Real-world networks often exhibit strong transitivity with nontrivial local clustering spectra and degree correlations. Such features are not easily modelled in tractable network models, creating an obstacle to the theoretical understanding of such complex network structures. Here, we address this problem using a model for strongly clustered random graphs in which each triad of a random backbone is closed with a certain probability. Despite the intricate loopy local structure of the graphs obtained, we provide exact expressions for the local clustering spectrum and the degree correlations, filling the gap in the theoretical description of this model for random graphs. In particular, we find positive degree assortativity accompanies high transitivity, and non-trivial structure in the clustering spectrum. Exact asymptotic analytical results are complemented with extensive numerical characterization of finite size effects.

Many real-world phenomena can be modeled as dynamical processes on networks, a prominent example being the spread of infectious diseases such as COVID-19. Mean-field approximations are a widely used tool to analyze such dynamical processes on networks, but these are typically derived using plausible probabilistic reasoning, introducing uncontrolled errors that may lead to invalid mathematical conclusions. In this paper, we present a rigorous approach to derive mean-field approximations from the exact description of Markov chain dynamics on networks through a process of averaging called approximate lumping. We consider a general class of Markov chain dynamics on networks in which each vertex can adopt a finite number of "vertex-states"" (e.g. susceptible, infected, recovered), and transition rates depend on the number of neighbors of each type. Our approximate lumping is based on counting the number of each type of vertex-state in subsets of vertices, and this results in a density dependent population process. In the large graph limit, this reduces to a low dimensional system of ordinary differential equations, special cases of which are well known mean-field approximations. Our approach provides a general framework for the derivation of mean-field approximations of dynamics on networks that unifies previously disconnected approaches and highlights the sources of error.

Triadic closure, the formation of a connection between two nodes in a network sharing a common neighbor, is considered a fundamental mechanism determining the clustered nature of many real-world topologies. In this work we define a static triadic closure (STC) model for clustered networks, whereby starting from an arbitrary fixed backbone network, each triad is closed independently with a given probability. Assuming a locally treelike backbone we derive exact expressions for the expected number of various small, loopy motifs (triangles, 4-loops, diamonds and 4-cliques) as a function of moments of the backbone degree distribution. In this way we determine how transitivity and its suitably defined generalizations for higher-order motifs depend on the heterogeneity of the original network, revealing the existence of transitions due to the interplay between topologically inequivalent triads in the network. Furthermore, under reasonable assumptions for the moments of the backbone network, we establish approximate relationships between motif densities, which we test in a large dataset of real-world networks. We find a good agreement, indicating that STC is a realistic mechanism for the generation of clustered networks, while remaining simple enough to be amenable to analytical treatment.

Percolation processes on random networks have been the subject of intense research activity over the last decades: the overall phenomenology of standard percolation on uncorrelated and unclustered topologies is well known. Still some critical properties of the transition, in particular for heterogeneous substrates, have not been fully elucidated and contradictory results appear in the literature. In this paper we present, by means of a generating functions approach, a thorough and complete investigation of percolation critical properties in uncorrelated locally treelike random networks. We determine all critical exponents, the associated critical amplitude ratios, and the form of the cluster size distribution for networks of any level of heterogeneity. We uncover, in particular for highly heterogeneous networks, subtle crossover phenomena, nontrivial scaling forms, and violations of hyperscaling. In this way we clarify the origin of inconsistencies in the previous literature.

Decision tree and random forest classification and regression are some of the most widely used in machine learning approaches. Binary decision tree implementations commonly use conditioning in the form 'feature ≤ (or <) threshold', with the threshold being the midpoint between two observed feature values. In this paper, we investigate the bias introduced by the choice of conditioning operator (an intrinsic property of implementations) in the presence of features with lattice characteristics. We propose techniques to eliminate this bias, requiring an additional prediction with decision trees and incurring no cost for random forests. Using 20 classification and 20 regression datasets, we demonstrate that the bias can lead to statistically significant differences in terms of AUC and r^2 scores. The proposed techniques successfully mitigate the bias, compared to the worst-case scenario, statistically significant improvements of up to 0.1-0.2 percentage points of AUC and r^2 scores were achieved and the improvement of 1.5 percentage points of r^2 score was measured in the most sensitive case of random forest regression. The implementation of the study is available on GitHub at the following repository: .

Classical percolation theory underlies many processes of information transfer along the links of a network. In these standard situations, the requirement for two nodes to be able to communicate is the presence of at least one uninterrupted path of nodes between them. In a variety of more recent data transmission protocols, such as the communication of noisy data via error-correcting repeaters, both in classical and quantum networks, the requirement of an uninterrupted path is too strict: two nodes may be able to communicate even if all paths between them have interruptions or gaps consisting of nodes that may corrupt the message. In such a case a different approach is needed. We develop the theoretical framework for extended-range percolation in networks, describing the fundamental connectivity properties relevant to such models of information transfer. We obtain exact results, for any range R, for infinite random uncorrelated networks and we provide a message-passing formulation that works well in sparse real-world networks. The interplay of the extended range and heterogeneity leads to novel critical behavior in scale-free networks.

The nonbacktracking matrix and the related nonbacktracking centrality (NBC) play a crucial role in models of percolation-type processes on networks, such as nonrecurrent 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 neighborhood 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.

Message-passing theories have proved to be invaluable tools in studying percolation, nonrecurrent 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 neighboring nodes. Our findings suggest that in most networks, degree-degree correlations beyond nearest neighbor 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.

The outcome of SIR epidemics with heterogeneous infective lifetimes, or heterogeneous susceptibilities, can be mapped onto a directed percolation process on the underlying contact network. In this paper we study SIR models where heterogeneity is a result of the degree dependence of disease transmission rates. We develop numerical methods to determine the epidemic threshold, the epidemic probability and epidemic size close to the threshold for configuration model contact networks with arbitrary degree distribution and an arbitrary matrix of transmission rates (dependent on transmitting and receiving node degree). For the special case of separable transmission rates we obtain analytical expressions for these quantities. We propose a categorization of spreading processes based on the ratio of the probability of an epidemic and the expected size of an epidemic, and demonstrate that this ratio has a complex dependence on the degree distribution and the degree-dependent transmission rates. For scale-free contact networks and transmission rates that are power functions of transmitting and receiving node degrees, the epidemic threshold may be finite even when the degree distribution powerlaw exponent is below γ < 3. We give an expression, in terms of the degree distribution and transmission rate exponents, for the limit at which the epidemic threshold vanishes. We find that the expected epidemic size and the probability of an epidemic may grow nonlinearly above the epidemic threshold, with exponents that depend not only on the degree distribution powerlaw exponent, but on the parameters of the transmission rate degree dependence functions, in contrast to ordinary directed percolation and previously studied variations of the SIR model.

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: Nodes with dependency neighbors may continue to function if at least one of their dependency neighbors is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erdős-Rényi and random scale-free networks with an underlying Erdős-Rényi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erdős-Rényi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity S-S_{c}∝(p-p_{c})^{1/4}. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.

In many social networks it is a useful assumption that, regarding a given quality, an underlying hierarchy of the connected individuals exists, and that the outcome of interactions is to some extent determined by the relative positions in the hierarchy. We consider a simple and broadly applicable method of estimating individual positions in a linear hierarchy, and the corresponding uncertainties. The method relies on solving a linear system characterized by a modified Laplacian matrix of the underlying network of interactions, and is equivalent to finding the equilibrium configuration of a system of directed linear springs. We provide a simple first-order approximation to the exact solution, which can be evaluated in linear time. The uncertainty of the hierarchy estimate is determined by the network structure and the potantial energy of the corresponding spring system in equilibrium. The method generalizes straightforwardly to multidimensional hierarchies and higher-order, non-pairwise interactions.

The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history of growing trees, based on exact results on root finding. We show that our linear-time algorithm produces the exact stepwise most probable history in a broad class of tree growth models. Our formulation is able to treat very large trees and therefore allows us to make reliable numerical observations regarding the possibility of root inference and history reconstruction in growing trees. We obtain the general formula 〈lnN〉≅NlnN-cN for the size dependence of the mean logarithmic number of possible histories of a given tree, a quantity that largely determines the reconstructability of tree histories. We also reveal an uncertainty principle: a relationship between the inferability of the root and that of the complete history, indicating that there is a tradeoff between the two tasks; the root and the complete history cannot both be inferred with high accuracy at the same time.

By recourse of computational mechanics, back-stresses are unveiled as a major source for the increase in work hardening during forming of hexagonal close-packaged (hcp) metals. Polycrystalline visco-plastic self-consistent (VPSC) and crystal plasticity finite element modelling (CPFEM) simulations of tensile uniaxial experiments were used along with experimental texture information. Simulations took into account the analogous variation in the critical resolved shear stress (CRSS) values of each slip family that could result from an increase in the test temperature. As the CRSS ratio between secondary and primary slip families increased, two different contributions to the variation of the work hardening rate were observed depending on the simulation framework: (i) a decrease in the work hardening rate in VPSC simulations attributed to texture evolution or geometrical hardening and (ii) an increase in the work hardening rate in CPFEM simulations due to back-stresses. While geometrical hardening is present in both simulation frameworks, only CPFEM is able to capture the influence of back-stresses on the increase of the work hardening rate with temperature. The results provided here contribute to a better understanding of the deformation mechanisms present in warm forming of hcp metals, showing also that CPFEM is a better simulation framework to study warm forming of hcp metals.

The giant mutually connected component (GMCC) of an interdependent or multiplex network collapses with a discontinuous hybrid transition under random damage to the network. If the nodes to be damaged are selected in a targeted way, the collapse of the GMCC may occur significantly sooner. Finding the minimal damage set which destroys the largest mutually connected component of a given interdependent network is a computationally prohibitive simultaneous optimization problem. We introduce a simple heuristic strategy -- Effective Multiplex Degree -- for targeted attack on interdependent networks that leverages the indirect damage inherent in multiplex networks to achieve a damage set smaller than that found by any other non computationally intensive algorithm. We show that the intuition from single layer networks that decycling (damage of the $2$-core) is the most effective way to destroy the giant component, does not carry over to interdependent networks, and in fact such approaches are worse than simply removing the highest degree nodes.

Directed networks have various topologically different extensive components, in contrast to a single giant component in undirected networks. We study the sensitivity (response) of the sizes of these extensive components in directed complex networks to the addition and pruning of edges and vertices. We introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the Caenorhabditis elegans neural network, directed Erdős-Rényi graphs, and others. We reveal a nonmonotonic dependence of the sensitivity to random pruning of edges or vertices in the case of C. elegans and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.

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 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.

Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.

Recent reports of discontinuous yielding in the uniaxial compression of extruded magnesium suggest that twinning is accompanied by a significant stress decrease in the parent grain, leading to corresponding stress concentrations. We performed crystal plasticity finite element simulations using a simple twinning implementation that includes this "softening" effect for tensile twinning. With this method we were able to reproduce the experimentally observed yield point elongation and the Lüders-like propagation of strain and twinning over the sample. Twin variant selection was analysed in detail at various stages of deformation. Although lower rank variants were found to be activated, most of the twinning activity occurred on variants with the highest global Schmid factor. The stress fluctuations arising from the stress relaxation due to twinning did not appear to alter the dominance of the highest Schmid factor variant significantly. However, when twin softening is included, the twinning strain is dominated by a single variant. Despite giving rise to the highest stress fluctuations during yielding, simulations using the twin stress relaxation implementation actually showed less heterogeneity in stress after yielding. The results of our simple model suggest that "twin softening" could be key to the development of discontinuous yielding in extruded magnesium alloys.

A majority of studied models for scale-free networks have degree distributions with exponents greater than two. Real networks, however, can demonstrate essentially more heavy-tailed degree distributions. We explore two models of scale-free equilibrium networks that have the degree distribution exponent γ=1, P(q)∼q^{-γ}. Such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high. Our models exploit a rewiring mechanism. They are local in the sense that no knowledge of the network structure, apart from the immediate neighborhood of the vertices, is required. These models generate uncorrelated networks in the infinite size limit, where they are solved explicitly. We investigate finite size effects by the use of simulations. We find that both models exhibit disassortative degree-degree correlations for finite network sizes. In addition, we observe a markedly degree-dependent clustering in the finite networks. We indicate a real-world network with a similar degree distribution.

A majority of studied models for scale-free networks have degree distributions with exponents greater than $2$. Real networks, however, can demonstrate essentially more heavy-tailed degree distributions. We explore two models of scale-free equilibrium networks that have the degree distribution exponent $\gamma = 1$, $P(q) \sim q^{-\gamma}$. Such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high. Our models exploit a rewiring mechanism. They are local in the sense that no knowledge of the network structure, apart from the immediate neighbourhood of the vertices, is required. These models generate uncorrelated networks in the infinite size limit, where they are solved explicitly. We investigate finite size effects by the use of simulations. We find that both models exhibit disassortative degree-degree correlations for finite network sizes. In addition, we observe a markedly degree-dependent clustering in the finite networks. We indicate a real-world network with a similar degree distribution.