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Frequency Assignment Problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Furt ..."
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Cited by 42 (3 self)
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The ever growing number of wireless communications systems deployed around the globe have made the optimal assignment of a limited radio frequency spectrum a problem of primary importance. At issue are planning models for permanent spectrum allocation, licensing, regulation, and network design. Further at issue are online algorithms for dynamically assigning frequencies to users within an established network. Applications include aeronautical mobile, land mobile, maritime mobile, broadcast, land fixed (pointto point), and satellite systems. This paper surveys research conducted by theoreticians, engineers, and computer scientists regarding the frequency assignment problem (FAP) in all of its guises. The paper begins by defining some of the more common types of FAPs. It continues with a discussion on measures of optimality relating to the use of spectrum, models of interference, and mathematical representations of the many FAPs, both in graph theoretic terms, and as mathematical pro...
First passage percolation on random graphs with finite mean degrees
, 2009
"... We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the numb ..."
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Cited by 26 (7 self)
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We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the socalled hopcount. We analyze the configuration model with degree powerlaw exponent τ> 2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of powerlaw form with exponent τ − 1> 1, or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while the variance is finite for τ> 3, but infinite for τ ∈ (2, 3). We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to α log n, where α ∈ (0, 1) for τ ∈ (2, 3), while α> 1 for τ> 3. Here n denotes the size of the graph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path, and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it is proved that the hopcount remains uniformly bounded and converges in distribution.
Extreme value theory, PoissonDirichlet distributions and FPP on random networks
, 2009
"... We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end, we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of ..."
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Cited by 15 (4 self)
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We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end, we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal weight path, which can be computed in terms of the PoissonDirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, namely, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, as well as the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the meanfield setting or the locally treelike setting, which occurs as τ> 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that one can transfer information remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models, see [3, 5, 6].
Edge Flows in the Complete RandomLengths Network
, 2007
"... Consider the complete nvertex graph whose edgelengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the n → ∞ limit we find explicitly the emp ..."
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Cited by 9 (4 self)
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Consider the complete nvertex graph whose edgelengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some random total flow. In the n → ∞ limit we find explicitly the empirical distribution of these edgeflows, suitably normalized.
First Passage percolation on locally tree like networks I: Dense random graphs
, 2007
"... We study various properties of least cost paths under iid disorder for the complete graph and dense ErdosRenyii random graphs in the connected phase, with iid exponential and uniform weights on edges. Using a simple heuristic, we compute explicitly, limiting distributions for (properly recentered) ..."
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Cited by 5 (4 self)
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We study various properties of least cost paths under iid disorder for the complete graph and dense ErdosRenyii random graphs in the connected phase, with iid exponential and uniform weights on edges. Using a simple heuristic, we compute explicitly, limiting distributions for (properly recentered) lengths of shortest paths between typical nodes, as well as multiple source destination pairs; we also derive asymptotics for the number of edges on the shortest path, namely the hopcount and find that the addition of edge weights converts these graphs from ultrasmall world networks to small world networks. Finally we study the VickreyClarkeGrooves measure of overpayment for the complete graph with exponential edge weights and show that the complete graph is far from monopolistic for large n. Key words. VickreyClarkeGrooves measure of overpayment, flow, random graph, random network, first passage percolation, Cox point process, hopcount, Yule process
Notes on random optimization problems
, 2008
"... These notes are under construction. They constitute a combination of what I have said in the lectures, what I will say in future lectures, and what I will not say due to time constraints. Some sections are very brief, and this is generally because they are not yet written. Some of the “problems and ..."
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These notes are under construction. They constitute a combination of what I have said in the lectures, what I will say in future lectures, and what I will not say due to time constraints. Some sections are very brief, and this is generally because they are not yet written. Some of the “problems and exercises” describe things that I am actually going to write down in detail in the text. This is because I have used the problems & exercises section in this way to take short notes of things I should not forget to mention.
DEGREE DISTRIBUTION OF SHORTEST PATH TREES AND BIAS OF NETWORK SAMPLING ALGORITHMS
"... Abstract. In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the powerlaw exponent of the degree distribution of this tree and compare it to the de ..."
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Abstract. In this article, we explicitly derive the limiting distribution of the degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the powerlaw exponent of the degree distribution of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic meanfield model of distance) as well as on the configuration model with edgeweights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree powerlaw exponent. We also consider random rregular graphs for large r, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean field model of distance. We use our results to explain an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [7, 8, 6] of analyzing the effect of attaching random edge lengths on the geometry of random network models. 1.