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The authors have declared that no competing interests exist.

Biological systems can share and collectively process information to yield emergent effects, despite inherent noise in communication. While man-made systems often employ intricate structural solutions to overcome noise, the structure of many biological systems is more amorphous. It is not well understood how communication noise may affect the computational repertoire of such groups. To approach this question we consider the basic collective task of rumor spreading, in which information from few knowledgeable sources must reliably flow into the rest of the population. We study the effect of communication noise on the ability of groups that lack stable structures to efficiently solve this task. We present an impossibility result which strongly restricts reliable rumor spreading in such groups. Namely, we prove that, in the presence of even moderate levels of noise that affect all facets of the communication, no scheme can significantly outperform the trivial one in which agents have to wait until directly interacting with the sources—a process which requires linear time in the population size. Our results imply that in order to achieve efficient rumor spread a system must exhibit either some degree of structural stability or, alternatively, some facet of the communication which is immune to noise. We then corroborate this claim by providing new analyses of experimental data regarding recruitment in

Biological systems must function despite inherent noise in their communication. Systems that enjoy structural stability, such as biological neural networks, could potentially overcome noise using simple redundancy-based procedures. However, when individuals have little control over who they interact with, it is unclear what conditions would prevent runaway error accumulation. This paper takes a general stance to investigate this problem, concentrating on the basic information-dissemination task of rumor spreading. Drawing on a theoretical model, we prove that fast rumor spreading can only be achieved if some part of the communication setting is either stable or reliable. We then provide empirical support for this claim by conducting new analyses of data from experiments on recruitment in desert ants.

Systems composed of tiny mobile components must function under conditions of unreliability. In particular, any sharing of information is inevitably subject to communication noise. The effects of communication noise in distributed living systems appears to be highly variable. While some systems disseminate information efficiently and reliably despite communication noise [

Computation under noise has been extensively studied in the computer science community. These studies suggest that different forms of error correction (

The impact of noise in stochastic systems with ephemeral connectivity patterns is far less understood. To study these, we focus on

A successful and efficient rumor spreading process is one in which a large group manages to quickly learn information initially held by one or a few informed individuals. Fast information flow to the whole group dictates that messages be relayed between individuals. Similar to the game of Chinese Whispers, this may potentially result in runaway buildup of noise and loss of any initial information [

In this paper we take a general stance to identify limitations under which reliable and fast rumor spreading cannot be achieved. Modeling a well-mixed population, we consider a passive communication scheme in which information flow occurs as one agent observes the cues displayed by another. If these interactions are perfectly reliable, the population could achieve extremely fast rumor spreading [

the system exhibits some degree of structural stability, or

some facet of the pairwise communication is immune to noise.

An intuitive description of the model follows. For more precise definitions, see, Section

Consider a population of

To achieve this goal, each agent continuously displays one of several _{m,m′} be the probability that, any time some agent _{m,m′} define the entries of the noise-matrix

The noise is characterized by a

_{m,m′} ≥

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

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, …

Complete knowledge of the system: Agents have access to all parameters of the system (including

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.

The purpose of this work is to identify limitations under which efficient rumor spreading would be impossible. Our main result is theoretical and, informally, states that when all components of communication are noisy fast rumor spreading through large populations is not feasible. In other words, our results imply that fast rumor spreading can only be achieved if the system either exhibits some degree of structural stability or that some facet of its communication is immune to noise. These results in hand, a next concern is how far our highly theoretical analysis can go in explaining actual biological systems.

Theoretical results with a high degree of generality may hold relevance to a wider range of biological systems. Lower bound and impossibility results follow this approach. Indeed, impossibility results from physics and information theory have previously been used to further the understanding of several biological systems [

While the generality of our lower bound results makes them relevant to a large number of biological systems it also constitutes a weakness. Namely, the assumptions on which such theorems are based are not tailored to describe a particular system. This implies that comparisons between the model assumptions and the actual details of a specific system will not be perfect. Nevertheless, we show how our theoretical results can shed light on some non-trivial behaviors in a specific biological system whose characteristics are close enough to the underlying theoretical assumptions (see Section

Distributed computing provides an effective means of studying biological groups [

In all the statements that follow we consider the parallel-

Observe that the lower bound we present loses relevance when

Our results suggest that, in contrast to systems that enjoy stable connectivity, structureless systems are highly sensitive to communication noise (see

On the left, we consider an example with non-uniform noise. Assume that the message vocabulary consists of 5 symbols, that is, Σ = {_{1}, _{2}, _{3}, _{4}, _{5}}, where _{1} = 0 and _{5} = 1, represent the opinions. Assume that noise can occur only between consecutive messages. For example, _{2} can be observed as either _{2}, _{3} or _{1}, all with positive constant probability, but can never be viewed as _{4} or _{5}. In this scenario, the population can quickly converge on the correct opinion by executing the following. The sources always display the correct opinion, _{1} or _{5}, and each other agent displays _{3} unless it has seen either _{1} or _{5} in which case it adopts the opinion it saw and displays it. In other words, _{3} serves as a default message for non-source agents, and _{1} and _{5} serve as attracting sinks. It is easy to see that the correct opinion will propagate quickly through the system without disturbance, and within

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

In our experimental setup, summarized in

It has been shown that recruitment in

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 [

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 (

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

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

Here, we provide the intuition for our main theoretical result, Theorem 1.1. For a formal proof please refer to the

Consider an efficient protocol

To establish the desired lower bound, we next show how the rumor spreading problem in the broadcast-

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 (

On the top there are two possible coins with slightly different distributions for yielding a head (_{j}(^{(t)} ∣ observations) for ^{(t = 6)}|_{1} measures how “far” the the ^{(6)}|_{1} = |_{0}(_{1}(_{0}(_{1}(^{(t)}|_{1} from above.

Despite this apparent complexity, we show that the difficulty of this distinguishing task can be captured by two scalar parameters, denoted ^{(<t)}, the next observation has the same probability to be attained in each process, up to an ^{(<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

The last step of the proof shows that at least Ω(^{2}) samples are required in order to solve any distinguishing task with parameters ^{2} to make the error less than, say 1/3. This bound translates to a lower bound of Ω(^{2}^{2}(1 − ^{2})) steps for the broadcast-^{2}(1 − ^{2})) rounds for the parallel-

Several of the assumptions discussed earlier for the parallel-^{2}) in the case where

Our lower bounds on the parallel-

Perhaps the main reason why these two models are often considered similar is that with an extra bit in the message, a

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

Synaptic connectivity in the mammalian brain is known to be highly noisy [

Animal groups can also benefit from stable connectivity to enhance the reliability of rumor spreading. An example comes from house-hunting rock ants [

When the physical structure of a group is not well defined, the importance of reliable messaging schemes grows. In flocks of birds and schools of fish, changes in the behavior of a single individual can be relayed across a series of local interactions [

As noted above, push-type communication is another route which may potentially add sufficient reliability to support rumor spreading. In this sense, what distinguishes push from pull is the trait by which a non-message cannot be confused with a message. An example for the usefulness of an active push behavior comes from alarm behavior in ants. A single ant sensing danger can actively excrete discrete volatile alarm pheromones that are sensed by a large number of group members and elicit panic or attack responses [

The difficulty of spreading information fast, as indicated by our theoretical results, is further consistent with the fact that, even in fully-cooperative groups, such as ants or bees, an animal that receives information from a conspecific will often not transfer it further before obtaining its own independent first-hand knowledge [

Finally, we note that given the aforementioned discussion, our insight regarding the difficulty of functioning under uniform noise can serve an evolutionary explanation for the emergence of new communication signals (

All experimental results presented in this manuscript are re-analyses of data obtained in

The reaction of the ants to this manipulation was filmed and the locations, speeds and interactions of all participating ants were extracted from the resulting videos.

To estimate the noise parameter

An alphabet of three messages was used since the average responses of

Assuming equal priors to all messages in Σ, and given specific speed of the receiver ant, _{i}(_{k∈Σ} _{v} _{i}(

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The authors would like to thank Brieuc Guinard for helpful comments on an early version of this draft. Part of this work was done while Emanuele Natale was a fellow of the Simons Institute for the Theory of Computing.