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Concept: Vertex-transitive graph


Currently, most paired link based scaffolding algorithms intrinsically mask the sequences between two linked contigs and bypass their direct link information embedded in the original de Bruijn assembly graph. Such disadvantage substantially complicates the scaffolding process and leads to the inability of resolving repetitive contig assembly. Here we present a novel algorithm, inGAP-sf, for effectively generating high-quality and continuous scaffolds. inGAP-sf achieves this by using a new strategy based on the combination of direct link and paired link graphs, in which direct link is used to increase graph connectivity and to decrease graph complexity and paired link is employed to supervise the traversing process on the direct link graph. Such advantage greatly facilitates the assembly of short-repeat enriched regions. Moreover, a new comprehensive decision model is developed to eliminate the noise routes accompanying with the introduced direct link. Through extensive evaluations on both simulated and real datasets, we demonstrated that inGAP-sf outperforms most of the genome scaffolding algorithms by generating more accurate and continuous assembly, especially for short repetitive regions.

Concepts: Algorithm, Graph theory, Graph, Discrete mathematics, Planar graph, Vertex-transitive graph


A key issue in face recognition is to seek an effective descriptor for representing face appearance. In the context of considering the face image as a set of small facial regions, this paper presents a new face representation approach coined spatial feature interdependence matrix (SFIM). Unlike classical face descriptors which usually use a hierarchically organized or a sequentially concatenated structure to describe the spatial layout features extracted from local regions, SFIM is dedicated to the exploitation of the underlying feature interdependences regarding local region pairs inside a class specific face. According to SFIM, the face image is projected onto an undirected connected graph in a manner which explicitly encodes feature interdependence based relationships between local regions. We calculate the pair-wise interdependence strength as the weighted discrepancy between two feature sets extracted in a hybrid feature space fusing histograms of intensity, local binary pattern (LBP) and oriented gradients. To achieve face recognition goal, our SFIM based face descriptor is embedded in three different recognition frameworks, namely nearest neighbor search, subspace based classification and linear optimization based classification. Extensive experimental results on four well-known face databases and comprehensive comparisons with the state of the art results are provided to demonstrate the efficacy of the proposed SFIM based descriptor.

Concepts: Mathematics, Graph theory, Graph, Nearest neighbor search, Planar graph, Dean Koontz, Branch and bound, Vertex-transitive graph


Accurate diagnosis of Alzheimer’s disease and its prodromal stage, i.e., mild cognitive impairment, is very important for early treatment. Over the last decade, various machine learning methods have been proposed to predict disease status and clinical scores from brain images. It is worth noting that many features extracted from brain images are correlated significantly. In this case, feature selection combined with the additional correlation information among features can effectively improve classification/regression performance. Typically, the correlation information among features can be modeled by the connectivity of an undirected graph, where each node represents one feature and each edge indicates that the two involved features are correlated significantly. In this paper, we propose a new graph-guided multi-task learning method incorporating this undirected graph information to predict multiple response variables (i.e., class label and clinical scores) jointly. Specifically, based on the sparse undirected feature graph, we utilize a new latent group Lasso penalty to encourage the correlated features to be selected together. Furthermore, this new penalty also encourages the intrinsic correlated tasks to share a common feature subset. To validate our method, we have performed many numerical studies using simulated datasets and the Alzheimer’s Disease Neuroimaging Initiative dataset. Compared with the other methods, our proposed method has very promising performance.

Concepts: Alzheimer's disease, Scientific method, Machine learning, Graph theory, Graph, Planar graph, Vertex-transitive graph


LetGbe a connected graph of ordern. The remoteness ofG, denoted byρ, is the maximum average distance from a vertex to all other vertices. Let [Formula: see text], [Formula: see text] and [Formula: see text] be the distance, distance Laplacian and distance signless Laplacian eigenvalues ofG, respectively. In this paper, we give lower bounds on [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] and the corresponding extremal graphs are also characterized.

Concepts: Graph theory, Critical point, Graph, Planar graph, Laplacian matrix, Vertex, Vertex-transitive graph


This paper investigates the mean square consensus problem of dynamical networks of leader-following multi-agent systems with measurement noises and time-varying delays. We consider that the fixed undirected communication topologies are connected. A neighbor-based tracking algorithm together with distributed estimators are presented. Using tools of algebraic graph theory and the Gronwall-Bellman-Halanay type inequality, we establish sufficient conditions to reach consensus in mean square sense via the proposed consensus protocols. Finally, a numerical simulation is provided to demonstrate the effectiveness of the obtained theoretical result.

Concepts: Mathematics, Topology, Real number, Graph theory, Graph, Glossary of graph theory, Planar graph, Vertex-transitive graph


In this paper, we investigate Confluent Drawings (CD), a technique for bundling edges in node-link diagrams based on network connectivity. Edge-bundling techniques are designed to reduce edge clutter in node-link diagrams by coalescing lines into common paths or bundles. Unfortunately, traditional bundling techniques introduce ambiguity since edges are only bundled by spatial proximity, rather than network connectivity; following an edge from its source to its target can lead to the perception of incorrect connectivity if edges are not clearly separated within the bundles. Contrary, CDs bundle edges based on common sources or targets. Thus, a smooth path along a confluent bundle indicates precise connectivity. While CDs have been described in theory, practical investigation and application to real-world networks (i.e., networks beyond those with certain planarity restrictions) is currently lacking. Here, we provide the first algorithm for constructing CDs from arbitrary directed and undirected networks and present a simple layout method, embedded in a sand box environment providing techniques for interactive exploration. We then investigate patterns and artifacts in CDs, which we compare to other common edge-bundling techniques. Finally, we present the first user study that compares edge-compression techniques, including CD, power graphs, metro-style, and common edge bundling. We found that users without particular expertise in visualization or network analysis are able to read small CDs without difficulty. Compared to existing bundling techniques, CDs are more likely to allow people to correctly perceive connectivity.

Concepts: Graph theory, Path, Graph, Network theory, Planar graph, Allan Holdsworth, Vertex-transitive graph, Power graph analysis


This paper establishes several bounds on the algebraic connectivity and spectral radius of graphs. Before deriving these bounds, a directed graph with a leader node is first investigated, for which some bounds on the spectral radius and the smallest real part of all the eigenvalues of M=L+D are obtained using the properties of M-matrix and non-negative matrix under a mild assumption, where $L$ is the Laplacian matrix of the graph and D=diag{d₁,d₂,…dN} with dᵢ>0 if node i can access the information of the leader node and 0 otherwise. Subsequently, by virtue of the results on directed graphs, the bounds on the algebraic connectivity and spectral radius of an undirected connected graph are provided. Besides establishing these bounds, another important feature is that all these bounds are distributed in the sense of only knowing the information of edge weights' bounds and the number of nodes in a graph, without using any information of inherent structures of the graph. Therefore, these bounds can be in some sense applied to agent networks for reducing the conservatism where control gains in control protocols depend on the eigenvalues of matrices M or L, which are global information. Also some examples are provided for corroborating the feasibility of the theoretical results.

Concepts: Graph theory, Matrices, Graph, Planar graph, Connectivity, Vertex-transitive graph, Graph property, Algebraic connectivity


Detecting functional motifs in biological networks is one of the challenging problems in systems biology. Given a multiset of colors as query and a list-colored graph (an undirected graph with a set of colors assigned to each of its vertices), the problem is reduced to finding connected subgraphs, which best cover the multiset of query. To solve this NP-complete problem, we propose a new color-based centrality measure for list-colored graphs. Based on this newly-defined measure of centrality, a novel polynomial time algorithm is developed to discover functional motifs in list-colored graphs, using a greedy strategy. This algorithm, called CeFunMO, has superior running time and acceptable accuracy in comparison with other well-known algorithms, such as RANGI and GraMoFoNe.

Concepts: Algorithm, Graph theory, Computational complexity theory, Graph, Discrete mathematics, Greedy algorithm, Planar graph, Vertex-transitive graph


This paper revisits the distributed adaptive control problem for synchronization of multiagent systems where the dynamics of the agents are nonlinear, nonidentical, unknown, and subject to external disturbances. Two communication topologies, represented, respectively, by a fixed strongly-connected directed graph and by a switching connected undirected graph, are considered. Under both of these communication topologies, we use distributed neural networks to approximate the uncertain dynamics. Decentralized adaptive control protocols are then constructed to solve the cooperative tracker problem, the problem of synchronization of all follower agents to a leader agent. In particular, we show that, under the proposed decentralized control protocols, the synchronization errors are ultimately bounded, and their ultimate bounds can be reduced arbitrarily by choosing the control parameter appropriately. Simulation study verifies the effectiveness of our proposed protocols.

Concepts: Mathematics, Graph theory, Graph, Agent-based model, Glossary of graph theory, Planar graph, Vertex-transitive graph


In order to accomplish cooperative tasks, decentralized systems are required to communicate among each other. Thus, maintaining the connectivity of the communication graph is a fundamental issue. Connectivity maintenance has been extensively studied in the last few years, but generally considering undirected communication graphs. In this paper, we introduce a decentralized control and estimation strategy to maintain the strong connectivity property of directed communication graphs. In particular, we introduce a hierarchical estimation procedure that implements power iteration in a decentralized manner, exploiting an algorithm for balancing strongly connected directed graphs. The output of the estimation system is then utilized for guaranteeing preservation of the strong connectivity property. The control strategy is validated by means of analytical proofs and simulation results.

Concepts: Mathematics, Graph theory, Communication, Graph, Discrete mathematics, Planar graph, Vertex-transitive graph