556 research outputs found
On Spectral Graph Embedding: A Non-Backtracking Perspective and Graph Approximation
Graph embedding has been proven to be efficient and effective in facilitating
graph analysis. In this paper, we present a novel spectral framework called
NOn-Backtracking Embedding (NOBE), which offers a new perspective that
organizes graph data at a deep level by tracking the flow traversing on the
edges with backtracking prohibited. Further, by analyzing the non-backtracking
process, a technique called graph approximation is devised, which provides a
channel to transform the spectral decomposition on an edge-to-edge matrix to
that on a node-to-node matrix. Theoretical guarantees are provided by bounding
the difference between the corresponding eigenvalues of the original graph and
its graph approximation. Extensive experiments conducted on various real-world
networks demonstrate the efficacy of our methods on both macroscopic and
microscopic levels, including clustering and structural hole spanner detection.Comment: SDM 2018 (Full version including all proofs
Theories for influencer identification in complex networks
In social and biological systems, the structural heterogeneity of interaction
networks gives rise to the emergence of a small set of influential nodes, or
influencers, in a series of dynamical processes. Although much smaller than the
entire network, these influencers were observed to be able to shape the
collective dynamics of large populations in different contexts. As such, the
successful identification of influencers should have profound implications in
various real-world spreading dynamics such as viral marketing, epidemic
outbreaks and cascading failure. In this chapter, we first summarize the
centrality-based approach in finding single influencers in complex networks,
and then discuss the more complicated problem of locating multiple influencers
from a collective point of view. Progress rooted in collective influence
theory, belief-propagation and computer science will be presented. Finally, we
present some applications of influencer identification in diverse real-world
systems, including online social platforms, scientific publication, brain
networks and socioeconomic systems.Comment: 24 pages, 6 figure
Effects of Backtracking on PageRank
In this paper, we consider three variations on standard PageRank:
Non-backtracking PageRank, -PageRank, and -PageRank, all of which
alter the standard formula by adjusting the likelihood of backtracking in the
algorithm's random walk. We show that in the case of regular and bipartite
biregular graphs, standard PageRank and its variants are equivalent. We also
compare each centrality measure and investigate their clustering capabilities
Centrality metrics and localization in core-periphery networks
Two concepts of centrality have been defined in complex networks. The first
considers the centrality of a node and many different metrics for it has been
defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality,
etc). The second is related to a large scale organization of the network, the
core-periphery structure, composed by a dense core plus an outlying and
loosely-connected periphery. In this paper we investigate the relation between
these two concepts. We consider networks generated via the Stochastic Block
Model, or its degree corrected version, with a strong core-periphery structure
and we investigate the centrality properties of the core nodes and the ability
of several centrality metrics to identify them. We find that the three measures
with the best performance are marginals obtained with belief propagation,
PageRank, and degree centrality, while non-backtracking and eigenvector
centrality (or MINRES}, showed to be equivalent to the latter in the large
network limit) perform worse in the investigated networks.Comment: 15 pages, 8 figure
Centralities in complex networks
In network science complex systems are represented as a mathematical graphs
consisting of a set of nodes representing the components and a set of edges
representing their interactions. The framework of networks has led to
significant advances in the understanding of the structure, formation and
function of complex systems. Social and biological processes such as the
dynamics of epidemics, the diffusion of information in social media, the
interactions between species in ecosystems or the communication between neurons
in our brains are all actively studied using dynamical models on complex
networks. In all of these systems, the patterns of connections at the
individual level play a fundamental role on the global dynamics and finding the
most important nodes allows one to better understand and predict their
behaviors. An important research effort in network science has therefore been
dedicated to the development of methods allowing to find the most important
nodes in networks. In this short entry, we describe network centrality measures
based on the notions of network traversal they rely on. This entry aims at
being an introduction to this extremely vast topic, with many contributions
from several fields, and is by no means an exhaustive review of all the
literature about network centralities.Comment: 10 pages, 3 figures. Entry for the volume "Statistical and Nonlinear
Physics" of the Encyclopedia of Complexity and Systems Science, Chakraborty,
Bulbul (Ed.), Springer, 2021 Updated versio
Assessing Percolation Threshold Based on High-Order Non-Backtracking Matrices
Percolation threshold of a network is the critical value such that when nodes
or edges are randomly selected with probability below the value, the network is
fragmented but when the probability is above the value, a giant component
connecting large portion of the network would emerge. Assessing the percolation
threshold of networks has wide applications in network reliability, information
spread, epidemic control, etc. The theoretical approach so far to assess the
percolation threshold is mainly based on spectral radius of adjacency matrix or
non-backtracking matrix, which is limited to dense graphs or locally treelike
graphs, and is less effective for sparse networks with non-negligible amount of
triangles and loops. In this paper, we study high-order non-backtracking
matrices and their application to assessing percolation threshold. We first
define high-order non-backtracking matrices and study the properties of their
spectral radii. Then we focus on 2nd-order non-backtracking matrix and
demonstrate analytically that the reciprocal of its spectral radius gives a
tighter lower bound than those of adjacency and standard non-backtracking
matrices. We further build a smaller size matrix with the same largest
eigenvalue as the 2nd-order non-backtracking matrix to improve computation
efficiency. Finally, we use both synthetic networks and 42 real networks to
illustrate that the use of 2nd-order non-backtracking matrix does give better
lower bound for assessing percolation threshold than adjacency and standard
non-backtracking matrices.Comment: to appear in proceedings of the 26th International World Wide Web
Conference(WWW2017
Model of Brain Activation Predicts the Neural Collective Influence Map of the Brain
Efficient complex systems have a modular structure, but modularity does not
guarantee robustness, because efficiency also requires an ingenious interplay
of the interacting modular components. The human brain is the elemental
paradigm of an efficient robust modular system interconnected as a network of
networks (NoN). Understanding the emergence of robustness in such modular
architectures from the interconnections of its parts is a long-standing
challenge that has concerned many scientists. Current models of dependencies in
NoN inspired by the power grid express interactions among modules with fragile
couplings that amplify even small shocks, thus preventing functionality.
Therefore, we introduce a model of NoN to shape the pattern of brain
activations to form a modular environment that is robust. The model predicts
the map of neural collective influencers (NCIs) in the brain, through the
optimization of the influence of the minimal set of essential nodes responsible
for broadcasting information to the whole-brain NoN. Our results suggest new
intervention protocols to control brain activity by targeting influential
neural nodes predicted by network theory.Comment: 18 pages, 5 figure
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