Abstract We examine two greedy heuristics — wiring and rewiring — for constructing maximum assortative graphs over all simple connected graphs with a target degree sequence. Counterexamples show that natural greedy rewiring heuristics do not necessarily return a maximum assortative graph, even though it is known that the meta-graph of all simple connected graphs with given degree is connected under rewiring. Counterexamples show an elegant greedy graph wiring heuristic from the literature may fail to achieve the target degree sequence or may fail to wire a maximally assortative graph.
Graph enumeration and generation. The results in this paper where achieved using geng , a tool in the nauty package created by McKay @cite_0 , to generate all simple connected graphs of a given order.
@cite_0
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This paper reformulates the theory of graph mining on the technical basis of graph matching, and extends its scope of applications to computer vision. Given a set of attributed relational graphs (ARGs), we propose to use a hierarchical And-Or Graph (AoG) to model the pattern of maximal-size common subgraphs embedded in the ARGs, and we develop a general method to mine the AoG model from the unlabeled ARGs. This method provides a general solution to the problem of mining hierarchical models from unannotated visual data without exhaustive search of objects. We apply our method to RGB RGB-D images and videos to demonstrate its generality and the wide range of applicability. The code will be available at https: sites.google.com site quanshizhang mining-and-or-graphs. Realistic workloads are essential in evaluating middleware for computational grids. One important component is the raw grid itself: a network topology graph annotated with the hardware and software available on each node and link. This paper defines our requirements for grid generation and presents GridG, our extensible generator. We describe GridG in two steps: topology generation and annotation. For topology generation, we have both model and mechanism. We extend Tiers, an existing tool from the networking community, to produce graphs that obey recently discovered power laws of Internet topology. We also contribute to network topology theory by illustrating a contradiction between two laws and proposing a new version of one of them. For annotation, GridG captures intra- and inter-host correlations between attributes using conditional probability rules. We construct a set of rules, including one based on empirical evidence of OS concentration in subnets, that produce sensible host annotations. We present a simple generation procedure which turns out to be an effective source of very hard cases for graph 3-colorability. The graphs distributed according to this generation procedure are much denser in very hard cases than previously reported for the same problem size. The coloring cost for these instances is also orders of magnitude bigger. This ability is issued from the fact that the procedure favors-inside the class of graphs with given connectivity and free of 4-cliques-the generation of graphs with relatively few paths of length three (that we call 3-paths). There is a critical value of the ratio between the number of 3-paths and the number of edges, independent of the number of nodes, which separates the graphs having the same connectivity in two regions: one contains almost all graphs free of 4-cliques while the other contains almost no such graphs. The generated very hard cases are near this phase transition, and have a regular structure, witnessed by the low variance in node degrees, as opposite to the random graphs. This regularity in the graph structure seems to confuse the coloring algorithm by inducing an uniform search space, with no clue for the search. Given a connected graphG=(V, E) with |V|=n and maximum degree Δ such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with Δ colors. We generalize this as follows: letG-v be Δ-colored; then,v can be colored by considering the vertices in anO(logΔn) radius aroundv and by recoloring anO(logΔn) length “augmenting path” inside it. Using this, we show that Δ-coloringG is reducible inO(log3n logΔ) time to (Δ+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for Δ-coloring.