3 research outputs found
Cell Attention Networks
Since their introduction, graph attention networks achieved outstanding
results in graph representation learning tasks. However, these networks
consider only pairwise relationships among nodes and then they are not able to
fully exploit higher-order interactions present in many real world data-sets.
In this paper, we introduce Cell Attention Networks (CANs), a neural
architecture operating on data defined over the vertices of a graph,
representing the graph as the 1-skeleton of a cell complex introduced to
capture higher order interactions. In particular, we exploit the lower and
upper neighborhoods, as encoded in the cell complex, to design two independent
masked self-attention mechanisms, thus generalizing the conventional graph
attention strategy. The approach used in CANs is hierarchical and it
incorporates the following steps: i) a lifting algorithm that learns {\it edge
features} from {\it node features}; ii) a cell attention mechanism to find the
optimal combination of edge features over both lower and upper neighbors; iii)
a hierarchical {\it edge pooling} mechanism to extract a compact meaningful set
of features. The experimental results show that CAN is a low complexity
strategy that compares favorably with state of the art results on graph-based
learning tasks.Comment: Preprint, under revie
Succinct Data Structures for Chordal Graphs
We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time.
We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries:
whether two vertices u,v are adjacent in time f(n) for any f(n) \in \omega(1).
the degree of a vertex in O(1) time.
the vertices adjacent to u in O(f(n)^2) time per neighbour
the length of the shortest path from u to v in O(n f(n)) tim
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version