Stability and Scalability of Blockchain Systems

This paper studies distributed consensus dynamics in blockchain networks and establishes the conditions under which such consensus can occur. The contributions of this paper are threefold.

Armed with these results, we analyze two natural constructions of blockchain DAGs, which we refer to as the tree and throughput-optimal policies. The tree policy is implemented in the Bitcoin and Ethereum blockchains, the two most widely-adopted blockchain systems [39, 12]; the throughput-optimal policy is introduced in [34]. We show that, if the network is stable, then any blockchain constructed under these two policies has a one-ended limiting DAG. Thus, if the network is stable, these two constructions are able to confirm infinitely many blocks in the limit as time $t\to\infty$.

Following the stability analysis, we use our bounds to assess the scalability of blockchain systems. A sequence of peer-to-peer networks $(H_{k})_{k\in\mathbb{N}}$ is scalable if there exists a positive block arrival rate $\lambda^{*}$ such that each network is stable with arrival rate $\lambda^{*}$. We determine a necessary condition for scalability, which in turn provides a sufficient condition for the lack thereof. We show as an example that sequences of peer-to-peer networks containing large stars not scalable.

Using measurements of the Bitcoin peer-to-peer network taken by [14] and [25], and traces of the Bitcoin peer-to-peer network taken by [10] we compare the performance of our model with a simulated Poisson block arrival input and with an input of traces of the Bitcoin blockchain system. We find that the assumption of Poisson block arrivals is a good approximation of the real process.

In Section 2, we present our stochastic network model and relevant definitions for our analysis. In Section 3 we identify a structural relationship between confirmed blocks and provide a sufficient condition for the existence of infinitely many confirmed blocks as time $t\to\infty$. In Section 4, we show that stable blockchain systems using the tree and throughput-optimal policies confirm infinitely many blocks as time $t\to\infty$. In Section 5, we derive bounds on the stability region for our model. We also use our stability bounds to deduce the non-scalability of networks containing large stars and show that the per-peer block arrival rate decreases to $0$ with network size, even in a scalable sequence of networks. In Section 6, we interpret our theoretical results and identify new network metrics to characterize stable blockchain dynamics. In Section 7, we conduct numerical experiments. In Section 8, we discuss related work. We provide concluding remarks in Section 9. Proofs are deferred to appendices.

From henceforth and for clarity, $G_{p}(t)$, the blockchain graph at a peer $p$ at time $t$, and the union $G(t):=\bigcup_{0\leq s\leq t}\bigcup_{p\in\{1,\ldots,N\}}G_{p}(s)$ are referred to as DAGs. The vertices of any DAG are referred to as blocks and the (directed) edges are referred to as references. Similarly, the graph $H$, which represents the communication structure among the $N$ peers is referred to as a network. The vertices of any network are referred to as peers and the (undirected) edges are referred to as links.

At time $0$, we assume that $G_{p}(0)$ is a single vertex indexed $0$, for all peers $p\in\{1,\ldots,N\}$. We denote by $B(t)=\bigcup_{p\in\{1,\ldots,N\}}B_{p}(t)$ and $E(t)=\bigcup_{p\in\{1,\ldots,N\}}E_{p}(t)$. The DAG $G(t)$ is the graph on the vertex set $B(t)$ with edge set $E(t)$.

Note that the DAGs $G_{p}(t),p\in\{1,\ldots,N\}$ and $G(t)$ are random DAGs as their growth is governed by a stochastic process. These graphs are parameterized by the P2P network $H$ but we do not include this in our notation as the context will always be clear.

Observe that P2P network dynamics are a continuous time rumor-spreading process with exogenous arrivals [23]. Here, rumors represent blocks which are disseminated on the network. For simplicity and without loss of generality, we assume that each peer $p$ has unit communication bandwidth, i.e., the process $T_{p}$ is rate $1$ block per second. We relax this assumption in Remark 2. As a peer $p$ can communicate at most a single block at the epochs of $T_{p}$, the block dissemination is bandwidth-limited.

In this paper, we consider the following two reference selection policies. In both policies, we fix a block $i\in\mathbb{N}$, that arrives at a (random) peer $p\in\{1,\ldots,N\}$, at time $t\in\mathbb{R}_{+}$.

1. 1.

   Tree Policy - $\mathcal{O}_{i}\subseteq\mathcal{L}_{p}(t),$ such that $|\mathcal{O}_{i}|=1$. Every block has exactly one outgoing reference, chosen according to a deterministic rule from the set $\mathcal{L}_{p}(t)$. We assume without loss of generality that each block $i$ has an outgoing reference to the least indexed block in $\mathcal{L}_{p}(t)$ in the event that $|\mathcal{L}_{p}(t)|>1$.

2. 2.

   Throughput Optimal Policy -
   $\mathcal{O}_{i}=\{b\in B_{p}(t):b\text{ is a leaf in }G_{p}(t)\}.$ Every block connects to all leaves in $G_{p}(t^{-})$. We explain after Corollary 4 why this policy is called throughput-optimal.

In this paper, we only analyze blockchain systems that use the tree and throughput-optimal policies. Bitcoin [39] and Ethereum [12], the two most widely adopted blockchain implementations, both use the tree policy, and the throughput-optimal policy is studied in [34].

We now define stability and scalability for blockchain-like systems, which are motivated from distributed agreement among peers.

In Figure 1, subfigures (b), (h), and (j) are times of consistency. We discuss consistency and its relation to analogous concepts in the queueing theory literature in Sections 6 and 7.

In other words, the blockchain system is stable if there are infinitely many times of consistency with at least one block arrival between subsequent times of consistency. We now define scalability.

Stability is a property of the blockchain system for a fixed P2P network, in the limit as time goes to infinity. Notice that stability is an asymptotic concept and can only be assessed in the limit as time goes to infinity. In addition, stability is a property only of the block communication dynamics on a P2P network and the block arrival rate and is not dependent on the reference selection policy by which individual peers add blocks. This follows as stability only requires that all peers be aware of the same set of blocks infinitely often, and the communication among peers is governed only by block indices.

Scalability, on the other hand, is a double limit, where for a fixed P2P system, we take time to infinity, and then subsequently, take the size of the P2P system to infinity. Thus, the definition of scalability is with respect to a particular sequence of P2P networks, whose vertex sets grow to infinity. Precisely, we deem an infinite sequence of P2P networks scalable if there exists a non-zero arrival rate such that a blockchain system on every P2P network of the sequence is stable for that rate.

In subsection, we provide definitions and key properties of infinite DAGs. These will be used to describe the blockchain DAG later in the paper, where the vertices will be blocks and the (directed) edges will be references.

Note that maximal paths exist for all vertices in finite DAGs; but in general this need not be true for infinite DAGs. In general, a vertex may have more than one maximal path.

Halin [28] introduces the concept of ends in infinite graphs, which we re-state for infinite DAGs below.

Note that any finite or infinite path in a DAG does not contain any repeated vertices and has either a first or a last vertex $v_{1}\in V$.

The equivalence relation in Definition 6 partitions the set of all infinite paths in a DAG into equivalence classes called ends.

Note that under Definition 7, all finite DAGs have 0 ends.

We give the following examples to illustrate the number of ends in various graphs. All of these examples are on the vertex set $\mathbb{N}$.

1. 1.

   There is an edge from vertex $i$ to vertex $1$ for all $i>1$, and no other edges. As there are no infinite paths, this graph is $0$ ends.

2. 2.

   There is an edge from vertex $i$ to vertex $i+1$ for all $i\in\mathbb{N}$. Here, any infinite path is necessarily of the form $j,j+1,\ldots$ for some $j\in\mathbb{N}$. For any two infinite paths beginning at $j,k\in\mathbb{N}$, the path $\max(j,k),\max(j,k)+1,\ldots$ intersects both paths infinitely often; hence this graph is one-ended.

3. 3.

   There is an edge from vertex $1$ to vertex $2$. In addition, there is an edge from vertex $i$ to vertex $i+2$. The following infinite paths are not equivalent: $1,3,5,\ldots$ and $2,4,6,\ldots$. Thus this graph is $2$-ended.

Further work on ends in random graphs can be found in [2, 4, 7].

Note that any finite DAG is also locally finite; however an infinite DAG need not be locally finite.

The proof is given in Appendix B.

Under the tree policy, the distinguished path in $G_{p}(t)$, for any peer $p$ and time $t\geq 0$, is the longest maximal path beginning from the vertex $i:=\inf\{b\in\mathcal{L}_{p}(t)\}$.

Our analyses in this paper involve studying an infinite DAG which is constructed by iteratively adding a single vertex to a finite DAG. We denote by the limiting blockchain DAG $G(\infty):=\bigcup_{t\geq 0}G(t)=\bigcup_{t\geq 0}\bigcup_{p=1}^{N}G_{p}(t)$. The DAG $G(\infty)$ is infinite, as its vertex set consists of all blocks, i.e., the vertex set of $G(\infty)$ is $\mathbb{N}$. Observe from the definition of $G(t)$, that the map $t\rightarrow G(t)$ is monotone non-decreasing. This follows as the outgoing edges for a block are never changed after it is created at the instant of block arrival, and blocks (vertices) are never removed. Notice that for all times $0\leq t<\infty$, $G(t)$ is a almost surely finite, as almost surely, in any finite interval of time, only finitely many blocks arrive to the system. Notice that almost surely, for all times $0\leq t<\infty$, $G(t)$ is finite, as only finitely many blocks arrive to the system before any finite $t$. Thus, $G(\infty):=\bigcup_{t\geq 0}G(t)$ is well-defined as it can almost surely be expressed as a countable union of DAGs. In Appendix A, we give additional technical details and establish that $G(\infty)$ as an appropriate limit of a sequence of finite graphs, and thus we call it the limiting blockchain DAG.

In this subsection give a precise definition of a confirmed block. In the sequel we identify the relationship between confirmed blocks in $G(\infty)$ and identify a sufficient condition such that infinitely many confirmed blocks exist.

Observe that the Definition 10 is an asymptotic property; namely it can only be verified in the limit as $t\to\infty$ and not at any finite time.

Note that the notion of trust in Definition 11 is one of distributed agreement. This is motivated from the fact that building on an existing block requires that a peer has verified that block’s content.

The proof is given in Appendix C.1.

The Bitcoin whitepaper suggests that as the number of blocks with a directed path to any particular block $b$ increases, a peer can be increasingly confident that all other peers are aware of and have added blocks that reference block $b$. Definitions 10 and 11 and Proposition 2 aim to capture this notion by looking at the asymptotic structure of the blockchain DAG.

Observe that a natural requirement on the performance of blockchain systems is that there are infinitely many confirmed blocks. Otherwise, the peers consume infinite bandwidth in the limit as $t\to\infty$, yet they only confirm finitely many blocks.

For the rest of this section we assume that the limiting DAG $G(\infty)$ is locally finite. Recall that a DAG is locally finite if each vertex has finite degree. This is done without loss of generality, as we show in Section 4 that the two most natural polices of interest, the tree and throughput-optimal policies, lead to locally finite limiting DAGs.

The proof is given in Appendix C.2.

Lemma 1 shows that if there are infinitely many confirmed blocks, then $G(\infty)$ has at least one end. Moreover, the (infinite) subgraph of all confirmed blocks in $G(\infty)$ is one ended. In the following lemma we establish that the one-endedness of $G(\infty)$ is a sufficient condition for the existence of infinitely many confirmed blocks.

The proof is given in Appendix C.3.

The results in Lemmas 1 and 2 are asymptotic guarantees in the limit as $t\to\infty$; like stability and confirmation, these results cannot be determined at any finite time $t$. Lemmas 1 and 2 show that having a one-ended limiting DAG $G(\infty)$ is a desirable property in a blockchain system, as it ensures that the number of confirmed blocks (evaluated in the limit as $t\to\infty$) is infinite. In Section 4, we show that under the assumption of stability, both the tree and throughput-optimal policies produce one-ended limiting DAGs.

The proof is given in Appendix D.2.

The proof is given in Appendix D.3.

The proof is given in Appendix D.4.

Corollary 2 shows that for stable blockchain DAGs constructed using the Tree policy, a peer only needs a finite amount of time determine whether or not a particular block will be eventually confirmed. If there exists a time of consistency such that $b$ is on the distinguished path in $G(C)$, and additionally the system parameters imply stability (we give conditions for this in Theorem 3), then it follows that block $b$, in the limit as $t\rightarrow\infty$, will be confirmed. Moreover, in stable blockchain systems using the tree policy, Corollary 2 provides a necessary and sufficient condition for the confirmation of a block $b$. In particular, it shows that confirmation is equivalent to the existence of an infinite sequence of blocks $(b_{k})_{k\in\mathbb{N}}$ such that there is a path in $G(\infty)$ from $b_{k}$ to $b$ for all $k\in\mathbb{N}$. In general, this condition is only a necessary condition, but not sufficient. Examples of blockchains using the tree policy are Bitcoin and Ethereum, the two most commonly used blockchain implementations [39, 12]. In Sections 6 and 7 we use the fact that for stable blockchain systems using the tree policy, confirmation can be determined in finite time in order to identify and numerically estimate several network parameters related to stability of the Bitcoin P2P network.

The proof is given in Appendix D.5.

This theorem does not follow from Theorem 1, since in general, it is not true that adding or removing countably many edges to or from a one-ended DAG results again in a one-ended DAG. A counterexample in each direction is given in Figure 2.

The proof is given in Appendix D.6.

The proof is given in Appendix D.7.

As a result of Corollary 4, we note that all blocks will eventually be confirmed. Hence, we denote this policy as throughput-optimal.

Lemma 3 and Theorems 1 and 2 establish that stable
blockchain systems using the tree and throughput-optimal policies have one-ended limiting DAGs. In particular, Corollary 2 shows that under the tree policy, there exists an infinite path consisting of all (infinitely many) confirmed blocks; Corollary 4 shows that under the throughput-optimal policy, stability implies that all blocks will eventually be confirmed.

In this section, we provide an illustrative example using Theorem 3 to reject the scalability of a sequence of peer-to-peer networks (note that Theorem 3 cannot be used to ensure scalability). In addition, we show that no network on $N$ peers can support an arrival rate greater than $\frac{N}{N-1}$; hence even if a sequence of networks is scalable, it is impossible to support a block arrival rate of greater than $2$.

The goal of the blockchain paradigm is to enable a set of anonymous peers to agree on a distributed ledger, where information is stored in discrete units called blocks. In particular, as an infinite amount of bandwidth is consumed in the limit as $t\to\infty$, a natural requirement is that the number of confirmed blocks also tends to infinity as $t\to\infty$.

We show in Lemma 2 that the one-endedness and local finiteness of the limiting DAG $G(\infty)$ is a sufficient condition for the existence of infinitely many confirmed blocks. We then show, in Theorems 1 and 2, that under the assumption of stability, blockchains using the tree and throughput-optimal policies have one-ended limiting DAGs. For the throughput-optimal policy, we find that all blocks are eventually confirmed. For the tree policy, we find that an external observer who is aware of the states of $G_{p}(t)$ for all peers $p\in\{1,\ldots,N\}$ can determine, in finite time, whether or not any particular block will be eventually confirmed. In particular, such determinations occur at times of consistency. This ties the confirmation of blocks under the tree policy to the network dynamics on the peer-to-peer network $H$.

We thus identify the following metrics on the network $H$. As the underlying peer-to-peer networks in the Bitcoin and Ethereum blockchain implementations ([39, 12]) use the tree policy, these metrics provide insight into the behavior of those systems. In Section 7, we use these metrics in a simulation environment to numerically estimate key network properties of the Bitcoin peer-to-peer network.

In this section, we identify some quantitative system performance metrics which further characterize the performance of the blockchain system when it is stable. Recall Theorem 3 provides bounds on the maximum block arrival rate $\mu$ to guarantee stability.

Observe that as the cycle length is at least the mean idle time, it goes to infinity as the block arrival rate goes to zero. Thus, the cycle length metric captures the trade-off between the time to consistency (which goes to $0$ as the block arrival rate goes to $0$) and the block arrival rate.

Our simulation results indicate that the cycle length may be a convex function of block arrival rates (Figure 6); thus on a given P2P network $H$ there may be a unique optimal block arrival rate that minimizes the cycle length. Of note, the results in Figure 6 identify that the cycle length may satisfy two key robustness properties. The first is that for a fixed $N$, there is a wide range of block arrival rates for which the cycle length is approximately constant. The second is that there exists a range of block arrival rates for which the cycle length is nearly invariant to the number of peers in the network.

Our simulation results (Figure 8) indicate that despite a monotonically decreasing consistency fraction, the growth rate of the distinguished path increases to a maximum before decaying. This suggests that there may be a unique block arrival rate which maximizes the growth rate of the distinguished path. We note that the arrival rate which minimizes the cycle length appears to be less than the arrival rate which maximizes the growth rate of the distinguished path.

When the block arrival rate is small, almost all blocks are on the distinguished path and very few are orphaned (not on the distinguished path). Therefore, a small increase in arrival rate increases the throughput. On the contrary, for large block arrival rates, nearly all blocks are orphaned; hence increasing the block arrival rate decreases throughput.

We note that there are various trade-offs between the metrics identified above, which are confirmed by the simulations in Section 7. In particular, we note that the time to consistency, consistency fraction, and age of information are all optimized as the block rate decreases to zero. However, at the expense of performance with respect to these metrics, increasing the block rate can decrease the cycle length and increase the growth rate of the distinguished path in tree policy blockchains, as shown in Figures 6 and 8.

The cycle length and growth rate are more sophisticated metrics and characterize the temporal dynamics of a blockchain system. The growth rate of the distinguished path also characterizes the generation rate of orphaned blocks. Minimizing the rate of orphaned blocks is desirable as orphaned blocks may pose security threats in applications such as cryptocurrencies [26]. Nevertheless, a detailed analysis of the security of blockchains is application-specific and therefore out of the scope of this paper [42, 24], as our goal is to study aspects of blockchain which are universal to all applications.

For the core blockchain protocol, the block arrival rate is the only system parameter that can be chosen by a system designer. This underscores the need for further study on the performance of blockchain systems with regard to the metrics identified in this work in order to improve current blockchain systems and design future ones.

We numerically study our stochastic network model and further characterize its performance whenever it is stable. In particular, we use the metrics identified in Section 6 to gain further insights into the network behavior. This complements our theoretical result which gives bounds on the stability region. We analyze the metrics identified in Section 6 with respect to varying block arrival rates, using synthetic parameters for the network. This is only possible with synthetic data, as the real data set consists of a single arrival process of blocks and thereby we cannot use it to directly assess the impact of varying the block arrival rate on system performance.

Figure 5 shows the effect of increasing the block arrival rate on the time to consistency. As expected, we observe that the the time to consistency grows monotonically with block arrival rate.

In Figure 6 we depict the average cycle length, which is the sum of the mean time to consistency and the mean length of consistency. Observe that the cycle length has asymptotes to infinity at both $0$ and $\mu$, which are observed in Figure 6. We observe that the cycle length appears to be a convex function of block arrival rates for complete networks and is nearly flat near its infimum. This suggests that when $H$ is a complete network, there is a wide range of block arrival rates for which the cycle length is approximately constant, indicating a sense of robustness for block arrival rates for the designer of the system to choose. The figure also suggests there is a reasonably large set of block rates for which the cycle length is fairly robust to changes in the number of peers.

Figure 7 captures the relationship between increasing block rates and the mean fraction of consistent peers. There appears to be an inflection point when one half of peers are consistent – above this point the graph is concave; below it is convex.

Figure 8 shows the relationship between the block arrival rate and the growth rate of the distinguished path. As with consistency fraction, the growth rate curve appears to have an inflection point. We observe that the growth rate of the distinguished path appears to have a unique maximum. However, the block arrival rate that produces this maximum is not equal to the rate that minimizes the cycle length or the rate that produces an inflection point for the consistency fraction.

Figure 9 depicts, for a typical peer $p$, the mean number of blocks behind consistency. As expected, the age of information at a typical peer tends to infinity as the block arrival rate approaches the critical value.

In this section, we analyze the performance of the blockchain system using real data from the Bitcoin Blockchain network [10, 39]. We do not consider the growth rate metric as the data given in [10] does not include the blockchain DAG.

We observe from our results in Tables 1, 2, 3 and 4 that, for the metrics we consider, the estimates on the real block arrival data and the equivalent Poisson data are similar, thereby providing another validation for the Poisson block arrival model typically used in the literature [39, 8, 52, 41, 35, 36, 22].

From the results in Tables 1, 2, 3,4, it appears that the impact of the network topology on the performance of the Bitcoin network (using the data in [14, 25]) is minimal. This is in large part due to the fact that block propagation delay is much lower than the block arrival rate [14]. However, the results from our synthetic simulations suggest that with increasing network sizes (and thus with increasing block propagation delay), the effect of network topology should become more pronounced. The current trend of blockchain adoption suggests that that the size of blockchain networks is rapidly increasing [14, 25], and thus understanding the impact of network size and topology on performance is crucial for designing future blockchain systems.