Degree sums and dense spanning trees

Li, Tao; Gao, Yingqi; Dong, Qiankun; Wang, Hua

doi:10.1371/journal.pone.0184912

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…Common generalizations of this optimization problem for the undirected and unweighted case include the dense, sparse, and minimum routing cost spanning tree problems. Many other variations of optimal spanning trees exist [15,16,[27][28][29][30][31][32][33][34][35][36], and such trees are relevant to an ever widening diversity of applications from optical network design [14,37] to networked oscillator synchronization [38].…”

Section: Introductionmentioning

confidence: 99%

Spanning Trees of Recursive Scale-Free Graphs

Diggans,

Bollt,

ben-Avraham

2021

Preprint

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We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties of the ensemble of spanning trees to be analytically solved exactly. We show how a judicious application of the prescribed growth rules selects for certain subsets of the spanning trees with particular desired properties (small-world or extended diameter, degree distribution, etc.), and thus provides solutions to several optimization problems. The analysis of spanning trees enhances the usefulness of recursive graphs as sophisticated models for everyday life complex networks.

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Section: Introductionmentioning

confidence: 99%

Spanning Trees of Recursive Scale-Free Graphs

Diggans,

Bollt,

ben-Avraham

2021

Preprint

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“…Seeking the spanning tree of a given graph structure is a classic problem that has numerous applications and variations. For some examples of such study one may see [1,2,3,4,5,6,7,8,9]. In a weighted graph, finding the spanning tree with minimum total weight is known as the minimum spanning tree problem and is probably one of the most extensively studied problems.…”

Section: Introductionmentioning

confidence: 99%

Globally Optimal Dense and Sparse Spanning Trees, and their Applications

Ozen

Lešaja

Wang

2020

Stat., optim. inf. comput.

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Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of dense ( sparse ) tree, and in particular spanning tree (DST and SST respectively), is introduced as the structure that have a large (small) number of subtrees, or small (large) sum of distances between vertices. We show that finding DST and SST reduces to solving the discrete optimization problems. New and efficient approaches to find such spanning trees is achieved by imposing certain conditions on the vertex degrees which are then used to define an objective function that is minimized over all spanning trees of the graph under consideration. Solving this minimization problem exactly may be prohibitively time consuming for large graphs. Hence, we propose to use genetic algorithm (GA) which is one of well known metaheuristics methods to solve DST and SST approximately. As far as we are aware this is the first time GA has been used in this context.We also demonstrate on a number of applications that GA approach is well suited for these types of problems both in computational efficiency and accuracy of the approximate solution. Furthermore, we improve the efficiency of the proposed method by using Kruskal s algorithm in combination with GA. The application of our methods to several practical large graphs and networks is presented. Computational results show that they perform faster than previously proposed heuristic methods and produce more accurate solutions. Furthermore, the new feature of the proposed approach is that it can be applied recursively to sub-trees or spanning trees with additional constraints in order to further investigate the graphical properties of the graph and/or network. The application of this methodology on the gene network of a cancer cell led to isolating key genes in a network that were not obvious from previous studies.

show abstract

“…Seeking the spanning tree of a given graph structure is a classic problem that has numerous applications and variations. For some examples of such study one may see [1][2][3][4][5][6][7][8][9]. In a weighted graph, finding the spanning tree with minimum total weight is known as the minimum spanning tree problem and is probably one of the most extensively studied problems.…”

Section: Introductionmentioning

confidence: 99%

“…Alternatively, minimizing a condition such as C j can be used to find a Sparse Spanning Tree (SST) in the given graph. In [9], through computational analysis, it was observed that for j = 4, 2, 0, 0 or 4, 2, 2, 0 or 4, 2, 2, 2 the corresponding objective functions…”

Section: Introductionmentioning

confidence: 99%

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Globally optimal dense and sparse spanning trees, and their applications

Ozen,

Lesaja,

Wang

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Finding spanning trees under various constraints is a classic problem with applications in many fields. Recently, a novel notion of "dense" ("sparse") tree, and in particular spanning tree (DST and SST respectively), is introduced as the structure that have a large (small) number of subtrees, or small (large) sum of distances between vertices. We show that finding DST and SST reduces to solving the discrete optimization problems. New and efficient approaches to find such spanning trees is achieved by imposing certain conditions on the vertex degrees which are then used to define an objective function that is minimized over all spanning trees of the graph under consideration. Solving this minimization problem exactly may be prohibitively time consuming for large graphs. Hence, we propose to use genetic algorithm (GA) which is one of well known metaheuristics methods to solve DST and SST approximately. As far as we are aware this is the first time GA has been used in this context. We also demonstrate on a number of applications that GA approach is well suited for these types of problems both in computational efficiency and accuracy of the approximate solution. Furthermore, we improve the efficiency of the proposed method by using Kruskal's algorithm in combination with GA.The application of our methods to several practical large graphs and networks is presented. Computational results show that they perform faster than previously proposed heuristic methods and produce more accurate solutions. Furthermore, the new feature of the proposed approach is that it can be applied recursively to sub-trees or spanning trees with additional constraints in order to further investigate the graphical properties of the graph and/or network. The application of this methodology on the gene network of a cancer cell led to isolating key genes in a network that were not obvious from previous studies.

show abstract

Degree sums and dense spanning trees

Cited by 4 publications

References 14 publications

Spanning Trees of Recursive Scale-Free Graphs

Spanning Trees of Recursive Scale-Free Graphs

Globally Optimal Dense and Sparse Spanning Trees, and their Applications

Globally optimal dense and sparse spanning trees, and their applications

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