We consider a scenario of broadcasting information over a network of nodes connected by noiseless communication links. A source node in the network has some data packets to broadcast. It encodes these data packets into n coded packets in such a way that any node in the network that receives any k out of the n coded packets will be able to retrieve all the original data packets. The source transmits the n coded packets to its one-hop neighbours. Every other node in the network follows a probabilistic forwarding protocol, in which it forwards a previously unreceived packet to all its neighbours with a certain probability p. We say that the information from the source undergoes a "near-broadcast" if the expected fraction of nodes that receive at least k of the n coded packets is close to 1. The forwarding probability p is chosen so as to minimize the expected total number of transmissions needed for a nearbroadcast. We study how, for a given k, this minimum forwarding probability and the associated expected total number of packet transmissions varies with n. We specifically analyze the probabilistic forwarding of coded packets on two network topologies: binary trees and square grids. For trees, our analysis shows that for fixed k, the expected total number of transmissions increases with n. On the other hand, on grids, a judicious choice of n significantly reduces the expected total number of transmissions needed for a near-broadcast. Behaviour similar to that of the grid is also observed in other well-connected network topologies such as random geometric graphs and random regular graphs.Index Terms-Broadcast, ad-hoc networks, probabilistic forwarding, coded packets, site percolation. I. MOTIVATIONA N AD-HOC network is a network of nodes which communicate with each other without relying on any centralized infrastructure. A classical example of ad-hoc networks is wireless sensor networks (WSNs) which have sensors measuring temperature, humidity etc. connected with each other. The Internet of Things (IoT) network, which involves different types of physical devices-sensors, actuators, routers, mobiles etc.-communicating with each other over a network can be thought of as an ad-hoc network.Broadcast mechanisms on such distributed networks are crucial in order to disburse key network-related information throughout the network. In the applications mentioned above,

This paper is concerned with the problem of broadcasting information from a source node to every node in an ad-hoc network. Flooding, as a broadcast mechanism, involves each node forwarding any packet it receives to all its neighbours. This results in excessive transmissions and thus a high energy expenditure overall. Probabilistic forwarding or gossiping involves each node forwarding a received packet to all its neighbours only with a certain probability p. In this paper, we study the effect of introducing redundancy, in the form of coded packets, into a probabilistic forwarding protocol. Specifically, we assume that the source node has k data packets to broadcast, which are encoded into n ≥ k coded packets, such that any k of these coded packets are sufficient to recover the original k data packets. Our interest is in determining the minimum forwarding probability p for a "successful broadcast", which we take to be the event that the expected fraction of network nodes that receive at least k of the n coded packets is close to 1. We examine, via simulations and analysis of a number of different network topologies (e.g., trees, grids, random geometric graphs), how this minimum forwarding probability, and correspondingly, the expected total number of packet transmissions varies with the amount of redundancy added. Our simulation results indicate that over network topologies that are highly connected, the introduction of redundancy into the probabilistic forwarding protocol is useful, as it can significantly reduce the expected total number of transmissions needed for a successful broadcast. On the other hand, for trees, our analysis shows that the expected total number of transmissions needed increases with redundancy.

Introduction The COVID‐19 pandemic produced an unprecedented crisis across the world. Long‐term cohort studies were stalled, including our longitudinal aging cohort study in rural India. Methods We describe approaches undertaken to engage with our cohort ( n = 1830) through multiple rounds of calls and how we provided useful services to our subjects during the lockdown period. Consenting subjects also underwent telephonic assessments for depression and anxiety using validated, self‐report questionnaires. Results Subjects reported benefitting from our telephonic engagement strategies, including the COVID‐related safety awareness and counselling service. The proportion of subjects with depression increased from 7.42% pre‐COVID to 28.97% post‐COVID. Discussion We envisage that such engagement strategies would improve subject rapport and cohort retention, and thus, could be adopted by similar cohort studies across the world. This marginalized, rural Indian community had severe, adverse psychological impact in this pandemic. Urgent public health measures are needed to mitigate this impact and develop appropriate preventive strategies.

In this paper, we discuss the preparation of Li-doped ZnO nanostructures through combustion and report on their structural, morphological, optical, and electrocatalysis properties. X-ray diffraction analyses show that the samples have a structure crystallized into the usual hexagonal wurtzite ZnO structure according to the P63mc space group. The scanning electron microscope images conceal all samples’ nanosphere bundles and aggregates. The reflectance spectra analysis showed that the direct bandgap values varied from 3.273 eV (for pure ZnO, i.e., ZnL1) to 3.256 eV (for high Li-doped ZnO). The measured capacitance concerning frequency has estimated the variation of dielectric constant, dielectric loss, and AC conductivity against AC electric field frequency. The dielectric constant variations and AC conductivity are analyzed and discussed by well-known models such as Koop’s phenomenological theory and Jonscher’s law. The Raman spectra have been recorded and examined for the prepared samples. Rhodamine B was electro-catalytically degraded in all prepared samples, with the fastest time for ZnL5 being 3 min.

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