76 research outputs found
Towards an approximate graph entropy measure for identifying incidents in network event data
A key objective of monitoring networks is to identify potential service threatening outages from events within the network before service is interrupted. Identifying causal events, Root Cause Analysis (RCA), is an active area of research, but current approaches are vulnerable to scaling issues with high event rates. Elimination of noisy events that are not causal is key to ensuring the scalability of RCA. In this paper, we introduce vertex-level measures inspired by Graph Entropy and propose their suitability as a categorization metric to identify nodes that are a priori of more interest as a source of events. We consider a class of measures based on Structural, Chromatic and Von Neumann Entropy. These measures require NP-Hard calculations over the whole graph, an approach which obviously does not scale for large dynamic graphs that characterise modern networks. In this work we identify and justify a local measure of vertex graph entropy, which behaves in a similar fashion to global measures of entropy when summed across the whole graph. We show that such measures are correlated with nodes that generate incidents across a network from a real data set
Random local algorithms
Consider the problem when we want to construct some structure on a bounded
degree graph, e.g. an almost maximum matching, and we want to decide about each
edge depending only on its constant radius neighbourhood. We show that the
information about the local statistics of the graph does not help here. Namely,
if there exists a random local algorithm which can use any local statistics
about the graph, and produces an almost optimal structure, then the same can be
achieved by a random local algorithm using no statistics.Comment: 9 page
An example of graph limits of growing sequences of random graphs
We consider a class of growing random graphs obtained by creating vertices
sequentially one by one: at each step, we choose uniformly the neighbours of
the newly created vertex; its degree is a random variable with a fixed but
arbitrary distribution, depending on the number of existing vertices. Examples
from this class turn out to be the ER random graph, a natural random threshold
graph, etc. By working with the notion of graph limits, we define a kernel
which, under certain conditions, is the limit of the growing random graph.
Moreover, for a subclass of models, the growing graph on any given n vertices
has the same distribution as the random graph with n vertices that the kernel
defines. The motivation stems from a model of graph growth whose attachment
mechanism does not require information about properties of the graph at each
iteration.Comment: 12 page
Limits of kernel operators and the spectral regularity lemma
We study the spectral aspects of the graph limit theory. We give a
description of graphon convergence in terms of converegnce of eigenvalues and
eigenspaces. Along these lines we prove a spectral version of the strong
regularity lemma. Using spectral methods we investigate group actions on
graphons. As an application we show that the set of isometry invariant graphons
on the sphere is closed in terms of graph convergence however the analogous
statement does not hold for the circle. This fact is rooted in the
representation theory of the orthogonal group
Regularity lemmas in a Banach space setting
Szemer\'edi's regularity lemma is a fundamental tool in extremal graph
theory, theoretical computer science and combinatorial number theory. Lov\'asz
and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst,
Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space
interpretation of the lemma and an interpretation in terms of compact- ness of
the space of graph limits. In this paper we prove several compactness results
in a Banach space setting, generalising results of Lov\'asz and Szegedy as well
as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and
Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random
graph models, and power law distributions, arXiv preprint arXiv:1401.2906
(2014)].Comment: 15 pages. The topological part has been substantially improved based
on referees comments. To appear in European Journal of Combinatoric
Stochastic blockmodel approximation of a graphon: Theory and consistent estimation
Non-parametric approaches for analyzing network data based on exchangeable
graph models (ExGM) have recently gained interest. The key object that defines
an ExGM is often referred to as a graphon. This non-parametric perspective on
network modeling poses challenging questions on how to make inference on the
graphon underlying observed network data. In this paper, we propose a
computationally efficient procedure to estimate a graphon from a set of
observed networks generated from it. This procedure is based on a stochastic
blockmodel approximation (SBA) of the graphon. We show that, by approximating
the graphon with a stochastic block model, the graphon can be consistently
estimated, that is, the estimation error vanishes as the size of the graph
approaches infinity.Comment: 20 pages, 4 figures, 2 algorithms. Neural Information Processing
Systems (NIPS), 201
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