We study the dynamics of information (or virus) dissemination by @math mobile agents performing independent random walks on an @math -node grid. We formulate our results in terms of two scenarios: broadcasting and gossiping. In the broadcasting scenario, the mobile agents are initially placed uniformly at random among the grid nodes. At time 0, one agent is informed of a rumor and starts a random walk. When an informed agent meets an uninformed agent, the latter becomes informed and starts a new random walk. We study the broadcasting time of the system, that is, the time it takes for all agents to know the rumor. In the gossiping scenario, each agent is given a distinct rumor at time 0 and all agents start random walks. When two agents meet, they share all rumors they are aware of. We study the gossiping time of the system, that is, the time it takes for all agents to know all rumors. We prove that both the broadcasting and the gossiping times are @math w.h.p., thus achieving a tight characterization up to logarithmic factors. Previous results for the grid provided bounds which were weaker and only concerned average times. In the context of virus infection, a corollary of our results is that static and dynamically moving agents are infected at about the same speed.
With the advent of mobile ad-hoc networks there has been growing interest in studying information dissemination in dynamic scenarios, where a number of agents move either in a continuous space or along the nodes of some underlying graph and exchange information when their positions satisfy a specified proximity constraint. In @cite_15 @cite_2 the authors study the time it takes to broadcast information from one of @math mobile agents to all others. The agents move on a square grid of @math nodes and in each time step, an agent can (a) exchange information with all agents at distance at most @math from it, and (b) move to any random node at distance at most @math from its current position. The results in these papers only apply to a very dense scenario where the number of agents is linear in the number of grid nodes (i.e., @math ). They show that the broadcasting time is @math w.h.p., when @math and @math @cite_15 , and it is @math w.h.p., when @math @cite_2 . These results crucially rely on @math , which implies that the range of agents' communications or movements at each step defines a connected graph.
@cite_15 @cite_2
2027036473 1590585877 1981696935 2087097376
In this paper, we study information propagation in a 2D mobile ad-hoc network, where mobile nodes are randomly and independently distributed on a torus following a homogeneous Poisson process with a given density. Nodes in the network move following a random direction mobility model. A piece of information is broadcast from a source node to all other nodes in the network, using a Susceptible-Infectious-Recovered (SIR) epidemic routing protocol. A distinguishing feature of the SIR algorithm, which leverages the mobility of mobile users, is that a relay node carries and forwards a piece of information for a specified amount of time. We first propose a metric fundamentally characterizing the information propagation in mobile ad-hoc networks. Then analytical results are derived for the probability that a non-zero fraction of nodes receive the information in the limit of large network size and for the expected fraction of nodes that receive the information. The analytical results are verified using simulations. The research provides useful insights on the design of mobile ad-hoc networks. Designing robust algorithms for mobile agents with reliable communication is difficult due to the distributed nature of computation, in mobile ad hoc networks (MANETs) the matter is exacerbated by the need to ensure connectivity. Existing distributed algorithms provide coordination but typically assume connectivity is ensured by other means. We present a connectivity service that encapsulates an arbitrary motion planner and can refine any plan to preserve connectivity (the graph of agents remains connected) and ensure progress (the agents advance towards their goal). The service is realized by a distributed algorithm that is modular in that it makes no assumptions of the motion-planning mechanism except the ability for an agent to query its position and intended goal position, local in that it uses 1-hop broadcast to communicate with nearby agents but doesn't need any network routing infrastructure, and oblivious in that it does not depend on previous computations. We prove the progress of the algorithm in one round is at least Ω(min(d, r)), where d is the minimum distance between an agent and its target and r is the communication radius. We characterize the worst case configuration and show that when d ≥ r this bound is tight and the algorithm is optimal, since no algorithm can guarantee greater progress. Finally we show all agents get Ɛ-close to their targets within O(D0 r+n2 Ɛ) rounds where n is the number of agents and D0 is the sum of the initial distances to the targets. We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step. Bar-Yehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104--126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. Bar-Yehuda, O. Goldreich, and A. Itai, Errata Regarding "On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization," http: www.wisdom.weizmann.ac.il mathusers oded p .html, 2002]). We construct an algorithm that broadcasts in logarithmic time on all graphs from the Bar-Yehuda, Goldreich, and Itai paper (BGI). Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter @math . On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time @math on these graphs. In view of the randomized algorithm from BGI, running in expected time @math on all @math -node graphs of diameter D (cf. also a recent @math -time algorithm from [D. Kowalski and A. Pelc, Proceedings of the 22nd Annual ACM Symposium on Principles of Distributed Computing, Boston, 2003, pp. 73--82; A. Czumaj and W. Rytter, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 492--501]), our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting, under the considered model of radio communication. Consider a team of mobile software agents deployed to capture a (possibly hostile) intruder in a network. All agents, including the intruder move along the network links; the intruder could be arbitrarily fast, and aware of the positions of all the agents. The problem is to design the agents' strategy for capturing the intruder. The main efficiency parameter is the size of the team. This is an instance of the well known graph-searching problem whose many variants have been extensively studied in the literature. In all existing solutions, and in all the variants of the problem, it is assumed that agents can be removed from their current location and placed in another network site arbitrarily and at any time. As a consequence, the existing optimal strategies cannot be employed in situations for which agents cannot access the network at any point, or cannot "jump" across the network, or cannot reach an arbitrary point of the network via an internal travel through insecure zones. This motivates the contiguous search problem in which agents cannot be removed from the network, and clear links must form a connected sub-network at any time, providing safety of movements. This new problem is NP-complete in general. We study it for tree networks, and we consider its more general version, the weighted case, which arises naturally when considering networks whose nodes and links are of different nature and thus require a different number of agents to be explored. We give a linear-time algorithm that computes, for any tree @math , the minimum number of agents to capture the intruder, and the corresponding search strategy. Beside its optimality in time, our algorithm is naturally distributed: if @math is a processor-network, then the minimal search strategy for @math can be computed by @math in a decentralized manner, using a linear number of messages.