240 research outputs found
Sizing the length of complex networks
Among all characteristics exhibited by natural and man-made networks the
small-world phenomenon is surely the most relevant and popular. But despite its
significance, a reliable and comparable quantification of the question `how
small is a small-world network and how does it compare to others' has remained
a difficult challenge to answer. Here we establish a new synoptic
representation that allows for a complete and accurate interpretation of the
pathlength (and efficiency) of complex networks. We frame every network
individually, based on how its length deviates from the shortest and the
longest values it could possibly take. For that, we first had to uncover the
upper and the lower limits for the pathlength and efficiency, which indeed
depend on the specific number of nodes and links. These limits are given by
families of singular configurations that we name as ultra-short and ultra-long
networks. The representation here introduced frees network comparison from the
need to rely on the choice of reference graph models (e.g., random graphs and
ring lattices), a common practice that is prone to yield biased interpretations
as we show. Application to empirical examples of three categories (neural,
social and transportation) evidences that, while most real networks display a
pathlength comparable to that of random graphs, when contrasted against the
absolute boundaries, only the cortical connectomes prove to be ultra-short
Evolutionary Algorithms for Community Detection in Continental-Scale High-Voltage Transmission Grids
Symmetry is a key concept in the study of power systems, not only because the admittance and Jacobian matrices used in power flow analysis are symmetrical, but because some previous studies have shown that in some real-world power grids there are complex symmetries. In order to investigate the topological characteristics of power grids, this paper proposes the use of evolutionary algorithms for community detection using modularity density measures on networks representing supergrids in order to discover densely connected structures. Two evolutionary approaches (generational genetic algorithm, GGA+, and modularity and improved genetic algorithm, MIGA) were applied. The results obtained in two large networks representing supergrids (European grid and North American grid) provide insights on both the structure of the supergrid and the topological differences between different regions. Numerical and graphical results show how these evolutionary approaches clearly outperform to the well-known Louvain modularity method. In particular, the average value of modularity obtained by GGA+ in the European grid was 0.815, while an average of 0.827 was reached in the North American grid. These results outperform those obtained by MIGA and Louvain methods (0.801 and 0.766 in the European grid and 0.813 and 0.798 in the North American grid, respectively)
Proximity Drawings of High-Degree Trees
A drawing of a given (abstract) tree that is a minimum spanning tree of the
vertex set is considered aesthetically pleasing. However, such a drawing can
only exist if the tree has maximum degree at most 6. What can be said for trees
of higher degree? We approach this question by supposing that a partition or
covering of the tree by subtrees of bounded degree is given. Then we show that
if the partition or covering satisfies some natural properties, then there is a
drawing of the entire tree such that each of the given subtrees is drawn as a
minimum spanning tree of its vertex set
Some recent developments on the Steklov eigenvalue problem
The Steklov eigenvalue problem, first introduced over 125 years ago, has seen
a surge of interest in the past few decades. This article is a tour of some of
the recent developments linking the Steklov eigenvalues and eigenfunctions of
compact Riemannian manifolds to the geometry of the manifolds. Topics include
isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the
case of surfaces and then in higher dimensions), stability and instability of
eigenvalues under deformations of the Riemannian metric, optimisation of
eigenvalues and connections to free boundary minimal surfaces in balls, inverse
problems and isospectrality, discretisation, and the geometry of
eigenfunctions. We begin with background material and motivating examples for
readers that are new to the subject. Throughout the tour, we frequently compare
and contrast the behavior of the Steklov spectrum with that of the Laplace
spectrum. We include many open problems in this rapidly expanding area.Comment: 157 pages, 7 figures. To appear in Revista Matem\'atica Complutens
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
Self-Assembly of Geometric Space from Random Graphs
We present a Euclidean quantum gravity model in which random graphs
dynamically self-assemble into discrete manifold structures. Concretely, we
consider a statistical model driven by a discretisation of the Euclidean
Einstein-Hilbert action; contrary to previous approaches based on simplicial
complexes and Regge calculus our discretisation is based on the Ollivier
curvature, a coarse analogue of the manifold Ricci curvature defined for
generic graphs. The Ollivier curvature is generally difficult to evaluate due
to its definition in terms of optimal transport theory, but we present a new
exact expression for the Ollivier curvature in a wide class of relevant graphs
purely in terms of the numbers of short cycles at an edge. This result should
be of independent intrinsic interest to network theorists. Action minimising
configurations prove to be cubic complexes up to defects; there are indications
that such defects are dynamically suppressed in the macroscopic limit. Closer
examination of a defect free model shows that certain classical configurations
have a geometric interpretation and discretely approximate vacuum solutions to
the Euclidean Einstein-Hilbert action. Working in a configuration space where
the geometric configurations are stable vacua of the theory, we obtain direct
numerical evidence for the existence of a continuous phase transition; this
makes the model a UV completion of Euclidean Einstein gravity. Notably, this
phase transition implies an area-law for the entropy of emerging geometric
space. Certain vacua of the theory can be interpreted as baby universes; we
find that these configurations appear as stable vacua in a mean field
approximation of our model, but are excluded dynamically whenever the action is
exact indicating the dynamical stability of geometric space. The model is
intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice
Finding multiple core-periphery pairs in networks
With a core-periphery structure of networks, core nodes are densely
interconnected, peripheral nodes are connected to core nodes to different
extents, and peripheral nodes are sparsely interconnected. Core-periphery
structure composed of a single core and periphery has been identified for
various networks. However, analogous to the observation that many empirical
networks are composed of densely interconnected groups of nodes, i.e.,
communities, a network may be better regarded as a collection of multiple cores
and peripheries. We propose a scalable algorithm to detect multiple
non-overlapping groups of core-periphery structure in a network. We illustrate
our algorithm using synthesised and empirical networks. For example, we find
distinct core-periphery pairs with different political leanings in a network of
political blogs and separation between international and domestic subnetworks
of airports in some single countries in a world-wide airport network.Comment: 11 figures and 9 tables. MATLAB codes are available at
www.naokimasuda.net/cp_codes.zi
Mining complex trees for hidden fruit : a graph–based computational solution to detect latent criminal networks : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Information Technology at Massey University, Albany, New Zealand.
The detection of crime is a complex and difficult endeavour. Public and private organisations – focusing on law enforcement, intelligence, and compliance – commonly apply the rational isolated actor approach premised on observability and materiality. This is manifested largely as conducting entity-level risk management sourcing ‘leads’ from reactive covert human intelligence sources and/or proactive sources by applying simple rules-based models. Focusing on discrete observable and material actors simply ignores that criminal activity exists within a complex system deriving its fundamental structural fabric from the complex interactions between actors - with those most unobservable likely to be both criminally proficient and influential. The graph-based computational solution developed to detect latent criminal networks is a response to the inadequacy of the rational isolated actor approach that ignores the connectedness and complexity of criminality.
The core computational solution, written in the R language, consists of novel entity resolution, link discovery, and knowledge discovery technology. Entity resolution enables the fusion of multiple datasets with high accuracy (mean F-measure of 0.986 versus competitors 0.872), generating a graph-based expressive view of the problem. Link discovery is comprised of link prediction and link inference, enabling the high-performance detection (accuracy of ~0.8 versus relevant published models ~0.45) of unobserved relationships such as identity fraud. Knowledge discovery uses the fused graph generated and applies the “GraphExtract” algorithm to create a set of subgraphs representing latent functional criminal groups, and a mesoscopic graph representing how this set of criminal groups are interconnected. Latent knowledge is generated from a range of metrics including the “Super-broker” metric and attitude prediction.
The computational solution has been evaluated on a range of datasets that mimic an applied setting, demonstrating a scalable (tested on ~18 million node graphs) and performant (~33 hours runtime on a non-distributed platform) solution that successfully detects relevant latent functional criminal groups in around 90% of cases sampled and enables the contextual understanding of the broader criminal system through the mesoscopic graph and associated metadata. The augmented data assets generated provide a multi-perspective systems view of criminal activity that enable advanced informed decision making across the microscopic mesoscopic macroscopic spectrum
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