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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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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 “smallworld 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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Random graph theory is used to examine the “smallworld 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 2colourings (equal numbers of red and blue). We show that the general problem is NPhard but there is a polynomialtime 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 scalefree networks, where the probability of linking to a node is proportional to the number of its preexisting links. In both case ..."
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cases, new nodes are linked to m = 1 nodes. Average nodenode 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 nodenode distance d for large N
Minimum Average Distance Triangulations
"... 5. The (NP) hard problem Given X = {p1,...,pn} points in the plane and weights w: X 2 → R, find a geometric, crossingfree graph T embedded on X with edge weights given by w, such as to minimize: W(T) = ∑ 1≤i<j≤n where d T is the graphtheoretic distance using T. dT(pi,pj) The solution is always ..."
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5. The (NP) hard problem Given X = {p1,...,pn} points in the plane and weights w: X 2 → R, find a geometric, crossingfree graph T embedded on X with edge weights given by w, such as to minimize: W(T) = ∑ 1≤i<j≤n where d T is the graphtheoretic distance using T. dT(pi,pj) The solution
Counterexample to regularity in averagedistance problem
 OF MINIMIZERS OF AVERAGEDISTANCE PROBLEM 21
, 2012
"... The averagedistance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a onedimensional 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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The averagedistance problem is to find the best way to approximate (or represent) a given measure µ on Rd by a onedimensional 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
AVERAGEDISTANCE PROBLEM FOR PARAMETERIZED CURVES
"... ABSTRACT. 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 i ..."
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in R, Γγ = γ(I), and d(x,Γγ) is the distance of x to Γγ. The problem is closely related to the averagedistance 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
Average Distance, Diameter, and Clustering in Social Networks with Homophily
 IN PROCEEDINGS OF THE WORKSHOP ON INTERNET AND NETWORK ECONOMICS
, 2008
"... I examine a random network model where nodes are categorized by type and linking probabilities can differ across types. I show that as homophily increases (so that the probability to link to other nodes of the same type increases and the probability of linking to nodes of some other types decreases) ..."
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) the average distance and diameter of the network are unchanged, while the average clustering in the network increases.
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