Sur une caractérisation des vidanges fortes: corrections and extensions

1978 ◽  
Vol 15 (2) ◽  
pp. 268-279
Author(s):  
Jean-Guy Dion
Keyword(s):  
Stochastic Process ◽  
Weighted Graph ◽  
Complete Graphs ◽  
Weighted Graphs ◽  
Urn Scheme

A draining is a stochastic process defined by an urn scheme where the successive drawings are made without replacement and according to a drawing algorithm associated with a weighted graph. A draining is said to be a strong one if the number of balls which can be drawn from the urn (under the algorithm) converges in probability to the total number of balls in the urn at the beginning of the drawing when it goes to infinity. In particular, drainings associated with complete graphs having equal weights are found to be strong and for some others associated weighted graphs, strong drainings exist.

Sur une caractérisation des vidanges fortes: corrections and extensions

1978 ◽  
Vol 15 (02) ◽  
pp. 268-279
Author(s):  
Jean-Guy Dion
Keyword(s):  
Stochastic Process ◽  
Weighted Graph ◽  
Complete Graphs ◽  
Weighted Graphs ◽  
Urn Scheme

A draining is a stochastic process defined by an urn scheme where the successive drawings are made without replacement and according to a drawing algorithm associated with a weighted graph. A draining is said to be a strong one if the number of balls which can be drawn from the urn (under the algorithm) converges in probability to the total number of balls in the urn at the beginning of the drawing when it goes to infinity. In particular, drainings associated with complete graphs having equal weights are found to be strong and for some others associated weighted graphs, strong drainings exist.

scholarly journals EDGE IDEALS OF WEIGHTED GRAPHS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250223 ◽  
Author(s):  
CHELSEY PAULSEN ◽  
SEAN SATHER-WAGSTAFF
Keyword(s):  
Weighted Graph ◽  
Polynomial Rings ◽  
Complete Graphs ◽  
Weighted Graphs ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Weighted Vertex

We study weighted graphs and their "edge ideals" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of "weighted vertex covers". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen–Macaulay, and we prove that all weighted complete graphs are Cohen–Macaulay.

scholarly journals Spread of Epidemic Disease on Edge-Weighted Graphs from a Database: A Case Study of COVID-19

2021 ◽  
Vol 18 (9) ◽  
pp. 4432
Author(s):  
Ronald Manríquez ◽  
Camilo Guerrero-Nancuante ◽  
Felipe Martínez ◽  
Carla Taramasco
Keyword(s):  
Public Health ◽  
Infectious Diseases ◽  
Preventive Measures ◽  
Weighted Graph ◽  
Real Data ◽  
Sir Model ◽  
Weighted Graphs ◽  
Epidemic Disease ◽  
The Real

The understanding of infectious diseases is a priority in the field of public health. This has generated the inclusion of several disciplines and tools that allow for analyzing the dissemination of infectious diseases. The aim of this manuscript is to model the spreading of a disease in a population that is registered in a database. From this database, we obtain an edge-weighted graph. The spreading was modeled with the classic SIR model. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics. Moreover, a deterministic approximation is provided. With database COVID-19 from a city in Chile, we analyzed our model with relationship variables between people. We obtained a graph with 3866 vertices and 6,841,470 edges. We fitted the curve of the real data and we have done some simulations on the obtained graph. Our model is adjusted to the spread of the disease. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics, in this case with real data of COVID-19. This valuable information allows us to also include/understand the networks of dissemination of epidemics diseases as well as the implementation of preventive measures of public health. These findings are important in COVID-19’s pandemic context.

Out-of-equilibrium random walks

2020 ◽  
Vol 52 (3) ◽  
pp. 772-797
Author(s):  
Leonardo A. Videla
Keyword(s):  
Stochastic Process ◽  
Random Walks ◽  
Complete Graphs ◽  
Knowledge Process ◽  
Random Walker ◽  
Underlying Graph ◽  
Graph Sequence

AbstractWe study the long-term behaviour of a random walker embedded in a growing sequence of graphs. We define a (generally non-Markovian) real-valued stochastic process, called the knowledge process, that represents the ratio between the number of vertices already visited by the walker and the current size of the graph. We mainly focus on the case where the underlying graph sequence is the growing sequence of complete graphs.

WEIGHTED-EDGE-COLORING OF k-DEGENERATE GRAPHS AND BIN-PACKING

2011 ◽  
Vol 12 (01n02) ◽  
pp. 109-124
Author(s):  
FLORIAN HUC
Keyword(s):  
Bin Packing ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Specific Class ◽  
Weighted Graphs ◽  
Outerplanar Graphs ◽  
Underlying Graph ◽  
Multiple Edges

The weighted-edge-coloring problem of an edge-weighted graph whose weights are between 0 and 1, consists in finding a coloring using as few colors as possible and satisfying the following constraints: the sum of weights of edges with the same color and incident to the same vertex must be at most 1. In 1991, Chung and Ross conjectured that if G is bipartite, then [Formula: see text] colors are always sufficient to weighted-edge-color (G,w), where [Formula: see text] is the maximum of the sums of the weights of the edges incident to a vertex. We prove this is true for edge-weighted graphs with multiple edges whose underlying graph is a tree. We further generalise this conjecture to non-bipartite graphs and prove the generalised conjecture for simple edge-weighted outerplanar graphs. Finally, we introduce a list version of this coloring together with the list-bin-packing problem, which allows us to obtain new results concerning the original coloring for a specific class of graphs, namely the k-weight-degenerate weighted graph.

Isoperimetric properties of weighted graphs related to the Laplacian spectrum and canonical correlations

2002 ◽  
Vol 39 (3-4) ◽  
pp. 425-441 ◽  
Author(s):  
M. Bolla ◽  
G. Molnár-Sáska
Keyword(s):  
Volume Ratio ◽  
Weighted Graph ◽  
Weighted Graphs ◽  
Laplacian Spectrum ◽  
Minimum Requirement ◽  
Cheeger Constant ◽  
Canonical Correlations ◽  
Laplacian Spectra ◽  
Weighted Laplacian ◽  
Similar Cluster

The relation between isoperimetric properties and Laplacian spectra of weighted graphs is investigated. The vertices are classified into k clusters with „few" inter-cluster edges of „small" weights (area) and „similar" cluster sizes (volumes). For k=2 the Cheeger constant represents the minimum requirement for the area/volume ratio and it is estimated from above by v?1(2-?1), where ?1 is the smallest positive eigenvalue of the weighted Laplacian. For k?2 we define the k-density of a weighted graph that is a generalization of the Cheeger constant and estimated from below by Si=1k-1?i and from above by c2 Si=1k-1 ?i, where 0<?1=…=Sk-1 are the smallest Laplacian eigenvalues and the constant c?1 depends on the metric classification properties of the corresponding eigenvectors. Laplacian spectra are also related to canonical correlations in a probabilistic setup.

scholarly journals Characteristic polynomials of some weighted graph bundles and its application to links

1994 ◽  
Vol 17 (3) ◽  
pp. 503-510 ◽  
Author(s):  
Moo Young Sohn ◽  
Jaeun Lee
Keyword(s):  
Characteristic Polynomial ◽  
Weighted Graph ◽  
Double Covering ◽  
Characteristic Polynomials ◽  
Weighted Graphs

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weightedK2(K¯2)-bundles over a weighted graphG?can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs areGAs an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

scholarly journals Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

2021 ◽  
pp. 23-38
Author(s):  
Václav Blažej ◽  
Pratibha Choudhary ◽  
Dušan Knop ◽  
Jan Matyáš Křišt’an ◽  
Ondřej Suchý ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Fault Tolerant ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Constant Factor ◽  
Weighted Graphs ◽  
Feedback Vertex Set ◽  
Fixed Integer ◽  
Vertex Set

AbstractConsider a vertex-weighted graph G with a source s and a target t. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from s to t is unique. In this work, we derive a factor 66-approximation algorithm for Tracking Paths in weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related r-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer r and a given vertex-weighted graph G, the task is to find a minimum weight set of vertices intersecting every cycle of G in at least $$r+1$$ r + 1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O ( r 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.

Clustering Vertices in Weighted Graphs

2011 ◽  
pp. 285-298
Author(s):  
Derry Tanti Wijaya ◽  
Stephane Bressan
Keyword(s):  
Graph Algorithms ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Weighted Graph ◽  
Graph Clustering ◽  
Weighted Graphs ◽  
Similarity Relations ◽  
Natural Notion ◽  
Cluster Density ◽  
Markov Clustering

Clustering is the unsupervised process of discovering natural clusters so that objects within the same cluster are similar and objects from different clusters are dissimilar. In clustering, if similarity relations between objects are represented as a simple, weighted graph where objects are vertices and similarities between objects are weights of edges; clustering reduces to the problem of graph clustering. A natural notion of graph clustering is the separation of sparsely connected dense sub graphs from each other based on the notion of intra-cluster density vs. inter-cluster sparseness. In this chapter, we overview existing graph algorithms for clustering vertices in weighted graphs: Minimum Spanning Tree (MST) clustering, Markov clustering, and Star clustering. This includes the variants of Star clustering, MST clustering and Ricochet.

Universal Mixing of Quantum Walk on Graphs

2007 ◽  
Vol 7 (8) ◽  
pp. 738-751
Author(s):  
W. Carlson ◽  
A. Ford ◽  
E. Harris ◽  
J. Rosen ◽  
C. Tamon ◽  
...  
Keyword(s):  
Probability Distributions ◽  
Weighted Graph ◽  
Quantum Walk ◽  
Weighted Graphs ◽  
New Family ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Time Quantum ◽  
Average Probability ◽  
Complete Multipartite

We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph $G$ is {\em universal mixing} if the instantaneous or average probability distribution of the quantum walk on $G$ ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is {\em uniform} mixing if it visits the uniform distribution. Our results include the following: 1) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no {\em unweighted} complete multipartite graphs are uniform mixing (except for the four-cycle $K_{2,2}$). 2) For all $n \ge 1$, the weighted claw $K_{1,n}$ is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted $K_{1,n}$ is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. 3) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. 4) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.

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