Abstract: How does a ‘normal’ computer (or social) network look like? How can we spot ‘abnormal’ sub-networks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal money-laundering rings), forecasting, and simulations (“how will a computer virus spread?”). The heart of the problem is finding the properties of real graphs that seem to persist over multiple disciplines. We list such “laws” and, more importantly, we propose a simple, parsimonious model, the “recursive matrix” (R-MAT) model, which can quickly generate realistic graphs, capturing the essence of each graph in only a few parameters. Contrary to existing generators, our model can trivially generate weighted, directed and bipartite graphs; it subsumes the celebrated Erdős-Renyi model as a special case; it can match the power law behaviors, as well as the deviations from them (like the “winner does not take it all” model of Pennock et al. [20]). We present results on multiple, large real graphs, where we show that our parameter fitting algorithm (AutoMAT-fast) fits them very well.

Abstract: We present algorithms for solving symmetric, diagonally-dominant linear systems to accuracy e in time linear in their number of non-zeros and log (κf (A) e), where κf (A) is the condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with preconditioners designed using nearly-linear time algorithms for graph sparsification and graph partitioning.

Abstract: A critical problem in cluster ensemble research is how to combine multiple clusterings to yield a final superior clustering result. Leveraging advanced graph partitioning techniques, we solve this problem by reducing it to a graph partitioning problem. We introduce a new reduction method that constructs a bipartite graph from a given cluster ensemble. The resulting graph models both instances and clusters of the ensemble simultaneously as vertices in the graph. Our approach retains all of the information provided by a given ensemble, allowing the similarity among instances and the similarity among clusters to be considered collectively in forming the final clustering. Further, the resulting graph partitioning problem can be solved efficiently. We empirically evaluate the proposed approach against two commonly used graph formulations and show that it is more robust and achieves comparable or better performance in comparison to its competitors.

Abstract: Networks of companies can be constructed by using return correlations. A crucial issue in this approach is to select the relevant correlations from the correlation matrix. In order to study this problem, we start from an empty graph with no edges where the vertices correspond to stocks. Then, one by one, we insert edges between the vertices according to the rank of their correlation strength, resulting in a network called asset graph. We study its properties, such as topologically different growth types, number and size of clusters and clustering coefficient. These properties, calculated from empirical data, are compared against those of a random graph. The growth of the graph can be classified according to the topological role of the newly inserted edge. We find that the type of growth which is responsible for creating cycles in the graph sets in much earlier for the empirical asset graph than for the random graph, and thus reflects the high degree of networking present in the market. We also find the number of clusters in the random graph to be one order of magnitude higher than for the asset graph. At a critical threshold, the random graph undergoes a radical change in topology related to percolation transition and forms a single giant cluster, a phenomenon which is not observed for the asset graph. Differences in mean clustering coefficient lead us to conclude that most information is contained roughly within 10% of the edges.

Abstract: We describe a framework for managing network attack graph complexity through interactive visualization, which includes hierarchical aggregation of graph elements. Aggregation collapses non-overlapping subgraphs of the attack graph to single graph vertices, providing compression of attack graph complexity. Our aggregation is recursive (nested), according to a predefined aggregation hierarchy. This hierarchy establishes rules at each level of aggregation, with the rules being based on either common attribute values of attack graph elements or attack graph connectedness. The higher levels of the aggregation hierarchy correspond to higher levels of abstraction, providing progressively summarized visual overviews of the attack graph. We describe rich visual representations that capture relationships among our semantically-relevant attack graph abstractions, and our views support mixtures of elements at all levels of the aggregation hierarchy. While it would be possible to allow arbitrary nested aggregation of graph elements, it is better to constrain aggregation according to the semantics of the network attack problem, i.e., according to our aggregation hierarchy. The aggregation hierarchy also makes efficient automatic aggregation possible. We introduce the novel abstraction of protection domain as a level of the aggregation hierarchy, which corresponds to a fully-connected subgraph (clique) of the attack graph. We avoid expensive detection of attack graph cliques through knowledge of the network configuration, i.e. protection domains are predefined. While significant work has been done in automatically generating attack graphs, this is the first treatment of the management of attack graph complexity for interactive visualization. Overall, computation in our framework has worst-case quadratic complexity, but in practice complexity is greatly reduced because users generally interact with (often negligible) subsets of the attack graph. We apply our framework to a real network, using a software system we have developed for generating and visualizing network attack graphs.

Abstract: ggA multivariate Gaussian graphical Markov model for an undirected graph G, also called a covariance selection model or concentration graph model, is defined in terms of the Markov properties, i.e. conditional independences associated with G, which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients. By means of Fisher's z-transformation and Sidak's correlation inequality, conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained, leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion. The simultaneous p-values corresponding to the partial correlations are partitioned into three disjoint sets, a significant set S, an indeterminate set I and a nonsignificant set N. Our model selection method selects two graphs, a graph G SI whose edges correspond to the set S∪I, and a more conservative graph G S whose edges correspond to S only. Similar considerations apply to covariance graph models, which are defined in terms of marginal independence rather than conditional independence. The method is applied to some well-known examples and to simulated data.

Abstract: Many machine learning algorithms for clustering or dimensionality reduction take as input a cloud of points in Euclidean space, and construct a graph with the input data points as vertices. This graph is then partitioned (clustering) or used to redefine metric information (dimensionality reduction). There has been much recent work on new methods for graph-based clustering and dimensionality reduction, but not much on constructing the graph itself. Graphs typically used include the fully-connected graph, a local fixed-grid graph (for image segmentation) or a nearest-neighbor graph. We suggest that the graph should adapt locally to the structure of the data. This can be achieved by a graph ensemble that combines multiple minimum spanning trees, each fit to a perturbed version of the data set. We show that such a graph ensemble usually produces a better representation of the data manifold than standard methods; and that it provides robustness to a subsequent clustering or dimensionality reduction algorithm based on the graph.

Abstract: This paper investigates the stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified (undirected) communication graph, G. The feedback control is based only on relative information about vehicle states shared via the communication links. We prove that a linear stabilizing feedback always exists provided that G is connected. Moreover, we show how the rate of convergence to formation is governed by the size of the smallest positive eigenvalue of the Laplacian of G. Several numerical simulations are used to illustrate the results.

Abstract: In Van den Nest et al. [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action of local Clifford operations on graph states in terms of a graph transformation rule, known in graph theory as local complementation. It was shown that two graph states are equivalent under the local Clifford group if and only if there exists a sequence of local complementations which relates their associated graphs. In this Brief Report we report the existence of a polynomial time algorithm, published in A. Bouchet [Combinatorica 11, 315 (1991)], which decides whether two given graphs are related by a sequence of local complementations. Hence an efficient algorithm to detect local Clifford equivalence of graph states is obtained.

Abstract: We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. For H=K3 (the triangle) we give an O(22klog k+1.869kn2) algorithm, and for general H an O(2k|H|logk+2k|H|log |H|n|H|) algorithm. We introduce a preprocessing (kernelization) technique based on crown decompositions of an auxiliary graph. For H=K3 this leads to a preprocessing algorithm that reduces an arbitrary input graph of the problem to a graph on O(k3) vertices in polynomial time.

Abstract: Symbolic model-checking usually includes two steps: the building of a compact representation of a state graph and the evaluation of the properties of the system upon this data structure. In case of properties expressed with a linear time logic, it appears that the second step is often more time consuming than the first one. In this work, we present a mixed solution which builds an observation graph represented in a non symbolic way but where the nodes are essentially symbolic set of states. Due to the small number of events to be observed in a typical formula, this graph has a very moderate size and thus the complexity time of verification is neglectible w.r.t. the time to build the observation graph. Thus we propose different symbolic implementations for the construction of the nodes of this graph. The evaluations we have done on standard examples show that our method outperforms the pure symbolic methods which makes it attractive.

Abstract: Recently many new "scale-free" random graph models have been introduced, motivated by the power-law degree sequences observed in many large-scale real-world networks. The most studied of these is the Barabasi-Albert growth with "preferential attachment" model, made precise as the LCD model by the present authors. Here we use coupling techniques to show that in certain ways the LCD model is not too far from a standard random graph; in particular, the fractions of vertices that must be retained under an optimal attack in order to keep a giant component are within a constant factor for the scale-free and classical models.

Abstract: A finite automaton, simply referred to as a robot, has to explore a graph whose nodes are unlabeled and whose edge ports are locally labeled at each node. The robot has no a priori knowledge of the topology of the graph or of its size. Its task is to traverse all the edges of the graph. We first show that, for any K-state robot and any d > 3, there exists a planar graph of maximum degree d with at most K + 1 nodes that the robot cannot explore. This bound improves all previous bounds in the literature. More interestingly, we show that, in order to explore all graphs of diameter D and maximum degree d, a robot needs Ω(D log d) memory bits, even if we restrict the exploration to planar graphs. This latter bound is tight. Indeed, a simple DFS at depth D + 1 enables a robot to explore any graph of diameter D and maximum degree d using a memory of size O(D log d) bits. We thus prove that the worst case space complexity of graph exploration is Θ(D log d) bits.

Abstract: We propose a framework where behavioural properties of finite-state systems modelled as graph transformation systems can be expressed and verified. The technique is based on the unfolding semantics and it generalises McMillan’s complete prefix approach, originally developed for Petri nets, to graph transformation systems. It allows to check properties of the graphs reachable in the system, expressed in a monadic second order logic.

Abstract: We propose a method for drawing AS graph data using 2.5D graph visualization. In order to bring out the pure graph structure of the AS graph we consider its core hierarchy. The k-cores are represented by 2D layouts whose interdependence for increasing k is displayed by the third dimension. For the core with maximum value a spectral layout is chosen thus emphasizing on the most important part of the AS graph. The lower cores are added iteratively by force-based methods. In contrast to alternative approaches to visualize AS graph data, our method illustrates the entire AS graph structure. Moreover, it is generic with regard to the hierarchy displayed by the third dimension.

Abstract: A cyclic graph may be partitioned. The partitions of the cyclic graph may include an acyclic component of the cyclic graph and a set of partitioned links. The partitions of the cyclic graph may have a particular order. The elements of the cyclic graph may be serialized with a particular serialization order. The serialization order of the elements of the cyclic graph may correspond to the order of the partitions of the cyclic graph. The elements of the acyclic component of the cyclic graph may be serialized before the elements of the set of partitioned links. A computer system may include a graph synchronization component configured to partition the cyclic graph and determine the serialization order of the elements of the cyclic graph. A serialization of the cyclic graph does serialize the elements of the cyclic graph in the determined serialization order.

Abstract: Space-variant sampling of visual input is ubiquitous in the higher vertebrate brain, because a large input space may be processed with high peak precision without requiring an unacceptably large brain mass. Space-variant sampling has been studied in computer vision for decades. A major obstacle to exploiting this architecture in machines, and understanding its role in biology, is the lack of algorithms that generalize beyond regular samplings. Most image processing algorithms implicitly assume a Cartesian grid underlying the sensor. This thesis generalizes image processing to a sensor architecture described by an arbitrary graph. This data structure separates the sensor topology, expressed by the graph edge structure, from its geometry, represented by coordinates of the vertex set. The combinatorial Laplacian of the sensor graph is a key operator underlying a series of novel image processing algorithms. First, a new graph partitioning algorithm for segmentation is presented that heuristically minimizes the ratio of the perimeter of the partition border and the area of the partitions, under a suitable definition of graph-theoretic area. This approach produces high quality image segmentations. Interpolation of missing data on graphs is developed, using a combinatorial version of the Dirichlet Problem, i.e., minimizing the average gradients of the interpolated values while maintaining fixed boundary conditions. This leads to the solution of the Laplace Equation, which represents the steady-state of the diffusion process for stated boundary conditions. Results compare favorably to both isotropic and anisotropic diffusion for filling-in of missing data. A pyramid graph is defined by connecting vertical and horizontal levels of the Laplacian pyramid data structure. The isoperimetric algorithm, run on the graph pyramid, yields an improved segmentation at little extra computational cost. Finally, a small-world graph topology is employed by randomly introducing a few new edges to the image graph. This results in a large speed-up in computation time, with identical final results. The algorithms developed in this thesis do not require that the data associated with the graph are embedded in two-dimensions or even have a metric structure. Therefore, this approach to generalized image processing may find wider application in other areas of discrete data processing.

Abstract: For a given acyclic graph G, an important problem is to characterize all of the eigenvalues over all symmetric matrices with graph G. Of particular interest is the connection between this standard inverse eigenvalue problem and describing all the possible associated ordered multiplicity lists, along with determining the minimum number of distinct eigenvalues for a symmetric matrix with graph G. In this note two important open questions along these lines are resolved, both in the negative.

Abstract: The small world phenomenon, that consistently occurs in numerous exist- ing networks, refers to two similar but different properties — small average distance and the clustering effect. We consider a hybrid graph model that incorporates both properties by combining a global graph and a local graph. The global graph is modeled by a random graph with a power law degree distribution, while the local graph has specified local connectivity. We will prove that the hybrid graph has average distance and diameter close to that of random graphs with the same degree distribution (under certain mild conditions). We also give a simple decomposition algorithm which, for any given (real) graph, identifies the global edges and extracts the local graph (which is uniquely determined depending only on the local connectivity). We can then apply our theoretical results for analyzing real graphs, provided the parameters of the hybrid model can be appropriately chosen.

Abstract: This paper surveys the theory of bidimensional graph problems We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems

Abstract: ACG (adjacent constraint graph) is invented as a general floorplan representation. It has advantages of both adjacency graph and constraint graph of a floorplan: edges in an ACG are between modules close to each other, thus the physical distance of two modules can be measured directly in the graph; since an ACG is a constraint graph, the floorplan area and module positions can be simply found by longest path computations. A natural combination of horizontal and vertical relations within one graph renders a beautiful data structure with full symmetry. The direct correspondence between geometrical positions of modules and ACG structures also makes it easy to incrementally change a floorplan and evaluate the result. Experimental results show the superiority of this representation.

Abstract: The present invention is a method and system for identifying words, text fragments, or concepts of interest in a corpus of text. A graph is built which covers the corpus of text. The graph includes nodes and links, where nodes represent a word or a concept and links between the nodes represent directed relation names. A score is then computed for each node in the graph. Scores can also be computed for larger sub-graph portions of the graph (such as tuples) The scores are used to identify desired sub-graph portions of the graph, those sub-graph portions being referred to as graph fragments.

Abstract: We consider the problem of searching a randomly growing graph by a random walk. In particular we consider two simple models of "web-graphs." Thus at each time step a new vertex is added and it is connected to the current graph by randomly chosen edges. At the same time a "spider" S makes a number of steps of a random walk on the current graph. The parameter we consider is the expected proportion of vertices that have been visited by S up to time t.

Abstract: Since the state space of most games is a directed graph, many game-playing systems detect repeated positions with a transposition table. This approach can reduce search effort by a large margin. However, it suffers from the so-called Graph History Interaction (GHI) problem, which causes errors in games containing repeated positions. This paper presents a practical solution to the GHI problem that combines and extends previous techniques. Because our scheme is general, it is applicable to different game tree search algorithms and to different domains. As demonstrated with the two algorithms αβ and df-pn in the two games checkers and Go, our scheme incurs only a very small overhead, while guaranteeing the correctness of solutions.

Abstract: Lexicographic Breadth First Search, introduced by Rose, Tarjan and Lueker for the recognition of chordal graphs is currently the most popular graph algorithmic search paradigm, with applications in recognition of restricted graph families, diameter approximation for restricted families and determining a dominating pair in an AT-free graph. This paper surveys this area and provides new directions for further research in the area of graph searching.

Abstract: In this work we devise algorithmic techniques to compare the interconnection structure of the Internet AS graph with that of graphs produced by topology generators that match the power-law degree distribution of the AS graph. We are guided by the existing notion that nodes in the AS graph can be placed in tiers with the resulting graph having an hierarchical structure. Our techniques are based on identifying graph nodes at each tier, decomposing the graph by removing such nodes and their incident edges, and thus explicitly revealing the interconnection structure of the graph. We define quantitative metrics to analyze and compare the decomposition of synthetic power-law graphs with the Internet-AS graph. Through experiments, we observe qualitative similarities in the decomposition structure of the different families of power-law graphs and explain any quantitative differences based on their generative models. We believe our approach provides insight into the interconnection structure of the AS graph and finds continuing applications in evaluating the representativeness of synthetic topology generators.

Abstract: A system for exchanging routing information over a communications network constructs a connectivity graph that indicates connectivity between a first node and a first set of nodes in the network. The system constructs an adjacency graph that indicates a second set of nodes with which the first node will exchange routing data, where the adjacency graph is distinct from the connectivity graph. The system exchanges routing data between the first node and each node of the second set of nodes based on the adjacency graph.