Linear Approximate Groups

This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as \SL_n(k). For example, generalising a result of Helfgott (who handled the cases $n = 2$ and 3), we show that any approximate subgroup of \SL_n(\F_q) which generates the group must be either very small or else nearly all of \SL_n(\F_q). The argument is valid for all Chevalley groups G(\F_q).Comment: 11 pages. Submitted, Electronic Research Announcements. Small change

Traffic Analysis in Random Delaunay Tessellations and Other Graphs

In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erd\"os-Renyi random graphs, geometric random graphs, expanders and random $k$-regular graphs. Moreover we show that adding a random matching to the original graph can considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

Robustness of Random Graphs Based on Natural Connectivity

Recently, it has been proposed that the natural connectivity can be used to efficiently characterise the robustness of complex networks. Natural connectivity quantifies the redundancy of alternative routes in a network by evaluating the weighted number of closed walks of all lengths and can be regarded as the average eigenvalue obtained from the graph spectrum. In this article, we explore the natural connectivity of random graphs both analytically and numerically and show that it increases linearly with the average degree. By comparing with regular ring lattices and random regular graphs, we show that random graphs are more robust than random regular graphs; however, the relationship between random graphs and regular ring lattices depends on the average degree and graph size. We derive the critical graph size as a function of the average degree, which can be predicted by our analytical results. When the graph size is less than the critical value, random graphs are more robust than regular ring lattices, whereas regular ring lattices are more robust than random graphs when the graph size is greater than the critical value.Comment: 12 pages, 4 figure

Decompositions of Triangle-Dense Graphs

High triangle density -- the graph property stating that a constant fraction of two-hop paths belong to a triangle -- is a common signature of social networks. This paper studies triangle-dense graphs from a structural perspective. We prove constructively that significant portions of a triangle-dense graph are contained in a disjoint union of dense, radius 2 subgraphs. This result quantifies the extent to which triangle-dense graphs resemble unions of cliques. We also show that our algorithm recovers planted clusterings in approximation-stable k-median instances.Comment: 20 pages. Version 1->2: Minor edits. 2->3: Strengthened {\S}3.5, removed appendi

Random Metric Spaces and Universality

WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S_{\infty}-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so called anarchical uniformly distributed sequences. We give examples of other categories in which the randomness and universality coincide (graph, etc.).Comment: 38 PAGE

Deep Expander Networks: Efficient Deep Networks from Graph Theory

Efficient CNN designs like ResNets and DenseNet were proposed to improve accuracy vs efficiency trade-offs. They essentially increased the connectivity, allowing efficient information flow across layers. Inspired by these techniques, we propose to model connections between filters of a CNN using graphs which are simultaneously sparse and well connected. Sparsity results in efficiency while well connectedness can preserve the expressive power of the CNNs. We use a well-studied class of graphs from theoretical computer science that satisfies these properties known as Expander graphs. Expander graphs are used to model connections between filters in CNNs to design networks called X-Nets. We present two guarantees on the connectivity of X-Nets: Each node influences every node in a layer in logarithmic steps, and the number of paths between two sets of nodes is proportional to the product of their sizes. We also propose efficient training and inference algorithms, making it possible to train deeper and wider X-Nets effectively. Expander based models give a 4% improvement in accuracy on MobileNet over grouped convolutions, a popular technique, which has the same sparsity but worse connectivity. X-Nets give better performance trade-offs than the original ResNet and DenseNet-BC architectures. We achieve model sizes comparable to state-of-the-art pruning techniques using our simple architecture design, without any pruning. We hope that this work motivates other approaches to utilize results from graph theory to develop efficient network architectures.Comment: ECCV'1