Version Of Graph Research Articles

Uniformity in Association Schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations. Perhaps the most important subclass of these objects is the class of cometric Q-antipodal association schemes. The concept of a cometric association scheme (the dual version of a distance-regular graph) is well-known; however, until recently it has not been studied well outside the area of distance-regular graphs. Uniformity is a concept introduced by Higman, but this likewise has not been well-studied. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain - uniform - coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a (\non-exceptional) strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples.

Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

We define a block-reduced version of the crystal graph of spin representations of the symmetric and alternating groups, and separate it into layers, each obtained by translating the previous layer and, possibly, adding new defect zero blocks. We demonstrate that each layer has weight-preserving central symmetry, and study the sequence of weights occurring in the maximal strings. The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks. This article is part of a study of the spin block case.

Line graphs, link partitions, and overlapping communities

In this paper, we use a partition of the links of a network in order to uncover its community structure. This approach allows for communities to overlap at nodes so that nodes may be in more than one community. We do this by making a node partition of the line graph of the original network. In this way we show that any algorithm that produces a partition of nodes can be used to produce a partition of links. We discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this.

Markov clustering versus affinity propagation for the partitioning of protein interaction graphs

BackgroundGenome scale data on protein interactions are generally represented as large networks, or graphs, where hundreds or thousands of proteins are linked to one another. Since proteins tend to function in groups, or complexes, an important goal has been to reliably identify protein complexes from these graphs. This task is commonly executed using clustering procedures, which aim at detecting densely connected regions within the interaction graphs. There exists a wealth of clustering algorithms, some of which have been applied to this problem. One of the most successful clustering procedures in this context has been the Markov Cluster algorithm (MCL), which was recently shown to outperform a number of other procedures, some of which were specifically designed for partitioning protein interactions graphs. A novel promising clustering procedure termed Affinity Propagation (AP) was recently shown to be particularly effective, and much faster than other methods for a variety of problems, but has not yet been applied to partition protein interaction graphs.ResultsIn this work we compare the performance of the Affinity Propagation (AP) and Markov Clustering (MCL) procedures. To this end we derive an unweighted network of protein-protein interactions from a set of 408 protein complexes from S. cervisiae hand curated in-house, and evaluate the performance of the two clustering algorithms in recalling the annotated complexes. In doing so the parameter space of each algorithm is sampled in order to select optimal values for these parameters, and the robustness of the algorithms is assessed by quantifying the level of complex recall as interactions are randomly added or removed to the network to simulate noise. To evaluate the performance on a weighted protein interaction graph, we also apply the two algorithms to the consolidated protein interaction network of S. cerevisiae, derived from genome scale purification experiments and to versions of this network in which varying proportions of the links have been randomly shuffled.ConclusionOur analysis shows that the MCL procedure is significantly more tolerant to noise and behaves more robustly than the AP algorithm. The advantage of MCL over AP is dramatic for unweighted protein interaction graphs, as AP displays severe convergence problems on the majority of the unweighted graph versions that we tested, whereas MCL continues to identify meaningful clusters, albeit fewer of them, as the level of noise in the graph increases. MCL thus remains the method of choice for identifying protein complexes from binary interaction networks.

A polynomial-time approximation scheme for embedding hypergraph in a cycle

We consider the problem of embedding hyperedges of a hypergraph as paths in a cycle such that the maximum congestion, namely the maximum number of paths that use any single edge in a cycle, is minimized. The minimum congestion hypergraph embedding in a cycle problem is known to be NP-hard and its graph version, the minimum congestion graph embedding in a cycle , is solvable in polynomial-time. Furthermore, for the graph problem, a polynomial-time approximation scheme for the weighted version is known. For the hypergraph model, several approximation algorithms with a ratio of two have been previously published. A recent paper reduced the approximation ratio to 1.5. We present a polynomial-time approximation scheme in this article, settling the debate regarding whether the problem is polynomial-time approximable.

Modeling the web as a hypergraph to compute page reputation

In this work we propose a model to represent the web as a directed hypergraph (instead of a graph), where links connect pairs of disjointed sets of pages. The web hypergraph is derived from the web graph by dividing the set of pages into non-overlapping blocks and using the links between pages of distinct blocks to create hyperarcs. A hyperarc connects a block of pages to a single page, in order to provide more reliable information for link analysis. We use the hypergraph model to create the hypergraph versions of the Pagerank and Indegree algorithms, referred to as HyperPagerank and HyperIndegree, respectively. The hypergraph is derived from the web graph by grouping pages by two different partition criteria: grouping together the pages that belong to the same web host or to the same web domain. We compared the original page-based algorithms with the host-based and domain-based versions of the algorithms, considering a combination of the page reputation, the textual content of the pages and the anchor text. Experimental results using three distinct web collections show that the HyperPagerank and HyperIndegree algorithms may yield better results than the original graph versions of the Pagerank and Indegree algorithms. We also show that the hypergraph versions of the algorithms were slightly less affected by noise links and spamming.

Modified risk graph method using fuzzy rule-based approach

The risk graph is one of the most popular methods used to determine the safety integrity level for safety instrumented functions. However, conventional risk graph as described in the IEC 61508 standard is subjective and suffers from an interpretation problem of risk parameters. Thus, it can lead to inconsistent outcomes that may result in conservative SILs. To overcome this difficulty, a modified risk graph using fuzzy rule-based system is proposed. This novel version of risk graph uses fuzzy scales to assess risk parameters and calibration may be made by varying risk parameter values. Furthermore, the outcomes which are numerical values of risk reduction factor (the inverse of the probability of failure on demand) can be compared directly with those given by quantitative and semi-quantitative methods such as fault tree analysis (FTA), quantitative risk assessment (QRA) and layers of protection analysis (LOPA).

Contact replacement for NMR resonance assignment

Motivation: Complementing its traditional role in structural studies of proteins, nuclear magnetic resonance (NMR) spectroscopy is playing an increasingly important role in functional studies. NMR dynamics experiments characterize motions involved in target recognition, ligand binding, etc., while NMR chemical shift perturbation experiments identify and localize protein–protein and protein–ligand interactions. The key bottleneck in these studies is to determine the backbone resonance assignment, which allows spectral peaks to be mapped to specific atoms. This article develops a novel approach to address that bottleneck, exploiting an available X-ray structure or homology model to assign the entire backbone from a set of relatively fast and cheap NMR experiments.Results: We formulate contact replacement for resonance assignment as the problem of computing correspondences between a contact graph representing the structure and an NMR graph representing the data; the NMR graph is a significantly corrupted, ambiguous version of the contact graph. We first show that by combining connectivity and amino acid type information, and exploiting the random structure of the noise, one can provably determine unique correspondences in polynomial time with high probability, even in the presence of significant noise (a constant number of noisy edges per vertex). We then detail an efficient randomized algorithm and show that, over a variety of experimental and synthetic datasets, it is robust to typical levels of structural variation (1–2 AA), noise (250–600%) and missings (10–40%). Our algorithm achieves very good overall assignment accuracy, above 80% in α-helices, 70% in β-sheets and 60% in loop regions.Availability: Our contact replacement algorithm is implemented in platform-independent Python code. The software can be freely obtained for academic use by request from the authors.Contact: gopal@cs.purdue.edu; cbk@cs.dartmouth.edu

Fuzzy Risk Graph Model for Determining Safety Integrity Level

The risk graph is one of the most popular methods used to determine the safety integrity level for safety instrumented functions. However, conventional risk graph as described in the IEC 61508 standard is subjective and suffers from an interpretation problem of risk parameters. Thus, it can lead to inconsistent outcomes that may result in conservative SIL's. To overcome this difficulty, a modified risk graph using fuzzy rule-based system is proposed. This novel version of risk graph uses fuzzy scales to assess risk parameters, and calibration may be made by varying risk parameter values. Furthermore, the outcomes which are numerical values of risk reduction factor (the inverse of the probability of failure on demand) can be compared directly with those given by quantitative and semiquantitative methods such as fault tree analysis (FTA), quantitative risk assessment (QRA), and layers of protection analysis (LOPA).

What is a complex graph?

Many papers published in recent years show that real-world graphs G ( n , m ) ( n nodes, m edges) are more or less “complex” in the sense that different topological features deviate from random graphs. Here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs. We present different measures that quantify this complexity, for instance C 1 e , the relative number of non-isomorphic one-edge-deleted subgraphs (i.e. DECK size). However, because these different subgraph measures are computationally demanding, we also study simpler complexity measures focussing on slightly different aspects of graph complexity. We consider heuristically defined “product measures”, the products of two quantities which are zero in the extreme cases of a path and clique, and “entropy measures” quantifying the diversity of different topological features. The previously defined network/graph complexity measures Medium Articulation and Offdiagonal complexity ( OdC) belong to these two classes. We study OdC measures in some detail and compare it with our new measures. For all measures, the most complex graph G C max has a medium number of edges, between the edge numbers of the minimum and the maximum connected graph n − 1 < m ( G C max ) < n ( n − 1 ) / 2 . Interestingly, for some measures C ̃ this number scales exactly with the geometric mean of the extremes: m ( G C ̃ max ) = n / 2 ( n − 1 ) ∼ n 1.5 . All graph complexity measures are characterized with the help of different example graphs. For all measures the corresponding time complexity is given. Finally, we discuss the complexity of 33 real-world graphs of different biological, social and economic systems with the six computationally most simple measures (including OdC). The complexities of the real graphs are compared with average complexities of two different random graph versions: complete random graphs (just fixed n , m ) and rewired graphs with fixed node degrees.

MicroBusiness - Cellular Industry Competes for Broadband Business

Our previous column on WiMAX business prospects included a graph of the mobile WiMAX competitive environment. Many different versions of such graphs exist. What these graphs have in common is that they are all populated by a progression of cellular systems toward ever-higher broadband capabilities. The 4G target peak data rates agreed upon in ITU-R Working Party 8F for "Systems Beyond IMT-2000" (renamed "IMT-Advanced") are "up to approximately 100 Mb/s for high mobility such as mobile access and up to approximately 1 Gb/s for low mobility such as nomadic wireless access". The cellular industry is pursuing mobile broadband development toward these objectives along two distinct 3G/4G branches, one evolving from WCDMA and the other from CDMA2000. The former is spearheaded by the 3rd Generation Partnership Project (3GPP), and the latter by 3GPP2.

Combinatorics of transformations from standard to non-standard bases in Brauer algebras

Transformation coefficients between standard bases for irreducible representations of the Brauer centralizer algebra and split bases adapted to the subalgebra (f1 + f2 = f) are considered. After providing the suitable combinatorial background, based on the definition of the i-coupling relation on nodes of the subduction grid, we introduce a generalized version of the subduction graph which extends the one given in Chilla (2006 J. Phys. A: Math. Gen. 39 7657) for symmetric groups. Thus, we can describe the structure of the subduction system arising from the linear method and give an outline of the form of the solution space. An ordering relation on the grid is also given and then, as in the case of symmetric groups, the choices of the phases and of the free factors governing the multiplicity separations are discussed.

A linear time algorithm for max-min length triangulation of a convex polygon

We consider the following planar max-min length triangulation problem: given a set of n points in the Euclidean plane, find a triangulation such that the length of the shortest edge in the triangulation is maximized. In this paper, a linear time algorithm is proposed for computing the max-min length triangulation of a set of points in convex position. In addition, an O ( n log n ) time algorithm is proposed for computing the max-min length k-set triangulation of a set of points in convex position, where we are to compute a set of k vertices such that the max-min length triangulation on them is minimized over all possible k-set. We further show that the graph version of max-min length triangulation is NP-complete, and some common heuristics such as greedy algorithm are in general not able to give a bounded-ratio approximation to the max-min length triangulation.

An ant colony algorithm for the pos/neg weighted p-median problem

Let a graph G = (V, E) with vertex set V and edge set E be given. The classical graph version of the p-median problem asks for a subset \(X\subseteq V\) of cardinality p, so that the (weighted) sum of the minimum distances from X to all other vertices in V is minimized. We consider the semi-obnoxious case, where every vertex has either a positive or a negative weight. This gives rise to two different objective functions, namely the weighted sum of the minimum distances from X to the vertices in V\X and, differently, the sum over the minimum weighted distances from X to V\X. In this paper an Ant Colony algorithm with a tabu restriction is designed for both problems. Computational results show its superiority with respect to a previously investigated variable neighborhood search and a tabu search heuristic.

Deza digraphs

We consider a directed version of Deza graphs. A digraph is said to be a Deza digraph if it is regular and the number of common out-neighbors of any two distinct vertices takes on at most two values. We introduce some constructions and develop some basic theory. In particular, constructions from Cayley digraphs, graph lexicograph products and graph directed products are described. Moreover, we characterize a family of normal Deza digraphs, and list all cyclic strictly Deza digraphs with at most six vertices. Finally, connections to weakly distance-regular digraphs are discussed, and all strictly Deza digraphs which are also weakly distance-regular digraphs with at most ten vertices are listed.

Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property pi--e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property pi. Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

On the expansion rate of Margulis expanders

In this note we determine exactly the expansion rate of an infinite 4-regular expander graph which is a variant of an expander due to Margulis. The vertex set of this graph consists of all points in the plane. The point ( x , y ) is adjacent to the points S ( x , y ) , S - 1 ( x , y ) , T ( x , y ) , T - 1 ( x , y ) where S ( x , y ) = ( x , x + y ) and T ( x , y ) = ( x + y , y ) . We show that the expansion rate of this 4-regular graph is 2. The main technical result asserts that for any compact planar set A of finite positive measure, | S ( A ) ∪ S - 1 ( A ) ∪ T ( A ) ∪ T - 1 ( A ) ∪ A | | A | ≥ 2 , where | B | is the Lebesgue measure of B. The proof is completely elementary and is based on symmetrization—a classical method in the area of isoperimetric problems. We also use symmetrization to prove a similar result for a directed version of the same graph.

Tracking a detected face with dynamic programming

In this paper, we consider the problem of tracking a moving human face in front of a video camera in real-time for a model-based coding application. The 3D head tracking in a MBC system could be implemented sequentially as 2D location tracking, coarse 3D orientation estimation and accurate 3D motion estimation. This work focuses on the 2D location tracking of one face object through continuously using a face detector. The face detection scheme is based on a boosted cascade of simple Haar-like feature classifiers. Although such a detector demonstrated rapid processing speed, high detection rate can only be achieved for rather strictly near front faces. This introduces the ‘loss of tracking’ problem when used in 2D tracking. This paper suggests an easy method of solving the pose problem by using the technique of Dynamic Programming. The Haar-like facial features used in the 2D face detector are spatially arranged into a 1D deformable face graph and the Dynamic Programming matching is used to handle the ‘loss of track’ problem. Dynamic Programming matches the deformed version of the face graph extracted from a rotated face with the template taken online before ‘loss of tracking’ happens. Since the deformable face graph covers a big pose variation, the developed technique is robust in tracking rotated faces. Embedding Haar-like facial features into a deformable face graph is the key feature of our tracking scheme. A real time tracking system based on this technique has been set up and tested. Encouraging results have been got and are reported.

On scheduling mesh-structured computations for Internet-based computing

Advances in technology have rendered the Internet a viable medium for employing multiple independent computers collaboratively in the solution of a single computational problem. A variety of mechanisms - e.g., Web-based computing, peer-to-peer computing, and grid computing - have been developed for such Internet-based computing (IC). Scheduling a computation for IC presents challenges that were not encountered with earlier modalities of parallel or distributed computing, especially when the computation's constituent tasks have interdependencies that constrain their order of execution. The process of scheduling such computations for IC is studied via a "pebble game" that abstracts the process of orchestrating the allocation of a computation's interdependent tasks to participating computers. A quality measure for plays of this game is developed that addresses the danger of "gridlock" in IC when a computation stalls because (due to dependencies) no tasks are eligible for execution. This measure rewards schedules that maximize the number of tasks that are eligible for execution at every step of the computation, one avenue for minimizing the likelihood of "gridlock." The resulting formal setting is illustrated via the problem of scheduling computations whose intertask dependencies have the structure of "evolving" meshes of finite dimensionalities. Within an idealized setting, a simple scheduling strategy is shown to be optimal when the dependencies have the structure of a two-dimensional mesh and within a constant factor of optimal for meshes of higher dimensionalities. The strategy remains optimal for a generalization of two-dimensional meshes whose structures are determined by abelian monoids (a monoid-based version of Cayley graphs). The optimality results for the idealized setting provide scheduling guidelines for real settings.

Graph Imperfection with a Co-Site Constraint

We are interested in a version of graph coloring where there is a "co-site" constraint value k. Given a graph G with a nonnegative integral demand xv at each node v, we must assign xv positive integers (colors) to each node v such that the same integer is never assigned to adjacent nodes, and two distinct integers assigned to a single node differ by at least k. The aim is to minimize the span, that is, the largest integer assigned to a node. This problem is motivated by radio channel assignment where one has to assign frequencies to transmitters so as to avoid interference. We compare the span with a clique-based lower bound when some of the demands are large. We introduce the relevant graph invariant, the k-imperfection ratio, give equivalent definitions, and investigate some of its properties. The k-imperfection ratio is always at least 1: we call a graph k-perfect when it equals 1. Then 1-perfect is the same as perfect, and we see that for many classes of perfect graphs, each graph in the class is k-perfect for all k. These classes include bipartite graphs and more generally comparability graphs, co-comparability graphs, and line-graphs of bipartite graphs.