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Average distance in weighted graphs
- Discrete Math
"... Dedicated to Gert Sabidussi on the occasion of his 80th birthday We consider the following generalisation of the average distance of a graph. Let G be a connected, finite graph with a nonnegative vertex weight function c. Let N be the total weight of the vertices. If N ̸ = 0, 1, then the weighted av ..."
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Cited by 3 (0 self)
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Dedicated to Gert Sabidussi on the occasion of his 80th birthday We consider the following generalisation of the average distance of a graph. Let G be a connected, finite graph with a nonnegative vertex weight function c. Let N be the total weight of the vertices. If N ̸ = 0, 1, then the weighted
The Average Distance in a Random Graph with Given Expected Degrees
"... Random graph theory is used to examine the “small-world phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d whe ..."
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Cited by 289 (13 self)
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Random graph theory is used to examine the “small-world phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d
Average Distance in Coloured Graphs
"... For a graph G where the vertices are coloured, the coloured distance of G is defined as the sum of the distances between all unordered pairs of vertices having different colours. Then for a fixed supply s of colours, ds(G) is defined as the minimum coloured distance over all colourings with s. This ..."
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. This generalizes the concepts of median and average distance. In this paper we explore bounds on this parameter especially a natural lower bound and the particular case of balanced 2-colourings (equal numbers of red and blue). We show that the general problem is NP-hard but there is a polynomial-time algorithm
Abstract Average Distances of Pyramid Networks
"... For an interconnection network, calculating average distance of it is in general more difficult than determining its diameter. Diameters of pyramid networks are well known. This study calculates average distances of pyramid networks. ..."
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For an interconnection network, calculating average distance of it is in general more difficult than determining its diameter. Diameters of pyramid networks are well known. This study calculates average distances of pyramid networks.
Average distance in growing trees
, 2003
"... Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barabási–Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both case ..."
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cases, new nodes are linked to m = 1 nodes. Average node-node distance d is calculated numerically in evolving trees as dependent on the number of nodes N. The results for N not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance d for large N
Counterexample to regularity in average-distance problem
- OF MINIMIZERS OF AVERAGE-DISTANCE PROBLEM 21
, 2012
"... The average-distance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure µ, minimize E(Σ) = d(x, Σ)dµ(x) + λH 1 (Σ ..."
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Cited by 7 (2 self)
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The average-distance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure µ, minimize E(Σ) = d(x, Σ)dµ(x) + λH 1
AVERAGE-DISTANCE PROBLEM FOR PARAMETERIZED CURVES
, 2014
"... We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure µ, with µ(Rd)> 0 for p ≥ 1 and λ> 0 we consider the functional E(γ) = Rd d(x,Γγ) pdµ(x) + λLength(γ) where γ: I → Rd, I is an interval in R, Γγ ..."
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Cited by 1 (1 self)
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= γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form
AVERAGE-DISTANCE PROBLEM FOR PARAMETERIZED CURVES
"... ABSTRACT. We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure µ, for p ≥ 1 and λ> 0 we consider the functional E(γ) = Rd d(x,Γγ) pdµ(x) + λLength(γ) where γ: I → Rd, I is an interval in R, Γγ ..."
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= γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form
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