14,691 research outputs found
VoG: Summarizing and Understanding Large Graphs
How can we succinctly describe a million-node graph with a few simple
sentences? How can we measure the "importance" of a set of discovered subgraphs
in a large graph? These are exactly the problems we focus on. Our main ideas
are to construct a "vocabulary" of subgraph-types that often occur in real
graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the
most succinct description of a graph in terms of this vocabulary. We measure
success in a well-founded way by means of the Minimum Description Length (MDL)
principle: a subgraph is included in the summary if it decreases the total
description length of the graph.
Our contributions are three-fold: (a) formulation: we provide a principled
encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop
\method, an efficient method to minimize the description cost, and (c)
applicability: we report experimental results on multi-million-edge real
graphs, including Flickr and the Notre Dame web graph.Comment: SIAM International Conference on Data Mining (SDM) 201
{VoG}: {Summarizing} and Understanding Large Graphs
How can we succinctly describe a million-node graph with a few simple sentences? How can we measure the "importance" of a set of discovered subgraphs in a large graph? These are exactly the problems we focus on. Our main ideas are to construct a "vocabulary" of subgraph-types that often occur in real graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the most succinct description of a graph in terms of this vocabulary. We measure success in a well-founded way by means of the Minimum Description Length (MDL) principle: a subgraph is included in the summary if it decreases the total description length of the graph. Our contributions are three-fold: (a) formulation: we provide a principled encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop \method, an efficient method to minimize the description cost, and (c) applicability: we report experimental results on multi-million-edge real graphs, including Flickr and the Notre Dame web graph
Moving to Extremal Graph Parameters
Which graphs, in the class of all graphs with given numbers n and m of edges
and vertices respectively, minimizes or maximizes the value of some graph
parameter? In this paper we develop a technique which provides answers for
several different parameters: the numbers of edges in the line graph, acyclic
orientations, cliques, and forests. (We minimize the first two and maximize the
third and fourth.)
Our technique involves two moves on the class of graphs. A compression move
converts any graph to a form we call fully compressed: the fully compressed
graphs are split graphs in which the neighbourhoods of points in the
independent set are nested. A second consolidation move takes each fully
compressed graph to one particular graph which we call H(n,m). We show
monotonicity of the parameters listed for these moves in many cases, which
enables us to obtain our results fairly simply.
The paper concludes with some open problems and future directions
StructMatrix: large-scale visualization of graphs by means of structure detection and dense matrices
Given a large-scale graph with millions of nodes and edges, how to reveal
macro patterns of interest, like cliques, bi-partite cores, stars, and chains?
Furthermore, how to visualize such patterns altogether getting insights from
the graph to support wise decision-making? Although there are many algorithmic
and visual techniques to analyze graphs, none of the existing approaches is
able to present the structural information of graphs at large-scale. Hence,
this paper describes StructMatrix, a methodology aimed at high-scalable visual
inspection of graph structures with the goal of revealing macro patterns of
interest. StructMatrix combines algorithmic structure detection and adjacency
matrix visualization to present cardinality, distribution, and relationship
features of the structures found in a given graph. We performed experiments in
real, large-scale graphs with up to one million nodes and millions of edges.
StructMatrix revealed that graphs of high relevance (e.g., Web, Wikipedia and
DBLP) have characterizations that reflect the nature of their corresponding
domains; our findings have not been seen in the literature so far. We expect
that our technique will bring deeper insights into large graph mining,
leveraging their use for decision making.Comment: To appear: 8 pages, paper to be published at the Fifth IEEE ICDM
Workshop on Data Mining in Networks, 2015 as Hugo Gualdron, Robson Cordeiro,
Jose Rodrigues (2015) StructMatrix: Large-scale visualization of graphs by
means of structure detection and dense matrices In: The Fifth IEEE ICDM
Workshop on Data Mining in Networks 1--8, IEE
The Erdős-Ko-Rado properties of various graphs containing singletons
Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR.
Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′
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