16,536 research outputs found
Convergence Speed of the Consensus Algorithm with Interference and Sparse Long-Range Connectivity
We analyze the effect of interference on the convergence rate of average
consensus algorithms, which iteratively compute the measurement average by
message passing among nodes. It is usually assumed that these algorithms
converge faster with a greater exchange of information (i.e., by increased
network connectivity) in every iteration. However, when interference is taken
into account, it is no longer clear if the rate of convergence increases with
network connectivity. We study this problem for randomly-placed
consensus-seeking nodes connected through an interference-limited network. We
investigate the following questions: (a) How does the rate of convergence vary
with increasing communication range of each node? and (b) How does this result
change when each node is allowed to communicate with a few selected far-off
nodes? When nodes schedule their transmissions to avoid interference, we show
that the convergence speed scales with , where is the
communication range and is the number of dimensions. This scaling is the
result of two competing effects when increasing : Increased schedule length
for interference-free transmission vs. the speed gain due to improved
connectivity. Hence, although one-dimensional networks can converge faster from
a greater communication range despite increased interference, the two effects
exactly offset one another in two-dimensions. In higher dimensions, increasing
the communication range can actually degrade the rate of convergence. Our
results thus underline the importance of factoring in the effect of
interference in the design of distributed estimation algorithms.Comment: 27 pages, 4 figure
Resolving Structure in Human Brain Organization: Identifying Mesoscale Organization in Weighted Network Representations
Human brain anatomy and function display a combination of modular and
hierarchical organization, suggesting the importance of both cohesive
structures and variable resolutions in the facilitation of healthy cognitive
processes. However, tools to simultaneously probe these features of brain
architecture require further development. We propose and apply a set of methods
to extract cohesive structures in network representations of brain connectivity
using multi-resolution techniques. We employ a combination of soft
thresholding, windowed thresholding, and resolution in community detection,
that enable us to identify and isolate structures associated with different
weights. One such mesoscale structure is bipartivity, which quantifies the
extent to which the brain is divided into two partitions with high connectivity
between partitions and low connectivity within partitions. A second,
complementary mesoscale structure is modularity, which quantifies the extent to
which the brain is divided into multiple communities with strong connectivity
within each community and weak connectivity between communities. Our methods
lead to multi-resolution curves of these network diagnostics over a range of
spatial, geometric, and structural scales. For statistical comparison, we
contrast our results with those obtained for several benchmark null models. Our
work demonstrates that multi-resolution diagnostic curves capture complex
organizational profiles in weighted graphs. We apply these methods to the
identification of resolution-specific characteristics of healthy weighted graph
architecture and altered connectivity profiles in psychiatric disease.Comment: Comments welcom
Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees
In this paper, we analyze the limiting spectral distribution of the adjacency
matrix of a random graph ensemble, proposed by Chung and Lu, in which a given
expected degree sequence
is prescribed on the ensemble. Let if there is an edge
between the nodes and zero otherwise, and consider the normalized
random adjacency matrix of the graph ensemble: . The empirical spectral distribution
of denoted by is the empirical
measure putting a mass at each of the real eigenvalues of the
symmetric matrix . Under some technical conditions on the
expected degree sequence, we show that with probability one,
converges weakly to a deterministic
distribution . Furthermore, we fully characterize this
distribution by providing explicit expressions for the moments of
. We apply our results to well-known degree distributions,
such as power-law and exponential. The asymptotic expressions of the spectral
moments in each case provide significant insights about the bulk behavior of
the eigenvalue spectrum
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