4,947 research outputs found
Multifractal Network Generator
We introduce a new approach to constructing networks with realistic features.
Our method, in spite of its conceptual simplicity (it has only two parameters)
is capable of generating a wide variety of network types with prescribed
statistical properties, e.g., with degree- or clustering coefficient
distributions of various, very different forms. In turn, these graphs can be
used to test hypotheses, or, as models of actual data. The method is based on a
mapping between suitably chosen singular measures defined on the unit square
and sparse infinite networks. Such a mapping has the great potential of
allowing for graph theoretical results for a variety of network topologies. The
main idea of our approach is to go to the infinite limit of the singular
measure and the size of the corresponding graph simultaneously. A very unique
feature of this construction is that the complexity of the generated network is
increasing with the size. We present analytic expressions derived from the
parameters of the -- to be iterated-- initial generating measure for such major
characteristics of graphs as their degree, clustering coefficient and
assortativity coefficient distributions. The optimal parameters of the
generating measure are determined from a simple simulated annealing process.
Thus, the present work provides a tool for researchers from a variety of fields
(such as biology, computer science, biology, or complex systems) enabling them
to create a versatile model of their network data.Comment: Preprint. Final version appeared in PNAS
Spectral Graph Forge: Graph Generation Targeting Modularity
Community structure is an important property that captures inhomogeneities
common in large networks, and modularity is one of the most widely used metrics
for such community structure. In this paper, we introduce a principled
methodology, the Spectral Graph Forge, for generating random graphs that
preserves community structure from a real network of interest, in terms of
modularity. Our approach leverages the fact that the spectral structure of
matrix representations of a graph encodes global information about community
structure. The Spectral Graph Forge uses a low-rank approximation of the
modularity matrix to generate synthetic graphs that match a target modularity
within user-selectable degree of accuracy, while allowing other aspects of
structure to vary. We show that the Spectral Graph Forge outperforms
state-of-the-art techniques in terms of accuracy in targeting the modularity
and randomness of the realizations, while also preserving other local
structural properties and node attributes. We discuss extensions of the
Spectral Graph Forge to target other properties beyond modularity, and its
applications to anonymization
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
Replica methods for loopy sparse random graphs
I report on the development of a novel statistical mechanical formalism for
the analysis of random graphs with many short loops, and processes on such
graphs. The graphs are defined via maximum entropy ensembles, in which both the
degrees (via hard constraints) and the adjacency matrix spectrum (via a soft
constraint) are prescribed. The sum over graphs can be done analytically, using
a replica formalism with complex replica dimensions. All known results for
tree-like graphs are recovered in a suitable limit. For loopy graphs, the
emerging theory has an appealing and intuitive structure, suggests how message
passing algorithms should be adapted, and what is the structure of theories
describing spin systems on loopy architectures. However, the formalism is still
largely untested, and may require further adjustment and refinement.Comment: 11 pages, no figures. To be published in Proceedings of The
International Meeting on High-Dimensional Data-Driven Science (HD3-2015),
Kyoto, Japan, on 14-17 December, 201
Universal transient behavior in large dynamical systems on networks
We analyze how the transient dynamics of large dynamical systems in the
vicinity of a stationary point, modeled by a set of randomly coupled linear
differential equations, depends on the network topology. We characterize the
transient response of a system through the evolution in time of the squared
norm of the state vector, which is averaged over different realizations of the
initial perturbation. We develop a mathematical formalism that computes this
quantity for graphs that are locally tree-like. We show that for unidirectional
networks the theory simplifies and general analytical results can be derived.
For example, we derive analytical expressions for the average squared norm for
random directed graphs with a prescribed degree distribution. These analytical
results reveal that unidirectional systems exhibit a high degree of
universality in the sense that the average squared norm only depends on a
single parameter encoding the average interaction strength between the
individual constituents. In addition, we derive analytical expressions for the
average squared norm for unidirectional systems with fixed diagonal disorder
and with bimodal diagonal disorder. We illustrate these results with numerical
experiments on large random graphs and on real-world networks.Comment: 19 pages, 7 figures. Substantially enlarged version. Submitted to
Physical Review Researc
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