3,691 research outputs found
Laplacian Dynamics and Multiscale Modular Structure in Networks
Most methods proposed to uncover communities in complex networks rely on
their structural properties. Here we introduce the stability of a network
partition, a measure of its quality defined in terms of the statistical
properties of a dynamical process taking place on the graph. The time-scale of
the process acts as an intrinsic parameter that uncovers community structures
at different resolutions. The stability extends and unifies standard notions
for community detection: modularity and spectral partitioning can be seen as
limiting cases of our dynamic measure. Similarly, recently proposed
multi-resolution methods correspond to linearisations of the stability at short
times. The connection between community detection and Laplacian dynamics
enables us to establish dynamically motivated stability measures linked to
distinct null models. We apply our method to find multi-scale partitions for
different networks and show that the stability can be computed efficiently for
large networks with extended versions of current algorithms.Comment: New discussions on the selection of the most significant scales and
the generalisation of stability to directed network
SVD, discrepancy, and regular structure of contingency tables
We will use the factors obtained by correspondence analysis to find
biclustering of a contingency table such that the row-column cluster pairs are
regular, i.e., they have small discrepancy. In our main theorem, the constant
of the so-called volume-regularity is related to the SVD of the normalized
contingency table. Our result is applicable to two-way cuts when both the rows
and columns are divided into the same number of clusters, thus extending partly
the result of Butler estimating the discrepancy of a contingency table by the
second largest singular value of the normalized table (one-cluster, rectangular
case), and partly a former result of the author for estimating the constant of
volume-regularity by the structural eigenvalues and the distances of the
corresponding eigen-subspaces of the normalized modularity matrix of an
edge-weighted graph (several clusters, symmetric case)
Consistency of Spectral Hypergraph Partitioning under Planted Partition Model
Hypergraph partitioning lies at the heart of a number of problems in machine
learning and network sciences. Many algorithms for hypergraph partitioning have
been proposed that extend standard approaches for graph partitioning to the
case of hypergraphs. However, theoretical aspects of such methods have seldom
received attention in the literature as compared to the extensive studies on
the guarantees of graph partitioning. For instance, consistency results of
spectral graph partitioning under the stochastic block model are well known. In
this paper, we present a planted partition model for sparse random non-uniform
hypergraphs that generalizes the stochastic block model. We derive an error
bound for a spectral hypergraph partitioning algorithm under this model using
matrix concentration inequalities. To the best of our knowledge, this is the
first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl
Mesoscopic Community Structure of Financial Markets Revealed by Price and Sign Fluctuations
The mesoscopic organization of complex systems, from financial markets to the
brain, is an intermediate between the microscopic dynamics of individual units
(stocks or neurons, in the mentioned cases), and the macroscopic dynamics of
the system as a whole. The organization is determined by "communities" of units
whose dynamics, represented by time series of activity, is more strongly
correlated internally than with the rest of the system. Recent studies have
shown that the binary projections of various financial and neural time series
exhibit nontrivial dynamical features that resemble those of the original data.
This implies that a significant piece of information is encoded into the binary
projection (i.e. the sign) of such increments. Here, we explore whether the
binary signatures of multiple time series can replicate the same complex
community organization of the financial market, as the original weighted time
series. We adopt a method that has been specifically designed to detect
communities from cross-correlation matrices of time series data. Our analysis
shows that the simpler binary representation leads to a community structure
that is almost identical with that obtained using the full weighted
representation. These results confirm that binary projections of financial time
series contain significant structural information.Comment: 15 pages, 7 figure
Searching for network modules
When analyzing complex networks a key target is to uncover their modular
structure, which means searching for a family of modules, namely node subsets
spanning each a subnetwork more densely connected than the average. This work
proposes a novel type of objective function for graph clustering, in the form
of a multilinear polynomial whose coefficients are determined by network
topology. It may be thought of as a potential function, to be maximized, taking
its values on fuzzy clusterings or families of fuzzy subsets of nodes over
which every node distributes a unit membership. When suitably parametrized,
this potential is shown to attain its maximum when every node concentrates its
all unit membership on some module. The output thus is a partition, while the
original discrete optimization problem is turned into a continuous version
allowing to conceive alternative search strategies. The instance of the problem
being a pseudo-Boolean function assigning real-valued cluster scores to node
subsets, modularity maximization is employed to exemplify a so-called quadratic
form, in that the scores of singletons and pairs also fully determine the
scores of larger clusters, while the resulting multilinear polynomial potential
function has degree 2. After considering further quadratic instances, different
from modularity and obtained by interpreting network topology in alternative
manners, a greedy local-search strategy for the continuous framework is
analytically compared with an existing greedy agglomerative procedure for the
discrete case. Overlapping is finally discussed in terms of multiple runs, i.e.
several local searches with different initializations.Comment: 10 page
- …