scholarly journals Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs

2021 ◽  
Vol 182 (3) ◽  
pp. 219-242
Author(s):  
Mostafa Haghir Chehreghani ◽  
Albert Bifet ◽  
Talel Abdessalem
Keyword(s):  
Betweenness Centrality ◽  
Directed Graphs ◽  
Randomized Algorithm ◽  
Exact Algorithm ◽  
Directed Network ◽  
Weighted Graphs ◽  
Single Vertex ◽  
Small Constant ◽  
Positive Weights ◽  
Real World Datasets

Graphs (networks) are an important tool to model data in different domains. Realworld graphs are usually directed, where the edges have a direction and they are not symmetric. Betweenness centrality is an important index widely used to analyze networks. In this paper, first given a directed network G and a vertex r ∈ V (G), we propose an exact algorithm to compute betweenness score of r. Our algorithm pre-computes a set ℛ𝒱(r), which is used to prune a huge amount of computations that do not contribute to the betweenness score of r. Time complexity of our algorithm depends on |ℛ𝒱(r)| and it is respectively Θ(|ℛ𝒱(r)| · |E(G)|) and Θ(|ℛ𝒱(r)| · |E(G)| + |ℛ𝒱(r)| · |V(G)| log |V(G)|) for unweighted graphs and weighted graphs with positive weights. |ℛ𝒱(r)| is bounded from above by |V(G)| – 1 and in most cases, it is a small constant. Then, for the cases where ℛ𝒱(r) is large, we present a simple randomized algorithm that samples from ℛ𝒱(r) and performs computations for only the sampled elements. We show that this algorithm provides an (ɛ, δ)-approximation to the betweenness score of r. Finally, we perform extensive experiments over several real-world datasets from different domains for several randomly chosen vertices as well as for the vertices with the highest betweenness scores. Our experiments reveal that for estimating betweenness score of a single vertex, our algorithm significantly outperforms the most efficient existing randomized algorithms, in terms of both running time and accuracy. Our experiments also reveal that our algorithm improves the existing algorithms when someone is interested in computing betweenness values of the vertices in a set whose cardinality is very small.

scholarly journals A note on compact and compact circular edge-colorings of graphs

2008 ◽  
Vol Vol. 10 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Dariusz Dereniowski ◽  
Adam Nadolski
Keyword(s):  
Approximation Algorithm ◽  
Decision Problem ◽  
Edge Coloring ◽  
Exact Algorithm ◽  
Weighted Graphs ◽  
Edge Colorings ◽  
Odd Cycle ◽  
International Audience ◽  
Np Complete ◽  
Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

Virtual-Reality graph visualization based on Fruchterman-Reingold using Unity and SteamVR

2021 ◽  
pp. 147387162110603
Author(s):  
Gerd Kortemeyer
Keyword(s):  
Virtual Reality ◽  
Directed Graphs ◽  
Dynamic Visualization ◽  
Graph Visualization ◽  
Weighted Graphs ◽  
Virtual Reality System ◽  
Newtonian Physics ◽  
Dynamic Extrusion ◽  
Relationship Of ◽  
The Relationship

The paper describes a method for the immersive, dynamic visualization of undirected, weighted graphs. Using the Fruchterman-Reingold method, force-directed graphs are drawn in a Virtual-Reality system. The user can walk through the data, as well as move vertices using controllers, while the network display rearranges in realtime according to Newtonian physics. In addition to the physics behind the employed method, the paper explains the most pertinent computational mechanisms for its implementation, using Unity, SteamVR, and a Virtual-Reality system such as HTC Vive (the source package is made available for download). It was found that the method allows for intuitive exploration of graphs with on the order of [Formula: see text] vertices, and that dynamic extrusion of vertices and realtime readjustment of the network structure allows for developing an intuitive understanding of the relationship of a vertex to the remainder of the network. Based on this observation, possible future developments are suggested.

Passenger Mobility Prediction via Representation Learning for Dynamic Directed and Weighted Graphs

2022 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Yuandong Wang ◽  
Hongzhi Yin ◽  
Tong Chen ◽  
Chunyang Liu ◽  
Ben Wang ◽  
...  
Keyword(s):  
Fundamental Problem ◽  
Route Planning ◽  
Representation Learning ◽  
Data Representation ◽  
Weighted Graphs ◽  
Spatial And Temporal Patterns ◽  
Graph Representations ◽  
Demand Prediction ◽  
Passenger Demand ◽  
Real World Datasets

In recent years, ride-hailing services have been increasingly prevalent, as they provide huge convenience for passengers. As a fundamental problem, the timely prediction of passenger demands in different regions is vital for effective traffic flow control and route planning. As both spatial and temporal patterns are indispensable passenger demand prediction, relevant research has evolved from pure time series to graph-structured data for modeling historical passenger demand data, where a snapshot graph is constructed for each time slot by connecting region nodes via different relational edges (origin-destination relationship, geographical distance, etc.). Consequently, the spatiotemporal passenger demand records naturally carry dynamic patterns in the constructed graphs, where the edges also encode important information about the directions and volume (i.e., weights) of passenger demands between two connected regions. aspects in the graph-structure data. representation for DDW is the key to solve the prediction problem. However, existing graph-based solutions fail to simultaneously consider those three crucial aspects of dynamic, directed, and weighted graphs, leading to limited expressiveness when learning graph representations for passenger demand prediction. Therefore, we propose a novel spatiotemporal graph attention network, namely Gallat ( G raph prediction with all at tention) as a solution. In Gallat, by comprehensively incorporating those three intrinsic properties of dynamic directed and weighted graphs, we build three attention layers to fully capture the spatiotemporal dependencies among different regions across all historical time slots. Moreover, the model employs a subtask to conduct pretraining so that it can obtain accurate results more quickly. We evaluate the proposed model on real-world datasets, and our experimental results demonstrate that Gallat outperforms the state-of-the-art approaches.

Efficient Batched Distance, Closeness and Betweenness Centrality Computation in Unweighted and Weighted Graphs

2017 ◽  
Vol 17 (2) ◽  
pp. 169-182 ◽  
Author(s):  
Manuel Then ◽  
Stephan Günnemann ◽  
Alfons Kemper ◽  
Thomas Neumann
Keyword(s):  
Betweenness Centrality ◽  
Weighted Graphs

scholarly journals MAXIMUM WEIGHT CYCLE PACKING IN DIRECTED GRAPHS, WITH APPLICATION TO KIDNEY EXCHANGE PROGRAMS

2009 ◽  
Vol 01 (04) ◽  
pp. 499-517 ◽  
Author(s):  
PÉTER BIRÓ ◽  
DAVID F. MANLOVE ◽  
ROMEO RIZZI
Keyword(s):  
Directed Graphs ◽  
Exact Algorithm ◽  
Maximum Size ◽  
Practical Experience ◽  
Optimal Solutions ◽  
Maximum Weight ◽  
Packing Problems ◽  
Exchange Programs ◽  
Kidney Exchange ◽  
Cycle Packing

Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration.

scholarly journals Fixation probabilities for simple digraphs

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120676 ◽  
Author(s):  
Burton Voorhees ◽  
Alex Murray
Keyword(s):  
Complete Bipartite Graph ◽  
Directed Graphs ◽  
Closed Form Solution ◽  
Form Solution ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Fixation Probability ◽  
Single Vertex ◽  
Fixation Probabilities ◽  
Birth Death

The problem of finding birth–death fixation probabilities for configurations of normal and mutants on an N -vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the single-vertex fixation probabilities. This also yields a closed-form solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r >1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.

scholarly journals Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2001
Author(s):  
Konstantin Gorbunov ◽  
Vassily Lyubetsky
Keyword(s):  
Linear Time ◽  
Exact Algorithm ◽  
Time Algorithm ◽  
Additive Constant ◽  
Weighted Graphs ◽  
Total Cost ◽  
Operation Costs ◽  
Double Cut And Join ◽  
Vertex Degrees ◽  
Directed Weighted Graphs

We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.

scholarly journals Analysis of Regular Patterns in Un-weighted Directed Graphs

2022 ◽  
Vol 12 (1) ◽  
pp. 0-0
Keyword(s):  
Real World ◽  
Directed Graphs ◽  
Degree Centrality ◽  
Relative Significance ◽  
Evolving Networks ◽  
Graph Metrics ◽  
Mining Algorithm ◽  
Real World Datasets ◽  
Regular Patterns

Time evolving networks tend to have an element of regularity. This regularity is characterized by existence of repetitive patterns in the data sequences of the graph metrics. As per our research, the relevance of such regular patterns to the network has not been adequately explored. Such patterns in certain data sequences are indicative of properties like popularity, activeness etc. which are of vital significance for any network. These properties are closely indicated by data sequences of graph metrics - degree prestige, degree centrality and occurrence. In this paper, (a) an improved mining algorithm has been used to extract regular patterns in these sequences, and (b) a methodology has been proposed to quantitatively analyse the behavior of the obtained patterns. To analyze this behavior, a quantification measure coined as "Sumscore" has been defined to compare the relative significance of such patterns. The patterns are ranked according to their Sumscores and insights are then drawn upon it. The efficacy of this method is demonstrated by experiments on two real world datasets.

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