8,107 research outputs found
Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks
Using methods from algebraic graph theory and convex optimization, we study
the relationship between local structural features of a network and spectral
properties of its Laplacian matrix. In particular, we derive expressions for
the so-called spectral moments of the Laplacian matrix of a network in terms of
a collection of local structural measurements. Furthermore, we propose a series
of semidefinite programs to compute bounds on the spectral radius and the
spectral gap of the Laplacian matrix from a truncated sequence of Laplacian
spectral moments. Our analysis shows that the Laplacian spectral moments and
spectral radius are strongly constrained by local structural features of the
network. On the other hand, we illustrate how local structural features are
usually not enough to estimate the Laplacian spectral gap.Comment: IEEE Automatic Control, accepted for publicatio
Metrics for Graph Comparison: A Practitioner's Guide
Comparison of graph structure is a ubiquitous task in data analysis and
machine learning, with diverse applications in fields such as neuroscience,
cyber security, social network analysis, and bioinformatics, among others.
Discovery and comparison of structures such as modular communities, rich clubs,
hubs, and trees in data in these fields yields insight into the generative
mechanisms and functional properties of the graph.
Often, two graphs are compared via a pairwise distance measure, with a small
distance indicating structural similarity and vice versa. Common choices
include spectral distances (also known as distances) and distances
based on node affinities. However, there has of yet been no comparative study
of the efficacy of these distance measures in discerning between common graph
topologies and different structural scales.
In this work, we compare commonly used graph metrics and distance measures,
and demonstrate their ability to discern between common topological features
found in both random graph models and empirical datasets. We put forward a
multi-scale picture of graph structure, in which the effect of global and local
structure upon the distance measures is considered. We make recommendations on
the applicability of different distance measures to empirical graph data
problem based on this multi-scale view. Finally, we introduce the Python
library NetComp which implements the graph distances used in this work
Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities
We study structure, eigenvalue spectra and diffusion dynamics in a wide class
of networks with subgraphs (modules) at mesoscopic scale. The networks are
grown within the model with three parameters controlling the number of modules,
their internal structure as scale-free and correlated subgraphs, and the
topology of connecting network. Within the exhaustive spectral analysis for
both the adjacency matrix and the normalized Laplacian matrix we identify the
spectral properties which characterize the mesoscopic structure of sparse
cyclic graphs and trees. The minimally connected nodes, clustering, and the
average connectivity affect the central part of the spectrum. The number of
distinct modules leads to an extra peak at the lower part of the Laplacian
spectrum in cyclic graphs. Such a peak does not occur in the case of
topologically distinct tree-subgraphs connected on a tree. Whereas the
associated eigenvectors remain localized on the subgraphs both in trees and
cyclic graphs. We also find a characteristic pattern of periodic localization
along the chains on the tree for the eigenvector components associated with the
largest eigenvalue equal 2 of the Laplacian. We corroborate the results with
simulations of the random walk on several types of networks. Our results for
the distribution of return-time of the walk to the origin (autocorrelator)
agree well with recent analytical solution for trees, and it appear to be
independent on their mesoscopic and global structure. For the cyclic graphs we
find new results with twice larger stretching exponent of the tail of the
distribution, which is virtually independent on the size of cycles. The
modularity and clustering contribute to a power-law decay at short return
times
Random matrix analysis of network Laplacians
We analyze eigenvalues fluctuations of the Laplacian of various networks
under the random matrix theory framework. Analyses of random networks,
scale-free networks and small-world networks show that nearest neighbor spacing
distribution of the Laplacian of these networks follow Gaussian orthogonal
ensemble statistics of random matrix theory. Furthermore, we study nearest
neighbor spacing distribution as a function of the random connections and find
that transition to the Gaussian orthogonal ensemble statistics occurs at the
small-world transition.Comment: 14 pages, 5 figures, replaced with the final versio
Symmetry-based coarse-graining of evolved dynamical networks
Networks with a prescribed power-law scaling in the spectrum of the graph
Laplacian can be generated by evolutionary optimization. The Laplacian spectrum
encodes the dynamical behavior of many important processes. Here, the networks
are evolved to exhibit subdiffusive dynamics. Under the additional constraint
of degree-regularity, the evolved networks display an abundance of symmetric
motifs arranged into loops and long linear segments. Exploiting results from
algebraic graph theory on symmetric networks, we find the underlying backbone
structures and how they contribute to the spectrum. The resulting
coarse-grained networks provide an intuitive view of how the anomalous
diffusive properties can be realized in the evolved structures.Comment: 6 pages, 5 figure
Graph theoretical approaches for the characterization of damage in hierarchical materials
We discuss the relevance of methods of graph theory for the study of damage
in simple model materials described by the random fuse model. While such
methods are not commonly used when dealing with regular random lattices, which
mimic disordered but statistically homogeneous materials, they become relevant
in materials with microstructures that exhibit complex multi-scale patterns. We
specifically address the case of hierarchical materials, whose failure, due to
an uncommon fracture mode, is not well described in terms of either damage
percolation or crack nucleation-and-growth. We show that in these systems,
incipient failure is accompanied by an increase in eigenvector localization and
a drop in topological dimension. We propose these two novel indicators as
possible candidates to monitor a system in the approach to failure. As such,
they provide alternatives to monitoring changes in the precursory avalanche
activity, which is often invoked as a candidate for failure prediction in
materials which exhibit critical-like behavior at failure, but may not work in
the context of hierarchical materials which exhibit scale-free avalanche
statistics even very far from the critical load.Comment: 12 pages, 6 figure
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