# Preferential imitation of vaccinating behavior can invalidate the targeted
subsidy on complex networks

Hai-Feng Zhang  School of Mathematical Science, Anhui University, Hefei
230039, P. R. China  Key Laboratory of Computer Network and Information
Integration (Southeast University), Ministry of Education  Department of
Communication Engineering, North University of China, Taiyuan, Shan’xi 030051,
P. R. China  Pan-Pan Shu  Ming Tang<EMAIL_ADDRESS>Web Sciences
Center, University of Electronic Science and Technology of China, Chengdu
611731, P. R. China  Michael Small<EMAIL_ADDRESS>School of
Mathematics and Statistics, The University of Western Australia, Crawley,
Western Australia 6009, Australia

###### Abstract

We consider the effect of inducement to vaccinate during the spread of an
infectious disease on complex networks. Suppose that public resources are
finite and that only a small proportion of individuals can be vaccinated
freely (complete subsidy), for the remainder of the population vaccination is
a voluntary behavior — and each vaccinated individual carries a perceived
cost. We ask whether the classical targeted subsidy strategy is definitely
better than the random strategy: does targeting subsidy at individuals
perceived to be with the greatest risk actually help? With these questions, we
propose a model to investigate the _interaction effects_ of the subsidy
policies and individuals responses when facing subsidy policies on the
epidemic dynamics on complex networks. In the model, a small proportion of
individuals are freely vaccinated according to either the targeted or random
subsidy policy, the remainder choose to vaccinate (or not) based on voluntary
principle and update their vaccination decision via an imitation rule. Our
findings show that the targeted strategy is only advantageous when individuals
prefer to imitate the subsidized individuals’ strategy. Otherwise, the effect
of the targeted policy is worse than the random immunization, since
individuals preferentially select non-subsidized individuals as the imitation
objects. More importantly, we find that under the targeted subsidy policy,
increasing the proportion of subsidized individuals may increase the final
epidemic size. We further define social cost as the sum of the costs of
vaccination and infection, and study how each of the two policies affect the
social cost. Our result shows that there exist some optimal intermediate
regions leading to the minimal social cost.

###### pacs:

89.75.-k, 87.23.Ge, 02.50.Le, 05.65.+b

> ##  I Introduction

Vaccination is the most effective response to large-scale epidemic outbreaks,
such as seasonal influenza and influenza like epidemics. Vaccination both
limits the extent of the outbreak and reduces morbidity and mortality Reluga
_et al._ (2006); Mbah  _et al._ (2012). However, electing to take vaccination
brings certain side effects or inconveniences; while protecting not only those
who are vaccinated but also their neighbors, and leading to the so-called
effect of “herd-immunity”. This scenario naturally leads to the problem of
“free-riding” commonly observed in public goods studies Wu  _et al._ (2011).
Recently, by incorporating evolutionary game theory into traditional
epidemiological models, many researchers have demonstrated that a voluntary
vaccination policy without incentives may be unable to eradicate a vaccine-
preventable disease Zhang  _et al._ (2013a); Reluga and Galvani (2011); Zhang
_et al._ (2010); Bauch  _et al._ (2003); Bauch and Earn (2004); Bauch (2005);
Vardavas  _et al._ (2007). One possible way of solving the social dilemma with
respect to vaccination is for the government or health organizations to offer
some incentive programs, e.g., subsidy and insurance policies Zhang  _et al._
(2013b); Wells  _et al._ (2013).

Since many diseases spread through human populations by physical contact
between infected individuals and susceptible individuals, the pattern of these
disease-causing contact forms a network. In the past decades, network-based
epidemiological models have attracted myriad attention and many significant
results have been achieved Newman (2002); Moreno  _et al._ (2002); Barthélemy
_et al._ (2005); Castellano and Pastor-Satorras (2010); Yang  _et al._ (2008);
Boccaletti  _et al._ (2014); Granell  _et al._ (2013). To control the epidemic
spreading on complex networks, many well-studied immunization strategies have
been proposed, such as targeted immunization Pastor-Satorras and Vespignani
(2002) and acquaintance immunization Cohen  _et al._ (2003). Compared to a
random immunization strategy, these two immunization strategies have been
proven to be more efficient in controlling the spread of epidemic diseases on
complex networks. However, the results were obtained by the two implicit
assumptions: 1) there are enough public resources to vaccinate many
individuals; 2) the behavioral responses of the remaining individuals are not
considered. In reality, public resource limitations mean that government
agencies can usually only freely vaccinate a small proportion of people. The
remaining population decide whether or not to take vaccination according to a
voluntary principle — and subject to an associated vaccination cost. In this
case, the vaccination decision of the remaining people is dependent on the
risk of infection, prevalence of infection, the cost of vaccination, public
opinion and so on Funk  _et al._ (2009). In particular, their vaccination
decisions are sensitively influenced by the proportion of subsidized
individuals as well as which kinds of individuals are subsidized. For example,
if some hub nodes are freely immunized, the nodes with small degrees have
small probability of being infected due to the herd immunity effect from these
hub nodes, as a result, many nodes are unwilling to take vaccination, leading
to an increase in the final epidemic size. Therefore, we need to reconsider
whether the classical targeted subsidy policy (i.e., targeted immunization
under which certain individuals with the highest degree are freely immunized)
is still better than the random case Wells  _et al._ (2013); Rat-Aspert and
Fourichon (2010).

In this paper, a vaccination decision model with a preferential selection
mechanism based on game theory is established to investigate the interplay of
the subsidy policy, the voluntary vaccination decision and the spreading
dynamics on networks. We then compare the effects of the targeted subsidy
policy with the random policy on the vaccination and epidemic dynamics. We
find that the preferential selection mechanism plays a vital role on whether
the targeted subsidy policy yields ideal results. We show that the advantage
of the targeted policy is greatly increased when the non-subsidized
individuals prefer to select the subsidized individuals as imitation objects.
However, the control effect of the targeted policy is even worse than that of
the random policy when non-subsidized individuals randomly or preferentially
select other non-subsidized nodes as their imitation objects. More
importantly, we find that, for the targeted policy, increasing the subsidy
proportion may increase _rather than_ reduce the final epidemic size when non-
subsidized individuals are preferentially selected, it is because that a small
subsidy proportion may greatly reduce the vaccination willingness of the non-
subsidized individuals. In addition, by defining the social cost as the sum of
the costs of vaccination and infection, we find that the social cost non-
monotonically depends on the subsidized proportion $p$ for both subsidy
policies. Our findings indicate that the effects of well-designed subsidy
policies may be undermined or counter-productive if the policies are
implemented without considering the intricate human behavioral responses, such
as neglecting the non-subsidized individuals’ responses on the subsidy
policies.

In Sec. II, we describe our model of epidemic spreading on complex networks by
integrating game theory. In Sec. III, we present our main results concerning
the effects of the two subsidy policies on the vaccination coverage, the final
epidemic size and the social cost. Meanwhile, an example of influenza case is
given in Sec. IV to demonstrate the applicability of these results. In Sec. V,
we present conclusion and discussions.

##  II Descriptions of the model

Taking into account seasonal flu-like diseases and the limited effectiveness
of vaccines and following previous studies Fu  _et al._ (2011); Zhang  _et
al._ (2013a, 2014a), we model the vaccination dynamics and the epidemic as a
two-stage process: the first stage is the vaccination decision stage, which is
implemented before each epidemic season; the second stage is the epidemic
season stage, where non-vaccinated individuals may be infected by diseases
described by the standard susceptible-infected-removed (SIR) model.
Specifically, at the vaccination stage, except for a proportion $p$ of
individuals who are freely vaccinated (according to either the targeted
strategy or the random strategy), each individual needs to decide whether or
not to take vaccination. Choosing to take vaccination will incur a cost
$C_{V}$ [the cost of vaccination can account for the monetary cost of the
vaccine, the perceived vaccine risks, side effects, long-term healthy impacts,
and so forth] regardless of the non-subsidized or subsidized individuals (the
vaccination cost of the subsidized individuals is $C_{V}$ though they need not
pay the cost themselves 111Of course, this assumes that the non-monetary as
well as monetary cost of vaccination can be reduced to zero for the subsidised
vaccinations.). We also assume that the vaccine can perfectly protect
vaccinated individuals from infection in the following epidemic season. The
second stage is the epidemic season stage, during which each unvaccinated
individual has a probability of being infected during the season. In this
stage the epidemic strain infects an initial number $I_{0}$ of individuals,
and the disease spreads according to SIR dynamics with daily transmission rate
$\lambda$ and recovery rate $\mu$. The epidemic continues until there are no
more newly infected individuals. An individual who gets infected during the
epidemic season pays a cost $C_{I}$. Without loss of generality, we set
$C_{I}=1$ and let $c=C_{V}/C_{I}$ describe the relative cost of vaccination,
whose value is restricted in the region of $[0,1]$ Fu  _et al._ (2011); Liu
_et al._ (2012). Those individuals who are neither vaccinated nor infected pay
no cost, so they are the free riders.

Let $P_{i}(t)$ be the current payoff of individual $i$ at season $t$.
According to the above descriptions, $i$ takes one of the three cost values as
follows

$\displaystyle P_{i}(t)=$
$\displaystyle-c,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{vaccination};$
$\displaystyle P_{i}(t)=$
$\displaystyle-1,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{infected};$ (1)
$\displaystyle P_{i}(t)=$ $\displaystyle
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{free-rider}.$

After the last epidemic season, individuals can update their strategies again
in the following vaccination decision stage based on their previous
experiences. Considering that imitation phenomena commonly exist in the real
world Tennie  _et al._ (2006); Heyes and Galef Jr (1996), many studies have
investigated the impact of an imitation rule from the game theoretical
perspective as well as in the vaccination decision-making epidemiological
models Fu  _et al._ (2011); Mbah  _et al._ (2012). Hence, we also assume
individuals update their strategies based on an imitation rule. Specifically,
if an individual $i$ updates her vaccination strategy, she _randomly_ chooses
an immediate neighbor $j$ with which to compare their expected payoff, and
then adopts $j$’s strategy with a probability dependent on their payoff
difference Fu  _et al._ (2011); Szabó and Tőke (1998):

$W(s_{i}\leftarrow s_{j})=\frac{1}{1+\exp[-\beta(P_{j}(t)-P_{i}(t))]},$ (2)

where $s_{i}=1\ \mbox{or}\ 0$ denotes the vaccination choice for individual
$i$: either vaccinated or not, where $\beta$ describes the rationality of the
individuals. Unless otherwise stated, we let $\beta=10$ in this paper,
implying that better-performing individuals are readily imitated, but it is
still possible to adopt the behavior of an individual performing worse.

When the subsidy policies from the governments or health organizations are
included, it is difficult to be definitive over whether individuals choose to
imitate subsidized or non-subsidized individuals. For example, some
individuals may prefer to imitate the subsidized individuals’ strategies since
these subsidized individuals are chosen by authoritative organizations.
Conversely, some individuals are willing to imitate non-subsidized
individuals’ strategies since their strategies come from their self-choices
but not from the external environments. In view of this, we introduce a
selection parameter $\alpha$ to differentiate the preferential selection rule.
In detail, we first give a different weight value $w=1$ ($w=5$) for non-
subsidized (subsidized) individuals222Here we use 1 and 5 to distinguish the
non-subsidized and subsidized individuals, respectively. Obviously, the other
two different positive values have the same effects., then the probability of
node $i$ choosing neighbor $j$ is given as:

$f(i\leftarrow j)=\frac{e^{\alpha w_{j}}}{\sum_{k\in\Lambda_{i}}e^{\alpha
w_{k}}},$ (3)

here $\Lambda_{i}$ denotes the neighborhood of node $i$ and $w_{i}$ is the
weight of node $i$. For $\alpha<0$, non-subsidized neighbors are more likely
to be selected as the potential strategy objects. On the contrary, for
$\alpha>0$, individuals tend to select subsidized neighbors as potential
imitation objects. Neighbors are randomly selected as $\alpha=0$, which
returns to the original model. Our main aim is to explore how the value of
$\alpha$ affects the vaccination coverage, the final epidemic size and the
social cost in regard with the two different subsidy policies.

##  III main results

In our simulations, for targeted subsidy policy, a proportion $p$ of
individuals with the highest degrees are freely vaccinated, for the random
subsidy policy, a proportion $p$ of individuals of individuals is randomly
selected and freely vaccinated. Then equal fractions of vaccinated and non-
vaccinated individuals are randomly distributed among a proportion $1-p$ of
individuals to start the iterative process. Meanwhile, the number of initially
infected individuals in each epidemic season is $I_{0}=5$. The long run
equilibrium results are obtained by averaging the last 1000 iterations of a
total of 3000 iterations in 100 independent realizations. Our results are
implemented on networks generated from the standard configuration model Newman
_et al._ (2001) with degree distribution $P(k)\sim k^{-3}$. The size of the
network studied is $N=2000$, the minimal and maximal degrees are $k_{min}=3$
and $k_{max}=\sqrt{N}$, respectively. Finally, in Sec. IV we also study a
practical case by using the email network to validate the generality of the
results and demonstrate the application of this approach.

Before comparing the two subsidy policies’ effects on the vaccination coverage
($V$) and the final epidemic size ($R$), we first investigate how the pure
immunization strategy (we also say subsidy policy in the following context,
only $p$ proportion of individuals is immunized, the remaining individuals do
nothing) and the pure vaccination decision dynamics (without the subsidy
policy, _i.e._ , $p=0$) affect the epidemic dynamics, respectively. The
targeted immunization strategy (labelled TAR in the following figures) and the
random immunization strategy (labelled RAN in the following figures) are
compared in Fig. 1(a). As provided in previous work, Fig. 1(a) shows that the
targeted immunization strategy can better control the outbreaks of epidemics
when the resources are sufficient, and can be eliminated completely once
$p\approx 0.18$. Furthermore, the vaccination coverage $V$ and the final
epidemic size $R$ versus the cost of vaccination $c$ are given in Fig. 1(b),
one can find that the value of $V$ decreases with the value of $c$, leading to
a corresponding increase of $R$.

Figure 1: (Color online) (a) For the pure immunization strategy, comparison of
the effects the targeted immunization (or subsidy) strategy and the random
immunization strategy on the final epidemic size $R$; (b) the final epidemic
size $R$ and the vaccination coverage $V$ as functions of the cost of
vaccination $c$ when the subsidy proportion $p=0$. In this and the following
figures, unless otherwise specified, we let $\lambda=0.2$ and $\mu=0.25$ to
calibrate the epidemic size near $0.9$ when there are no protective measures.

In the following, we want to know what new phenomena can be produced when the
subsidy policies and the vaccination decisions from an individual’s voluntary
behaviors are integrated. Fig. 2 gives the final epidemic size (see Fig. 2(a))
and the vaccination coverage (including the subsidized individuals and the
vaccinated individuals decided by themselves, see Fig. 2(b)) as functions of
the parameter $\alpha$ for different subsidy policies as well as the different
values of $p$. As shown in Fig. 2(b), for $\alpha>0$, the vaccination coverage
is enhanced and the larger value of $p$ gives rise to the greater enhancement
of $V$ for the both subsidy policies. However, one can see that the
vaccination coverage for the targeted policy is enhanced more significantly,
leading to almost the elimination of the epidemic for the targeted policy (see
Fig. 2(a)). On the contrary, for $\alpha\leq 0$, a non-trivial phenomenon
appears: the vaccination coverage for the targeted policy is much lower than
the random policy, meanwhile, for the targeted policy, larger value of $p$
yields lower level of vaccination coverage. The non-trivial phenomenon becomes
more remarkable as the value of $\alpha$ is further decreased.
Correspondingly, the control effect of the targeted policy becomes worse than
the random case. This result suggests that the feel good subsidy policy
(targeted subsidy policy) may produce the opposite effect when the
individuals’ responses on subsidy policies are neglected.

Figure 2: (Color online) Effects of $\alpha$ on the control effects of the
random subsidy policy and the targeted subsidy policy for different values of
$p$: (a) The final epidemic size $R$ versus the select parameter $\alpha$ for
different subsidy policies; (b) the vaccination coverage $V$ versus the select
parameter $\alpha$ for different subsidy policies. Here the cost of
vaccination $c=0.5$.

To systematically analyze the above phenomena, we first define the voluntary
vaccination probability $V_{k}$ (excluding the subsidized individuals) for the
nodes with the same degree $k$ as: $V_{k}=N^{v}_{k}/N_{k}$, where $N^{v}_{k}$
and $N_{k}$ are the _number_ of voluntarily vaccinated nodes with degree $k$
and the number of nodes with degree $k$, respectively. In Fig. 3(a), we plot
$V_{k}$ as a function of $k$ for the case of $p=0$, _i.e._ , the subsidy
policy is omitted. As illustrated in Fig. 3(a), the probability $V_{k}$
increases with the degree $k$ until to a upper bound ($\approx 0.65$), that is
to say, the nodes with larger degrees (‘large nodes’ or hub nodes) are more
likely to take vaccination than the ‘small nodes’ (the nodes with small
degrees) even when there is no any subsidy policy. The reason comes from that
the ‘large nodes’ have higher risk of infection Zhang  _et al._ (2010).

We then consider the cases of $\alpha=-3$, $0$ and $3$ in Fig. 3(b), (c) and
(d) with $p=0.05$. One can observe that, when $\alpha<0$ (Fig. 3(b)) or
$\alpha=0$ (Fig. 3(c)), the value of $V_{k}$ for the random subsidy policy is
larger than the targeted case. Note that, for the targeted case, $V_{k\geq
13}=0$ since the nodes with degree $k\geq 13$ have been freely vaccinated. As
analyzed in Fig. 3(a), the voluntary vaccination probability for hub nodes is
high even without the subsidy policy, in this situation, the governments or
health organizations still use the limited public resource to subsidize these
nodes rather than the nodes who are not willing to take vaccination
voluntarily, meaning that the public resources are somewhat wasted and cannot
as fully stimulate the vaccination enthusiasm of ‘small nodes’. Meanwhile, for
the targeted policy, the non-subsidized nodes are ‘small nodes’, and they are
not willing to take voluntarily vaccination owing to the low risk of
infection. In this case, the non-vaccinated neighbor nodes have higher
probability of being selected if $\alpha<0$, which leads to the low
vaccination coverage for the targeted policy, _i.e._ , the lower value of
$V_{k}$. However, for the random subsidy policy, each node has the same
probability to be freely vaccinated, in this case, the hub nodes with the
higher voluntarily vaccination probability [see Fig. 3(a)] are more likely to
be selected by others owing to their large connectivity even when $\alpha\leq
0$, so their vaccination decision is more likely to be imitated by others.
Combining the above explanations, we can understand why the vaccination
coverage for the targeted policy is much lower than the random case as
$\alpha\leq 0$. When $\alpha>0$ (Fig. 3(d)), firstly, subsidized individuals
are likely to be selected as the imitation objects, secondly, for the targeted
subsidy policy, the subsidized nodes are the hub nodes who have more
connectivity and can be selected by others with larger probability. As a
result, the $V_{k}$ of the targeted policy is larger than the random case.

Figure 3: (Color online) The voluntary vaccination probability $V_{k}$ as a
function of degree $k$ for the random subsidy policy and the targeted subsidy
policy. (a) the subsidy policy is not considered, i.e., $p=0$. (b) $\alpha=-3$
and $p=0.05$; (c) $\alpha=0$ and $p=0.05$; (d) $\alpha=3$ and $p=0.05$. Here
the cost of vaccination $c=0.5$.

Furthermore, Fig. 4 plots the contour of $R$ (left column of Fig. 4) and the
contour of $V$ (right column of Fig. 4) versus the parameters $\alpha$ and $p$
for the two subsidy policies. For the random subsidy policy, one can find that
the vaccination coverage $V$ increases with $p$ and $\alpha$ (see Fig. 4(b)),
leading to the reduction of $R$ when $p$ or $\alpha$ is increased (see Fig.
4(a)). For the targeted case, a very interesting result can be observed: for
$\alpha<0$, the vaccination coverage first decreases with the parameter $p$
and to a minimal level as $p\in(0.03,0.17)$, and then the vaccination coverage
increases again with further increase in the value of $p$ (see Fig. 4(d)).
Fig. 2(b) and Fig. 3(b) have shown the targeted subsidy policy cannot improve
but will inhibit the vaccination coverage as $\alpha\leq 0$, which can explain
why the vaccination coverage is reduced at first. With the further increase of
$p$, there are more vaccinated individuals even when the number of the
voluntarily vaccinated individuals is negligible, so the vaccination coverage
is increased again. The existence of the minimal vaccination coverage leads to
the maximal final epidemic size as $p\in(0.03,~{}0.08)$ and $\alpha<-2$(see
Fig. 4(c)), which means that subsidy policy ($p>0$) does not reduce the final
epidemic size but makes the final epidemic size larger than the case of $p=0$.
Moreover, one can see that the maximal final epidemic size region (see Fig.
4(c)) is much smaller than the minimal vaccination coverage region (see Fig.
4(c)). The reason can be deduced from Fig. 1(a), which indicates that the
epidemic has been controlled to a rather low level as $p=0.1$.

Figure 4: (Color online) Comparison between the random subsidy policy and the
targeted subsidy policy for the final epidemic size $R$ and the vaccination
coverage $V$ in the full phase diagram ($\alpha,p$).(a) The final epidemic
size $R$ versus $\alpha$ and $p$ for random policy; (b) the vaccination
coverage $V$ versus $\alpha$ and $p$ for random policy; (c) the final epidemic
size $R$ versus $\alpha$ and $p$ for targeted policy; and, (d) the vaccination
coverage $V$ versus $\alpha$ and $p$ for targeted policy. Here the cost of
vaccination $c=0.5$.

From the perspective of group interest, the purpose of introducing subsidy
policies is to minimize the total social cost at the population level. We thus
want to know how the social cost is affected by the two subsidy policies as
well as the selection parameter $\alpha$. To this end, we first define the
social cost as Dybiec  _et al._ (2004); Zhang  _et al._ (2013b):
$SC=N_{R}\times 1.0+N_{V}\times c$ with $N_{R}$ and $N_{V}$ be the _number_ of
infected and vaccinated individuals.

In Fig. 5(c) we study the effect of the parameter $p$ on the value of $SC$ for
the random subsidy policy. From Fig. 5(c) we can find that the value of $SC$
decreases with $p$ as $\alpha\leq 0$ (we here restrict $p\leq 0.3$ given that
the public resources are limited.). However, there exists an optimal value of
$p$ leading to the minimal social cost when $\alpha>0$ (2.5 and 5.0). To
explain such a phenomenon, we present the values of $R$ and $V$ as the
functions of $p$ for different values of $\alpha$ in Fig. 5(a) and Fig. 5(b).
One can see that the vaccination coverage $V$ is not increased remarkably as
$\alpha<0$, which leads to the final epidemic size $R$ not being significantly
reduced. Thus we need to further increase $p$ to reduce the social cost since
the cost of infection is larger than the cost of vaccination. Yet, for
$\alpha>0$, from Fig. 5(b) and Fig. 5(a) we can observe that the vaccination
coverage is greatly improved and the $R\rightarrow 0$ once $p>0.1$, which
implies that the proportion of subsidy is sufficient to control the outbreak
of epidemic, and further increase the value of $p$ just gives rise to the
overload of subsidy.

The effect of subsidy proportion $p$ on the social cost $SC$ for the targeted
policy is illustrated in Fig. 5(f). Note that, compared with Fig. 5(c), there
also exists an optimal value of $p$ leading to the minimal social cost even
for $\alpha<0$. As we know, unlike the random subsidy policy, Fig. 1(a) shows
that the epidemic is almost eliminated by the targeted subsidy policy as
$p>0.15$ even without voluntarily vaccinated individuals. So further increase
of $p$ definitely causes a rise in the social cost. For $\alpha>0$, the
vaccination coverage increases with $p$ (see Fig. 5(e)), leading to the rapid
decrease of the final epidemic size, especially, when $p\approx 0.05$, the
epidemic is almost controlled (see Fig. 5(d)). As a result, for $\alpha>0$,
there exists an optimal value of $p$ around 0.05 causing the minimal social
cost. Meanwhile, Fig. 5(e) suggests that the optimal value of $p$ for
$\alpha>0$ is smaller than the case of $\alpha\leq 0$.

Figure 5: (Color online) Given different values of $\alpha$, the final
epidemic size $R$, the vaccination coverage $V$ and the social cost $SC$
respectively as a function of $p$ for the random policy and the targeted
policy. (a) $R$ versus $p$ for the random policy; (b) $V$ versus $p$ for the
random policy; (c) $SC$ versus $p$ for the random policy; (d) $R$ versus $p$
for the targeted policy; (e) $V$ versus $p$ for the targeted policy; (f) $SC$
versus $p$ for the targeted policy. Here the cost of vaccination $c=0.5$.

The systematic effects of the parameters $p$ and $\alpha$ for the random
policy and the targeted policy on the social cost are presented in Fig. 6(a)
and (b), respectively. As shown in Fig. 6(a), the optimal phenomenon only
evident when $\alpha>0$. However, for the targeted case, there always exists
an optimal value of $p$ inducing the minimal social cost. Meanwhile, the
optimal value of $p$ becomes larger as the value of $\alpha$ is gradually
reduced.

Figure 6: (Color online) The contour of $SC$ as a function of ($\alpha,p$) for
the random subsidy policy (a) and the targeted subsidy policy, respectively.
Here the cost of vaccination $c=0.5$.

##  IV application: A practical example of influenza

To demonstrate the universality of this phenomenon, we calibrate the
parameters of the model based on the scenario of the 2009 H1N1 influenza
epidemic, in which a reproduction number $R_{0}$ was estimated as $R_{0}=1.6$
Xia and Liu (2013); Yang  _et al._ (2009); Poletti  _et al._ (2011) and
recovery rate was set to $\mu=1.0/3$ (_i.e._ , the mean infectious period is 3
days) Fu  _et al._ (2011). It is difficult to obtain the real contact networks
on which the influenza spreads, so without loss of generality, we use an
online social network—an email network as a proxy from which to study the
spreading of influenza on social networks. The email network is composed of
$N=1133$ nodes and 5451 links. In addition, the cluster coefficient, the
assortative coefficient and the average shortest distance are respectively
$C=0.11$, $r=0.078$ and $D=3.716$ Leskovec  _et al._ (2009); Zhao  _et al._
(2014). We further choose the transmission rate $\lambda$ such that the final
epidemic size is that of the well-mixed population without vaccination. For
this purpose, we let $\lambda\approx 0.085$ for the email network. According
to the Refs. Cornforth  _et al._ (2011); Luce  _et al._ (2008); Galvani  _et
al._ (2007), the cost of vaccination is set $c_{V}=\$27$ Luce  _et al._ (2008)
and the cost of infection is set $c_{I}=\$73$ Galvani  _et al._ (2007).

With the above preparations, we compare the two subsidy policies with regard
to the vaccination coverage, the final epidemic size as well as the social
cost in Fig. 7. As shown in Fig. 7, all the phenomena discussed in main text
can be observed. Such as, for the targeted policy, decreasing the value of
$\alpha$ leads to the rapid reduction of vaccination coverage and yields worse
control outcomes (see Fig. 7(a) and (b)), increasing the value of $p$ can
increase the final epidemic size (see Fig. 7(d)) as $\alpha<0$; For the random
subsidy policy, the optimal value of $p$ leading to the minimal social cost
appears only when $\alpha>0$, but for the targeted case, the optimal
phenomenon can appear for $\alpha>0$ as well as $\alpha\leq 0$. Furthermore,
one case is worthy to be stressed: for the targeted case, there exists a worst
region in which the social cost is the maximal as $p\in(0.03,0.08)$ and
$\alpha\in[-5,-2]$.

Figure 7: (Color online) Comparison between the random subsidy policy and the
targeted subsidy policy by considering an influenza case on an email network.
(a) the final epidemic size $R$ as a function of $\alpha$; (b) the vaccination
coverage $V$ as a function of $\alpha$; (c) the final epidemic size $R$ as a
function of ($\alpha,p$) for the random case; (d) the final epidemic size $R$
as a function of ($\alpha,p$) for the targeted case; (e) the social cost $SC$
as a function of ($\alpha,p$) for the random case; (f) the social cost $SC$ as
a function of ($\alpha,p$) for the targeted case. The detailed definitions and
the values of the other parameters are given in the main text.

##  V Conclusions

Targeted immunization strategies have proven to be an efficient measure to
control the spread of epidemics on complex networks with heterogenous degree
distribution Pastor-Satorras and Vespignani (2002); Zhang  _et al._ (2010,
2014b). However, the strategy assumes that the vaccination campaign is
compulsively enforced and the public resources required for vaccination are
sufficient. A more realistic case is that the public resources are limited and
the government or health organizations can only subsidize a small proportion
of individuals — with the remaining individuals left to decide to vaccinate or
not based on their self-interest Bhattacharyya and Bauch (2012). In this
situation, we want to know whether the targeted subsidy policy is still better
than the random subsidy policy. In this paper, by introducing a preferential
selection rule into the epidemiological model, and via a game-theoretic
framework, we have studied how the preferential selection rule affects the
impact of the targeted subsidy policy and the random subsidy policy on the
vaccination coverage, the final epidemic size as well as the social cost. Our
findings have shown that, when the selection parameter $\alpha>0$, the
targeted subsidy policy can produce positive effects on controlling the
spreading of epidemic since these subsidized hubs’ vaccination strategy is
more likely to be imitated by others. Otherwise, owing to the facts that, on
the one hand, the targeted subsidy policy _indirectly_ reduces the risk of the
remaining nodes and yields the lower vaccination willingness of these nodes,
on the other hand, the non-subsidized nodes who are unwilling to take
vaccination are more likely to be imitated when the selection parameter
$\alpha\leq 0$, the combing effects cause the control effect of the targeted
policy is worse than the random case. In particular, the introduction of the
targeted policy may increase but not reduce the final epidemic size as
$\alpha<0$. When considering the impacts of the two subsidy policies on social
cost, we found that, for the random subsidy policy, there exists an optimal
value of subsidy proportion $p$ leading to the minimal social cost when the
selection parameter $\alpha<0$. However, for the targeted case, there always
exists an optimal value of $p$ inducing the minimal social cost irrespective
of whether $\alpha\leq 0$ or $\alpha>0$, and the optimal value of $p$ becomes
larger when the value of $\alpha$ is gradually decreased. To illustrate this
model we provide an example based on realistic spread of influenza. The
results imply that, on the one hand, how to distribute limited public resource
to yield the best outcomes sensitively depends on human behavioral responses;
whilst on the other hand, more subsidized individuals do not necessarily
guarantee better control effect.

## Acknowledgments

This work is funded the National Natural Science Foundation of China (Grant
Nos. 61473001, 91324002, 11331009) and HZ is partially supported by the
Doctoral Research Foundation of Anhui University (Grant No. 02303319). MS is
funded by an Australian Research Council Future Fellowship (FT110100896).

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# Twisted Edge Laplacians on finite graphs from a Kähler structure

Soumalya Joardar  Department of Mathematics and Statistics, Indian Institute
of Science Education and Research Kolkata, Mohanpur - 741246, West Bengal,
India<EMAIL_ADDRESS>and  Atibur Rahaman  Department of Mathematics
and Statistics, Indian Institute of Science Education and Research Kolkata,
Mohanpur - 741246, West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

This is a continuation of the work done by the authors in [Sou_Ati1]. In
[Sou_Ati1] an almost complex structure on finitely many points from bidirected
polygon was introduced. In this paper we study a Kähler structure on finite
points. In particular, we study the edge Laplacian of a graph twisted by the
Kähler structure introduced in this paper. We also discuss a metric aspect
from a twisted holomorphic Dolbeault-Dirac spectral triple and show that the
points have a finite diameter with respect to Connes’ distance.

###### 1991 Mathematics Subject Classification:

46L87, 05C25

##  1\. Introduction

One of the exciting problems in recent times in the context of noncommutative
geometry has been to extend the classical complex geometric ideas into the
realm of noncommutative geometry. Although the interplay between the algebraic
and analytic aspects of noncommutative complex geometry is far from well
understood, the study of noncommutative complex geometry has gained some
momentum in recent years. The axioms have been mainly made keeping the quantum
flag manifolds in sight. The reader can see [Kahler_Buachalla, Somberg, Beggs]
for familiarizing himself/herself with such axioms and set up. Although it is
worth mentioning that there are other examples and approaches to
noncommuatative complex geometry (see [Khalkhali-NCGQspace, Schwartz, Staford,
NCTori] for example). After setting up basic complex geometric notions in the
realm of the noncommutative geometry, keeping the success of the Kähler
structure in classical complex geometry in mind, the noncommutative Kähler
structure has been explored mainly in the context of quantum flag manifolds by
various authors (see [Kahler_Buachalla, Das_Kahler, Kodaira] among many
others). It is worth mentioning that whether metrics have a key role to play
in noncommutative complex geometry is still not very clear. The study of
noncommutative complex geometry being in its relatively early days, it has
become imperative to construct new examples to test any hypothesis. The
authors of this paper have recently explored an almost complex structure on
finitely many points in [Sou_Ati1]. There it has been established that given a
bidirected graph over finitely many points, there is a natural almost complex
structure associated to it. Furthermore such almost complex structure is
naturally integrable. Then a concrete example of such an almost complex
structure was studied where the bomodule of one-forms came from a bidirected
polygon. In this paper we continue exploring the almost complex structure and
we introduce a Kähler structure. But we don’t discuss the Hodge theory here.
Rather we study a twisted holomorphic Dolbeault-Dirac operator on a
holomorphic hermitian module and an associated twisted edge Laplacian in the
spirit of [Das_Kahler]. At this point it is worth mentioning that an almost
complex structure has also been explored on a three point space using the
spectral triple formalism in [debashish]. But unfortunately there is no
compatible Kähler structure in their set up. Coming back to the twisted
Laplacian, the twist arises from a left module on finite points which comes
from a directed graph on the points. It is shown that the left module coming
from a directed graph is a finitely generated projective module admitting a
canonical hermitian structure (see Lemma 3.3). A general holomorphic structure
on such a module is indexed by a class of left module maps (see the discussion
following Corollary 3.4). one can restrict the Laplacian on the corresponding
left module and the left module being the function space over the edges, can
rightfully termed as a twisted edge Laplacian on the corresponding graph. We
explore the spectrum of such a twisted edge Laplacian for regular graphs, in
particular regular $k$-gons. Then we go on to consider a twisted holomorphic
Dolbeaut-Dirac operator coming from a holomorphic hermitian module. Due to the
finite dimensionality of the Hilbert spaces involved the adjointabilty of such
a Dirac operator (any other such analytical issue) does not pose any problem
and we are content with considering Connes’ distance between points coming
from the twisted Dolbeault Dirac spectral triple. In particular we show that
$n$-many points have finite diameter$\leq[\frac{n}{2}]$ (Corollary 5.2).

##  2\. Kähler structure on finite points from polygon graph

We start off by briskly recalling the rudiments of noncommutative complex
geometry. We mostly follow the definitions and conventions of [Somberg,
Sou_Ati1, Kahler_Buachalla, Kodaira].

######  Definition 2.1.

([Khalkhali]) Let ${\cal A}$ be a $\ast$-algebra over the field of complex
numbers $\mathbb{C}$. Then a $\ast$-differential calculus over ${\cal A}$ is a
datum $(\Omega^{\bullet}({\cal A}),\textup{d},\ast)$ where

  1. (i)

$\Omega^{\bullet}({\cal A})=\oplus_{n\geq 0}\Omega^{n}({\cal A})$ is a graded
$\ast$-algebra such that $\Omega^{\bullet}({\cal A})$ is an ${\cal A}$–${\cal
A}$-bimodule with $\Omega^{0}({\cal A})={\cal A}$;

  2. (ii)

$\textup{d}:\Omega^{\bullet}({\cal A})\rightarrow\Omega^{\bullet+1}({\cal A})$
is a graded derivation i.e. d is a $\mathbb{C}$-linear map satisfying the
graded Leibniz rule

$\textup{d}(\omega\wedge\eta)=\textup{d}\omega\wedge\eta+(-1)^{{\rm
deg}(\omega)}\omega\wedge\textup{d}\eta,$

for $\omega\in\Omega^{n}({\cal A}),\eta\in\Omega^{\bullet}({\cal A})$ and
$\textup{d}^{2}=0$;

  3. (iii)

for all $\omega\in\Omega^{\bullet}({\cal A})$,
$\textup{d}(\omega^{\ast})=(\textup{d}\omega)^{\ast}$.

######  Remark 2.2.

As usual practice, we have denoted the product of the graded product by
$\wedge$ and with respect to the product $\wedge$, $\ast$ is antilinear graded
involution meaning that

$(\omega\wedge\eta)^{\ast}=(-1)^{{\rm deg}(\omega){\rm
deg}(\eta)}\eta^{\ast}\wedge\omega^{\ast}.$

######  Definition 2.3.

Let $(\Omega^{\bullet}({\cal A}),\textup{d},\ast)$ be a $\ast$-differential
calculus over a $\ast$-algebra ${\cal A}$. An almost complex structure on
$(\Omega^{\bullet}({\cal A}),\textup{d},\ast)$ is a degree zero derivation
$J:\Omega^{\bullet}({\cal A})\rightarrow\Omega^{\bullet}({\cal A})$ such that

  1. 1.

$J$ is identically zero on ${\cal A}$ and hence an ${\cal A}$–${\cal
A}$-bimodule endomorphism of $\Omega^{\bullet}({\cal A})$;

  2. 2.

$J^{2}=-{\rm Id}$ on $\Omega^{1}({\cal A})$; and

  3. 3.

$J(\omega^{\ast})=(J\omega)^{\ast}$ for all $\omega\in\Omega^{1}({\cal A})$.

Given an almost complex structure on a $\ast$-differential calculus, the
bimodule of one-forms has the following decomposition into eigenspaces of $J$:

$\Omega^{1}({\cal A})=\Omega^{1,0}({\cal A})\oplus\Omega^{0,1}({\cal A}),$

where $\Omega^{1,0}({\cal A}):=\\{\omega\in\Omega^{1}({\cal
A}):J(\omega)=i\omega\\}$, $\Omega^{0,1}:=\\{\omega\in\Omega^{1}({\cal
A}):J(\omega)=-i\omega\\}$. $\Omega^{1,0}({\cal A})$ and $\Omega^{0,1}({\cal
A})$ are again ${\cal A}$–${\cal A}$-bimodules and called the bimodules of
$(1,0)$ and $(0,1)$-forms respectively. More generally, one has the higher
dimensional analogues of the $(1,0)$ and $(0,1)$-forms. For a fixed $n$, the
bimodule $\Omega^{n}({\cal A})$ has the following decomposition into bimodules
of $(p,q)$-forms for $p+q=n$, where $p,q$ are non negative
integers([Beggs]*Sec. 2.5):

$\Omega^{n}({\cal A})=\bigoplus_{p+q=n}\Omega^{p,q}({\cal A}),$

where $\Omega^{p,q}({\cal A}):=\\{\omega\in\Omega^{n}({\cal
A}):J(\omega)=i(p-q)\omega\\}$. In the following definition we are going to
use the Lefschetz operators. For the definition of the Lefschetz operators the
reader is referred to [Kahler_Buachalla] (see Definition 4.1).

######  Definition 2.4.

Given an almost complex structure $(\Omega^{\bullet}({\cal
A}),\textup{d},\ast,J)$, a Kähler form $\kappa$ is a central closed $(1,1)$
form such that with respect to Lefschetz map $L:\Omega^{\bullet}({\cal
A})\rightarrow\Omega^{\bullet}({\cal A})$ given by
$\omega\mapsto\kappa\wedge\omega$, isomorphisms are given by

$L^{n-k}:\Omega^{k}\rightarrow\Omega^{2n-k},\ for\ all\ 0\leq k<n.$

Given a Kähler-structure with the associated Kähler-form $\kappa$, one can
define the Hodge $\ast$-map $\ast_{\kappa}$ which we do not recall here. The
reader is referred to Section 2.3 of [Das_Kahler] for details of the Hodge
$\ast$-map on a Kähler structure. Now a metric can be defined using the Hodge
star map.

######  Definition 2.5.

The canonical Hermitian metric associated to the Kähler structure
$(\Omega^{\bullet}({\cal A}),\textup{d},J,\kappa)$ is the unique map
$g_{\kappa}:\Omega^{\bullet}({\cal A})\times\Omega^{\bullet}({\cal
A})\rightarrow{\cal A}$ such that $g_{\kappa}(\Omega^{k},\Omega^{l})=0$ for
$k\neq l$ and

$g_{\kappa}(\omega,\eta)=\ast_{\kappa}(\omega\wedge\ast_{\kappa}(\eta^{\ast})).$

The Hermitian metric $g_{\kappa}$ is said to be positive definite if
$g_{\kappa}(\omega,\omega)$ belongs to the positive cone ${\cal A}^{+}:={\rm
Sp}\\{a^{\ast}a:a\in{\cal A}\\}$ for all $\omega\in\Omega^{\bullet}({\cal
A})$.

We are interested in a Kähler structure on finitely many points. For that
purpose we recall the almost complex structure coming from the bidirected
polygon on $n$-many vertices from [Sou_Ati1]. We follow the notations and
conventions as in [Sou_Ati1]. So we label the vertices by natural numbers
modulo $n$. We denote the function on $E$ which takes the value $1$ on an edge
$\mu\rightarrow\nu$ and zero on the rest of the vertices by
$\xi_{\nu\rightarrow\nu}$. Then recall that with respect to a canonical
prolongation procedure using a bimodule map $\sigma$, the differential
calculus coming from the polygon is a $2$-dimensional orientable calculus. It
has an almost complex structure which determines the following spaces of
forms:

$\displaystyle\Omega^{1,0}({\cal A})={\rm
Sp}_{\mathbb{C}}\\{\xi_{\mu\rightarrow\mu+1}\\}_{\mu=1,2,\ldots,n}$
$\displaystyle\Omega^{0,1}({\cal A})={\rm
Sp}_{\mathbb{C}}\\{\xi_{\mu+1\rightarrow\mu}\\}_{\mu=1,2,\ldots,n}$
$\displaystyle\Omega^{2,0}({\cal A})=\Omega^{0,2}({\cal A})=0$
$\displaystyle\Omega^{1,1}({\cal A})={\rm
Sp}_{\mathbb{C}}\\{\xi_{\mu\rightarrow\mu-1}\wedge\xi_{\mu-1\rightarrow\mu}\\}_{\mu=1,2,\ldots,n}.$

######  Lemma 2.6.

On the almost complex structure on finitely many points coming from the
bidirected polygon graph the form $\kappa=i\sum_{\mu\in
V}\xi_{\mu\rightarrow\mu-1}\wedge\xi_{\mu-1\rightarrow\mu}$ is a Kähler form.

###### Proof.

By construction $\kappa$ is an $(1,1)$-form and central. As $\Omega^{3}({\cal
A})=0$, $\kappa$ is closed i.e. $d\kappa=0$. Recalling that for any edge
$\mu\rightarrow\nu$, $\xi_{\mu\rightarrow\nu}^{\ast}=-\xi_{\nu\rightarrow\mu}$
and $\xi_{\mu\rightarrow\mu-1},\xi_{\mu-1\rightarrow\mu}\in\Omega^{1}({\cal
A})$, a straightforward checking establishes that $\kappa$ is real i.e.
$\kappa^{\ast}=\kappa$. As for the isomorphisms
$L_{\kappa}^{n-k}:\Omega^{k}\rightarrow\Omega^{2n-k}$, as $n=1$, there is only
one isomorphism to settle viz. $L_{\kappa}:{\cal A}\rightarrow\Omega^{2}({\cal
A})$ given by $f\mapsto f\kappa$. But this is contained in the proof of
Theorem 3.2 of [Sou_Ati1]. This establishes that $\kappa$ is a Kähler form.
$\Box$

Now working through various definitions from [Kahler_Buachalla] (in particular
Definition 4.11), one can write down the formulae for the Hodge $\ast$-map on
forms. This is relatively easy and we collect them in the following lemma
leaving the straightforward proof to the reader. The reader is referred to
[Kahler_Buachalla] (see Definition 4.2) for the definition of the spaces of
primitive forms $P^{(a,b)}$ which are defined using the Lefschetz map $L$.

######  Lemma 2.7.

With the notations of [Kahler_Buachalla], the spaces $P^{(a,b)}$ are given by
$P^{0}={\cal A}$, $P^{1,0}=\Omega^{1,0}({\cal A})$,
$P^{0,1}=\Omega^{0,1}({\cal A})$, $P^{2}=0$. The Hodge $\ast$-map
$\ast_{\kappa}$ is given by the following formulae:

(2.1) $\displaystyle\ast_{\kappa}(f)=L_{\kappa}(f)=f\kappa\ for\ f\in{\cal A}$
(2.2)
$\displaystyle\ast_{\kappa}(L_{\kappa}^{0}(\omega))=\ast_{\kappa}(\omega)=-i\omega\
for\ \omega\in\Omega^{1,0}({\cal A})$ (2.3)
$\displaystyle\ast_{\kappa}(L_{\kappa}^{0}(\omega))=\ast_{\kappa}(\omega)=i\omega\
for\ \omega\in\Omega^{0,1}({\cal A})$ (2.4)
$\displaystyle\ast_{\kappa}(\omega)=f\ for\ \omega\in\Omega^{2}({\cal A})\
with\ L_{\kappa}^{-1}(\omega)=f.$

######  Lemma 2.8.

The canonical Hermitian metric $g_{\kappa}$ corresponding to the Kähler form
$\kappa$ is positive definite.

###### Proof.

Clearly it suffices to show that for any $\omega\in\Omega^{k}({\cal A})$,
$g_{\kappa}(\omega,\omega)\in{\cal A}_{+}$ for $k=0,1,2$. For $k=0$ and
$f\in{\cal A}$,
$g_{\kappa}(f,f)=\ast_{\kappa}(f\wedge\ast_{\kappa}(f^{\ast}))=|f|^{2}\in{\cal
A}^{+}$. For an $\omega=\sum
c_{\mu}\xi_{\mu\rightarrow\mu+1}\wedge\xi_{\mu+1\rightarrow\mu}\in\Omega^{1,1}({\cal
A})$,

$\displaystyle g_{\kappa}(\omega,\omega)$ $\displaystyle=$
$\displaystyle\ast_{\kappa}(\omega\wedge\ast_{\kappa}(\omega^{\ast}))$
$\displaystyle=$ $\displaystyle f,$

where $f(\mu)=|c_{\mu}|^{2}$ so that $f\in{\cal A}^{+}$. As
$\Omega^{2,0}({\cal A})=\Omega^{0,2}({\cal A})=0$, we have
$g_{\kappa}(\omega,\omega)\in{\cal A}^{+}$ for all $\omega\in\Omega^{2}({\cal
A})$.
Now note that for any $\omega\in\Omega^{1,0}({\cal A})$ and
$\eta\in\Omega^{0,1}({\cal A})$,
$g_{\kappa}(\omega,\eta)=0=g_{\kappa}(\eta,\omega)$. With this observation for
one forms it is enough to show that $g_{\kappa}(\omega,\omega)\in{\cal A}^{+}$
for $\omega\in\Omega^{1,0}({\cal A})$ or $\omega\in\Omega^{0,1}({\cal A})$. So
let $\omega=\sum{c_{\mu}}\xi_{\mu\rightarrow\mu+1}\in\Omega^{1,0}({\cal A})$
where $c_{\mu}\in\mathbb{C}$,

$\displaystyle g_{\kappa}(\omega,\omega)$ $\displaystyle=$
$\displaystyle\ast_{\kappa}(\omega\wedge\ast_{\sigma}(\omega^{\ast}))$
$\displaystyle=$
$\displaystyle\ast_{\kappa}(-i\sum|c_{\mu}|^{2}\xi_{\mu\rightarrow\mu+1}\wedge\xi_{\mu+1\rightarrow\mu})$
$\displaystyle=$ $\displaystyle f,$

where $f(\mu)=|c_{\mu}|^{2}$ and therefore $g_{\kappa}(\omega,\omega)\in{\cal
A}^{+}$. For $\omega=\sum
c_{\mu}\xi_{\mu\rightarrow\mu-1}\in\Omega^{0,1}({\cal A})$,

$\displaystyle g_{\kappa}(\omega,\omega)$ $\displaystyle=$
$\displaystyle\ast_{\kappa}(\omega\wedge\ast_{\kappa}(\omega^{\ast}))$
$\displaystyle=$
$\displaystyle\ast_{\kappa}(i\sum|c_{\mu}|^{2}\xi_{\mu\rightarrow\mu-1}\wedge\xi_{\mu-1\rightarrow\mu})$
$\displaystyle=$ $\displaystyle f,$

where $f(\mu)=|c_{\mu}|^{2}$ and therefore $g_{\kappa}(\omega,\omega)\in{\cal
A}^{+}$ for all $\omega\in\Omega^{1}({\cal A})$. $\Box$

##  3\. The twisted edge Laplacian

In this section we shall consider Hermitian holomorphic modules over finitely
many points. In particular we shall focus on such modules arising from finite
directed graphs on finitely many points. To that end we start by recalling the
definitions of holomorphic and hermitian modules over a $\ast$-algebra.

######  Definition 3.1.

Let ${\cal A}$ be a $\ast$-algebra admitting an almost complex structure. A
holomorphic module over ${\cal A}$ is a pair
$(\cal{E},\bar{\partial}_{\cal{E}})$ where $\cal{E}$ is a finitely generated
projective left module over ${\cal A}$ and
$\bar{\partial}_{\cal{E}}:\cal{E}\rightarrow\Omega^{0,1}({\cal A})\bar{\
\otimes\ }\cal{E}$ is a flat $(0,1)$-connection.

In the following definition and throughout the rest of the paper, for a left
module $\cal{E}$ over an algebra ${\cal A}$, we denote the set of all left
module maps from $\cal{E}\rightarrow{\cal A}$ by
${}^{\vee}\\!\\!\\!\mathcal{E}$ which has a natural right module structure
(see [Das_Kahler]). For the next definition the reader is urged to consult
section 2.6 of [Das_Kahler] for the definition (as well as the right module
structure) of the conjugate module $\bar{\cal{E}}$.

######  Definition 3.2.

A hermitian module over an algebra ${\cal A}$ is a pair
$(\cal{E},h_{\cal{E}})$ where $\cal{E}$ is a finitely generated projective
left module and a right ${\cal A}$-isomorphism
$h_{\cal{E}}:\bar{\cal{E}}\rightarrow^{\vee}\\!\\!\\!\mathcal{E}$ where
$\bar{\cal{E}}$ is the conjugate right ${\cal A}$-module such that for the
associated sesquilinear pairing
$h_{\cal{E}}(-,-):\cal{E}\times\cal{E}\rightarrow{\cal A}$ given by
$(e,k)\mapsto h_{\cal{E}}(\bar{k})(e)$ satisfies

$h_{\cal{E}}(e,k)=h_{\cal{E}}(k,e)^{\ast}\quad\text{ and }\quad
h_{\cal{E}}(e,e)\in{\cal A}^{+}.$

Given a holomorphic, hermitian module over a $\ast$-algebra ${\cal A}$
admitting a Kähler structure $(\Omega^{\bullet}({\cal
A}),\textup{d},J,\kappa)$, recall the map $g_{\cal{E}}:\Omega^{\bullet}({\cal
A})\bar{\ \otimes\ }\cal{E}\times\Omega^{\bullet}({\cal A})\bar{\ \otimes\
}\cal{E}\rightarrow{\cal A}$ from [Das_Kahler] (see discussion after
Definition 2.3). If the $\ast$-algebra ${\cal A}$ admits a faithful state
$\tau:{\cal A}\rightarrow\mathbb{C}$, then there is a $\mathbb{C}$-valued
inner product $\langle-,-\rangle_{\cal{E}}$ on $\Omega^{\bullet}({\cal
A})\bar{\ \otimes\ }\cal{E}$ given by $\tau\circ g_{\cal{E}}$ provided
$g_{\cal{E}}$ is positive definite.

######  Lemma 3.3.

Every left module coming from a finite directed graph on finite points is a
finitely generated, projective Hermitian module. Moreover, the Hermitian
structure is positive definite.

###### Proof.

Let us denote the left $C(V)$-module $C(E)$ by $\cal{E}$. For any finite graph
$(V,E)$, we denote the function on $E$ which takes the value $1$ on a
particular edge $e\in E$ and $0$ on other edges by $\xi_{e}$. Then
$\\{\xi_{e}\\}_{e\in E}$ forms a $\mathbb{C}$-linear basis of $C(E)$. As
$\cal{E}$ is a finite dimensional complex vector space and therefore finitely
generated as a left module. To prove that $\cal{E}$ is projective, note that
the by Prposition 3.2 of [Majid] complete graph on the vertex set $V$ is a
free left module over $C(V)$. If the number of vertices is $n$, let us denote
the complete graph on $V$ by $K_{n}$ and the corresponding edge set by
$E_{n}$. We define a surjective idempotent $I:C(E_{n})\rightarrow\cal{E}$ by
the following formula:

$I(\sum_{e\in E_{n}}\lambda_{e}\xi_{e})=\sum_{e^{\prime}\in
E}\lambda_{e^{\prime}}\xi_{e^{\prime}}.$

Clearly $I$ is a left module morphism and $I(C(E_{n}))\cong\cal{E}$ as left
modules and $C(E_{n})\cong\cal{E}\oplus(1-I)(C(E_{n}))$, proving that
$\cal{E}$ is a projective module. Now we shall define a canonical Hermitian
structure on $\cal{E}$. To that end let us define a right $C(V)$-module
isomorphism $h:\bar{\cal{E}}\rightarrow^{\vee}\\!\\!\\!\mathcal{E}$ satisfying
the conditions Definition 3.2. We claim that ${}^{\vee}\\!\\!\\!\mathcal{E}$
is a finite dimensional vector space with dimension equal to the number of
edges in the graph $(V,E)$. For $e\in E$ and
$\Phi\in^{\vee}\\!\\!\\!\mathcal{E}$, we have

$\Phi(\xi_{e})=\sum_{v\in V}\lambda_{v}\delta_{v}.$

For any $w\neq s(e)$, $\delta_{w}\xi_{e}=0$ and consequently
$\Phi(\delta_{w}\xi_{e})=0$. On the other hand, as $\Phi$ is a left
$C(V)$-module map,

$\displaystyle\Phi(\delta_{w}\xi_{e})$ $\displaystyle=$
$\displaystyle\delta_{w}\Phi(\xi_{e})$ $\displaystyle=$
$\displaystyle\delta_{w}\sum_{v\in V}\lambda_{v}\delta_{v}$ $\displaystyle=$
$\displaystyle\lambda_{w}\delta_{w}.$

Therefore for any vertex $w\neq s(e)$, $\lambda_{w}=0$. Hence
$\Phi(\xi_{e})=\lambda_{s(e)}\delta_{s(e)}$ for some
$\lambda_{s(e)}\in\mathbb{C}$. For each $e\in E$, we define
$\Phi^{e}\in^{\vee}\\!\\!\\!\mathcal{E}$ by the following:

$\displaystyle\Phi^{e}(\xi_{f}):=\delta_{e,f}\delta_{s(e)}.$

Then it can be shown that $\\{\Phi^{e}:e\in E\\}$ is a $\mathbb{C}$-linear
basis for ${}^{\vee}\\!\\!\\!\mathcal{E}$. Now we define
$h:\bar{\cal{E}}\rightarrow^{\vee}\\!\\!\\!\mathcal{E}$ by
$h(\bar{\xi_{e}})=\Phi^{e}$. Clearly this is a vector space isomorphism. To
show that this is a right $C(V)$-isomorphism, it suffices to show that $h$ is
a right $C(V)$-linear map. To that end let $\phi\in C(V)$. Then
$h(\bar{\xi_{e}}.\phi)=h(\bar{\phi^{\ast}\xi_{e}})=\phi(s(e))\Phi^{e}$. But
for any $f\in E$, $(\Phi^{e}.\phi)(\xi_{f})=\delta_{e,f}\phi\delta_{s(e)}$. On
the other hand,

$\phi(s(e))\Phi^{e}(\xi_{f})=\delta_{e,f}\phi(s(e))\delta_{s(e)},$

and therefore

$h(\bar{\xi_{e}}.\phi)=\Phi^{e}.\phi=h(\bar{\xi_{e}}).\phi.$

Let $\xi=\sum_{e\in E}\lambda_{e}\xi_{e},\ \eta=\sum_{e\in E}\mu_{e}\xi_{e}\in
C(E)$. Then by definition of the seequiliniear pairing $h_{\cal{E}}(-,-)$,

$\displaystyle h_{\cal{E}}(\xi,\eta)$ $\displaystyle:=$ $\displaystyle
h(\bar{\eta})(\xi)$ $\displaystyle\ =$
$\displaystyle\sum_{e}\bar{\mu_{e}}\lambda_{e}\delta_{s(e)}.$

Now it is clear that $h_{\cal{E}}(\xi,\eta)=h_{\cal{E}}(\eta,\xi)^{\ast}$ and
$h_{\cal{E}}(\xi,\xi)=\sum_{e\in E}|\lambda_{e}|^{2}\delta_{s(e)}\in
C(V)^{+}$. $\Box$

Combining Lemma 2.8 and Lemma 3.3 with Proposition 5.11 of [Kodaira], we get
the following

######  Corollary 3.4.

The Hermitian metric $g_{\cal{E}}$ is positive definite.

Recall the canonical faithful state $\tau$ on $C(V)$ given by
$\tau(f):=\frac{1}{n}\sum_{\mu\in V}f(\mu)$. Then we have the
$\mathbb{C}$-valued inner product (see [Kodaira]) $\langle,\rangle_{\cal{E}}$
on $(\Omega^{\bullet}({\cal A})\bar{\ \otimes\ }\cal{E})$ given by $\tau\circ
g_{\cal{E}}$.
Now we shall consider holomorphic modules over $C(V)$ coming from finite
directed graph ${\cal G}=(V,E)$. We denote the left module coming from any
finite directed graph over the vertex set $V$ by $\cal{E}$. We consider the
holomorphicity with respect to the almost complex structure on $V$ coming from
the bidirected polygon over $V$. Note that since $\cal{E}$ is finitely
generated projective and $\Omega^{0,2}({\cal A})=0$, any left connection
$\bar{\partial}_{\cal{E}}:\cal{E}\rightarrow\Omega^{0,1}({\cal A})\bar{\
\otimes\ }\cal{E}$ gives a holomorphic module
$(\cal{E},\bar{\partial}_{\cal{E}})$ canonically. Recall that a left
connection $\bar{\partial}_{\cal{E}}$ is a $\mathbb{C}$-linear map from
$\cal{E}$ to $\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E}$ such that

$\bar{\partial}_{\cal{E}}(a.\omega)=\bar{\partial}_{\cal{E}}(a)\ \otimes\
\omega+a\bar{\partial}_{\cal{E}}(\omega)$

for $a\in{\cal A}$ and $\omega\in\cal{E}$. Also recall the element
$\theta=\sum_{\mu\in V}\xi_{\mu+1\rightarrow\mu}\in\Omega^{0,1}({\cal A})$
from [Sou_Ati1]. Then we have a base $(0,1)$-connection $\nabla_{0}$ on
$\cal{E}$ given by the following:

$\nabla_{0}(\omega)=\theta\ \otimes\ \omega,\ \omega\in\cal{E}.$

Any $(0,1)$-connection $\bar{\partial}_{\cal{E}}$ is determined by a left
module map $\zeta:\cal{E}\rightarrow\Omega^{0,1}({\cal A})\bar{\ \otimes\
}\cal{E}$ and can be written as

(3.1) $\displaystyle\bar{\partial}_{\cal{E}}=\nabla_{0}+\zeta.$

The $\mathbb{C}$-linear basis for $\Omega^{0,1}({\cal A})$ is given by
$\\{\xi_{\mu\rightarrow\mu-1}\\}_{\mu\in V}$ whereas we shall denote the
$\mathbb{C}$-linear basis of $\cal{E}$ by
$\\{\chi_{\mu\rightarrow\nu}\\}_{\mu\rightarrow\nu\in E}$. With these
notations, for a left-module map $\zeta$, there are scalars
$c_{\nu,\nu^{\prime}}^{\mu}\in\mathbb{C}$ such that

$\zeta(\chi_{\mu\rightarrow\nu})=\sum_{\mu,\nu,\nu^{\prime}\in
V:\mu-1\rightarrow\nu^{\prime}\in
E}c^{\mu}_{\nu,\nu^{\prime}}\xi_{\mu\rightarrow\mu-1}\ \otimes\
\chi_{\mu-1\rightarrow\nu^{\prime}}.$

As $\Omega^{0,2}({\cal A})\bar{\ \otimes\ }\cal{E}=0$, extension of the
operator $\bar{\partial}_{\cal{E}}$ on the Hilbert space
$L^{2}(\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E})$ becomes the zero map.
Thus we consider the operator $\bar{\partial}_{\cal{E}}$ as a
$\mathbb{C}$-linear operator in the hilbert space ${\cal
H}=L^{2}(\cal{E})\oplus L^{2}(\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E})$
where the hilbert space is equipped with the inner product
$\langle,\rangle_{\cal{E}}$. We quickly note the following lemma whose proof
is straightforward verification using the definition of the inner product
$\langle,\rangle_{\cal{E}}$ and is ommitted.

######  Lemma 3.5.

The hilbert space ${\cal H}$ has the following orthonormal basis:

(3.2) $\displaystyle\\{\sqrt{n}\chi_{\theta\rightarrow\nu}\\}_{\theta,\nu\in
V:\theta\rightarrow\nu\in E},\\{\sqrt{n}\xi_{\lambda+1\rightarrow\lambda}\
\otimes\ \chi_{\lambda\rightarrow\nu}\\}_{\lambda,\nu:\lambda\rightarrow\nu\in
E}.$

As all the modules involved here are finite dimensional vector spaces, the
analytic technicality of whether $\bar{\partial}_{\cal{E}}$ is adjointable or
not does not arise so that we can write down the adjoint of the map
$\bar{\partial}_{\cal{E}}$. We denote the adjoint by
$\bar{\partial}_{\cal{E}}^{\dagger}$. Then using the facts that
$\partial_{\cal{E}}^{2}=0=\partial_{\cal{E}}^{\dagger 2}$ and
$\partial_{\cal{E}}^{\dagger}$ is zero on $\cal{E}$, the twisted Laplacian
${\cal L}$ on $\cal{E}$ is given by

${\cal
L}:=(\bar{\partial}_{\cal{E}}+\bar{\partial}_{\cal{E}}^{\dagger})^{2}=\bar{\partial}_{\cal{E}}^{\dagger}\bar{\partial}_{\cal{E}}.$

As $\bar{\partial}_{\cal{E}}^{\dagger}=\nabla_{0}^{\dagger}+\zeta^{\dagger}$,
we shall determine $\nabla_{0}^{\dagger}$ and $\zeta^{\dagger}$ separately.

######  Lemma 3.6.

For any basis element $\chi_{\mu\rightarrow\nu}\in\cal{E}$,

(3.3) $\displaystyle\nabla_{0}^{\dagger}(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})=\chi_{\mu\rightarrow\nu}.$

###### Proof.

For $\chi_{\mu\rightarrow\nu}\in\cal{E}$,

$\displaystyle\langle\nabla_{0}(\chi_{\mu\rightarrow\nu}),(\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}$
$\displaystyle\qquad=\langle(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu}),(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}$ $\displaystyle\qquad=\tau\circ
g_{\cal{E}}\Big{(}(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu}),(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})\Big{)}$
$\displaystyle\qquad=\tau\Big{(}\ast_{\kappa}(\xi_{\mu+1\rightarrow\mu}\wedge
h(\bar{\chi}_{\mu\rightarrow\nu})(\chi_{\mu\rightarrow\nu})\wedge\ast_{\kappa}(\xi_{\mu+1\rightarrow\mu})^{\ast})\Big{)}$
$\displaystyle\qquad=\tau\Big{(}\ast_{\kappa}(\xi_{\mu+1\rightarrow\mu}\delta_{\mu}\wedge(i\xi_{\mu\rightarrow\mu+1})\Big{)}$

But $\ast_{\kappa}(i\xi_{\mu+1\rightarrow\mu}\wedge\xi_{\mu\rightarrow\mu+1})$
is the function which takes the value $1$ on the vertex $\mu$ and $0$ on the
rest of the vertices. Therefore

$\langle\nabla_{0}(\chi_{\mu\rightarrow\nu}),(\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}=\frac{1}{n}.$

For any other edge $\mu^{\prime}\rightarrow\nu^{\prime}$ than
$\mu\rightarrow\nu$, similar computation will yield

$\langle\nabla_{0}(\chi_{\mu^{\prime}\rightarrow\nu^{\prime}}),(\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}=0.$

Again by definition of $\langle,\rangle_{\cal{E}}$,
$\langle\chi_{\mu^{\prime}\rightarrow\nu^{\prime}},\chi_{\mu\rightarrow\nu}\rangle_{\cal{E}}=\delta_{\mu,\mu^{\prime}}\delta_{\nu,\nu^{\prime}}\frac{1}{n}$.
Consequently by $\mathbb{C}$-linearily of $\nabla_{0}$, for all
$\omega\in\cal{E}$,

$\langle\nabla_{0}(\omega),(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}=\langle\omega,\chi_{\mu\rightarrow\nu}\rangle_{\cal{E}}.$

Hence by non-degeneracy of $\langle,\rangle_{\cal{E}}$,
$\nabla_{0}^{\dagger}(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})=\chi_{\mu\rightarrow\nu}$. $\Box$

######  Lemma 3.7.

For any basis element $\chi_{\mu\rightarrow\nu}\in\cal{E}$,

(3.4) $\displaystyle\zeta^{\dagger}(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})=\sum\overline{c^{\mu+1}_{\nu^{\prime},\nu}}\chi_{\mu+1\rightarrow\nu^{\prime}}$

###### Proof.

Fix a vertex $\mu+1$. Then for any vertices $\nu^{\prime},\nu$ such that
$\chi_{\mu+1\rightarrow\nu^{\prime}},\chi_{\mu\rightarrow\nu}\in\cal{E}$,

$\displaystyle\langle\zeta(\chi_{\mu+1\rightarrow\nu^{\prime}}),(\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}$ $\displaystyle\qquad=\sum
c^{\mu+1}_{\nu^{\prime},\nu^{\prime\prime}}\langle\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu^{\prime\prime}},\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu}\rangle_{\cal{E}}$
$\displaystyle\qquad=c^{\mu+1}_{\nu^{\prime},\nu}\tau(\delta_{\mu})$
$\displaystyle\qquad=\frac{1}{n}c^{\mu+1}_{\nu^{\prime},\nu}.$

As for any other vertices $\theta,\theta^{\prime}$ such that $\theta\neq\mu+1$
and $\chi_{\theta\rightarrow\theta^{\prime}}\in\cal{E}$,

$\langle\zeta(\chi_{\theta\rightarrow\theta^{\prime}}),(\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu})\rangle_{\cal{E}}=0.$

But it is easy to see that

$\displaystyle\left\langle\chi_{\theta\rightarrow\nu^{\prime}},\sum\overline{c^{\mu+1}_{\nu^{\prime},\nu}}\chi_{\mu+1\rightarrow\nu^{\prime}}\right\rangle_{\cal{E}}=\begin{cases}\frac{1}{n}c^{\mu+1}_{\nu^{\prime},\nu},&\quad\theta=\mu+1\\\
0,&\quad\theta\neq\mu+1.\end{cases}$

Therefore, by non-degeneracy of the inner product $\langle,\rangle_{\cal{E}}$,
we conclude that

$\zeta^{\dagger}(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})=\sum\overline{c^{\mu+1}_{\nu^{\prime},\nu}}\chi_{\mu+1\rightarrow\nu^{\prime}}.$

$\Box$

Consequently, we have the following easy to see formulae:

(3.5) $\displaystyle\zeta^{\dagger}\zeta(\chi_{\mu\rightarrow\nu})$
$\displaystyle=$
$\displaystyle\sum_{\mu-1\rightarrow\nu^{\prime},\mu\rightarrow\nu^{\prime\prime}}c^{\mu}_{\nu,\nu^{\prime}}\overline{c^{\mu}_{\nu^{\prime\prime},\nu^{\prime}}}\chi_{\mu\rightarrow\nu^{\prime\prime}}$
(3.6) $\displaystyle\nabla_{0}^{\dagger}\nabla_{0}(\chi_{\mu\rightarrow\nu})$
$\displaystyle=$ $\displaystyle\chi_{\mu\rightarrow\nu}$ (3.7)
$\displaystyle\nabla_{0}^{\dagger}\zeta(\chi_{\mu\rightarrow\nu})$
$\displaystyle=$ $\displaystyle\sum
c^{\mu}_{\nu,\nu^{\prime}}\chi_{\mu-1\rightarrow\nu^{\prime}}$ (3.8)
$\displaystyle\zeta^{\dagger}\nabla_{0}(\chi_{\mu\rightarrow\nu})$
$\displaystyle=$
$\displaystyle\sum\overline{c^{\mu+1}_{\nu^{\prime},\nu}}\chi_{\mu+1\rightarrow\nu^{\prime}}$

Then for any $\chi_{\mu\rightarrow\nu}\in\cal{E}$,

$\displaystyle{\cal L}(\chi_{\mu\rightarrow\nu})$ $\displaystyle=$
$\displaystyle\bar{\partial}_{\cal{E}}^{\dagger}\bar{\partial}_{\cal{E}}(\chi_{\mu\rightarrow\nu})$
$\displaystyle=$
$\displaystyle(\nabla_{0}^{\dagger}+\zeta^{\dagger})(\nabla_{0}+\zeta)(\chi_{\mu\rightarrow\nu})\
$ $\displaystyle=$ $\displaystyle\chi_{\mu\rightarrow\nu}+\sum
c^{\mu}_{\nu,\nu^{\prime}}\chi_{\mu-1\rightarrow\nu^{\prime}}+\sum\overline{c^{\mu+1}_{\nu^{\prime},\nu}}\chi_{\mu+1\rightarrow\nu^{\prime}}$
$\displaystyle\qquad+$
$\displaystyle\sum_{\begin{subarray}{c}\mu-1\rightarrow\nu^{\prime},\\\
\mu\rightarrow\nu^{\prime\prime}\end{subarray}}c^{\mu}_{\nu,\nu^{\prime}}\overline{c^{\mu}_{\nu^{\prime\prime},\nu^{\prime}}}\chi_{\mu\rightarrow\nu^{\prime\prime}}.$

We may call the left module map $\zeta$ a potential for the Laplacian ${\cal
L}$ and the coefficients $c^{\mu}_{\nu,\nu^{\prime}}$ the potential
coefficients. We shall be interested in the spectrum of the twisted edge
Laplacian where all the potential coefficients are $1$. Then the Laplacian
${\cal L}$ acts on the basis elements by the following formula:

(3.11) $\displaystyle{\cal L}(\chi_{\mu\rightarrow\nu})$ $\displaystyle=$
$\displaystyle\Big{(}1+{\rm
deg}(\mu-1)\Big{)}\chi_{\mu\rightarrow\nu}+\sum_{\mu\rightarrow\nu^{\prime}\in
E:\nu\neq\nu^{\prime}}\Big{(}{\rm
deg}(\mu-1)\Big{)}\chi_{\mu\rightarrow\nu^{\prime}}$ $\displaystyle\qquad+$
$\displaystyle\sum_{\mu-1\rightarrow\nu\in
E}\chi_{\mu-1\rightarrow\nu}+\sum_{\mu+1\rightarrow\nu\in
E}\chi_{\mu+1\rightarrow\nu},$

where ${\rm deg}(\mu)$ is equal to the number of edges $e$ such that
$s(e)=\mu$. For a general $f\in C(E)$ , for an edge $e\in E$,

$\displaystyle{\cal L}(f)(e)=f(e)$ $\displaystyle+$
$\displaystyle\sum_{s(e)=s(e^{\prime})}{\rm
deg}\big{(}s(e)-1\big{)}f(e^{\prime})$ $\displaystyle+$
$\displaystyle\sum_{s(e)=s(e^{\prime\prime})+1}f(e^{\prime\prime})+\sum_{s(e)=s(e^{\prime\prime\prime})-1}f(e^{\prime\prime\prime})$

##  4\. Eigenvalues of twisted edge Laplacians: Examples

In this section we shall see a couple of examples of the twisted edge
Laplacians where the Laplacian is considered with respect to the potential
coefficients $c^{\mu}_{\nu,\nu^{\prime}}=1$. In this particular case, the
Laplacian ${\cal L}$ takes the form (3). We start with a $d$-regular graph.
1\. $d$-regular graph: Let $\mathcal{G}=(V,E)$ be a $d$-regular graph. By a
$d$-regular graph we mean that each vertex of the graph has degree $d$. We
index the vertices of $\mathcal{G}$ by $1,2,\ldots,n$. We take the following
ordered basis of $C(E)$ to write down the twisted edge Laplacian of the graph
$\mathcal{G}$:

$\\{\chi_{1\rightarrow\mu}\\}_{\mu\in\\{1,2,\ldots,n\\}},\\{\chi_{2\rightarrow\mu}\\}_{\mu\in\\{1,2,\ldots,n\\}},\ldots,\\{\chi_{n\rightarrow\mu}\\}_{\mu\in\\{1,2,\ldots,n\\}}.$

Note that for any $i=1,2,\ldots,n$, only those $\mu$’s appear in the above
list for which $i\rightarrow\mu$ is an edge. By formula (3.11), with respect
to the above basis of $C(E)$, ${\cal L}$ can be represented by a block matrix
of order $nd\times nd$ such that the diagonal blocks of of order $d$ are of
the form

$\begin{pmatrix}d+1&d&d&\cdots&d\\\ d&d+1&d&\cdots&d\\\
\vdots&\vdots&\ddots&&\vdots\\\ d&d&\cdots&&d+1\end{pmatrix}$

and $ij$-th block $B$ of order $d$ is of the form

$B=(b_{kl})=\begin{cases}b_{kl}=1\text{ for all }k,l\text{ iff i and j are
adjacent}\\\ b_{kl}=0\text{ for all }k,l\text{ otherwise}.\end{cases}$

In this case it can be seen easily that the sum of entries of each row and
column of the above matrix are equal to $(d+1)+d(d-1)+2d=(d+1)^{2}$.

######  Lemma 4.1.

For a $d$-regular graph, for any the eigenvalue $\lambda$ of the twisted edge
Laplacian ${\cal L}$, we have $0\leq\lambda\leq(d+1)^{2}$ with $(d+1)^{2}$
being an eigenvalue.

###### Proof.

As the sum of each row and each column of the matrix of ${\cal L}$ is equal to
$(d+1)^{2}$, it is trivial to see that $(d+1)^{2}$ is an eigenvalue with
eigenvector $(1,1,\ldots,1)$. To show that $|\lambda|\leq(d+1)^{2}$, we shall
apply Gershgorin circle theorem. Take any $i$-th row. Observe that each entry
of such a row is a non-negative integer (in fact $d,d+1$ or $1$). If we denote
the entries by $a_{ij}$, then $a_{ii}=(d+1)$. Therefore any eigenvalue of
${\cal L}$ lies within the disc $D(a_{ii},R_{i})$ where $R_{i}=\sum_{i\neq
j}a_{ij}=(d+1)^{2}-(d+1)=d(d+1)$. In fact for all $i$, $a_{ii}=(d+1)$ and
$R_{i}=d(d+1)$. This shows that any eigenvalue lies in the closed disc
$D((d+1),d(d+1))$ by the Gershgorin circle theorem. In particular it shows
that for any eigenvalue $\lambda$, $|\lambda|\leq(d+1)+d(d+1)=(d+1)^{2}$. Now
as ${\cal L}$ is a positive definite, $0\leq\lambda\leq(d+1)^{2}$. $\Box$

2\. directed regular $n$-gon:

$1$$2$$3$$4$$n$$5$$6$$n-1$

Directed polygon on $n$-points

######  Lemma 4.2.

The complete list of eigenvalues of the twisted edge Laplacian ${\cal L}$ is
given by

(4.1) $\displaystyle 2+2{\rm cos}\frac{2\pi j}{n},\ 0\leq j\leq n-1,$

where $\omega=e^{\frac{2\pi i}{n}}$ is a primitive $n$-th root of unity. In
particular, the eigenvalues lie between $0$ and $4$ with $4$ being an
eigenvalue. For $n=2m$, $0$ is an eigenvalue whereas for $n=2m+1$, $0$ is not
an eigenvalue of ${\cal L}$.

###### Proof.

As a regular $n$-gon is $1$-regular, by Lemma 4.1, for any eigenvalue
$\lambda$ of ${\cal L}$, $0\leq\lambda\leq(1+1)^{2}=4$ with $4$ being an
eigenvalue. In fact note that the matrix ${\cal L}$ with respect to the basis
introduced in the beginning of this section takes the following form:

$\begin{pmatrix}2&1&0&.&.&.&0&1\\\ 1&2&1&0&.&.&.&0\\\ 0&1&2&1&0&.&.&0\\\
.&0&1&2&1&.&.&0\\\ .\\\ .\\\ 1&0&.&.&.&0&1&2\end{pmatrix}$

which is an $n\times n$ circulant matrix. Let $\omega=e^{\frac{2\pi i}{n}}$.
Therefore the eigenvalues are given by, for $\ 0\leq j\leq n-1$,

$\displaystyle\lambda_{j}$ $\displaystyle=$ $\displaystyle
2+\omega^{j}+\omega^{(n-1)j}$ $\displaystyle=$ $\displaystyle 2+2{\rm
cos}\frac{2\pi j}{n}.$

Now let $n$ be an even integer say $2m$. Then for $j=m$, $\lambda=0$. Hence
$0$ is an eigenvalue of ${\cal L}$. In fact, $(1,-1,1,-1,\cdots,1,-1)^{T}$ is
an eigenvector for the eigenvalue $0$. If $n=2m+1$ for some integer $m$, it is
easy to see that the list of eigenvalues does not contain $0$. Therefore for
an odd integer $n$, the twisted edge Laplacian ${\cal L}$ of a regular
directed $n$-gon is invertible. $\Box$

##  5\. A twisted holomorphic Dolbeault Dirac spectral triple

In this section we briefly discuss Connes’ distance between the vertices of a
graph $(V,E)$ with respect to the spectral triple obtained from the twisted
holomorphic Dolbeault Dirac operator acting on the Hilbert space ${\cal
H}=L^{2}(\cal{E})\oplus L^{2}(\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E})$
where $\cal{E}$ is the left $C(V)$-module and the Hilbert space is formed with
respect to the inner product $\langle,\rangle_{\cal{E}}$ obtained in Section
3. The consideration of Dolbeault-Dirac spectral triple and its detailed study
(particularly the compact resolvent condition of the Dolbeault-Dirac operator)
in the context of quantum projective space appeared in [Das_Somberg]. It is
worth mentioning that in our case as the Hilbert space is finite dimensional,
the compact resolvent condition is automatic for our spectral triple. Recall
the twisted holomorphic Dolbeault-Dirac operator ${\cal
D}:=\bar{\partial}_{\cal{E}}+\bar{\partial}_{\cal{E}}^{\dagger}$ acting on
${\cal H}$ where $\bar{\partial}_{\cal{E}}=\nabla_{0}+\zeta$ as in Equation
number (3.1). Recall that $\nabla_{0}$ is the base $(0,1)$-connection and
$\zeta:\cal{E}\rightarrow\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E}$ is
some left module map. As the Hilbert space is finite dimensional, denoting
$C(V)$ by ${\cal A}$ as usual, $({\cal A},{\cal H},{\cal D})$ is canonically a
spectral triple. ${\cal A}=C(V)$ acts on $\cal{E}$ as well as
$\Omega^{0,1}({\cal A})\bar{\ \otimes\ }\cal{E}$ by its left module action.
Again as before, we label the vertices $V$ by natural numbers
$\mu=1,2,\ldots,n$ and adopt the convention of [Sou_Ati1]. The pure states of
${\cal A}$ correspond to the points in $V$. Let us briskly recall Connes’
distance between states on a $C^{\ast}$-algebra. If $({\cal A},{\cal H},{\cal
D})$ is a spectral triple where ${\cal A}\subset{\cal B}({\cal H})$ is a
$C^{\ast}$-algebra, then for any two states $\phi,\psi\in{\cal A}^{\ast}$, the
Connes’ distance between $\phi,\psi$ is given by the following formula (see
[Connes]):

$d(\phi,\psi)={\rm sup}_{f\in{\cal A}:||[{\cal D},f]||\leq
1}|\phi(f)-\psi(f)|.$

In the above formula we have supressed the notation of representation of
${\cal A}$ in ${\cal B}({\cal H})$ and will continue to do so throughout the
rest of the paper. It is known that Connes’ distance in particular recovers
the geodesic distance between points on a Riemannian manifold for the spinor
spectral triple. However, for a general spectral triple, Connes’ distance may
be infinite at times. In the following lemma $d$ stands for the Connes’
distance between two points seen as pure states on ${\cal A}$.

######  Lemma 5.1.

For all $\mu=1,2,\ldots,n$, $d(\mu,\mu+1)=1$ and the distance between
$\mu,\mu+1$ does not depend upon any choice of left module map $\zeta$ to
define the Dolbeault-Dirac operator ${\cal D}$.

###### Proof.

We denote both the Hilbert space norm and the operator norm by $||\cdot||$.
Note that $\\{\chi_{\mu\rightarrow\nu}\\}_{\mu\rightarrow\nu\in E}$ forms an
orthogonal set with respect to the inner product $\langle,\rangle_{\cal{E}}$.
Moreover
$\langle\chi_{\mu\rightarrow\nu},\chi_{\mu\rightarrow\nu}\rangle_{\cal{E}}=\frac{1}{n}$.
Therefore $\\{\sqrt{n}\chi_{\mu\rightarrow\nu}\\}_{\mu\rightarrow\nu\in E}$
forms an orthonormal $\mathbb{C}$-linear basis of $L^{2}(\cal{E})$ with
respect to the inner product $\langle,\rangle_{\cal{E}}$. With this
observation let $f\in{\cal A}$ such that $||[{\cal D},f]||\leq 1$. For any
$\mu\rightarrow\nu\in E$, $||\sqrt{n}\chi_{\mu\rightarrow\nu}||=1$ in ${\cal
H}$. Then $||[{\cal D},f](\sqrt{n}\chi_{\mu\rightarrow\nu})||\leq||[{\cal
D},f]||\leq 1$. Note that for any left module map
$\zeta:\cal{E}\rightarrow\Omega^{0,1}\bar{\ \otimes\ }\cal{E}$,
$[\zeta,f]=0=[\zeta^{\dagger},f]$ on appropriate subspaces. Also for any
$\chi_{\mu\rightarrow\nu}$,
$\bar{\partial}_{\cal{E}}^{\dagger}(\chi_{\mu\rightarrow\nu})=0$. With these
observations using the fact that $\langle\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu},\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu}\rangle_{\cal{E}}=\frac{1}{n}$, we get the following

$\displaystyle(\sqrt{n}\chi_{\mu\rightarrow\nu})$
$\displaystyle=\sqrt{n}\Big{(}f(\mu){\cal
D}(\chi_{\mu\rightarrow\nu})-f.(\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu})\Big{)}$
$\displaystyle=\sqrt{n}\Big{(}f(\mu)-f(\mu+1)\Big{)}\Big{(}\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu}\Big{)}.$ $\displaystyle||[{\cal
D},f](\sqrt{n}\chi_{\mu\rightarrow\nu})||$
$\displaystyle=\sqrt{\Big{\langle}}\sqrt{n}\Big{(}f(\mu)-f(\mu+1)\Big{)}\Big{(}\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu}\Big{)},$
$\displaystyle\qquad\sqrt{n}\Big{(}f(\mu)-f(\mu+1)\Big{)}\Big{(}\xi_{\mu+1\rightarrow\mu}\
\otimes\ \chi_{\mu\rightarrow\nu}\Big{)}\Big{\rangle}_{\cal{E}}$
$\displaystyle=|f(\mu)-f(\mu+1)|.$

Therefore for any $f\in{\cal A}$ such that $||[{\cal D},f]||\leq 1$,
$|f(\mu)-f(\mu+1)|\leq 1$. Hence by definition, $d(\mu,\mu+1)\leq 1$ for all
$\mu=1,2,\ldots,n$. Now we shall show that the upper bound is actually
attained. To that end for a fixed vertex $\mu$, choose a function $g$ on $V$
that takes the value $1$ on $\mu$ and $0$ on $\mu+1$. Recalling the
orthonormal basis of ${\cal H}$ from 3.2, for any element
$\Xi=\sum_{\theta,\nu:\theta\rightarrow\nu\in
E}c_{\theta}^{\nu}\sqrt{n}\chi_{\theta\rightarrow\nu}+\sum_{\lambda,\nu^{\prime}:\lambda\rightarrow\nu^{\prime}\in
E}d_{\lambda}^{\nu^{\prime}}\sqrt{n}\xi_{\lambda+1\rightarrow\lambda}\
\otimes\ \chi_{\lambda\rightarrow\nu^{\prime}}\in{\cal H}$ such that
$||\Xi||\leq 1$, we have

(5.1) $\sum_{\theta,\nu:\theta\rightarrow\nu\in
E}|c_{\theta}^{\nu}|^{2}+\sum_{\lambda,\nu^{\prime}:\lambda\rightarrow\nu^{\prime}\in
E}|d_{\lambda}^{\nu^{\prime}}|^{2}\leq 1.$

Then again using $[\zeta,g]=[\zeta^{\dagger},g]=0$ on appropriate subspaces,

$\displaystyle[{\cal D},g](\Xi)$
$\displaystyle=(\nabla_{0}+\nabla_{0}^{\dagger})g.\left(\sum_{\begin{subarray}{c}\theta,\nu\\\
\theta\rightarrow\nu\in
E\end{subarray}}c_{\theta}^{\nu}\sqrt{n}\chi_{\theta\rightarrow\nu}+\sum_{\begin{subarray}{c}\lambda,\nu^{\prime}\\\
\lambda\rightarrow\nu^{\prime}\in
E\end{subarray}}d_{\lambda}^{\nu^{\prime}}\sqrt{n}\xi_{\lambda+1\rightarrow\lambda}\
\otimes\ \chi_{\lambda\rightarrow\nu^{\prime}}\right)$
$\displaystyle\quad-g.(\nabla_{0}+\nabla_{0}^{\dagger})\left(\sum_{\begin{subarray}{c}\theta,\nu\\\
\theta\rightarrow\nu\in
E\end{subarray}}c_{\theta}^{\nu}\sqrt{n}\chi_{\theta\rightarrow\nu}+\sum_{\begin{subarray}{c}\lambda,\nu^{\prime}\\\
\lambda\rightarrow\nu^{\prime}\in
E\end{subarray}}d_{\lambda}^{\nu^{\prime}}\sqrt{n}\xi_{\lambda+1\rightarrow\lambda}\
\otimes\ \chi_{\lambda\rightarrow\nu^{\prime}}\right)$
$\displaystyle=(\nabla_{0}+\nabla_{0}^{\dagger})\left(\sum_{\begin{subarray}{c}\nu:\\\
\mu\rightarrow\nu\in
E\end{subarray}}c_{\mu}^{\nu}\sqrt{n}\chi_{\mu\rightarrow\nu}+\sum_{\begin{subarray}{c}\nu^{\prime}\\\
\mu-1\rightarrow\nu^{\prime}\in
E\end{subarray}}d_{\mu-1}^{\nu^{\prime}}\sqrt{n}\xi_{\mu\rightarrow\mu-1}\
\otimes\ \chi_{\mu-1\rightarrow\nu^{\prime}}\right)$
$\displaystyle\quad-g.\left(\sum
c^{\nu}_{\theta}\sqrt{n}\xi_{\theta+1\rightarrow\theta}\ \otimes\
\chi_{\theta\rightarrow\nu}+\sum
d_{\lambda}^{\nu^{\prime}}\sqrt{n}\chi_{\lambda\rightarrow\nu^{\prime}}\right)$
$\displaystyle=\left(\sum_{\nu:\mu\rightarrow\nu\in
E}c_{\mu}^{\nu}\sqrt{n}\xi_{\mu+1\rightarrow\mu}\ \otimes\
\chi_{\mu\rightarrow\nu}+\sum_{\nu^{\prime}:\mu-1\rightarrow\nu^{\prime}\in
E}d_{\mu-1}^{\nu^{\prime}}\sqrt{n}\chi_{\mu-1\rightarrow\nu^{\prime}}\right)$
$\displaystyle\quad-\left(\sum_{\nu:\mu-1\rightarrow\nu\in
E}c^{\nu}_{\mu-1}\sqrt{n}\xi_{\mu\rightarrow\mu-1}\ \otimes\
\chi_{\mu-1\rightarrow\nu}+\sum_{\nu^{\prime}:\mu\rightarrow\nu^{\prime}\in
E}d_{\mu}^{\nu^{\prime}}\sqrt{n}\chi_{\mu\rightarrow\nu^{\prime}}\right)$

Therefore

$||[{\cal D},g](\Xi)||^{2}=\sum_{\begin{subarray}{c}\nu:\\\
\mu\rightarrow\nu\in
E\end{subarray}}|c_{\mu}^{\nu}|^{2}+\sum_{\begin{subarray}{c}\nu:\\\
\mu-1\rightarrow\nu\in
E\end{subarray}}|c_{\mu-1}^{\nu}|^{2}+\sum_{\begin{subarray}{c}\nu:\\\
\mu\rightarrow\nu\in
E\end{subarray}}|d_{\mu}^{\nu}|^{2}+\sum_{\begin{subarray}{c}\nu:\\\
\mu-1\rightarrow\nu\in E\end{subarray}}|d_{\mu-1}^{\nu}|^{2}.$

Then by inequality (5.1),

$||[{\cal D},g](\Xi)||\leq 1,$

which implies that $||[{\cal D},g]||\leq 1$. But $|g(\mu)-g(\mu+1)|=1$. Hence
$d(\mu,\mu+1)=1$ for all $\mu=1,2,\ldots,n$. From the proof it is clear that
the distance does not depend upon any choice of left module map $\zeta$ to
define the holomorphic Dolbeault-Dirac operator ${\cal D}$. $\Box$

We have the following easy to see corollary:

######  Corollary 5.2.

For any two points $\mu,\nu\in V$, $d(\mu,\nu)\leq[\frac{n}{2}]$. In
particular, $V$ has finite diameter.

Acknowledgement: The first author is supported by SERB MATRICS grant (grant
number MTR/2022/000515), Government of India. The first author would like to
thank Md. Ali Zinna for fruitful discussion, particularly concerning the proof
of Lemma 3.3.

## References