This paper studies the problem of locating multiple diffusion sources in networks with partial observations. We propose a new source localization algorithm, named Optimal-Jordan-Cover (OJC). The algorithm first extracts a subgraph using a candidate selection algorithm that selects source candidates based on the number of observed infected nodes in their neighborhoods. Then, in the extracted subgraph, OJC finds a set of nodes that "cover" all observed infected nodes with the minimum radius. The set of nodes is called the Jordan cover, and is regarded as the set of diffusion sources. Considering the heterogeneous susceptible-infected-recovered (SIR) diffusion in the Erdos-Renyi (ER) random graph, we prove that OJC can locate all sources with probability one asymptotically with partial observations. OJC is a polynomial-time algorithm in terms of network size. However, the computational complexity increases exponentially in m; the number of sources. We further propose a low-complexity heuristic based on the K-Means for approximating the Jordan cover, named Approximate-Jordan-Cover (AJC). Simulations on random graphs and real networks demonstrate that both AJC and OJC significantly outperform other heuristic algorithms.

This paper studies information source detection in networks under the independent cascade (IC) model. Assume the spread of information starts from a single source in a network and a complete snapshot of the network is obtained at some time. The goal is to identify the source based on the observation. We derive the maximum a posterior (MAP) estimator of the source for tree networks and propose a Short-Fat Tree (SFT) algorithm for general networks based on the MAP estimator. The algorithm selects the Jordan infection center [1] and breaks ties according the degree of boundary infected nodes. Loosely speaking, the algorithm selects the node such that the breadth-first search (BFS) tree from it has the minimum depth but the maximum number of leaf nodes. On the Erdos-Renyi (ER) random graph, we establish the following possibility and impossibility results: (i) when the infection duration0.5, SFT identifies the source with probability 1 (w.p.1) asymptotically (as network size increases to infinity), where n is the network size and μ is the average node degree; (ii) when the infection duration > ⌈log n/ log μ ⌉ + 2, the probability of identifying the source approaches zero asymptotically under any algorithm; and (iii) when infection duration0, asymptotically, at least 1-δ fraction of the nodes on the BFS tree starting from the source are leaf-nodes, where δ = 3√log n/μ, i.e., the BFS tree starting from the actual source is a fat tree. 1Numerical experiments on tree networks, the ER random graphs and real world networks with different evaluation metrics show that the SFT algorithm outperforms existing algorithms.

MapReduce/Hadoop framework has been widely used to process large-scale datasets on computing clusters. Scheduling map tasks with data locality consideration is crucial to the performance of MapReduce. Many works have been devoted to increasing data locality for better efficiency. However, to the best of our knowledge, fundamental limits of MapReduce computing clusters with data locality, including the capacity region and theoretical bounds on the delay performance, have not been well studied. In this paper, we address these problems from a stochastic network perspective. Our focus is to strike the right balance between data locality and load balancing to simultaneously maximize throughput and minimize delay. We present a new queueing architecture and propose a map task scheduling algorithm constituted by the Join the Shortest Queue policy together with the MaxWeight policy. We identify an outer bound on the capacity region, and then prove that the proposed algorithm can stabilize any arrival rate vector strictly within this outer bound. It shows that the outer bound coincides with the actual capacity region, and the proposed algorithm is throughput-optimal. Furthermore, we study the number of backlogged tasks under the proposed algorithm, which is directly related to the delay performance based on Little's law. We prove that the proposed algorithm is heavy-traffic optimal, i.e., it asymptotically minimizes the number of backlogged tasks as the arrival rate vector approaches the boundary of the capacity region. Therefore, the proposed algorithm is also delay-optimal in the heavy-traffic regime. The proofs in this paper deal with random processing times with heterogeneous parameters and nonpreemptive task execution, which differentiate our work from many existing works on MaxWeight-type algorithms, so the proof techniques themselves for the stability analysis and the heavy-traffic analysis are also novel contributions.

This paper studies the problem of detecting the information source in a network in which the spread of information follows the popular Susceptible-Infected-Recovered (SIR) model. We assume all nodes in the network are in the susceptible state initially, except one single information source that is in the infected state. Susceptible nodes may then be infected by infected nodes, and infected nodes may recover and will not be infected again after recovery. Given a snapshot of the network, from which we know the graph topology and all infected nodes but cannot distinguish susceptible nodes and recovered nodes, the problem is to find the information source based on the snapshot and the network topology. We develop a sample-path-based approach where the estimator of the information source is chosen to be the root node associated with the sample path that most likely leads to the observed snapshot. We prove for infinite-trees, the estimator is a node that minimizes the maximum distance to the infected nodes. A reverse-infection algorithm is proposed to find such an estimator in general graphs. We prove that for g+1-regular trees such that gq > 1, where g+1 is the node degree and q is the infection probability, the estimator is within a constant distance from the actual source with a high probability, independent of the number of infected nodes and the time the snapshot is taken. Our simulation results show that for tree networks, the estimator produced by the reverse-infection algorithm is closer to the actual source than the one identified by the closeness centrality heuristic. We then further evaluate the performance of the reverse infection algorithm on several real-world networks.

The focus of this paper is on rate and power control algorithms in DSRC. We first design a utility maximization framework by leveraging the well-developed network congestion control, and formulate two subproblems, one on rate control with fixed transmit powers and the other on power control with fixed rates. Distributed rate control and power control algorithms are developed to solve these two subproblems, respectively, and are proved to be asymptotically optimal. Joint rate and power control can be done by using the two algorithms in an alternating fashion. The performance enhancement of our algorithms compared with a recent rate control algorithm, called EMBARC [1], is evaluated by using the network simulator ns2.

In this paper, we consider the problem of locating the information source with sparse observations. We assume that a piece of information spreads in a network following a heterogeneous susceptible-infected-recovered (SIR) model, where a node is said to beinfectedwhen it receives the information andrecoveredwhen it removes or hides the information. We further assume that a small subset of infected nodes are reported, from which we need to find the source of the information. We adopt the sample path based estimator developed in [1], and prove that on infinite trees, the sample path based estimator is a Jordan infection center with respect to the set of observed infected nodes. In other words, the sample path based estimator minimizes the maximum distance to observed infected nodes. We further prove that the distance between the estimator and the actual source is upper bounded by a constant independent of the number of infected nodes with a high probability on infinite trees. Our simulations on tree networks and real world networks show that the sample path based estimator is closer to the actual source than several other algorithms.

In standard clustering problems, data points are represented by vectors, and by stacking them together, one forms a data matrix with row or column cluster structure. In this paper, we consider a class of binary matrices, arising in many applications, which exhibit both row and column cluster structure, and our goal is to exactly recover the underlying row and column clusters by observing only a small fraction of noisy entries. We first derive a lower bound on the minimum number of observations needed for exact cluster recovery. Then, we study three algorithms with different running time and compare the number of observations needed by them for successful cluster recovery. Our analytical results show smooth time-data trade offs: one can gradually reduce the computational complexity when increasingly more observations are available.

The broadcast of periodic messages is a key functionality in vehicular ad hoc networks. In the emerging vehicular networks, IEEE 802.11p is the standard of choice to support the PHY and MAC layer functionalities. The broadcast process in IEEE 802.11p is based on the CSMA mechanism where a device transmitting a packet senses the channel for ongoing transmissions and performs a random back-off before accessing the channel. Without RTS/CTS mechanisms, carrier sensing is expected to provide a protection region around the transmitter where no other transmitters are allowed. The point process characterizing the concurrent transmitters is expected to enforce a minimum separation between concurrent transmitters. However, at increasing densities, the CSMA behavior breaks down to an ALOHA-like transmission pattern where concurrent transmitters are distributed as a Poisson point process, indicating the lack of protection around transmitters. In this paper, we model the CSMA mechanism as a slotted system and analytically characterize the critical node/packet arrival density where the CSMA mechanism approaches an ALOHAlike behavior. Further, we use tools from stochastic geometry to establish closed-form expressions for the performance metrics of the broadcast mechanism in the ALOHA regime. Finally, using ns2 (an unslotted asynchronous simulator), we compare the theoretical results with simulations.

Scheduling map tasks to improve data locality is crucial to the performance of MapReduce. Many works have been devoted to increasing data locality for better efficiency. However, to the best of our knowledge, fundamental limits of MapReduce computing clusters with data locality, including the capacity region and theoretical bounds on the delay performance, have not been studied. In this paper, we address these problems from a stochastic network perspective. Our focus is to strike the right balance between data-locality and load-balancing to simultaneously maximize throughput and minimize delay. We present a new queueing architecture and propose a map task scheduling algorithm constituted by the Join the Shortest Queue policy together with the MaxWeight policy. We identify an outer bound on the capacity region, and then prove that the proposed algorithm stabilizes any arrival rate vector strictly within this outer bound. It shows that the algorithm is throughput optimal and the outer bound coincides with the actual capacity region. Further, we study the number of backlogged tasks under the proposed algorithm, which is directly related to the delay performance based on Little's law. We prove that the proposed algorithm is heavy-traffic optimal, i.e., it asymptotically minimizes the number of backlogged tasks as the arrival rate vector approaches the boundary of the capacity region. Therefore, the proposed algorithm is also delay optimal in the heavy-traffic regime.