Motivated by the growing interest in mobile systems, we study the dynamics of information dissemination between agents moving independently on a plane. Formally, we consider @math mobile agents performing independent random walks on an @math -node grid. At time @math , each agent is located at a random node of the grid and one agent has a rumor. The spread of the rumor is governed by a dynamic communication graph process @math , where two agents are connected by an edge in @math iff their distance at time @math is within their transmission radius @math . Modeling the physical reality that the speed of radio transmission is much faster than the motion of the agents, we assume that the rumor can travel throughout a connected component of @math before the graph is altered by the motion. We study the broadcast time @math of the system, which is the time it takes for all agents to know the rumor. We focus on the sparse case (below the percolation point @math ) where, with high probability, no connected component in @math has more than a logarithmic number of agents and the broadcast time is dominated by the time it takes for many independent random walks to meet each other. Quite surprisingly, we show that for a system below the percolation point the broadcast time does not depend on the relation between the mobility speed and the transmission radius. In fact, we prove that @math for any @math , even when the transmission range is significantly larger than the mobility range in one step, giving a tight characterization up to logarithmic factors. Our result complements a recent result of (SODA 2011) who showed that above the percolation point the broadcast time is polylogarithmic in @math .
Recent work by @cite_22 studies a process in which agents follow independent Brownian motions in @math . They investigate several properties of the system, such as detection, coverage and percolation times, and characterize them as functions of the spatial density of the agents, which is assumed to be greater than the percolation point. Leveraging on these results, they show that the broadcast time of a message is polylogarithmic in the number of agents, under the assumption that a message spreads within a connected component of the communication graph instantaneously, before the graph is altered by agents' motion.
@cite_22
2019737634 2023019696 2104040866 1990476939
The dynamic behavior of the R-ALOHA packet broadcast system with multipacket messages is analyzed in this paper. It is assumed that each user handles one message at a time and the number of packets in a message is geometrically distributed. A Markovian model of the system is first formulated which explicitly contains the influence of the propagation delay of the broadcast channel. An approximate technique called equilibrium point analysis (EPA) is utilized to analyze the multidimensional Markov chain. The system stability behavior and the throughput-average message delay performance are demonstrated by the EPA. Numerical results from both analysis and simulation are given to assess the accuracy of the analytic results. Applying the analytic results to the slotted ALOHA with single packet messages, we prove mathematically that a method by Kleinrock and Lam for taking into account the influence of the propagation delay is an excellent approximation. Abstract Patterns of deliberate human activity and behavior are of utmost importance in areas as diverse as disease spread, resource allocation, and emergency response. Because of its widespread availability and use, e-mail correspondence provides an attractive proxy for studying human activity. Recently, it was reported that the probability density for the inter-event time τ between consecutively sent e-mails decays asymptotically as τ−α, with α ≈ 1. The slower-than-exponential decay of the inter-event time distribution suggests that deliberate human activity is inherently non-Poissonian. Here, we demonstrate that the approximate power-law scaling of the inter-event time distribution is a consequence of circadian and weekly cycles of human activity. We propose a cascading nonhomogeneous Poisson process that explicitly integrates these periodic patterns in activity with an individual's tendency to continue participating in an activity. Using standard statistical techniques, we show that our model is consistent with the empirical data. Our findings may also provide insight into the origins of heavy-tailed distributions in other complex systems. complex systems human activity hypothesis testing point process In this paper, we investigate distributed multi-agent tracking of a convex set specified by multiple moving leaders with unmeasurable velocities. Various jointly connected interaction topologies of the follower agents with uncertainties are considered in the study of set tracking. Based on the connectivity of the time-varying multi-agent system, necessary and sufficient conditions are obtained for set input-to-state stability and set integral input-to-state stability for a nonlinear neighbor-based coordination rule with switching directed topologies. Conditions for asymptotic set tracking are also proposed with respect to the polytope spanned by the leaders. The dynamic behavior of a network in which information is changing continuously over time requires robust and efficient mechanisms for keeping nodes updated about new information. Gossip protocols are mechanisms for this task in which nodes communicate with one another according to some underlying deterministic or randomized algorithm, exchanging information in each communication step. In a variety of contexts, the use of randomization to propagate information has been found to provide better reliability and scalability than more regimented deterministic approaches.In many settings, such as a cluster of distributed computing hosts, new information is generated at individual nodes, and is most "interesting" to nodes that are nearby. Thus, we propose distance-based propagation bounds as a performance measure for gossip mechanisms: a node at distance d from the origin of a new piece of information should be able to learn about this information with a delay that grows slowly with d, and is independent of the size of the network.For nodes arranged with uniform density in Euclidean space, we present natural gossip mechanisms, called spatial gossip, that satisfy such a guarantee: new information is spread to nodes at distance d, with high probability, in O(log1 + e d) time steps. Such a bound combines the desirable qualitative features of uniform gossip, in which information is spread with a delay that is logarithmic in the full network size, and deterministic flooding, in which information is spread with a delay that is linear in the distance and independent of the network size. Our mechanisms and their analysis resolve a conjecture of [1987].We further show an application of our gossip mechanisms to a basic resource location problem, in which nodes seek to rapidly learn the location of the nearest copy of a resource in a network. This problem, which is of considerable practical importance, can be solved by a very simple protocol using Spatial Gossip, whereas we can show that no protocol built on top of uniform gossip can inform nodes of their approximately nearest resource within poly-logarithmic time. The analysis relies on an additional useful property of spatial gossip, namely that information travels from its source to sinks along short paths not visiting points of the network far from the two nodes.