Lightning Network Economics: Topology

1. Introduction

As originally conceived, the Bitcoin payment system records every transaction on its blockchain. This design feature, coupled with the blockchain’s limited capacity, imposes a ceiling on the throughput of the Bitcoin payment system. That ceiling amounts to fewer than 10 transactions per second—a low number in comparison with other payment systems, such as Visa. However, that limited throughput can be expanded, even substantially so, by payers committing to transactions off-chain and only occasionally posting these commitments on-chain. The Lightning Network (LN) is such a device (Poon and Dryja 2015).

The Lightning Network consists of nodes and channels that connect pairs of nodes. Each channel is associated with a net payment flow through it. Two nodes are directly connected if there is a channel between them and are connected if there is a path of channels that enables flows between the nodes. Thus, any node on an LN can pay any other node in the same connected component of the LN. The LN is the leading example of a class of layer-two solutions called payment channel networks (PCNs). This paper’s results apply also to other PCNs.

The cost of a channel in the LN arises from locking funds in the channel and from the fees that occasionally need to be paid to rebalance it.¹ At the leading order, the cost of a channel is an increasing function of the average net flow.² In particular, in a symmetric channel, the flows offset each other, and the overall cost is negligible in comparison with a channel with nonzero net flow. Thus, at the leading order, the cost of a channel between Alice and Bob is approximately the same, whether (i) Alice, on average, sends Bob one unit annually or (ii) Alice, on average, receives one unit from Bob annually or (iii) Alice, on average, sends Bob 11 units annually and Bob, on average, sends her 10 units annually. The cost-minimizing network is always connected because symmetric channels, which have negligible costs, can be added to connect all components as needed. (Guasoni et al. (2024) offer the details.)

The LN is designed to support a given set of flows between a given set of nodes. In principle, these flows could all be posted on the blockchain, but directing them through the LN is less costly. Starting with an arbitrary LN, the paper considers cost-reducing modifications of that LN.

The present paper identifies channel configurations that are excessively costly and offers procedures to replace them with less costly configurations. The first part of the paper shows that a cost-reducing reconfiguration is available if there are two distinct channels attached to the same node, which is the origin in one of the channels and the destination in the other. Direct corollaries of this result are (i) the presence of a cycle with an odd number of channels implies the availability of a cost-reducing channel reconfiguration and (ii) a cost-minimizing LN is always a bipartite graph. The common intuition of both findings is that, in a cost-minimizing configuration, all the flow should be from net payers to net payees, avoiding intermediate steps.

The second part of the paper considers adding a new node to the network, connecting a set of existing nodes to that new node and deleting all the other preexisting channels connecting these nodes to each other. In this new star-like configuration, all flows among the set of nodes go through the star’s center, which is the newly added node. The paper’s second part articulates (i) the conditions that the set of nodes needs to meet for the reconfiguration to be cost-reducing; (ii) a polynomial-time algorithm to find such a set of nodes in a given LN; (iii) a proof that such a set must exist if the average number of channels involving each node is large enough; (iv) a proof that such a set is very likely to exist if the network flows are chosen randomly, according to a distribution whose tails are not thicker than a power law.

The third and last part of the paper establishes bounds on the minimum possible cost of an LN that supports a given set of flows. The upper bound is at most twice as large as the lower bound. A star, in which a single hub connects all nodes, in general, is not the least costly topology, but is no more than twice as costly. Finding the exact cost-minimizing topology is an NP-complete problem in general. However, the concluding result describes a quadratic-complexity algorithm to construct a two-factor approximation of the minimum, which is robust to the improvements identified in the paper.

The paper’s results consider the cost of LN topology from the viewpoint of aggregate welfare, abstracting from the issue of assigning such costs to individual nodes. The rationale for this analysis lies in the familiar Coase (1960) theorem, whereby aggregate cost savings can be achieved by rational participants who are willing to share them. Identifying protocols that encourage participants to share aggregate savings is an important topic that lies beyond the scope of this paper.

The next section briefly reviews the literature on the topology of the lightning network and its implications. The following section presents the model of the LN and the cost functions of channels. Section 3 contains the main results of the paper. Section 4 presents an algorithm that gives an LN that approximates the minimal cost possible, and Section 5 contains empirical results. Concluding remarks are in Section 6. Appendices A–D contain the proofs of the results. Appendix E provides an example of cost function that meets the definitions in the model, whereas Appendix F reports the statistics of the lightning network snapshots examined in the empirical section.

1.1. Related Work

Guasoni et al. (2024) study the economics of channels, (i) identifying conditions for two parties to optimally establish a channel, (ii) finding explicit formulas for channel costs, (iii) obtaining the optimal collateral and savings entailed, and (iv) deriving the resulting reduction in congestion of the blockchain. Brânzei et al. (2022) obtain partial results on the cost of channels.

Ersoy et al. (2020) discuss game-theoretic aspects of channel construction. In Sali and Zohar (2020), together with game-theoretic questions, the authors show that, under the assumption of all channels being symmetric, the cost of an LN with a star topology is at most twice the minimal possible cost. Theorem 4 and Corollary 4 in this paper generalize this result to any set of channels, symmetric or not.

Several papers study the current topology of the LN (e.g., Lin et al. 2020, Martinazzi and Flori 2020, Seres et al. 2020), observing that the topology is tending toward the centralized hub-and-spoke structure. In Bartolucci et al. (2020), the authors predict the future topology of the LN using tools from percolation theory, abstracting from economic incentives.

Network topology has significant security implications. Rohrer and Tschorsch (2020) introduce an attack that allows routing nodes to learn the destination of a payment—information meant to be private. Empirical results concerning privacy attacks in the LN are in Kappos et al. (2021). Channels may be rendered unusable by an adversary performing a denial-of-service attack. This and similar attacks are discussed in Rohrer et al. (2019). (For mitigation strategies, see Shikhelman and Tikhomirov 2022.) Attacks that aim to steal funds by flooding the base blockchain (Harris and Zohar 2020, Sguanci and Sidiropoulos 2023) become significantly easier under certain topologies. The general robustness of the network also heavily depends on its structure as shown in Lee and Kim (2020).

Liquidity management and throughput maximization in the LN influence both the costs of routing nodes and the success probability of payment routing. Several papers (e.g., Pickhardt and Nowostawski 2020; Sivaraman et al. 2020; Papadis and Tassiulas 2022, 2023) study the state of the current network and suggest potential improvements to the protocol.

Payments in the LN and in PCNs in general often use a route between two nodes instead of opening a direct channel. Contemporary routing algorithms take into account not only fees, but also privacy, efficiency, success probability, and other parameters. Examples of such algorithms include Grunspan et al. (2020), Roos et al. (2017), Tang et al. (2020), Varma and Maguluri (2021), Wang et al. (2019), and Yu et al. (2018). The performance of these algorithms depends on the topology of the network.

2. The Model

The starting point is a network of n nodes (or users) with some desired rates of flow. Each user i has some required payment rate ${\varphi }_{ij}$ (positive or negative) from each other user j. The flow $f_i={\sum }_{j=1}^n{\varphi }_{ij}$ represents the aggregate of such required payments. The problem is how to organize their execution so that each user’s net flow is $f_i$. In other words, though the problem’s original inputs may be the required payment rates, its solution only depends on aggregate net flows, which are the focus of the discussion that follows. For example, Alice might send at rate three to Bob and receive at rate one from Carol. Thus, her overall flow is $-3+1=-2$. If Bob receives three from Alice and does not send or receive from anyone else, his flow is three.

Denote the sequence of net flows as $\mathcal{F}={(f_i)}_{i=1}^n$. Because each payment is debited to the payer and credited to the payee, the sum of all the flows is zero, that is,

At the modeling level, this accounting identity means that $f_i$ represents the aggregate flow to node i within the network, excluding outside means of payments, such as on-chain transactions or physical transfers.

To facilitate these flows over the LN, channels must be established. A flow between two nodes may or may not be supported by a channel connecting them. Rather, it can be supported by a path of connected channels through which the flow is routed with the flow starting in one node (the origin) and ending in the other (the destination). Thus, a node need not have a channel with every other node with which it interacts. If Alice has channels with both Bob and Carol, Bob can send funds to Carol through Alice even in the absence of a direct channel between Bob and Carol.

A channel’s size is the amount locked in it. That amount entails a cost that depends on the expected flow through the channel. We assume that the parties to a channel choose the cost-minimizing size. Moreover, cost minimization implies that the amount locked by each party increases with the relative frequency of payments sent rather than received. (Guasoni et al. (2024) offer a formal development of these results.) Thus, the cost of the channel is a function of the flow. The cost function is general, representing the leading order of the cost of a channel and relying only on a few assumptions that capture some basic properties: it must be increasing in net flow, invariant to its direction, and null for channels with zero net flow (including absent channels).

2.1. Channel Network

Channels connect nodes to support the required net flows. For two nodes, say u and v, denote by ${\lambda }_{u,v}$ the net flow from u to v. If there is no channel between u and v, then ${\lambda }_{u,v}=0$. Thus, the matrix $\mathrm{\Lambda }={({\lambda }_{i,j})}_{i,j=1}^n$ is, by definition, antisymmetric as ${\lambda }_{u,v}=-{\lambda }_{v,u}$. For example, if the channel is between u and v with u sending three and v sending one, then the net flow of the channel is two. Denote the above channel as ${}^1{\overleftrightarrow{uv}}^3$.

A given LN ${({\lambda }_{i,j})}_{i,j=1}^n$ supports the underlying payment flow $\mathcal{F}$ if, for every node i, the sum of net flows in the channels involving i sums up to the net flow of i, that is,

In particular, the matrix ${({\lambda }_{i,j})}_{i,j=1}^n$ describes the flows along each edge, whereas the vector $\mathcal{F}={(f_i)}_{i=1}^n$ describes the net flows to each node.

2.2. Cost Function

In general, the cost of a channel may depend on a variety of properties, including the statistical properties of arrival times and sizes of payments (as well as the cost of opening and closing a channel, which is common to all channels). Yet both models of money demand (Baumol 1952, Tobin 1956, Miller and Orr 1966) and recent models of Lightning channels (Brânzei et al. 2022, Guasoni et al. 2024) highlight that, at first order, a sufficient statistic of a channel’s cost is its net average flow. In particular, the cost of channels with zero net flow is of second order.

Motivated by these observations, this paper offers a first order analysis in which a channel’s cost is modeled as a function of its average flow. Rather than providing an analysis based on a specific cost function, the results are established for a general cost function $c:\mathbb{R}\to {\mathbb{R}}_{+}$ that satisfies the following structural conditions:

• (MO) $c(x)>c(y)$ for $x>y\ge 0$ (monotonicity).

• (SY) $c(-x)=c(x)$ (symmetry).

• (CI) $c(0)=0$ (costless inaction).

• (SA) $c(x+y)<c(x)+c(y)$ for $x>y\ge 0$ (subadditivity).

Assumption (MO) stipulates that a channel’s cost increases in its average flow. When the flow is unidirectional, it is clear that a larger flow requires a bigger channel so as to reduce the frequency of resets, and a bigger channel implies a larger opportunity cost (creating a channel requires depositing its full amount, which cannot earn interest or be used for other purposes). Further, if the same average flow arises from bidirectional payments, the required channel is even bigger because a party’s balance may be depleted also because of random fluctuations in payments’ directions.

Assumption (SY) specifies that a channel’s cost depends on the average flow without regard to the identities of the two parties: if Alice pays Bob at unit rate or vice versa, the optimal channel’s cost is the same. This property stems from the decentralized nature of PCNs, whereby the same rules and protocol apply to all users without distinction. Note also that the cost function refers to the total cost of operating a channel regardless of how such costs are allocated between the two parties.

Assumption (CI) has a dual meaning. First, it recognizes that the absence of a channel does not generate any costs. Second, it implies that the cost of an average zero flow, such as a symmetric random walk, is negligible in comparison with an average positive flow. Intuitively, the cost of symmetric flows stems entirely from random fluctuations, which occasionally deplete a channel’s balance even in the absence of directional flow and are second order to average flow. This intuition originates from the comparison of unidirectional (Baumol 1952, Tobin 1956) and symmetric (Miller and Orr 1966) models of money demand and is confirmed in recent models of payment channels (Brânzei et al. 2022, Guasoni et al. 2024). This assumption is appropriate in this paper because it aims to study first order effects, which depend on average flows alone.

Assumption (SA) states that combining flows in a single channel is cheaper than handling each of them in a separate channel. This property is natural for three related reasons: First, merging the balances of two channels into one saves on reset costs, which are fixed, regardless of a channel’s size. (Because the total balance is the same, opportunity costs do not vary.) Second, a single channel may be able to handle payments of larger size (two channels of size one and two each cannot process a payment of 2.5, but a channel of size three can). Third, if channels’ balances are affected by random fluctuations, then a single channel can withstand more shocks before requiring a reset.³ In the cryptocurrency context, this property also reflects the fixed costs that creating a channel entails regardless of its size. Such costs are instead related to the space occupied on the blockchain and play the same role as withdrawal costs in models of money demand.

The results below on hermetization (Theorems 2 and 3) also require a slightly stronger property than subadditivity:

• (SA+) for some strictly increasing $g_{+}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$,

  $$\frac{c(kx)}{k}<\frac{c(x)}{g_{+}(k)}\hspace{1em}\hspace{1em}x,k>0.$$

Assumption (SA+) requires that channel costs exhibit economies of scale: as a channel’s flow is scaled by a factor, its average cost (defined as the channel’s cost divided by the factor) should decrease as the factor increases.⁴ In practice, in the typical setting of $c(x)\sim x^{\alpha }$ for some $\alpha \in (0,1)$, this property is satisfied for $g_{+}(x)=c{x}^{1-\beta }$ for $\beta >\alpha$ and some constant $c>0$. (Appendix E verifies that the cost function obtained in Guasoni et al. (2024) satisfies these properties.)

A channel ${}^{{\mu }_u}{\overleftrightarrow{uv}}^{{\mu }_v}$ maps to a flow of ${\lambda }_{u,v}={\mu }_u-{\mu }_v$. Denoting by $c{(}^{{\mu }_u}{\overleftrightarrow{uv}}^{{\mu }_v})$ the cost of the specific channel between u and v, it equals the cost function of the net flow:

Let G be an LN with net flows ${({\lambda }_{i,j})}_{i,j=1}^n$, the cost of G is defined as the sum of the costs of all of the channels:

3. Main Results

This section contains the statements of the main results. Given an underlying payment flow $\mathcal{F}={(f_i)}_{i=1}^n$ and an LN that supports it, consider different cost-reducing changes to the LN. The first change is to locally restructure channels among a subset of nodes in the LN. The second change is the addition of a new node that serves as a hub for several nodes that previously had channels between them. The problem is to understand which topology minimizes the overall cost among the topologies that support $\mathcal{F}$.

Note that the present analysis focuses on recurring costs, that is, costs per unit of time. Thus, the costs of replacing the channels of one topology with the channels of another topology, which are one off, are neglected. Of course, the rationale is that small recurring savings over time exceed their initial one-off costs.

3.1. Local Channel Reconfiguration

First, consider the case in which a given set of nodes reconfigures the channels between them to reduce the overall cost. In general, a smaller node set is more likely to reach an agreement on channel reconfiguration.

Theorem 1 states that, given three nodes u, v, and w, if there is a channel directed from u to v and a channel directed from v to w, then this topology is not cost minimizing. It offers a cost-reducing improvement, which is summarized in Figure 1.

An important message of this result is that cost minimization fosters disintermediation in that it is inefficient for a node to merely forward others’ payments. Instead, Figure 1 shows that a natural partner is someone with strong personal flow with one party who can benefit from offsetting some of this flow by forwarding payments in the opposite direction, thereby extending a channel’s life and abating reset costs. Such a partner is not a traditional intermediary for two reasons. First, payments forwarded on behalf of others are less than the partner’s own flow. Second, in a competitive equilibrium, the partner would not charge any fee to forward such payments and would even accept a small negative fee—if it were possible⁵—because forwarding payments to offset personal flows reduces one’s costs.

A simple example helps illustrate this point: imagine that Alice’s employer (u) pays Alice’s bank (v), which pays Alice (w). This arrangement is not optimal: if Alice’s salary ${\lambda }_1$ is greater than Alice’s net payments ${\lambda }_2$ (because she uses some of the salary to buy goods and services), then the employer should pay Alice in relation to her net flow, keeping a channel with her bank. However, if Alice’s net payments are higher than her salary (because she has additional income in excess of expenses), the employer is better off paying her gross salary, leaving her to independently manage unrelated flows.

Note also that Theorem 1 does not require the subadditivity condition (SA). In this sense, disintermediation is a very basic feature of payment-channel networks, independent of economies of scale (or lack thereof) that underpin subadditivity. Instead, disintermediation results from avoiding multiple transfers in the same direction.

Two corollaries follow from Theorem 1: (i) an odd cycle is not cost minimizing, and (ii) a cost-minimizing LN must be a bipartite graph.

A stylized implication of Corollary 1 is that a cost-minimizing topology should have no triangles, and therefore, its clustering coefficient should be zero. (The clustering coefficient is the fraction of connected triples that are also triangles. See Section 5 for details.) Section 5 examines empirically this testable prediction and finds that the LN has indeed an unusually low clustering coefficient in comparison with other payment networks.

3.2. Adding a Routing Node

Centralization is a natural means to simplify a network by routing flows through a single node, thereby requiring no more than a channel for each user. In the case of the Lighting Network, the question is whether a low number of channels also translates in low aggregate costs. A broader goal is to understand whether economic forces, such as collective cost minimization, may progressively drive the network’s topology toward a central structure. This section examines the extent to which centralization may reduce costs.

Consider a set of nodes that can benefit from deleting all the channels between them and connecting the nodes to a new node that serves as a single central hub. Such sets are defined as hermetizable. (See Figure 2.) An LN is hermetizable if it contains a hermetizable set.

Theorem 2 gives sufficient conditions for the existence of a hermetizable set in an LN and presents an efficient algorithm to find such a set if it exists. Theorem 3 proves that, under mild assumptions on the distribution of channel costs in the LN, the probability of the LN being hermetizable is high.

Note that, in principle, the central hub does not have to be a new node; it could be one of the existing nodes. When thinking about real-world examples, this prospect is not very likely. Nodes represent a specific individual or business and process the transactions to and from their represented entity plus the transactions that transit through them on behalf of others. Being a central hub is another specialization, which demands the continuous maintenance of several channels through refills, resets, and other liquidity-management measures. Thus, we conservatively assume that regular nodes are not amenable to take such a role, leaving it instead to specialized nodes. In principle, if one of the existing nodes became the hub, the cost of the LN would be slightly lower as its own traffic would not be required to pass through another hub. For this reason, both Theorems 2 and 3 hold even if an existing node were used as hub instead of a new node.

Note also that hermetized configurations are not optimal in general even if the central hub is chosen among existing nodes. Unless the central hub is the sole payer or the sole payee, one of the spokes is a payee and another a payer, resulting in payment forwarding, which is suboptimal according to Theorem 1. The main message of Theorems 2 and 3 is that, under mild assumptions, hermetization leads to lower—not minimal—costs.

To state Theorems 2 and 3 formally, the following definitions are necessary.

A natural question is whether the conditions of Theorem 2 are met by a typical LN. Theorem 3 shows that, if the LN is chosen randomly using a distribution with tails that are not thicker than a power, then, with high probability, its cost can be reduced by hermetization as in Theorem 2, indicating that cost reduction is the norm, not the exception.

Theorem 3 examines the situation in which flows ${\lambda }_{i,j}$ are chosen randomly (and independently) with the same distribution. If the distribution of the resulting costs $c({\lambda }_{i,j})$ in the LN has a tail thinner than power and each $c({\lambda }_{i,j})$ is either zero or has a uniform lower bound, then the LN is hermetizable with high probability.

3.3. Bounds on the Global Minimal Cost of an LN

Finally, consider the minimal possible cost of an LN that supports the flow $\mathcal{F}={(f_i)}_{i=1}^n$ and denote it by

The following bounds hold for any flow.

Recall that a star topology corresponds to a graph in which one node has edges with all others, and no other edges exist. The star topology is a natural candidate for the globally cost-minimal topology because it has the minimal number of edges for a connected graph. The global bounds in Theorem 1 clarify the conditions under which a star is optimal and the extent to which it is close to optimal.

Clearly, if there is only one net payer or only one net payee then a star with such a user as the star’s center minimizes costs because it achieves the lower bound in Theorem 1. The next corollary shows that this condition is also necessary, namely, if there are at least two payers and two payees, then the star does not minimize costs.

In addition, another implication of Theorem 4 is that the star topology is at most twice as costly as the minimum.

The results so far suggest that a few iterations of the improvements described above could lead to an exact minimal-cost topology. If so, the previous results would imply an exact optimization algorithm with quadratic complexity. In general, this is not the case. The main observation is that the cost-minimization problem for general cost functions nests combinatorial optimization problems that are known to be NP-complete.

The intuition of this result is relatively simple and is exemplified by a minimal example: if there are many payers and only two payees (or vice versa) and it is possible to assign each payer to a single payee so that both payees receive their flow, then finding the matching subset for a payee is equivalent to identifying (or determining the absence of) a subset of numbers with a given sum. This is the subset-sum problem, which is known to be NP-complete, because its solution requires us to examine nearly all subsets of payers (Karp 1972).

4. Algorithm

The NP-completeness of the exact cost-minimization problem and the previous results on cost reduction motivate the development of an algorithm that approximates the minimum, keeping computational complexity polynomial. In this spirit, the following quadratic-time algorithm yields a two-factor approximation: a network whose cost is less than twice the theoretical lower bound.

Algorithm 1 proceeds by first creating a star centered on node 1. Then, the function getpairs assigns to each other node a payer or payee number, depending on the sign of its flow. Then, payer–payee pairs are created (first payer with first payee, etc.): as each such pair is suboptimal by Theorem 1, the function disintermediate replaces it by a cheaper pair. The identification replacement continues until further improvements are no longer possible.

5. Lightning Network Structure

The preceding sections establish and study attributes of a cost-minimal Lightning Network, thus motivating the empirical examination of the visible parts of the actual LN. The goal is to understand the connectivity properties of the Lightning Network and evaluate the extent to which they are consistent with the implications of the above results.

The LN is decentralized. Its nodes use the gossip protocol to broadcast to each other newly created and deleted public channels. Thereby the information is synchronized across the nodes. (Synchronization delays of a few seconds are possible.) The Lightning Network does not allow entities that do not own a channel to learn anything beyond the channels’ existence and their size and fee policies. Enhanced privacy is one of the main features of the LN, and improvements are regularly implemented to reduce any leakage of information. Such built-in privacy features entail that nodes’ flows cannot be observed, thereby placing limits on testable implications. Yet a key necessary condition within the observable purview is the absence of cliques, which is evaluated quantitatively as low clustering. This section examines clustering in the LN over time and in comparison with other payment networks.

The data set in Table F.1 summarizes 23 monthly snapshots of public channels in the LN from October 2020 to July 2023 obtained from a Lightning node (Decker 2023). (Observations contain gaps from November 2021 to January 2022 and from August 2022 to April 2023. Data in the months immediately before and after the gaps may be incomplete but are, nevertheless, included for the sake of integrity.)

Publicly available information about each channel includes its size and its parties but not their respective balances or balance updates (i.e., payments from one party to another). Moreover, even nodes that act as payment intermediaries in chains of transfers do not know the origin and the destination of the payments they facilitate, but merely the identity of the two nodes from and to which they transfer funds.

Yet even the limited amount of publicly available information allows us to probe some implications of our results. Specifically, Corollary 1 states that a cost-minimizing network should not contain cycles of odd length and, in particular, triangles, that is, cycles of three nodes. In the parlance of graph theory, this property means that the clustering coefficient should be zero as now explained.

Defining a triplet as three nodes i, j, k connected by at least two channels, the clustering coefficient is the fraction of triangles (i.e., triplets connected by three channels) out of all triplets (i.e., connected by at least two channels). It is formally defined as⁶

Thus, in theory, Corollary 1 implies that it is suboptimal for three channels to exist in any triplet; hence, the numerator in this equation should be zero as should be the clustering coefficient.

In practice, the clustering coefficient need not be zero for multiple reasons. First, a channel may have its balance mostly shifted to one party, making the usual payer unable to make payments and indifferent to closing the channel. Yet the balance’s main owner may delay closure to times when on-chain transactions are cheaper (such as the weekend). Second, a channel may have already been closed but unilaterally, thereby requiring a delay (time lock) of several days to finalize. Third, the channel may serve other purposes than payments, such as liquidity management or redundancy (Papadis and Tassiulas 2020), which are not modeled here.

Notwithstanding these caveats, the broad message of Corollary 1 is that triangles are generally not cost-efficient and, therefore, should be infrequent in comparison with other payment networks with different cost structures. Figure 3 plots the clustering coefficient (along with the number of nodes and edges) of the LN snapshots over time. Two features are apparent despite the gaps in observations: the clustering coefficient is low and steadily decreases over time. (The slope of the regression line in Figure 3 is −0.98 and is significantly negative; $p<0.001$.)

The clustering coefficient never exceeds 6% in any of the snapshots considered: choosing three random nodes connected by two channels, it is very unlikely that also a third channel is present among them. In fact, even 6% is a rather low figure for a payment network. As a comparison, consider Venmo, a payment network owned by Paypal, in which users can send funds to each other. Venmo has both differences and similarities to the LN: as in the LN, funds held in Venmo accounts do not earn interest. However, unlike the LN, Venmo balances are held in a single user account rather than in the bilateral channels typical of the LN, and there is no cost in connecting to another user. Also, payments among users are handled centrally rather than routed though nodes.

Absent any incentive to minimize their number of connections, Venmo users create as many of them as their payees. Thus, the resulting network can be considered as a reference case in which channel costs are null. Empirically, Bhattacharya et al. (2020) and Zhang et al. (2017) report that the clustering coefficient in Venmo is 14.7%—more than twice the LN clustering of 6% in October 2020 and more than five times its 2.8% clustering in July 2023.

As, in the LN, clustering is several times smaller than it is in payment networks with costless connections, it is tempting to ascribe such difference to efficiency choices rather than to payment relationships. More broadly, one might interpret the declining clustering coefficient as a gradual evolution of the LN toward a more efficient structure, fueled by growth in its size, reach, and sophistication of participants. Yet these macroscopic network properties do not provide direct observation of the network’s payment efficiency per se because payment flows cannot be observed publicly in the LN.⁷ Thus, these topological changes could also be related to a natural lower clustering of active payment relationships in general, of Lightning adopters in particular, or may be due to other phenomena.

6. Conclusion

Blockchains are at the heart of multiple protocols designed to manipulate ledgers securely in a decentralized fashion. Low throughput is a weakness shared by many of these protocols. A way to address this weakness is to relegate transaction processing to a device outside the main blockchain, secure these transactions cryptographically off-chain, and periodically post a summary of the transactions on the blockchain, thereby incorporating these originally off-chain transactions into the consensus. The Lightning Network is one of the most popular layer-two solutions. It supports Bitcoin payments.

The term “Lightning Network” often refers to collections of paths that support payments between nodes that are not directly connected. Intermediate nodes of an LN can and often charge fees to facilitate the intermediation. Viability of the LN protocol requires cryptographically guaranteed communication between nodes along the payment route. (For details, see Guasoni et al. (2024).)

This paper focuses on cost reduction through modifications of the LN. LN modifications amount to modifications of the LN topology, which affects various performance aspects, such as payment success rate, node privacy, and censorship resistance. Thus, reconfiguration of the LN to reduce cost can have wide-ranging implications beyond mere cost reduction.

The paper shows that computing the general global minimum cost configuration is an NP-complete problem and then describes a quadratic-time algorithm that is guaranteed to achieve a two-factor approximation. It also addresses the intuitive conjecture that a star-like structure is cost minimal. It is not.

Although the LN protocol treats all nodes equally, each node can differentiate itself from others by offering competitive fee rates for forwarding payments, thereby introducing a layer of heterogeneity in the network and creating the potential for centralization.