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Abstract. Motivated by real world networks and use of algorithms based on random walks on these networks we study the simple random walks on dynamic undirected graphs with fixed underlying vertex set, i.e., graphs which are modified by inserting or deleting edges at every step of the walk. We are interested in the expected time needed to visit all the vertices of such a dynamic graph, the cover time, under the assumption that the graph is being modified by an oblivious adversary. It is well known that on connected static undirected graphs the cover time is polynomial in the size of the graph. On the contrary and somewhat counter-intuitively, we show that there are adversary strategies which force the expected cover time of a simple random walk on connected dynamic graphs to be exponential. We relate this result to the cover time of static directed graphs. In addition we provide a simple strategy, the lazy random walk, that guarantees polynomial cover time regardless of the changes made by the adversary.

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time -the expected time required to visit every node in a graph at least once -and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds. *

Many existing systems for sensor networks rely on state information stored in the nodes for proper operation (e.g., pointers to parent in a spanning tree, routing information, etc). In dynamic environments, such systems must adopt failure recovery mechanisms, which significantly increase the complexity and impact the overall performance. In this paper, we investigate alternative schemes for query processing based on random walk techniques. The robustness of this approach under dynamics follows from the simplicity of the process, which only requires the connectivity of the neighborhood to keep moving. In addition we show that visiting a constant fraction of sensor network, say 80%, using a random walk is efficient in number of messages and sufficient for answering many interesting queries with high quality. Finally, the natural behavior of a random walk, also provide the important properties of load-balancing and scalability.

The cover time and mixing time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r ) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r . The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r opt such that for any r ≥ r opt G(n, r ) has optimal cover time of Θ(n log n) with high probability, and, importantly, r opt = Θ(r con ) where r con denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r con ). On the other hand, the radius required for rapid mixing r rapid = ω(r con ), and, in particular, r rapid = Θ(1/poly(log n)). We are able to draw our results by giving a tight bound on the electrical resistance and conductance of G(n, r ) via certain constructed flows.

We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time-the expected time required to visit every node in a graph at least once-and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

The glass ceiling effect has been defined in a recent US Federal Commission report as "the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements". It is well documented that many societies and organizations exhibit a glass ceiling. In this paper we formally define and study the glass ceiling effect in social networks and propose a natural mathematical model, called the biased preferential attachment model, that partially explains the causes of the glass ceiling effect. This model consists of a network composed of two types of vertices, representing two sub-populations, and accommodates three well known social phenomena: (i) the "rich get richer" mechanism, (ii) a minority-majority partition, and (iii) homophily. We prove that our model exhibits a strong moment glass ceiling effect and that all three conditions are necessary, i.e., removing any one of them will prevent the appearance of a glass ceiling effect. Additionally, we present empirical evidence taken from a mentor-student network of researchers (derived from the DBLP database) that exhibits both a glass ceiling effect and the above three phenomena.

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(∆n) where ∆ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n 2 ) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n 2 ). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.

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