Biomedical subjects

A Krzywicki

Publications and source records attributed to A Krzywicki.

15 recordsLinked to original sources

PubMed2006

From simple to complex networks: inherent structures, barriers, and valleys in the context of spin glasses.

Given discrete degrees of freedom (spins) on a graph interacting via an energy function, what can be said about the energy local minima and associated inherent structures? Using the lid algorithm in the context of a spin glass energy function, we investigate the properties of the energy landscape for a variety of graph topologies. First, we find that the multiplicity N(s) of the inherent structures generically has a log-normal distribution. In addition, the large volume limit of ln / differs from unity, except for the Sherrington-Kirkpatrick model. Second, we find simple scaling laws for the growth of the height of the energy barrier between the two degenerate ground states and the size of the associated valleys. For finite connectivity models, changing the topology of the underlying graph does not modify qualitatively the energy landscape, but at the quantitative level the models can differ substantially.

Journal Article↗

PubMed2004

Perturbing general uncorrelated networks.

This paper is a direct continuation of an earlier work, where we studied Erdös-Rényi random graphs perturbed by an interaction Hamiltonian favoring 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 Erdös-Rényi graphs are generic, at the qualitative level. However, scale-free graphs are an exception to this general rule and exhibit a singular behavior, studied thoroughly in this paper, both analytically and numerically.

Journal Article↗

PubMed2004

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, analytic techniques to develop a static model with nontrivial clustering are introduced. Computer simulations complete the analytic discussion.

Cluster Analysis↗

PubMed2003

Tree networks with causal structure.

A geometry of networks endowed with a causal structure is discussed using the conventional framework of the equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree graphs, an analytically solvable case. General formulas are derived, describing the degree distribution, the ancestor-descendant correlation, and the probability that a randomly chosen node lives at a given geodesic distance from the root. It is shown that the Hausdorff dimension d(H) of the causal networks is generically infinite, in contrast to the maximally random trees where it is generically finite.

Journal Article↗

PubMed2003

Uncorrelated random networks.

We define a statistical ensemble of nondegenerate graphs, i.e., graphs without multiple-connections and self-connections between nodes. The node degree distribution is arbitrary, but the nodes are assumed to be uncorrelated. This completes our earlier publication [Phys. Rev. 64, 046118 (2001)] where trees and degenerate graphs were considered. An efficient algorithm generating nondegenerate graphs is constructed. The corresponding computer code is available on request. Finite-size effects in scale-free graphs, i.e., those where the tail of the degree distribution falls like n(-beta), are carefully studied. We find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like N(gamma), with gamma=min[1/2,1/(beta-1)], where N is the total number of nodes. The consequence is that, independently of any specific model, the internode correlations seem to be a necessary ingredient of the physics of scale-free networks observed in nature.

Journal Article↗

PubMed2001

Statistical ensemble of scale-free random graphs.

A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented. Methods borrowed from field theory are used to define the ensemble and to study analytically its properties. The ensemble is characterized by two global parameters, the fractal and the spectral dimensions, which are explicitly calculated. It is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified. The stability of the scale-free regime is also considered: when it breaks down, either a scale is spontaneously generated or else, a "singular" node appears and the graphs become crumpled. A new computer algorithm to generate these random graphs is proposed. Possible generalizations are also discussed. In particular, more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary (degenerate) scale-free random graphs.

Journal Article↗

PubMed1994