Vertex Weights Research Articles

A Hybrid Evolutionary Algorithm for the Diversified Top-k Weight Clique Search Problem (Student Abstract)

The diversified top-k weight clique (DTKWC) search problem is an important generalization of the diversified top-k clique search problem, which extends the DTKC search problem by taking into account the weight of vertices. This problem involves finding at most k maximal weighted cliques that cover maximum weight of vertices with low overlapping in a given graph. In this study, a mixed integer linear program constraint formulation is proposed to model DTKWC search problem and an efficient hybrid evolutionary algorithm (HEA-D) based on some heuristic strategies is proposed to tackle it. Experiments on two sets of 110 graphs show that HEA-D outperforms the state-of-art methods.

Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs

Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least $\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is the number of vertices of $G$. This ratio is almost tight for large values of $n$. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least $\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors $\frac{3}{4}-\epsilon$ and $\frac{3}{5}-\epsilon$ for the MaxwIST problem in cubic graphs and claw-free graphs having no degree-2 vertices, for any $\epsilon>0$. The new algorithms are short (compared to that of Chen et al.) and fairly simple as they employ a variant of the depth-first search algorithm. Moreover, they take linear time while previous algorithms for similar problem instances are super-linear.

Mean isoperimetry with control on outliers: Exact and approximation algorithms

Given a weighted graph G=(V,E) with weight functions c:E→R+ and π:V→R+, and a subset U⊆V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U. The mean isoperimetry problem, ISO1(G,k), for a weighted graph G is a generalization of the classical uniform sparsest cut problem in which, given a parameter k, the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer 0≤ρ≤|V|. The problem may also be considered as the normalized version of the classical k-multiway cut problem on edge-weighted graphs.Our main result states that ISO1(G,k), as well as its robust version, CRISO1(G,k,ρ), subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time O(k2ρ2π(V(T))3) on any weighted tree T, in which π(V(T)) is the sum of the vertex-weights. This result implies that ISO1(G,k) is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-hard for weighted trees in general. As far as applications are concerned, the connectivity requirement may be interpreted as an approach to model the practical consistency of the parts, which together with having control on the size of the outlier set and applying a smooth “mean” cost function (as opposed to, say, the “max” version), characterizes our solution to CRISO1(G,k,ρ) on trees as one of the most flexible and accurate procedures within the framework of isoperimetry-based clustering.Also, using this, we show that both mentioned problems, ISO1(G,k) and CRISO1(G,k,ρ) as well as the ordinary robust mean isoperimetry problem RISO1(G,k,ρ), admit polynomial-time O(log1.5⁡|V|log⁡log⁡|V|)-approximation algorithms for weighted graphs with polynomially bounded weights, using the Räcke-Shah tree cut sparsifier.

THE METHODS OF ENSURING FAULT TOLERANCE, SURVIVABILITY AND PROTECTION OF INFORMATION OF SPECIALIZED INFORMATION TECHNOLOGIES UNDER THE INFLUENCE OF MALICIOUS SOFTWARE

The paper examines the provision of fault tolerance, survivability and protection of IT information on the impact of malicious software and computer attacks. Each method is presented separately by its steps. The states of hardware and software on which the implemented methods are impelled in the corresponding systems are investigated. The common states are singled out and on the basis of them and together with the steps of the methods the synthesis of the method of ensuring fault tolerance, survivability and protection of IT information is carried out. It combines three developed methods. This method is represented by four generalized steps. All representations of the models are made by graphs with weight vertices, which specify either the states or steps of the methods. This representation made it possible to connect common vertices.Some methods of ensuring resilience, survivability and protection of IT information under the influence of malicious software were compared with one integrated method. Experimental studies confirm the effectiveness of both the proposed solution to ensure fault tolerance, survivability and protection of IT information and the effectiveness of the method, which combines the provision of fault tolerance, survivability and protection of IT information.

Local antimagic vertex coloring for generalized friendship graphs

Let G = (V, E) be a graph of order p and size q without isolated vertices. A bijection f: E → {1, 2, … , q} is called a local antimagic labeling if w(u) ≠ w(v) for all uv ∈ E, where the vertex weight w(u) = Ʃ e∈E(u) f(e) and E(u) is the set of edges incident to the vertex u ∈ V. The local antimagic chromatic number χ la (G) is defined to be the minimum number of colors(vertex weights) taken over all colorings of G induced by local antimagic labelings of G. In this paper, we study the local chromatic number for generalized friendship graph of complete graphs and cycles.

An averaging process on hypergraphs

AbstractConsider the following iterated process on a hypergraph H. Each vertex v starts with some initial weight $x_v$ . At each step, uniformly at random select an edge e in H, and for each vertex v in e replace the weight of v by the average value of the vertex weights over all vertices in e. This is a generalization of an interactive process on graphs which was first introduced by Aldous and Lanoue. In this paper we use the eigenvalues of a Laplacian for hypergraphs to bound the rate of convergence for this iterated averaging process.

Evolving Bipartite Model Reveals the Bounded Weights in Mobile Social Networks

Many realistic mobile social networks can be characterized by evolving bipartite graphs [1], [2], in which dynamically added elements are divided into two entities and connected by links between these two entities, such as users and items in recommendation networks, followees and followers in Twitter networks, authors and scientific topics in scholarly networks, etc. However, given the fact that connections between two entities are often weighted, how to mathematically model such weighted evolving bipartite relationships, along with quantitative characterizations, remains unexplored. Motivated by this, we develop a novel Evolving Bipartite Model (EBM), which, based on empirically validated power-law distribution on multiple realistic mobile social networks, discloses that the distribution of total weights of incoming and outgoing edges in networks is determined by the weighting scale and bounded by certain ceilings and floors. Based on these theoretical results, for evolving bipartite networks whose degree follows power-law distribution, their overall weights of vertices can be predicted by EBM. To illustrate, in recommendation networks, the evaluation of items, i.e., total rating scores, can be estimated through the given bounds; in Twitter social networks, the influence of users can be roughly measured; in scholarly networks

Comparing symbolic powers of edge ideals of weighted oriented graphs

Let D be a weighted oriented graph and I(D) be its edge ideal. If D contains an induced odd cycle of length $$2n+1$$ , under certain condition, we show that $$ {I(D)}^{(n+1)} \ne {I(D)}^{n+1}$$ . We give necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideals of weighted oriented graphs having each edge in some induced odd cycle of it. We characterize the weighted naturally oriented unicyclic graphs with unique odd cycles and weighted naturally oriented even cycles for the equality of ordinary and symbolic powers of their edge ideals. Let $$ D^{\prime } $$ be the weighted oriented graph obtained from D after replacing the weights of vertices with non-trivial weights which are sinks, by trivial weights. We show that the symbolic powers of I(D) and $$I(D^{\prime })$$ behave in a similar way. Finally, if D is any weighted oriented star graph, we prove that $$ {I(D)}^{(s)} = {I(D)}^s $$ for all $$s \ge 2.$$

Computing harmonic maps between Riemannian manifolds

AbstractIn our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.

Maximum Decoding Clique Based Maximum Weight Vertex Search Algorithm for Buffered Instantly Decodable Network Codes

The traditional instantly decodable network coding (IDNC) schemes discard the non-instantly decodable packets (NIDPs), which results in the loss of useful information. In this letter, we consider a network coding scheme named as buffered instantly decodable network coding (B-IDNC) where each receiving node can cache the NIDPs containing two unreceived packets for subsequent network decoding, and give a decoding algorithm by utilizing the maximum decoding clique. The maximum decoding clique is defined as a subset of vertices in the IDNC graph, which can be decoded together. We approximate the average packet decoding delay minimization problem as the maximum weight encoding clique problem, and a heuristic algorithm named as maximum decoding clique based maximum weight vertex (MDC-MWV) searching algorithm is proposed to solve the approximated problem. Simulation results show that our proposed algorithm can reduce the average packet decoding delay compared with the conventional IDNC schemes.

Trees with total Roman domination number equal to Roman domination number plus its domination number: complexity and structural properties

A Roman dominating function on a graph G is a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. A total Roman dominating function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function f is the value The total Roman domination number of G, is the minimum weight of a total Roman dominating function in G. In this paper we first study the complexity issue of the problem posed in [A. Cabrera Martinez, S. Cabrera Garcia, A. Carrión Garcia, Further results on the total Roman domination in graphs, Mathematics 8(3) (2020) 349], and then show that the problem of deciding whether is NP-complete. Also, we prove that for any tree T of order if and only if

On local vertex irregular reflexive coloring of graphs

Let χ(G) be a chromatic number of proper coloring on G. For an injection f : V (G) → {0, 2, . ; 2kυg } and f : E(G) → }1, 2, . , ke }, where k = max{ke , 2kχ } for kυ , ke are natural number. The associated weight of a vertex u, υ ∈ V (G) under f is w(u) = f(u) + ∑ uυ∈E(G)f(uυ). The function f is called a local vertex irregular reflexive k-labeling if every two adjacent vertices has distinct weight. When we assign each vertex of G with a color of the vertex weight w(uυ), thus we say the graph G admits a local vertex irregular reflexive coloring. The smallest number of vertex weights needed to color the vertices of G such that no two adjacent vertices share the same color is called a local vertex irregular reflexive chromatic number, denoted by χirvs (G). Furthermore, the minimum k required such that χlrvs (G) = χ(G) is called a local reflexive vertex color strength, denoted by lrvcs(G). In this paper, we will obtain the lrvcs(G) and characterize the existence of a graph with given its local reflexive vertex color strength.

Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

Differentiable surface triangulation

Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous representation is essentially a hybrid one that decouples continuous vertex locations from the discrete topological triangulation. Unfortunately, the combinatorial nature of the triangulation prevents taking derivatives over the space of possible meshings of any given surface. As a result, to date, mesh processing and optimization techniques have been unable to truly take advantage of modular gradient descent components of modern optimization frameworks. In this work, we present a differentiable surface triangulation that enables optimization for any per-vertex or per-face differentiable objective function over the space of underlying surface triangulations. Our method builds on the result that any 2D triangulation can be achieved by a suitably perturbed weighted Delaunay triangulation. We translate this result into a computational algorithm by proposing a soft relaxation of the classical weighted Delaunay triangulation and optimizing over vertex weights and vertex locations. We extend the algorithm to 3D by decomposing shapes into developable sets and differentiably meshing each set with suitable boundary constraints. We demonstrate the efficacy of our method on various planar and surface meshes on a range of difficult-to-optimize objective functions. Our code can be found online: https://github.com/mrakotosaon/diff-surface-triangulation.

Profiling tau accumulation with SPReAD: Sub-stages for propagation regions in Alzheimer's disease.

Monitoring pathophysiological stages of Alzheimer's Disease (AD) is important for understanding, forecasting and ultimately treating the disease. One prominent example of such staging is the spread of tau protein manifested in the stage-wise accumulation of neurofibrillary tangles [1]. With the development of high-affinity tau PET tracers, it is possible to track the protein accumulation in vivo [3]. Quantification of the spatial continuity of such spread longitudinally could help disentangle the tau propagation mechanism. We introduce "Sub-stages for Propagation Regions in Alzheimer's Disease" ("SPReAD") as a method to represent Braak stage related tau tracer uptake continuously as a spatial series. To achieve such mapping, we projected tau uptake data to surfaces [2] with discrete Braak stage parcels (Fig. 1, top left), and, by assigning vertex weights recursively according to the neighbourhood relationships of adjacent stages' regions, we made a sub-stage map (Fig. 1, bottom left); then, we sampled average uptake values within these sub-stages to obtain each subject's uptake profile. We validated SPReAD using [18F]MK6240 standardized uptake value ratio (SUVR) data from 20 AD subjects at 2 visits; 5 AD subjects at 3 visits; 2 AD subjects at 4 visits, with approximately 12-month inter-visit interval, all amyloid positive; Braak stage discordant cases were excluded. The resulting sub-stage profiles appeared highly correlated across visits for AD patients (r=0.9; p<0.01). The differences between baseline and follow-up SUVRs were generally less prominent at the early stages, with especially significantly greater SUVR (p<0.05, two-sample t-test) at the follow-up at the latest Braak stage for a given subject (Fig. 1, right shows two such subjects with Braak stage VI). The Braak sub-stage profile representation using SPReAD helps identify changes within a given Braak stage and across the stages, both in terms of magnitude and spatial progression. It reduces surface manifold data into more interpretable spatial series, at the same time providing information on subtle between-stage transitions which represents a novel systematic way to quantify tau propagation patterns.

Solving Robust Weighted Independent Set Problems on Trees and under Interval Uncertainty

The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

All graph in this paper are simple, finite, and connected. Let be a labeling of a graph . The function is called antimagic rainbow edge labeling if for any two vertices and , all internal vertices in path have different weight, where the weight of vertex is the sum of its incident edges label. The vertex weight denoted by for every . If G has a antimagic rainbow edge labeling, then is a antimagic rainbow vertex connection, where the every vertex is assigned with the color . The antimagic rainbow vertex connection number of , denoted by , is the minimum colors taken over all rainbow vertex connection induced by antimagic rainbow edge labeling of . In this paper, we determined the exact value of the antimagic rainbow vertex connection number of path ( ), wheel ( ), friendship ( ), and fan ( ).

A Fast Vertex Weighting-Based Local Search for Finding Minimum Connected Dominating Sets

The minimum connected dominating set (MCDS) problem consists of selecting a minimum set of vertices from an undirected graph, such that each vertex not in this set is adjacent to at least one of the vertices in it, and the subgraph induced by this vertex set is connected. This paper presents a fast vertex weighting (FVW) algorithm for solving the MCDS problem, which integrates several distinguishing features, such as a vertex weighting-based local search with tabu and perturbation strategies to help the search to jump out of the local optima, as well as a search space reduction strategy to improve the search efficiency. Computational experiments on four sets of 112 commonly used public benchmark instances, as well as 15 newly introduced sparse instances, show that FVW is highly competitive compared with the state-of-the-art algorithms in the literature despite its simplicity. FVW improves the previous best-known results for 20 large public benchmark instances while matching the best-known results for all but 2 of the remaining ones. Several ingredients of FVW are investigated to demonstrate the importance of the proposed ideas and techniques.Summary of Contribution: As a challenging classical NP-hard problem, the minimum connected dominating set (MCDS) problem has been studied for decades in the areas of both operations research and computer science, although there does not exist an exact polynomial algorithm for solving it. Thus, the new breakthrough on this classical NP-hard problem in terms of the computational results on classical benchmark instances is significant. This paper presents a new fast vertex weighting local search for solving the MCDS problem. Computational experiments on four sets of 112 commonly used public benchmark instances show that fast vertex weighting (FVW) is able to improve the previous best-known results for 20 large instances while matching the best-known results for all but 2 of the remaining instances. Several ingredients of FVW are also investigated to demonstrate the importance of the proposed ideas and techniques.

Clustering Vertex-Weighted Graphs by Spectral Methods

Spectral techniques are often used to partition the set of vertices of a graph, or to form clusters. They are based on the Laplacian matrix. These techniques allow easily to integrate weights on the edges. In this work, we introduce a p-Laplacian, or a generalized Laplacian matrix with potential, which also allows us to take into account weights on the vertices. These vertex weights are independent of the edge weights. In this way, we can cluster with the importance of vertices, assigning more weight to some vertices than to others, not considering only the number of vertices. We also provide some bounds, similar to those of Chegeer, for the value of the minimal cut cost with weights at the vertices, as a function of the first non-zero eigenvalue of the p-Laplacian (an analog of the Fiedler eigenvalue).