10,722 research outputs found
Ground States for Exponential Random Graphs
We propose a perturbative method to estimate the normalization constant in
exponential random graph models as the weighting parameters approach infinity.
As an application, we give evidence of discontinuity in natural parametrization
along the critical directions of the edge-triangle model.Comment: 12 pages, 3 figures, 1 tabl
Network Transitivity and Matrix Models
This paper is a step towards a systematic theory of the transitivity
(clustering) phenomenon in random networks. A static framework is used, with
adjacency matrix playing the role of the dynamical variable. Hence, our model
is a matrix model, where matrices are random, but their elements take values 0
and 1 only. Confusion present in some papers where earlier attempts to
incorporate transitivity in a similar framework have been made is hopefully
dissipated. Inspired by more conventional matrix models, new analytic
techniques to develop a static model with non-trivial clustering are
introduced. Computer simulations complete the analytic discussion.Comment: 11 pages, 7 eps figures, 2-column revtex format, print bug correcte
Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices
We solve the Edwards-Anderson model (EA) in different Husimi lattices. We
show that, at T=0, the structure of the solution space depends on the parity of
the loop sizes. Husimi lattices with odd loop sizes have always a trivial
paramagnetic solution stable under 1RSB perturbations while, in Husimi lattices
with even loop sizes, this solution is absent. The range of stability under
1RSB perturbations of this and other RS solutions is computed analytically
(when possible) or numerically. We compute the free-energy, the complexity and
the ground state energy of different Husimi lattices at the level of the 1RSB
approximation. We also show, when the fraction of ferromagnetic couplings
increases, the existence, first, of a discontinuous transition from a
paramagnetic to a spin glass phase and latter of a continuous transition from a
spin glass to a ferromagnetic phase.Comment: 20 pages, 10 figures (v3: Corrected analysis of transitions. Appendix
proof fixed
On the Universality of Matrix Models for Random Surfaces
We present an alternative procedure to eliminate irregular contributions in
the perturbation expansion of c=0-matrix models representing the sum over
triangulations of random surfaces, thereby reproducing the results of Tutte [1]
and Brezin et al. [2] for the planar model. The advantage of this method is
that the universality of the critical exponents can be proven from general
features of the model alone without explicit determination of the free energy
and therefore allows for several straightforward generalizations including
cases with non-vanishing central charge c< 1.Comment: 9 pages, 3 figure
Perturbing General Uncorrelated Networks
This paper is a direct continuation of an earlier work, where we studied
Erd\"os-R\'enyi random graphs perturbed by an interaction Hamiltonian favouring
the formation of short cycles. Here, we generalize these results. We keep the
same interaction Hamiltonian but let it act on general graphs with uncorrelated
nodes and an arbitrary given degree distribution. It is shown that the results
obtained for Erd\"os-R\'enyi graphs are generic, at the qualitative level.
However, scale-free graphs are an exception to this general rule and exhibit a
singular behaviour, studied thoroughly in this paper, both analytically and
numerically.Comment: 7 pages, 7 eps figures, 2-column revtex format, references adde
Random graph ensembles with many short loops
Networks observed in the real world often have many short loops. This
violates the tree-like assumption that underpins the majority of random graph
models and most of the methods used for their analysis. In this paper we sketch
possible research routes to be explored in order to make progress on networks
with many short loops, involving old and new random graph models and ideas for
novel mathematical methods. We do not present conclusive solutions of problems,
but aim to encourage and stimulate new activity and in what we believe to be an
important but under-exposed area of research. We discuss in more detail the
Strauss model, which can be seen as the `harmonic oscillator' of `loopy' random
graphs, and a recent exactly solvable immunological model that involves random
graphs with extensively many cliques and short loops.Comment: 18 pages, 10 figures,Mathematical Modelling of Complex Systems (Paris
2013) conferenc
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
Computational methods for finding long simple cycles in complex networks
© 2017 Elsevier B.V. Detection of long simple cycles in real-world complex networks finds many applications in layout algorithms, information flow modelling, as well as in bioinformatics. In this paper, we propose two computational methods for finding long cycles in real-world networks. The first method is an exact approach based on our own integer linear programming formulation of the problem and a data mining pipeline. This pipeline ensures that the problem is solved as a sequence of integer linear programs. The second method is a multi-start local search heuristic, which combines an initial construction of a long cycle using depth-first search with four different perturbation operators. Our experimental results are presented for social network samples, graphs studied in the network science field, graphs from DIMACS series, and protein-protein interaction networks. These results show that our formulation leads to a significantly more efficient exact approach to solve the problem than a previous formulation. For 14 out of 22 networks, we have found the optimal solutions. The potential of heuristics in this problem is also demonstrated, especially in the context of large-scale problem instances
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