arXiv subjects

Ido Tishby

Publications and source records attributed to Ido Tishby.

16 recordsLinked to original sources

arXiv2024

Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks

We present analytical results for the distribution of first return (FR) times of non-backtracking random walks (NBWs) on undirected configuration model networks consisting of $N$ nodes with degree distribution $P(k)$. We focus on the case in which the network consists of a single connected component. Starting from a random initial node $i$ at time $t=0$, an NBW hops into a random neighbor of $i$ at time $t=1$ and at each subsequent step it continues to hop into a random neighbor of its current node, excluding the previous node. We calculate the tail distribution $P ( T_{\rm FR} > t )$ of first return times from a random initial node to itself. It is found that $P ( T_{\rm FR} > t )$ is given by a discrete Laplace transform of the degree distribution $P(k)$. This result exemplifies the relation between structural properties of a network, captured by the degree distribution, and properties of dynamical processes taking place on the network. Using the tail-sum formula, we calculate the mean first return time ${\mathbb E}[ T_{\rm FR} ]$. Surprisingly, ${\mathbb E}[ T_{\rm FR} ]$ coincides with the result obtained from Kac's lemma that applies to simple random walks (RWs). We also calculate the variance ${\rm Var}(T_{\rm FR})$, which accounts for the variability of first return times between different NBW trajectories. We apply this formalism to Erd{\H o}s-R\'enyi networks, random regular graphs and configuration model networks with exponential and power-law degree distributions and obtain closed-form expressions for $P( T_{\rm FR} > t )$ as well as its mean and variance. These results provide useful insight on the advantages of NBWs over simple RWs in network exploration, sampling and search processes.

cond-mat.stat-mech↗

arXiv2023

The distribution of the number of cycles in directed and undirected random 2-regular graphs

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. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with $\ln N$, while the length of the longest cycle scales with $N$. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles $\langle S \rangle$ scales with $\ln N$. Here we present exact analytical results for the distribution $P_N(S=s)$ of the number of cycles $s$ in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large $N$ limit. The moments and cumulants of $P_N(S=s)$ are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of $N$ objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

cond-mat.stat-mech↗

arXiv2022

Analytical results for the distribution of first-passage times of random walks on random regular graphs

We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of $N$ nodes of degree $c \ge 3$. Starting from a random initial node at time $t=0$, at each time step $t \ge 1$ an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution $P( T_{\rm FP} = t )$ of first-passage times from a random initial node $i$ to a random target node $j$, where $j \ne i$. We distinguish between FP trajectories whose backbone follows the shortest path (SPATH) from the initial node $i$ to the target node $j$ and FP trajectories whose backbone does not follow the shortest path ($\lnot {\rm SPATH}$). More precisely, the SPATH trajectories from the initial node $i$ to the target node $j$ are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network. Moreover, the shortest path between $i$ and $j$ on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly when the length $\ell_{ij}$ of the shortest path between the initial node $i$ and the target node $j$ is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

cond-mat.stat-mech↗

arXiv2022

The mean and variance of the distribution of shortest path lengths of random regular graphs

The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of $N$ nodes of degree $c \ge 3$, the DSPL, denoted by $P(L=\ell)$, follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain closed-form expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by $\langle L \rangle = \frac{\ln N}{\ln (c-1)} + \frac{1}{2} - \frac{ \ln c - \ln (c-2) +γ}{\ln (c-1)} + \mathcal{O} \left( \frac{\ln N}{N} \right)$, where $γ$ is the Euler-Mascheroni constant. While the leading term is known, this result includes a novel correction term, which yields very good agreement with the results obtained from direct numerical evaluation of $\langle L \rangle$ via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of $\langle L \rangle$ as a function of $c$ or $N$. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance ${\rm Var}(L)$ of the DSPL, which captures the overall dependence of the variance on $c$ but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as $N$ is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.

cond-mat.stat-mech↗

arXiv2021

Analytical results for the distribution of cover times of random walks on random regular graphs

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of $N$ nodes of degree $c$ ($c \ge 3$). Starting from a random initial node at time $t=1$, at each time step $t \ge 2$ an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time $T_{\rm C}$ is the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution $P_t(S=s)$ of the number of distinct nodes $s$ visited by an RW up to time $t$ and solve it analytically. Inserting $s=N$ we obtain the cumulative distribution of cover times, namely the probability $P(T_{\rm C} \le t) = P_t(S=N)$ that up to time $t$ an RW will visit all the $N$ nodes in the network. Taking the large network limit, we show that $P(T_{\rm C} \le t)$ converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times $P( T_{{\rm PC},k} = t )$, which is the probability that at time $t$ an RW will complete visiting $k$ distinct nodes. We also calculate the distribution of random cover (RC) times $P( T_{{\rm RC},k} = t )$, which is the probability that at time $t$ an RW will complete visiting all the nodes in a subgraph of $k$ randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

cond-mat.dis-nn↗

arXiv2021

Analytical results for the distribution of first return times of random walks on random regular graphs

We present analytical results for the distribution of first return (FR) times of random walks (RWs) on random regular graphs (RRGs) consisting of $N$ nodes of degree $c \ge 3$. Starting from a random initial node $i$ at time $t=0$, at each time step $t \ge 1$ an RW hops into a random neighbor of its previous node. We calculate the distribution $P ( T_{\rm FR} = t )$ of first return times to the initial node $i$. We distinguish between first return trajectories in which the RW retrocedes its own steps backwards all the way back to the initial node $i$ and those in which the RW returns to $i$ via a path that does not retrocede its own steps. In the retroceding scenario, each edge that belongs to the RW trajectory is crossed the same number of times in the forward and backward directions. In the non-retroceding scenario the subgraph that consists of the nodes visited by the RW and the edges it has crossed between these nodes includes at least one cycle. In the limit of $N \rightarrow \infty$ the RRG converges towards the Bethe lattice. The Bethe lattice exhibits a tree structure, in which all the first return trajectories belong to the retroceding scenario. Moreover, in the limit of $N \rightarrow \infty$ the trajectories of RWs on RRGs are transient in the sense that they return to the initial node with probability $<1$. In this sense they resemble the trajectories of RWs on regular lattices of dimensions $d \ge 3$. The analytical results are found to be in excellent agreement with the results obtained from computer simulations.

cond-mat.stat-mech↗

arXiv2021

Analytical results for the distribution of first hitting times of random walks on random regular graphs

We present analytical results for the distribution of first hitting times of random walks (RWs) on random regular graphs (RRGs) of degree $c \ge 3$ and a finite size $N$. Starting from a random initial node at time $t=1$, at each time step $t \ge 2$ an RW hops randomly into one of the $c$ neighbors of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. The first time at which the RW enters a node that has already been visited before is called the first hitting time or the first intersection length. The first hitting event may take place either by backtracking (BT) to the previous node or by retracing (RET), namely stepping into a node which has been visited two or more time steps earlier. We calculate the tail distribution $P( T_{\rm FH} > t )$ of first hitting (FH) times as well as its mean $\langle T_{\rm FH} \rangle$ and variance ${\rm Var}(T_{\rm FH})$. We also calculate the probabilities $P_{\rm BT}$ and $P_{\rm RET}$ that the first hitting event will occur via the backtracking scenario or via the retracing scenario, respectively. We show that in dilute networks the dominant first hitting scenario is backtracking while in dense networks the dominant scenario is retracing and calculate the conditional distributions $P(T_{\rm FH}=t| {\rm BT})$ and $P(T_{\rm FH}=t| {\rm RET})$, for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the first hitting event as a termination mechanism of the RW trajectories, 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.

cond-mat.dis-nn↗

arXiv2021

The 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. Articulation points are nodes in a network which, if removed, would split the network component on which they are located into two or more separate components, while bredges are edges whose removal would split the network component on which they are located into two separate components. Both articulation points and bredges play an important role in processes of network dismantling and it is therefore useful to know the evolution of the probability of nodes or edges to be articulation points and bredges, respectively, when a fraction of edges is randomly removed from the network in a percolation process. Due to the heterogeneity of the 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, as well as a deconvolution of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. Our methods allow us to obtain these distributions both for large single instances of networks as well as for ensembles of networks in the configuration model class in the thermodynamic limit of infinite system size. We also derive closed form expressions for the large mean degree limit of Erdős-Rényi networks.

cond-mat.dis-nn↗

arXiv2020

Convergence towards an Erd{\H o}s-Rényi graph structure in network contraction processes

In a highly influential paper twenty years ago, Barabási and Albert [Science 286, 509 (1999)] showed that networks undergoing generic growth processes with preferential attachment evolve towards scale-free structures. In any finite system, the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures, attacks or epidemics. Using the master equation formulation and computer simulations we analyze the structural evolution of networks subjected to contraction processes via random, preferential and propagating node deletions. We show that the contracting networks converge towards an Erd{\H o}s-Rényi network structure whose mean degree continues to decrease as the contraction proceeds. This is manifested by the convergence of the degree distribution towards a Poisson distribution and the loss of degree-degree correlations.

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

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

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↗