The impact of agent density on scalability in collective systems: noise-induced versus majority-based bistability

Abstract

In this paper, we show that non-uniform distributions in swarms of agents have an impact on the scalability of collective decision-making. In particular, we highlight the relevance of noise-induced bistability in very sparse swarm systems and the failure of these systems to scale. Our work is based on three decision models. In the first model, each agent can change its decision after being recruited by a nearby agent. The second model captures the dynamics of dense swarms controlled by the majority rule (i.e., agents switch their opinion to comply with that of the majority of their neighbors). The third model combines the first two, with the aim of studying the role of non-uniform swarm density in the performance of collective decision-making. Based on the three models, we formulate a set of requirements for convergence and scalability in collective decision-making.

Appendices

Appendix 1: Agent density in terms of area

For a given swarm size N and a given area A the swarm density is given by $\rho =N/A$. For simplicity we set the area to $A=1$ [space unit]. We also require the concept of a critical swarm size $N_\mathrm{c}$ that corresponds to a critical swarm density $\rho _\mathrm{c}=N_\mathrm{c}/A=N_\mathrm{c}$.

The node degree $\lambda$, as mentioned above, defines the group size $\lambda +1$ of an agent. We compute the area $A_\mathrm{s}$ covered by an agent’s sensor as $A_\mathrm{s}=\pi u^2$ (for sensor range u), and we assume $A_\mathrm{s}\ll A$. We get the expected node degree

$$\begin{aligned} \lambda = \rho A_\mathrm{s}=NA_\mathrm{s}. \end{aligned}$$

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Hence, the swarm density is defined in terms of swarm size N and node degree $\lambda$ or group size $\lambda +1$ for fixed sensor range u, where N or $\rho$ can be varied. Note that the uniform distribution used for the agents approaches the imposed density $\rho$ averaged over big areas but may vary considerably within small areas because the agents are not distributed with equidistant positions. Hence, the local node degrees vary as well.

Appendix 2: Computation of MFPT of a Markov chain model

For a Markov chain model we can compute the MFPT from state i to state j (Hunter 2005, 2007) as

$$\begin{aligned} m_{i,j} = p_{i,j} + \sum _{k \ne j} p_{i,k} (m_{k,j} + 1), \end{aligned}$$

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for the transition probability matrix P of the Markov chain with entries $p_{i,j}$. Our Markov chain is ergodic; thus, the mean first passage time can be computed using the fundamental matrix F of the Markov chain which is defined as

$$\begin{aligned} F = (I - P + S)^{-1}, \end{aligned}$$

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where I is the identity matrix and $S=\lim _{t\rightarrow \infty }P^t$ is a matrix whose rows are equal to each other and given by the stationary distribution s. An entry $f_{i,j}$ of F gives the expected number of visits to transient state $s_j$ if the system is started in transient state $s_i$. Here we can compute M with entries $m_{i,j}$ giving the MFPT from state i to state j using the fundamental matrix of the ergodic chain by

$$\begin{aligned} m_{i,j} = \dfrac{f_{j,j}-f_{i,j}}{s_j}. \end{aligned}$$

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