2,088 research outputs found
Large Graph Analysis in the GMine System
Current applications have produced graphs on the order of hundreds of
thousands of nodes and millions of edges. To take advantage of such graphs, one
must be able to find patterns, outliers and communities. These tasks are better
performed in an interactive environment, where human expertise can guide the
process. For large graphs, though, there are some challenges: the excessive
processing requirements are prohibitive, and drawing hundred-thousand nodes
results in cluttered images hard to comprehend. To cope with these problems, we
propose an innovative framework suited for any kind of tree-like graph visual
design. GMine integrates (a) a representation for graphs organized as
hierarchies of partitions - the concepts of SuperGraph and Graph-Tree; and (b)
a graph summarization methodology - CEPS. Our graph representation deals with
the problem of tracing the connection aspects of a graph hierarchy with sub
linear complexity, allowing one to grasp the neighborhood of a single node or
of a group of nodes in a single click. As a proof of concept, the visual
environment of GMine is instantiated as a system in which large graphs can be
investigated globally and locally
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
Embedding of Cortical Representations by the Superficial Patch System
Pyramidal cells in layers 2 and 3 of the neocortex of many species collectively form a clustered system of lateral axonal projections (the superficial patch system—Lund JS, Angelucci A, Bressloff PC. 2003. Anatomical substrates for functional columns in macaque monkey primary visual cortex. Cereb Cortex. 13:15-24. or daisy architecture—Douglas RJ, Martin KAC. 2004. Neuronal circuits of the neocortex. Annu Rev Neurosci. 27:419-451.), but the function performed by this general feature of the cortical architecture remains obscure. By comparing the spatial configuration of labeled patches with the configuration of responses to drifting grating stimuli, we found the spatial organizations both of the patch system and of the cortical response to be highly conserved between cat and monkey primary visual cortex. More importantly, the configuration of the superficial patch system is directly reflected in the arrangement of function across monkey primary visual cortex. Our results indicate a close relationship between the structure of the superficial patch system and cortical responses encoding a single value across the surface of visual cortex (self-consistent states). This relationship is consistent with the spontaneous emergence of orientation response-like activity patterns during ongoing cortical activity (Kenet T, Bibitchkov D, Tsodyks M, Grinvald A, Arieli A. 2003. Spontaneously emerging cortical representations of visual attributes. Nature. 425:954-956.). We conclude that the superficial patch system is the physical encoding of self-consistent cortical states, and that a set of concurrently labeled patches participate in a network of mutually consistent representations of cortical inpu
Massively-Parallel Heat Map Sorting and Applications To Explainable Clustering
Given a set of points labeled with labels, we introduce the heat map
sorting problem as reordering and merging the points and dimensions while
preserving the clusters (labels). A cluster is preserved if it remains
connected, i.e., if it is not split into several clusters and no two clusters
are merged.
We prove the problem is NP-hard and we give a fixed-parameter algorithm with
a constant number of rounds in the massively parallel computation model, where
each machine has a sublinear memory and the total memory of the machines is
linear. We give an approximation algorithm for a NP-hard special case of the
problem. We empirically compare our algorithm with k-means and density-based
clustering (DBSCAN) using a dimensionality reduction via locality-sensitive
hashing on several directed and undirected graphs of email and computer
networks
IST Austria Thesis
We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
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