The fluid limit of a random graph model for a shared ledger

TL;DR

This paper analyzes the growth of the tangle, a shared ledger protocol used in IOTA, modeling it as a random directed acyclic graph and deriving a fluid limit to understand its behavior at high transaction rates.

Contribution

It introduces a fluid model for the tangle, proves convergence of the stochastic process to this model, and establishes its global stability using martingale techniques.

Findings

Derived a delay differential equation for the fluid model.

Proved convergence in probability of the tangle process to the fluid model.

Established global stability of the fluid model.

Abstract

A shared ledger is a record of transactions that can be updated by any member of a group of users. The notion of independent and consistent record-keeping in a shared ledger is important for blockchain and more generally for distributed ledger technologies. In this paper we analyze the growth of a model for the tangle, which is the shared ledger protocol used as the basis for the IOTA cryptocurrency. The model is a random directed acyclic graph, and its growth is described by a non-Markovian stochastic process. We derive a delay differential equation for the fluid model which describes the tangle at high arrival rate. We prove convergence in probability of the tangle process to the fluid model, and also prove global stability of the fluid model. The convergence proof relies on martingale techniques.

Equations

$$m=\lambda h$$

$$W_n$$

$$X_n$$

$$L_n$$

$$U_n$$

$$\displaystyle W_{n}=\sum_{j=n-m+1}^{n}U_{j}\quad\mbox{for all $n\geq m$}$$

$$X_{n+1}$$

$$L_{n+1}$$

$$L_n=W_n+X_n$$

$$X_{n+1}$$

$$L_{n+m+1}$$

$$L_{n+m+1}=X_{n+1}+L_{2m}-X_m$$

$$L_{2m}=W_{2m}+X_{2m}=\sum _{i=m+1}^{2m}{U}_i+X_m+\sum _{i=m}^{2m-1}(1-U_{i+1})=X_m+m$$

$$L_n=X_{n-m}+m$$

$$\mathcal{F}(\lambda ,n)=\sigma \left(U_1,\dots ,U_n\right)$$

$$\displaystyle{\cal F}(\lambda,n_{1})\subset{\cal F}(\lambda,n_{2})\quad\mbox{for all $n_{1}<n_{2}$}$$

$$\mathbb{P}(U_{n+1}=2|\mathcal{F}(\lambda ,n))$$

$$\mathbb{P}(U_{n+1}=0|\mathcal{F}(\lambda ,n))$$

$$\mathbb{P}(U_{n+1}=1|\mathcal{F}(\lambda ,n))$$

$$\mathbb{E}[U_{n+1}|\mathcal{F}(\lambda ,n)]=2\frac{X_n}{L_n}-\frac{X_n}{L_n^2}$$

$$U_i$$

$$W_m$$

$$X_m$$

$$L_m$$

$$W_{m+1}$$

$$X_{m+1}$$

$$L_{m+1}$$

$$L_{m+1}={\xi }_m+1+\sum _{i=2}^m{u}_i$$

$$\displaystyle L_{m+j}=\xi_{m}+j+\sum_{i=j+1}^{m}u_{i}\quad\mbox{for $j=0,\dots,m$}$$

$$\displaystyle A^{(\lambda)}(t)=\lambda^{-1}\,X_{n(t)},\qquad B^{(\lambda)}(t)=\lambda^{-1}\,L_{n(t)},\qquad\mbox{where $n(t)=\lfloor\lambda\,t\rfloor$}$$

$$\frac{A^{(\lambda )}(t_{n+1})-A^{(\lambda )}(t_n)}{t_{n+1}-t_n}=1-U_{n+1}$$

$$\frac{B^{(\lambda )}(t_{n+1})-B^{(\lambda )}(t_n)}{t_{n+1}-t_n}=1-U_{n-m+1}$$

$$\mathbb{E}[U_{n+1}|\mathcal{F}(\lambda ,n)]=2\frac{A^{(\lambda )}(t_n)}{B^{(\lambda )}(t_n)}-{\lambda }^{-1}\frac{A^{(\lambda )}(t_n)}{B^{(\lambda )}(t_n)}\simeq 2\frac{A^{(\lambda )}(t_n)}{B^{(\lambda )}(t_n)}\simeq 2\frac{a(t)}{b(t)}$$

$$\mathbb{E}[U_{n-m+1}|\mathcal{F}(\lambda ,n)]\simeq 2\frac{A^{(\lambda )}(t_n-h)}{B^{(\lambda )}(t_n-h)}\simeq 2\frac{a(t-h)}{b(t-h)}$$

$$\frac{da}{dt}=1-2\frac{a(t)}{b(t)},\frac{db}{dt}=1-2\frac{a(t-h)}{b(t-h)}$$

$$\displaystyle 0\leq u(t)\leq 2\quad\mbox{for all $0\leq t\leq h$}$$

$$b(h)=a(h)+{\int }_0^{h}u(s)ds,$$

Full text

1 Introduction

In this paper we analyze a stochastic growth process for a family of directed acyclic graphs, and show that the fluid limit of this process is described by a delay differential equation. This stochastic process describes a type of shared ledger which was introduced as the foundation of the cryptocurrency IOTA [7], and the result about the fluid limit was used previously to analyze the persistence of competing transaction records in this ledger [4]. The main contribution of this paper is to provide a precise formulation of the results about the fluid limit. We use martingale techniques to establish that the process converges weakly in the limit where the arrival rate goes to infinity, and show that the fluid limit is given as the solution of a suitable delay differential equation. We also prove a convergence result for the solutions of the delay differential equation.

The term ‘shared ledger’ refers to a record of transactions which may be amended independently by any member of a group of users. The goal of designing a shared ledger is to allow users to add transactions to the record without centralized control, while at the same time protecting the record against tampering by malicious agents. As a background to this topic we will review below the well-known blockchain protocol [6] which involves linking blocks (collections of transactions) by complicated hash function computations. If blocks are represented by vertices on a graph and the hashing link between two blocks is represented by a directed edge between those vertices, then the whole blockchain ledger can be viewed as a directed graph. This point of view leads to our random graph analysis, and will form the basis for the stochastic process that will be analyzed in this paper.

1.1 The blockchain protocol

The blockchain technology underlying Bitcoin is a well-known implementation of a shared ledger which provides security against malicious users [6], [8]. Recall that the blockchain is an ordered string of blocks, each containing several hundred transaction records; each block has a unique numerical ID (256 bits for Bitcoin) that satisfies a challenging constraint. The ID of a block is computed using a complicated hash function, and the input for the hash function involves the block’s own data, the ID of the previous block, and some extra bits which are chosen so that the output satisfies the constraint. Thus every block’s ID depends on the data of the previous block, and hence also on the data in all previous blocks. Therefore any change in the data of a block would change the IDs of all subsequent blocks, and the altered IDs would almost certainly not satisfy the tight constraint mentioned above. This failure would be a signal to all observers that the ledger had been altered, and so the existence of a ledger with valid IDs for all blocks is its own guarantee of security. The key mechanism for security is the difficulty of computing a valid ID for a block. This task is called the proof of work, and requires finding an input to a complicated hash function which will produce an output of the specified form. The blockchain miners compete to find this inverse, and the first successful one adds the new block to the chain.

One essential constraint in Bitcoin is that a new block can only be linked to the most recent block in the chain. This constraint ensures that the blockchain is a linear graph. It also ensures that every transaction record in the chain is linked to all subsequent records, and indeed the security of a transaction increases as later blocks are added (a typical rule of thumb is that a transaction record in blockchain is ‘safe’ after at least six subsequent blocks have been added). However this constraint leads to a ‘winner takes all’ rule for the miners, who must compete to be first to add a new block. Consequently mining has become a dedicated enterprise requiring specialized technology, and there is much effort wasted (and energy expended) by the miners.

1.2 Modifying the blockchain protocol

There have been many proposed modifications of the blockchain protocol. In this paper we consider one such proposal [7] which involves removing the constraint that a new block can only be linked to the most recent block in the chain. Removing this constraint has several immediate consequences. First, there is no competition between miners, hence each user can perform their own proof of work (which is much easier than in Bitcoin) and there are no rewards for adding a new block. Second, since a new block can link to any previous block, the graph of links for the ledger is no longer linear, and can be much more complicated than in Bitcoin. Furthermore, since there are many possible ways to link a new block to the ledger, it is reasonable to view the ledger as a randomly growing graph and to investigate its typical properties. We will pursue this point of view for the modification known as the tangle protocol which was introduced in [7].

1.3 The tangle protocol

In the tangle protocol [7] each new block contains just one transaction. A new transaction links to two existing transactions in the ledger (this is another change from the blockchain protocol), and the proof of work uses the ID’s of these two transactions as part of its input. Thus the ledger grows by the addition of transactions each with two directed edges which link to existing transactions in the ledger. The resulting graph of links is a connected directed acyclic graph (DAG). The proof of work lasts for some amount of time $h$, so there is a delay between the time when a new transaction starts its proof of work and the time when it is added to the DAG as a new vertex. This time $h$ is much shorter than in Bitcoin (where the proof of work on average lasts for ten minutes), however it plays a crucial role in the growth of the ledger.

As in the blockchain, the security of a transaction increases as later transactions are added which are linked either directly or indirectly to it. Although in principle a new transaction may choose to link to any existing transaction in the ledger, it is advantageous to select two recently arrived transactions for linking. Transactions which have not yet been linked by subsequent transactions are called tips; in the tangle protocol all users select tips for linking. Note that for blockchain the security relies on the community solving one exceedingly difficult hash function inversion for each new block, whereas for the tangle the security relies on a large community of users each performing relatively simple computations in parallel.

1.4 Summary of results

There are two time scales in this model, namely $\lambda^{-1}$, the average time between arrivals of new transactions, and $h$, the duration of the proof of work. We are interested in how the average number of tips in the tangle depends on these quantities as the arrival rate $\lambda$ goes to infinity. Let $L(t)$ be the number of tips at time $t$. Several approaches to this question [7], [4] have shown that $L(t)$ is roughly proportional to the product $\lambda h$, at least for large $\lambda$. We investigate in this paper the limit where $\lambda$ approaches infinity, so we define the rescaled variable $B^{(\lambda)}(t)=\lambda^{-1}L(t)$. The first result Theorem 5 shows that $B^{(\lambda)}(t)$ converges in probability to a deterministic function $b(t)$ as $\lambda\rightarrow\infty$. The bound in Theorem 5 also shows that the fluctuations in $|B^{(\lambda)}(t)-b(t)|$ are no larger than $O(\lambda^{-1/2})$. The second result Theorem 6 shows that the function $b(t)$ converges exponentially to $2h$ as $t\rightarrow\infty$. Putting these results together shows that for large $\lambda$ and large $t$, $L(t)$ can be written approximately as $2\lambda h+O(\lambda^{1/2})$.

1.5 Outline of the paper

In Section 2 we formulate a stochastic process for the number of tips on the DAG which represents the tangle. In Section 3 we describe the fluid limit of the rescaled process (the fluid limit refers to the limit where the arrival rate of new transactions goes to infinity) and also describe how initial conditions can be consistently formulated for the process and the fluid limit. The main results of the paper Theorems 5 and 6 are stated in Section 4, and are proved in Sections 5 and 6. Future directions of research on this topic are discussed in Section 7.

2 Definition of the DAG model

Let $G=(V,E)$ be a finite connected acyclic directed graph (DAG) where $V$ is the vertex set and $E$ is the edge set. If an edge $e\in E$ is directed from vertex $x$ to vertex $y$ we will write $e=\langle x,y\rangle$ and say that $y$ is the head and $x$ is the tail. A tangle is a DAG with two additional properties: first, there is a unique vertex which is not the tail in any edge – this is called the genesis vertex. Second, every vertex is the tail in at most two edges. The subset of vertices which are not the heads of any edges will be called the tips of the tangle.

We will define a stochastic growth model for the tangle. The arrival rate of new transactions is denoted by $\lambda$, and for simplicity we assume that new transactions are created at the deterministic sequence of times $\{t_{n}=\lambda^{-1}n\,:\,n=1,2,\dots\}$. At time $t_{n}$, two tips $x_{1}(n)$ and $x_{2}(n)$ on the tangle are selected for the proof of work by the new transaction (it is possible that $x_{1}(n)=x_{2}(n)$). The proof of work lasts for a fixed length of time $h$. For simplicity we will assume that $\lambda$ is always chosen so that $\lambda h$ is an integer:

[TABLE]

At time $t_{n}+h=t_{n+m}$ the new transaction is added to the tangle as a tip $y_{n}$, and the two directed edges $\langle y_{n},x_{1}(n)\rangle$ and $\langle y_{n},x_{2}(n)\rangle$ are also added to the graph. This is the only mechanism by which the tangle grows.

Obviously the vertices $x_{1}(n)$ and $x_{2}(n)$ are no longer tips after time $t_{n}+h$, however it may happen that these vertices had already ceased to be tips at an earlier time, due to their being linked to some other new transaction. We say that a tip is pending if it has been selected for proof of work by a transaction but has not yet been linked. We say that a tip is free if it is not pending.

Definition 1

[TABLE]

We have defined $U_{n}$ to be the number of the vertices $\{x_{1}(n),x_{2}(n)\}$ which are free at time $t_{n}$, so $U_{n}\in\{0,1,2\}$. After selection these free vertices immediately become pending vertices, hence they will never contribute to any of the subsequent values $U_{n+1},U_{n+2},\dots$. Furthermore at any time $n\geq m$ there are exactly $m$ new transactions which are each in the process of carrying out their proof of work on two vertices on the graph (this holds because $m=\lambda h$ and we assume that the value of $h$ is fixed and identical for all users). Therefore the total number of pending vertices at any time $t_{n}$ (with $n\geq m$) is the sum of $\{U_{n},U_{n-1},\dots,U_{n-m+1}\}$, that is

[TABLE]

We also have the following evolution relations:

[TABLE]

and the relation

[TABLE]

We will discuss shortly how the processes $X_{n}$ and $L_{n}$ can be defined using appropriate initial conditions. For this purpose it will be convenient to use the evolution relations (2) starting at $n=m$ and $n=2m$ respectively. These relations imply that

[TABLE]

Therefore

[TABLE]

and we have from (8), (6) and (2) that

[TABLE]

So we conclude form (10) that for all $n\geq 2m$,

[TABLE]

We will refer to $(W_{n},X_{n},L_{n})$ as the tangle process. It follows from (12) that the process $\{L_{n}\}$ (for $n\geq 2m$) is in fact fully determined by $\{X_{n}:n\geq m\}$.

2.1 The random tip growth model

The remaining ingredient in the definition of the DAG model is the method of choosing vertices $x_{1}(n)$ and $x_{2}(n)$. We will assume in this paper that the tips $x_{1}(n)$ and $x_{2}(n)$ are chosen independently and uniformly from the set of tips, and we call this the random tip growth (RTG) model for the tangle process. Thus the numbers $\{U_{n}\}$ are random variables whose distributions depend on the number of tips at time $t_{n}$. The RTG model is one of the tip selection algorithms discussed in [7], [5], and it is expected that the fluid limit results presented in this paper can be extended to those other tip selection algorithms.

We denote by ${\cal F}(\lambda,n)$ the $\sigma$-algebra generated by $\{U_{1},\dots,U_{n}\}$:

[TABLE]

It follows from (2) that $X_{n}-X_{0}$ and $L_{n+m}-L_{m}$ are measurable with respect to ${\cal F}(\lambda,n)$. We also have the filtration relation

[TABLE]

The conditional distribution of the random variable $U_{n+1}$ for the RTG model is

[TABLE]

Note that

[TABLE]

It is clear that $(X_{n},L_{n})$ is not a Markov process, as the distribution of $L_{n+1}$ depends on $U_{n-m+1}$, which in turn depends on $(X_{n-m},L_{n-m})$ through (2.1).

2.2 Generating the process from initial conditions

The stochastic process $(X_{n},L_{n})$ defined by (2) and (2.1) must be supplemented with initial conditions in order to be well-defined. This is done most easily by assigning values to the variables $(U_{1},\dots,U_{m})$ and $X_{m}$. Once these assignments have been made, the distribution of the process $(X_{n},L_{n})$ is determined for all $n\geq m$, as will be explained below. In particular the variables $X_{0},\dots,X_{m-1}$ and $L_{0},\dots,L_{m-1}$ do not play any role, and we will ignore their values.

Let $(u_{1},\dots,u_{m})$ be a sequence with $u_{i}\in\{0,1,2\}$ for all $i=1,\dots,m$, and let $\xi_{m}\geq 1$ be an integer. Then we assign as initial conditions

[TABLE]

To see how this generates the process for $n>m$, note from (2.1) that the distribution of $U_{m+1}$ is determined by (2.2), and is well-defined since $L_{m}\geq\xi_{m}\geq 1$. Given $U_{m+1}$, we compute

[TABLE]

We have $L_{m+1}\geq X_{m+1}\geq X_{m}-1$, and also the formulas (2.1) imply that if $X_{m}=1$ then $U_{m+1}\in\{0,1\}$, and thus $L_{m+1}\geq X_{m+1}\geq X_{m}$. Therefore $L_{m+1}\geq 1$, and so the distribution of $U_{m+2}$ is well-defined, and this random variable can be used to compute $W_{m+2},X_{m+2},L_{m+2}$. This bootstrap continues and allows us to generate the whole process starting from the initial conditions (2.2), and furthermore the resulting process satisfies $X_{n}\geq 1$ for all $n\geq m$. Hence from (12) we also deduce that $L_{n}\geq m+1$ for all $n\geq 2m$.

From (2.2) and (2.2) we also deduce that

[TABLE]

and so $L_{m+1}$ is also fixed by the initial conditions. The same is true for $L_{2},\dots,L_{m}$, and we have the formula

[TABLE]

3 The fluid limit

Given the process $\{X_{n},L_{n}\}$ we rescale variables and define for all $t>0$

[TABLE]

The variables $(A^{(\lambda)}(t),B^{(\lambda)}(t))$ are piecewise constant in the intervals $[t_{n},t_{n+1})$, and change by at most $\pm\lambda^{-1}$ at each time $t_{n}$. Thus it is reasonable that in the limit $\lambda\rightarrow\infty$ these variables will converge to continuous functions $a(t)$ and $b(t)$. Furthermore after rescaling (2) the evolution equations become

[TABLE]

The left sides of (3) are expected to converge to $a^{\prime}(t)$ and $b^{\prime}(t)$ as $\lambda\rightarrow\infty$, so it is reasonable to expect that the fast variations on the right side will be averaged out in the limit, leaving the expected values of the variables $U_{n+1}$ and $U_{n-m+1}$. From (16) we have

[TABLE]

and similarly

[TABLE]

Assuming that the right sides of (3) converge to these average values, we are led to the following pair of coupled delay differential equations for the fluid limit:

[TABLE]

3.1 Delay differential equations

The equations (25) must be supplemented with suitable initial conditions. We will say that the combination $\alpha=(a(h),\{u(t)\,:\,0\leq t\leq h\})$ is a DDE initial condition if $a(h)>0$, $u(t)$ is integrable and

[TABLE]

These initial conditions can be used to define a solution of the fluid equations (25) for $t\geq 2h$, in the same way as the initial conditions (2.2) were used to construct the tangle process. The idea is that the function $u(t)$ plays the same role as the initial sequence $\{u_{i}\}$ for the discrete process. Thus we first define the initial value $b(h)$ as

[TABLE]

and we then define $b(t)$ for $h\leq t\leq 2h$ as the solution of the delay equation

[TABLE]

This leads to the solution

[TABLE]

We then compute $a(t)$ for $t\in[h,2h]$ as the solution of the equation

[TABLE]

which gives

[TABLE]

where

[TABLE]

Note that (29) implies $b(t)\geq a(h)>0$ for all $h\leq t\leq 2h$, so (32) is well-defined for $(x,y)=(h,t)$ with $t$ in this interval, and (31) also implies that $a(t)>0$ for all $h\leq t\leq 2h$. The equation (29) also implies that $b(2h)=a(h)+h$. Having obtained the functions $(a(t),b(t))$ in the interval $[h,2h]$, we then extend the solutions to the interval $[2h,3h]$ by first defining

[TABLE]

and then solving the differential equation for $a(t)$ to obtain

[TABLE]

From (33) we have $b(t)\geq h$, and thus $P(2h,t)$ is well-defined for $2h\leq t\leq 3h$, and again implies positivity of $a(t)$. This construction can be continued in the same way for subsequent intervals $[3h,4h],\dots$, and produces a solution of the equations (25) for all $t>2h$. We collect together our results about this solution in the following lemma.

Lemma 2

Let $\alpha$ be a DDE initial condition. There are unique functions $(a(t),b(t))$ defined for all $t>h$ which satisfy the equations (29) and (31) in the interval $[h,2h]$, and which satisfy the differential equations (25) for all $t>2h$. For $t\geq 2h$ the solutions also satisfy the following conditions:

[TABLE]

Proof of Lemma 2: the formulas (29) and (31) show that $(a(t),b(t))$ is uniquely defined and differentiable in the interval $(h,2h)$, and is continuous at $t=2h$. The iterative construction outlined above produces a unique differentiable solution in every interval $(jh,(j+1)h)$ for $j=2,3,\dots$. The solution is clearly continuous at $t=jh$ for all $j\geq 2$. It is also differentiable at $t=jh$ for all $j\geq 3$ because it satisfies the differential equations (25) in both intervals $((j-1)h,jh)$ and $(jh,(j+1)h)$. Properties (1), (2), (3) follow by construction. To see that Property (4) holds, let $c(t)=b(t)-a(t)$ and consider first the interval $[h,2h]$, where we have

[TABLE]

Therefore for some constant $K$ we have

[TABLE]

Evaluating at $t=h$ we see from (27) that $K=0$, and hence we have at $t=2h$ the relation

[TABLE]

Now for $t\geq 2h$ we have

[TABLE]

and thus for some constant $K^{\prime}$

[TABLE]

Evaluating at $t=2h$ we deduce that $K^{\prime}=0$, and this establishes Property (4). Property (5) follows immediately.

3.2 Fluid limit: Initial conditions for the tangle from DDE initial condition

Let $\alpha$ be a DDE initial condition. As Lemma 2 shows, $\alpha$ provides the necessary information to generate a unique solution of the delay equations (25). We will now show that $\alpha$ also generates the initial conditions for a tangle process. Recall that $L_{m},\dots,L_{2m}$ are determined by the initial conditions $\xi_{m},u_{1},\dots,u_{m}$ through the relation (20), and that the function $\{b(t)\,:\,h\leq t\leq 2h\}$ is determined by $a(h),\{u(s):0\leq s\leq h\}$ through the formula (29). We will choose the initial values $u_{1},\dots,u_{m}$ for the tangle process depending on the function $u(s)$ in such a way that the difference $B^{(\lambda)}(t)-b(t)$ is small for all $t\in[h,2h]$, where $B^{(\lambda)}(t)$ is the rescaled variable defined in (21). Define the set of all initial value sequences:

[TABLE]

Definition 3

Let $\alpha=(a(h),\{u(s):0\leq s\leq h\})$ be a DDE initial condition, and let $b_{\alpha}(t)$ be defined for $h\leq t\leq 2h$ by the formula (29). Let $\xi_{\alpha}=\max(\lfloor\lambda a(h)\rfloor,1)$. Given an initial condition $(\xi_{\alpha},v)$ for the tangle process, where $v\in{\cal S}(m)$, let $\{B_{v}^{(\lambda)}(t)\,:\,h\leq t\leq 2h\}$ be given by (21) where $L_{m},\dots,L_{2m}$ are defined by the formula (20) with $\xi_{m}=\xi_{\alpha}$ and $u_{i}=v_{i}$. Define

[TABLE]

Lemma 4

The set $F(\alpha,\lambda)$ is non-empty.

Proof: from (20) and (29) we derive for $m\leq n\leq 2m$

[TABLE]

where

[TABLE]

We also have from the definition of $\xi_{\alpha}$

[TABLE]

We now introduce a product probability measure on ${\cal S}(m)$ so that the coordinates $v_{1},\dots,v_{m}$ are independent random variables: for any sequence $(u_{1},\dots,u_{m})$,

[TABLE]

The distribution $\mathbb{P}_{j}$ is chosen so that

[TABLE]

(since $0\leq x_{j}\leq 2$ this is always possible). Define

[TABLE]

Since the $\{v_{j}\}$ are independent with finite variances and (51) holds, we can apply Kolmogorov’s maximal inequality [2] and deduce that for any $\delta>0$

[TABLE]

Since $|v_{j}|\leq 2$ for all $j$, we have ${\rm VAR}[v_{j}-x_{j}]\leq 4$, and hence by independence

[TABLE]

Taking $\delta=4h^{1/2}\,\lambda^{1/2}$ we deduce that

[TABLE]

Therefore using (49) and the formula (47) we get

[TABLE]

and so we deduce that $F(\alpha,\lambda)$ is non-empty.

4 Statement of results

Theorem 5

Let $\alpha$ be a DDE initial condition, and let $(a_{\alpha}(t),b_{\alpha}(t))$ be the associated solutions of the fluid equations (25) as described in Lemma 2. Let $v\in F(\alpha,\lambda)$, and let $(A_{v}^{(\lambda)}(t),B_{v}^{(\lambda)}(t))$ be the rescaled tangle process with initial conditions $(\xi_{\alpha},v)$ as described in Sections 2.2 and 3.2. For all $T\geq 2h$, and for all $\delta>0$, there is a constant $C<\infty$ (depending on $T,\alpha$) and $\lambda_{0}<\infty$ (depending on $T,\delta,\alpha$) such that for all $\lambda\geq\lambda_{0}$

[TABLE]

Remark: Theorem 5 confirms that the rescaled processes $(A^{(\lambda)}(t),B^{(\lambda)}(t))$ converge in probability to the deterministic solutions of the delay equations as $\lambda\rightarrow\infty$. This kind of behavior is familiar for Markov jump processes. One novelty of Theorem 5 is that although the processes are not Markov, due to the delay time $h$, nevertheless the same kind of limiting behavior holds, albeit with the more complicated delay differential equation.

The proof of Theorem 5 relies on martingale techniques. The constants $C$ and $\lambda_{0}$ that appear in the Theorem depend on $\alpha$, the initial conditions for the process. Simulations of the tangle process [4] have shown that the delay equations (25) give an accurate representation of the tangle even for relatively small values of $\lambda$.

The next result shows that the solution of the delay equation (25) converges to a constant as $t\rightarrow\infty$.

Theorem 6

Let $\alpha$ be a DDE initial condition, and let $(a_{\alpha}(t),b_{\alpha}(t))$ be the associated solutions of the fluid equations (25) as described in Lemma 2. Define

[TABLE]

Then for all $t\geq 4h$,

[TABLE]

Theorem 6 shows that the solutions of the delay equation converge exponentially to their stationary values with rate at least $\mu$. This limiting behavior shows that the number of tips behaves as $2\lambda h$ to leading order for large arrival rates.

5 Proof of Theorem 5

Theorem 5 will be proved using standard martingale techniques as presented for example in [3]. For convenience we will drop the subscripts $v,\,\alpha$ on the variables. We assume that $\lambda$ is sufficiently large so that $\xi_{\alpha}=\lambda a(h)\geq 1$. Define

[TABLE]

The quantity $l$ will appear in many of the bounds derived later in this proof, and will represent the effect of the initial conditions on the constants $C$ and $\lambda_{0}$ appearing in Theorem 5. It follows from (20) and (12) that

[TABLE]

and from (29), (33) that

[TABLE]

Define for $t\in[h,T]$

[TABLE]

Note that the quantity of interest in Theorem 5 is $\mathbb{P}(g(T)>\delta)$. We next derive the first inequality in (57). Recall that $b(t)=a(t-h)+h$ for all $t\geq 2h$, and that $L_{n}=X_{n-m}+m$ for all $n\geq 2m$. Therefore if $t\geq 2h$ and $t\in[t_{n},t_{n+1})$ we have

[TABLE]

and therefore

[TABLE]

This establishes the first inequality in (57), and so reduces the result to deriving a bound for $\mathbb{P}(g(T)>\delta)$.

As a first step we will derive a uniform bound for the difference $B^{(\lambda)}(s)-b(s)$ is terms of the function $g$ and an error term coming from the initial conditions. Recall that by assumption $v\in F(\alpha,\lambda)$, therefore

[TABLE]

Furthermore if $t\in[h,2h]$ and $t\in[t_{n},t_{n+1})$ then

[TABLE]

Therefore (66) and (67) together imply that

[TABLE]

where we have used $h\lambda\geq 1$. Combining (65) and (68) we get the uniform bound

[TABLE]

Next we will derive a bound for the quantity $A^{(\lambda)}(t)-a(t)$. For all $j\geq m$ we define

[TABLE]

Then we have

[TABLE]

since $X_{m}=\lambda a(h)=\lambda a(t_{m})$. The sum $\sum_{j=m}^{n-1}G_{j+1}$ is a martingale (as will be explained below) and we will use this fact to bound the probability that it grows too large. The other sum $\sum_{j=m}^{n-1}H_{j+1}$ is treated as an error term, and will be controlled using coarse bounds. We will first derive the bounds for this sum, then return to the martingale estimates.

From (71), (2.2) and (25) we get

[TABLE]

We write

[TABLE]

Using the bounds $A^{(\lambda)}(s)\leq B^{(\lambda)}(s)$ and (61), (62), (63) and (69) we have from (74)

[TABLE]

Therefore we deduce from (5) that

[TABLE]

which gives the bound

[TABLE]

Next we use the martingale property to bound the first sum on the left side of (72). Using (70) we have

[TABLE]

It follows that $G_{j+1}$ is ${\cal F}(\lambda,j+1)$-measurable, and

[TABLE]

Furthermore $|G_{j+1}|\leq 2$, so $\{G_{j}\}$ is a bounded martingale difference series relative to the filtration ${\cal F}(\lambda,n)$. Therefore $\sum_{j=m}^{n-1}G_{j+1}$ is a martingale, and $\left(\sum_{j=m}^{n-1}G_{j+1}\right)^{2}$ is a submartingale, so we can apply Doob’s martingale inequality [2] to deduce that for any $N>m$ and any $\theta>0$

[TABLE]

Define the event

[TABLE]

so we have

[TABLE]

Combining (72), (5) and (82), it follows that the event $E$ implies that for any $m\leq n\leq N$,

[TABLE]

where

[TABLE]

Since $g(t)\geq 0$, and (84) holds for all $m\leq n\leq N$, this also implies that

[TABLE]

Furthermore if $t\in[2m,T]$ and $t\in[t_{k},t_{k+1})$ we have

[TABLE]

where we used the bound $|a^{\prime}(s)|\leq 1$ for all $s>h$ (which follows from (25)). Therefore on the event $E$

[TABLE]

Now applying the discrete Gronwall inequality [1] to (87) we deduce that for all $m\leq n\leq N$, the event $E$ implies that

[TABLE]

Given $\delta>0$ we choose

[TABLE]

Then for $\lambda\geq\lambda_{0}$ we have $\rho+\lambda^{-1}\leq 3\,\theta$ and

[TABLE]

and hence (88) implies that $g(T)\leq\delta$ on the event $E$. Therefore

[TABLE]

and this completes the proof with

[TABLE]

6 Proof of Theorem 6

Recall the delay equation (25) for $a(t)$. Applying Lemma 2 we get

[TABLE]

Given the solution $a(t)$ for $t\leq T$, (94) is a linear equation for $a(t)$ in the interval $[T,T+h]$, and we can write down an explicit solution in terms of the solution in the interval $[T-h,T]$. Then by translating coordinates the equation (94) can be viewed as providing a map from the space of functions on $[0,h]$ into itself. In order to prove (59) we will consider instead $a(t)-h$, so define for $t\in[0,h]$

[TABLE]

then from (94) we derive

[TABLE]

As explained above, we will view (96) as a map from $x$ to $y$. Define the functional ${\cal F}$ as the map which takes $x$ to the solution $y$ of the equation (96):

[TABLE]

with the norms

[TABLE]

We will prove the following bounds: for all differentiable $x$,

[TABLE]

where $\kappa$ was defined in (6). Before proving (6) we note that it implies the bound (59): indeed for $t\geq 4h$, there is integer $n\geq 2$ such that $(2n-1)h\leq t<(2n+1)h$. The first inequality in (6) implies that

[TABLE]

Define $x_{0}(s)=(a(s)-h)/2$ for $s\in[h,2h]$. Then for any $t\in[(2n-1)h,(2n+1)h]$ the inequalities (6) and (100) imply

[TABLE]

where we used $C_{1}=2\|x_{0}\|$, and also that $\kappa$ is an increasing function.

So we have reduced the proof to (6). Given $x$, let $y$ be the solution of (96), and let $t\in[0,h]$. There are three cases:

Case 1: $y^{\prime}(t)=0$ it follows from (96) that $y(t)=x(t)/2$ and hence

[TABLE]

Case 2: $y^{\prime}(t)>0$ Define

[TABLE]

Then $y(t_{1})<y(t)<y(t_{2})$. By assumption $y^{\prime}$ is continuous, so if $t_{1}>0$ then $y^{\prime}(t_{1})=0$, and so $y(t_{1})=x(t_{1})/2$. If $t_{1}=0$ then $y(t_{1})=y(0)=x(h)$. Thus in either case

[TABLE]

Similarly if $t_{2}<h$ then $y^{\prime}(t_{2})=0$, and so $y(t_{2})=x(t_{2})/2$. If $t_{2}=h$ then $y(t_{2})=y(h)$ and $y^{\prime}(h)>0$, so $y(h)<x(h)/2$. Thus in either case

[TABLE]

Therefore

[TABLE]

and so we deduce that

[TABLE]

Case 3: $y^{\prime}(t)<0$ Define

[TABLE]

Then $y(t_{3})>y(t)>y(t_{4})$. By assumption $y^{\prime}$ is continuous, so if $t_{3}>0$ then $y^{\prime}(t_{3})=0$, and so $y(t_{3})=x(t_{3})/2$. If $t_{3}=0$ then $y(t_{3})=y(0)=x(h)$. Thus in either case

[TABLE]

Similarly if $t_{4}<h$ then $y^{\prime}(t_{4})=0$, and so $y(t_{4})=x(t_{4})/2$. If $t_{4}=h$ then $y(t_{4})=y(h)>x(h)/2$, and thus in either case

[TABLE]

Therefore

[TABLE]

and so we deduce again that for this case

[TABLE]

Putting together these three cases we have the bound

[TABLE]

This immediately implies that $\|y\|\leq\|x\|$ which is the first inequality in (6). For the second inequality, we will provide a bound for $|y(h)|$ in terms of $\|x\|$, which will be combined with (113) to derive (6). Again we examine several cases.

Case 4: $y^{\prime}(h)=0$ In this case $y(h)=x(h)/2$ and so $|y(h)|\leq\|x\|/2$.

Case 5: $y^{\prime}(h)<0$ In this case $y(h)>x(h)/2$. We assume that $y^{\prime}(t)<0$ for all $t\in[0,h]$: if this is not true then as with Case 3 we deduce the existence of $t_{3}$ such that $y(h)<y(t_{3})=x(t_{3})/2$, and then we have $x(h)/2<y(h)<x(t_{3})/2$, which implies $|y(h)|\leq\|x\|/2$. We also assume that $y(h)>0$: if $y(h)\leq 0$ then the inequality $y(h)>x(h)/2$ implies $|y(h)|\leq\|x\|/2$. Since $y(t)$ is monotone decreasing and $y(h)>0$ this implies that $y(t)>0$ for all $t\in[0,h]$. Also $y^{\prime}(t)<0$ implies

[TABLE]

Suppose first that there is some $t\in[0,h)$ such that

[TABLE]

Then

[TABLE]

and therefore

[TABLE]

If no such $t$ exists then we have

[TABLE]

and hence (since by assumption $y(t)>0$)

[TABLE]

We immediately deduce that

[TABLE]

Putting together these two possibilities we get

[TABLE]

Case 6: $y^{\prime}(h)>0$ The analysis of this case is identical to Case 5 with some signs reversed, and the same conclusion holds.

Combining Cases 4,5,6 we conclude that the bound (121) holds in all cases. Together with (113) we conclude that

[TABLE]

Finally we return to the second inequality in (6), and deduce from (122) that

[TABLE]

7 Discussion and future directions

Theorem 5 confirms that the tangle process converges (in probability) to the solution of the delay differential equation (25). This convergence was explored using numerical simulations in the paper [4], and was observed to give an accurate representation of the behavior even for relatively small values of the arrival rate $\lambda$. There are several interesting questions which arise out of this result. One question is to describe fluctuations of the rescaled process $A^{(\lambda)}(t)$ around the deterministic solution $a(t)$ of the delay differential equation. Theorem 5 shows that the scale of fluctuations is not larger than $\lambda^{-1/2}$. This is also the scale of the central limit theorem, and it would be interesting to determine if the fluctuations are gaussian in leading order. Another interesting question concerns $h$, the duration of the proof of work. In this paper we assumed throughout that $h$ is constant, however it would be natural to consider $h$ as a random variable. Finally the convergence of the tangle model to its fluid limit for other tip selection algorithms is also an interesting problem.

Acknowledgements

The author thanks Robert Shorten and Pietro Ferraro for helpful discussions and suggestions.