42,768 research outputs found
On Large-Scale Graph Generation with Validation of Diverse Triangle Statistics at Edges and Vertices
Researchers developing implementations of distributed graph analytic
algorithms require graph generators that yield graphs sharing the challenging
characteristics of real-world graphs (small-world, scale-free, heavy-tailed
degree distribution) with efficiently calculable ground-truth solutions to the
desired output. Reproducibility for current generators used in benchmarking are
somewhat lacking in this respect due to their randomness: the output of a
desired graph analytic can only be compared to expected values and not exact
ground truth. Nonstochastic Kronecker product graphs meet these design criteria
for several graph analytics. Here we show that many flavors of triangle
participation can be cheaply calculated while generating a Kronecker product
graph. Given two medium-sized scale-free graphs with adjacency matrices and
, their Kronecker product graph has adjacency matrix . Such
graphs are highly compressible: edges are represented in memory and can be built in a distributed setting from
small data structures, making them easy to share in compressed form. Many
interesting graph calculations have worst-case complexity bounds and often these are reduced to
for Kronecker product graphs, when a Kronecker formula can be derived yielding
the sought calculation on in terms of related calculations on and .
We focus on deriving formulas for triangle participation at vertices, , a vector storing the number of triangles that every vertex is involved
in, and triangle participation at edges, , a sparse matrix storing
the number of triangles at every edge.Comment: 10 pages, 7 figures, IEEE IPDPS Graph Algorithms Building Block
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
Graph theory models the Internet mathematically, and a number of plausible mathematically intersecting network models for the Internet have been developed and studied. Simultaneously, Internet researchers have developed methodology to use real data to validate, or invalidate, proposed Internet models. The authors look at these parallel developments, particularly as they apply to scale-free network models of the preferential attachment type
Systematic Topology Analysis and Generation Using Degree Correlations
We present a new, systematic approach for analyzing network topologies. We
first introduce the dK-series of probability distributions specifying all
degree correlations within d-sized subgraphs of a given graph G. Increasing
values of d capture progressively more properties of G at the cost of more
complex representation of the probability distribution. Using this series, we
can quantitatively measure the distance between two graphs and construct random
graphs that accurately reproduce virtually all metrics proposed in the
literature. The nature of the dK-series implies that it will also capture any
future metrics that may be proposed. Using our approach, we construct graphs
for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing
accuracy, important properties of measured and modeled Internet topologies. We
find that the d=2 case is sufficient for most practical purposes, while d=3
essentially reconstructs the Internet AS- and router-level topologies exactly.
We hope that a systematic method to analyze and synthesize topologies offers a
significant improvement to the set of tools available to network topology and
protocol researchers.Comment: Final versio
Gravity-Inspired Graph Autoencoders for Directed Link Prediction
Graph autoencoders (AE) and variational autoencoders (VAE) recently emerged
as powerful node embedding methods. In particular, graph AE and VAE were
successfully leveraged to tackle the challenging link prediction problem,
aiming at figuring out whether some pairs of nodes from a graph are connected
by unobserved edges. However, these models focus on undirected graphs and
therefore ignore the potential direction of the link, which is limiting for
numerous real-life applications. In this paper, we extend the graph AE and VAE
frameworks to address link prediction in directed graphs. We present a new
gravity-inspired decoder scheme that can effectively reconstruct directed
graphs from a node embedding. We empirically evaluate our method on three
different directed link prediction tasks, for which standard graph AE and VAE
perform poorly. We achieve competitive results on three real-world graphs,
outperforming several popular baselines.Comment: ACM International Conference on Information and Knowledge Management
(CIKM 2019
Spatial and performance optimality in power distribution networks
(c) 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Complex network theory has been widely used in vulnerability analysis of power networks, especially for power transmission ones. With the development of the smart grid concept, power distribution networks are becoming increasingly relevant. In this paper, we model power distribution systems as spatial networks. Topological and spatial properties of 14 European power distribution networks are analyzed, together with the relationship between geographical constraints and performance optimization, taking into account economic and vulnerability issues. Supported by empirical reliability data, our results suggest that power distribution networks are influenced by spatial constraints which clearly affect their overall performance.Peer ReviewedPostprint (author's final draft
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