Unweighted Graphs Research Articles

A Max-Cut approximation using a graph based MBO scheme

The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph $ G $, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of $ G $ into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut.We introduce the signless Ginzburg–Landau functional and prove that this functional $ \Gamma $-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman–Bence–Osher (MBO) scheme, which uses a signless Laplacian. We derive a Lyapunov functional for the iterations of our signless MBO scheme. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of $ \mathcal{O}(|E|) $, where $ |E| $ is the total number of edges on $ G $.

Structural parameters, tight bounds, and approximation for [formula omitted]-center

In (k,r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: •For any r≥1, we show an algorithm that solves the problem in O∗((3r+1)cw) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm’s performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.•We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph’s vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.•We show that the complexity of the problem parameterized by tree-depth is 2Θ(td2), by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k,r. In particular, we give algorithms which, for any ϵ>0, run in time O∗((tw∕ϵ)O(tw)), O∗((cw∕ϵ)O(cw)) and return a (k,(1+ϵ)r)-center if a (k,r)-center exists, thus circumventing the problem’s W-hardness.

Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs

In this article, we show how to compute for n -vertex planar graphs in O ( n 11/6 polylog( n )) expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In O ( n 15/8 polylog( n )) expected time, we can also compute the number of pairs of vertices at distances smaller than a given threshold. These are the first algorithms for these problems using time O ( n c ) for some constant c < 2, even when restricted to undirected, unweighted planar graphs.

Improved Distance Queries and Cycle Counting by Frobenius Normal Form

Consider an unweighted, directed graph G with the diameter D. In this paper, we introduce the framework for counting cycles and walks of given length in matrix multiplication time widetilde {O}(n^{omega }). The framework is based on the fast decomposition into Frobenius normal form and the Hankel matrix-vector multiplication. It allows us to solve the All-Nodes Shortest Cycles, All-Pairs All Walks problems efficiently and also give some improvement upon distance queries in unweighted graphs.

Efficient Algorithms for Constructing Very Sparse Spanners and Emulators

Miller et al. [48] devised a distributed 1 algorithm in the CONGEST model that, given a parameter k = 1,2,…, constructs an O ( k )-spanner of an input unweighted n -vertex graph with O ( n 1+1/ k ) expected edges in O ( k ) rounds of communication. In this article, we improve the result of Reference [48] by showing a k -round distributed algorithm in the same model that constructs a (2 k −1)-spanner with O ( n 1+1/ k }/ϵ) edges, with probability 1−ϵ for any ϵ>0. Moreover, when k =ω(log n ), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n (1+ o (1)), with probability 1− o (1). To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that are sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large k , it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for (1+ϵ, β)-spanners and emulators. In particular, we show that for any unweighted n -vertex graph and any ϵ > 0, there exists a (1+ ϵ, (log log n / ϵ) log log n )-emulator with O ( n ) edges. All previous constructions of (1+ϵ, β)-spanners and emulators employ a superlinear number of edges for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths’ computation in unweighted graphs.

Sparse Hardware Embedding of Spiking Neuron Systems for Community Detection

We study the applicability of spiking neural networks and neuromorphic hardware for solving general opti- mization problems without the use of adaptive training or learning algorithms. We leverage the dynamics of Hopfield networks and spin-glass systems to construct a fully connected spiking neural system to generate synchronous spike responses indicative of the underlying community structure in an undirected, unweighted graph. Mapping this fully connected system to current generation neuromorphic hardware is done by embedding sparse tree graphs to generate only the leading-order spiking dynamics. We demonstrate that for a chosen set of benchmark graphs, the spike responses generated on a current generation neuromorphic processor can improve the stability of graph partitions and non-overlapping communities can be identified even with the loss of higher-order spiking behavior if the graphs are sufficiently dense. For sparse graphs, the loss of higher-order spiking behavior improves the stability of certain graph partitions but does not retrieve the known community memberships.

Robust system size reduction of discrete fracture networks: a multi-fidelity method that preserves transport characteristics

We propose a multi-fidelity system reduction technique that uses weighted graphs paired with three-dimensional discrete fracture network (DFN) modelling for efficient simulation of subsurface flow and transport in fractured media. DFN models are used to simulate flow and transport in subsurface fractured rock with low-permeability. One method to alleviate the heavy computational overhead associated with these simulations is to reduce the size of the DFN using a graph representation of it to identify the primary flow sub-network and only simulate flow and transport thereon. The first of these methods used unweighted graphs constructed solely on DFN topology and could be used for accurate predictions of first-passage times. However, these techniques perform poorly when predicting later stages of the mass breakthrough. We utilize a weighted-graph representation of the DFN where edge weights are based on hydrological parameters in the DFN that allows us to exploit the kinematic quantities derivable a posteriori from the flow solution obtained on the graph representation of the DFN to perform system reduction and predict the later stages of the breakthrough curve with high fidelity. We also propose and demonstrate the use of an adaptive pruning algorithm with error control that produces a pruned DFN sub-network whose predicted mass breakthrough agrees with the original DFN within a user-specified tolerance. The method allows for the level of accuracy to be a user-controlled parameter.

A fast divisive community detection algorithm based on edge degree betweenness centrality

Many complex systems in the real world such as social networks can be modeled by complex networks. The complex network analysis and especially community detection is an important research topic in graph analysis that aims to identify the structure of a graph and its similar groups of nodes. In recent years, various algorithms such as Girvan and Newman’s method (GN) is introduced which is based on a divisive approach for graph clustering. Although GN is a highly popular and widely used method, it suffers from scalability and computational complexity. GN needs O(m3) and O(m3 + m3logm) time to detects communities in unweighted and weighted graphs respectively. Hence, in this paper, a faster method is suggested that detects communities in O(m2) for both weighted and unweighted graphs. In this paper, firstly, we define degree for each edge and then we propose a new and fast approach for the calculation of edges betweenness that is based on edge degree centrality. Furthermore, in order to boost the speed of the algorithm, we suggest instead of just one edge, multiple edges can be removed in each iteration. Since the proposed method wants to enhance the GN method, in the evaluation section the quality of detected communities, the accuracy and speed of the suggested method are assessed by the comparison with the GN method. Results prove that our proposed method is extremely faster than plain GN and the detected communities often have better quality than the plain GN method. Furthermore, we compare our proposed method with meta-heuristic algorithms which are a novel approach for community detection. Results clarify that the suggested method is notably faster, scalable, stable, reliable, and efficient than meta-heuristic algorithms.

Local Measurement and Diffusion Reconstruction for Signals on a Weighted Graph

Bandlimited graph signals on an unweighted graph can be reconstructed by its local measurement, which is a generalization of decimation. Since most signals are weighted in real life, we extend and improve the iterative local measurement reconstruction (ILMR) by introducing the diffusion operators to reconstruct bandlimited signals on a weighted graph. We prove that the proposed reconstruction converges to the original signal. Moreover, the simulation results demonstrate that the improved algorithm has better convergence and has robustness against noise.

Parameterized approximation algorithms for some location problems in graphs

We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) r-Domination problem and the (Connected) p-Center problem for unweighted and undirected graphs. Given a graph G, we show how to construct a (connected) (r+O(μ))-dominating set D with |D|≤|D⁎| efficiently. Here, D⁎ is a minimum (connected) r-dominating set of G and μ is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of G. Additionally, we show that a +O(μ)-approximation for the (Connected) p-Center problem on G can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.

On dynamic network security: A random decentering algorithm on graphs

AbstractRandom Decentering Algorithm (RDA) on a undirected unweighted graph is defined and tested over several concrete scale-free networks. RDA introduces ancillary nodes to the given network following basic principles of minimal cost, density preservation, centrality reduction and randomness. First simulations over scale-free networks show that RDA gives a significant decreasing of both betweenness centrality and closeness centrality and hence topological protection of network is improved. On the other hand, the procedure is performed without significant change of the density of connections of the given network. Thus ancillae are not distinguible from real nodes (in a straightforward way) and hence network is obfuscated to potential adversaries by our manipulation.

A Sentence Similarity Computation for Restricted Domain

To improve the accuracy of sentence similarity computation in automatic question answering system for restricted domain, this paper proposed a new TextRank-RD algorithm based on vector space model. The algorithm assigns the node an initial value based on three factors: whether the domain dictionary contains the word, whether the word is a none, the position of the word. And it uses a weighted graph model that assigns weights according to the importance of nodes rather than an unweighted graph model that assigns weights equally. The experimental results show that the algorithm improved the accuracy of sentence similarity calculation compared with the TF-IDF algorithm based on vector space model, and this has important significance to improve the efficiency of the automatic question answering system for restricted domain.

Computationally Efficient Nonlinear Bell Inequalities for Quantum Networks.

The correlations in quantum networks have attracted strong interest with new types of violations of the locality. The standard Bell inequalities cannot characterize the multipartite correlations that are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for general quantum networks. In this work, we show a significant improvement by presenting new, explicit Bell-type inequalities for general networks including cyclic networks. These nonlinear inequalities are related to the matching problem of an equivalent unweighted bipartite graph that allows constructing a polynomial-time algorithm. For the quantum resources consisting of bipartite entangled pure states and generalized Greenberger-Horne-Zeilinger (GHZ) states, we prove the generic nonmultilocality of quantum networks with multiple independent observers using new Bell inequalities. The violations are maximal with respect to the presented Tsirelson's bound for Einstein-Podolsky-Rosen states and GHZ states. Moreover, these violations hold for Werner states or some general noisy states. Our results suggest that the presented Bell inequalities can be used to characterize experimental quantum networks.

Research on Robustness of Complex Networks Based on Genetic Algorithm

Complex network is widely used in networks, such as social networking services. The robustness is always a focus problem. Based on the analysis on influence of network node malfunction to the overall connectivity, we propose a node ranking strategy to optimize the robustness of complex network. This node ranking strategy is based on the Genetic Algorithm, which is applied to an unweighted graph in this paper, where the robustness value is an optimization target. Comparing the nodes in the network graph to the gene fragments, the cross mutation is realized, which ranking the nodes in the network. The simulation experiment based on the data from routing networks and a simulation of network, show the feasibility and practicability of this algorithm.

Dynamic Merging of Frontiers for Accelerating the Evaluation of Betweenness Centrality

Betweenness Centrality (BC) is a widely used metric of the relevance of a node in a network. The fastest-known algorithm for the evaluation of BC on unweighted graphs builds a tree representing information about the shortest paths for each vertex to calculate its contribution to the BC score. Actually, for specific vertices, the shortest-path trees of neighboring nodes could be leveraged to reduce the computational burden, but existing BC algorithms do not exploit that information and carry out redundant computations. We propose a new algorithm, called dynamic merging of frontiers, which makes use of such information to derive the BC score of degree-2 vertices by re-using the results of the sub-trees of the neighbors. We implemented our idea in parallel fashion exploiting Graphics Processing Units. Compared to state-of-the-art implementations, our approach achieves a linear improvement in the number of degree-2 vertices and an average improvement of × over a variety of real-world graphs.

Supermodularity in Unweighted Graph Optimization I: Branchings and Matchings

The main result of this paper is motivated by the following two apparently unrelated graph optimization problems: (A) As an extension of Edmonds’ disjoint branchings theorem, characterize digraphs comprising k disjoint branchings Bi each having a specified number μi of arcs. (B) As an extension of Ryser’s maximum term rank formula, determine the largest possible matching number of simple bipartite graphs complying with degree-constraints. The solutions to these problems and to their generalizations will be obtained from a new min-max theorem on covering a supermodular function by a simple degree-constrained bipartite graph. A specific feature of the result is that its minimum cost extension is already NP-hard. Therefore classic polyhedral tools themselves definitely cannot be sufficient for solving the problem, even though they make some good service in our approach.

Supermodularity in Unweighted Graph Optimization III: Highly Connected Digraphs

By generalizing a recent result of Hong, Liu and Lai on characterizing the degree-sequences of simple strongly connected directed graphs, a characterization is provided for degree-sequences of simple k-node-connected digraphs. More generally, we solve the directed node-connectivity augmentation problem when the augmented digraph is degree-specified and simple. As for edge-connectivity augmentation, we solve the special case when the edge-connectivity is to be increased by one and the augmenting digraph must be simple.

Cache-Oblivious Buffer Heap and Cache-Efficient Computation of Shortest Paths in Graphs

We present the buffer heap , a cache-oblivious priority queue that supports Delete-Min , Delete , and a hybrid Insert / Decrease-Key operation in O (1/ B log 2 N / M ) amortized block transfers from main memory, where M and B are the (unknown) cache size and block size, respectively, and N is the number of elements in the queue. We introduce the notion of a slim data structure that captures the situation when only a limited portion of the cache, which we call a slim cache , is available to the data structure to retain data between data structural operations. We show that a buffer heap automatically adapts to such an environment and supports all operations in O (1/λ + 1/ B log 2 N /λ) amortized block transfers each when the size of the slim cache is λ. Our results provide substantial improvements over known trivial cache performance bounds for cache-oblivious priority queues with Decrease-Keys . Using the buffer heap, we present cache-oblivious implementations of Dijkstra’s algorithm for undirected and directed single-source shortest path (SSSP) problems for graphs with non-negative real edge-weights. On a graph with n vertices and m edges, our algorithm for the undirected case performs O ( n + m / B log 2 n / M ) block transfers and for the directed case performs O (( n + m / B ) ċ log 2 n / B ) block transfers. These results give the first non-trivial cache-oblivious bounds for the SSSP problem on general graphs. For the all-pairs shortest path (APSP) problem on weighted undirected graphs, we incorporate slim buffer heaps into multi-buffer-buffer-heaps and use these to improve the cache-aware cache complexity. We also present a simple cache-oblivious APSP algorithm for unweighted undirected graphs that performs O ( mn / B log M / B n / B ) block transfers. This matches the cache-aware bound and is a substantial improvement over the previous cache-oblivious bound for the problem.

A Note on the Spectral Radius of Weighted Signless Laplacian Matrix

A weighted graph is a graph that has a numeric label associated with each edge, called the weight of edge. In many applications, the edge weights are usually represented by nonnegative integers or square matrices. The weighted signless Laplacian matrix of a weighted graph is defined as the sum of adjacency matrix and degree matrix of same weighted graph. In this paper, a brief overview of the notation and concepts of weighted graphs that will be used throughout this study is given. In Section 2, the weighted signless Laplacian matrix of simple connected weighted graphs is considered, some upper bounds for the spectral radius of the weighted signless Laplacian matrix are obtained and some results on weighted and unweighted graphs are found.

A Spectral Graph Theoretic Approach for Monitoring Multivariate Time Series Data From Complex Dynamical Processes

The objective of this paper is to monitor complex process dynamics manifest in multivariate (multidimensional) time series data using a spectral (algebraic) graph theoretic approach. We test the hypothesis that the spectral graph-based topological invariants detect incipient process drifts earlier [lower average run length (ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> )] and with higher fidelity (consistency of detection) when compared with the conventional statistics-based approaches. The presented approach maps a multidimensional sensor data stream X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N×d</sup> (visualize N as time and d as the number of sensors) as an unweighted and undirected network graph G(V, E), indexed by its vertices V and edges E, i.e., X → G(V, E). The rationale is that the graph-based topological invariants are surrogate representatives of the system state. We compare the monitoring performance of spectral graph theoretic invariants with conventional statistical features in an exponentially weighted moving average control chart setting. The practical utility of the approach is substantiated in the context of process monitoring in two advanced manufacturing scenarios, namely, ultraprecision machining (UPM) and semiconductor chemical mechanical planarization. These studies corroborate the hypothesis that graph theoretic invariants, when used as monitoring statistics, lead to lower ARL <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and more consistent detections in contrast to conventional statistical features. For instance, in the UPM case, the fault detection delay using graph theoretic invariants is less than 160 ms, compared with over 8 s of delay with statistical features.