The problem

An intuitive description of the model follows. For more precise definitions, see, Section The models in the Supplementary Information.

Consider a population of n agents. Thought of as computing entities, assume that each agent has a discrete internal state, and can execute randomized algorithms—by internally flipping coins. In addition, each agent has an opinion, which we assume for simplicity to be binary, i.e., either 0 or 1. A small number, s, of agents play the role of sources. Source agents are aware of their role and share the same opinion, referred to as the correct opinion. The goal of all agents is to have their opinion coincide with the correct opinion.

To achieve this goal, each agent continuously displays one of several messages taken from some finite alphabet Σ. Agents interact according to a random pattern, termed as the parallel- model: In each round , each agent u observes the message currently displayed by another agent v, chosen independently and uniformly at random from all agents. Importantly, communication is noisy, hence the message observed by u may differ from that displayed by v. More precisely, for any m, m′ ∈ Σ, let P_m,m′ be the probability that, any time some agent u observes an agent v holding some message m ∈ Σ, u actually receives message m′. The probabilities P_m,m′ define the entries of the noise-matrix P [21], which does not depend on time.

The noise is characterized by a noise parameter δ > 0. Our model encapsulates a large family of noise distributions, making our bounds highly general. Specifically, the noise distribution can take any form, as long as it satisfies the following criterion.

Definition 1 (δ-uniform noise) We say that the noise is δ-uniform if P_m,m′ ≥ δ for any m, m′ ∈ Σ.

When messages are noiseless, it is easy to see that the number of rounds that are required to guarantee that all agents hold the correct opinion with high probability is [16]. In what follows, we aim to show that when the δ-uniform noise criterion is satisfied, the number of rounds required until even one non-source agent can be moderately certain about the value of the correct opinion is very large. Specifically, thinking of δ and s as constants independent of the population size n, this number of rounds is at least Ω(n).

To prove the lower bound, we will bestow the agents with capabilities that far surpass those that are reasonable for biological entities. These include:

• Unique identities: Agents have unique identities in the range {1, 2, …n}. When observing agent v, its identity is received without noise.
• Complete knowledge of the system: Agents have access to all parameters of the system (including n, s, and δ) as well as to the full knowledge of the initial configuration except, of course, the correct opinion and the identity of the sources. In addition, agents have access to the results of random coin flips used internally by all other agents.
• Full synchronization: Agents know when the execution starts, and can count rounds.

We show that even given this extra computational power, fast convergence cannot be achieved. All the more so, fast convergence is impossible under more realistic assumptions.

Theoretical results

In all the statements that follow we consider the parallel- model satisfying the δ-uniform noise criterion, with cs/n < δ ≤ 1/|Σ| for some sufficiently large constant c, where the upper bound follows from the criterion given in Definition 1. Hence, the previous lower bound on δ implies a restriction on the alphabet size, specifically, |Σ| < n/(cs).

Theorem 1.1 Any rumor spreading protocol cannot converge in less than rounds.

Observe that the lower bound we present loses relevance when s is of order greater than , as our proof technique becomes uninformative in presence of a large number of sources (see Remark 2 in the Supplementary Information). Recall also that we assume that a source is aware that it is a source, but if it wishes to identify itself as such to agents that observe it, it must encode this information in a message, which is, in turn, subject to noise. We also consider the case in which an agent can reliably identify a source when it observes one (that is, this information is not noisy). For this case, the following lower bound, which is weaker than the previous one but still polynomial, apply (see also the S1 Text, Detectable sources):

Corollary 1.1 Assume that sources are reliably detectable. There is no rumor spreading protocol that converges in less than rounds.

Our results suggest that, in contrast to systems that enjoy stable connectivity, structureless systems are highly sensitive to communication noise (see Fig 1). More concretely, the two crucial assumptions that make our lower bounds applicable are: 1) stochastic interactions, and 2) δ-uniform noise (Fig 1, right hand panel). When agents can stabilize their interactions the first assumption is violated. In such cases, agents can overcome noise by employing simple error-correction techniques, e.g., using redundant messaging or waiting for acknowledgment before proceeding to the next action. As demonstrated in Fig 1, (left hand panel), when the noise is not uniform, it might be possible to overcome it with simple techniques based on using default neutral messages, and employing exceptional distinguishable signals only when necessary.

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Fig 1. Non-uniform noise vs. uniform noise.

On the left, we consider an example with non-uniform noise. Assume that the message vocabulary consists of 5 symbols, that is, Σ = {m₁, m₂, m₃, m₄, m₅}, where m₁ = 0 and m₅ = 1, represent the opinions. Assume that noise can occur only between consecutive messages. For example, m₂ can be observed as either m₂, m₃ or m₁, all with positive constant probability, but can never be viewed as m₄ or m₅. In this scenario, the population can quickly converge on the correct opinion by executing the following. The sources always display the correct opinion, i.e., either m₁ or m₅, and each other agent displays m₃ unless it has seen either m₁ or m₅ in which case it adopts the opinion it saw and displays it. In other words, m₃ serves as a default message for non-source agents, and m₁ and m₅ serve as attracting sinks. It is easy to see that the correct opinion will propagate quickly through the system without disturbance, and within number of rounds, where n is the size of the population, all agents will hold it with high probability. In contrast, in the case of δ-uniform noise as depicted on the right picture, if every message can be observed as any other message with some constant positive probability (for clarity, some of the arrows have been omitted from the sketch), then convergence cannot be achieved in less than Ω(n) rounds, as Theorem 1.1 dictates.

https://doi.org/10.1371/journal.pcbi.1006195.g001

Recruitment in desert ants

Our theoretical results assert that efficient rumor spreading in large groups could not be achieved without some degree of communication reliability. An example of a biological system whose communication reliability appears to be deficient in all of its components is recruitment in Cataglyphis niger desert ants. In this species, when a forager locates an oversized food item, she returns to the nest to recruit other ants to help in its retrieval [31, 32].

In our experimental setup, summarized in Fig 2, recruitment occurs in the small area of the nest’s entrance chamber (Fig 2a). We find that within this confined area, the interactions between ants follow a near uniformly random meeting pattern [33]. In other words, ants seem to have no control over which of their nest mates they will meet next (Fig 2b). This random meeting pattern approximates the first main assumption of our model. Another of the model’s assumptions is that ants interact in parallel. This implies that the interaction rate per ant be constant and independent of group size. Indeed, the empirical rate of interaction during the recruitment process was measured to be 0.82 ± 0.07 (mean ± sem, N = 44) interactions per minute per ant and induces a small increase with group size: 0.62 ± 0.13 for two ants (N = 8) and 1 ± 0.2 for a group sizes of 9-10 (N = 5).

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Fig 2. Unreliable communication and slow recruitment by desert ant (Cataglyphis niger).

a. The experimental setup. The recruiter ant (circled) returns to the nest’s entrance chamber (dark, 9cm diameter, disc) after finding the immobilized food item (arrow). Group size is ten. b. A pdf of the number of interactions that an ant experiences before meeting the same ant twice. The pdf is compared to uniform randomized interaction pattern. Data summarizes N = 671 interactions from seven experiments with a group size of 6 ants. c. Interactions of stationary ants with moving ants were classified into three different messages (‘a’ to ‘c’) depending on the moving ants’ speed. The noise at which messages were confused with each other was estimated according to the response of the recipient, initially stationary, ants (see Materials and methods). Gray scale indicates the estimated overlap between every two messages δ(i, j). Note δ = min(δ(i, j)) ≈ 0.3. Data collected over N = 278 interactions. d. The mean time it takes an ant that is informed about the food to recruit two nest-mates to exit the nest is presented for two group size ranges. Error bars represent standard error of the means over N = 24 experiments.

https://doi.org/10.1371/journal.pcbi.1006195.g002

It has been shown that recruitment in Cataglyphis niger ants relies on rudimentary alerting interactions [34, 35] which are subject to high levels of noise [32]. Moreover, the information an ant passes in an interaction can be attributed solely to her speed before the interaction [32]. Binning ant speeds into three discrete messages and measuring the responses of stationary ants to these messages, we can estimate the probabilities of one message to be mistakenly perceived as another one (see Estimating δ in the Methods). We find that this communication is extremely noisy which complies with the uniform-noise assumption with a δ of approximately 0.3 (Fig 2c). While artificially dividing the continuous speed signals into a large number of discrete messages (thus creating a larger alphabet) would inevitably decrease δ, this is not supported by our empirical data (see Section Methods).

Finally, the interaction scheme, as exhibited by the ants, can be viewed somewhere in-between the noisy-push and the noisy-pull models. Moving ants tend to initiate more interaction [32] and this may resemble, at first glance, a noisy-push interaction scheme. However, the ants’ interactions actually share characteristics with noisy-pull communication. Mainly, ants cannot reliably distinguish an ant that attempts to transmit information from any other non-communicating individual [32]. The fact that a receiver ant cannot be certain that a message was indeed communicated to her coincides with the lack of reliability in information transmission in line with our theoretical assumptions (see more details on this point in the Section Separation between and ).

Given the coincidence between the communication patterns in this ant system and the requirements of our lower bound we expect long delays before any uninformed ant can be relatively certain that a recruitment process is occurring. We therefore measured the time it takes an ant, that has been at the food source, to recruit the help of two nest-mates for different total group size. One might have expected this time to be independent of the group size or even to decrease as two ants constitute a smaller fraction of larger groups. To the contrary, we find that the time until the second ant is recruited increases with group size (p < 0.05 Kolmogorov-Smirnov test over N = 24 experiments, see Fig 2d).

Our theoretical results set a lower bound on the minimal time it takes uninformed ants to be recruited. Note that our lower bounds actually correspond to the time until any individual can be sure with more than 2/3 probability of the rumor. In the context of the ant recruitment experiment this means that if an ant goes out of the nest only if she is sure with some probability that there is a reason to exit, then the lower bounds correspond to the time until the first, and similarly the second (see Fig 2d), ants exit the nest.

Our lower bound is linear in the group size (Theorem 1.1). Note that this does not imply that the ants’ biological algorithm matches the lower bound and must be linear as well. Rather, our theoretical results qualitatively predict that as group size grows, recruitment times must eventually grow as well. This stands in agreement with Fig 2d. Thus, in this system, inherently noisy interactions on the microscopic level have direct implications on group level performance.

Overview of the main lower bound proof

Here, we provide the intuition for our main theoretical result, Theorem 1.1. For a formal proof please refer to the S1 Text, The lower bounds. The proof can be broken into three parts and, below, we refer to each of them separately.

Part II. From broadcast- to a statistical inference problem.

To establish the desired lower bound, we next show how the rumor spreading problem in the broadcast- model relates to a statistical inference test. That is, from the perspective of a given agent, the rumor spreading problem can be understood as the following: Based on a sequence of noisy observations, the agent should be able to tell whether the correct opinion is 0 or 1. We formulate this problem as a specific task of distinguishing between two random processes, one originated by running the protocol assuming the correct opinion is 0 and the other assuming it is 1.

One of the main difficulties lies in the fact that these processes may have a memory. At different time steps, they do not necessarily consist of independent draws of a given random variable. In other words, the probability distribution of an observation not only depends on the correct opinion, on the initial configuration and on the underlying randomness used by agents, but also on the previous noisy observation samples and (consequently) on the messages agents themselves choose to display on that round. An intuitive version of this problem is the task of distinguishing between two (multi-valued) biased coins, whose bias changes according to the previous outcomes of tossing them (e.g., due to wear). See Fig 3 for an illustration.

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Fig 3. Distinguishing between two types of coins.

On the top there are two possible coins with slightly different distributions for yielding a head (H) or a tail (T). (We depicted two possible outcomes but our model can account for more.) Given a sequence of observations (corresponding to the random outcomes of coin tosses), the goal of the observer is to guess the coin type being used (either 0 or 1). The wear induced by tossing the coins may, with time, change the probability that they land on either heads or tails in a way that depends on the coin type as well as on the previous toss outcomes (observations). In particular, notice that without a change in the probability of heads in our example we would not obtain a posterior probability 0.51 starting with a prior of 0.45 after six coin flips involving two heads. P_j(H^(t) ∣ observations) for j ∈ {0, 1} denotes the probability of Coin of type j to yield H given the particular sequence of t observations. Here, this sequence is H, T, T, H, T, T. At the beginning of the next round, i.e., the 7th round, |ε^{(t = 6)}|₁ measures how “far” the the H vs T distribution generated by the worn down coin 1 is from the same distribution as generated by the worn down coin 2. More precisely, for the case above, |ε⁽⁶⁾|₁ = |P₀(H ∣ observations) − P₁(H ∣ observations)| + |P₀(T ∣ observations) − P₁(T ∣ observations)| = |0.51 − 0.43| + |0.49 − 0.57| = 0.16. The parameter ε bounds all possible |ε^(t)|₁ from above.

https://doi.org/10.1371/journal.pcbi.1006195.g003

Despite this apparent complexity, we show that the difficulty of this distinguishing task can be captured by two scalar parameters, denoted ε and δ. Parameter δ lower bounds the probability for any observation to be attained (given any sequence of observations). Parameter ε captures the extent to which the processes are ‘similar’. More specifically, at round t, given a sequence of previous observations, denoted x^(<t), the next observation has the same probability to be attained in each process, up to an ε additive term (see Fig 3). A crucial observation is that ε is very small, precisely, ε = Θ(s(1 − δ|Σ|)/n). This follows from the fact that given x^(<t), the behavior of non-source agents in the two processes is the same, regardless of the value of the correct opinion. Indeed, internally, an agent is only affected by its initial knowledge, the randomness it uses, and the sequence of observations it sees. This means that at round t, the processes would differ only if the agent to be observed on that round happens to be a source (which happens with probability s/n) and, on top of that, the observed message is not the changed by noise (which accounts for the factor (1 − δ|Σ|)). However, a small value of ε is not enough to ensure slow running time. Indeed, even though the t’th observation may be distributed almost the same, if it happens that some observation can be attained only in one process, then seeing such an observation would immediately allow the observer to distinguish the two processes. A sufficiently large δ prevents the aforementioned scenario.

Generalizations

Several of the assumptions discussed earlier for the parallel- model were made for the sake of simplicity of presentation. In fact, our results can be shown to hold under more general conditions, that include: 1) different rate for sampling a source, and 2) a more relaxed noise criterion. In addition, our theorems were stated with respect to the parallel- model. In this model, at every round, each agent samples a single agent u.a.r. In fact, for any integer k, our analysis can be applied to the model in which, at every round, each agent observes k agents chosen u.a.r. In this case, the lower bound would simply reduce by a factor of k. Our analysis can also apply to a sequential variant, in which in each time step, two agents u and v are chosen u.a.r from the population and u observes v. In this case, our lower bounds would multiply by a factor of n, yielding, for example, a lower bound of Ω(n²) in the case where δ and s are constants. Observe that the latter increase is not surprising as each round in the parallel- model consists of n observations, while the sequential model consists of only one observation in each time step. See more details in the Supplementary Information.

Exponential separation between and

Our lower bounds on the parallel- model (where agents observe other agents) should be contrasted with known results in the parallel- model (this is the push equivalent to parallel- model, where in each round each agent may or may not actively push a message to another agent chosen u.a.r.). Although never proved, and although their combination is known to achieve more power than each of them separately [16], researchers often view the parallel- and parallel- models as very similar on complete communication topologies. Our lower bound result, however, undermines this belief, proving that in the context of noisy communication, there is an exponential separation between the two models. Indeed, when the noise level is constant for instance, convergence (and in fact, a much stronger convergence than we consider here) can be achieved in the parallel- using only logarithmic number of rounds [20, 21], by a simple strategy composed of two stages. The first stage consists of providing all agents with a guess about the source’s opinion, in such a way that ensures a non-negligible bias toward the correct guess. The second stage then boosts this bias by progressively amplifying it. A crucial aspect in the first stage is that agents remain silent until a certain point in time that they start sending out messages. This prevents agents from starting to spread information before they have sufficiently reliable knowledge and allows for a balanced control of the rumor spread. More specifically, marking an edge corresponding to a message received for the first time by an agent, the set of marked edges forms a spanning tree of low depth, rooted at the source. The depth of such tree can be interpreted as the deterioration of the message’s reliability. On the other hand, as shown here, in the parallel- model, even with the synchronization assumption, rumor spreading cannot be achieved in less than a linear number of rounds.

Perhaps the main reason why these two models are often considered similar is that with an extra bit in the message, a protocol can be approximated in the model, by letting this bit indicate whether the agent in the model was aiming to push its message. However, for such a strategy to work, this extra bit has to be reliable. Yet, in the noisy model, no bit is safe from noise, and hence, as we show, such an approximation cannot work. In this sense, the extra power that the noisy model gains over the noisy model, is that the very fact that one node attempts to communicate with another is reliable. This, seemingly minor, difference carries significant consequences.

Strategies to overcome noise in biological systems

Communication in man-made computer networks is often based on reliable signals which are typically transferred over highly defined structures. These allow for ultra-fast and highly reliable calculations. Biological networks are very different from this and often lack reliable messaging, well defined connectivity patterns or both. Our theoretical results seem to suggest that, under such circumstances, efficient spread of information would not be possible. Nevertheless, many biological groups disseminate and share information, and, often, do so reliably. Next, we discuss information sharing in biological systems within the general framework of our lower-bounds.

The correctness of the lower bounds relies on two major assumptions: 1) stochastic interactions, and 2) uniform noise. Communication during desert ant recruitment complies with both these assumptions (see Fig 2b and 2c) and indeed the speed at which messages travel through the group (see Fig 2d) is low. Below, we discuss several biological examples where efficient rumor spreading is achieved. We expect that, in these examples, at least one of the assumptions mentioned above should break adding some degree of reliability to the overall communication. The group can then utilize this reliability and follow one of the strategies mentioned in Section Theoretical results, in order to yield reliable collective performance. We begin by discussing examples that violate the first assumption, namely, that of stochastic interactions, and then discuss examples that violate the second assumption, namely, uniform noise.

Calculation of δ

To estimate the noise parameter δ we used interactions between ants moving at three different speed ranges (measured in cm/sec), namely, ‘a’: 1-10, ‘b’: 10-20, and ‘c’: over 20 and “receiver” ants. Only interactions in which the receiver ant was initially stationary were used as to ensure that the state of these ants before the interaction is as similar as possible. The message alphabet is then assumed to be Σ = {a, b, c}. The response of a stationary ant v to the interaction was quantified in terms of her speed after the interaction.

An alphabet of three messages was used since the average responses of v to any two messages were significantly different (all p-values smaller than 0.01) justifying the fact that these are not artificial divisions of a continuous speed signal into a large number of overlapping messages. On the other hand, dividing the bins further (say, each bin divided into 2 equal bins) yielded statistically indistinguishable responses from the receiver (all p-values larger than 0.11). Therefore, our current data best supports a three letter alphabet.

Assuming equal priors to all messages in Σ, and given specific speed of the receiver ant, v, the probability that it was the result of a specific message i ∈ Σ was calculated as p_i(v) = p(v ∣ i)/∑_k∈Σ p(v ∣ k), where p(v ∣ j) is the probability of responding in speed v after “observing” j. The probability δ(i, j) that message i was perceived as message j was then estimated as the weighted sum over the entire probability distribution measured as a response to j: δ(i, j) = ∑_v p(v ∣ j) ⋅ p_i(v). The parameter δ can then be calculated using δ = min{δ(i, j) ∣ i, j ∈ Σ}.