NTNU Gjøvik

# Imbalance measure and proactive channel rebalancing algorithm for the
Lightning Network

Rene Pickhardt and Mariusz Nowostawski

###### Abstract

Making a payment in a privacy-aware payment channel network is achieved by
trying several payment paths until one succeeds. With a large network, such as
the Lightning Network, a completion of a single payment can take up to several
minutes. We introduce a network imbalance measure and formulate the
optimization problem of improving the balance of the network as a sequence of
rebalancing operations of the funds within the channels along circular paths
within the network. As the funds and balances of channels are not globally
known, we introduce a greedy heuristic with which every node despite the
uncertainty can improve its own local balance. In an empirical simulation on a
recent snapshot of the Lightning Network we demonstrate that the imbalance
distribution of the network has a Kolmogorov-Smirnoff distance of $0.74$ in
comparison to the imbalance distribution after the heuristic is applied. We
further show that the success rate of a single unit payment increases from
$11.2\%$ on the imbalanced network to $98.3\%$ in the balanced network.
Similarly, the median possible payment size across all pairs of participants
increases from $0$ to $0.5$ mBTC for initial routing attempts on the cheapest
possible path. We provide an empirical evidence that routing fees should be
dropped for proactive rebalancing operations. Executing $4$ different
strategies for selecting rebalancing cycles lead to similar results indicating
that a collaborative approach within the friend of a friend network might be
preferable from a practical point of view.

###### keywords:

Imbalance, rebalancing, Optimization, Bitcoin, Lightning Network, payment
channel networks, path finding, routing, liquidity, flow control, congestion
control, game theory, uncertainty, simulations, collaborative problem solving,
privacy

##  1 Introduction

The Lightning Network has been introduced in order to mitigate the scaling
issues of blockchain technologies such as Bitcoin[9]. With the help of smart
contracts, a payment channel can be established to allow the cryptographically
secure transfer of value without the necessity to record the transaction to
the central ledger. This yields enhanced privacy properties for the
participants of the network. In order to protect that privacy while routing a
payment through several payment channels privacy aware payment channel
networks use a source-based onion routing scheme, like the Sphinx Mix format
[2]. This prevents routing nodes from learning who paid whom.

While the Lightning Network, for example, shares the capacity of public
channels with its participants through its gossip protocol the local split of
the capacity of the channels into the balance of its participants is not
shared with the rest of the network for two reasons: (1) this would compromise
the privacy as one could collect all changes of the channel balances and
reconstruct the flow of payments; (2) propagating this information would
essentially mean that every node in the network is made aware of every payment
which would have the same poor scaling properties as blockchain technologies
and other broadcast networks. The decision to use source-based routing
together with the unknown channel balances of the network results in a
challenge for finding a path of payment channels so that all channels along
the path have enough liquidity to be able to forward an attempted payment.
Currently, this challenge is met by probing paths with a brute force approach
until one such path succeeds to route the payment. It has been shown that
probing for paths can take more than 3 minutes for more than $5\%$ of the
attempted payments [3]. This leads to a poor payment latency and poor user
experience. Imagine a grocery store in which for every $20^{th}$ customer the
cash register would have to wait three minutes until the payment was received.

In this paper we examine the consequences of nodes proactively and
collaboratively distributing their funds evenly across their channels. We
introduce the notation of an imbalanced network. The imbalance is measured as
the average of the imbalance scores of its nodes. The node’s imbalance of
payment channels is defined as the Gini coefficient of a node’s channel
balance coefficients which are the relative amount of funds a node owns in a
channel in comparison to the capacity of that channel. Note, nodes on both
ends of the channel already have all the information they need to compute
their imbalance. Nodes can reduce their imbalance either by initiating a
rebalancing their own channels themselves, or, by collaboratively rebalancing
the channels with the help of their channel partners by conducting circular
payments. Another option is to use a submarine
swap111https://github.com/submarineswaps/swaps-service as an off-chain/on-
chain swapping service.

We formulate an optimization problem of finding a sequence of rebalancing
operations that minimize the network’s imbalance. As the information necessary
to solve the optimization problem is not publicly known in privacy-aware
payment channel networks, we provide a greedy heuristic for its participants
to find a minimum for the problem. In an empirical study, we show that the
greedy heuristic will lead to a success rate of $98.3\%$ for a small payment
between two arbitrary selected nodes. Also the median possible payment size
between all pairs of nodes increases from $0$ in the imbalanced network to
$0.5$ mBTC in the best balanced networks that we found, when the cheapest
possible path is selected for the payment.

While single nodes can already execute the algorithm in currently existing
payment channel networks they have to pay routing fees for the rebalancing
operations. Thus economically, the fees disincentivize the nodes to conduct
rebalancing operations. We compared the earned fees for being participant of
another nodes rebalancing attempt and the paid fees for initiating rebalancing
operations. Showing that these numbers are normally distributed around $0$ we
propose to omit fees for proactive rebalancing operations to incentivize nodes
to execute our algorithm which appears to be beneficial for the entire
network. Looking at different strategies of finding rebalancing cycles we
suggest, for the sake of speed, that nodes share with their neighbors on which
local channels they would like to have inbound or outbound capacity. This
information could easily be probed anyway and will not to worsen the privacy
of the nodes involved in rebalancing.

The remainder of this paper is organized as follows. In Section 2 we give a
short overview of related work in this field. We then introduce the notation
and definition of the imbalance score in Section 3 and formulate the
optimization problem to reduce the imbalance. We propose a greedy and
collaborative algorithm in Section 3.1 to address the optimization problem. We
also propose different strategies for the algorithm to probe for potential
rebalancing cycles. In Section 4 we introduce the experimental setup and the
data set that we used. After showing our empirical results in Section 5 we
discuss the results, we propose extensions and future work in the last two
sections.

##  2 Related Work

While the Lightning Network white paper [9] does not discuss path finding and
states routing as an easy problem it is generally recognized that pathfinding
on the Lightning Network is a difficult problem [6, 10, 1, 8, 4, 11]. There is
already research conducted in the field of rebalancing channels [5] which was
more about the cryptographic protocols used to make sure that participants can
enforce the rebalancing that was agreed upon. There are rebalancing operations
for c-lightning222https://github.com/lightningd/plugins/tree/master/rebalance
and for lnd333https://github.com/bitromortac/lndmanage. In particular, the
idea of just in time rebalancing while fulfilling routing requests [7] has
been implemented as JIT-routing for
c-lightning444https://github.com/lightningd/plugins/pull/66.

##  3 Formalization and Assumptions

Let $N=(V,E,c)$ be a payment channel network with a finite set of nodes. The
payment channels are the edges in the network such that $E\subset V\times V$.
Additionally, we have a publicly known capacity function
$c:E\longrightarrow\mathbb{N}$ that assigns a capacity to every edge of the
network. For every edge $e=(u,v)$ we denote $e_{u}:=(e,u)$ as the first
participant of the channel and $e_{v}=(e,v)$ as the second participant.
Naturally, the capacity of every channel $e=(u,v)$ is privately split into the
local balances with the balance function $b:E\times
V\longrightarrow\mathbb{N}$ such that
$b(e_{u})+b(e_{v})\stackrel{{\scriptstyle!}}{{=}}c(e)$.

We define the channel balance coefficient for $u$ on the channel $e=(u,v)$ as
$\zeta_{(u,v)}=\frac{b(e_{u})}{c(e)}$. This is just the relative amount of
funds that the participant $u$ has in the channel $e$. Since
$b(e_{u})+b(e_{v})\stackrel{{\scriptstyle!}}{{=}}c(e)$ we also have
$\zeta_{(u,v)}+\zeta_{(v,u)}=1$. In general, for an imbalanced chanel we have
$\zeta_{(u,v)}\neq\zeta_{(v,u)}$.

The neighbor function $n:V\longrightarrow 2^{E}$ assigns every node a set of
all the channels it is part of. To make the following formulas easier to read
let us introduce the neighborhood $U:=n(u)$.

The total funds of a participant $u$ are denoted as
$\tau_{u}:=\displaystyle{\sum_{e\in U}b(e_{u})}$. The value of $\tau_{u}$ is
constant while no payments are made and no channels are being opened and
closed. In contrast the balance function $b$ can vary subject to re-allocation
of funds. The total capacity of a participant $u$ is denoted as
$\kappa_{u}:=\displaystyle{\sum_{e\in U}c(e)}$. Using the last two definitions
let us define the node balance coefficient for a participant $u$ as
$\nu_{u}=\frac{\tau_{u}}{\kappa_{u}}$.

We call a node $u$ balanced if its channel balance coefficients
$\zeta_{(u,v_{1})},\dots,\zeta_{(u,v_{d})}$ have the same value.555This value
does by no means have to take a value of $0.5$ though it certainly seems to be
a reasonable goal. This means that the distribution of a node’s relative funds
across all its channels is the same. Consequently, we consider a node
unbalanced, if it local channel balance coefficients are unequal.
Statistically, inequality of a distribution can be measured with the Gini
coefficient. Thus, for a node $u$ with channel balance coefficients
$\zeta_{(u,v_{1})},\dots,\zeta_{(u,v_{d})}$ we define
$G_{u}=\frac{\displaystyle{\sum_{i\in U}\sum_{j\in
U}}|\zeta_{i}-\zeta_{j}|}{2\displaystyle{\sum_{i\in U}\sum_{j\in
U}\zeta_{j}}}$ If $G_{u}=0$ this means that the channel balance coefficients
are equal. In contrast, if $G_{u}=1$ the channel balance coefficients are
distributed in the most unequal way.

Note, that the Gini coefficient of a channel balance coefficients of a node
$u$ takes the value $0$ if and only if its channel balance coefficients all
take the same value. This value will be exactly the same as the node’s balance
coefficient $\nu_{u}$. Nodes which have channels with large differences in
capacities will not be able to get a Gini coefficient of $0$ if we used
absolute balances for the definition of the $\zeta$ values as smaller channels
might be drained by larger ones during rebalancing operations.

Finally, $G$ denotes the imbalance of the network.
$G=\displaystyle{\frac{1}{|V|}\sum_{v\in V}G_{v}}$. It is the mean of the
imbalance values of all nodes in the network. A perfectly balanced network
would be achieved if $G$ takes the value of $0$ whereas the balance is poor if
the value of $G$ is close to 1.

Our goal is to find a balance function $b$ which minimizes $G$ given a privacy
aware payment channel network with initial distribution of funds
$\tau_{u_{1}},\dots,\tau_{u_{n}}$. The constraint to this optimization problem
is that the total funds $\tau_{u}$ are fixed for every node $u\in V$ and any
choice of the balance function $b$. In a privacy aware payment channel network
the distribution of funds $\tau_{u_{1}}\dots,\tau_{u_{n}}$ is not publicly
known.666Even though the initial distribution could be guessed from the
funding transactions in the case of the Bitcoin Lightning Network (c.f.:
https://github.com/lightningnetwork/lightning-rfc/issues/720) for public
channels it will change with the first payment that is not a rebalancing
operation. In the same way the initial balance function is not publicly known.
As we lack knowledge about the global network state, we cannot apply standard
optimization techniques such as gradient descent, conjugate gradient methods
or simulated annealing. We suggest to use an heuristic in which every
participant executes some operations to improve its own balance which others
support if it also improves their balance.

###  3.1 Description of the rebalancing algorithm

The developed rebalancing algorithm uses a heuristic in which participants use
the local knowledge and make local adjustments to estimate the optimization
problem of finding $b$ such that $G$ is minimized. The algorithm therefore is
privacy-aware. It executes as follows:

  1. 1.

A node $u$ computes its node balance coefficient $\nu_{u}$.

  2. 2.

$u$ then computes the channel balance coefficients
$\zeta_{(u,v_{1})},\dots,\zeta_{(u,v_{d})}$ for its $d$ channels.

  3. 3.

The node selects all channels $e=(u,v_{i})$ for which its channel balance
coefficient is higher than its node balance coefficient, i.e.
$C=\\{(u,v_{i})|\zeta_{(u,v_{i})}-\nu_{u}\ >0\\}$777 Note, that we do not need
to take absolute values as $u$ will only be able to initiate a rebalancing
operation by sending money which means decreasing its channel balance
coefficient of $\zeta_{(u,v_{i})}$ towards $\nu_{u}$..

  4. 4.

From the candidate set $C$ a random channel $e=(u,v)$ is selected.

  5. 5.

Now the node searches for a circular payment to itself along $e=(u,v)$ by
choosing a path $p=[v,x_{1},\dots,x_{n},u]$. The amount of that payment should
decrease the value of $\zeta_{(u,v)}$ to that of $\nu_{u}$ and can be computed
as $a=c(e)\cdot(\zeta_{(u,v)}-\nu_{u})$. The end of the circle should be a
channel $(x_{n},u)$ for which the channel balance coefficient
$\zeta_{(u,x_{n})}$ is smaller than the node balance coefficient
$\nu_{u}$.888In one of the experiments we weaken this strong criteria and show
empirically that it makes sense to drop it for the benefit of easier path
finding at the cost of small oscillations of the algorithm.

  6. 6.

The node conducts the payment if all the nodes on the path $p$ agree to
participate. It could happen that some nodes will only participate with a
value smaller than $a$. As this is already progress $u$ will accept the
suggested amount instead of being stubborn.

  7. 7.

Repeat all steps as long as the local balance coefficients are not even enough
and as long as paths are found.

When a circular path exists, it is easy to see that making a payment along
this path will make the distribution of local balance coefficients for node
$u$ and those of all participants more even as that was the condition under
which they participated in the rebalancing operation. If no such circular path
can be found the local balance coefficients and thus the Network health stays
constant.

##  4 Experimental Setup

We evaluated $4$ different strategies for finding rebalancing cycles. cycles4
tests all cycles of length $4$ or smaller. In the same way cycle5 tests all
cycles of length $5$ or smaller. foaf tests most cycles of the friend of a
friend network.999This approximation is achieved by using cycle4 as a cycle
base as all cycles of length 4 and smaller certainly are part of the friend of
a friend network. mpp is short for multi-path payments and uses the same
heuristic as foaf but will only take a $20^{th}$ of the maximum possible
amount for rebalancing operation since it is supposed to split the rebalancing
over various cycles. Our approximation for foaf and mpp explains why the final
results for these two strategies are not entirely monotonic but oscillating a
bit.

We conducted our experiment on a snapshot of the public Lightning Network from
October 2019. That network view was retrieved from the gossip store of our
lightning node which is accessible at https://ln.rene-pickhardt.de: As
channels are currently almost always opened by one side101010Dual funded
channels are not part of the protocol and no implementation is merged to any
of the standard nodes we randomly guessed who opened the channel by a coin
flip and allocated the entire channel capacity to that node. This had to be
done as the gossip protocol does neither propagate any information about who
is opening a channel nor who owns which funds.

Figure 1: Probability mass of path-lengths of shortest paths between all pairs
of nodes. As this is computed on the strongly connected component there will
always be a path between each pair of nodes

After we randomly chose the founder of each channel we compute the largest
strongly connected component to remove nodes which had the funds allocated in
a way that would not allow any rebalancing. The strongly connected component
consists of $2707$ nodes and $24161$ edges. The diameter of the strongly
connected component had a value of 10 and the cumulative distance distribution
of shortest paths between all pairs can be seen in Figure 1. Note that the
notion of shortest paths with respect to fees depends on the amount that is
being transferred as the fees on channels have a fixed base fee and a variable
fee rate. For the computation of shortest paths we have assumed a fixed
payment size and, essentially, computed the shortest paths by only respecting
the fixed base fee. Shortest paths correspond to the cheapest paths.

Note that almost $40\%$ of shortest paths are longer than the cycles that our
strategies have tested. This demonstrates that the single rebalancing
operations which we conducted were not covering the entire network but only
taking place within local parts of the network. This is particularly true for
the rebalancing in the friend of a friend network.

Figure 2: The histogram shows the distribution of relative funds $\nu$ for all
nodes of the network.

From Figure 2 we can see the result of our random process of allocating funds
to the nodes and selecting the largest strongly connected component. It is
plausible that $\nu=0.5$ is the mode of the histogram as the coin flip will
statistically lead to as many channels with inbound capacity as channels with
outbound capacity.

Figure 3: The distribution of Gini coefficients for each node over its
distribution of channel balance coefficients $\zeta$ of the input network
before any rebalancing experiment was taking place.

Figure 3 depicts the initial distribution of Gini coefficients for each node
of the examined network. This plot also reflects the random nature of the
allocated funds already resulting in an average value as the total network
imbalance very close to $0.5$.

In order to conduct the experiments we simulated the proposed algorithm and
strategies in the following way. For each channel we computed up to $5000$
rebalancing cycles following the defined strategy. The number $5000$ was
chosen to be as large as possible so that we were able to conduct the
simulation within the $16$ GB of main memory of our machine. In most cases
there were less than $5000$ cycles available. However sometimes we had to cut
off the precomputed cycles as the hard cap of $5000$ was hit.

For each cycle we checked if a rebalancing was possible without having nodes
on the path worsening their channel balance coefficients. As this assumption
is very strong the vast majority in rebalancing attempts along a cycle would
fail or only work with an amount smaller than the one proposed by the
initiator.

##  5 Results

Figure 4: Comparing how the imbalance score of the network behaves with the
amount of successfull rebalancing operations. As the rebalancing amounts are
much smaller with multi path payments the x-axis has a logarithmic scale.

Figure 4 confirms that the imbalance of the network is decreasing over time
when running our simulation. This means that the network is becoming more
balanced. We see that most improvement is happening quite fast with $10$k to
$100$k rebalancing operations being necessary. This is not more than $37$
rebalancing operations per node. Taking into account that the average node
degree is $9$ this means that on average each node needs to successfully
execute about $4$ rebalancing operations per channel. This seams plausible in
the sense that every channel gets rebalanced at least once.111111Note, that
due to the collaborative behaviour of nodes a node might also get its channels
rebalanced in the rebalancing attempts of other nodes. The multi-path
rebalancing needs more operations which makes sense as we only rebalanced for
a $20^{th}$ of the possible amount in the multi-path case since we wanted to
have multiple other cycles to rebalance that particular channel. While cycle4
seems to improve the balance more quickly it does not reach the minimum
imbalance as well as the other strategies. In particular the algorithm
converges to a local minimum rather quickly. Our experimental runs resulted in
states that are not identical when we shuffled the order of cycles that we
probed, however, the differences were negligible. We guess that the local
minimum might be close to global minimum of the optimization problem.

Figure 5: Every time the imbalance score reached a new low to a two digit
decimal we computed the probability for a random payment of the smallest
possible amount to succeed on the cheapest path.

In Figure 5 we see that a lower imbalance score $G$ yields a higher success
rate of payments as one would expect. The success rate is measured by
attempting a payment of $1$ Satoshi between all pairs of nodes and counting
the relative amount of payment attempts for which the cheapest path from the
source to the destination was successful. We computed the success rate of
every time the balance score hits a new score rounded to two decimals. It is
the first experiment that indicates that the defined imbalance score was well
chosen. Again the multi-path payment strategy sticks out as it achieves a
rather high success rate while the network is still rather imbalanced. A
success rate of $80\%$ is reached for an imbalance score of $0.43$. Comparing
this to Figure 4 we see that about $250$k rebalance operations are necessary
to achieve this success rate. Even the worst performing algorithm achieves
this success rate with far less than $100$k rebalancing operations and a much
better overall imbalance score.

This effect - while expected as multi-path rebalancing uses smaller amounts -
becomes more visible in the next experiment. As $1$ Satoshi payments are not
too useful in a real world setting we checked if the overall amounts that can
be routed increase in a better balanced network.

Figure 6: The imbalance of the network is plotted against the median value of
payments that can be fulfilled between all pairs of nodes on the first attempt
along the cheapest path. Note that failed payments are part of the statistics
and counted as payments which can send $0$ Satoshi.

In Figure 6 we compare the imbalance of the network with the median possible
payment amount which is computed as follows: For all pairs of shortest paths
on the base fee graph we look at the amount that could be forwarded along that
path. Finally we take the median of those values. This means that at least
$50\%$ of payment pairs are able to forward this amount. Again we see that the
lower the imbalance score becomes the higher the median possible payment size
gets. This suggests that statistically a more balanced network is able to
successfully route higher payment amounts on the first try. All four
strategies are achieving a median possible payment amount of roughly $50000$
Satoshi. This means that in a balanced network $50\%$ of payment pairs are
able to successfully conduct a payment of $0.5$ mBTC along the cheapest path.
The result also indicates that the strategy for probing rebalancing cycles
seems to have little to no impact to the final abilities of the network to
perform payments.

Figure 7: The maximal payable amounts on all pairs of cheapest paths on the
initial imbalanced network and on the network after applying the friend of a
friend rebalancing strategy.

We look a little bit closer at the data point of the most balanced network
with the friend of a friend strategy and compare it to the imbalanced network.
Instead of just looking at the median we study the cumulative distribution
function of the histogram in Figure 7. We can see that in the imbalanced
network almost $88.8\%$ of the paths are not able to forward a single Satoshi
(meaning a $11.2\%$ success rate). This number drops below $1.7\%$ (meaning a
$98.3\%$ success rate) for the balanced network.

For the friend of a friend network we have also plotted the final distribution
of Gini coefficients in Figure 8.

Figure 8: The distribution of Gini coefficients for each node over its
distribution of channel balance coefficients $\zeta$ of the network after
applying the foaf rebalancing strategy.

For a better comparison with Figure 3 we provide the cumulative distribution
function of both distributions with a smaller bin size in Figure 9.

Figure 9: Comparing the distribution of the node’s Gini coefficients before
and after rebalancing with the friend of a friend strategy.

The Kolmogorov-Smirnoff distance takes a value of $0.74$. This verifies the
statistical significance of the rebalancing heuristic in comparison to staying
with the imbalanced network. The median value of the Gini coefficients in the
best balanced network we could acquire is about $0.15$. In comparison on the
imbalanced network the median Gini coefficient takes the value of $0.5$.

Finally, we tracked the routing fees that nodes would have to pay for their
rebalancing operations and that they would earn while participating in the
rebalancing operation of others.

Figure 10: Total fees earned by nodes during all rebalancing operations.
Negative fees mean that the node paid more fees than it earned.

Figure 10 shows that the overall fees are normally distributed around $0$.
This means that the expected value of fees that need to be paid or are being
earned if all nodes participate in proactive rebalancing is $0$. An expected
value of $0$ is a strong indicator that fees should not play a role for
rebalancing operations if those are beneficial for all nodes that are asked to
take part in the rebalancing operation. We zoomed heavily in so that only
$50\%$ of the nodes have been depicted in this view due to the large variance
of this distribution.

##  6 Discussion

We empirically verified that the overall ability of the network participants
to route payments on random paths is improved a lot in a balanced network. As
privacy aware payment channel networks use source-based path finding without
knowing the constraints of the channels along an attempted path, the
likelihood of conducting a successful payment with few attempts becomes
considerably higher in a balanced network. Our results show that nodes do not
need to strive for $\nu_{u}=0.5$ and perfect local balance of their channels
in order to produce high success rate of random payments. It is sufficient for
the overall network to be balanced if every node tries to optimize their
channel balance coefficients be almost equal. This can be achieved by making
circular rebalancing payments.

We have proposed a greedy rebalancing heuristic which we have tested with $4$
different strategies to select rebalancing cycles. We could see that the
results of these four strategies do not significantly differ for the overall
performance of the network. While multi-path rebalancing achieves the actual
best values on both success rates and median payment size the lead is
negligible and stands in no ratio to the extra amount of rebalancing
operations that need to take place.

Computing the rebalancing cycles is the most expensive operation in our
simulation since we had to do it for every node of the network. If the greedy
heuristic is implemented in a real payment channel network this is not too
expensive for a single node. This holds in particular true if those cycles
will only be searched within the friend of a friend network which is typically
small enough. As the results of the different strategies for finding cycles
did not vary and for the previous reason we propose to stick to rebalancing in
the friend of a friend network. In a real payment channel network this
computation would only have to be conducted if the distribution of channel
balance coefficients becomes too uneven, which is easily monitored after every
payment or routing attempt without additional performance overhead.

Finally, we empirically showed that the earned and paid fees for rebalancing
operations are normally distributed around $0$ Satoshi suggesting that this is
a zero sum game if every node participates in proactive rebalancing. Routing
fees might discourage nodes from proactively rebalancing as they might hope to
earn fees if they wait for others to rebalance their channels. This behaviour
is referred to as the tragedy of the commons121212c.f.
https://en.wikipedia.org/wiki/Tragedy_of_the_commons and would be mitigated
with a fee free rebalancing protocol. As fees are empirically a zero sum game
and to incentivize the participation of the rebalancing it seems reasonable to
omit fees for rebalancing operations if those are beneficial for every
participant as in the experiment.131313This would of course need a protocol
change in the lightning network protocol as the current onion routing could
not guarantee to the routing noes that they process a fee free rebalancing
payment.

In a similar way as Internet routers collaboratively solve the path finding
problem by exchanging routing tables, we furthermore suggest for the network
to work collaboratively towards achieving a good balance by locally sharing
rebalancing hints. As balances of channels in the friend of a friend network
can easily be probed, it seems plausible in privacy aware payment channel
networks to include a communication protocol with which a node signals to
their partners that it wants to rebalance a certain channel with a certain
amount. Instead of a certain amount a node could just signal if it needs
inbound or outbound liquidity on a channel. We think that our suggestion for
sharing channels that need a rebalancing operation does not reveal more
information as can currently be collected with probing attacks. However,
sharing rebalancing hints gives out perfect information to compute available
rebalancing cycles.

The rebalancing is particularly important as privacy aware payment channel
networks use source based onion routing and technically have to guess a path
for making a payment. When those initially guessed paths provide a high
success rate with high payment amounts there are considerable benefits for the
overall functioning of the payment channel network. The user experience is
improved, the payment delays are lowered, and the amount of messaging and
coordination is minimised.

##  7 Future Work

As the simulation was already consuming a lot of computational resources we
did not check if the algorithm greedy heuristic is stable under concurrent
payments and rebalancing operations taking place. In a similar gist
simulations on larger networks would be interesting but require more
significant computational resources and time to be conducted.

The current experiments have assumed a stable network topology. However, every
channel opening or closure, as well as every single payment, will change the
imbalance of the network. It is therefore necessary to study how it adapts if
the topology changes in real time due to opening or closing channels as well
as a reallocation of funds due to payments that are taking place. It would be
in particular interesting to see how this heuristic works in combination with
JIT-Routing as this would unbalance nodes to provide liquidity which our
heuristic would then redistribute over the friend of a friend network. This
would occur along all nodes on a path in intersecting friend of a friend
networks. It would be interesting to see how this algorithm behaves together
with Atomic Multipath Payments which are in the style of [1] redundant but
along our results carry at most $50000$ Satoshi.

Another extension of the work is to check how the heuristic performs if only a
fraction of nodes participate in the sharing of rebalancing hints and fee free
rebalancing protocol.

Current implementations of the Lightning Network provide autopilots to the
users which help the users to automatically open channels. The results from
this research should be used in those autopilots and an investigation how
exactly this might be useful would be of high interest.

The balance metric used could be used as a measure of network health in the
context of random path payments. However, random path payments are only one of
the possible scenarios of payments flowing in a payment network. Thus, a
natural extension of this work is to study the relationship between the
proposed balance and network health. Can nodes define their local health in
various ways and what would happen with the greedy algorithm if different
health definitions are followed by individual nodes? For example routing nodes
or liquidity providers might very well want to have high values of $\zeta$ on
some of their channels where as they might be willing to accept low values for
$\zeta$ on other channels. In a similar way merchants will probably prefer
many channels with low values of $\zeta$. The proposed algorithm could
potentially be exchanged to an updated version in which nodes decide their own
health metric, instead of using balance as the measure of network health.

##  8 Acknowledgements

We are grateful to Vivek Bagaria and Joachim Neu for an interesting discussion
about their and our work that initiated this particular research. We thank
Stefan Richter for helpful comments on early drafts of this work. A special
thank goes to Thomas Gottron, Christian Decker and Dmytro Piatkivskyi. Finally
we thank the donors to https://tallyco.in/s/lnbook and Patreons supporting
open knowledge.

## References

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  * [7] Rene Pickhardt.  Just in time routing (jit-routing) and a channel rebalancing heuristic as an add on for improved routing success in bolt 1.0.  https://lists.linuxfoundation.org/pipermail/lightning-dev/2019-March/001891.html, 2019\.
  * [8] Rene Pickhardt.  Path finding in the lightning network.  2019\.  https://youtu.be/MeEFUaRnMak?t=2010.
  * [9] Joseph Poon and Thaddeus Dryja.  The bitcoin lightning network: Scalable off-chain instant payments, 2016\.
  * [10] Pavel Prihodko, Slava Zhigulin, Mykola Sahno, Aleksei Ostrovskiy, and Olaoluwa Osuntokun.  Flare: An approach to routing in lightning network.  White Paper, 2016.
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# Thermodynamic properties, thermal image and phase transition of Einstein-
Gauss-Bonnet black hole coupled with nonlinear electrodynamics

Dharm Veer Singh<EMAIL_ADDRESS>Department of Physics, Institute of
Applied Science and Humanities, GLA University, Mathura, 281406 India.  Vinod
Kumar Bhardwaj<EMAIL_ADDRESS>Department of Mathematics,
Institute of Applied Science and Humanities, GLA University, Mathura, 281406
India.  Sudhaker Upadhyay111Visiting Associate at IUCAA, Pune, Maharashtra
411007, India<EMAIL_ADDRESS>Department of Physics, K. L. S.
College, Nawada, Bihar 805110, India  Department of Physics, Magadh
University, Bodh Gaya, Bihar 824234, India  School of Physics, Damghan
University, Damghan 3671641167, Iran

###### Abstract

We obtain an exact solution of $AdS$ black hole solution in Einstein-Gauss-
Bonnet (EGB) gravity coupled with nonlinear electrodynamics. It interpolates
with the $AdS$ regular black hole and $AdS$ EGB black hole in the absence of
the Gauss-Bonnet coupling constant and both magnetic monopole charge and
deviation parameter, respectively. Based on horizon thermodynamics, we study
the thermodynamic properties of the obtained solution (e.g. mass, temperature,
entropy, heat capacity and free energy). The Hawking temperature of the
nonsingular black hole gets the maximum value at the point where specific heat
diverges and the second-order phase transition occurs at the same point. We
find that the smaller nonsingular black holes are stable due to positive heat
capacity and negative free energy. We explicitly trace the relations between
the black hole shadow and the critical behavior of charged EGB $AdS$ regular
black hole in the extended phase space.

Einstein-Gauss-Bonnet gravity; Nonlinear electrodynamics; Thermodynamics.

##  I Overview and motivation

Black holes, one of the most fascinating objects of nature, were predicted by
Einstein’s gravity in general relativity (GR). This is because the geometric
properties of black holes are described by elegant mathematical equations. The
recent (astrophysical) observations and experiments associated with
gravitational waves 1 ; 2  and (supermassive) black hole shadows 3  can guide
us to test the different gravity models. Due to the limitations of Einstein’s
gravity, modifications to Einstein’s gravity have been studied widely in the
cosmological context. In this connection, the modifications in the form of
higher orders of the Riemann curvature (and of its contractions) had been
discussed for the sake of answer to the problem of the dark energy dominance
in the evolution of our universe as well as the hierarchy problem. Lee-Wick
gravity theories, on the other hand, are another example of modified gravity
that removes the conflict between unitarity and renormalizability in quantum
gravity l ; l1 ; sd .

In general, gravity theories with higher derivative curvature terms had
consistency issues due to negative norm states (ghost) in their spectra at the
quantum level. Lovelock gravity is one of the higher derivative gravity
theories that maintain second-order equations of motion for the metric and
makes the theory ghost-free lav . Gauss-Bonnet (GB) or Einstein-Gauss-Bonnet
(EGB) gravity is particular Lovelock gravity that gets relevance in higher
dimensions 4 . Interestingly, higher orders of the Riemann curvature
corrections appear naturally in the low-energy limit of string theory 5 . This
gave rise to considerable interest in EGB gravity theories (especially, the
black-hole solutions of these theories) 6 ; 7 . The GB term is not dynamical
in $4d$ due to total derivative. It becomes dynamical in $4d$ only when GB
term must be coupled to a matter field otherwise dimension should be greater
than four. Quite interestingly, by re-scaling the GB coupling constant
$\alpha\to\alpha/d-4$, the EGB the black hole solutions in $4d$ have been
studied gla  where the GB term contributes to the (local) dynamics gla . The
consistency of field-equation is rather questioned me . The limit $d\to 4$
costs us by either partial breaking of the diffeomorphism or leading to
additional degrees of freedom.

In Refs. gl1 ; gl2 , a consistent EGB gravity in $4d$ has been proposed that
admits spatial covariance only but not the temporal covariance. Recently,
black hole solution and their thermal properties have been studied for 4D
$AdS$ EGB gravity coupled with Yang-Mills field Singh:2021xbk . Moreover, a
black hole solution along with its thermodynamics for $4D$ EGB gravity fran ,
$4D$ EGB massive gravity is presented Upadhyay:2022axg , $4D$ EGB wormhole
solutions Godani:2022jwz ; Jusufi:2020yus , and $4D$ EGB quark star
Tangphati:2021tcy ; Pretel:2021czp ; Tangphati:2021wng ; Panotopoulos:2022dbu
.

From the perspective of GR and its extensions, nonlinear electrodynamics is
one of the most crucial material sources of gravity as it leads to many
interesting geometries, in particular, regular black holes. The nonlinear
electrodynamics is an extension of the Maxwell electrodynamics which came into
the picture due to Born and Infeld (BI) in removing the central singularity of
a point charge be . A generalized BI action appears naturally in superstrings
scenario 8 ; 9  induces interest in nonlinear electrodynamics. Black holes
with nonlinear electrodynamics as a matter source are of great importance in
astrophysical observations and, therefore, studied extensively 10 ; 11 ; nnn ;
ab ; abg11 ; abg ; 12 ; 13 ; sabir ; dvs1 ; kumar1 ; dvs2 . However, the first
regular black hole was proposed by Bardeen 14 . In the context of nonlinear
electrodynamics, it is shown that a regular black hole only describes the
formation of a black hole from an initial vacuum region. The preference for
nonlinear electrodynamics over Maxwell’s theory is only because, along with
the conformal breaking, the infinite electric field and self-energy also
appear at the position of a point charge.

After Bekenstein and Hawking’s proposal of black hole entropy 16  and black
hole radiation 17 , black holes are being considered as a thermal system.
Bekenstein stated that the addition of black hole entropy and the the entropy
of matter is a non-decreasing function of time. The generalized second law of
thermodynamics suggests that black holes and self-gravitating systems comprise
a larger thermodynamic space. This leads to the possibility of phase
transitions between black holes and self-gravitating systems. Understanding
such phase transitions is itself important. Such analysis may provide
significantly insights to quantum theories of gravity 15 .

The plan of the paper is as follows. In Sec. II, we get an exact black hole
solution for the EGB gravity coupled with nonlinear electrodynamics and
negative cosmological constant. In Sec. III, we provide a thermal description
of the obtained solution by computing mass, temperature and entropy. Within
section, we also discuss the local and global stability of the system. In Sec.
IV, we study the $P-v$ criticality and phase diagram of the resulting black
hole solution. In Sec. V, we outline the relation between phase transition and
Shadow radius. Within section, we discuss the effects of various parameter on
black hole shadow. The discussion of results is reported in the last section.

##  II $4D$ regular EGB Black hole solution

The action of EGB gravity coupled with nonlinear electrodynamics and negative
cosmological constant can be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{d}x\sqrt{-g}\left[{R}-2\Lambda+\frac{\alpha}{D-4}{\cal
L_{GB}}-2{L}(F)\right],$ (1)

where $\alpha$ represents the dimensionless GB coupling constant, and ${R}$
denotes $D$-dimensional curvature scalar, the GB Lagrangian density is given
by $\mathcal{L_{GB}}:=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}$. Here, ${L}(F)$
corresponds to matter Lagrangian which depends on $F=\frac{1}{4}F_{ab}F^{ab}$.
Variation the action (1) with respect to $g_{ab}$ and the potential $A_{a}$,
we obtained the following equations of motion:

$\displaystyle R_{ab}-\frac{1}{2}\tilde{g}_{ab}R+\Lambda
g_{ab}+\frac{\alpha}{d-4}H_{ab}=T_{ab}\equiv
2\left[\frac{\partial{{L(F)}}}{\partial F}F_{ac}F_{\
b}^{c}-\tilde{g}_{ab}{{L(F)}}\right],$ (2)
$\displaystyle\nabla_{a}\left(\frac{\partial{{L(F)}}}{\partial
F}F^{ab}\right)=0\qquad\text{and}\qquad\nabla_{a}(*F^{ab})=0,$ (3)

where GB tensor is expressed as

$\displaystyle H_{ab}=-\frac{1}{2}\left[8R^{cd}R_{acbd}-4R_{a}^{\
cde}R_{bcde}-4RR_{ab}+8R_{ac}R_{\
b}^{c}+g_{ab}\left(R_{cdef}R^{cdef}-4R_{cd}R^{cd}+R^{2}\right)\right].$ (4)

Here, we note that the GB invariant in $d=4$ does not contribute to the field
equations. So, let us briefly explain the idea of how GB term contributes to
the dynamics of gravitational field by re-scaling the GB coupling constant to
$\alpha/(d-4)$. For maximally symmetric solution, Riemann tensor is given by
gla  $R^{ab}_{\ \ cd}=(\delta^{a}_{\ c}\delta^{b}_{\ d}-\delta^{a}_{\
d}\delta^{b}_{\ c})\Lambda_{eff}/(d-1)$, where $\Lambda_{eff}$ is an effective
cosmological constant. To estimate Gauss-Bonnet contribution, one calculates
gla

$\displaystyle\frac{g_{bc}}{\sqrt{-g}}\frac{{\cal L_{GB}}}{\delta
g_{ac}}=\frac{(d-2)(d-3)(d-4)}{2(d-1)}\Lambda_{eff}\delta^{a}_{b}.$ (5)

Here, the appeared vanishing factor $(d-4)$ due to Gauss-Bonnet term in $4D$
gets canceled by the divergent factor $1/(d-4)$ coming from the re-scaling GB
coupling constant $\alpha/(d-4)$.

Here, we are interested in a particular form of the nonlinear electrodynamics
Lagrangian $L(F)$ which admits the freedom of the duality rotations as
discussed in Refs. nnn ; ab ; abg11 . This is given by

${{L(F)}}=\frac{Fe^{-s(2g^{2}F)^{1/4}}}{(1+\sqrt{2g^{2}F})^{3/2}}\left(1+\frac{3}{s}\frac{(2g^{2}F)^{1/4}}{(1+\sqrt{2g^{2}F)}}\right),$
(6)

where $s=|g|/2M$ with the free parameters $g$ and $M$ associated with magnetic
charge and mass, respectively. For weak fields, this nonlinear electrodynamics
correspondence to Maxwell theory.

To obtain a static spherically symmetric black hole solution in $4D$, we
consider the following line element:

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Omega_{2}^{2},$ (7)

where $f(r)$ is the metric and $d\Omega_{2}^{2}=d\theta^{2}+\sin^{2}\theta
d\phi^{2}$ is the metric of a $2$-dimensional sphere.

To determine the metric function, we consider the following magnetic choice
for the Maxwell’s field-strength tensor $F_{ab}$ abg ; Singh:2022xgi

$F_{ab}=2\delta^{\theta}_{[a}\delta^{\phi}_{b]}Y(r,\theta).$ (8)

Here, we must stress that there exists spherically symmetric solutions with a
globally regular metric for gravity coupled to nonlinear electrodynamics with
the Lagrangian $L(F)$, possessing a correct weak field limit, for the magnetic
case only Bronnikov:2017tnz ; Bronnikov:2000vy . However, the electric analogs
of magnetic solutions can be found with different Lagrangian (using a Legendre
transformation in Hamiltonian formalism) in different ranges of the radial
coordinate Bronnikov:2000vy .

With ansatz (8), the integration of equation (3) give rise to

$F_{ab}=2\delta^{\theta}_{[a}\delta^{\phi}_{b]}g\sin\theta.$ (9)

Here, $g$ is constant (independent of $r$) is confirmed by using exterior
derivative of differential 2-form (8). Now, the field-strength tensor and,
therefore, matter Lagrangian can be simplified to

$F_{\theta\phi}=g\sin\theta,\qquad
F=\frac{1}{2}\frac{g^{2}}{r^{4}},\qquad\text{and}\qquad{L(F)}=\frac{g^{2}}{2r^{2}}\frac{e^{-sg/r}}{(r^{2}+g^{2})^{3/2}}\left[1+\frac{g}{s}\frac{r}{(r^{2}+g^{2})}\right].$
(10)

The non-zero components of energy momentum tensor are given by

$\displaystyle T^{t}_{t}=T^{r}_{r}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2r^{2}}\frac{e^{-sg/r}}{(r^{2}+g^{2})^{3/2}}\left[1+\frac{g}{s}\frac{r}{(r^{2}+g^{2})}\right],$
(11) $\displaystyle T^{\theta}_{\theta}=T^{\phi}_{\phi}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4r^{2}}\frac{e^{-sg/r}}{(r^{2}+g^{2})^{3/2}}\left[1+\frac{g}{s}\frac{r}{(r^{2}+g^{2})}-\frac{g(3g^{2}+8r^{2})}{sr}\right.$
(12) $\displaystyle-$
$\displaystyle\left.\frac{(3g^{2}+7r^{4})}{r^{2}(r^{2}+g^{2})^{5/2}}+\frac{gs}{r^{3}(r^{2}+g^{2})^{3/2}}\right].$

The $(r,r)$ components of Eq. (2) in the limit $d\to 4$ gives

$\displaystyle
r^{5}-2r^{3}\alpha(f-1)\frac{df}{dr}+r^{4}(f-1)+r^{2}\alpha(f-1)^{2}-\Lambda
r^{2}=\frac{g^{2}}{4r^{2}}\frac{e^{-sg/r}}{(r^{2}+g^{2})^{3/2}}\left[1+\frac{g}{s}\frac{r}{(r^{2}+g^{2})}\right].$
(13)

The solution of above equation is given by

$\displaystyle
f_{\pm}(r)=1+\frac{r^{2}}{2\alpha}\pm\frac{r^{2}}{2\alpha}\sqrt{1+\frac{8M\alpha\,e^{-k/r}}{(r^{2}+g^{2})^{3/2}}-\frac{4\alpha}{l^{2}}},$
(14)

where $k\,(=g^{2}/2M)$ is called as deviation parameter that measures the
deviation from Schwarzschild black hole solution. For $AdS$, the cosmological
constant ($\Lambda$) can be expressed in terms of length scale $l$ as
$\Lambda=-3/l^{2}$. Here, $M$ is a constant of integration related to the
total mass of the black hole. The obtained black hole solution (14) is
characterized by the mass, deviation parameter, magnetic charge, GB coupling
constant ($\alpha$) and cosmological constant. In the absence of deviation
parameter, this solution (14) identifies to the following $4D$ EGB Bardeen
black hole Singh:2020xju :

$\displaystyle
f_{\pm}(r)=1+\frac{r^{2}}{2\alpha}\pm\frac{r^{2}}{2\alpha}\sqrt{1+\frac{8M\alpha\,}{(r^{2}+g^{2})^{3/2}}-\frac{4\alpha}{l^{2}}}.$
(15)

In the absence of magnetic charge, this solution identifies to following $4D$
EGB regular black hole Singh:2020mty :

$\displaystyle
f_{\pm}(r)=1+\frac{r^{2}}{2\alpha}\pm\frac{r^{2}}{2\alpha}\sqrt{1+\frac{8M\alpha\,e^{-k/r}}{r^{3}}-\frac{4\alpha}{l^{2}}}.$
(16)

In the absence of GB coupling parameter, this identifies to solution obtained
in Ref. abg

$\displaystyle
f(r)=1+\frac{2Mr^{2}e^{-k/r}}{(r^{2}+g^{2})^{3/2}}+\frac{r^{2}}{l^{2}}.$ (17)

Moreover, our solution reduces to $AdS$ EGB black hole gla , Bardeen black
hole, $AdS$ regular black hole tzi  and $AdS$ Schwarzschild black hole in the
limit of $g=k=0$ and $\alpha=0$, respectively.

The solutions (14) behaves asymptotically as

$\displaystyle
f_{-}=1-\frac{2Me^{-k/r}}{(r^{2}+g^{2})^{3/2}}+\frac{r^{2}}{l^{2}}+\frac{r^{2}}{2\alpha}+{{\cal
O}\left(\frac{1}{r^{3}}\right)},$ (18) $\displaystyle
f_{+}=1+\frac{2Me^{-k/r}}{(r^{2}+g^{2})^{3/2}}-\frac{r^{2}}{l^{2}}+\frac{r^{2}}{2\alpha}+{{\cal
O}\left(\frac{1}{r^{3}}\right)}.$ (19)

Here, one can see that the $+$ve branch of the solutions is not physical as
the positive mass term indicates graviton instabilities and, therefore, this
solution will be ruled out. However, $-$ve branch of the solutions matches
with the physical $4D$ $AdS$ EGB black hole solution, so we must stick with
it. In the limit $r\to\infty$ ($M=0$), the EGB black hole solution (18)
becomes asymptotically flat.

In the limit $k=g=0$, the solution (14) reduces to

$\displaystyle
f_{\pm}(r)=1+\frac{r^{2}}{2\alpha}\left(1\pm\sqrt{1+\frac{8M\alpha}{r^{3}}-\frac{4\alpha}{l^{2}}}\,\right).$
(20)

This solution coincides with the one obtained by Glavan and Lin in Ref. gla .
This even matches further with $4D$ $AdS$ Schwarzschild black hole solution in
limit $\alpha\to 0$.

As we know that $f(r)=0$ determines the horizons. Since Eq. (14) is a
transcendental equation and can not be solved analytically. So, to determine
the horizons of the black hole, we solve it numerically and the numerical
results are tabulated in the TAB. 1.

| $\alpha=0.1$ |  |  |  |  | $\alpha=0.2$ |
---|---|---|---|---|---|---|---
$g$ | $r_{+}$ | $r_{-}$ | $\delta$ | $g$ | $r_{+}$ | $r_{-}$ | $\delta$
0.20 | 1.77 | 0.25 | 1.52 | 0.20 | 1.71 | 0.33 | 1.38
0.40 | 1.64 | 0.45 | 1.19 | 0.40 | 1.56 | 0.57 | 0.99
0.63 | 1.08 | 1.08 | 0.00 | 0.57 | 1.11 | 1.11 | 0.00
| $\alpha=0.1$ |  |  |  |  | $\alpha=0.2$ |
$k$ | $r_{+}$ | $r_{-}$ | $\delta$ | $g$ | $r_{+}$ | $r_{-}$ | $\delta$
0.10 | 1.72 | 0.33 | 1.39 | 0.10 | 1.65 | 0.44 | 1.21
0.30 | 1.44 | 0.51 | 0.93 | 0.30 | 1.31 | 0.66 | 0.65
0.47 | 0.92 | 0.92 | 0.00 | 0.385 | 0.97 | 0.97 | 0.00
Table 1: Cauchy horizon ($r_{-}$) and event horizon ($r_{+}$) and their
deviation $\delta=r_{+}-r_{-}$ for different values of $g$ and $k$ and fixed
value of $l$ and $M$.
---
Figure 1: Metric function $f(r)$ versus horizon radius for different magnetic
monopole charge $(g)$ and deviation parameter ($k$) with fixed values of GB
parameter $\alpha=0.1$ and $\alpha=0.2$. Here, we set $M=1$.

In the Fig. 1, we notice how the horizon structure depends on parameters $k$
and $g$. The zeros of metric function declares that there is no horizon if
$k>k_{c}$ (or $g>g_{c}$), one horizon if $k=k_{c}$ (or $g=g_{c}$), and are two
horizon if $k<k_{c}=0.47$ (or $g<g_{c}=0.60$). It is worthwhile to mention
that the size of horizon increases with decreasing magnetic charge $(g)$ and
deviation parameter $(k)$. The same effect also appears for the GB coupling.
The TABLE 1 reflects the values of event (outer) horizon and Cauchy (inner)
horizon.

In order to study the nature of singularities of $AdS$ black hole (14),
curvature invariants of Ricci square ($R_{\mu\nu}R^{\mu\nu}$) and Kretshmann
scalars ($R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}$) are important. The
behaviors of the scalar invariants are given by

$\displaystyle
R=f^{\prime\prime}(r)+\frac{6f^{\prime}(r)}{r}-\frac{6(1-f(r))}{r^{2}},$ (21)
$\displaystyle
R^{\mu\nu}R_{\mu\nu}=\frac{1}{2}{{f^{\prime\prime}(r)^{2}}}+\frac{15}{2}{\left({\frac{{f^{\prime}(r)}}{{r}}}\right)^{2}}+\frac{2}{r}f^{\prime\prime}(r)f(r)+\frac{12}{r^{4}}f(r)^{2}-\frac{24}{r^{4}}f(r)$
$\displaystyle\qquad\qquad-\frac{24}{r^{4}}(rf^{\prime}(r)+1)+\frac{24}{r^{3}}f(r)f^{\prime}(r),$
(22) $\displaystyle
R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}={{f^{\prime\prime}(r)^{2}}}+6{\left({\frac{{f^{\prime}(r)}}{{r}}}\right)^{2}}+12{\left({\frac{{f(r)-1}}{{{r^{2}}}}}\right)^{2}}.$
(23)

These invariants are smooth in $r\to 0$ limit and, therefore, signify regular
space-time.

##  III Thermodynamics

In this section, we investigate the thermal properties of $4D$ $AdS$ EGB
regular black hole. In this connection, we discuss, particularly, mass,
temperature, entropy, specific heat, and Gibbs free energy at the black hole
horizon. The mass of the black hole can be estimated from the metric function
(14) on the horizon as

$\displaystyle
M=\left(g^{2}+r_{+}^{2}\right)^{3/2}e^{k/r_{+}}\Big{[}\frac{(\alpha+r_{+}^{2})}{2r_{+}^{4}}+\frac{1}{2l^{2}}\Big{]}.$
(24)

This expression represents the mass of regular black hole in $4D$ EGB gravity
(24). From the above expression, it is clear that, for $k=0$ it reduces to the
mass of the $AdS$ EGB Bardeen black hole Singh:2020xju , for $g=0$ it reduces
to $AdS$ EGB regular black hole Singh:2020mty  and for both $k=0,g=0$ it
reduces to $4D$ EGB black hole gla . For vanishing $\alpha$ and $k$, the above
expressing reduces to the mass of Bardeen black hole tzi  and for vanishing
$\alpha$ and $g=0$ it reduces to the mass of $AdS$ regular black hole.
Finally, it reduces to mass of the $AdS$ Schwarzschild black hole for
vanishing all three parameters: $g$, $\alpha$ and $k$.

The standard formula to calculate Hawking temperature is given by

$T_{+}=\frac{1}{2\pi}{\sqrt{-\frac{1}{2}\nabla_{\mu}\xi_{\nu}\nabla^{\mu}\xi^{\nu}}}=\frac{1}{4\pi}f^{\prime}(r_{+}).$
(25)

In our case, this leads to

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{+}}\bigg{(}\frac{3r_{+}^{7}+l^{2}r_{+}^{2}(\alpha+r_{+}^{2})(9r-2k)-k(g^{2}+r_{+}^{2})(\alpha
l^{2}+l^{2}r_{+}^{2}+r_{+}^{4})}{l^{2}r_{+}(g^{2}+r_{+}^{2})(r_{+}^{2}+2\alpha)}\bigg{)}.$
(26)

From the expression of Hawking temperature it is clear that when $k=0$ it
identifies to the Hawking temperature of the $AdS$ EGB Bardeen black hole
Singh:2020xju  and and to the $AdS$ EGB regular black hole Singh:2020mty  for
$g=0$. In the limit of $\alpha=0$, the Hawking temperature reduces to

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi
r_{+}}\bigg{(}\frac{3r_{+}^{7}+l^{2}r_{+}^{4}(9r-2k)-k(g^{2}+r_{+}^{2})(l^{2}r_{+}^{2}+r_{+}^{4})}{l^{2}r^{3}_{+}(g^{2}+r_{+}^{2})(r_{+}^{2}+2)}\bigg{)}.$
(27)

In order to analyze the behavior of temperature for the various values of
magnetic charge $g$ and the deviation parameter $k$, we plot Fig. 2.

---
Figure 2: Temperature versus horizon radius for different valued of magnetic
monopole charge $(g)$ and deviation parameter ($k$) with fixed values of GB
parameter $\alpha=0.1$ and $\alpha=0.2$. Here, we set $l=10$.

Here, we see that the temperature first grows to a maximum value of
$T_{+}^{max}$ and then it drops to a minimum value, and then it starts growing
again.

The maximum values of temperature for different values of parameters are
tabulated in TABLES 2 and 3.

|  $\alpha=0.1$, $k=0.1$ |  |  |  $\alpha=0.2$, $k=0.1$ |
---|---|---|---|---|---
$g$ | $r$ | $T_{+}^{max}$ | $g$ | $r$ | $T_{+}^{max}$
0.00 | 0.89 | 0.055 |  0.00 | 1.22 | 0.042
0.20 | 1.12 | 0.047 |  0.20 | 1.42 | 0.038
0.40 | 1.59 | 0.036 |  0.40 | 1.86 | 0.032
Table 2: The values of maximum Hawking temperature for the $4D$ AdS EGB regular black hole with different values of parameters and fixed value of $l=10$. |  $\alpha=0.1$, $g=0.1$ |  |  |  $\alpha=0.2$, $g=0.1$ |
---|---|---|---|---|---
$k$ | $r$ | $T_{+}^{max}$ | $k$ | $r$ | $T_{+}^{max}$
0.10 | 0.95 | 0.052 |  0.10 | 1.27 | 0.040
0.30 | 1.25 | 0.039 |  0.30 | 1.62 | 0.035
0.50 | 1.60 | 0.032 |  0.50 | 2.00 | 0.029
Table 3: The values of maximum Hawking temperature for the $4D$ AdS EGB
regular black hole with different values of parameters and fixed value of
$l=10$.

From these TABLES, we see that the values of maximum temperature decrease with
an increase in the values of $g$ and $k$. The maximum temperature diverges
when the horizon radius shrinks to zero. Fig. 2 also suggests that the Hawking
temperature vanishes when the two horizons (Cauchy and event) merge.

The thermodynamical quantities must follow the first law of thermodynamics

$dM=T_{+}dS_{+}.$ (28)

Since we have explicit expressions of mass and Hawking temperature, then the
first law gives the value of entropy as

$\displaystyle S_{+}=\pi r_{+}^{2}+4\alpha\log[r_{+}].$ (29)

Now, we are interested in the stability of this black hole. We are aware that
local stability is decided by the signature of heat capacity. The positive
indicates that the system is stable and the negative indicates the unstable
one sabir ; dvs1 ; kumar1 . However, the global stability can be studied from
the Gibbs free energy.

Now, we first calculate from the given formula

$C_{+}=\frac{\partial M}{\partial T_{+}}=\bigg{(}\frac{\partial M}{\partial
r_{+}}\bigg{)}\bigg{(}\frac{\partial r_{+}}{\partial T_{+}}\bigg{)}.$ (30)

This gives

$C_{+}=-\frac{2\pi\left(g^{2}+r_{+}^{2}\right)^{5/2}e^{k/r_{+}}\left(2\alpha+r_{+}^{2}\right)^{2}\bigg{(}-k\left(g^{2}+r_{+}^{2}\right)\left(l^{2}\left(\alpha+r_{+}^{2}\right)+r_{+}^{4}\right)+\mathcal{X}\bigg{)}}{r_{+}^{3}\bigg{(}2l^{2}g^{4}r_{+}\left(2\alpha+r_{+}^{2}\right)^{2}+2k\left(g^{2}+r_{+}^{2}\right)^{2}\mathcal{W}+\mathcal{Y}+\mathcal{Z}\bigg{)}},$
(31)

where

$\displaystyle\mathcal{W}=l^{2}\left(2\alpha^{2}+r_{+}^{4}+2\alpha
r_{+}^{2}\right)-2\alpha r_{+}^{4},$ (32)
$\displaystyle\mathcal{X}=-2g^{2}l^{2}r_{+}\left(2\alpha+r_{+}^{2}\right)+l^{2}r_{+}^{3}\left(r_{+}^{2}-\alpha\right)+3r_{+}^{7},$
(33)
$\displaystyle\mathcal{Y}=g^{2}r_{+}^{2}\left(l^{2}\left(7r_{+}^{5}+31\alpha
r_{+}^{3}+22\alpha^{2}r_{+}\right)+9r_{+}^{7}+30\alpha r_{+}^{5}\right),$ (34)
$\displaystyle\mathcal{Z}=r_{+}^{4}\left(l^{2}\left(-r_{+}^{5}+5\alpha
r_{+}^{3}+2\alpha^{2}r_{+}\right)+3\left(r_{+}^{7}+6\alpha
r_{+}^{5}\right)\right).$ (35)

From the Eq. (31), it is evident that the heat capacity depends on all the
parameters $g$, $k$, $\alpha$, and $\Lambda$. For $k=0$, it coincides with the
mass of the heat capacity of $AdS$ EGB Bardeen black hole Singh:2020xju . The
heat capacity in absence of the GB parameter reduces to

$\displaystyle
C_{+}=-\frac{2\pi\left(g^{2}+r_{+}^{2}\right)^{5/2}e^{k/r_{+}}r_{+}^{4}\bigg{(}-k\left(g^{2}+r_{+}^{2}\right)\left(l^{2}\left(r_{+}^{2}\right)+r_{+}^{4}\right)+\mathcal{X}\bigg{)}}{r_{+}^{3}\bigg{(}2l^{2}g^{4}r_{+}\left(r_{+}^{2}\right)^{2}+2k\left(g^{2}+r_{+}^{2}\right)^{2}\mathcal{W}+\mathcal{Y}+\mathcal{Z}\bigg{)}},$
(36)

with

$\displaystyle\mathcal{W}=l^{2}r_{+}^{4},\qquad\qquad\mathcal{X}=\left(-2g^{2}l^{2}r_{+}^{3}+l^{2}r_{+}^{5}+3r_{+}^{7}\right),$
(37)
$\displaystyle\mathcal{Y}=g^{2}r_{+}^{2}\left(7r_{+}^{5}l^{2}+9r_{+}^{7}\right),\qquad\text{and}\qquad\mathcal{Z}=r_{+}^{4}\left(-l^{2}r_{+}^{5}+3r_{+}^{7}\right).$
(38)

Here, we observe that the expression of heat capacity we obtain is the most
general one. The various limiting case can be studied just by switching
various parameters off.

Now, the stability can be checked when one observes the plots of heat capacity
for fixing the parameters as depicted in FIG. 3.

---
Figure 3: The heat capacity $C_{+}$ versus horizon radius with the different
values of $g$ and $k$ for two fixed value of GB parameter $\alpha=0.1$ and
$\alpha=0.2$.

The heat capacity is showing discontinuity at a particular point (say critical
radius $r_{+}=r_{C}$) which signifies a second-order phase transition.
Remarkably, at this point of critical radius, the temperature attains the
maximum value $T_{+}^{max}$. Hence, a phase transition occurs in-between
stable and unstable phases when the black hole gets bigger.

Now we turn to the global stability of the black hole, which is characterized
by Gibbs free energy. The Gibbs free energy can be estimated by standard
definition $F_{+}=M-T_{+}S_{+}$, as follows

$\displaystyle F_{+}$ $\displaystyle=$
$\displaystyle\frac{\left(g^{2}+r_{+}^{2}\right)^{3/2}\left(\alpha
l^{2}+l^{2}r_{+}^{2}+r_{+}^{4}\right)e^{k/r_{+}}}{2l^{2}r_{+}^{4}}$ (39)
$\displaystyle+$ $\displaystyle\frac{k(g^{2}+r_{+}^{2})(\alpha
l^{2}+l^{2}r_{+}^{2}+r_{+}^{4})-3r_{+}^{7}-l^{2}r_{+}^{2}(\alpha+r_{+}^{2})(9r_{+}-2k)}{4l^{2}(g^{2}+r_{+}^{2})(r_{+}^{2}+2\alpha)}.$

Since, global stability of the black hole is confirmed by condition $F_{+}\leq
0$. Now, to analyze the behavior of this expression, we plot FIG. 4.

---
Figure 4:  Gibbs free energy versus horizon radius for different values of $g$
and $k$ with fixed values of GB parameter $\alpha=0.1$ and $\alpha=0.2$.

Here, we observe that there exists a local minimum ($r_{min}$) and maximum
($r_{max}$) which are in agreement with the extremal points of Hawking
temperature. At this point free energy change nature. After this point
${r_{min}}$, the free energy increases with horizon radius $r$ and attains the
maximum value at $r_{max}$. After this point, the free energy decreases with
the horizon radius.

##  IV  $P-v$ Criticality and Phase Diagrams

Now, we plan to explore the critical behavior of the present black hole
solution in an extended phase space, by considering $\Lambda$ as the
thermodynamic pressure. The cosmological constant appears in the pressure as
$P=-\frac{\Lambda}{8\pi}=\frac{3}{8\pi l^{2}}$. Plugging the value of $l$ from
Eq. (26), the form of pressure becomes

$\displaystyle
P=\frac{3T(r_{+}^{6}+2r_{+}^{4}\alpha+g^{2}r_{+}^{2}(r_{+}^{2}+2\alpha))}{2r_{+}^{4}(g^{2}k+(k-3r_{+})r_{+}^{2})}+\frac{3(r_{+}^{5}-r_{+}^{2}+\alpha+(r_{+}^{2}+\alpha)[k(r_{+}^{2}+g^{2})+2g^{2}r_{+}])}{8\pi
r_{+}^{4}(g^{2}k+(k-3r_{+})r_{+}^{2})}.$ (40)

The enthalpy in the case of the black hole system is described by the total
mass of the system. Therefore, the thermodynamic volume can be calculated as

$\displaystyle V=\left(\frac{\partial M}{\partial P}\right)_{S_{+}}=\frac{4\pi
r_{+}^{3}}{3}.$ (41)

The critical temperature and critical pressure can be determined from the
conditions:

$\left.\frac{\partial P_{+}}{\partial
r_{+}}\right|_{T_{+}}=\left.\frac{\partial^{2}P_{+}}{\partial
r_{+}^{2}}\right|_{T_{+}}=0.$ (42)

For the given pressure (40), these conditions lead to the following equation:

$\displaystyle
r^{7}(45g^{4}k+36g^{4}r+30kg^{2}r^{2}+45g^{2}r^{3}+9kr^{4}-3r^{5})+r^{3}\alpha(4g^{6}k+156r^{4}kr^{2}+204g^{4}r^{3}+$
$\displaystyle
84g^{2}kr^{4}+360g^{2}r^{5})+\alpha^{2}(132kg^{4}r^{2}-16g^{2}k^{2}-24g^{6}kr-48g^{4}k^{2}r^{2}+360g^{4}r^{4}-48g^{2}k^{2}r^{4}+$
$\displaystyle 144kg^{2}r^{5}+588g^{2}r^{6}-16k^{2}r^{6}+84kr^{7}+36r^{8})=0.$
(43)

The critical points and the horizon radius can be determined by solving the
above equation. However, one can not solve this equation analytically but can
determine the critical radius $r_{C}$, critical pressure $P_{C}$, and
temperature $T_{C}$ numerically. We write the numerical results in TABLE 4 for
various values of deviation parameter and magnetic charge. From the table, we
find that the value of critical radius increases along with magnetic charge
$g$. However, the value of the critical radius decreases along with the
deviation parameter $k$. Interestingly, the universal ratio
$\frac{P_{C}r_{C}}{T_{C}}$ increases along with both the magnetic charge $g$
and deviation parameter $k$.

$g$ | $r_{C}$ | $T_{C}$ | $P_{C}$ | ${P_{C}\,r_{C}}/{T_{C}}$
---|---|---|---|---
0.1 | 0.8636 | 0.0884 | 0.0153 | 0.1494
0.2 | 1.1144 | 0.0609 | 0.0108 | 0.1976
0.3 | 1.4071 | 0.0373 | 0.0075 | 0.2829
0.4 | 1.717 | 0.0195 | 0.0054 | 0.4754
0.5 | 2.039 | 0.0063 | 0.0040 | 1.2946
$k$ | $r_{C}$ | $T_{C}$ | $P_{C}$ | ${P_{C}\,r_{C}}/{T_{C}}$
0.1 | 0.9527 | 0.0841 | 0.0153 | 0.1732
0.2 | 1.2151 | 0.0501 | 0.0096 | 0.2328
0.3 | 1.4022 | 0.0342 | 0.0073 | 0.2993
0.4 | 1.5798 | 0.0245 | 0.0061 | 0.3858
Table 4: Values of critical temperature $T_{C}$, critical pressure $P_{C}$ and
universal ration $P_{C}\,r_{C}/T_{C}$ corresponding to different values of $g$
and $k$ for $\alpha=0.1$.

In order to see the behavior of pressure, we plot expression of pressure (40)
in FIG. 5.

---
Figure 5: The plot of pressure versus horizon radius for different values of
$T_{+}$ and $k$ with $\alpha=0.1$ and $\alpha=0.2$ respectively.

##  V Connection between phase transition and Shadow radius

To compute the geodesics in the spacetime, we first focus to the motion of
photon in the the horizon of black hole solution (14). The Lagrangian for a
photon constrained to move in the equatorial plane $(\theta=\pi/2)$ can be
expressed as

${\cal{L}}=-f(r){dt}^{2}+g_{rr}{\dot{r}}^{2}+\frac{1}{f(r)}{d\theta^{2}}+r^{2}{d\phi}^{2}.$
(44)

The corresponding Hamiltonian is written as

${\cal{H}}=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}=\frac{1}{2}\left[-\frac{p_{t}^{2}}{f(r)}+{p_{r}}^{2}f(r)+\frac{p_{\phi}^{2}}{r^{2}}\right].$
(45)

The equations of motion can be calculated using the Hamiltonian formalism as
follows

${\dot{t}}=\frac{\partial{\cal H}}{\partial
p_{t}}=-\frac{p_{t}}{f(r)},\qquad\qquad{\dot{\phi}}=\frac{\partial{\cal
H}}{\partial
p_{\phi}}=\frac{p_{\phi}}{r^{2}},\quad\text{and}\quad{\dot{r}}=\frac{\partial{\cal
H}}{\partial p_{r}}={p_{r}}f(r).$ (46)

where dot denotes the derivative with respect to affine parameter. Using
equations of motion and conserved quantities, the null geodesics equation is
given by

${\dot{r}}^{2}+V_{eff}(r)=0,\qquad\text{with}\qquad
V_{eff}=f(r)\left(\frac{J^{2}}{r^{2}}-\frac{E^{2}}{f(r)}\right).$ (47)

For a circular null geodesics which describes the radius of the photon sphere,
the effective potential must follow

$V_{eff}=0,\qquad\text{and}\qquad\frac{\partial V_{eff}}{\partial r}=0.$ (48)

The equation of the photon radius ($r_{p}$) can be estimated by Eq. (48). But
this equation (48) can not be solve analytically, so we solve it numerically
and the values are tabulated in the TABLE 5.

g |  | 0.1 |  | 0.3 |  | 0.5 |  | 0.7
---|---|---|---|---|---|---|---|---
$r_{p}$ |  | 0.12 |  |  0.099 |  |  0.097 |  |  0.095
$r_{s}$ |  | 0.142 |  |  0.098 |  |  0.097 |  |  0.096
$k$ |  |  0.1 |  |  0.3 |  |  0.5 |  |  0.7
$r_{p}$ |  | 0.12 |  |  0.491 |  |  0.711 |  |  0.
$r_{s}$ |  | 0.142 |  |  0.605 |  |  0.844 |  |  0.
$\alpha$ |  |  0.1 |  |  0.3 |  |  0.5 |  |  0.7
$r_{p}$ |  | 0.12 |  |  0.111 |  |  0.108 |  |  0.
$r_{s}$ |  | 0.142 |  |  0.122 |  |  0.116 |  |  0.
Table 5: The numerical values of photon radius and shadow radius corresponding
to different values of magnetic charge, deviation parameter and GB coupling
constant.

We can obtain the shadow radius of the black hole. The shadow radius of the
black hole is determined by

$r_{s}=\sqrt{\alpha^{2}+\beta^{2}}=\frac{L_{p}}{E}=\frac{r}{\sqrt{f(r)}}|_{r=r_{p}}.$
(49)

Here, we plug the value of $f(r)$ (14) into above equation (49) and we
substitute the numerical values of $r_{p}$ as presented in TABLE 5. From the
TABLE 5, we notice that the photon radius increases with the deviation
parameter and decreases with the $g$ and $\alpha$. The celestial coordinates
$\alpha$ and $\beta$ Kumar:2019pjp ; Singh:2017vfr ; Ahmed:2022qge ;
Ahmed:2020dzj  for our black hole solution are given by

$\displaystyle{\bf
x}=\lim_{r\to\infty}\left(\frac{rp^{\phi}}{p^{t}}\right),\qquad\text{and}\qquad{\bf
y}=\lim_{r\to\infty}\left(\frac{rp^{\theta}}{p^{t}}\right).$ (50)

The shadow of the obtained solution for different values of magnetic charge
$(g)$ and deviation parameter $(k)$ is depicted in FIG. 6.

---
Figure 6: Plot of shadow with different values of deviation parameter $(k)$,
magnetic charge $(g)$, GB coupling $(\alpha)$ and length parameter $l$.

From the figure, we observe the effects of different values of deviation
parameter $k$, magnetic charge $g$, GB coupling $\alpha$ and cosmological
$(\Lambda=-3/l^{2})$ the on the black hole shadow. Here, we find that

1) the size of the shadow increases with increases in the deviation parameter,

2) the size of the shadow decreases with an increase in the magnetic charge
and GB coupling constant,

3) and the size of the shadow does not change significantly with increase in
the value of $AdS$ length.

##  VI Conclusions

We have obtained a general $AdS$ regular black hole solution for EGB gravity
coupled with non-linear electrodynamics. Interestingly, in the limiting cases,
our solution coincides with AdS EGB black hole, Bardeen black hole, AdS
regular black hole and AdS Schwarzschild black hole. The horizons of this
black hole solution are estimated numerically. We have also analysed the
effects of deviation parameter, magnetic charge and GB parameter on the
horizon structure. The nature of singularities of obtained solution is also
checked by calculating curvature invariants which in turn signifies regular
space-time.

Thermodynamics of this general $AdS$ regular black hole solution is also
investigated. First of all, we have computed total mass of the system and
found that (a) switching off deviation parameter the calculated mass of our
system coincides with the mass of the $AdS$ EGB Bardeen black hole
Singh:2020xju , (b) switching off magnetic charge this coincides with the mass
of $AdS$ EGB regular black hole Singh:2020mty , (c) switching off both the
deviation parameter and magnetic charge this identifies to the mass of $4D$
EGB black hole gla , (d) switching off both the GB parameter and deviation
parameter this reduces to the mass of Bardeen black hole tzi , (e) switching
off both the GB parameter and magnetic charge this reduces to the mass of
$AdS$ regular black hole and finally (f) switching all three GB parameter,
deviation parameter and magnetic charge this identifies with the mass of $AdS$
Schwarzschild black hole. Furthermore, we have determined Hawking temperature
and entropy of the system. The behavior of temperature is also depicted in
plots. Moreover, we have studied stability of the system by estimating heat
capacity. From the plot, we have observed that heat capacity has discontinuity
at a critical radius which signifies a second-order phase transition. At
critical radius, the temperature gains its maximum value. This signifies that
a phase transition occurs when the size gets bigger.

In addition, $P-v$ criticality of the obtained black hole solution is also
discussed. Here, in summery, we found that, even though the critical radius is
an increasing function of magnetic charge and decreasing function of deviation
parameter, the universal ratio $\frac{P_{C}r_{C}}{T_{C}}$ increases along with
both the magnetic charge $g$ and deviation parameter $k$.

Ultimately, in order to discuss black hole shadow, we have estimated the
photon radius and shadow radius and their dependence on the different values
of magnetic charge and deviation parameter. To study the behavior of black
hole shadow, we have plotted diagrams as well.

###### Acknowledgements.

One of us (D.V.S.) acknowledges University Grant Commission for the start-up
grant (No.30-600/2021(BSR)/1630).

## Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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