arXiv subjects

S. N. Dorogovtsev

Publications and source records attributed to S. N. Dorogovtsev.

At least 19 recordsLinked to original sources

arXiv2026

Early warning signals for phase transitions in networks

The percolation phase transition in complex network systems attracts much attention and has numerous applications in various research fields. Finite size effects smooth the transition and make it difficult to predict the critical point of appearance or disappearance of the giant connected component. For this end, we introduce the susceptibility of arbitrary random undirected and directed networks and show that a strong increase of the susceptibility is the early warning signal of approaching the transition point. Our method is based on the introduction of `observers', which are randomly chosen nodes monitoring the local connectivity of a network. To demonstrate efficiency of the method, we derive explicit equations determining the susceptibility and study its critical behavior near continuous and mixed-order phase transitions in uncorrelated random undirected and directed networks, networks with dependency links, and $k$-cores of networks. The universality of the critical behavior is supported by the phenomenological Landau theory of phase transitions.

cond-mat.dis-nn↗

arXiv2025

Deterministic simplicial complexes

We investigate simplicial complexes deterministically growing from a single vertex. In the first step, a vertex and an edge connecting it to the primordial vertex are added. The resulting simplicial complex has a 1-dimensional simplex and two 0-dimensional faces (the vertices). The process continues recursively: On the $n$-th step, every existing $d-$dimensional simplex ($d\leq n-1$) joins a new vertex forming a $(d+1)-$dimensional simplex; all $2^{d+1}-2$ new faces are also added so that the resulting object remains a simplicial complex. The emerging simplicial complex has intriguing local and global characteristics. The number of simplices grows faster than $n!$, and the upper-degree distributions follow a power law. Here, the upper degree (or $d$-degree) of a $d$-simplex refers to the number of $(d{+}1)$-simplices that share it as a face. Interestingly, the $d$-degree distributions evolve quite differently for different values of $d$. We compute the Hodge Laplacian spectra of simplicial complexes and show that the spectral and Hausdorff dimensions are infinite. We also explore a constrained version where the dimension of the added simplices is fixed to a finite value $m$. In the constrained model, the number of simplices grows exponentially. In particular, for $m=1$, the spectral dimension is $2$. For $m=2$, the spectral dimension is finite, and the degree distribution follows a power law, while the $1$-degree distribution decays exponentially.

math.CO↗

arXiv2023

Statistics of remote regions of networks

We delve into the statistical properties of regions within complex networks that are distant from vertices with high centralities, such as hubs or highly connected clusters. These remote regions play a pivotal role in shaping the asymptotic behaviours of various spreading processes and the features of associated spectra. We investigate the probability distribution $P_{\geq m}(s)$ of the number $s$ of vertices located at distance $m$ or beyond from a randomly chosen vertex in an undirected network. Earlier, this distribution and its large $m$ asymptotics $1/s^2$ were obtained theoretically for undirected uncorrelated networks [S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Nucl. Phys. B 653 (2003) 307]. Employing numerical simulations and analysing empirical data, we explore a wide range of real undirected networks and their models, including trees and loopy networks, and reveal that the inverse square law is valid even for networks with strong correlations. We observe this law in the networks demonstrating the small-world effect and containing vertices with degree $1$ (so-called leaves or dead ends). We find the specific classes of networks for which this law is not valid. Such networks include the finite-dimensional networks and the networks embedded in finite-dimensional spaces. We notice that long chains of nodes in networks reduce the range of $m$ for which the inverse square law can be spotted. Interestingly, we detect such long chains in the remote regions of the undirected projection of a large Web domain.

physics.soc-ph↗

arXiv2023

Localization of nonbacktracking centrality on dense subgraphs of sparse networks

The nonbacktracking matrix, and the related nonbacktracking centrality (NBC) play a crucial role in models of percolation-type processes on networks, such as non-recurrent epidemics. Here we study the localization of NBC in infinite sparse networks that contain an arbitrary finite subgraph. Assuming the local tree-likeness of the enclosing network, and that branches emanating from the finite subgraph do not intersect at finite distances, we show that the largest eigenvalue of the nonbacktracking matrix of the composite network is equal to the highest of the two largest eigenvalues: that of the finite subgraph and of the enclosing network. In the localized state, when the largest eigenvalue of the subgraph is the highest of the two, we derive explicit expressions for the NBCs of nodes in the subgraph and other nodes in the network. In this state, nonbacktracking centrality is concentrated on the subgraph and its immediate neighbourhood in the enclosing network. We obtain simple, exact formulas in the case where the enclosing network is uncorrelated. We find that the mean NBC decays exponentially around the finite subgraph, at a rate which is independent of the structure of the enclosing network, contrary to what was found for the localization of the principal eigenvector of the adjacency matrix. Numerical simulations confirm that our results provide good approximations even in moderately sized, loopy, real-world networks.

physics.soc-ph↗

arXiv2022

Weak percolation on multiplex networks with overlapping edges

We solve the weak percolation problem for multiplex networks with overlapping edges. In weak percolation, a vertex belongs to a connected component if at least one of its neighbors in each of the layers is in this component. This is a weaker condition than for a mutually connected component in interdependent networks, in which any two vertices must be connected by a path within each of the layers. The effect of the overlaps on weak percolation turns out to be opposite to that on the giant mutually connected component. While for the giant mutually connected component, overlaps do not change the critical phenomena, our theory shows that in two layers any (nonzero) concentration of overlaps drives the weak percolation transition to the ordinary percolation universality class. In three layers, the phase diagram of the problem contains two lines -- of a continuous phase transition and of a discontinuous one -- connected in various ways depending on how the layers overlap. In the case of only doubled overlapped edges, two of the end points of these lines coincide, resulting in a tricritical point like that seen in heterogeneous $k$-core percolation.

cond-mat.dis-nn↗

arXiv2022

Hidden transition in multiplex networks

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, $P(0)=0$. Above a critical value of a control parameter, the removal of a tiny fraction $Δ$ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, $Π=P(1)$. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point $Π_c$, respectively. In the limit $Δ\to0$ the total collapse for $Π>Π_\text{c}$ takes a time $T \propto 1/(Π-Π_\text{c})$, while there is an exponential relaxation below $Π_\text{c}$ with a relaxation time $τ\propto 1/[Π_\text{c}-Π]$.

physics.soc-ph↗

arXiv2021

Approximating nonbacktracking centrality and localization phenomena in large networks

Message-passing theories have proved to be invaluable tools in studying percolation, non-recurrent epidemics and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix whose largest eigenvalue, the corresponding eigenvector, and the closely related nonbacktracking centrality play a central role in determining how the given dynamical model behaves. Here we propose a degree-class-based method to approximate these quantities using a smaller matrix related to the joint degree-degree distribution of neighbouring nodes. Our findings suggest that in most networks degree-degree correlations beyond nearest neighbour are actually not strong, and our first-order description already results in accurate estimates, particularly when message-passing itself is a good approximation to the original model in question, that is when the number of short cycles in the network is sufficiently low. We show that localization of the nonbacktracking centrality is also captured well by our scheme, particularly in large networks. Our method provides an alternative to working with the full nonbacktracking matrix in very large networks where this may not be possible due to memory limitations.

physics.soc-ph↗

arXiv2020

Effect of the initial configuration of weights on the training and function of artificial neural networks

The function and performance of neural networks is largely determined by the evolution of their weights and biases in the process of training, starting from the initial configuration of these parameters to one of the local minima of the loss function. We perform the quantitative statistical characterization of the deviation of the weights of two-hidden-layer ReLU networks of various sizes trained via Stochastic Gradient Descent (SGD) from their initial random configuration. We compare the evolution of the distribution function of this deviation with the evolution of the loss during training. We observed that successful training via SGD leaves the network in the close neighborhood of the initial configuration of its weights. For each initial weight of a link we measured the distribution function of the deviation from this value after training and found how the moments of this distribution and its peak depend on the initial weight. We explored the evolution of these deviations during training and observed an abrupt increase within the overfitting region. This jump occurs simultaneously with a similarly abrupt increase recorded in the evolution of the loss function. Our results suggest that SGD's ability to efficiently find local minima is restricted to the vicinity of the random initial configuration of weights.

cs.LG↗

arXiv2020

Weak Multiplex Percolation

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. We describe a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. We fully describe the critical phenomena of this process. In particular, in two layers, with finite second moments of the degree distributions, an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers we find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+ 1/(M-1)$ for $M$ layers.

cond-mat.dis-nn↗

arXiv2020

Filtering Statistics on Networks

We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable simple example of information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring we obtained an exact solution of the problem and described numerically more difficult filter setups. For each of the filter patterns and networks we found a few numbers essentially describing the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures appear to be essentially determined by two factors: whether the graphs structure is deterministic or random, and the vertex degree. We find that filtering in random graphs produces a much richer statistics than in deterministic graphs. This statistical richness is reduced by increasing the graph's degree.

nlin.CG↗

arXiv2020

Exotic Critical Behavior of Weak Multiplex Percolation

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several non-typical phenomena. In two layers, the giant component emerges with a continuos phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent $γ$, the discontinuity vanishes but at $γ=1.5$ in three layers, more generally at $γ= 1+ 1/(M-1)$ in $M$ layers.

cond-mat.dis-nn↗

arXiv2020

Complex distributions emerging in filtering and compression

In filtering, each output is produced by a certain number of different inputs. We explore the statistics of this degeneracy in an explicitly treatable filtering problem in which filtering performs the maximal compression of relevant information contained in inputs (arrays of zeroes and ones). This problem serves as a reference model for the statistics of filtering and related sampling problems. The filter patterns in this problem conveniently allow a microscopic, combinatorial consideration. This allows us to find the statistics of outputs, namely the exact distribution of output degeneracies, for arbitrary input sizes. We observe that the resulting degeneracy distribution of outputs decays as $e^{-c\log^α\!d}$ with degeneracy $d$, where $c$ is a constant and exponent $α>1$, i.e. faster than a power law. Importantly, its form essentially depends on the size of the input data set, appearing to be closer to a power-law dependence for small data set sizes than for large ones. We demonstrate that for sufficiently small input data set sizes typical for empirical studies, this distribution could be easily perceived as a power law. We extend our results to filter patterns of various sizes and demonstrate that the shortest filter pattern provides the maximum informative representations of the inputs.

cond-mat.dis-nn↗

arXiv2019

Generalization of core percolation on complex networks

We introduce a $k$-leaf removal algorithm as a generalization of the so-called leaf removal algorithm. In this pruning algorithm, vertices of degree smaller than $k$, together with their first nearest neighbors and all incident edges are progressively removed from a random network. As the result of this pruning the network is reduced to a subgraph which we call the Generalized $k$-core ($Gk$-core). Performing this pruning for the sequence of natural numbers $k$, we decompose the network into a hierarchy of progressively nested $Gk$-cores. We present an analytical framework for description of $Gk$-core percolation for undirected uncorrelated networks with arbitrary degree distributions (configuration model). To confirm our results, we also derive rate equations for the $k$-leaf removal algorithm which enable us to obtain the structural characteristics of the $Gk$-cores in another way. Also we apply our algorithm to a number of real-world networks and perform the $Gk$-core decomposition for them.

cond-mat.dis-nn↗

arXiv2018

Finding the optimal nets for self-folding Kirigami

Three-dimensional shells can be synthesized from the spontaneous self-folding of two-dimensional templates of interconnected panels, called nets. However, some nets are more likely to self-fold into the desired shell under random movements. The optimal nets are the ones that maximize the number of vertex connections, i.e., vertices that have only two of its faces cut away from each other in the net. Previous methods for finding such nets are based on random search and thus do not guarantee the optimal solution. Here, we propose a deterministic procedure. We map the connectivity of the shell into a shell graph, where the nodes and links of the graph represent the vertices and edges of the shell, respectively. Identifying the nets that maximize the number of vertex connections corresponds to finding the set of maximum leaf spanning trees of the shell graph. This method allows not only to design the self-assembly of much larger shell structures but also to apply additional design criteria, as a complete catalog of the maximum leaf spanning trees is obtained.

cs.DS↗

arXiv2017

Mapping the Structure of Directed Networks: Beyond the "Bow-tie" Diagram

We reveal a hierarchical, multilayer organization of finite components -- i.e., tendrils and tubes -- around the giant connected components in directed networks and propose efficient algorithms allowing one to uncover the entire organization of key real-world directed networks, such as the World Wide Web, the neural network of \emph{Caenorhabditis elegans}, and others. With increasing damage, the giant components decrease in size while the number and size of tendril layers increase, enhancing the susceptibility of the networks to damage.

cond-mat.dis-nn↗

arXiv2016

A unified approach to percolation processes on multiplex networks

Many real complex systems cannot be represented by a single network, but due to multiple sub-systems and types of interactions, must be represented as a multiplex network. This is a set of nodes which exist in several layers, with each layer having its own kind of edges, represented by different colours. An important fundamental structural feature of networks is their resilience to damage, the percolation transition. Generalisation of these concepts to multiplex networks requires careful definition of what we mean by connected clusters. We consider two different definitions. One, a rigorous generalisation of the single-layer definition leads to a strong non-local rule, and results in a dramatic change in the response of the system to damage. The giant component collapses discontinuously in a hybrid transition characterised by avalanches of diverging mean size. We also consider another definition, which imposes weaker conditions on percolation and allows local calculation, and also leads to different sized giant components depending on whether we consider an activation or pruning process. This 'weak' process exhibits both continuous and discontinuous transitions.

cond-mat.dis-nn↗

arXiv2016

Metastable Localization of Diseases in Complex Networks

We describe the phenomenon of localization in the epidemic SIS model on highly heterogeneous networks in which strongly connected nodes (hubs) play the role of centers of localization. We find that in this model the localized states below the epidemic threshold are metastable. The longevity and scale of the metastable outbreaks do not show a sharp localization transition, instead there is a smooth crossover from localized to delocalized states as we approach the epidemic threshold from below. Analyzing these long-lasting local outbreaks for a random regular graph with a hub, we show how this localization can be detected from the shape of the distribution of the number of infective nodes.

physics.soc-ph↗

arXiv2015

Ranking scientists

Currently the ranking of scientists is based on the $h$-index, which is widely perceived as an imprecise and simplistic though still useful metric. We find that the $h$-index actually favours modestly performing researchers and propose a simple criterion for proper ranking.

cs.DL↗