35 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Applications of Centrality Measures and Extremal Combinatorics

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    Centrality measures assign numbers or rankings to network nodes that reflect their importance. There are many types of centrality measures, each suitable for different types of networks and applications. In Chapter 2, we consider a model of astronaut health during a space mission. Katz centrality is commonly used to measure the influence of nodes in social and biological networks. We motivate its use in this application to estimate the expected quality time lost due to the progression of medical conditions. In Chapter 3, we find dominating sets in satellite networks. To do this, we use the Shapley value, a centrality measure that originates in game theory and is computed based only on local network information. We demonstrate that the Shapley value is an effective and time-efficient tool for finding small dominating sets in various random graph families and a set of real-world networks. In Chapter 4, we introduce a novel algorithm for categorizing which nodes are nearest the boundary, called boundary nodes, in a network that uses Chvátal’s definition of a line in a graph. We test this algorithm on random geometric graphs and compare its effectiveness to other known methods for boundary node detection. In Chapter 5, for certain sets S and equations eq, we look for the smallest number of colors rb(S, eq) such that for every surjective rb(S, eq)-coloring of S, there exists a solution to eq where every element of the solution set is assigned a distinct color. We show that rb([n], x_1 + x_2 = x_3) = ⌊log_2(m) + 2⌋ and rb([m] × [n], x_1 + x_2 = x_3) = m + n + 1 for m, n \u3e 1. In Chapter 6, a graph G is H-semi-saturated if adding an edge e to G that is not currently in G causes H to appear as a subgraph in G that contains e. We say that G is H-saturated if G does not contain H as a subgraph before adding e. The smallest number of edges in an H-semi-saturated (H-saturated) graph is called the semi-saturation number of H (saturation number of H). We show that the saturation and semi-saturation numbers differ by at least 1 for a disjoint union of paths called a linear forest. Additionally, we find graph families for which the saturation number is monotonic with respect to the subgraph relation

    Sampling uniform hypergraphs with given degrees

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    Graphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple k-uniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a black-box bipartite graph sampler. This algorithm was presented in a paper by myself and co-authors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple k-uniform hypergraphs with degree sequence d. Our algorithm uses a black-box sampler A for producing (possibly non-simple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is near-regular. This extends the range of degree sequences for which efficient sampling schemes are known

    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum

    Embedding problems in graphs and hypergraphs

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    In this thesis, we explore several mathematical questions about substructures in graphs and hypergraphs, focusing on algorithmic methods and notions of regularity for graphs and hypergraphs. We investigate conditions for a graph to contain powers of paths and cycles of arbitrary specified linear lengths. Using the well-established graph regularity method, we determine precise minimum degree thresholds for sufficiently large graphs and show that the extremal behaviour is governed by a family of explicitly given extremal graphs. This extends an analogous result of Allen, Böttcher and Hladký for squares of paths and cycles of arbitrary specified linear lengths and confirms a conjecture of theirs. Given positive integers k and j with j < k, we study the length of the longest j-tight path in the binomial random k-uniform hypergraph Hk(n, p). We show that this length undergoes a phase transition from logarithmic to linear and determine the critical threshold for this phase transition. We also prove upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm finds a long j-tight path. Finally, we investigate the embedding of bounded degree hypergraphs into large sparse hypergraphs. The blow-up lemma is a powerful tool for embedding bounded degree spanning subgraphs with wide-ranging applications in extremal graph theory. We prove a sparse hypergraph analogue of the blow-up lemma, showing that large sparse partite complexes with sufficiently regular small subcomplex counts and no atypical vertices behave as if they were complete for the purpose of embedding complexes with bounded degree and bounded partite structure

    Proceedings of the Fourth Russian Finnish Symposium on Discrete Mathematics

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Global hypercontractivity and its applications

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    The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general pp-biased measures. However, simple examples show that when p=o(1)p = o(1), there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general pp that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a pp-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556
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