A. Krzywicki

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.

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.

Distribution of essential interactions in model gene regulatory networks under mutation-selection balance

Gene regulatory networks typically have low in-degrees, whereby any given gene is regulated by few of the genes in the network. They also tend to have broad distributions for the out-degree. What mechanisms might be responsible for these degree distributions? Starting with an accepted framework of the binding of transcription factors to DNA, we consider a simple model of gene regulatory dynamics. There, we show that selection for a target expression pattern leads to the emergence of minimum connectivities compatible with the selective constraint. As a consequence, these gene networks have low in-degree and functionality is parsimonious, i.e., is concentrated on a sparse number of interactions as measured for instance by their essentiality. Furthermore, we find that mutations of the transcription factors drive the networks to have broad out-degrees. Finally, these classes of models are evolvable, i.e., significantly different genotypes can emerge gradually under mutation-selection balance.

Statistical mechanics of random graphs

We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.

Network of inherent structures in spin glasses: scaling and scale-free distributions.

The local minima (inherent structures) of a system and their associated transition links give rise to a network. Here we consider the topological and distance properties of such a network in the context of spin glasses. We use steepest descent dynamics, determining for each disorder sample the transition links appearing within a given barrier height. We find that differences between linked inherent structures are typically associated with local clusters of spins; we interpret this within a framework based on droplets in which the characteristic length scale grows with the barrier height. We also consider the network connectivity and the degrees of its nodes. Interestingly, for spin glasses based on random graphs, the degree distribution of the network of inherent structures exhibits a nontrivial scale-free tail.

Adaptive networks of trading agents

Multiagent models have been used in many contexts to study generic collective behavior. Similarly, complex networks have become very popular because of the diversity of growth rules giving rise to scale-free behavior. Here we study adaptive networks where the agents trade wealth when they are linked together while links can appear and disappear according to the wealth of the corresponding agents; thus the agents influence the network dynamics and vice versa. Our framework generalizes a multiagent model of Bouchaud and Mézard [Physica A 282, 536 (2000)], and leads to a steady state with fluctuating connectivities. The system spontaneously self-organizes into a critical state where the wealth distribution has a fat tail and the network is scale free; in addition, network heterogeneities lead to enhanced wealth condensation.

WILSON FERMIONS ON A RANDOMLY TRIANGULATED MANIFOLD

A general method of constructing the Dirac operator for a randomly triangulated manifold is proposed. The fermion field and the spin connection live, respectively, on the nodes and on the links of the corresponding dual graph. The construction is carried out explicitly in 2-d, on an arbitrary orientable manifold without boundary. It can be easily converted into a computer code. The equivalence, on a sphere, of Majorana fermions and Ising spins in 2-d is rederived. The method can, in principle, be extended to higher dimensions.

The anomalous behaviour of nuclear structure functions revisited

The anomalous behaviour of the nuclear structure functions is discussed in the framework of a simple statistical parton model, where the nucleus is treated as a bag of uncorrelated partons. We show that the model reproduces correctly the main features of the effect and, to some extent, it is even successful at the numerical level. The characteristic prediction of the model (to be tested experimentally) is a saturation law: for largeA (=nuclear mass number) the anomalous nuclear behaviour of the structure functions is described by a universal (i.e.A-independent) function of the Bjorken variable.

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.

Gluon bremsstrahlung by confined sources

We study the effect of confinement on gluon bremsstrahlung. A natural infrared cutoff emerges both at small gluon momenta and at small angles. If the confinement potential is of the linear “string” type, the cutoff is controlled by the tension parameter and is thus about 1GeV for the transverse momentum of a hard gluon relative to its parent quark. We propose that this confinement effect may remove the necessity for introducing ad hoc cutoffs by a large “intrinsic partonpT” in phenomenological applications of perturbative QCD.