Abstract : Graphs change over time, and typically variations on the small multiples or animation pattern is used to convey this dynamism visually. However, both of these classical techniques have significant drawbacks, so a new approach, Storyline Visualization of Events on a Network (SVEN) is proposed. SVEN builds on storyline techniques, conveying nodes as contiguous lines over time. SVEN encodes time in a natural manner, along the horizontal axis, and optimizes the vertical placement of storylines to decrease clutter (line crossings, straightness, and bends) in the drawing. This paper demonstrates SVEN on several different flavors of real-world dynamic data, and outlines the remaining near-term future work.
To the author's knowledge, the earliest portrayal of events on a network with a spatial encoding of time are sequence diagrams,'' developed in the 1970s, which are used to understand timing and synchronization in distributed systems @cite_6 . These visualizations were originally hand drawn; nodes were represented as vertical parallel lines with time moving from the bottom to the top of the page. Communication events between processes were represented as wavy lines connecting connecting the processes sending and receiving the message at the local times the message was sent and received. Sequence diagrams are effective for understanding trivially small networks, but become difficult to scale up as more nodes are added. The ordering of nodes in the diagram can be chosen to minimize crossings, but this is the Traveling Salesman Problem, and even if an optimal solution was found, there is no guarantee that the resulting diagram would be interpretable for large or dense networks.
@cite_6
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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. 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. It has been known for some time that larger graphs can be interpreted if laid out in 3D and displayed with stereo and or motion depth cues to support spatial perception. However, prior studies were carried out using displays that provided a level of detail far short of what the human visual system is capable of resolving. Therefore, we undertook a graph comprehension study using a very high resolution stereoscopic display. In our first experiment, we examined the effect of stereoscopic display, kinetic depth, and using 3D tubes versus lines to display the links. The results showed a much greater benefit for 3D viewing than previous studies. For example, with both motion and stereoscopic depth cues, unskilled observers could see paths between nodes in 333 node graphs with less than a 10p error rate. Skilled observers could see up to a 1000-node graph with less than a 10p error rate. This represented an order of magnitude increase over 2D display. In our second experiment, we varied both nodes and links to understand the constraints on the number of links and the size of graph that can be reliably traced. We found the difference between number of links and number of nodes to best account for error rates and suggest that this is evidence for a “perceptual phase transition.” These findings are discussed in terms of their implications for information display. We present the communication model, which assumes nodes have minimal knowledge about their environment and severely limited communication capabilities. Specifically, nodes have no information regarding the local or global structure of the network, don't have access to synchronized clocks and are woken up by an adversary. Moreover, instead on communicating through messages they rely solely on carrier sensing to exchange information. We study the problem of , a variant of vertex coloring specially suited for the studied beeping model. Given a set of resources, the goal of interval coloring is to assign every node a large contiguous fraction of the resources, such that neighboring nodes share no resources. To highlight the importance of the discreteness of the model, we contrast it against a continuous variant described in [17]. We present an O(1 @math (T ) @math T @math O( n) @math (T ) @math O( n) @math ( n)$ on the time required to solve interval coloring for this model against randomized algorithms. This lower bound implies that our algorithm is asymptotically optimal for constant degree graphs.