Localization and centrality in networks

Perceptions of elder financial abuse in the courtroom

In the present study, male and female community members (N = 194) read a fictional civil or criminal trial summary describing an 80-year-old female victim of a lottery scam (losing $10,000 or $100,000). The results yielded a significant main effect of amount stolen: Higher (vs. lower) amounts stolen led to more plaintiff decisions/guilty verdicts and higher pro-victim rating scores. This was qualified by a three-way interaction between type of trial, amount of money, and participant gender. Follow-up analyses showed no differences between conditions for women, but a type of trial x amount stolen interaction for men. Men rendered the most plaintiff decisions/guilty verdicts and higher pro-victim rating scores in civil cases that involved a high amount of money. Analyses on the verdict/decision data revealed that reasons for pro-victim decisions/verdicts centered around a "scam" taking place. Reasons for a verdict/decision supporting the defendant focused on "not enough evidence."

Network Models of BACE-1 Inhibitors: Exploring Structural and Biochemical Relationships

This study investigates the clustering patterns of human β-secretase 1 (BACE-1) inhibitors using complex network methodologies based on various distance functions, including Euclidean, Tanimoto, Hamming, and Levenshtein distances. Molecular descriptor vectors such as molecular mass, Merck Molecular Force Field (MMFF) energy, Crippen partition coefficient (ClogP), Crippen molar refractivity (MR), eccentricity, Kappa indices, Synthetic Accessibility Score, Topological Polar Surface Area (TPSA), and 2D/3D autocorrelation entropies are employed to capture the diverse properties of these inhibitors. The Euclidean distance network demonstrates the most reliable clustering results, with strong agreement metrics and minimal information loss, indicating its robustness in capturing essential structural and physicochemical properties. Tanimoto and Hamming distance networks yield valuable clustering outcomes, albeit with moderate performance, while the Levenshtein distance network shows significant discrepancies. The analysis of eigenvector centrality across different networks identifies key inhibitors acting as hubs, which are likely critical in biochemical pathways. Community detection results highlight distinct clustering patterns, with well-defined communities providing insights into the functional and structural groupings of BACE-1 inhibitors. The study also conducts non-parametric tests, revealing significant differences in molecular descriptors, validating the clustering methodology. Despite its limitations, including reliance on specific descriptors and computational complexity, this study offers a comprehensive framework for understanding molecular interactions and guiding therapeutic interventions. Future research could integrate additional descriptors, advanced machine learning techniques, and dynamic network analysis to enhance clustering accuracy and applicability.

Topological and spectral properties of random digraphs

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs of size n and connection probability p, D(n,p). In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices V_{x}(D) as well as two vertex-degree-based topological indices: the Randić index R(D) and sum-connectivity index χ(D). First, by performing a scaling analysis, we show that the average degree 〈k〉 serves as a scaling parameter for the average values of V_{x}(D), R(D), and χ(D). Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to (n,p), the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius sqrt[np(1-p)]. Subsequently, we compute six different invariants related to the eigenvalues of D(n,p) and observe that these quantities also scale with sqrt[np(1-p)]. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function (n,p). Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.

Quantifying External Information in Social Network Analysis: An Application to Comparative Mythology

Social network analysis is a popular tool to understand the relationships between interacting agents by studying the structural properties of their connections. However, this kind of analysis can miss some of the domain-specific knowledge available in the original information domain and its propagation through the associated network. In this work, we develop an extension of classical social network analysis to incorporate external information from the original source of the network. With this extension we propose a new centrality measure, the semantic value, and a new affinity function, the semantic affinity, that establishes fuzzy-like relationships between the different actors in the network. We also propose a new heuristic algorithm based on the shortest capacity problem to compute this new function. As an illustrative case study, we use the novel proposals to analyze and compare the gods and heroes from three different classical mythologies: 1) Greek; 2) Celtic; and 3) Nordic. We study the relationships of each individual mythology and those of the common structure that is formed when we fuse the three of them. We also compare our results with those obtained using other existing centrality measures and embedding approaches. In addition, we test the proposed measures on a classical social network, the Reuters terror news network, as well as in a Twitter network related to the COVID-19 pandemic. We found that the novel method obtains more meaningful comparisons and results than previous existing approaches in every case.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Ranking cliques in higher-order complex networks

Traditional network analysis focuses on the representation of complex systems with only pairwise interactions between nodes. However, the higher-order structure, which is beyond pairwise interactions, has a great influence on both network dynamics and function. Ranking cliques could help understand more emergent dynamical phenomena in large-scale complex networks with higher-order structures, regarding important issues, such as behavioral synchronization, dynamical evolution, and epidemic spreading. In this paper, motivated by multi-node interactions in a topological simplex, several higher-order centralities are proposed, namely, higher-order cycle (HOC) ratio, higher-order degree, higher-order H-index, and higher-order PageRank (HOP), to quantify and rank the importance of cliques. Experiments on both synthetic and real-world networks support that, compared with other traditional network metrics, the proposed higher-order centralities effectively reduce the dimension of a large-scale network and are more accurate in finding a set of vital nodes. Moreover, since the critical cliques ranked by the HOP and the HOC are scattered over a complex network, the HOP and the HOC outperform other metrics in ranking cliques that are vital in maintaining the network connectivity, thereby facilitating network dynamical synchronization and virus spread control in applications.

Influence maximization: Divide and conquer

The problem of influence maximization, i.e., finding the set of nodes having maximal influence on a network, is of great importance for several applications. In the past two decades, many heuristic metrics to spot influencers have been proposed. Here, we introduce a framework to boost the performance of such metrics. The framework consists in dividing the network into sectors of influence, and then selecting the most influential nodes within these sectors. We explore three different methodologies to find sectors in a network: graph partitioning, graph hyperbolic embedding, and community structure. The framework is validated with a systematic analysis of real and synthetic networks. We show that the gain in performance generated by dividing a network into sectors before selecting the influential spreaders increases as the modularity and heterogeneity of the network increase. Also, we show that the division of the network into sectors can be efficiently performed in a time that scales linearly with the network size, thus making the framework applicable to large-scale influence maximization problems.

Network analysis of the autophagy biochemical network in relation to various autophagy-targeted proteins found among SARS-CoV-2 variants of concern

Autophagy is an important cellular process that triggers a coordinated action involving multiple individual proteins and protein complexes while SARS-CoV-2 (SARS2) was found to both hinder autophagy to evade host defense and utilize autophagy for viral replication. Interestingly, the possible significant stages of the autophagy biochemical network in relation to the corresponding autophagy-targeted SARS2 proteins from the different variants of concern (VOC) were never established. In this study, we performed the following: autophagy biochemical network design and centrality analyses; generated autophagy-targeted SARS2 protein models; and superimposed protein models for structural comparison. We identified 2 significant biochemical pathways (one starts from the ULK complex and the other starts from the PI3P complex) within the autophagy biochemical network. Similarly, we determined that the autophagy-targeted SARS2 proteins (Nsp15, M, ORF7a, ORF3a, and E) are structurally conserved throughout the different SARS2 VOC suggesting that the function of each protein is preserved during SARS2 evolution. Interestingly, among the autophagy-targeted SARS2 proteins, the M protein coincides with the 2 significant biochemical pathways we identified within the autophagy biochemical network. In this regard, we propose that the SARS2 M protein is the main determinant that would influence autophagy outcome in regard to SARS2 infection.

Message passing methods on complex networks

Networks and network computations have become a primary mathematical tool for analyzing the structure of many kinds of complex systems, ranging from the Internet and transportation networks to biochemical interactions and social networks. A common task in network analysis is the calculation of quantities that reside on the nodes of a network, such as centrality measures, probabilities or model states. In this perspective article we discuss message passing methods, a family of techniques for performing such calculations, based on the propagation of information between the nodes of a network. We introduce the message passing approach with a series of examples, give some illustrative applications and results and discuss the deep connections between message passing and phase transitions in networks. We also point out some limitations of the message passing approach and describe some recently introduced methods that address these limitations.

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 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.

Phenotypic Disease Network-Based Multimorbidity Analysis in Idiopathic Cardiomyopathy Patients with Hospital Discharge Records

BACKGROUND

Idiopathic cardiomyopathy (ICM) is a rare disease affecting numerous physiological and biomolecular systems with multimorbidity. However, due to the small sample size of uncommon diseases, the whole spectrum of chronic disease co-occurrence, especially in developing nations, has not yet been investigated. To grasp the multimorbidity pattern, we aimed to present a multidimensional model for ICM and differences among age groups.

METHODS

Hospital discharge records were collected from a rare disease centre of ICM inpatients (n = 1036) over 10 years (2012 to 2021) for this retrospective analysis. One-to-one matched controls were also included. First, by looking at the first three digits of the ICD-10 code, we concentrated on chronic illnesses with a prevalence of more than 1%. The ICM and control inpatients had a total of 71 and 69 chronic illnesses, respectively. Second, to evaluate the multimorbidity pattern in both groups, we built age-specific cosine-index-based multimorbidity networks. Third, the associated rule mining (ARM) assessed the comorbidities with heart failure for ICM, specifically.

RESULTS

The comorbidity burden of ICM was 78% larger than that of the controls. All ages were affected by the burden, although those over 50 years old had more intense interactions. Moreover, in terms of disease connectivity, central, hub, and authority diseases were concentrated in the metabolic, musculoskeletal and connective tissue, genitourinary, eye and adnexa, respiratory, and digestive systems. According to the age-specific connection, the impaired coagulation function was required for raising attention (e.g., autoimmune-attacked digestive and musculoskeletal system disorders) in young adult groups (ICM patients aged 20-49 years). For the middle-aged (50-60 years) and older (≥70 years) groups, malignant neoplasm and circulatory issues were the main confrontable problems. Finally, according to the result of ARM, the comorbidities and comorbidity patterns of heart failure include diabetes mellitus and metabolic disorder, sleeping disorder, renal failure, liver, and circulatory diseases.

CONCLUSIONS

The main cause of the comorbid load is aging. The ICM comorbidities were concentrated in the circulatory, metabolic, musculoskeletal and connective tissue, genitourinary, eye and adnexa, respiratory, and digestive systems. The network-based approach optimizes the integrated care of patients with ICM and advances our understanding of multimorbidity associated with the disease.

Comparing measures of centrality in bipartite patient-prescriber networks: A study of drug seeking for opioid analgesics

Visiting multiple prescribers is a common method for obtaining prescription opioids for nonmedical use and has played an important role in fueling the United States opioid epidemic, leading to increased drug use disorder and overdose. Recent studies show that centrality of the bipartite network formed by prescription ties between patients and prescribers of opioids is a promising indicator for drug seeking. However, node prominence in bipartite networks is typically estimated with methods that do not fully account for the two-mode topology of the underlying network. Although several algorithms have been proposed recently to address this challenge, it is unclear how these algorithms perform on real-world networks. Here, we compare their performance in the context of identifying opioid drug seeking behaviors by applying them to massive bipartite networks of patients and providers extracted from insurance claims data. We find that two variants of bipartite centrality are significantly better predictors of subsequent opioid overdose than traditional centrality estimates. Moreover, we show that incorporating non-network attributes such as the potency of the opioid prescriptions into the measures can further improve their performance. These findings can be reproduced on different datasets. Our results demonstrate the potential of bipartiteness-aware indices for identifying patterns of high-risk behavior.

The rise and fall of countries in the global value chains

Countries become global leaders by controlling international and domestic transactions connecting geographically dispersed production stages. We model global trade as a multi-layer network and study its power structure by investigating the tendency of eigenvector centrality to concentrate on a small fraction of countries, a phenomenon called localization transition. We show that the market underwent a significant drop in power concentration precisely in 2007 just before the global financial crisis. That year marked an inflection point at which new winners and losers emerged and a remarkable reversal of leading role took place between the two major economies, the US and China. We uncover the hierarchical structure of global trade and the contribution of individual industries to variations in countries’ economic dominance. We also examine the crucial role that domestic trade played in leading China to overtake the US as the world’s dominant trading nation. There is an important lesson that countries can draw on how to turn early signals of upcoming downturns into opportunities for growth. Our study shows that, despite the hardships they inflict, shocks to the economy can also be seen as strategic windows countries can seize to become leading nations and leapfrog other economies in a changing geopolitical landscape.

Age- and Sex-Specific Differences in Multimorbidity Patterns and Temporal Trends on Assessing Hospital Discharge Records in Southwest China: Network-Based Study

Background

Multimorbidity represents a global health challenge, which requires a more global understanding of multimorbidity patterns and trends. However, the majority of studies completed to date have often relied on self-reported conditions, and a simultaneous assessment of the entire spectrum of chronic disease co-occurrence, especially in developing regions, has not yet been performed.

Objective

We attempted to provide a multidimensional approach to understand the full spectrum of chronic disease co-occurrence among general inpatients in southwest China, in order to investigate multimorbidity patterns and temporal trends, and assess their age and sex differences.

Methods

We conducted a retrospective cohort analysis based on 8.8 million hospital discharge records of about 5.0 million individuals of all ages from 2015 to 2019 in a megacity in southwest China. We examined all chronic diagnoses using the ICD-10 (International Classification of Diseases, 10th revision) codes at 3 digits and focused on chronic diseases with ≥1% prevalence for each of the age and sex strata, which resulted in a total of 149 and 145 chronic diseases in males and females, respectively. We constructed multimorbidity networks in the general population based on sex and age, and used the cosine index to measure the co-occurrence of chronic diseases. Then, we divided the networks into communities and assessed their temporal trends.

Results

The results showed complex interactions among chronic diseases, with more intensive connections among males and inpatients ≥40 years old. A total of 9 chronic diseases were simultaneously classified as central diseases, hubs, and bursts in the multimorbidity networks. Among them, 5 diseases were common to both males and females, including hypertension, chronic ischemic heart disease, cerebral infarction, other cerebrovascular diseases, and atherosclerosis. The earliest leaps (degree leaps ≥6) appeared at a disorder of glycoprotein metabolism that happened at 25-29 years in males, about 15 years earlier than in females. The number of chronic diseases in the community increased over time, but the new entrants did not replace the root of the community.

Conclusions

Our multimorbidity network analysis identified specific differences in the co-occurrence of chronic diagnoses by sex and age, which could help in the design of clinical interventions for inpatient multimorbidity.

Influential nodes identification using network local structural properties

With the rapid development of information technology, the scale of complex networks is increasing, which makes the spread of diseases and rumors harder to control. Identifying the influential nodes effectively and accurately is critical to predict and control the network system pertinently. Some existing influential nodes detection algorithms do not consider the impact of edges, resulting in the algorithm effect deviating from the expected. Some consider the global structure of the network, resulting in high computational complexity. To solve the above problems, based on the information entropy theory, we propose an influential nodes evaluation algorithm based on the entropy and the weight distribution of the edges connecting it to calculate the difference of edge weights and the influence of edge weights on neighbor nodes. We select eight real-world networks to verify the effectiveness and accuracy of the algorithm. We verify the infection size of each node and top-10 nodes according to the ranking results by the SIR model. Otherwise, the Kendall [Formula: see text] coefficient is used to examine the consistency of our algorithm with the SIR model. Based on the above experiments, the performance of the LENC algorithm is verified.

A network analysis of adolescent mental well-being during the coronavirus pandemic: Evidence for cross-cultural differences in central features

The COVID-19 pandemic continues to pose unprecedented threat globally. Adolescents and youth may be especially susceptible to the long-term impact of these stressors, thus intervening early is an important priority. However, it is also crucial to understand how young people maintain psychological well-being in the face of adversity, particularly given that many nations are experiencing further waves of the pandemic. The understanding of such resilient outcomes could inform the development of programs to encourage positive mental health.We explored adolescents' resilience to the COVID-19 pandemic stress by examining core aspects of well-being across countries using network analysis. Using the short Warwick-Edinburgh Mental Wellbeing Scale, cross-sectional data was collected online from adolescents from India (N = 310; Males = 159, Females = 151, aged 12-18 years), Israel (N = 306; Males = 154, Females = 152, aged 12-18 years) and the United Kingdom (UK; N = 1666; Males = 598, Females = 1068, aged 12-25 years). Two highly similar network clusters were identified for UK and Israel, with three clusters emerging for India. UK and Israeli networks centred on "dealing with problems well" while the Indian network centred on "feeling useful". As central items highlight aspects of well-being that influence or are influenced by other aspects, these findings may inform interventions to safeguard adolescent mental health during future phases of the pandemic.

Generating functions for message passing on weighted networks: Directed bond percolation and susceptible, infected, recovered epidemics

We study the SIR (susceptible, infected, removed/recovered) model on directed graphs with heterogeneous transmission probabilities within the message-passing approximation. We characterize the percolation transition, predict cluster size distributions, and suggest vaccination strategies. All predictions are compared to numerical simulations on real networks. The percolation threshold that we predict is a rigorous lower bound to the threshold on real networks. For large, locally treelike networks, our predictions agree very well with the numerical data.

Approximating nonbacktracking centrality and localization phenomena in large 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.

Adjustable reach in a network centrality based on current flows

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. Most prominent centrality measures can be expressed as an aggregation of influence flows between pairs of nodes. As there are many ways in which influence can be defined, many different centrality measures are in use. Parametrized centralities allow further flexibility and utility by tuning the centrality calculation to the regime most appropriate for a given purpose and network. Here we identify two categories of centrality parameters. Reach parameters control the attenuation of influence flows between distant nodes. Grasp parameters control the centrality's tendency to send influence flows along multiple, often nongeodesic paths. Combining these categories with Borgatti's centrality types [Borgatti, Soc. Networks 27, 55 (2005)0378-873310.1016/j.socnet.2004.11.008], we arrive at a classification system for parametrized centralities. Using this classification, we identify the notable absence of any centrality measures that are radial, reach parametrized, and based on acyclic, conserved flows of influence. We therefore introduce the ground-current centrality, which is a measure of precisely this type. Because of its unique position in the taxonomy, the ground-current centrality differs significantly from similar centralities. We demonstrate that, compared to other conserved-flow centralities, it has a simpler mathematical description. Compared to other reach-parametrized centralities, it robustly preserves an intuitive rank ordering across a wide range of network architectures, capturing aspects of both the closeness and betweenness centralities. We also show that it produces a consistent distribution of centrality values among the nodes, neither trivially equally spread (delocalization) nor overly focused on a few nodes (localization). Other reach-parametrized centralities exhibit both of these behaviors on regular networks and hub networks, respectively. We compare the properties of the ground-current centrality with several other reach-parametrized centralities on four artificial networks and seven real-world networks.

Impact of presymptomatic transmission on epidemic spreading in contact networks: A dynamic message-passing analysis

Infectious diseases that incorporate presymptomatic transmission are challenging to monitor, model, predict, and contain. We address this scenario by studying a variant of a stochastic susceptible-exposed-infected-recovered model on arbitrary network instances using an analytical framework based on the method of dynamic message passing. This framework provides a good estimate of the probabilistic evolution of the spread on both static and contact networks, offering a significantly improved accuracy with respect to individual-based mean-field approaches while requiring a much lower computational cost compared to numerical simulations. It facilitates the derivation of epidemic thresholds, which are phase boundaries separating parameter regimes where infections can be effectively contained from those where they cannot. These have clear implications on different containment strategies through topological (reducing contacts) and infection parameter changes (e.g., social distancing and wearing face masks), with relevance to the recent COVID-19 pandemic.

Belief propagation for networks with loops

Belief propagation is a widely used message passing method for the solution of probabilistic models on networks such as epidemic models, spin models, and Bayesian graphical models, but it suffers from the serious shortcoming that it works poorly in the common case of networks that contain short loops. Here, we provide a solution to this long-standing problem, deriving a belief propagation method that allows for fast calculation of probability distributions in systems with short loops, potentially with high density, as well as giving expressions for the entropy and partition function, which are notoriously difficult quantities to compute. Using the Ising model as an example, we show that our approach gives excellent results on both real and synthetic networks, improving substantially on standard message passing methods. We also discuss potential applications of our method to a variety of other problems.

Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks

Simple models of infectious diseases tend to assume random mixing of individuals, but real interactions are not random pairwise encounters: they occur within various types of gatherings such as workplaces, households, schools, and concerts, best described by a higher-order network structure. We model contagions on higher-order networks using group-based approximate master equations, in which we track all states and interactions within a group of nodes and assume a mean-field coupling between them. Using the susceptible-infected-susceptible dynamics, our approach reveals the existence of a mesoscopic localization regime, where a disease can concentrate and self-sustain only around large groups in the network overall organization. In this regime, the phase transition is smeared, characterized by an inhomogeneous activation of the groups. At the mesoscopic level, we observe that the distribution of infected nodes within groups of the same size can be very dispersed, even bimodal. When considering heterogeneous networks, both at the level of nodes and at the level of groups, we characterize analytically the region associated with mesoscopic localization in the structural parameter space. We put in perspective this phenomenon with eigenvector localization and discuss how a focus on higher-order structures is needed to discern the more subtle localization at the mesoscopic level. Finally, we discuss how mesoscopic localization affects the response to structural interventions and how this framework could provide important insights for a broad range of dynamics.

Model-free hidden geometry of complex networks

The fundamental idea of embedding a network in a metric space is rooted in the principle of proximity preservation. Nodes are mapped into points of the space with pairwise distance that reflects their proximity in the network. Popular methods employed in network embedding either rely on implicit approximations of the principle of proximity preservation or implement it by enforcing the geometry of the embedding space, thus hindering geometric properties that networks may spontaneously exhibit. Here we take advantage of a model-free embedding method explicitly devised for preserving pairwise proximity and characterize the geometry emerging from the mapping of several networks, both real and synthetic. We show that the learned embedding has simple and intuitive interpretations: the distance of a node from the geometric center is representative for its closeness centrality, and the relative positions of nodes reflect the community structure of the network. Proximity can be preserved in relatively low-dimensional embedding spaces, and the hidden geometry displays optimal performance in guiding greedy navigation regardless of the specific network topology. We finally show that the mapping provides a natural description of contagion processes on networks, with complex spatiotemporal patterns represented by waves propagating from the geometric center to the periphery. The findings deepen our understanding of the model-free hidden geometry of complex networks.

Percolation in random graphs with higher-order clustering

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally treelike and does not contain short-range loops between neighbors. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalization to the Molloy-Reed criterion for these networks to describe their critical properties and derive an approximate analytical description of the size of the giant component, providing solutions for Poisson and power-law networks. We find that networks comprising larger simple cycles behave increasingly more treelike. Conversely, clustering composed of larger cliques increasingly deviate from the treelike solution, although the behavior is strongly dependent on the degree-assortativity.

Localization and Universality of Eigenvectors in Directed Random Graphs

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, if the fourth moment of the degree distribution is finite, then the critical mean degree of the localization transition is independent of the degree fluctuations, which is different from localization in undirected graphs that is governed by degree fluctuations. We also show that in the high connectivity limit the distribution of the right eigenvector components is solely determined by the degree distribution. For delocalized eigenvectors, we recover in this limit the universal results from standard random matrix theory that are independent of the degree distribution, while for localized eigenvectors the eigenvector distribution depends on the degree distribution.

Spacing ratio characterization of the spectra of directed random networks

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdős-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

The localization of non-backtracking centrality in networks and its physical consequences

The spectrum of the non-backtracking matrix plays a crucial role in determining various structural and dynamical properties of networked systems, ranging from the threshold in bond percolation and non-recurrent epidemic processes, to community structure, to node importance. Here we calculate the largest eigenvalue of the non-backtracking matrix and the associated non-backtracking centrality for uncorrelated random networks, finding expressions in excellent agreement with numerical results. We show however that the same formulas do not work well for many real-world networks. We identify the mechanism responsible for this violation in the localization of the non-backtracking centrality on network subgraphs whose formation is highly unlikely in uncorrelated networks, but rather common in real-world structures. Exploiting this knowledge we present an heuristic generalized formula for the largest eigenvalue, which is remarkably accurate for all networks of a large empirical dataset. We show that this newly uncovered localization phenomenon allows to understand the failure of the message-passing prediction for the percolation threshold in many real-world structures.

Network analysis reveals essential proteins that regulate sodium-iodide symporter expression in anaplastic thyroid carcinoma

Anaplastic thyroid carcinoma (ATC) is the most rare and lethal form of thyroid cancer and requires effective treatment. Efforts have been made to restore sodium-iodide symporter (NIS) expression in ATC cells where it has been downregulated, yet without complete success. Systems biology approaches have been used to simplify complex biological networks. Here, we attempt to find more suitable targets in order to restore NIS expression in ATC cells. We have built a simplified protein interaction network including transcription factors and proteins involved in MAPK, TGFβ/SMAD, PI3K/AKT, and TSHR signaling pathways which regulate NIS expression, alongside proteins interacting with them. The network was analyzed, and proteins were ranked based on several centrality indices. Our results suggest that the protein interaction network of NIS expression regulation is modular, and distance-based and information-flow-based centrality indices may be better predictors of important proteins in such networks. We propose that the high-ranked proteins found in our analysis are expected to be more promising targets in attempts to restore NIS expression in ATC cells.

Heterogeneous node responses to multi-type epidemics on networks

Having knowledge of the contact network over which an infection is spreading opens the possibility of making individualized predictions for the likelihood of different nodes to become infected. When multiple infective strains attempt to spread simultaneously we may further ask which strain, or strains, are most likely to infect a particular node. In this article we investigate the heterogeneity in likely outcomes for different nodes in two models of multi-type epidemic spreading processes. For models allowing co-infection we derive message-passing equations whose solution captures how the likelihood of a given node receiving a particular infection depends on both the position of the node in the network and the interaction between the infection types. For models of competing epidemics in which co-infection is impossible, a more complicated analysis leads to the simpler result that node vulnerability factorizes into a contribution from the network topology and a contribution from the infection parameters.

Susceptible-infected-susceptible model on networks with eigenvector localization

It is a longstanding debate on the absence of threshold for susceptible-infected-susceptible (SIS) model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network gives rise to the inactive Griffiths phase featuring slow decay of the activity localized around highly connected nodes due to the dynamical fluctuation. We show how it dramatically changes our understanding of the SIS model, opening up new possibilities for the debate. We derive the critical condition for Griffiths to active phase transition: on average, an infected node can further infect another one in the characteristic lifespan of the star subgraph composed of the node and its nearest neighbors. The system approaches the critical point of avoiding the irreversible dynamical fluctuation and the trap of absorbing state. As a signature of the phase transition, the infection density of a node is not only proportional to its degree, but also proportional to the exponentially growing lifespan of the star. And the divergence of the average lifespan of the stars is responsible for the vanishing threshold in the thermodynamic limit. The eigenvector localization exponentially reinforces the infection of highly connected nodes, while it inversely suppresses the infection of small-degree nodes.

Influencer identification in dynamical complex systems

The integrity and functionality of many real-world complex systems hinge on a small set of pivotal nodes, or influencers. In different contexts, these influencers are defined as either structurally important nodes that maintain the connectivity of networks, or dynamically crucial units that can disproportionately impact certain dynamical processes. In practice, identification of the optimal set of influencers in a given system has profound implications in a variety of disciplines. In this review, we survey recent advances in the study of influencer identification developed from different perspectives, and present state-of-the-art solutions designed for different objectives. In particular, we first discuss the problem of finding the minimal number of nodes whose removal would breakdown the network (i.e. the optimal percolation or network dismantle problem), and then survey methods to locate the essential nodes that are capable of shaping global dynamics with either continuous (e.g. independent cascading models) or discontinuous phase transitions (e.g. threshold models). We conclude the review with a summary and an outlook.

Estimating degree-degree correlation and network cores from the connectivity of high-degree nodes in complex networks

Many of the structural characteristics of a network depend on the connectivity with and within the hubs. These dependencies can be related to the degree of a node and the number of links that a node shares with nodes of higher degree. In here we revise and present new results showing how to construct network ensembles which give a good approximation to the degree-degree correlations, and hence to the projections of this correlation like the assortativity coefficient or the average neighbours degree. We present a new bound for the structural cut-off degree based on the connectivity within the hubs. Also we show that the connections with and within the hubs can be used to define different networks cores. Two of these cores are related to the spectral properties and walks of length one and two which contain at least on hub node, and they are related to the eigenvector centrality. We introduce a new centrality measured based on the connectivity with the hubs. In addition, as the ensembles and cores are related by the connectivity of the hubs, we show several examples how changes in the hubs linkage effects the degree-degree correlations and core properties.

Network-based methods for predicting essential genes or proteins: a survey

Genes that are thought to be critical for the survival of organisms or cells are called essential genes. The prediction of essential genes and their products (essential proteins) is of great value in exploring the mechanism of complex diseases, the study of the minimal required genome for living cells and the development of new drug targets. As laboratory methods are often complicated, costly and time-consuming, a great many of computational methods have been proposed to identify essential genes/proteins from the perspective of the network level with the in-depth understanding of network biology and the rapid development of biotechnologies. Through analyzing the topological characteristics of essential genes/proteins in protein-protein interaction networks (PINs), integrating biological information and considering the dynamic features of PINs, network-based methods have been proved to be effective in the identification of essential genes/proteins. In this paper, we survey the advanced methods for network-based prediction of essential genes/proteins and present the challenges and directions for future research.

Beyond non-backtracking: non-cycling network centrality measures

Walks around a graph are studied in a wide range of fields, from graph theory and stochastic analysis to theoretical computer science and physics. In many cases it is of interest to focus on non-backtracking walks; those that do not immediately revisit their previous location. In the network science context, imposing a non-backtracking constraint on traditional walk-based node centrality measures is known to offer tangible benefits. Here, we use the Hashimoto matrix construction to characterize, generalize and study such non-backtracking centrality measures. We then devise a recursive extension that systematically removes triangles, squares and, generally, all cycles up to a given length. By characterizing the spectral radius of appropriate matrix power series, we explore how the universality results on the limiting behaviour of classical walk-based centrality measures extend to these non-cycling cases. We also demonstrate that the new recursive construction gives rise to practical centrality measures that can be applied to large-scale networks.

Predicting the Speed of Epidemics Spreading in Networks

Global transport and communication networks enable information, ideas, and infectious diseases to now spread at speeds far beyond what has historically been possible. To effectively monitor, design, or intervene in such epidemic-like processes, there is a need to predict the speed of a particular contagion in a particular network, and to distinguish between nodes that are more likely to become infected sooner or later during an outbreak. Here, we study these quantities using a message-passing approach to derive simple and effective predictions that are validated against epidemic simulations on a variety of real-world networks with good agreement. In addition to individualized predictions for different nodes, we find an overall sudden transition from low density to almost full network saturation as the contagion progresses in time. Our theory is developed and explained in the setting of simple contagions on treelike networks, but we are also able to show how the method extends remarkably well to complex contagions and highly clustered networks.

Experimental datasets of networks of nonlinear oscillators: Structure and dynamics during the path to synchronization

The analysis of the interplay between structural and functional networks require experiments where both the specific structure of the connections between nodes and the time series of the underlying dynamical units are known at the same time. However, real datasets typically contain only one of the two ways (structural or functional) a network can be observed. Here, we provide experimental recordings of the dynamics of 28 nonlinear electronic circuits coupled in 20 different network configurations. For each network, we modify the coupling strength between circuits, going from an incoherent state of the system to a complete synchronization scenario. Time series containing 30000 points are recorded using a data-acquisition card capturing the analogic output of each circuit. The experiment is repeated three times for each network structure allowing to track the path to the synchronized state both at the level of the nodes (with its direct neighbours) and at the whole network. These datasets can be useful to test new metrics to evaluate the coordination between dynamical systems and to investigate to what extent the coupling strength is related to the correlation between functional and structural networks.

Analysis of the dynamics and topology dependencies of small perturbations in electric transmission grids

We study the phase dynamics in power grids in response to small disturbances and how this depends on the grid topology. To this end, we consider the swing equations in linear order in phase disturbances and solve the resulting linear wave equation, deriving the eigenmodes of the weighted graph Laplacian. A linear response expression for the deviation of frequency is given in terms of these eigenvalues and eigenvectors, which it is argued to be the basis for future power system stabilizers and other control measures in power systems. As an example, we present results for random networks based on the Watts-Strogatz model, where we observe a transition to localized eigenstates as the randomness in the degree distribution grows. Moreover, it is found that localization leads to faster decay rates. Thereby, disturbances are found to remain localized on a few nodes where they decay faster. Finally, we also consider the German transmission grid topology, where the eigenstate of the lowest eigenfrequency, the Fiedler vector, is found to be extended, with large intensities at the northwestern and southern boundaries.

Eigenvalue repulsion and eigenvector localization in sparse non-Hermitian random matrices

Complex networks with directed, local interactions are ubiquitous in nature and often occur with probabilistic connections due to both intrinsic stochasticity and disordered environments. Sparse non-Hermitian random matrices arise naturally in this context and are key to describing statistical properties of the nonequilibrium dynamics that emerges from interactions within the network structure. Here we study one-dimensional (1D) spatial structures and focus on sparse non-Hermitian random matrices in the spirit of tight-binding models in solid state physics. We first investigate two-point eigenvalue correlations in the complex plane for sparse non-Hermitian random matrices using methods developed for the statistical mechanics of inhomogeneous two-dimensional interacting particles. We find that eigenvalue repulsion in the complex plane directly correlates with eigenvector delocalization. In addition, for 1D chains and rings with both disordered nearest-neighbor connections and self-interactions, the self-interaction disorder tends to decorrelate eigenvalues and localize eigenvectors more than simple hopping disorder. However, remarkable resistance to eigenvector localization by disorder is provided by large cycles, such as those embodied in 1D periodic boundary conditions under strong directional bias. The directional bias also spatially separates the left and right eigenvectors, leading to interesting dynamics in excitation and response. These phenomena have important implications for asymmetric random networks and highlight a need for mathematical tools to describe and understand them analytically.

Co-prescription network reveals social dynamics of opioid doctor shopping

This paper examines network prominence in a co-prescription network as an indicator of opioid doctor shopping (i.e., fraudulent solicitation of opioids from multiple prescribers). Using longitudinal data from a large commercially insured population, we construct a network where a tie between patients is weighted by the number of shared opioid prescribers. Given prior research suggesting that doctor shopping may be a social process, we hypothesize that active doctor shoppers will occupy central structural positions in this network. We show that network prominence, operationalized using PageRank, is associated with more opioid prescriptions, higher predicted risk for dangerous morphine dosage, opioid overdose, and opioid use disorder, controlling for number of prescribers and other variables. Moreover, as a patient's prominence increases over time, so does their risk for these outcomes, compared to their own average level of risk. Results highlight the importance of co-prescription networks in characterizing high-risk social dynamics.

Systematic comparison between methods for the detection of influential spreaders in complex networks

Influence maximization is the problem of finding the set of nodes of a network that maximizes the size of the outbreak of a spreading process occurring on the network. Solutions to this problem are important for strategic decisions in marketing and political campaigns. The typical setting consists in the identification of small sets of initial spreaders in very large networks. This setting makes the optimization problem computationally infeasible for standard greedy optimization algorithms that account simultaneously for information about network topology and spreading dynamics, leaving space only to heuristic methods based on the drastic approximation of relying on the geometry of the network alone. The literature on the subject is plenty of purely topological methods for the identification of influential spreaders in networks. However, it is unclear how far these methods are from being optimal. Here, we perform a systematic test of the performance of a multitude of heuristic methods for the identification of influential spreaders. We quantify the performance of the various methods on a corpus of 100 real-world networks; the corpus consists of networks small enough for the application of greedy optimization so that results from this algorithm are used as the baseline needed for the analysis of the performance of the other methods on the same corpus of networks. We find that relatively simple network metrics, such as adaptive degree or closeness centralities, are able to achieve performances very close to the baseline value, thus providing good support for the use of these metrics in large-scale problem settings. Also, we show that a further 2–5% improvement towards the baseline performance is achievable by hybrid algorithms that combine two or more topological metrics together. This final result is validated on a small collection of large graphs where greedy optimization is not applicable.

Coevolution spreading in complex networks

The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.

Spectra of networks containing short loops

The spectrum of the adjacency matrix plays several important roles in the mathematical theory of networks and network data analysis, for example in percolation theory, community detection, centrality measures, and the theory of dynamical systems on networks. A number of methods have been developed for the analytic computation of network spectra, but they typically assume that networks are locally treelike, meaning that the local neighborhood of any node takes the form of a tree, free of short loops. Empirically observed networks, by contrast, often have many short loops. Here we develop an approach for calculating the spectra of networks with short loops using a message passing method. We give example applications to some previously studied classes of networks and find that the presence of loops induces substantial qualitative changes in the shape of network spectra, generating asymmetries, multiple spectral bands, and other features.

Multifractality in random networks with power-law decaying bond strengths

In this paper we demonstrate numerically that random networks whose adjacency matrices A are represented by a diluted version of the power-law banded random matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one-dimensional samples with random long-range bonds. The bond strengths of the model, which decay as a power-law, are tuned by the parameter μ as A_{mn}∝|m-n|^{-μ}; while the sparsity is driven by the average network connectivity α: for α=0 the vertices in the network are isolated and for α=1 the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value μ=μ_{c}=1, we clearly show [from the scaling of the relative fluctuation of the participation number I_{2} as well as the scaling of the probability distribution functions P(lnI_{2})] the existence of the critical value μ_{c}≡μ_{c}(α) for α<1. Moreover, we characterize the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions D_{q}, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers exp〈lnI_{q}〉.

Effect of network clustering on mutually cooperative coinfections

The spread of an infectious disease can be promoted by previous infections with other pathogens. This cooperative effect can give rise to violent outbreaks, reflecting the presence of an abrupt epidemic transition. As for other diffusive dynamics, the topology of the interaction pattern of the host population plays a crucial role. It was conjectured that a discontinuous transition arises when there are relatively few short loops and many long loops in the contact network. Here we focus on the role of local clustering in determining the nature of the transition. We consider two mutually cooperative pathogens diffusing in the same population: An individual already infected with one disease has an increased probability of getting infected by the other. We look at how a disease obeying the susceptible-infected-removed dynamics spreads on contact networks with tunable clustering. Using numerical simulations we show that for large cooperativity the epidemic transition is always abrupt, with the discontinuity decreasing as clustering is increased. For large clustering strong finite-size effects are present and the discontinuous nature of the transition is manifest only in large networks. We also investigate the problem of influential spreaders for cooperative infections, revealing that both cooperativity and clustering strongly enhance the dependence of the spreading influence on the degree of the initial seed.