a smaller than that for the
Moran process at three values of $r<1$, i.e., $r\in\\{0.7,0.8,0.9\\}$. Note
that the fixation probability for the Moran process with $N$ individuals is
given by (see e.g. [2])

$\rho=\frac{1-\frac{1}{r}}{1-\frac{1}{r^{N}}}.$ (24)

Similarly, we say that a network is suppressor of selection if the fixation
probability is smaller than for the Moran process at the same six values of
$r$ larger than $1$ and larger than for the Moran process at the three values
of $r$ smaller than $1$. It is known that some static networks are neither
amplifier nor suppressor of selection [35].

We note that the Moran process is equivalent to the switching network in which
both $G_{1}$ and $G_{2}$ are the complete graph. In this manner, one can
regard that the comparison between a general switching network and the Moran
process is that between two temporal networks instead of that between a
temporal network and a static network.

###  3.6 Isothermal theorem

A network is called isothermal if its fixation probability is the same as that
for the Moran process, i.e., if Eq. (24) holds true [1]. A static undirected
network, which may be weighted, is isothermal if and only if all the nodes
have the same (weighted) degree [1, 60, 61]. One can easily construct
isothermal switching networks as follows.

######  Theorem 2.

If $G_{1}$ and $G_{2}$ are isothermal networks, then the switching network
$\left(G_{1},G_{2},\tau\right)$ is an isothermal network.

###### Proof.

The proof is exactly the same as in the static network case as shown in [1,
2]. We denote by $p_{m,m-1}$ the probability that the state of the network
moves from a state with $m$ mutants to a state with $m-1$ mutants in one time
step. Similarly, we denote by $p_{m,m+1}$ the probability that the state moves
from one with $m$ mutants to one with $m+1$ mutants in one time step. We
observe that $p_{m,m-1}/p_{m,m+1}=1/r$ at every time step $t$ because the
static network at any $t$, which is either $G_{1}$ or $G_{2}$, is isothermal.
Therefore, the fixation probability for $\left(G_{1},G_{2},\tau\right)$ is
given by Eq. (24). ∎

######  Remark 3.

The theorem including the present proof holds true both when we initially use
$G_{1}$ for time $\tau$ and when the mutant arises at a time selected
uniformly at random.

##  4 Fixation probability in various switching networks

In this section, we analyze the fixation probability in three types of
switching networks, i.e., networks with six nodes, larger switching networks
in which $G_{1}$ and $G_{2}$ have symmetry (i.e., complete graph, star graph,
and bipartite networks), and empirical networks.

###  4.1 Six-node networks

We first analyzed the fixation probability in switching networks that are
composed of two undirected and unweighted connected networks with $6$ nodes.
There are 112 non-isomorphic undirected connected networks on 6 nodes. We
switched between any ordered pair of different networks, giving us a total of
$112\times 111=12432$ switching networks. It should be noted that swapping the
order of $G_{1}$ and $G_{2}$ generally yields different fixation
probabilities. We randomly permuted the node labels in $G_{2}$. We did not
consider all possible labeling of nodes because there would be at most
$112\cdot 111\cdot 6!=8951040$ switching networks on 6 nodes if we allow
shuffling of node labeling, although the symmetry reduces this number.

In Fig. 2(a), we show two arbitrarily chosen static networks on six nodes,
$G_{1}$ and $G_{2}$, which are amplifiers of selection as static networks. In
Fig. 2(b), we plot the fixation probability as a function of the fitness of
the mutant, $r$, for the switching network $(G_{1},G_{2},\tau=1)$, the static
networks $G_{1}$ and $G_{2}$, the aggregate weighted static network generated
from $G_{1}$ and $G_{2}$, and the Moran process (i.e., complete graph on six
nodes). The aggregated weighted static network is the superposition of $G_{1}$
and $G_{2}$ such that the weight of the edge is either 1 or 2. It is
equivalent to the average of $G_{1}$ and $G_{2}$ over time. All these static
and switching networks yield $\rho=1/N=1/6$ at $r=1$, as expected (see Theorem
1). In addition, there exist differences in $\rho$ between the different
networks and the Moran process although the difference is small. In fact,
$G_{1}$ and $G_{2}$ are amplifiers of selection, with their fixation
probability being larger than that for the Moran process when $r>1$ and vice
versa when $r<1$, confirming the known result [27, 29]. Figure 2(b) also
indicates that the aggregate network is an amplifier of selection. However,
the switching network is suppressor of selection.

We reconfirm these results in Fig. 2(c), in which we show the difference in
the fixation probability between a given static or switching network and the
Moran process. If the difference is negative for $r<1$ and positive for $r>1$,
then the network is an amplifier of selection. If the difference is positive
for $r<1$ and negative for $r>1$, then the network is a suppressor of
selection. Figure 2(c) shows that $G_{1}$ is a stronger amplifier than $G_{2}$
and that $G_{2}$ is a stronger amplifier than the aggregate network. In
contrast, the switching network $(G_{1},G_{2},1)$ is a suppressor of
selection, while $(G_{1},G_{2},10)$ and $(G_{1},G_{2},50)$ are amplifiers of
selection. The result for $(G_{1},G_{2},50)$ is close to that for static
network $G_{1}$, which is because the evolutionary dynamics on
$(G_{1},G_{2},\tau)$ is equivalent to that on $G_{1}$ in the limit
$\tau\to\infty$. In practice, fixation for networks on six nodes occurs within
$50$ time steps in many cases, which renders $(G_{1},G_{2},50)$ close to
$G_{1}$. However, we have included the results for $\tau=50$ because fixation
does not occur within 50 time steps in many other cases. When the number of
nodes, $N$, is large, $(G_{1},G_{2},\tau)$ is a genuine switching network
because the fixation times are typically much longer than $N$ [53]. We
conclude that switching networks composed of two amplifiers can be a
suppressor, in particular when $\tau$ is small. We emphasize that this
counterintuitive result is not due to the property of the aggregate network
because the aggregate network, which is the time average of $G_{1}$ and
$G_{2}$, is also an amplifier.

We show the results for the switching network with the order of $G_{1}$ and
$G_{2}$ reversed and those for random initialization time in Appendix A. We
find that both $(G_{2},G_{1},1)$ and the switching network with $\tau=1$ and
random initialization time are suppressors of selection. The fixation
probability for the switching network with $\tau=1$ and random initialization
time is the average of that for $(G_{1},G_{2},1)$ and $(G_{2},G_{1},1)$.
Therefore, the fixation probability for the former lies between that for
$(G_{1},G_{2},1)$ and $(G_{2},G_{1},1)$ at each value of $r$. Switching
networks $(G_{2},G_{1},\tau)$ and those with random initialization time are
amplifiers of selection when $\tau$ is larger (i.e., $\tau\in\\{10,50\\}$);
this result is qualitatively the same as that for $(G_{1},G_{2},\tau)$.

To investigate the generality of this finding to other six-node networks, we
calculated the fixation probability for the switching networks derived from
all possible pairs of six-node networks. Table 1 shows the number of switching
networks on six nodes that are amplifiers of selection, that of suppressors of
selection, and that of networks that are neither amplifier or suppressor, for
four values of $\tau$. The table indicates that a majority of the six-node
switching networks investigated are suppressors of selection when $\tau=1$ and
$\tau=3$. This result is in stark contrast to the fact that there is only 1
suppressor of selection among 112 six-node static unweighted networks under
the birth-death process [29, 27]. Out of the 111 static networks that are not
suppressor, 100 networks are amplifiers, five are isothermal, and the other
six networks are neither amplifier, suppressor, nor isothermal [35, 62]. Most
switching networks are amplifiers when $\tau=50$, which is presumably because
most static networks are amplifiers and the birth-death process on
$(G_{1},G_{2},\tau)$ converges to that on $G_{1}$ in the limit
$\tau\to\infty$, as we discussed above.

We also examined the fixation probability for six-node stochastic switching
networks introduced in section 3.4. As in the case with deterministic
switching, we considered $112\times 111$ ordered pairs of networks and
permuted the node labels of $G_{2}$ uniformly at random. We show in Table 2
the number of amplifier of selection, suppressor of selection, and neither
type, assuming random initialization time, for $p\in\\{0.3,0.5,0.8\\}$. We
find that a substantial fraction of these stochastic switching networks is
suppressors of selection for each of the three $p$ values (i.e., 36.5% for
$p=0.3$; 47.3% for $p=0.5$; 24.3% for $p=0.8$). These results suggest that the
abundance of suppressing switching networks among six-node switching networks
is not due to the periodic switching nature of our switching network model.

Figure 2: A suppressing switching network composed of two amplifying static
networks on six nodes. (a) A switching network composed of six nodes. Both
$G_{1}$ and $G_{2}$ are amplifiers of selection. (b) Fixation probability in
the static and switching networks as a function of $r$. Moran refers to the
Moran process. Note that $G_{1}$, $G_{2}$, the aggregate network, and the
Moran process represent static networks. (c) Difference between the fixation
probability for the given network and that for the Moran process.

###  4.2 Larger symmetric switching networks

In this section, we assume symmetry in $G_{1}$ and $G_{2}$ to calculate the
fixation probability for larger switching networks. Specifically, we set
$G_{1}$ to be the star graph and $G_{2}$ to be either the complete graph or
complete bipartite graph.

$\tau$ | Amplifier | Suppressor | Neither
---|---|---|---
1 | 3636 | 8177 | 619
3 | 5190 | 6347 | 895
10 | 11102 | 629 | 701
50 | 12038 | 262 | 132
Table 1: Number of amplifiers and suppressors of selection among $112\cdot 111=12432$ periodically switching networks on six nodes. $p$ | Amplifier | Suppressor | Neither
---|---|---|---
0.3 | 7346 | 4536 | 550
0.5 | 5979 | 5880 | 573
0.8 | 8881 | 3023 | 528
Table 2: Number of amplifiers and suppressors of selection among $112\cdot
111=12432$ stochastically switching networks on six nodes.

####  4.2.1 Combination of the star graph and the complete graph

Consider switching networks in which $G_{1}$ is the star graph and $G_{2}$ is
the complete graph. For this switching network, we can reduce the dimension of
the transition probability matrix from $2^{N}\times 2^{N}$ to $2N\times 2N$ by
exploiting the symmetry in $G_{1}$ and $G_{2}$. Therefore, one can reduce the
number of equations from $2^{N}-2$ to $2N-2$. Specifically, one can uniquely
describe the state of the network by $(i,j)$, where $i\in\\{0,1\\}$ and
$j\in\\{0,\ldots,N-1\\}$. We set $i=0$ and $i=1$ when the hub node of $G_{1}$
is occupied by a resident or mutant, respectively. We set
$j\in\\{0,1,\ldots,N-1\\}$ to the number of mutants in the other $N-1$ nodes,
which we refer to as the leaves. Tuple $(i,j)$ is a valid expression of the
state of the network because the $N-1$ leaves are structurally equivalent to
each other in both $G_{1}$ and $G_{2}$. Tuples $(0,0)$ and $(1,N-1)$
correspond to the fixation of the resident and mutant type, respectively.

The transition probability from state $\left(i,j\right)$ to state
$\left(i^{\prime},j^{\prime}\right)$ in a single time step of the birth-death
process is nonzero if and only if $(i^{\prime},j^{\prime})=(i,j+1)$ and $i=1$,
$(i^{\prime},j^{\prime})=(i,j-1)$ and $i=0$,
$(i^{\prime},j^{\prime})=(1-i,j)$, or $(i^{\prime},j^{\prime})=(i,j)$. Let
$T^{(1)}$ denote the transition probability matrix for the star graph. We
obtain

$T^{(1)}_{\left(i,j\right)\rightarrow\left(i^{\prime},j^{\prime}\right)}=\begin{cases}\frac{rj}{C_{1}}&\text{if
}i=0\text{ and }i^{\prime}=1,\\\\[2.84526pt] \frac{N-1-j}{C_{2}}&\text{if
}i=1\text{ and }i^{\prime}=0,\\\\[2.84526pt]
\frac{1}{C_{1}}\cdot\frac{j}{N-1}&\text{if }i^{\prime}=i=0\text{ and
}j^{\prime}=j-1,\\\\[2.84526pt] \frac{r}{C_{2}}\cdot\frac{N-1-j}{N-1}&\text{if
}i^{\prime}=i=1\text{ and }j^{\prime}=j+1,\\\\[2.84526pt]
1-\sum\limits_{\mathclap{\begin{subarray}{c}(i^{\prime\prime},j^{\prime\prime})\neq\\\
(i,j)\end{subarray}}}T^{\left(1\right)}_{(i,j)\rightarrow(i^{\prime\prime},j^{\prime\prime})}&\text{if
}(i^{\prime},j^{\prime})=(i,j),\\\\[2.84526pt] 0&\text{
otherwise,}\end{cases}$ (25)

where $C_{1}=rj+N-j$ and $C_{2}=r(j+1)+N-\left(j+1\right)$ [1]. The first line
of Eq. (25) represents the probability that the type of the hub changes from
the resident to mutant. For this event to occur, one of the $j$ leaf nodes
occupied by the mutant must be chosen as parent, which occurs with probability
$rj/\left(rj+N-j\right)$. Because every leaf node is only adjacent to the hub
node, the hub node is always selected for death if a leaf node is selected as
parent. Therefore, the probability of $i$ changing from $0$ to $1$ is equal to
$rj/\left(rj+N-j\right)$, which is shown in the first line of Eq. (25). As
another example, consider state $\left(1,j\right)$, in which the hub has a
mutant, $j$ leaf nodes have mutants, and the other $N-1-j$ leaf nodes have
residents. For the state to change from $\left(1,j\right)$ to
$\left(1,j+1\right)$, the hub node must be selected as parent with probability
$r/\left[r\left(j+1\right)+N-\left(j+1\right)\right]$, and a leaf node of the
resident type must be selected for death, which occurs with probability
$(N-1-j)/(N-1)$. The fourth line of Eq. (25) is equal to the product of these
two probabilities. One can similarly derive the other lines of Eq. (25).

The transition probability matrix for $G_{2}$, which is the complete graph, is
given by

$T^{(2)}_{\left(i,j\right)\rightarrow\left(i^{\prime},j^{\prime}\right)}=\begin{cases}\frac{rj}{C_{1}}\cdot\frac{1}{N-1}&\text{if
}i=0\text{ and }i^{\prime}=1,\\\\[2.84526pt]
\frac{N-1-j}{C_{2}}\cdot\frac{1}{N-1}&\text{if }i=1\text{ and
}i^{\prime}=0,\\\\[2.84526pt] \frac{N-j}{C_{1}}\cdot\frac{j}{N-1}&\text{if
}i^{\prime}=i=0\text{ and }j^{\prime}=j-1,\\\\[2.84526pt]
\frac{rj}{C_{1}}\cdot\frac{N-1-j}{N-1}&\text{if }i^{\prime}=i=0\text{ and
}j^{\prime}=j+1,\\\\[2.84526pt] \frac{N-1-j}{C_{2}}\cdot\frac{j}{N-1}&\text{if
}i^{\prime}=i=1\text{ and }j^{\prime}=j-1,\\\\[2.84526pt]
\frac{r(j+1)}{C_{2}}\cdot\frac{N-1-j}{N-1}&\text{if }i^{\prime}=i=1\text{ and
}j^{\prime}=j+1,\\\\[2.84526pt]
1-\sum\limits_{\mathclap{\begin{subarray}{c}(i^{\prime\prime},j^{\prime\prime})\neq\\\
(i,j)\end{subarray}}}T^{\left(1\right)}_{(i,j)\rightarrow(i^{\prime\prime},j^{\prime\prime})}&\text{if
}(i^{\prime},j^{\prime})=(i,j),\\\\[2.84526pt] 0&\text{
otherwise.}\end{cases}$ (26)

For example, for the transition from state $\left(0,j\right)$ to
$\left(1,j\right)$ to occur, one of the $j$ mutant leaf nodes must be first
selected as parent, which occurs with probability $rj/\left(rj+N-j\right)$.
Then, the hub node must be selected for death, which occurs with probability
$1/\left(N-1\right)$. The first line of Eq. (26) is equal to the product of
these two probabilities. As another example, for the state to change from
$\left(1,j\right)$ to $\left(1,j+1\right)$, one of the mutant nodes, which may
be the hub or a leaf, must be first selected as parent, which occurs with
probability
$r\left(j+1\right)/\left[r\left(j+1\right)+N-\left(j+1\right)\right]$. Then, a
leaf node of the resident type must be selected for death, which occurs with
probability $\left(N-1-j\right)/\left(N-1\right)$. The right-hand side on the
sixth line of Eq. (26) is equal to the product of these two probabilities. One
can similarly derive the other lines of Eq. (26). It should be noted that
single-step moves from $\left(1,j\right)$ to $\left(1,j-1\right)$ and those
from $\left(0,j\right)$ to $\left(0,j+1\right)$ are possible in $G_{2}$,
whereas they do not occur in $G_{1}$.

In Fig. 3(a), we plot the fixation probability as a function of $r$ for
switching network $(G_{1},G_{2},\tau)$ in which $G_{1}$ is the star graph and
$G_{2}$ is the complete graph on four nodes. In this figure, we compare
$(G_{1},G_{2},\tau)$ with $\tau=1$, $10$, and $50$, the static star graph, the
aggregate network, and the Moran process. Figure 3(a) indicates that
$(G_{1},G_{2},10)$ and $(G_{1},G_{2},50)$ are amplifiers of selection and that
$(G_{1},G_{2},1)$ is a suppressor. We plot the difference in the fixation
probability between the switching networks and the Moran process in Fig. 3(b).
When $\tau=1$, the difference is positive for $r<1$ and negative for $r>1$,
which verifies that $(G_{1},G_{2},1)$ is a suppressor. This result is
surprising because $G_{1}$ is an amplifier of selection and $G_{2}$ is
equivalent to the Moran process and therefore not a suppressor of selection.
In contrast, when $\tau=10$ and $\tau=50$, the difference from the Moran
process is negative for $r<1$ and positive for $r>1$, which verifies that
$(G_{1},G_{2},10)$ and $(G_{1},G_{2},50)$ are amplifiers of selection. The
result for $\tau=50$ is close to that for the star graph. This is presumably
because the first $\tau=50$ steps with $G_{1}$ are sufficient to induce
fixation with a high probability given the small network size (i.e., $N=4$).

Figures 3(a) and 3(b) also indicate that the aggregate network is a weak
suppressor of selection. However, the aggregate network is a considerably
weaker suppressor of selection than $(G_{1},G_{2},1)$. Therefore, we conclude
that the suppressing effect of the switching network mainly originates from
the time-varying nature of the network rather than the structure of the
weighted aggregate network.

We show in Figs. 3(c) and 3(d) the fixation probability and its difference
from the case of the Moran process, respectively, as a function of $r$ for
$N=50$. We observe that the switching network is an amplifier of selection for
all the values of $\tau$ that we considered, i.e., $\tau=1$, $10$, and $50$.
In contrast, the aggregate network is a suppressor of selection albeit an
extremely weak one. The amplifying effect of the switching network is stronger
for a larger value of $\tau$. Unlike in the case of four nodes (see Figs. 3(a)
and 3(b)), the switching networks with 50 nodes are far less amplifying than
the star graph even with $\tau=50$. This phenomenon is expected because
fixation in a static network with 50 nodes usually needs much more than 50
steps.

These results for the switching networks with $N=4$ and $N=50$ nodes remain
similar for $(G_{2},G_{1},\tau)$, i.e., when we swap the order of $G_{1}$ and
$G_{2}$ (see Figs. 6(a) and 6(b)).

The present switching network is a suppressor of selection when $N=4$ and
$\tau=1$ and an amplifier of selection when $N=50$ or $\tau\in\\{10,50\\}$. To
examine the generality of these results with respect to the number of nodes,
$N$, we show in Figs. 3(e) and 3(f) the fixation probability relative to that
for the Moran process at $\tau=1$ and $\tau=50$, respectively, as a function
of $N$. In both figures, we show the fixation probabilities at $r=0.9$ and
$r=1.1$. Figure 3(e) indicates that the switching network is a suppressor of
selection for $N\leq 4$ and an amplifier of selection for $N\geq 5$ when
$\tau=1$. We have confirmed that this switching network with $N=3$ nodes is a
suppressor of selection by calculating the fixation probability across a range
of $r$ values in (see Fig. 7(a)  in Appendix B). Figure 3(f) indicates that
$(G_{1},G_{2},50)$ is an amplifier of selection for any $N$.

Figure 3: Fixation probability for switching networks in which $G_{1}$ is the
star graph and $G_{2}$ is the complete graph. (a) Fixation probability for
$N=4$. (b) Difference in the fixation probability from the case of the Moran
process for $N=4$. (c) Fixation probability for $N=50$. (d) Difference in
fixation probability from the case of the Moran process for $N=50$. In
(a)–(d), we also show the results for $G_{1}$ (i.e., star graph) and the
aggregate network, and the vertical lines at $r=1$ are a guide to the eyes.
The insets magnify selected ranges of $r<1$. (e) and (f): Difference in the
fixation probability for the switching network relative to the Moran process
as a function of $N$ at $r=0.9$ and $1.1$. We set $\tau=1$ in (e) and
$\tau=50$ in (f). In (e) and (f), the smallest value of $N$ is three.

####  4.2.2 Combination of the star graph and the complete bipartite graph

In this section, we analyze the switching network in which $G_{1}$ is the star
graph and $G_{2}$ is the complete bipartite graph $K_{N_{1},N_{2}}$. By
definition, $K_{N_{1},N_{2}}$ has two disjoint subsets of nodes $V_{1}$ and
$V_{2}$, and $V_{1}$ and $V_{2}$ contain $N_{1}$ and $N_{2}$ nodes,
respectively. Every node in $V_{1}$ is adjacent to every node in $V_{2}$ by an
edge. Therefore, every node in $V_{2}$ is adjacent to every node in $V_{1}$.
Without loss of generality, we assume that the hub node in $G_{1}$ is one of
the $N_{1}$ nodes in $V_{1}$.

Because of the symmetry, we do not need to distinguish among the $N_{1}-1$
nodes that are leaf nodes in $G_{1}$ and belong to $V_{1}$ in $G_{2}$, or
among the $N_{2}$ nodes that belong to $V_{2}$ in $G_{2}$. Therefore, one can
specify the state of this switching network by a tuple $(i,j,k)$, where
$i\in\\{0,1\\}$ represents whether the hub is occupied by a resident,
corresponding to $i=0$, or mutant, corresponding to $i=1$; variable
$j\in\\{0,\ldots,N_{1}-1\\}$ represents the number of mutants among the
$N_{1}-1$ nodes that are leaves in $G_{1}$ and belong to $V_{1}$ in $G_{2}$;
variable $k\in\\{0,\ldots,N_{2}\\}$ represents the number of mutants among the
$N_{2}$ nodes in $V_{2}$. Tuples $(0,0,0)$ and $(1,N_{1}-1,N_{2})$ correspond
to the fixation of the resident and mutant type, respectively. Using this
representation of the states, we reduce the $2^{N}\times 2^{N}$ transition
probability matrix to a $2N_{1}\left(N_{2}+1\right)\times
2N_{1}\left(N_{2}+1\right)$ transition probability matrix. We show the
transition probabalities $T^{(1)}$ and $T^{(2)}$ in Appendix C.

(27)

(28)

In Figs. 4(a) and 4(b), we show the fixation probability and its difference
from the case of the Moran process, respectively, for the switching network in
which $G_{1}$ is the star on $N=4$ nodes and $G_{2}$ is the complete bipartite
graph $K_{N_{1},N_{2}}$ with $N_{1}=N_{2}=2$. We set $\tau=1,10$, and $50$,
and varied $r$. We also show the results for $G_{1}$, $G_{2}$, and the
aggregate network in these figures for comparison. We find that
$(G_{1},G_{2},1)$ is a suppressor of selection. In contrast, $G_{1}$ is an
amplifier of selection, and $G_{2}$ is neutral (i.e., equivalent to the Moran
process). In fact, no static unweighted network with five nodes or less is a
suppressor of selection [29]. Because the aggregate network is an amplifier of
selection, albeit a weak one, the suppressing effect of $(G_{1},G_{2},1)$ owes
to the time-varying nature of the switching network. Similar to the case in
which $G_{2}$ is the complete graph shown in Fig. 3, $(G_{1},G_{2},10)$ and
$(G_{1},G_{2},50)$ are amplifiers of selection, and the behavior of
$(G_{1},G_{2},50)$ is close to that for $G_{1}$, i.e., the star graph.

In Figs. 4(c) and 4(d), we show the fixation probability and its difference
from the case of the Moran process, respectively, for $N_{1}=N_{2}=20$. We
have set $N=N_{1}+N_{2}=40$ as opposed to $N=50$, which we used for the
switching network analyzed in section 4.2.1, because of the computational
cost. In contrast to the case of $N_{1}=N_{2}=2$, the switching network with
$N_{1}=N_{2}=20$ is an amplifier of selection for the three values of $\tau$.
Furthermore, in contrast to when $N_{1}=N_{2}=2$, the fixation probabilities
for the switching networks are closer to those for the Moran process than to
those for the star graph. To explore the case $N_{1}\neq N_{2}$, we show the
results for $\left(N_{1},N_{2}\right)=\left(4,2\right)$ and
$\left(N_{1},N_{2}\right)=\left(30,10\right)$ in Appendix D. The switching
network with $\left(N_{1},N_{2}\right)=\left(4,2\right)$ is neither amplifier
or suppressor of selection. However, its fixation probabilities are close to
those for the Moran process than to those for the star or bipartite complete
graph. The switching network with
$\left(N_{1},N_{2}\right)=\left(30,10\right)$ is an amplifier of selection and
behaves similarly to the switching network with
$\left(N_{1},N_{2}\right)=\left(20,20\right)$.

These results for the switching networks with $N=4$ and $N=40$ nodes remain
similar for switching networks $(G_{2},G_{1},\tau)$, as we show in Figs. 6(c)
and 6(d).

To examine the dependence of the fixation probability on the number of nodes,
we show in Fig. 4(e) the difference between the fixation probability for the
present switching network and that for the Moran process as we vary $N$. We
set $\tau=1$ and $N_{1}=N_{2}=N/2\geq 2$, and compute the fixation probability
at $r=0.9$ and $r=1.1$. Figure 4(e) indicates that the switching network is a
suppressor of selection only when $N_{1}=N_{2}=2$ (i.e., $N=4$) and amplifier
of selection for any larger $N$. When we allow $N_{1}\neq N_{2}$, we found
just one additional suppressor of selection apart from $(N_{1},N_{2})=(2,2)$
under the constraints $\tau=1$ and $2\leq N_{1},N_{2}\leq 10$, which is
$(N_{1},N_{2})=(3,2)$ (see Fig. 7(b) in Appendix B). With $\tau=50$, this
switching network is amplifier of selection for any $N$ (see Fig. 4(f)).

Figure 4: Fixation probability for switching networks in which $G_{1}$ is the
star graph and $G_{2}$ is the complete bipartite graph. (a) Fixation
probability for $N_{1}=N_{2}=2$. (b) Difference in the fixation probability
from the case of to the Moran process for $N_{1}=N_{2}=2$. (c) Fixation
probability for $N_{1}=N_{2}=20$. (d) Difference in the fixation probability
from the case of the Moran process for $N_{1}=N_{2}=20$. (e) and (f):
Difference in the fixation probability for the switching network relative to
the Moran process as a function of $N$ at $r=0.9$ and $1.1$. We set $\tau=1$
in (e) and $\tau=50$ in (f). In (e) and (f), the smallest value of $N$ is
four.

###  4.3 Empirical temporal networks

####  4.3.1 Construction of switching networks

Finally, we numerically simulate the birth-death process on four switching
networks informed by empirical temporal network data. We split each of the
temporal network data set into two static networks $(V_{1},E_{1})$ and
$(V_{2},E_{2})$, where $(V_{1},E_{1})$ contains the first half of the time-
stamped edges in terms of the time, $(V_{2},E_{2})$ containing the second half
of the time-stamped edges, $V_{1}$ and $V_{2}$ are sets of nodes, and $E_{1}$
and $E_{2}$ are sets of edges. For simplicity, we regard $(V_{1},E_{1})$ and
$(V_{2},E_{2})$ as unweighted networks. We note that the purpose of studying
these empirical networks is not to examine how fixation occurs in real contact
sequences but to explore the generality of the results obtained in the
previous sections in asymmetric and large switching networks.

For two of the four empirical switching networks, both $V_{1}$ and $V_{2}$
contain all nodes. In this case, we switch between $G_{1}\equiv(V_{1},E_{1})$
and $G_{2}\equiv(V_{2},E_{2})$. For the other two empirical switching
networks, either $V_{1}$ or $V_{2}$ misses some nodes in the original temporal
network. In this case, we construct switching networks in the following two
manners. With the first method, we only use the nodes in $V_{1}\cap V_{2}$ and
the edges that exist between pairs of nodes in $V_{1}\cap V_{2}$ as $G_{1}$
and $G_{2}$. For each of the two empirical data sets for which $V_{1}$ or
$V_{2}$ misses some nodes, we have confirmed that the first and second halves
of the static networks induced on $V_{1}\cap V_{2}$ created with this method
are connected networks. With the second method, we use all nodes for both
$G_{1}$ and $G_{2}$. In other words, we set $G_{1}=(V_{1}\cup V_{2},E_{1})$
and $G_{2}=(V_{1}\cup V_{2},E_{2})$. Therefore, if $v\in V_{1}$ and $v\notin
V_{2}$, for example, then $v$ is an isolated node in $G_{2}$. Except with
special initial conditions, the fixation of either type never occurs in a
static network with isolated nodes. However, the fixation does occur in the
switching network if the aggregate network is connected, which we have
confirmed to be the case for all our empirical data sets.

####  4.3.2 Simulation procedure

As the initial condition, we place a mutant on one node selected uniformly at
random and all the other $N-1$ nodes are of the resident type. Then, we run
the birth-death process until all nodes were of the same type. We carried out
$2\times 10^{5}$ such runs in parallel on $56$ cores, giving us a total of
$112\times 10^{5}$ runs, for each network and each value of $r$. We
numerically calculated the fixation probability as the fraction of runs in
which the mutant fixates. We simulated the switching networks with
$\tau\in\\{1,10,50\\}$ and
$r\in\\{0.7,0.8,0.9,1,1.1,1.2,1.3,1.4,1.5,1.6,1.7\\}$ for all the networks
except the hospital network of $75$ nodes. For the hospital network, we
omitted $r=1.6$ and $1.7$ due to high computational cost.

####  4.3.3 Data

The ants’ colony data, which we abbreviate as ant [63], has 39 nodes and 330
time-stamped edges. Each node represents an ant in a colony. An edge
represents a trophallaxis event, which was recorded when the two ants were
engaged in mandible-to-mandible contact for greater than one second. The first
and second halves of the data have 34 nodes each.

The second data is the contacts between members of five households in the
Matsangoni sub-location within the Kilifi Health and Demographic Surveillance
Site (KHDSS) in coastal Kenya [64]. A household was defined as the group of
individuals who ate from the same kitchen [64]. Each participant in the study
had a wearable sensor that detected the presence of another sensor within
approximately 1.5 meters. Each node is an individual in a household. An edge
represents a time-stamped contact between two individuals. There were 47
nodes. There were 219 time-stamped edges representing contacts between pairs
of individuals in different households and $32,426$ time-stamped edges between
individuals of the same households. Both the first and second halves contain
all the 47 nodes and are connected networks as static network owing to the
relatively large number of time-stamped edges.

The third data is a mammalian temporal network based on interaction between
raccoons [65]. A node represents a wild raccoon. The time-stamped events were
recorded whenever two raccoons came within approximately 1 to 1.5 meters for
more than one second, using proximity logging collars that were placed on
raccoons. The recording was made in Ned Brown Forest Preserve in suburban Cook
County, Illinois, USA, from July 2004 to July 2005. There are 24 nodes and
$2,000$ time-stamped edges. Both the first and second halves of the data
contain all the 24 nodes and are connected networks as static network.

The fourth data is a contact network in a hospital [66]. The data were
recorded in a geriatric unit of a university hospital in Lyon, France, from
December 6, 2010 at 1 pm to December 10, 2010 at 2 pm. The unit contained 19
out of the $1,000$ beds in the hospital. During the recording period, 50
professionals worked in the unit, and 31 patients were admitted. Fourty-six
among the 50 professionals and 29 among the 31 patients participated in the
study. Therefore, the network had 75 nodes in total. The professionals
comprised of 27 nurses or nurses’ aides, 11 medical doctors, and 8
administrative staff members. An edge represents a time-stamped contact
between two individuals; there are $32,424$ time-stamped edges. The first and
second halves of the data contain 50 nodes each.

We obtained the ant, raccoon, and hospital data from
https://networkrepository.com/ [67]. We obtained the Kilifi data from
http://www.sociopatterns.org/.

####  4.3.4 Numerical results

We investigate the fixation probability on the switching networks with
$\tau=1$, $10$, and $50$, static networks $G_{1}$ and $G_{2}$, and the
aggregate network. We remind that the aggregate network is a static weighted
network, whereas $G_{1}$ and $G_{2}$ are unweighted networks. For the ant and
hospital data, the switching networks constructed with the second method are
different from those constructed with the first method. For these two data
sets, fixation does not occur on $G_{1}$ and $G_{2}$ because they miss some
nodes. Therefore, we do not analyze the fixation probability on $G_{1}$ and
$G_{2}$ for these data sets.

We show in Figs. 5(a) and 5(b) the fixation probability on the ant switching
networks constructed with the first and second methods, respectively. Because
we are interested in whether the switching networks are amplifiers or
suppressors of selection, we only show the difference between the fixation
probability on the given network and that for the Moran process in Fig. 5.
Figure 5(a) indicates that the switching networks are amplifiers of selection
but less amplifying than each of its constituent static networks, $G_{1}$ and
$G_{2}$. Another observation is that the fixation probability on the static
aggregate network is close to that on the switching networks. In this sense,
the switching networks do not yield surprising results. The switching networks
are more strongly amplifying when $\tau$ is larger. Moreover, the fixation
probability on the switching network is closer to that on $G_{1}$ when $\tau$
is larger. This result is expected because the evolutionary dynamics is the
same between the switching networks and $G_{1}$ in the first $\tau$ time
steps. For the switching networks constructed with the second method, Fig.
5(b) shows that the switching networks are amplifiers and more amplifying than
the static aggregate network. This result is qualitatively different from that
for the switching networks constructed with the first method shown in Fig.
5(a).

We show the results for the Kilifi networks in Fig. 5(c). Because the first
and second methods yield the same $G_{1}$ and $G_{2}$ for the Kilifi data, we
only present the results for the first method for this data set and also for
the next one (i.e., racoon networks). The figure indicates that the switching
networks are amplifiers but less amplifying than $G_{1}$ and $G_{2}$ and
similarly amplifying compared to the aggregate network. These results are
similar to those for the ant networks shown in Fig. 5(a).

We show the results for the raccoon networks in Fig. 5(d). We find that the
switching networks are amplifiers but less amplifying than $G_{1}$ and
$G_{2}$, similar to the case of the ant and Kilifi networks. We also find that
the switching networks are more amplifying than the aggregate network.

We show the results for the hospital switching networks in Figs. 5(e) and
5(f). The results for the switching networks constructed with the first method
(see Fig. 5(e)) are similar to those for the raccoon networks shown in Fig.
5(d). The switching networks constructed with the second method (see Fig.
5(f)) are more amplifying than the aggregate network, similar to the case of
the ant networks generated by the same method (see Fig. 5(b)).

In sum, for these empirical temporal networks, we did not find a surprising
result that the fixation probability for the switching networks is not an
interpolation of those for the two static networks $G_{1}$ and $G_{2}$.
However, the fixation probability for the empirical switching networks depends
on the $\tau$ value and deviates from the prediction from the aggregate
network in multiple ways.

Figure 5: Fixation probability on empirical switching networks. In each panel,
we show the difference in the fixation probability from the case of the Moran
process as a function of $r$. (a) Ant networks constructed with the first
method. (b) Ant networks constructed with the second method. (c) Kilifi
switching networks. (d) Raccoon networks. (e) Hospital networks constructed
with the first method. (f) Hospital networks constructed with the second
method. We compared the fixation probability on switching networks with
$\tau\in\\{1,10,50\\}$, $G_{1}$, $G_{2}$, and the aggregate network in each
panel.

##  5 Discussion

We have shown that, under the birth-death updating rule and uniform
initialization, a majority of the switching networks on six nodes are
suppressors of selection. This result contrasts with the case of static
networks, for which there exists only one suppressor of selection on six nodes
[29]. We also found that switching networks alternating between the star graph
and the complete graph and those alternating between the star graph and the
complete bipartite graph are suppressors of selection when the number of
nodes, $N$, is small. When $N$ is larger, the same switching networks are
amplifiers of selection but less amplifying than the star graph. Among the
empirical networks that we analyzed, we did not find any suppressors. However,
these switching networks were notably less amplifying than the constituent
static networks $G_{1}$ and $G_{2}$. In fact, the less amplifying nature of
switching networks is largely explained by the aggregate weighted network, or
the static network obtained by the superposition of $G_{1}$ and $G_{2}$.
Therefore, our results for the empirical switching networks are not
surprising. The result that the switching network composed of two amplifying
static networks can be suppressor is our main finding. Because all the
instances that we have found are small networks, searching suppressing
switching networks with larger $N$ including systematically constructing such
instances remains future work.

Our choices of the larger networks are primarily driven by computational
feasibility. The complete graph, star graph, and complete bipartite graph are
convenient families of networks owing to their highly symmetric nature, which
drastically reduces the number of the unknowns to be determined. Similarly,
all the empirical networks that we used had at most 75 nodes due to
computational cost. Additionally, we avoided disconnected and sparse networks
because fixation requires a network to be connected, and splitting a sparse
network into two networks often resulted in disconnected components.
Nevertheless, by studying small networks, larger symmetric networks, and the
empirical examples, we tried to provide a broader picture of the evolutionary
dynamics on switching networks. However, there remains ample room for future
work. For instance, faster algorithms for approximate computation for larger
switching networks, such as those assuming weak selection [28], remain to be
explored. We could also attempt to reduce simulation times. In [68], instead
of sampling every time step of the evolutionary dynamics, only ‘effective’
steps are sampled. Effective steps are defined as those in which the network
state changes. The steps in which a resident replaces a resident or a mutant
replaces a mutant are deemed as ineffective steps, which one does not sample
in their algorithm, hence accelerating the overall simulation time.
Additionally, we studied switching networks with only two snapshots. It is
straightforward to extend the present computational framework to the case of
switching networks with more than two snapshots. Last, many temporal network
data are provided as a list of time-stamped events between pairs of nodes.
Evolutionary dynamics driven by such event-based temporal network data is also
worth studying.

We considered exogenous changes of the network over time in this study.
Another opportunity of research is to assume that the change of the network
structure over time is driven by the state of the system, which is referred to
as adaptive networks [69, 70]. The recent modeling framework inspired by
biological examples in which the residents and mutants use different static
networks defined on the same node set [71, 72] can be interpreted as an
example of fixation dynamics on adaptive networks. Allowing nodes to
stochastically sever and create edges they own as the node’s type flips from
the resident to mutant and vice versa may lead to new phenomena in fixation
dynamics. Such models have been extensively studied for evolutionary games on
dynamic networks [22, 23, 24].

We recently found that most hypergraphs are suppressors of selection under the
combination of a birth-death process and uniform initialization, which are the
conditions under which most of conventional networks are amplifiers of
selection [56]. It has been longer known that most undirected networks are
suppressors of selection under the death-birth process [27], and in directed
networks under various imitation rules including birth-death processes [73].
The degree of amplification and suppression also depends on the initialization
[26, 33]. For example, non-uniform initializations can make the star, which is
a strong amplifier of selection under the birth-death process and uniform
initialization, a suppressor of selection [26]. Furthermore, it has been shown
that the amplifiers of selection are transient and bounded [74]. Our results
suggest that small temporal networks are another major case in which
suppressors of selection are common. These results altogether encourage us to
explore different variants of network models and evolutionary processes to
clarify how common amplifiers of selection are. This task warrants future
research.

## Funding

N.M. acknowledges support from AFOSR European Office (under Grant No.
FA9550-19-1-7024), the Japan Science and Technology Agency (JST) Moonshot
R$\&$D (under Grant No. JPMJMS2021), and the National Science Foundation
(under Grant No. 2052720 and 2204936).

## Appendices

### A. Switching networks with the order of the static networks reversed and
with random initialization time.

In this section, we consider switching networks in which $G_{2}$ rather than
$G_{1}$ appear first and those with uniformly random initialization time.

In Fig. 6(a), we show the results for the six-node switching networks in which
$G_{1}$ and $G_{2}$ are given in Fig. 2(a). We find that
$\left(G_{2},G_{1},1\right)$ and the switching network with $\tau=1$ and the
random initialization time are both suppressors of selection. These variants
of switching networks are amplifiers of selection when $\tau=10$ and
$\tau=50$. These results are qualitatively the same as those for
$\left(G_{1},G_{2},\tau\right)$.

Next, we consider switching networks
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},\tau)$
in which $G_{1}$ is the star graph and $G_{2}$ is the complete graph. We show
the difference in the fixation probability from the case of the Moran process
for the switching networks with $N=4$ and $N=50$ in Figs. 6(b) and 6(c),
respectively. With $N=4$, we find that
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},10)$
and
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},50)$
are amplifiers of selection and that
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},1)$
is a suppressor of selection (see Fig. 6(b). The aggregate network is a weak
suppressor of selection. With $N=50$, we find that
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},\tau)$
for all the three $\tau$ values (i.e., $\tau\in\\{1,10,50\\}$) are amplifiers
of selection and that the aggregate network is a weak suppressor of selection
(see Fig. 6(c)). These results are qualitatively the same as those for
$(G_{1},G_{2},\tau)$ shown in Fig. 3. A main difference is that, when
$\tau=50$, the fixation probability is reasonably close to that for the Moran
process in the case of the present switching network because the initially
used static network, i.e., $G_{2}$, is a regular graph and therefore
equivalent to the Moran process. In contrast, in Fig. 3, the switching network
is much more amplifying because the initially used static network is the star
graph, which is a strong amplifier of selection. As expected, the results in
the case of the random initialization time are between those for
$(G_{1},G_{2},\tau)$ and those for $(G_{2},G_{1},\tau)$.

We show in Figs. 6(d) and 6(e) the results for $(G_{2},G_{1},\tau)$ with $N=4$
and $N=40$, respectively, in which $G_{1}$ is the star graph and $G_{2}$ is
the complete bipartite graph. With $N=4$, we find that
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},1)$
is a suppressor of selection,
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},10)$
and
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},50)$
are amplifiers of selection , and the aggregate network is a weak amplifier of
selection (see Fig. 6(d)). With $N=40$, we find that
$({\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{2}},{\color[rgb]{1,0,0}}{\color[rgb]{0,0,0}G_{1}},\tau)$
with $\tau\in\\{1,10,50\\}$ is an amplifier of selection and that the
aggregate network is a weak amplifier of selection (see Fig. 6(e)). These
results are similar to those for $(G_{1},G_{2},\tau)$ shown in Figs. 4(a) and
4(b). Similar to Figs. 6(b) and 6(c), with $\tau=50$, the present switching
networks are close in behavior to the Moran process because the initially used
static network, i.e., $G_{2}$, is a regular network. This result contrasts to
the corresponding result for $(G_{1},G_{2},50)$ , which is a relatively strong
amplifier of selection because the initially used static network is the star
graph (see Figs. 4(a) and 4(b)). Again, the results in the case of the random
initialization time are between those for $(G_{1},G_{2},\tau)$ and those for
$(G_{2},G_{1},\tau)$.

Figure 6: Fixation probability for switching networks with the order of
$G_{1}$ and $G_{2}$ being swapped and with random initialization time. In each
panel, we show the difference in the fixation probability from the case of the
Moran process. (a) $G_{1}$ and $G_{2}$ given in Fig. 2(a). In (b) and (c),
$G_{1}$ is the star graph, and $G_{2}$ is the complete graph. (b) $N=4$. (c)
$N=50$. In (d) and (e), $G_{1}$ is the star graph, and $G_{2}$ is the complete
bipartite graph. (d) $N_{1}=N_{2}=2$. (e) $N_{1}=N_{2}=20$. In (a), the
results for all the three switching networks with $\tau=1$, shown by the blue
lines, heavily overlap with each other, and those for $(G_{1},G_{2},10)$,
shown by the gray solid line, and those for the random initialization with
$\tau=10$, shown by the gray dashed line, heavily overlap on top of each
other. In (b)–(e), $G_{2}$ is equivalent to the Moran process. Therefore, the
results for $G_{2}$, shown by the magenta solid line, completely overlap with
the horizontal axis. Similarly, in (b) and (d), the results for
$(G_{2},G_{1},50)$, shown by the green dotted lines, heavily overlap with the
horizontal axis and are almost hidden behind the magenta solid lines. In
addition, in (b), (c), and (e), the results for the aggregate network, shown
by the red solid lines, almost completely overlap with or are very close to
the horizontal axis.

### B. Further examples of small switching networks in which $G_{1}$ is the
star graph

In Fig. 7(a), we show the difference in the fixation probability from the case
of the Moran process for the switching networks in which $G_{1}$ is the star
graph and $G_{2}$ is the complete graph on $N=3$ nodes. We also plot the
results for $G_{1}$, $G_{2}$, and the aggregate network. It is known that
$G_{1}$ is an amplifier of selection [1] and that $G_{2}$ is equivalent to the
Moran process. In contrast, the switching network with $\tau=1$ and the
aggregate network are suppressors of selection. The aggregate network is much
less suppressing than the switching network. The switching networks with
$\tau\in\\{10,50\\}$ are amplifiers of selection.

In Fig. 7(b), we show the results for the switching networks in which $G_{1}$
is the star graph and $G_{2}$ is the complete bipartite graph,
$K_{\left(3,2\right)}$, on $N=5$ nodes. Note that both $G_{1}$ (i.e., star)
[1] and $G_{2}$ (i.e., complete bipartite graph $K_{\left(3,2\right)}$) [75]
are amplifiers of selection. In contrast, as in Fig. 7(a), the switching
network with $\tau=1$ (but not with $\tau\in\\{10,50\\}$) and the aggregate
network are suppressors of selection, and the aggregate network is only weakly
suppressing.

Figure 7: Fixation probability as a function of $r$ for two small switching
networks. (a) Switching network with $N=3$ in which $G_{1}$ is the star graph
and $G_{2}$ is the complete graph. (b) Switching network with $N=5$ in which
$G_{1}$ is the star graph and $G_{2}$ is the complete bipartite graph
$K_{\left(3,2\right)}$. In (a), because $G_{2}$ is the complete graph, its
plot is exactly on the horizontal axis.

### C. Transition probabilities for the combination of the star graph and the
complete bipartite graph

The transition probability matrix for the star graph is given by

$T^{\left(1\right)}_{\left(i,j,k\right)\rightarrow\left(i^{\prime},j^{\prime},k^{\prime}\right)}=\begin{cases}\frac{r(j+k)}{C_{3}}&\text{if
}i=0\text{ and }i^{\prime}=1,\\\\[2.84526pt] \frac{N-1-j-k}{C_{4}}&\text{if
}i=1\text{ and }i^{\prime}=0,\\\\[2.84526pt]
\frac{1}{C_{3}}\cdot\frac{j}{N-1}&\text{if }i=0\text{ and
}j^{\prime}=j-1,\\\\[2.84526pt] \frac{1}{C_{3}}\cdot\frac{k}{N-1}&\text{if
}i=0\text{ and }k^{\prime}=k-1,\\\\[2.84526pt]
\frac{r}{C_{4}}\cdot\frac{N_{1}-1-j}{N-1}&\text{if }i=1\text{ and
}j^{\prime}=j+1,\\\\[2.84526pt]
\frac{r}{C_{4}}\cdot\frac{N_{2}-j}{N-1}&\text{if }i=1\text{ and
}k^{\prime}=k+1,\\\\[2.84526pt]
1-\sum\limits_{\mathclap{\begin{subarray}{c}(i^{\prime\prime},j^{\prime\prime},k^{\prime\prime})\neq\\\
(i,j,k)\end{subarray}}}T^{\left(1\right)}_{(i,j,k)\rightarrow(i^{\prime\prime},j^{\prime\prime},k^{\prime\prime})}&\text{if
}(i^{\prime},j^{\prime},k^{\prime})=(i,j,k),\\\\[2.84526pt] 0&\text{
otherwise,}\end{cases}$ (29)

where

${\color[rgb]{0,0,0}C_{3}=r(j+k)+(N-j-k)}$ (30)

and

${\color[rgb]{0,0,0}C_{4}=r\left(j+k+1\right)+\left(N-j-k-1\right).}$ (31)

The first line of Eq. (29) represents the probability that the type of the hub
changes from the resident to mutant. For this event to occur, one of the $j+k$
leaf nodes occupied by the mutant must be chosen as parent, which occurs with
probability $r(j+k)/C_{3}$. Then, because any leaf node is only adjacent to
the hub node, the hub node is always selected for death. Therefore, the
probability of $i$ changing from $0$ to $1$ is equal to $r(j+k)/C_{3}$. As
another example, consider state $\left(1,j,k\right)$. For the state to change
from $\left(1,j,k\right)$ to $\left(1,j+1,k\right)$, the hub node, which the
mutant type currently inhabits, must be selected as parent with probability
$r/C_{4}$. Then, one of the $j$ leaf nodes of the resident type in $V_{1}$
must be selected for death, which occurs with probability
$\left[(N_{1}-1)-j\right]/\left(N-1\right)$. The fifth line of Eq. (29) is
equal to the product of these two probabilities. One can similarly derive the
other lines of Eq. (29).

The transition probability matrix for the complete bipartite graph is given by

$T^{\left(2\right)}_{\left(i,j,k\right)\rightarrow\left(i^{\prime},j^{\prime},k^{\prime}\right)}=\begin{cases}\frac{rk}{C_{3}}\cdot\frac{1}{N_{1}}&\text{if
}i=0\text{ and }i^{\prime}=1,\\\\[2.84526pt]
\frac{N_{2}-k}{C_{4}}\cdot\frac{1}{N_{1}}&\text{if }i=1\text{ and
}i^{\prime}=0,\\\\[2.84526pt]
\frac{N_{2}-k}{C_{3}}\cdot\frac{j}{N_{1}}&\text{if }i=0\text{ and
}j^{\prime}=j-1,\\\\[2.84526pt]
\frac{rk}{C_{3}}\cdot\frac{N_{1}-1-j}{N_{1}}&\text{if }i=0\text{ and
}j^{\prime}=j+1,\\\\[2.84526pt]
\frac{N_{1}-j}{C_{3}}\cdot\frac{k}{N_{2}}&\text{if }i=0\text{ and
}k^{\prime}=k-1,\\\\[2.84526pt]
\frac{rj}{C_{3}}\cdot\frac{N_{2}-k}{N_{2}}&\text{if }i=0\text{ and
}k^{\prime}=k+1,\\\\[2.84526pt]
\frac{N_{2}-k}{C_{4}}\cdot\frac{j}{N_{1}}&\text{if }i=1\text{ and
}j^{\prime}=j-1,\\\\[2.84526pt]
\frac{rk}{C_{4}}\cdot\frac{N_{1}-1-j}{N_{1}}&\text{if }i=1\text{ and
}j^{\prime}=j+1,\\\\[2.84526pt]
\frac{N_{1}-1-j}{C_{4}}\cdot\frac{k}{N_{2}}&\text{if }i=1\text{ and
}k^{\prime}=k-1,\\\\[2.84526pt]
\frac{r\left(j+1\right)}{C_{4}}\cdot\frac{N_{2}-k}{N_{2}}&\text{if }i=1\text{
and }k^{\prime}=k+1,\\\\[2.84526pt]
1-\sum\limits_{\mathclap{\begin{subarray}{c}(i^{\prime\prime},j^{\prime\prime},k^{\prime\prime})\neq\\\
(i,j,k)\end{subarray}}}T^{\left(2\right)}_{\left(i,j,k\right)\rightarrow\left(i^{\prime\prime},j^{\prime\prime},k^{\prime\prime}\right)}&\text{if
}(i^{\prime},j^{\prime},k^{\prime})=(i,j,k).\\\\[2.84526pt] 0&\text{
otherwise.}\end{cases}$ (32)

The first line of Eq. (32) represents the probability that the type of the hub
changes from the resident to mutant. For this event to occur, one of the $k$
mutant nodes in $V_{2}$ must be selected as parent with probability
$rk/C_{3}$. Then, the hub node must be selected for death with probability
$1/N_{1}$ because each node in $V_{2}$ is only adjacent to all the $N_{1}$
nodes in $V_{1}$. Therefore, the probability of $i$ changing from $0$ to $1$
is equal to $\left(rk/C_{3}\right)\cdot\left(1/N_{1}\right)$. As another
example, consider state $\left(1,j,k\right)$, in which there are $j+k+1$
mutants in total. For the state to change from $\left(1,j,k\right)$ to
$\left(1,j+1,k\right)$, one of the $k$ mutant nodes in $V_{2}$ must first be
selected as parent with probability $rk/C_{4}$. Then, one of the $j$ leaf
nodes in $V_{1}$ of the resident type must be selected for death, which occurs
with probability $(N_{1}-1-j)/N_{1}$. The eighth line of Eq. (32) is equal to
the product of these two probabilities. One can similarly derive the other
lines of Eq. (32).

### D. Examples in which $G_{1}$ is the star graph, $G_{2}$ is the complete
bipartite graph, and $N_{1}\neq N_{2}$

In this section we consider switching networks when $G_{1}$ is the star graph
and $G_{2}$ is the complete bipartite graph. In Fig. 4, we have shown the
results for $N_{1}=N_{2}$. The complete bipartite graph
$K_{\left(N_{1},N_{2}\right)}$ is isothermal when $N_{1}=N_{2}$, whereas it is
an amplifier of selection when $N_{1}\neq N_{2}$ [75]. Therefore, the fixation
probability for $(G_{1},G_{2},\tau)$ may be qualitatively different between
the cases $N_{1}=N_{2}$ and $N_{1}\neq N_{2}$. In this section, we examine two
switching networks when $N_{1}\neq N_{2}$.

In Fig. 8(a), we show the difference in the fixation probability from the case
of the Moran process for the switching networks in which $G_{1}$ is the star
graph on $N=6$ nodes and $G_{2}$ is the complete bipartite graph
$K_{\left(4,2\right)}$. We also plot the fixation probability for $G_{1}$,
$G_{2}$, and the aggregate network. Although $G_{1}$ and $G_{2}$ are both
amplifiers of selection [1, 75], $(G_{1},G_{2},1)$ is neither an amplifier nor
a suppressor of selection. It transitions approximately at $r=2.062$ as $r$
increases, similar to static networks analyzed before [35]. Specifically,
$(G_{1},G_{2},1)$ is amplifying when $r$ is approximately smaller than $2.062$
and suppressing when $r$ is approximately greater than $2.062$. The switching
networks with $\tau\in\\{10,50\\}$ as well as the aggregate network are
amplifiers of selection.

In Fig. 8(b), we show the results for $(G_{1},G_{2},\tau)$ in which $G_{1}$ is
the star graph on $N=40$ nodes and $G_{2}$ is the complete bipartite graph
$K_{\left(30,10\right)}$. We note that both $G_{1}$ and $G_{2}$ are amplifiers
of selection. In this case, all the switching networks and the static networks
are amplifiers of selection, which is qualitatively the same result as that
for $N_{1}=N_{2}=20$ (see Figs. 4(c) and (d)).

Figure 8: Fixation probability as a function of $r$ when $G_{1}$ is the star
graph, $G_{2}$ is the complete graph, and $N_{1}\neq N_{2}$. (a)
$\left(N_{1},N_{2}\right)=\left(4,2\right)$. (b)
$\left(N_{1},N_{2}\right)=$$\left(30,10\right)$.

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