547 research outputs found
Computing Vertex Centrality Measures in Massive Real Networks with a Neural Learning Model
Vertex centrality measures are a multi-purpose analysis tool, commonly used
in many application environments to retrieve information and unveil knowledge
from the graphs and network structural properties. However, the algorithms of
such metrics are expensive in terms of computational resources when running
real-time applications or massive real world networks. Thus, approximation
techniques have been developed and used to compute the measures in such
scenarios. In this paper, we demonstrate and analyze the use of neural network
learning algorithms to tackle such task and compare their performance in terms
of solution quality and computation time with other techniques from the
literature. Our work offers several contributions. We highlight both the pros
and cons of approximating centralities though neural learning. By empirical
means and statistics, we then show that the regression model generated with a
feedforward neural networks trained by the Levenberg-Marquardt algorithm is not
only the best option considering computational resources, but also achieves the
best solution quality for relevant applications and large-scale networks.
Keywords: Vertex Centrality Measures, Neural Networks, Complex Network Models,
Machine Learning, Regression ModelComment: 8 pages, 5 tables, 2 figures, version accepted at IJCNN 2018. arXiv
admin note: text overlap with arXiv:1810.1176
Embedding Graphs under Centrality Constraints for Network Visualization
Visual rendering of graphs is a key task in the mapping of complex network
data. Although most graph drawing algorithms emphasize aesthetic appeal,
certain applications such as travel-time maps place more importance on
visualization of structural network properties. The present paper advocates two
graph embedding approaches with centrality considerations to comply with node
hierarchy. The problem is formulated first as one of constrained
multi-dimensional scaling (MDS), and it is solved via block coordinate descent
iterations with successive approximations and guaranteed convergence to a KKT
point. In addition, a regularization term enforcing graph smoothness is
incorporated with the goal of reducing edge crossings. A second approach
leverages the locally-linear embedding (LLE) algorithm which assumes that the
graph encodes data sampled from a low-dimensional manifold. Closed-form
solutions to the resulting centrality-constrained optimization problems are
determined yielding meaningful embeddings. Experimental results demonstrate the
efficacy of both approaches, especially for visualizing large networks on the
order of thousands of nodes.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphic
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
Kirchhoff Index As a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms
Most previous work of centralities focuses on metrics of vertex importance
and methods for identifying powerful vertices, while related work for edges is
much lesser, especially for weighted networks, due to the computational
challenge. In this paper, we propose to use the well-known Kirchhoff index as
the measure of edge centrality in weighted networks, called -Kirchhoff
edge centrality. The Kirchhoff index of a network is defined as the sum of
effective resistances over all vertex pairs. The centrality of an edge is
reflected in the increase of Kirchhoff index of the network when the edge
is partially deactivated, characterized by a parameter . We define two
equivalent measures for -Kirchhoff edge centrality. Both are global
metrics and have a better discriminating power than commonly used measures,
based on local or partial structural information of networks, e.g. edge
betweenness and spanning edge centrality.
Despite the strong advantages of Kirchhoff index as a centrality measure and
its wide applications, computing the exact value of Kirchhoff edge centrality
for each edge in a graph is computationally demanding. To solve this problem,
for each of the -Kirchhoff edge centrality metrics, we present an
efficient algorithm to compute its -approximation for all the
edges in nearly linear time in . The proposed -Kirchhoff edge
centrality is the first global metric of edge importance that can be provably
approximated in nearly-linear time. Moreover, according to the
-Kirchhoff edge centrality, we present a -Kirchhoff vertex
centrality measure, as well as a fast algorithm that can compute
-approximate Kirchhoff vertex centrality for all the vertices in
nearly linear time in
Graph Theory and Networks in Biology
In this paper, we present a survey of the use of graph theoretical techniques
in Biology. In particular, we discuss recent work on identifying and modelling
the structure of bio-molecular networks, as well as the application of
centrality measures to interaction networks and research on the hierarchical
structure of such networks and network motifs. Work on the link between
structural network properties and dynamics is also described, with emphasis on
synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape
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