arXiv subjects
Publications and source records attributed to Bo Soderberg.
We investigate a class of random graph ensembles based on the Feynman graphs of multidimensional integrals, representing statistical-mechanical partition functions. We show that the resulting ensembles of random graphs strongly resemble those defined in random graphs with hidden color (CDRG), generalizing the known relation of the Feynman graphs of simple one-dimensional integrals to random graphs with a given degree distribution.
We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom, color, applied to vertices or stubs (half-edges). The color is assumed unobservable, but is allowed to affect edge probabilities. This serves as a convenient method to define very general classes of models within a common unifying formalism, and allowing for a non-trivial edge correlation structure.
We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global structural properties of graphs from the resulting ensembles is demonstrated. Cluster size statistics are derived with generating function techniques, yielding a well-defined percolation threshold. Explicit rules are derived for the enumeration of small subgraphs. Duality and redundancy is discussed, and subclasses corresponding to commonly studied models are identified.
We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its terminal vertices. This approach provides a general framework for the analysis of a large class of models. The generic phase structure is derived using generating function techniques, and relations to other classes of models are pointed out.
A novel artificial neural network approach to constraint satisfaction problems is presented. Based on information-theoretical considerations, it differs from a conventional mean-field approach in the form of the resulting free energy. The method, implemented as an annealing algorithm, is numerically explored on a testbed of K-SAT problems. The performance shows a dramatic improvement to that of a conventional mean-field approach, and is comparable to that of a state-of-the-art dedicated heuristic (Gsat+Walk). The real strength of the method, however, lies in its generality -- with minor modifications it is applicable to arbitrary types of discrete constraint satisfaction problems.
For combinatorial optimization problems that can be formulated as Ising or Potts spin systems, the Mean Field (MF) approximation yields a versatile and simple ANN heuristic, Deterministic Annealing. For assignment problems the situation is more complex -- the natural analog of the MF approximation lacks the simplicity present in the Potts and Ising cases. In this article the difficulties associated with this issue are investigated, and the options for solving them discussed. Improvements to existing Potts-based MF-inspired heuristics are suggested, and the possibilities for defining a proper variational approach are scrutinized.
There exists a generic relationship between the thermodynamics of a continuous polymer with a generic self-interaction and the two-point function of an interacting field-theory. In addition, the (2N)-point function of the resulting field theory is similarly related to a system of N interacting polymers. In the present paper, this relation is explored for the special case of a polyelectrolyte, characterized by a screened Coulomb pair potential. The corresponding field theory can be recast in a particularly simple form, corresponding to a quantum-mechanical particle, self-interacting via the emission and absorption of a massive scalar field. This is particularly useful in a perturbative treatment: perturbative expansions for the polyelectrolyte can be simply derived from the loop expansion for the related field-theoretical two-point function, for which established computational methods exist.