The population protocol model [AAD06] is a theoretical framework for analyzing distributed computation in ad hoc networks of anonymous, mobile agents. Through pairwise interactions among the finite-state nodes of a system, population protocols can solve numerous problems in distributed computing, including majority (which is also referred to as comparison or consensus) [AAE08, CHKM19, ATU20], source detection [ADK17, DK18], and leader election [AG15, GS18, GSU19]. Originally motivated by passively mobile sensor networks [AAD06], more recent research in population protocols has been used to study computation by chemical reaction networks [CDS14], DNA strand displacement [CDS13, TWS15], and biochemical networks [CCN12]. Applications of population protocols in chemistry have inspired various adaptations of the model, including the introduction of transient leaks, or spontaneous faulty state transitions [ADK17, ATU20]. In brief, a “leak” simulates the event that a molecule undergoes a reaction that typically takes place in the presence of a catalyst, where the catalyst is a molecule that enables the transformation of the other molecule but does not itself change state. Generally, leaks can be dealt with using error-correcting codes [WTE18]; however, for certain problems there are often more efficient, specialized solutions [ADK17, ATU20]. Interestingly, some of these solutions have also considered the behavior of protocols on populations containing a small number of persistent-state catalysts, but without the presence of leaks. For example, Alistarh et al. [ATU20] showed a state protocol for the comparison problem, which is equivalent to determining the majority value among the set of catalytic agents. Our contribution In this work, motivated by the recent interest in population models with catalytic agents and with transient leaks, we study the majority problem in the presence of these two variants. In particular, we consider the effect of catalysts and leaks on the behavior of the well-known third-state dynamics protocols [AAE08, PVV09] for computing approximate majority, which is the problem of determining which of two input values is initially more prevalent in a population, subject to a lower constraint on the size of the initial input value margin. To begin, we formalize a catalytic input (CI) model consisting of catalytic input agents which, in accordance with their namesake, do not ever change state, and worker agents that wish to compute some function on the configuration of catalytic agents. While conceptually similar to other models considering these types of catalytic agents [ADK17, ATU20, dCN20], introducing the distinction between the two (possibly unrelated) population sizes provides a new level of generality for designing and analyzing protocols in this setting, both with and without leaks. Although the CI and original population models share similarities, we show a strong separation between the computational power of the two: when , we prove a lower bound showing that cannot be computed in fewer than interactions with high probability in the CI model. On the other hand, the result of [KU18] shows that this predicate is computable within total steps in the original model with high probability111We define “high probability” to mean with probability at least where is the total number of agents and .. However, we show that not all predicates face the same hardness result in the CI model: by adapting the third-state dynamics process [AAE08], we present a constant-state protocol for approximate majority with catalytic inputs. The protocol converges with high probability to a correct output in total steps (where ) when the initial input margin is and . A similar lower bound technique is used to show that this input margin is optimal in the CI model up to a factor when . Moreover, in the presence of transient leak faults, we show that both the third-state dynamics protocol in the original model and our adapted protocol in the CI model exhibit a strong robustness to these spurious events: when the probability of a leak event is bounded, we show that both protocols still quickly reach a configuration where nearly all agents share the correct input majority value with high probability. Relation Between Third-State Dynamics in CI and Original Model The three-state protocol of Angluin et al. [AAE08] was proven to solve approximate majority with high probability in the original population model when the input margin (the initial difference between the counts of the two input values) is sufficiently large. Their protocol, which we refer to as Single-B Approximate Majority (or ), assumes one-way communication (where at most one agent per interaction can change state). Condon et al. [CHKM19] introduced a slight variation of the original protocol assuming two-way communication (where both agents can update their state following an interaction), which we refer to as Double-B Approximate Majority (or ). Condon et al. simplified the analysis for these approximate majority algorithms by introducing a tri-molecular chemical reaction network (CRN) called Tri (and referred to here as ). The protocol uses a molecule-count-preserving transition function for 3-way interactions to solve approximate majority in steps in populations where the initial margin is . The authors proved that is correct and efficient with high probability using random walk analysis techniques. Then, the authors showed how the correctness and efficiency of both the and protocols can be reduced to the analysis, as long as the initial margin in the population is at least . Given the simplicity of two-way communication, our protocol for approximate majority in the CI model adapts the behavior of the protocol. We refer to our variant of this protocol in the CI model as . Here, each catalytic input agent holds a persistent value of or , and each worker agent holds either an undecided, or blank value , or an or value corresponding to a belief in an or input majority, respectively. The transition rules for the , , and protocols are given in Figure 1. Structure of the Paper The structure of the remainder of the paper is as follows: in the next subsection we review related work, and in Section 2 we introduce notation and definitions central to our results. Section 3 presents our lower bounds over the CI model, which demonstrates the separation between the CI and original population models. In Section 4, we analyze the correctness and efficiency of the protocol for approximate majority in the CI model, and in Section 5 we demonstrate the leak-robustness of both the and original protocols. Then in Section 6, we compare the notion of transient leaks with the adversarial Byzantine model, demonstrating parallels between previous results examining Byzantine behavior and our work.
are both di-molecular CRNs where all rate constants are 1; these are also classified aspopulation protocols.
1.1 Related Work
The notion of a persistent source state was introduced by [DK18], where sources are used to solve source detection (the detection of a source in the population) and bit broadcast (the broadcast of a 0 or 1 message from a set of source agents). An accompanying work [ADK17] introduces the concept of leaks, or spontaneous faulty state changes, and investigates source detection in their presence. The work demonstrates that source detection in the presence of leaks (up to rate ) can be solved with high probability using states, where is the number of sources in the population. More recently, [ATU20] examines the problem of comparison in the presence of leaks. Comparison is a generalization of the majority problem, where some possibly small subset of the population is in input state or and the task of the population is to determine which of the two states is more prevalent. This work solves comparison in interactions using states per agent, assuming that for some constant , and that . The protocol is self-stabilizing, meaning that it dynamically responds to changes in the counts of input states. Our protocol is an adaptation of the third-state dynamics process introduced by [AAE08] and [PVV09] to the CI model. The approximate majority problem in the catalytic input model is equivalent to the comparison problem considered by [ATU20], so we demonstrate how our protocol compares to the results of this work. We show that our protocol for approximate majority in the CI model converges correctly within the same time complexity of total steps, while only using constant state space (compared to the logarithmic state used by the protocols in their work). Moreover, in populations where , our protocol tolerates a less restrictive bound on the input margin compared to [ATU20] ( compared to ). In the presence of transient leaks, our protocol also shows robustness to a higher adversarial leak rate of . However, unlike [ATU20], our protocol is not self-stabilizing and requires that the number of inputs be at least a constant fraction of the total population for our main results. In order to achieve these results, we leverage the random walk analysis techniques and analysis structure introduced by [CHKM19]. The third-state dynamics protocols considered in this work are also related more generally to opinion or consensus dynamics studied in other models of distributed computation and multi-agent systems (see [BCN20] for a recent survey). In particular, d’Amore et al. [dCN20] recently analyzed an analogous version of the protocol in the synchronous PULL model with a complete communication graph, where at each round all agents update their state in parallel after observing (pulling) the state of a random node in the network. In their work, the authors considered systems with stubborn agents (as in [YOA13]) and in the presence of probabilistic communication noise, which are similar to the persistent-state catalytic agents and transient leaks we consider in the present work, respectively. In the presence of either stubborn agents or communication noise (which the authors show are equivalent under certain technical conditions), the protocol of [dCN20] reaches a configuration where agents support the majority opinion within parallel rounds with high probability, so long as the initial input margin in the population is . The parallel, synchronous scheduling model considered in [dCN20] is fundamentally distinct from the sequential, pairwise scheduling used in population protocols. Additionally, while the consideration of stubborn agents in [dCN20] originally stems from the analysis of opinion dynamics on networks where some agents are unwilling to change states [YOA13], the presence of catalytic agents in our work was motivated by the transient leak events that are studied in chemical reaction networks. However, comparing the results of [dCN20] with our own (which share similar convergence guarantees) highlights the general robustness of third-state dynamics-like protocols in various distributed models to the presence of persistent-state agents and also transient leaks and other modes of communication failure. To this end, we also compare the impact of leaks in our protocols with that of faulty Byzantine agents that have been considered in previous work on population protocols. The fast robust approximate majority protocol of [AAE08] is proven to be robust to Byzantine agents. In [CHKM19], the same protocol is proven to reach a relaxed consensus (where agents converge to the correct output) in the presence of Byzantine agents, where is the initial margin. We show that is robust to leak rate with sampling error for and for . This second bound matches the result from [CHKM19], where the probability of sampling a non-convergent agent is .
We begin with some definitions. Denote by the number of agents in the population. Population Protocols
Population protocols are a class of algorithms which model interactions between mobile agents with limited communication range. Agents only interact with one another if they are within close enough proximity of each other. In order to model this type of system in an asynchronous setting, interactions between pairs of agents are executed in sequence. The interaction pattern of these agents is dictated by a scheduler, which may be random or adversarial. In this work we will assume that the scheduler is uniformly random, meaning that an ordered pair of agents is chosen to interact at each time step independently and uniformly at random from allordered pairs of agents in the system. As defined by the seminal paper which first introduced population protocols [AAD06], a population protocol consists of a state set , a rule set , an output alphabet , and an output function
. The output function computes the evaluation of some function on the population locally at each agent. The configuration of the population is denoted as a vectorsuch that each is equal to the number of agents in the population in state , from which it follows that . For convenience, we denote by the number of agents in the population in state . At each point in time, the scheduler chooses an ordered pair of agents , where is the initiator and is the responder [AAD06]. The agents interact and update their state according to the corresponding rule in . In general, a rule in is written as to convey that two agents, an initiator in state and a responder in state , interact and update their states to be and , respectively. By convention, interactions make one unit of parallel time [AAE08]. This convention is equivalent to assuming every agent interacts once per time unit on average. An execution is the sequence of configurations of a run of the protocol, which converges when the population arrives at a configuration such that all configurations chronologically after have the same output at each agent as those in [AAD06]. In order to determine the success or failure of an execution of , we will consider a sample of the population to signify the outcome of the protocol [ADK17]. After the expected time to converge, one agent is selected at random and its state is observed. The output associated with the agent’s state is considered the output of the protocol. The probability of sampling an agent whose state does not reflect the desired output of the protocol is called the sample error rate. Multiple samples can be aggregated to improve the rate of success. Catalysts and Leaks Following [ADK17], in an interaction of the form , we say catalyzes the transformation of the agent in state to be in state . If catalyzes every interaction it participates in, is referred to as a catalyst. In chemistry, a reaction that occurs in the presence of a catalyst also occurs at a lower rate in the absence of that catalyst. For this reason, recent work in DNA strand displacement, chemical reactions networks, and population protocols [TWS15, ADK17, ATU20] have studied the notion of leakage: When a catalytic reaction is possible, then there is some probability that a transition can occur without interacting with at all. This type of event, called a leak, was introduced in [TWS15]. The probability with which the non-catalyzed variation of a reaction takes place is the leak rate, which we denote by . We simulate a leak as follows: At each step in time, with probability , the scheduler samples an ordered pair of agents to interact with one another as described in the beginning of the Section; the rest of the time (i.e. with probability ) one agent is chosen uniformly at random from all possible agents and the leak function is applied to update this agent’s state. Catalytic Input Model In this work, we formalize a catalytic input (CI) model consisting of catalytic agents that supply the input and worker agents that perform the computation and produce output. We define to be the total number of agents in the population. At each time step, the scheduler samples any two agents in the population to interact with one another. If two catalysts are chosen to interact, then the interaction is considered to be null as no nontrivial state transition occurs. When , the probability that two catalysts are chosen to interact is upper bounded by a constant, and so the total running time of the protocol is asymptotically equivalent to the number of non-null interactions needed to reach convergence. In the CI model, we consider convergence to be a term that refers to the states of the worker agents only, as the catalytic agents never change state. Namely, for the approximate majority problem, successful convergence equates to all worker agents being in the majority-accepting state. In general, we wish to obtain results that hold with high probability with respect to the total number of agents .
3 Catalytic Input Model Lower Bounds
In this section, we give initial results characterizing the computational power of the CI population protocol model. Using information-theoretic arguments, we prove two lower bounds over the catalytic model when the number of input agents is a constant fraction of the total population:  In the catalytic input model with input agents and worker agents, any protocol that computes the of the inputs with probability at least requires at least total steps for any .  In the catalytic input model with input agents and worker agents, any protocol that computes the of the inputs within total steps requires an input margin of at least to be correct with probability at least for any . The first result can be viewed as a separation between the CI and original population models: since it is shown in [KU18] that the of agents can be computed in the original model within parallel time with high probability, our result indicates that not all semi-linear predicates over the input population in the CI model can be computed in sub-linear parallel time with high probability. Additionally, this rules out the possibility of designing fast protocols for exact majority in the CI model when the input size is a constant fraction of the entire population. On the other hand, the second result indicates the existence of a predicate — approximate — that does not require a large increase in convergence time to be computed with high probability in this new model. Sampling Catalytic Inputs One key characteristic of a CI population is the inability for worker agents to distinguish which inputs have previously interacted with a worker. Instead, every worker-input interaction acts like a random sample with replacement from the input population. For proving lower bounds in this model, this characteristic of a CI population leads to the following natural argument: consider a population of catalytic input agents and a worker population consisting of a single super-agent. Here, we assume the super-agent has unbounded state and computational power, and it is thus able to simulate the entire worker population of any protocol with more workers. In this simulation, any interaction between a worker and an input agent is equivalent to the super-agent interacting with an input chosen uniformly at random: in other words, as a sample with replacement from the input population. If the super-agent needs samples to compute some predicate over the inputs with high probability, then so does any multi-worker protocol in the CI model. We denote this information-theoretic model as the Super CI model, and restate the above argument more formally in the following lemma. Consider a population with catalytic input agents and a worker population consisting of a single super-agent . Let be a predicate over the input population that requires total interactions between and the input population in order for to correctly compute with probability . Then for a CI population with catalytic inputs and worker agents, computing correctly with probability requires at least total interactions.
3.1 Proof of Theorem 3
We now develop the proof of Theorem 3, which says that in a CI model population with input agents and worker agents where , computing the of the inputs requires at least total interactions to be correct with high probability. To do this, we will show that in the Super CI model described in the previous section, a computationally unbounded super-agent requires at least samples of the input population to correctly compute the input parity with high probability. Applying Lemma 3 then gives Theorem 3. Reduction from to For an input population of agents, each with input value or , the of is said to be
if an odd number of agents have input value, and otherwise. Now, consider separately the predicate over , which is simply the majority value of the input population. Letting denote the number of -inputs, and the number of -inputs, we refer to the input margin of the population as the quantity . Suppose that is odd and the input majority of is . Then and are either and or vice versa. These two cases can be distinguished either by computing the predicate or the predicate, making both of these problems equivalent to distinguishing the two cases under this constraint on the input. We will now argue that distinguishing these cases in the Super CI model requires samples. Optimality of the Sample Majority Map Recall that in the Super CI model, a predicate over the input population is computed by a single super agent worker with unbounded computational power. Thus, the output of can be viewed as a mapping between a string of input values obtained from interactions with between and the input population and the output set . We will refer to interactions between and the input population as samples of the input, and for a fixed number of samples , we refer to ’s output as its strategy. To begin, we will show that for a fixed distribution over the input values of , the strategy that maximizes the ’s probability of correctly outputting the majority value of is simply to output the majority value of its samples. Let be the sample string representing the independent samples with replacement taken by , and let denote the set of all possible sample strings. We model the population of input agents as being generated by an adversary. Specifically, let denote the majority value (0 or 1) of the input population, where we treat
as a a random variable whose distribution is unknown. In any realization of, we assume a fixed fraction of the inputs hold the majority value. Given an input population, the objective of the worker agent is to correctly determine the value of through its input sample string . By Yao’s principle [Yao77], the worst case expected error of the worker’s strategy when is chosen according to some arbitrary distribution is no more than the error of the best deterministic strategy when is chosen according to some fixed distribution. Thus, assuming is chosen according to some fixed distribution, we model the worker’s strategy as a fixed map . Letting denote the set of all such maps, then faces the following optimization problem: . For a given , let , and let denote the map that outputs the majority value of the input sample string . In the following lemma, we show that for any adversarially chosen input population, setting maximizes . In other words, to maximize the probability of correctly guessing the input population majority value, the worker’s optimal strategy is to simply guess the majority value of its
independent samples. The proof of the Lemma simply uses and expands the definitions of conditional probability and the Law of Total Probability to obtain the result.  Letbe a sample string of size drawn from an input population with majority value and majority ratio . Then for all maps , where is the map that outputs the majority value of the sample string .
Recall that we model the majority value of the input population as a 0-1 random variable whose distribution is chosen adversarially. By Yao’s Lemma, it is enough to prove the optimality of for the fixed average case, where . For any map , we can compute by
where the last inequality follows from assuming . Recall that is the input string of independent samples from the input population, and is the set of all possible values of . Thus for any , the law of total probability gives
Since the events and are independent, for every . Additionally, given that every is a deterministic map, we can rewrite
is the indicator random variable of the event. Thus for any and every we have
It can be similarly shown that
for every and a fixed . Thus substituting back into (7) gives
Now, let denote the set of of input sample strings with a 0-majority, and let denote the set of sample strings with a 1-majority. Without loss of generality, assume and are disjoint and that . Additionally, for a fixed and any define by
Thus for a fixed we can again rewrite
Recall that is the map that outputs the majority value of the input sample string . Fix any other map . Since , there exists at least one string such that , and assume without loss of generality that . By definition, this means and . Using the definition of , and recalling that and are the probabilities that a single sample of is 0 or 1 respectively, we have
where since . Meanwhile, for the same , using the majority sample map gives
where again since . Since by definition , it follows that for any where , and for any where . It can similarly be shown that for any with . By the definition of from (11), it follows that for any , thus proving the claim. ∎
Sample Lower Bound for with Input Margin 1 We have established by Lemma 3.1 that to correctly output the input population majority, the super worker agent’s error-minimizing strategy is to output the majority of its samples. Now the following lemma shows that when the input margin of the population is 1, this strategy requires at least samples in order to output the input population majority with probability at least for some constant
. The proof uses a tail bound on the Binomial distribution to show the desired trade off between the error of probability and the requisite number of samples needed to achieve this error.  Letbe a Super CI population of agents with majority value and input margin 1, and consider an input sample string obtained by a super worker agent . Then for any , letting denote the sample majority of , only holds when .
We will assume without loss of generality, meaning that . Since , we will prove that is a necessary constraint to satisfy . Here is just the lower tail of the CDF of a binomial distribution with parameter . Thus when for , we have the following lower bound on (see [Ash90]):
Here, denotes the Kullback-Leibler (KL) divergence between a fair coin and a Bernoulli random variable with bias . This can be rewritten as
and since we are assuming , we have
where for all . Thus to ensure , it is necessary to have . Taking natural logarithms then yields the following constraint on :
We now want to show is needed to satisfy (26). To do this, consider any . Observe then that and . It follows that
where the final inequality necessarily holds for all for large enough . Thus no value can satisfy the necessary condition of (26), which means that we must have in order to ensure holds for any . ∎
By Lemmas 3.1 and 3.1, in the Super CI model with an input population of agents and input margin 1, a single super-agent worker can only compute the majority value of with high probability by taking at least input samples. By Lemma 3, this means that in the regular CI model with an input population of agents, any protocol for with input margin 1 requires at least total steps to be computed correctly with probability at least . Because computing on a population with input margin 1 reduces to computing over this input, we have that any protocol for in the CI model requires at least total steps in the worst-case to be correct with probability at least . When the size of the worker population is , this means that . Thus for an appropriate choice of , computing on such populations requires at least samples to be correct with probability at least for any . ∎
3.2 Proof of Theorem 3
As mentioned, the result from Theorem 3 implies a strong separation between the CI model and original population model, as [KU18] has shown that is computable with high probability within total steps in the original model. Thus, the persistent-state nature of input agents in the CI model may seem to pose greater challenges than in the original model for computing predicates quickly with high probability. However, using the same sampling-based lower bound techniques developed in the preceding section, we show that when restricted only to total steps, any protocol computing in the CI model when requires an input margin of at least to be correct with high probability in . Moreover, in the Section 4 we present a protocol for in the CI model that converges correctly with high probability within total steps, so long as the initial input margin is . Thus, the existence of such a protocol indicates that the lower bound on the input margin is nearly tight (up to factors) for protocols limited to total steps when . We now proceed with a formal proof of Theorem 3, which follows similarly to the proof of Theorem 3.
Input Margin Lower Bound for
We return to the Super CI model and use the same notation developed in Section 3.1. We want to show that when , we must have in order to ensure . This lower bound on (the proportion of 1-agents, wlog, in the input population) corresponds to an input margin lower bound of . Assume a 0-1 population of agents with majority value and majority proportion , and consider an input sample string where . Then for any , only holds when , where is the map that outputs the majority value of .
Again wlog assume . Since , we will show that is a necessary condition to have when . Recall the lower bound on from Lemma 3.1:
which holds for . (Note that when , for all ). Setting for some lets us write (27) as
which means it is necessary to have to ensure that . Again by taking natural logarithms, we find that we require
where the final equality will hold for all and large enough . Thus if , then the necessary condition (29) will be violated, meaning that that we must have to ensure holds for any when . Since we defined , this corresponds to requiring an input margin of at least when . ∎
By Lemmas 3.1 and 3.2, in the Super CI model with an input population of size , computing correctly in samples with probability at least requires an input margin of at least . By an argument similar to Lemma 3, note that this implies that any protocol that computes in the regular CI model within total steps also requires an input margin of to be correct with probability at least . Now consider that the size of the worker population is , which means that . This implies that for an appropriate choice of constant and taking only total steps, the input margin must be at least in order for to be computed correctly with probability at least for any . ∎
4 Approximate Majority with Catalytic Inputs
We now present a protocol, , for computing approximate majority in the CI model. As mentioned earlier, the protocol is a natural adaptation of a third-state dynamics from the original model, where we now account for the behavior of catalytic input agents and worker agents. Using the CI model notation introduced in Section 2, we consider a population with total agents. Each input agent begins (and remains) in state or , and we assume each worker agent begins in a blank state , but may transition to states or according to the transition rules found in Figure 1. Letting and (and similarly and ) be random variables denoting the number of agents in states and (and respectively , and ), we denote the input margin of the population by . Throughout the section, we assume without loss of generality that . As described in Section 1, the protocol is a natural adaptation of the protocol for the CI model. Intuitively, an undecided (blank) worker agent adopts the state of a decided agent (either an input or worker), but decided workers only revert back to a blank state upon interactions with other workers of the opposite opinion. Thus the protocol shares the opinion-spreading behavior of the original protocol, but note that the inability for decided worker agents to revert back to the blank state upon subsequent interactions with an input allows the protocol to converge to a configuration where all workers share the same or opinion. Main Result The main result of the section characterizes the convergence behavior of the protocol when the input margin is sufficiently large. Assuming that , recall that we say the protocol correctly computes the of the inputs if we reach a configuration where . The following theorem then says that, subject to mild constraints on the population sizes, when the input margin is , the protocol correctly computes the majority value of the inputs in roughly logarithmic parallel time with high probability.  There exists some constant such that, for a population of inputs, workers, and initial input margin , the protocol correctly computes the majority value of the inputs within total interactions with probability at least for any when and is sufficiently large. Because the CI model allows for distinct (and possibly unrelated) input and worker population sizes, we aim to characterize all error and success probabilities with respect to the total population size . Intuitively, the result of Theorem 4 says that with input margin at least , the protocol correctly computes approximate majority within total steps. However, our analysis and proof of Theorem 4 characterize the convergence behavior for a broad range of input and worker population sizes, and thus the result includes higher-order multiplicative terms. On the other hand, in the case when — which is an assumption used to provide lower bounds over the CI model from Section 3 — we have as a corollary of the previous result that the protocol correctly computes the majority of the inputs within total steps with probability at least . This result is formally stated as Corollary 4.2.5 further below.
4.1 Analysis Overview
The proof of the main result leverages and applies the random walk tools from [CHKM19] (in their analysis of the original protocol) to the protocol. Given the uniformly-random behavior of the interaction scheduler, the random variables and (which represent the count of , , and worker agents in the population) each behave according to some one-dimensional random walk, where the biases in the walks change dynamically as the values of these random variables fluctuate. Based on the coupling principle that an upper bound on the number of steps for a random walk with success probability to reach a certain position is an upper bound on the step requirement for a second random walk with probability to reach the same position, we make use of several progress measures that give the behavior of the protocol a natural structure. First, we define , , and . The use of and match the analysis from [CHKM19], where it can be easily seen that will hold throughout the protocol. On the other hand, the progress measure captures the collective gap between the majority and non-majority opinions in the population. Observe that the protocol has correctly computed the input majority value when and . Now, similar again to the analysis of [CHKM19], we define the following Phases and stages of the protocol.
Phase 1 of the protocol starts with and completes correctly once . Each stage of Phase 1 begins with and completes correctly once or when , where .
Phase 2 of the protocol starts with (equivalent to ) and completes correctly once . Each stage of Phase 2 begins with and completes correctly once or when , where .
Phase 3 of the protocol starts with and completes correctly once .
Intuitively, every correctly-completed stage of Phase 1 results in the progress measure doubling, and every correctly-completed stage of Phase 2 results in the progress measure decreasing by a factor of two. Note that of the protocol’s non-null transitions (see Figure 1), only the interactions , , , and change the value of either progress measure. For this reason, we refer to the set of non-null transitions (which includes interactions) as productive steps, and the subset of interactions that change our progress measures as the set of blank-consuming productive steps (note we sometimes refer to these as productive-b or prod-b steps in the analysis). The strategy for every stage and Phase is to employ a combination of standard Chernoff bounds and martingale techniques (in general, see [GS01] and [Fel68]
) to obtain with-high-probability estimates of (1) the number of productive steps needed to complete each Phase/stage correctly, and (2) the number of total steps needed to obtain the productive step requirements. Given an input margin that is sufficiently large (that also places a mild restriction on the relative size of the input population), and also assuming a population where the number of worker agents is at least a small constant fraction of the input size, we can then sum over the error probabilities of each Phase/stage and apply a union bound to yield the final result of Theorem4. We note that while the protocol is conceptually similar to the original protocol, the presence of persistent-state catalysts whose opinions never change requires a careful analysis of the convergence behavior. Moreover, simulation results presented in Section 5 show interesting differences in the evolution of the protocol for varying input and worker population sizes.
4.2 Proof of Theorem 4
In this section we develop the tools used to prove Theorem 4. Similar to the techniques developed in [CHKM19], we divide the analysis into two main components: correctness and efficiency. For correctness, we derive with-high-probability bounds on the number of productive steps needed to correctly complete each stage of Phases 1 and 2, and Phase 3. For efficiency, we subsequently derive the number of total steps needed to obtain these productive-step counts with high probability. To begin, we state the following standard probabilistic tools used throughout the analysis: absorption probabilities for one-dimensional random walks, and standard upper and lower Chernoff bounds. [Fel68] If we run an arbitrarily long sequence of independent trials, each with success probability at least , then the probability that the number of failures ever exceeds the number of successes by is at most . [Che52] If we run independent Bernoulli trials, each with success probability , then the number of successes has expected value , and for , , and . The following subsections proceed to prove the correctness and efficiency of the stages and phases of the protocol.
4.2.1 Phase 1: Blank-Consuming Step Bounds
For a population with an initial input margin , the following lemma gives an upper bound on the number of productive-b (blank-consuming) steps needed to complete each stage of Phase 1 correctly. Recall that each stage of Phase 1 completes correctly when the progress measure doubles from its initial value. During Phase 1 of the protocol on a population with input margin for some , starting at , within productive-b steps will increase to with probability at least .
Observe that from the time at which until the point (if ever) decreases below , is strictly greater than . Therefore, until reaches , the probability of a successful interaction (conditioned on having a blank-consuming interaction) is at least
where the final inequality holds because , , and . The change in can thus be viewed as a biased random walk starting at with success probability . Now, by Lemma 4.2, starting at , the probability of ever having an excess margin of between or steps to or steps is at most
where the final inequality holds given the assumption that . Thus with high probability, will never drop below when starting initially from . Now in a sequence of productive-b steps, in order for to reach , it is sufficient to ensure that the number of or steps within the sequence (which we denote by ) exceeds the number of or steps within the sequence (which we denote by ) by at least . Assuming that holds, we can see that in expectation over the sequence of productive-b steps that . As long as it follows that , and thus applying an upper Chernoff bound shows that
where again the penultimate inequality is due to the assumption that . Now, summing over all error probabilities and taking a union bound shows that will increase to within productive-b steps with probability at least . ∎
4.2.2 Phases 2 and 3: Blank-Consuming Step Bounds
The following lemma gives analogous bounds on the number of productive-b steps needed to complete stages of Phase 2, and Phase 3, correctly with high probability. Recall that each stage of Phase 2 of the protocol begins with , where , and ends correctly when decreases by a factor of 2 from its original value (similarly, Phase 3 starts with and ends correctly once reaches 0). Note that the following lemma proves a slightly stronger result by showing the number of productive-b steps needed to bring to , which will always be an upper bound on the number of steps needed to complete a stage of Phase 2, or Phase 3 correctly. Say for during Phase 2 of the protocol on a population with input margin for some . Assuming that remains below , then after at most productive-b steps, goes to 0 with probability at least for .
Recall from Lemma 4.2.1 that throughout the execution of on a population with input margin , the probability of a or step (conditioned on having a blank-consuming step) is bounded from below by , where . This lower bound holds as long as never drops below , which is ensured given the starting conditions of Phase 2 and by the assumption that . Now, given that is invariant throughout the execution, can be rewritten as . Also, since we assume starts at and never exceeds with high probability, it follows that
Denoting and steps as succeeding and and steps as failing, then the probability of a succeeding, productive-b interaction (conditioned on a blank-consuming step) can be rewritten as