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Edited by: Xin Jin, National Renewable Energy Laboratory, USA

Reviewed by: Subramanian Ramamoorthy, The University of Edinburgh, UK; Soheil Bahrampour, Pennsylvania State University, USA

This article was submitted to Sensor Fusion and Machine Perception, a section of the journal Frontiers in Robotics and AI.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper proposes a novel event-triggered formulation as an extension of the recently developed generalized gossip algorithm for decision/awareness propagation in mobile sensor networks modeled as proximity networks. The key idea is to expend energy for communication (message transmission and reception) only when there is any event of interest in the region of surveillance. The idea is implemented by using an agent’s belief about presence of a hotspot as feedback to change its probability of (communication) activity. In the original formulation, the evolution of network topology and the dynamics of decision propagation were completely decoupled, which is no longer the case as a consequence of this feedback policy. Analytical results and numerical experiments are presented to show a significant gain in energy savings with no change in the first moment characteristics of decision propagation. However, numerical experiments show that the second moment characteristics may change and theoretical results are provided for upper and lower bounds for second moment characteristics. Effects of false alarms on network formation and communication activity are also investigated.

Application of mobile sensor networks for monitoring environment is increasingly ubiquitous for both military applications such as undersea mine hunting, anti-submarine warfare, and non-military applications such as weather monitoring and prediction (Lehning et al.,

In a recent work Sarkar et al. (

The idea is implemented by using a simple feedback policy where an agent’s communication activity gets determined by its belief regarding the occurrence of an event of interest in the region of surveillance. However, under this policy the network system can be susceptible to false alarms raised by the sensors (agents). The study therefore introduces a notion of false alarm to investigate its impact on energy saving. The specific contributions of this paper beyond the existing work by the authors are

An event-triggered formulation of generalized gossip policy to save energy and increase agents’ lives;

Analytical results and simulation-based validation regarding impacts of event-triggered policy on decision propagation characteristics;

Analytical results and simulation-based validation regarding impacts of false alarms on decision propagation and energy consumption.

The paper is organized in five sections including the present one. A brief background of the generalized gossip algorithm and key results are reviewed in Section

This section briefly describes the formulation and key results of the generalized gossip policy proposed in Sarkar et al. (_{m}_{m}_{m}_{m}

Definition 2.1. (Adjacency Matrix Patterson et al. (_{ij}^{th}_{m}_{ij}

Definition 2.2. (Laplacian Matrix Patterson et al. (^{i}^{th}^{i}

Definition 2.3. (Interaction Matrix Patterson et al. (_{2}(Π)| < 1.

In the context of proximity networks, the requirement of keeping Π as a stochastic matrix in Definition 2.3 is achieved by setting _{m}

Finally, a hotspot is modeled as a map for probability of detecting the threat in the region of interest. In this specific setup, such a 2-D probability distribution is assumed to be radially symmetric (i.e., highest detection probability at the center of a hotspot and decaying proportionally with the distance from the center). However, the event driven formulation in this paper is valid for any probability distribution of hotspot. Note, such modeling scheme inherently captures the miss-detection phenomena. However, it does not consider false alarms.

The generalized gossip strategy involves two characteristic variables associated with each agent, namely, the _{m}

Thus, this policy is simply a gossip algorithm with varying input _{τ}_{τ}

Owing to the inherent stochastic nature of the problem, even in the steady state, _{θ}_{θ}_{a}_{a}_{θ}_{θ}_{τ}_{a}_{θ}_{τ}_{θ}_{τ}_{θ}_{τ}_{a}_{a}_{1,} _{2, …,} _{N}^{T}

With these definitions, the first result is that the steady-state expected measure average (over agents) converges to the steady-state expected state average (over agents), i.e.,

The physical significance is that the detection decision of a hotspot by few agents is being redistributed as awareness over a (possibly) larger number of agents, where the total awareness measure is conserved. Also, the convergence time increases with an decrease in _{τ}_{τ}_{τ}

Apart from measure average, it is also interesting to know the nature of measure distribution in the agent population and measure variance (over agents) provides an insight in this aspect. For example, zero measure variance signifies

The first scenario is where the time scales for mobility and information dynamics are comparable, which means that, at each slow-time epoch _{2} denotes the second largest Eigenvalue of _{τ}^{T}_{τ}

On the other end of this spectrum, one can consider a situation where the two time scales are very different such that, the network evolution and the agent state updating can be treated independently as it is done in the _{2} denotes the second largest Eigenvalue of Π.

The original formulation presented above considers network topology evolution based solely upon proximity of agents. Although this formulation efficiently facilitates distributed decision propagation in a group of mobile agents, this is particularly not efficient in terms of energy consumption by the agents due to communication (i.e., transmission or reception of messages). This aspect is critical for applications, such as undersea surveillance and remote environment monitoring using mobile sensor networks. In such applications, the events of interest are typically rare and long deployment life-cycles are preferred. In the current context, this calls for a modification of the generalized gossip policy to make it energy-efficient. To this end, the basic idea proposed in this paper is to have the agents communicate only when there is an event of interest in the region of surveillance. This is implemented by using a simple feedback policy where an agent’s communication activity gets determined by its belief regarding the presence of a hotspot. The modification is formally introduced below.

Let the probability of communication (i.e., for both transmission and reception of messages) activity for an agent _{bias}_{θ}_{bias}_{bias}_{bias}_{bias}

Algorithm for Event-triggered Network evolution |
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_{end}_{end} _{m}_{bias} |

_{bias}

_{bias}

Based on the algorithm above, the vector form of information dynamics (see equation (

Note, under the original formulation an element _{ij}_{ij}_{m}_{m}_{m}_{ij}_{m}

Hence,

An extreme case is observed when the adjacency matrix is most sparse or all the agents are at their least possible level of activity, i.e., _{bias}

Therefore, with the least activity level of agents the modified Laplacian matrix (denoted by _{min}

Furthermore, in general

For the case of congruous time scale (CTS), where the agents’ mobility and information dynamics are comparable, the smallest possible value of _{m}

On the contrary, for the disparate time scale (DTS), the agents’ positions are invariant over the message lifetime. This implies that if two agents are within communication radius, they would remain so during the lifetime of the message (i.e., _{m}_{m}

Recall that in original formulation, the steady-state expected measure average (over agents) converges to the steady-state expected state average (over agents) as given by equation (_{τ}

Furthermore, following equation (_{bias}_{a}_{activity}

Using equation (

From this result, it is evident that without the presence of any hotspot (i.e., _{a}_{bias}_{m}

Hence, using equation (

Similar to the original formulation, it is also clear from equation (

Based on the discussion above, it appears that the gain in energy saving does not have any penalizing effect on measure average or its temporal convergence property. However, the measure variance may get affected by the energy aware formulation. Note (from equations (_{τ}^{T}_{τ}_{min}_{bias}

As discussed earlier, the modified Laplacian matrix _{bias}

Using the equation above, the second largest Eigenvalue of _{2} as

With this setup, it is straightforward to obtain variance upper bound in the Disparate Time Scale (DTS) case of agent motion and agent communication. Following equation (

Furthermore, using equation (_{bias}_{m}_{2}) as

To obtain similar bound for the case of congruous time scale (CTS), Eigenvalues of

Taking expectation on both sides

Let _{2} be the unity norm Eigenvector corresponding to the second largest Eigenvalue of ^{T}_{2} is orthogonal to [1, 1, …, 1]^{T}_{2},

Taking norms on both sides of equation (

However, the left hand side of equation (

Also,
_{2} is the second largest Eigenvalue of ^{T}

Therefore, the upper bound of the second largest Eigenvalue

The above expression simplifies to

Following equation (

Thus, using equation (_{bias}^{T}

Or, in terms of _{bias}

This section validates the analytical results obtained in the previous section via numerical simulation experiments. The simulation scenario considers a surveillance and reconnaissance mission for a region of area ^{2} performed by _{m}_{m}^{i}^{i}_{m}_{activity}_{bias}_{bias}_{activity}

The first analysis performed numerically verifies the relationship obtained between steady state expected activity and false alarm rate as given by equation (_{bias}_{bias}

_{bias}_{bias}

The individual mission goal of each agent is to detect the existence of any possible hotspot or event in the region and communicate the information to their neighboring agents. Although the state characteristic function _{i}_{thresh}_{bias}^{−3}, respectively.

Figure _{thresh}

Remark 4.1: In the above described simulation, the magnitude of the hotspot (in terms of magnitude and physical range) is an important factor for the ROC. The ROC curve in Figure ^{−3}, which is in fact lower than the false alarm rate.

Numerical experiments are performed to verify the analytical results related to convergence of statistical moments of measure values. Results are shown for _{bias}_{bias}_{bias}_{bias}

_{bias}

Finally, simulations are performed to analyze the effect of the spatial distribution of the agents on the ratio of the variances (belief _{2}(Π). For the simulation, the value of Λ_{2}(Π) is varied by modifying the communication radius of the agents. As the communication radius is reduced, the agent interaction reduces leading to a larger second Eigenvalue Λ_{2} (slower dynamics). The result shows that the nature of distributions of agents has little effect on the ratio of variances given that the second Eigenvalue of the agent interaction matrix is held fixed.

This paper proposes an energy aware formulation of the generalized gossip policy for distributed decision propagation in mobile sensor networks. The key idea is to conserve energy spent on communication (message transmission and reception) when there is no event of interest in the region of surveillance. The idea is implemented by using an agent’s belief measure as feedback to change its probability of (communication) activity. As a consequence, the time-varying network remains extremely sparse when there is no hotspot and becomes heavily connected when a hotspot appears in the region of surveillance. Analysis and numerical experiments show that the first moment characteristics of agent belief propagation remain same as it was in the original formulation. However, the second moment characteristics changes and as expected information propagation suffers from reduced activity of agents. Still a minimum required level of propagation characteristics can be maintained by choosing a proper value of _{bias}

It should be noted that the collaborative decision framework presents a different problem formulation and solution approach compared to those exist in literature today. Specifically, (i) typical source seeking problems (Atanasov et al.,

While detail validation experiments to verify the bounds on variance ratio for the event-triggered formulation under both CTS and DTS cases are currently being pursued, a couple of other important research directions are mentioned below.

In the original formulation, it was found that the generalization parameter

Mobile

The present formulation uses a feedback of agent belief as a mechanism to control the underlying dynamic communication graph. While this study focuses on how such feedback has the potential to save power consumption in the network, future work will investigate the other (potentially negative) impacts of the belief dynamics, such as network fragmentation.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.