44 research outputs found
Bounding the Number of Hyperedges in Friendship -Hypergraphs
For , an -uniform hypergraph is called a friendship
-hypergraph if every set of vertices has a unique 'friend' - that
is, there exists a unique vertex with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
We also obtain a new upper bound on the number of hyperedges in a friendship
-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when .Comment: 14 page
ON THE NUMBER OF CYCLES OF GRAPHS AND VC-DIMENSION
The number of the cycles in a graph is an important well-known parameter in graph theory and there are a lot of investigations carried out in the literature for finding suitable bounds for it. In this paper, we delve into studying this parameter and the cycle structure of graphs through the lens of the cycle hypergraphs and VC-dimension and find some new bounds for it, where the cycle hypergraph of a graph is a hypergraph with the edges of the graph as its vertices and the edge sets of the cycles as its hyperedges respectively. Note that VC-dimension is an important notion in extremal combinatorics, graph theory, statistics and machine learning. We investigate cycle hypergraph from the perspective of VC-theory, specially the celebrated Sauer-Shelah lemma, in order to give our upper and lower bounds for the number of the cycles in terms of the (dual) VC-dimension of the cycle hypergraph and nullity of graph. We compute VC-dimension and the mentioned bounds in some graph classes and also show that in certain classes, our bounds are sharper than many previous ones in the literature
Many copies in -free graphs
For two graphs and with no isolated vertices and for an integer ,
let denote the maximum possible number of copies of in an
-free graph on vertices. The study of this function when is a
single edge is the main subject of extremal graph theory. In the present paper
we investigate the general function, focusing on the cases of triangles,
complete graphs, complete bipartite graphs and trees. These cases reveal
several interesting phenomena. Three representative results are:
(i)
(ii) For any fixed , and ,
and
(iii) For any two trees and , where
is an integer depending on and (its precise definition is
given in Section 1).
The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri.
The proofs combine combinatorial and probabilistic arguments with simple
spectral techniques
Counting and Sampling Small Structures in Graph and Hypergraph Data Streams
In this thesis, we explore the problem of approximating the number of elementary substructures called simplices in large k-uniform hypergraphs. The hypergraphs are assumed to be too large to be stored in memory, so we adopt a data stream model, where the hypergraph is defined by a sequence of hyperedges.
First we propose an algorithm that (ε, δ)-estimates the number of simplices using O(m1+1/k / T) bits of space. In addition, we prove that no constant-pass streaming algorithm can (ε, δ)- approximate the number of simplices using less than O( m 1+1/k / T ) bits of space. Thus we resolve the space complexity of the simplex counting problem by providing an algorithm that matches the lower bound.
Second, we examine the triangle counting question –a hypergraph where k = 2. We develop and analyze an almost optimal O (n+m 3/2 / T) triangle-counting algorithm based on ideas introduced in [KMPT12]. The proposed algorithm is subsequently used to establish a method for uniformly sampling triangles in a graph stream using O(m 3/2 / T) bits of space, which beats the state-of-the-art O(mn / T) algorithm given by [PTTW13
Tensor Recovery in High-Dimensional Ising Models
The -tensor Ising model is an exponential family on a -dimensional
binary hypercube for modeling dependent binary data, where the sufficient
statistic consists of all -fold products of the observations, and the
parameter is an unknown -fold tensor, designed to capture higher-order
interactions between the binary variables. In this paper, we describe an
approach based on a penalization technique that helps us recover the signed
support of the tensor parameter with high probability, assuming that no entry
of the true tensor is too close to zero. The method is based on an
-regularized node-wise logistic regression, that recovers the signed
neighborhood of each node with high probability. Our analysis is carried out in
the high-dimensional regime, that allows the dimension of the Ising model,
as well as the interaction factor to potentially grow to with the
sample size . We show that if the minimum interaction strength is not too
small, then consistent recovery of the entire signed support is possible if one
takes samples, where denotes
the maximum degree of the hypernetwork in question. Our results are validated
in two simulation settings, and applied on a real neurobiological dataset
consisting of multi-array electro-physiological recordings from the mouse
visual cortex, to model higher-order interactions between the brain regions.Comment: 28 pages, 7 figure
Alignment and integration of complex networks by hypergraph-based spectral clustering
Complex networks possess a rich, multi-scale structure reflecting the
dynamical and functional organization of the systems they model. Often there is
a need to analyze multiple networks simultaneously, to model a system by more
than one type of interaction or to go beyond simple pairwise interactions, but
currently there is a lack of theoretical and computational methods to address
these problems. Here we introduce a framework for clustering and community
detection in such systems using hypergraph representations. Our main result is
a generalization of the Perron-Frobenius theorem from which we derive spectral
clustering algorithms for directed and undirected hypergraphs. We illustrate
our approach with applications for local and global alignment of
protein-protein interaction networks between multiple species, for tripartite
community detection in folksonomies, and for detecting clusters of overlapping
regulatory pathways in directed networks.Comment: 16 pages, 5 figures; revised version with minor corrections and
figures printed in two-column format for better readability; algorithm
implementation and supplementary information available at Google code at
http://schype.googlecode.co
On structural and temporal credit assignment in reinforcement learning
Reinforcement learning, or learning how to map situations to actions that maximise a numerical reward signal, poses two fundamental interdependent problems: exploration and credit assignment. The exploration problem concerns an agent's ability to discover useful experiences. The credit assignment problem pertains to an agent's ability to incorporate the discovered experiences. The latter comprises two distinct subproblems itself: structural and temporal credit assignment. The structural credit assignment problem involves determining how to assign credit for the outcome of an action to the many component structures, or internal decisions, that could have been involved in producing that action. The temporal credit assignment problem has to do with determining how to assign credit for outcomes of a sequence of experiences to the actions that could have contributed to those outcomes. In this thesis, we broadly study the credit assignment problem in reinforcement learning, making contributions to each of its subproblems in isolation.
In the first part of this thesis we address the reinforcement learning problem in environments with multi-dimensional discrete action spaces, a problem setting that plagues structural credit assignment, or generalisation, due to the Bellman's curse of dimensionality. We argue that leveraging the combinatorial structure of such action spaces is crucial for achieving rapid generalisation from limited data. To this end, we introduce two approaches for estimating action values that feature a capacity for leveraging such structures, in each case empirically validating that significant performance improvements in sample complexity can be gained. Furthermore, we demonstrate that our approaches unleash significant benefits concerning space and time complexity, thus allowing them to successfully scale to high-dimensional discrete action spaces where the conventional approach becomes computationally intractable.
In the second part of this thesis we address the temporal credit assignment problem. Specifically, we identify and analyse general training scenarios where appropriate temporal credit assignment is hindered by the mishandling of time limits or by the choice of discount factor. To address the first matter, we formalise the ways in which time limits may be interpreted in reinforcement learning and how they should be handled in each case accordingly. To address the second matter, we produce a possible explanation for why the performance of low discount factors tends to fall flat when used in conjunction with function approximation. In turn, this leads us to develop a method that enables a much larger range of discount factors by rectifying the hypothesised root cause.Open Acces