Distribution of shortest path lengths on trees of a given size in subcritical Erdős-Rényi networks.

In the subcritical regime Erdős-Rényi (ER) networks consist of finite tree components, which are nonextensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O.

Distribution of the number of cycles in directed and undirected random regular graphs of degree 2.

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles.

Structure of networks that evolve under a combination of growth and contraction.

We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random attachment step takes place with probability P_{add} and a random node deletion step takes place with probability P_{del}=1-P_{add}. The balance between the growth and contraction processes is captured by the parameter η=P_{add}-P_{del}.

Fate of articulation points and bredges in percolation.

We investigate the statistics of articulation points and bredges (bridge edges) in complex networks in which bonds are randomly removed in a percolation process. Because of the heterogeneous structure of a complex network, the probability of a node to be an articulation point or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem.

Statistical analysis of edges and bredges in configuration model networks.

A bredge (bridge-edge) in a network is an edge whose deletion would split the network component on which it resides into two separate components. Bredges are vulnerable links that play an important role in network collapse processes, which may result from node or link failures, attacks, or epidemics. Therefore, the abundance and properties of bredges affect the resilience of the network to these collapse scenarios.

Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy.

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion, and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution P_{0}(k) and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy S_{t}=S[P_{t}(k)||π(k|〈K〉_{t})] of the degree distribution P_{t}(k) of the contracting network at time t with respect to the corresponding Poisson distribution π(k|〈K〉_{t}) with the same mean degree 〈K〉_{t} as a distance measure between P_{t}(k) and Poisson.

Convergence towards an Erdős-Rényi graph structure in network contraction processes.

In a highly influential paper twenty years ago, Barabási and Albert [Science 286, 509 (1999)SCIEAS0036-807510.1126/science.286.

Generating random networks that consist of a single connected component with a given degree distribution.

We present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution. This approach extends the construction toolbox of random networks beyond the configuration model framework, in which one controls the degree distribution but not the number of components and their sizes. Unlike configuration model networks, which are completely uncorrelated, the resulting single-component networks exhibit degree-degree correlations.

Distribution of shortest path lengths in subcritical Erdős-Rényi networks.

Networks that are fragmented into small disconnected components are prevalent in a large variety of systems. These include the secure communication networks of commercial enterprises, government agencies, and illicit organizations, as well as networks that suffered multiple failures, attacks, or epidemics. The structural and statistical properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are nonextensive.

Revealing the microstructure of the giant component in random graph ensembles.

The microstructure of the giant component of the Erdős-Rényi network and other configuration model networks is analyzed using generating function methods. While configuration model networks are uncorrelated, the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree-degree correlations of all orders. We present exact analytical results for the degree distributions as well as higher-order degree-degree correlations on the giant components of configuration model networks.

Distribution of shortest cycle lengths in random networks.

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before.

Distribution of shortest path lengths in a class of node duplication network models.

We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks, and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication.

Distance distribution in configuration-model networks.

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial, and power-law degree distributions. The mean, mode, and variance of the distribution of shortest path lengths are also evaluated.

Stochastic analysis of bistability in coherent mixed feedback loops combining transcriptional and posttranscriptional regulations.

Mixed feedback loops combining transcriptional and posttranscriptional regulations are common in cellular regulatory networks. They consist of two genes, encoding a transcription factor and a small noncoding RNA (sRNA), which mutually regulate each other's expression. We present a theoretical and numerical study of coherent mixed feedback loops of this type, in which both regulations are negative.

A defense-offense multi-layered regulatory switch in a pathogenic bacterium.

Cells adapt to environmental changes by efficiently adjusting gene expression programs. Staphylococcus aureus, an opportunistic pathogenic bacterium, switches between defensive and offensive modes in response to quorum sensing signal. We identified and studied the structural characteristics and dynamic properties of the core regulatory circuit governing this switch by deterministic and stochastic computational methods, as well as experimentally.

Interactions between distant ceRNAs in regulatory networks.

Competing endogenous RNAs (ceRNAs) were recently introduced as RNA transcripts that affect each other's expression level through competition for their microRNA (miRNA) coregulators. This stems from the bidirectional effects between miRNAs and their target RNAs, where a change in the expression level of one target affects the level of the miRNA regulator, which in turn affects the level of other targets. By the same logic, miRNAs that share targets compete over binding to their common targets and therefore also exhibit ceRNA-like behavior.

Global regulation of transcription by a small RNA: a quantitative view.

Small RNAs are integral regulators of bacterial gene expression, the majority of which act posttranscriptionally by basepairing with target mRNAs, altering translation or mRNA stability. 6S RNA, however, is a small RNA that is a transcriptional regulator, acting by binding directly to σ(70)-RNA polymerase (σ(70)-RNAP) and preventing its binding to gene promoters. At the transition from exponential to stationary phase, 6S RNA accumulates and globally downregulates the transcription of hundreds of genes.

Stochastic analysis of complex reaction networks using binomial moment equations.

The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network.

Competition between small RNAs: a quantitative view.

Two major classes of small regulatory RNAs--small interfering RNAs (siRNAs) and microRNA (miRNAs)--are involved in a common RNA interference processing pathway. Small RNAs within each of these families were found to compete for limiting amounts of shared components, required for their biogenesis and processing. Association with Argonaute (Ago), the catalytic component of the RNA silencing complex, was suggested as the central mechanistic point in RNA interference machinery competition.

Binomial moment equations for stochastic reaction systems.

A highly efficient formulation of moment equations for stochastic reaction networks is introduced. It is based on a set of binomial moments that capture the combinatorics of the reaction processes. The resulting set of equations can be easily truncated to include moments up to any desired order.

Dimensional reduction of the master equation for stochastic chemical networks: The reduced-multiplane method.

Chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers. The analysis of these networks requires stochastic methods, which come in two forms: direct integration of the master equation and Monte Carlo simulations. The master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species.

Diffusion-limited reactions on a two-dimensional lattice with binary disorder.

Reaction-diffusion systems where transition rates exhibit quenched disorder are common in physical and chemical systems. We study pair reactions on a periodic two-dimensional lattice, including continuous deposition and spontaneous desorption of particles. Hopping and desorption are taken to be thermally activated processes.

Interaction of atomic and molecular hydrogen with tholin surfaces at low temperatures.

We study the interaction of atomic and molecular hydrogen with a surface of tholin, a man-made polymer considered to be an analogue of aerosol particles present in Titan's atmosphere, using thermal programmed desorption at low temperatures below 30 K. The results are fitted and analyzed using a fine-grained rate equation model that describes the diffusion, reaction, and desorption processes. We obtain the energy barriers for diffusion and desorption of atomic and molecular hydrogen.

Regulation of phenotypic variability by a threshold-based mechanism underlies bacterial persistence.

In the face of antibiotics, bacterial populations avoid extinction by harboring a subpopulation of dormant cells that are largely drug insensitive. This phenomenon, termed "persistence," is a major obstacle for the treatment of a number of infectious diseases. The mechanism that generates both actively growing as well as dormant cells within a genetically identical population is unknown.

Quantifying the connectivity of a network: the network correlation function method.

Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small-world networks), and a power-law degree distribution (scale-free networks).