130 research outputs found
Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution
The analysis of complex networks has so far revolved mainly around the role
of nodes and communities of nodes. However, the dynamics of interconnected
systems is commonly focalised on edge processes, and a dual edge-centric
perspective can often prove more natural. Here we present graph-theoretical
measures to quantify edge-to-edge relations inspired by the notion of flow
redistribution induced by edge failures. Our measures, which are related to the
pseudo-inverse of the Laplacian of the network, are global and reveal the
dynamical interplay between the edges of a network, including potentially
non-local interactions. Our framework also allows us to define the embeddedness
of an edge, a measure of how strongly an edge features in the weighted cuts of
the network. We showcase the general applicability of our edge-centric
framework through analyses of the Iberian Power grid, traffic flow in road
networks, and the C. elegans neuronal network.Comment: 24 pages, 6 figure
Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes
When network and graph theory are used in the study of complex systems, a
typically finite set of nodes of the network under consideration is frequently
either explicitly or implicitly considered representative of a much larger
finite or infinite region or set of objects of interest. The selection
procedure, e.g., formation of a subset or some kind of discretization or
aggregation, typically results in individual nodes of the studied network
representing quite differently sized parts of the domain of interest. This
heterogeneity may induce substantial bias and artifacts in derived network
statistics. To avoid this bias, we propose an axiomatic scheme based on the
idea of node splitting invariance to derive consistently weighted variants of
various commonly used statistical network measures. The practical relevance and
applicability of our approach is demonstrated for a number of example networks
from different fields of research, and is shown to be of fundamental importance
in particular in the study of spatially embedded functional networks derived
from time series as studied in, e.g., neuroscience and climatology.Comment: 21 pages, 13 figure
Scalable Algorithms for Laplacian Pseudo-inverse Computation
The pseudo-inverse of a graph Laplacian matrix, denoted as , finds
extensive application in various graph analysis tasks. Notable examples include
the calculation of electrical closeness centrality, determination of Kemeny's
constant, and evaluation of resistance distance. However, existing algorithms
for computing are often computationally expensive when dealing with
large graphs. To overcome this challenge, we propose novel solutions for
approximating by establishing a connection with the inverse of a
Laplacian submatrix . This submatrix is obtained by removing the -th
row and column from the original Laplacian matrix . The key advantage of
this connection is that exhibits various interesting combinatorial
interpretations. We present two innovative interpretations of based
on spanning trees and loop-erased random walks, which allow us to develop
efficient sampling algorithms. Building upon these new theoretical insights, we
propose two novel algorithms for efficiently approximating both electrical
closeness centrality and Kemeny's constant. We extensively evaluate the
performance of our algorithms on five real-life datasets. The results
demonstrate that our novel approaches significantly outperform the
state-of-the-art methods by several orders of magnitude in terms of both
running time and estimation errors for these two graph analysis tasks. To
further illustrate the effectiveness of electrical closeness centrality and
Kemeny's constant, we present two case studies that showcase the practical
applications of these metrics
THE DISTANCE CENTRALITY: MEASURING STRUCTURAL DISRUPTION OF A NETWORK
This research provides an innovative approach to identifying the influence of vertices on the topology of a graph by introducing and exploring the neighbor matrix and distance centrality. The neighbor matrix depicts the “distance profile” of each vertex, identifying the number of vertices at each shortest path length from the given vertex. From the neighbor matrix, we can derive 11 oft-used graph invariants. Distance centrality uses the neighbor matrix to identify how much influence a given vertex has over graph structure by calculating the amount of neighbor matrix change resulting from vertex removal. We explore the distance centrality in the context of three synthetic graphs and three graphs representing actual social networks. Regression analysis enables the determination that the distance centrality contains different information than four current centrality measures (betweenness, closeness, degree, and eigenvector). The distance centrality proved to be more robust against small changes in graphs through analysis of graphs under edge swapping, deletion, and addition paradigms than betweenness and eigenvector centrality, though less so than degree and closeness centralities. We find that the neighbor matrix and the distance centrality reliably enable the identification of vertices that are significant in different and important contexts than current measures.http://archive.org/details/thedistancecentr1094559576Lieutenant Colonel, United States ArmyApproved for public release; distribution is unlimited
The Physics of Communicability in Complex Networks
A fundamental problem in the study of complex networks is to provide
quantitative measures of correlation and information flow between different
parts of a system. To this end, several notions of communicability have been
introduced and applied to a wide variety of real-world networks in recent
years. Several such communicability functions are reviewed in this paper. It is
emphasized that communication and correlation in networks can take place
through many more routes than the shortest paths, a fact that may not have been
sufficiently appreciated in previously proposed correlation measures. In
contrast to these, the communicability measures reviewed in this paper are
defined by taking into account all possible routes between two nodes, assigning
smaller weights to longer ones. This point of view naturally leads to the
definition of communicability in terms of matrix functions, such as the
exponential, resolvent, and hyperbolic functions, in which the matrix argument
is either the adjacency matrix or the graph Laplacian associated with the
network. Considerable insight on communicability can be gained by modeling a
network as a system of oscillators and deriving physical interpretations, both
classical and quantum-mechanical, of various communicability functions.
Applications of communicability measures to the analysis of complex systems are
illustrated on a variety of biological, physical and social networks. The last
part of the paper is devoted to a review of the notion of locality in complex
networks and to computational aspects that by exploiting sparsity can greatly
reduce the computational efforts for the calculation of communicability
functions for large networks.Comment: Review Article. 90 pages, 14 figures. Contents: Introduction;
Communicability in Networks; Physical Analogies; Comparing Communicability
Functions; Communicability and the Analysis of Networks; Communicability and
Localization in Complex Networks; Computability of Communicability Functions;
Conclusions and Prespective
Fundamentals of spreading processes in single and multilayer complex networks
Spreading processes have been largely studied in the literature, both
analytically and by means of large-scale numerical simulations. These processes
mainly include the propagation of diseases, rumors and information on top of a
given population. In the last two decades, with the advent of modern network
science, we have witnessed significant advances in this field of research. Here
we review the main theoretical and numerical methods developed for the study of
spreading processes on complex networked systems. Specifically, we formally
define epidemic processes on single and multilayer networks and discuss in
detail the main methods used to perform numerical simulations. Throughout the
review, we classify spreading processes (disease and rumor models) into two
classes according to the nature of time: (i) continuous-time and (ii) cellular
automata approach, where the second one can be further divided into synchronous
and asynchronous updating schemes. Our revision includes the heterogeneous
mean-field, the quenched-mean field, and the pair quenched mean field
approaches, as well as their respective simulation techniques, emphasizing
similarities and differences among the different techniques. The content
presented here offers a whole suite of methods to study epidemic-like processes
in complex networks, both for researchers without previous experience in the
subject and for experts.Comment: Review article. 73 pages, including 24 figure
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