arXiv subjects

Ofer Biham

Publications and source records attributed to Ofer Biham.

At least 37 records · Page 2Linked to original sources

arXiv2019

Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication

We present exact results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p. In addition, the daughter node forms a directed link to the mother node itself. We obtain analytical results for the in-degree distribution $P_t(K_{in}=k)$, and for the out-degree distribution $P_t(K_{out}=k)$ at time t. The in-degrees follow a shifted power-law, so the network is asymptotically scale free. In contrast, the out-degree distribution is narrow, and converges to a Poisson distribution in the sparse network limit and to a Gaussian distribution in the dense network limit. Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using this we calculate the mean degree $\langle K_{in}\rangle_t=\langle K_{out}\rangle_t$, which converges to $1/(1-p)$ in the large network limit, for the whole range of $0 1/2$ the mean degree diverges in the large network limit. We also present analytical results for the distribution of the number of upstream and downstream nodes from a random node. The mean values $\langle N_{up}\rangle_t=\langle N_{down}\rangle_t$ scale logarithmically with the network size, implying that only a small fraction of pairs of nodes are connected by directed paths, unlike the undirected ND case that consists of a single component, hence not a small-world network.

physics.soc-ph↗

arXiv2019

Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication

We present exact analytical results for the distribution of shortest path lengths (DSPL) in a directed network model that grows by node duplication. Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability $p$. In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. In this network not all pairs of nodes are connected by directed paths, in spite of the fact that the corresponding undirected network consists of a single connected component. More specifically, in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths. To calculate the DSPL between those pairs of nodes that are connected by directed paths we derive a master equation for the time evolution of the probability $P_t(L=\ell)$, $\ell=1,2,\dots$, where $\ell$ is the length of the shortest directed path. Solving the master equation, we obtain a closed form expression for $P_t(L=\ell)$. It is found that the DSPL at time $t$ consists of a convolution of the initial DSPL $P_0(L=\ell)$, with a Poisson distribution and a sum of Poisson distributions. The mean distance ${\mathbb E}_t[L|L<\infty]$ between pairs of nodes which are connected by directed paths is found to depend logarithmically on the network size $N_t$. However, since in the large network limit the fraction of pairs of nodes that are connected by directed paths is diminishingly small, the corded DND network is not a small-world network, unlike the corresponding undirected network.

physics.soc-ph↗

arXiv2019

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. Moreover, they are found to be disassortative, namely high-degree nodes tend to connect to low-degree nodes and vice versa. We demonstrate the method for single-component networks with ternary, exponential and power-law degree distributions.

cond-mat.dis-nn↗

arXiv2018

Statistical analysis of articulation points in configuration model networks

An articulation point (AP) in a network is a node whose deletion would split the network component on which it resides into two or more components. APs are vulnerable spots that play an important role in network collapse processes, which may result from node failures, attacks or epidemics. Therefore, the abundance and properties of APs affect the resilience of the network to these collapse scenarios. We present analytical results for the statistical properties of APs in configuration model networks. In order to quantify their abundance, we calculate the probability $P(i \in {\rm AP})$, that a random node, i, in a configuration model network with P(K=k), is an AP. We also obtain the conditional probability $P(i \in {\rm AP}|k)$ that a random node of degree k is an AP, and find that high degree nodes are more likely to be APs than low degree nodes. Using Bayes' theorem, we obtain the conditional degree distribution, $P(K=k|{\rm AP})$, over the set of APs and compare it to P(K=k). We propose a new centrality measure based on APs: each node can be characterized by its articulation rank, r, which is the number of components that would be added to the network upon deletion of that node. For nodes which are not APs the articulation rank is $r=0$, while for APs $r \ge 1$. We obtain a closed form expression for the distribution of articulation ranks, P(R=r). Configuration model networks often exhibit a coexistence between a giant component and finite components. To examine the distinct properties of APs on the giant and on the finite components, we calculate the probabilities presented above separately for the giant and the finite components. We apply these results to ensembles of configuration model networks with a Poisson, exponential and power-law degree distributions. The implications of these results are discussed in the context of common attack scenarios and network dismantling processes.

cond-mat.dis-nn↗

arXiv2018

The 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 properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are non-extensive. Surprisingly, such networks do not exhibit the small-world property that is typical in supercritical random networks, where the mean distance between pairs of nodes scales logarithmically with the network size. Unlike supercritical networks whose structure has been studied extensively, subcritical networks have attracted little attention. A special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies. The overall statistics of the network can be obtained by a summation over all the components with suitable weights. We use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths (DSPL) on components of given sizes and topologies in subcritical Erdos-Renyi (ER) networks. From this expansion we obtain an exact analytical expression for the DSPL of the entire subcritical network, in the asymptotic limit. The DSPL, which accounts for all the pairs of nodes that reside on the same finite component (FC), is found to follow a geometric distribution of the form $P_{\rm FC}(L=\ell|L<\infty)=(1-c)c^{\ell-1}$, where $c<1$ is the mean degree. We confirm the convergence to this asymptotic result using computer simulations. Using the duality relations between subcritical and supercritical ER networks, we obtain the DSPL on the non-giant components above the percolation transition.

cond-mat.dis-nn↗

arXiv2018

Revealing the Micro-Structure of the Giant Component in Random Graph Ensembles

The micro-structure of the giant component of the Erd{\H o}s-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. We show that the degree-degree correlations are essential for the integrity of the giant component, in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component. To demonstrate the importance and broad applicability of these results, we apply them to the study of the distribution of shortest path lengths on the giant component, percolation on the giant component and the spectra of sparse matrices defined on the giant component. We show that by using the degree distribution on the giant component, one obtains high quality results for these properties, which can be further improved by taking the degree-degree correlations into account. This suggests that many existing methods, currently used for the analysis of the whole network, can be adapted in a straightforward fashion to yield results conditioned on the giant component.

cond-mat.stat-mech↗

arXiv2017

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. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdos-Rényi network), degenerate distribution (random regular graph) and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.

cond-mat.dis-nn↗

arXiv2017

The 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. Its daughter node is added to the network, forming a link to the mother node, and with probability $p$ to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability $P_t(L=\ell)$, $\ell=1,2,\dots$, where $L$ is the distance between a pair of nodes and $t$ is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for $P_t(L=\ell)$. The mean distance, $\langle L \rangle_t$, and the diameter, $Δ_t$, are found to scale like $\ln t$, namely the ND network is a small world network. The variance of the DSPL is also found to scale like $\ln t$. Interestingly, the mean distance and the diameter exhibit properties of a small world network, rather than the ultrasmall world network behavior observed in other scale-free networks, in which $\langle L \rangle_t \sim \ln \ln t$.

physics.soc-ph↗

arXiv2017

The distribution of first hitting times of random walks on directed Erdős-Rényi networks

We present analytical results for the distribution of first hitting times of random walkers (RWs) on directed Erdős-Rényi (ER) networks. Starting from a random initial node, a random walker hops randomly along directed edges between adjacent nodes in the network. The path terminates either by the retracing scenario, when the walker enters a node which it has already visited before, or by the trapping scenario, when it becomes trapped in a dead-end node from which it cannot exit. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$. The distribution $P(d>\ell)$ can be expressed as a product of an exponential distribution and a Rayleigh distribution. We obtain expressions for the mean, median and standard deviation of this distribution in terms of the network size and its mean degree. We also calculate the distribution of last hitting times, namely the path lengths of self-avoiding walks on directed ER networks, which do not retrace their paths. The last hitting times are found to be much longer than the first hitting times. The results are compared to those obtained for undirected ER networks. It is found that the first hitting times of RWs in a directed ER network are much longer than in the corresponding undirected network. This is due to the fact that RWs on directed networks do not exhibit the backtracking scenario, which is a dominant termination mechanism of RWs on undirected networks. It is shown that our approach also applies to a broader class of networks, referred to as semi-ER networks, in which the distribution of in-degrees is Poisson, while the out-degrees may follow any desired distribution with the same mean as the in-degree distribution.

cond-mat.dis-nn↗

arXiv2017

The distribution of first hitting times of non-backtracking random walks on Erdős-Rényi networks

We present analytical results for the distribution of first hitting times of non-backtracking random walks on finite Erdős-Rényi networks of $N$ nodes. The walkers hop randomly between adjacent nodes on the network, without stepping back to the previous node, until they hit a node which they have already visited before or get trapped in a dead-end node. At this point, the path is terminated. The length, $d$, of the resulting path, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,\dots$, of non-backtracking random walks starting from a random initial node. It turns out that the distribution $P(d>\ell)$ is given by a product of a discrete Rayleigh distribution and an exponential distribution. It is found that the paths of non-backtracking random walks, up to their termination at the first hitting time, are longer, on average, than those of the corresponding simple RWs. However, they are shorter than those of SAWs on the same network, which terminate at the last hitting time. We obtain analytical results for the probabilities, $p_{\rm ret}$ and $p_{\rm trap}$, that a path will terminate by retracing, namely stepping into an already visited node, or by trapping, namely entering a node of degree $k=1$, which has no exit link, respectively. It is shown that in dilute networks the dominant termination scenario is trapping while in dense networks most paths terminate by retracing. We obtain expressions for the conditional tail distributions of path lengths, $P(d>\ell|{\rm ret})$ and $P(d>\ell|\rm{trap})$, for those paths which terminate by retracing or by trapping, respectively. We also study a class of generalized non-backtracking random walk models which not only avoid the backtracking step into the previous node but avoid stepping into the last $S$ visited nodes.

cond-mat.dis-nn↗

arXiv2017

The distribution of first hitting times of random walks on Erdős-Rényi networks

Analytical results for the distribution of first hitting times of random walks on Erdős-Rényi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has already visited before. At this point, the path terminates. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, $P(d > \ell)$. The results are found to be in excellent agreement with numerical simulations. It is found %turns out that the distribution $P(d > \ell)$ follows a product of an exponential distribution and a Rayleigh distribution. The mean, median and standard deviation of this distribution are also calculated, in terms of the network size and its mean degree. The termination of an RW path may take place either by backtracking to the previous node or by retracing of its path, namely stepping into a node which has been visited two or more time steps earlier. We obtain analytical results for the probabilities, $p_b$ and $p_r$, that the cause of termination will be backtracking or retracing, respectively. It is shown that in dilute networks the dominant termination scenario is backtracking while in dense networks most paths terminate by retracing. We also obtain expressions for the conditional distributions $P(d=\ell | b)$ and $P(d=\ell | r)$, for those paths which are terminated by backtracking or by retracing, respectively. These results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

physics.soc-ph↗

arXiv2016

The distribution of path lengths of self avoiding walks on Erdős-Rényi networks

We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, $p_t(k)$, with an average degree, $\langle k \rangle_t$, that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, $n_T(\ell)$, increases dramatically as a function of $\ell$. We also obtain analytical results for the path-length distribution, $P(\ell)$, of the SAW paths which are actually pursued, starting from a random initial node. It turns out that $P(\ell)$ follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

cond-mat.dis-nn↗

arXiv2016

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. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.

cond-mat.dis-nn↗

arXiv2015

Analytical results for the distribution of shortest path lengths in random networks

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Rényi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case that the mean degree scales as $N^α$ with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance $d=\lfloor 1/α\rfloor$ from each other. The distribution of shortest path lengths between nodes of degree $m$ and the rest of the network is calculated. Its average is shown to be a monotonically decreasing function of $m$, providing an interesting relation between a local property and a global property of the network. The methodology presented here can be applied to more general classes of networks.

cond-mat.dis-nn↗

arXiv2014

Stochastic analysis of bistability in coherent mixed feedback loops combining transcriptional and post-transcriptional regulations

Mixed feedback loops combining transcriptional and post-transcriptional regulations are common in cellular regulatory networks. They consist of two genes, encoding a transcription factor and a small non-coding 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. Under suitable conditions, these feedback loops are expected to exhibit bistability, namely two stable states, one dominated by the transcriptional repressor and the other dominated by the sRNA. We use deterministic methods based on rate equation models, in order to identify the range of parameters in which bistability takes place. However, the deterministic models do not account for the finite lifetimes of the bistable states and the spontaneous, fluctuation-driven transitions between them. Therefore, we use stochastic methods to calculate the average lifetimes of the two states. It is found that these lifetimes strongly depend on rate coefficients such as the transcription rates of the transcriptional repressor and the sRNA. In particular, we show that the fraction of time the system spends in the sRNA dominated state follows a monotonically decreasing sigmoid function of the transcriptional repressor transcription rate. The biological relevance of these results is discussed in the context of such mixed feedback loops in {\it Escherichia coli}.

q-bio.MN↗

arXiv2010

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 runcated to include moments up to any desired order. The number of equations is dramatically reduced compared to the master equation. This formulation enables the simulation of complex reaction networks, involving a large number of reactive species much beyond the feasibility limit of any existing method. It provides an equation-based paradigm to the analysis of stochastic networks, complementing the commonly used Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2010

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. These barriers are found to be in the range of 30 to 60 meV, indicating that atom/molecule-surface interactions in this temperature range are dominated by weak adsorption forces. The implications of these results for the understanding of the atmospheric chemistry of Titan are discussed.

astro-ph.EP↗

arXiv2010

Entanglement of Periodic States, the Quantum Fourier Transform and Shor's Factoring Algorithm

The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.

quant-ph↗