198,148 research outputs found
Transformers Meet Directed Graphs
Transformers were originally proposed as a sequence-to-sequence model for
text but have become vital for a wide range of modalities, including images,
audio, video, and undirected graphs. However, transformers for directed graphs
are a surprisingly underexplored topic, despite their applicability to
ubiquitous domains, including source code and logic circuits. In this work, we
propose two direction- and structure-aware positional encodings for directed
graphs: (1) the eigenvectors of the Magnetic Laplacian - a direction-aware
generalization of the combinatorial Laplacian; (2) directional random walk
encodings. Empirically, we show that the extra directionality information is
useful in various downstream tasks, including correctness testing of sorting
networks and source code understanding. Together with a data-flow-centric graph
construction, our model outperforms the prior state of the art on the Open
Graph Benchmark Code2 relatively by 14.7%.Comment: 29 page
Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities
Many complex networks display a mesoscopic structure with groups of nodes
sharing many links with the other nodes in their group and comparatively few
with nodes of different groups. This feature is known as community structure
and encodes precious information about the organization and the function of the
nodes. Many algorithms have been proposed but it is not yet clear how they
should be tested. Recently we have proposed a general class of undirected and
unweighted benchmark graphs, with heterogenous distributions of node degree and
community size. An increasing attention has been recently devoted to develop
algorithms able to consider the direction and the weight of the links, which
require suitable benchmark graphs for testing. In this paper we extend the
basic ideas behind our previous benchmark to generate directed and weighted
networks with built-in community structure. We also consider the possibility
that nodes belong to more communities, a feature occurring in real systems,
like, e. g., social networks. As a practical application, we show how
modularity optimization performs on our new benchmark.Comment: 9 pages, 13 figures. Final version published in Physical Review E.
The code to create the benchmark graphs can be freely downloaded from
http://santo.fortunato.googlepages.com/inthepress
Nonasymptotic Convergence Rates for Cooperative Learning Over Time-Varying Directed Graphs
We study the problem of distributed hypothesis testing with a network of
agents where some agents repeatedly gain access to information about the
correct hypothesis. The group objective is to globally agree on a joint
hypothesis that best describes the observed data at all the nodes. We assume
that the agents can interact with their neighbors in an unknown sequence of
time-varying directed graphs. Following the pioneering work of Jadbabaie,
Molavi, Sandroni, and Tahbaz-Salehi, we propose local learning dynamics which
combine Bayesian updates at each node with a local aggregation rule of private
agent signals. We show that these learning dynamics drive all agents to the set
of hypotheses which best explain the data collected at all nodes as long as the
sequence of interconnection graphs is uniformly strongly connected. Our main
result establishes a non-asymptotic, explicit, geometric convergence rate for
the learning dynamic
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