Random graphs with arbitrary degree distributions and their applications

BAB-GSL: Using Bayesian influence with attention mechanism to optimize graph structure in basic views

In recent years, Graph Neural Networks (GNNs) have garnered significant attention, with a notable focus on Graph Structure Learning (GSL), a branch dedicated to optimizing graph structures to enhance network training performance. Current GSL methods primarily involve constructing optimized graph representations by analyzing one or more initial graph sources to improve performance in subsequent application tasks. Despite these advancements, achieving high-quality graphs that accurately and robustly reflect node relationships remains challenging. This paper introduces a novel approach, termed BAB-GSL, designed to approximate an ideal graph structure through a systematic process. Specifically, two basic views are extracted from the original graph and utilized as inputs for the model, where the preliminary optimized view is generated through the view fusion module. The Attention mechanism is then applied to the optimized view to improve nodes' connectivity and expressiveness. Subsequently, the trained view is re-structured using a Bayesian optimizer to produce the final graph structure. Extensive experiments were conducted across multiple datasets, both in undisturbed and attacked scenarios, to thoroughly evaluate the proposed method, demonstrating the effectiveness and robustness of the BAB-GSL approach.

Targeting attack activity-driven networks

Real-world complex systems demonstrated temporal features, i.e., the network topology varies with time and should be described as temporal networks since the traditional static networks cannot accurately characterize. To describe the deliberate attack events in the temporal networks, we propose an activity-based targeted attack on the activity-driven network to investigate temporal networks' temporal percolation properties and resilience. Based on the node activity and network mapping framework, the giant component and temporal percolation threshold are solved according to percolation theory and generating function. The theoretical results coincide with the simulation results near the thresholds. We find that targeted attacks can affect the temporal network, while random attacks cannot. As the probability of a highly active node being deleted increases, the temporal percolation threshold increases, and the giant component increases, thus enhancing robustness. When the network's activity distribution is extremely heterogeneous, network robustness decreases consequently. These findings help us to analyze and understand real-world temporal networks.

Broken detailed balance and entropy production in directed networks

The structure of a complex network plays a crucial role in determining its dynamical properties. In this paper , we show that the the degree to which a network is directed and hierarchically organized is closely associated with the degree to which its dynamics break detailed balance and produce entropy. We consider a range of dynamical processes and show how different directed network features affect their entropy production rate. We begin with an analytical treatment of a two-node network followed by numerical simulations of synthetic networks using the preferential attachment and Erdös-Renyi algorithms. Next, we analyze a collection of 97 empirical networks to determine the effect of complex real-world topologies. Finally, we present a simple method for inferring broken detailed balance and directed network structure from multivariate time series and apply our method to identify non-equilibrium dynamics and hierarchical organisation in both human neuroimaging and financial time series. Overall, our results shed light on the consequences of directed network structure on non-equilibrium dynamics and highlight the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.

Resilience of weighted networks with dynamical behavior against multi-node removal

In many real-world networks, interactions between nodes are weighted to reflect their strength, such as predator-prey interactions in the ecological network and passenger numbers in airline networks. These weighted networks are prone to cascading effects caused by minor perturbations, which can lead to catastrophic outcomes. This vulnerability highlights the importance of studying weighted network resilience to prevent system collapses. However, due to many variables and weight parameters coupled together, predicting the behavior of such a system governed by a multi-dimensional rate equation is challenging. To address this, we propose a dimension reduction technique that simplifies a multi-dimensional system into a one-dimensional state space. We applied this methodology to explore the impact of weights on the resilience of four dynamics whose weights are assigned by three weight assignment methods. The four dynamical systems are the biochemical dynamical system (B), the epidemic dynamical system (E), the regulatory dynamical system (R), and the birth-death dynamical system (BD). The results show that regardless of the weight distribution, for B, the weights are negatively correlated with the activities of the network, while for E, R, and BD, there is a positive correlation between the weights and the activities of the network. Interestingly, for B, R, and BD, the change in the weights of the system has little impact on the resilience of the system. However, for the E system, the greater the weights the more resilient the system. This study not only simplifies the complexity inherent in weighted networks but also enhances our understanding of their resilience and response to perturbations.

The Hierarchical Coevolutionary Units of Ecological Networks

In ecological networks, cohesive groups of species may shape the evolution of interactions, serving as coevolutionary units. Ranging across network scales, from motifs to isolated components, elucidating which cohesive groups are more determinant for coevolution remains a challenge in ecology. We address this challenge by integrating 376 empirical mutualistic and antagonistic networks and coevolutionary models. We identified cohesive groups at four network scales containing a significant proportion of potential direct coevolutionary effects. Cohesive groups displayed hierarchical organisation, and potential coevolutionary effects overflowing lower-scale groups were contained by higher-scale groups, underscoring the hierarchy's impact. However, indirect coevolutionary effects blurred group boundaries and hierarchy, particularly under strong selection from ecological interactions. Thus, under strong selection, indirect effects render networks themselves, and not cohesive groups, as the likely coevolutionary units of ecological systems. We hypothesise hierarchical cohesive groups to also shape how other forms of direct and indirect effects propagate in ecological systems.

Network compression with configuration models and the minimum description length

Random network models, constrained to reproduce specific statistical features, are often used to represent and analyze network data and their mathematical descriptions. Chief among them, the configuration model constrains random networks by their degree distribution and is foundational to many areas of network science. However, configuration models and their variants are often selected based on intuition or mathematical and computational simplicity rather than on statistical evidence. To evaluate the quality of a network representation, we need to consider both the amount of information required to specify a random network model and the probability of recovering the original data when using the model as a generative process. To this end, we calculate the approximate size of network ensembles generated by the popular configuration model and its generalizations, including versions accounting for degree correlations and centrality layers. We then apply the minimum description length principle as a model selection criterion over the resulting nested family of configuration models. Using a dataset of over 100 networks from various domains, we find that the classic configuration model is generally preferred on networks with an average degree above 10, while a layered configuration model constrained by a centrality metric offers the most compact representation of the majority of sparse networks.

Entropy-induced phase transitions in a hidden Potts model

A hidden state in which a spin does not interact with any other spin contributes to the entropy of an interacting spin system. We explore the q-state Potts model with extra r hidden states using the Ginzburg-Landau formalism in the mean-field limit. We analytically demonstrate that when 1

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Affiliation(s)

Cook Hyun Kim Center for Complex Systems, KI of Grid Modernization, Korea Institute of Energy Technology, Naju, Jeonnam 58330, Korea
D-S Lee School of Computational Sciences, and Center for AI and Natural Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea
B Kahng Center for Complex Systems, KI of Grid Modernization, Korea Institute of Energy Technology, Naju, Jeonnam 58330, Korea

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Intervention effect of targeted workplace closures may be approximated by single-layered networks in an individual-based model of COVID-19 control

Individual-based models of infectious disease dynamics commonly use network structures to represent human interactions. Network structures can vary in complexity, from single-layered with homogeneous mixing to multi-layered with clustering and layer-specific contact weights. Here we assessed policy-relevant consequences of network choice by simulating different network structures within an established individual-based model of SARS-CoV-2 dynamics. We determined the clustering coefficient of each network structure and compared this to several epidemiological outcomes, such as cumulative and peak infections. High-clustered networks estimate fewer cumulative infections and peak infections than less-clustered networks when transmission probabilities are equal. However, by altering transmission probabilities, we find that high-clustered networks can essentially recover the dynamics of low-clustered networks. We further assessed the effect of workplace closures as a layer-targeted intervention on epidemiological outcomes and found in this scenario a single-layered network provides a sufficient approximation of intervention effect relative to a multi-layered network when layer-specific contact weightings are equal. Overall, network structure choice within models should consider the knowledge of contact weights in different environments and pathogen mode of transmission to avoid over- or under-estimating disease burden and impact of interventions.

The relationship between clustering and networked Turing patterns

Networked Turing patterns often manifest as groups of nodes distributed on either side of the homogeneous equilibrium, exhibiting high and low density. These pattern formations are significantly influenced by network topological characteristics, such as the average degree. However, the impact of clustering on them remains inadequately understood. Here, we investigate the relationship between clustering and networked Turing patterns using classical prey-predator models. Our findings reveal that when nodes of high and low density are completely distributed on both sides of the homogeneous equilibrium, there is a linear decay in Turing patterns as global clustering coefficients increase, given a fixed node size and average degree; otherwise, this linear decay may not always hold due to the presence of high-density nodes considered as low-density nodes. This discovery provides a qualitative assessment of how clustering coefficients impact the formation of Turing patterns and may contribute to understanding why using refuges in ecosystems could enhance the stability of prey-predator systems. The results link network topological structures with the stability of prey-predator systems, offering new insights into predicting and controlling pattern formations in real-world systems from a network perspective.

Tale of two emergent games: Opinion dynamics in dynamical directed networks

Unidirectional social interactions are ubiquitous in real social networks whereas undirected interactions are intensively studied. We establish a voter model in a dynamical directed network. We analytically obtain the degree distribution of the evolving network at any given time. Furthermore, we find that the average degree is captured by an emergent game. However, we find that the fate of opinions is captured by another emergent game. Beyond expectation, the two emergent games are typically different due to the unidirectionality of the evolving networks. The Nash equilibrium analysis of the two games facilitates us to give the criterion under which the minority opinion with few disciples initially takes over the population eventually for in-group bias. Our work fosters the understanding of opinion dynamics ranging from methodology to research content.

An agent-based model of the spread of behavioural risk-factors for cardiovascular disease in city-scale populations

Cardiovascular disease (CVD) is the leading cause of mortality globally, and is the second main cause of mortality in the UK. Four key modifiable behaviours are known to increase CVD risk, namely: tobacco use, unhealthy diet, physical inactivity and harmful use of alcohol. Behaviours that increase the risk of CVD can spread through social networks because individuals consciously and unconsciously mimic the behaviour of others they relate to and admire. Exploiting these social influences may lead to effective and efficient public health interventions to prevent CVD. This project aimed to construct and validate an agent-based model (ABM) of how the four major behavioural risk-factors for CVD spread through social networks in a population, and examine whether the model could be used to identify targets for public health intervention and to test intervention strategies. Previous ABMs have typically focused on a single risk factor or considered very small populations. We created a city-scale ABM to model the behavioural risk-factors of individuals, their social networks (spousal, household, friendship and workplace), the spread of behaviours through these social networks, and the subsequent impact on the development of CVD. We compared the model output (predicted CVD events over a ten year period) to observed data, demonstrating that the model output is realistic. The model output is stable up to at least a population size of 1.2M agents (the maximum tested). We found that there is scope for the modelled interventions targeting the spread of these behaviours to change the number of CVD events experienced by the agents over ten years. Specifically, we modelled the impact of workplace interventions to show that the ABM could be useful for identifying targets for public health intervention. The model itself is Open Source and is available for use or extension by other researchers.

Choice of friction coefficient deeply affects tissue behaviour in stochastic epithelial vertex models

To understand the mechanisms that coordinate the formation of biological tissues, the use of numerical implementations is necessary. The complexity of such models involves many assumptions and parameter choices that result in unpredictable consequences, obstructing the comparison with experimental data. Here, we focus on vertex models, a family of spatial models used extensively to simulate the dynamics of epithelial tissues. Usually, in the literature, the choice of the friction coefficient is not addressed using quasi-static deformation arguments that generally do not apply to realistic scenarios. In this manuscript, we discuss the role that the choice of friction coefficient has on the relaxation times and consequently in the conditions of cell cycle progression and division. We explore the effects that these changes have on the morphology, growth rate and topological transitions of the tissue dynamics. These results provide a deeper understanding of the role that an accurate mechanical description plays in the use of vertex models as inference tools. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Catastrophic cascade of failures in interdependent hypergraphs

The failures of individual agents can significantly impact the functionality of associated groups in interconnected systems. To reveal these impacts, we develop a threshold model to investigate cascading failures in double-layer hypergraphs with interlayer interdependence. We hypothesize that a hyperedge disintegrates when the proportion of failed nodes within it exceeds a threshold. Due to the interdependence between a node and its replica in the other layer, the disintegrations of these hyperedges could trigger a cascade of events, leading to an iterative collapse across these two layers. We find that double-layer hypergraphs undergo abrupt, discontinuous first-order phase transitions during systemic collapse regardless of the specific threshold value. Additionally, the connectivity measured by average cardinality and hyperdegree plays a crucial role in shaping system robustness. A higher average hyperdegree always strengthens system robustness. However, the relationship between system robustness and average cardinality exhibits non-monotonic behaviors. Specifically, both excessively small and large average cardinalities undermine system robustness. Furthermore, a higher threshold value can boost the system's robustness. In summary, our study provides valuable insights into cascading failure dynamics in double-layer hypergraphs and has practical implications for enhancing the robustness of complex interdependent systems across domains.

Effect of the degree of an initial mutant in Moran processes in structured populations

We study effects of the mutant's degree on the fixation probability, extinction, and fixation times in Moran processes on Erdös-Rényi and Barabási-Albert graphs. We performed stochastic simulations and used mean-field-type approximations to obtain analytical formulas. We showed that the initial placement of a mutant has a significant impact on the fixation probability and extinction time, while it has no effect on the fixation time. In both types of graphs, an increase in the degree of an initial mutant results in a decreased fixation probability and a shorter time to extinction. Our results extend previous ones to arbitrary fitness values.

An integrative dynamical perspective for graph theory and the analysis of complex networks

Built upon the shoulders of graph theory, the field of complex networks has become a central tool for studying real systems across various fields of research. Represented as graphs, different systems can be studied using the same analysis methods, which allows for their comparison. Here, we challenge the widespread idea that graph theory is a universal analysis tool, uniformly applicable to any kind of network data. Instead, we show that many classical graph metrics-including degree, clustering coefficient, and geodesic distance-arise from a common hidden propagation model: the discrete cascade. From this perspective, graph metrics are no longer regarded as combinatorial measures of the graph but as spatiotemporal properties of the network dynamics unfolded at different temporal scales. Once graph theory is seen as a model-based (and not a purely data-driven) analysis tool, we can freely or intentionally replace the discrete cascade by other canonical propagation models and define new network metrics. This opens the opportunity to design-explicitly and transparently-dedicated analyses for different types of real networks by choosing a propagation model that matches their individual constraints. In this way, we take stand that network topology cannot always be abstracted independently from network dynamics but shall be jointly studied, which is key for the interpretability of the analyses. The model-based perspective here proposed serves to integrate into a common context both the classical graph analysis and the more recent network metrics defined in the literature which were, directly or indirectly, inspired by propagation phenomena on networks.

Clustering coefficients for networks with higher order interactions

We introduce a clustering coefficient for nondirected and directed hypergraphs, which we call the quad clustering coefficient. We determine the average quad clustering coefficient and its distribution in real-world hypergraphs and compare its value with those of random hypergraphs drawn from the configuration model. We find that real-world hypergraphs exhibit a nonnegligible fraction of nodes with a maximal value of the quad clustering coefficient, while we do not find such nodes in random hypergraphs. Interestingly, these highly clustered nodes can have large degrees and can be incident to hyperedges of large cardinality. Moreover, highly clustered nodes are not observed in an analysis based on the pairwise clustering coefficient of the associated projected graph that has binary interactions, and hence higher order interactions are required to identify nodes with a large quad clustering coefficient.

Anti-modularization for both high robustness and efficiency including the optimal case

Although robustness of connectivity and modular structures in networks have been attracted much attentions in complex networks, most researches have focused on those two features in Erdos-Renyi random graphs and Scale-Free networks whose degree distributions follow Poisson and power-law, respectively. This paper investigates the effect of modularity on robustness in a modular d-regular graphs. Our results reveal that high modularity reduces the robustness even from the optimal robustness of a random d-regular graph in the pure effect of degree distributions. Moreover, we find that a low modular d-regular graph exhibits small-world property that average path length is O(logN). These results indicate that low modularity on modular structures leads to coexistence of both high robustness and efficiency of paths.

Validity of annealed approximation in a high-dimensional system

This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations governing dense-network systems converge to those of the complete-graph version in the thermodynamic limit, where link disorder fluctuations vanish entirely. Consequently, the annealed-network systems, where fluctuations are attenuated, also exhibit the same dynamic behavior in the thermodynamic limit. However, a significant discrepancy arises in the incoherent (disordered) phase wherein the finite-size behavior becomes critical in determining the steady-state pattern. To explicitly elucidate this discrepancy, we focus on identical oscillators subject to competitive attractive and repulsive couplings. In the incoherent phase of dense networks, we observe the manifestation of random irregular states. In contrast, the annealed approximation yields a symmetric (regular) incoherent state where two oppositely coherent clusters of oscillators coexist, accompanied by the vanishing order parameter. Our findings imply that the annealed approximation should be employed with caution even in dense-network systems, particularly in the disordered phase.

How individuals' opinions influence society's resistance to epidemics: an agent-based model approach

BACKGROUND

Protecting public health from infectious diseases often relies on the cooperation of citizens, especially when self-care interventions are the only viable tools for disease mitigation. Accordingly, social aspects related to public opinion have been studied in the context of the recent COVID-19 pandemic. However, a comprehensive understanding of the effects of opinion-related factors on disease spread still requires further exploration.

METHODS

We propose an agent-based simulation framework incorporating opinion dynamics within an epidemic model based on the assumption that mass media channels play a leading role in opinion dynamics. The model simulates how opinions about preventive interventions change over time and how these changes affect the cumulative number of cases. We calibrated our simulation model using YouGov survey data and WHO COVID-19 new cases data from 15 different countries. Based on the calibrated models, we examine how different opinion-related factors change the consequences of the epidemic. We track the number of total new infections for analysis.

RESULTS

Our results reveal that the initial level of public opinion on preventive interventions has the greatest impact on the cumulative number of cases. Its normalized permutation importance varies between 69.67% and 96.65% in 15 models. The patterns shown in the partial dependence plots indicate that other factors, such as the usage of the pro-intervention channel and the response time of media channels, can also bring about substantial changes in disease dynamics, but only within specific ranges of the dominant factor.

CONCLUSIONS

Our results reveal the importance of public opinion on intervention during the early stage of the pandemic in protecting public health. The findings suggest that persuading the public to take actions they may be hesitant about in the early stages of epidemics is very costly because taking early action is critical for mitigating infectious diseases. Other opinion-related factors can also lead to significant changes in epidemics, depending on the average level of public opinion in the initial stage. These findings underscore the importance of media channels and authorities in delivering accurate information and persuading community members to cooperate with public health policies.

Strongly clustered random graphs via triadic closure: An exactly solvable model

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.

Nonrandom behavior in the projection of random bipartite networks

There are two main categories of networks studied in the complexity physics community: Monopartite and bipartite networks. In this paper, we present a general framework that provides insights into the connection between these two classes. When a random bipartite network is projected into a monopartite network, under quite general conditions, the result is a nonrandom monopartite network, the features of which can be studied analytically. Unlike previous studies in the physics literature on complex networks, which rely on sparse-network approximations, we provide a complete analysis, focusing on the degree distribution and the clustering coefficient. Our findings primarily offer a technical contribution, adding to the current body of literature by enhancing the understanding of bipartite networks within the community of physicists. In addition, our model emphasizes the substantial difference between the information that can be extracted from a network measuring its degree distribution, or using higher-order metrics such as the clustering coefficient. We believe that our results are general and have broad real-world implications.

Hubs and Bottlenecks in Protein-Protein Interaction Networks

Protein-protein interaction networks (PPINs) represent the physical interactions among proteins in a cell. These interactions are critical in all cellular processes, including signal transduction, metabolic regulation, and gene expression. In PPINs, centrality measures are widely used to identify the most critical nodes. The two most commonly used centrality measures in networks are degree and betweenness centralities. Degree centrality is the number of connections a node has in the network, and betweenness centrality is the measure of the extent to which a node lies on the shortest paths between pairs of other nodes in the network. In PPINs, proteins with high degree and betweenness centrality are referred to as hubs and bottlenecks respectively. Hubs and bottlenecks are topologically and functionally essential proteins that play crucial roles in maintaining the network's structure and function. This article comprehensively reviews essential literature on hubs and bottlenecks, including their properties and functions.

Theory of percolation on hypergraphs

Hypergraphs capture the higher-order interactions in complex systems and always admit a factor graph representation, consisting of a bipartite network of nodes and hyperedges. As hypegraphs are ubiquitous, investigating hypergraph robustness is a problem of major research interest. In the literature the robustness of hypergraphs so far only has been treated adopting factor-graph percolation, which describes well higher-order interactions which remain functional even after the removal of one of more of their nodes. This approach, however, fall short to describe situations in which higher-order interactions fail when any one of their nodes is removed, this latter scenario applying, for instance, to supply chains, catalytic networks, protein-interaction networks, networks of chemical reactions, etc. Here we show that in these cases the correct process to investigate is hypergraph percolation, with is distinct from factor graph percolation. We build a message-passing theory of hypergraph percolation, and we investigate its critical behavior using generating function formalism supported by Monte Carlo simulations on random graph and real data. Notably, we show that the node percolation threshold on hypergraphs exceeds node percolation threshold on factor graphs. Furthermore we show that differently from what happens in ordinary graphs, on hypergraphs the node percolation threshold and hyperedge percolation threshold do not coincide, with the node percolation threshold exceeding the hyperedge percolation threshold. These results demonstrate that any fat-tailed cardinality distribution of hyperedges cannot lead to the hyper-resilience phenomenon in hypergraphs in contrast to their factor graphs, where the divergent second moment of a cardinality distribution guarantees zero percolation threshold.

Edge-based compartmental modeling for the spread of cholera on random networks: A case study in Somalia

Cholera remains a major public health problem that threatens human health worldwide and its severity is continuing. In this paper, an edge-based model for cholera transmission on random networks is proposed and investigated. The model assumes that two communities share a common water source and includes three transmission routes, namely intra- and inter-community human-to-human transmission as well as water-to-human transmission. Intra-community human-to-human contacts are modeled through a random contact network, while both inter-community and water-to-human transmission are modeled through external nodes that reach each individual in the network to the same extent. The basic reproduction number and the equations of the final epidemic size are obtained. In addition, our study considers the cholera situation in Banadir, which is one of the most severely infected regions in Somalia, during the period (2019-2021). According to the geographical location, two adjacent districts are selected and our model fits well with the real data on the monthly cumulative cholera cases of these two districts during the above-mentioned period. From the perspective of network topology, cutting off high-risk contacts by supervising, isolating, quarantining and closing places with high-degree cholera-infected individuals to reduce degree heterogeneity is an effective measure to control cholera transmission. Our findings might offer some useful insights on cholera control.

Nearest-neighbor directed random hyperbolic graphs

Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbors. We introduce also a canonical version of this network (which we call "network with varied connection radius"), where maximal length of outgoing bond is space dependent and is determined by fixing the average out-degree to m. We study local bond length, in-degree, and reciprocity in these networks as a function of spacial coordinates of the nodes and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power-law distribution. We demonstrate that reciprocity of the network can be regulated by adjusting an additional temperature-like parameter without changing other global properties of the network.

Standardized local assortativity in networks and systemic risk in financial markets

The study of assortativity allows us to understand the heterogeneity of networks and the implication of network resilience. While a global measure has been predominantly used to characterize this network feature, there has been little research to suggest a local coefficient to account for the presence of local (dis)assortative patterns in diversely mixed networks. We build on existing literature and extend the concept of assortativity with the proposal of a standardized scale-independent local coefficient to observe the assortative characteristics of each entity in networks that would otherwise be smoothed out with a global measure. This coefficient provides a lens through which the granular level of details can be observed, as well as capturing possible pattern (dis)formation in dynamic networks. We demonstrate how the standardized local assortative coefficient discovers the presence of (dis)assortative hubs in static networks on a granular level, and how it tracks systemic risk in dynamic financial networks.

Distribution of shortest path lengths on trees of a given size in subcritical Erdős-Rényi networks

In the subcritical regime Erdős-Rényi (ER) networks consist of finite tree components, which are nonextensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O. Biham, and A. K. Hartmann, Phys. Rev. E 98, 012301 (2018)2470-004510.1103/PhysRevE.98.012301]. The DSPL, which accounts for the distance ℓ between any pair of nodes that reside on the same finite tree component, was found to follow a geometric distribution of the form P(L=ℓ|L<∞)=(1-c)c^{ℓ-1}, where 0 Collapse

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Affiliation(s)

Barak Budnick Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel
Ofer Biham Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel
Eytan Katzav Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel

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Tracing the birth of structural domains from loops during protein evolution

The structures and functions of proteins are embedded into the loop scaffolds of structural domains. Their origin and evolution remain mysterious. Here, we use a novel graph-theoretical approach to describe how modular and non-modular loop prototypes combine to form folded structures in protein domain evolution. Phylogenomic data-driven chronologies reoriented a bipartite network of loops and domains (and its projections) into 'waterfalls' depicting an evolving 'elementary functionome' (EF). Two primordial waves of functional innovation involving founder 'p-loop' and 'winged-helix' domains were accompanied by an ongoing emergence and reuse of structural and functional novelty. Metabolic pathways expanded before translation functionalities. A dual hourglass recruitment pattern transferred scale-free properties from loop to domain components of the EF network in generative cycles of hierarchical modularity. Modeling the evolutionary emergence of the oldest P-loop and winged-helix domains with AlphFold2 uncovered rapid convergence towards folded structure, suggesting that a folding vocabulary exists in loops for protein fold repurposing and design.

Functional alterations in bipartite network of white and grey matters during aging

The effects of normal aging on functional connectivity (FC) within various brain networks of gray matter (GM) have been well-documented. However, the age effects on the networks of FC between white matter (WM) and GM, namely WM-GM FC, remains unclear. Evaluating crucial properties, such as global efficiency (GE), for a WM-GM FC network poses a challenge due to the absence of closed triangle paths which are essential for assessing network properties in traditional graph models. In this study, we propose a bipartite graph model to characterize the WM-GM FC network and quantify these challenging network properties. Leveraging this model, we assessed the WM-GM FC network properties at multiple scales across 1,462 cognitively normal subjects aged 22-96 years from three repositories (ADNI, BLSA and OASIS-3) and investigated the age effects on these properties throughout adulthood and during late adulthood (age ≥70 years). Our findings reveal that (1) heterogeneous alterations occurred in region-specific WM-GM FC over the adulthood and decline predominated during late adulthood; (2) the FC density of WM bundles engaged in memory, executive function and processing speed declined with age over adulthood, particularly in later years; and (3) the GE of attention, default, somatomotor, frontoparietal and limbic networks reduced with age over adulthood, and GE of visual network declined during late adulthood. These findings provide unpresented insights into multi-scale alterations in networks of WM-GM functional synchronizations during normal aging. Furthermore, our bipartite graph model offers an extendable framework for quantifying WM-engaged networks, which may contribute to a wide range of neuroscience research.

Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks

We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.

Bifractality of fractal scale-free networks

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

Normalizing Input-Output Relationships of Cancer Networks for Reversion Therapy

Accumulated genetic alterations in cancer cells distort cellular stimulus-response (or input-output) relationships, resulting in uncontrolled proliferation. However, the complex molecular interaction network within a cell implicates a possibility of restoring such distorted input-output relationships by rewiring the signal flow through controlling hidden molecular switches. Here, a system framework of analyzing cellular input-output relationships in consideration of various genetic alterations and identifying possible molecular switches that can normalize the distorted relationships based on Boolean network modeling and dynamics analysis is presented. Such reversion is demonstrated by the analysis of a number of cancer molecular networks together with a focused case study on bladder cancer with in vitro experiments and patient survival data analysis. The origin of reversibility from an evolutionary point of view based on the redundancy and robustness intrinsically embedded in complex molecular regulatory networks is further discussed.

A hierarchical intervention scheme based on epidemic severity in a community network

As there are no targeted medicines or vaccines for newly emerging infectious diseases, isolation among communities (villages, cities, or countries) is one of the most effective intervention measures. As such, the number of intercommunity edges ([Formula: see text]) becomes one of the most important factor in isolating a place since it is closely related to normal life. Unfortunately, how [Formula: see text] affects epidemic spread is still poorly understood. In this paper, we quantitatively analyzed the impact of [Formula: see text] on infectious disease transmission by establishing a four-dimensional [Formula: see text] edge-based compartmental model with two communities. The basic reproduction number [Formula: see text] is explicitly obtained subject to [Formula: see text] [Formula: see text]. Furthermore, according to [Formula: see text] with zero [Formula: see text], epidemics spread could be classified into two cases. When [Formula: see text] for the case 2, epidemics occur with at least one of the reproduction numbers within communities greater than one, and otherwise when [Formula: see text] for case 1, both reproduction numbers within communities are less than one. Remarkably, in case 1, whether epidemics break out strongly depends on intercommunity edges. Then, the outbreak threshold in regard to [Formula: see text] is also explicitly obtained, below which epidemics vanish, and otherwise break out. The above two cases form a severity-based hierarchical intervention scheme for epidemics. It is then applied to the SARS outbreak in Singapore, verifying the validity of our scheme. In addition, the final size of the system is gained by demonstrating the existence of positive equilibrium in a four-dimensional coupled system. Theoretical results are also validated through numerical simulation in networks with the Poisson and Power law distributions, respectively. Our results provide a new insight into controlling epidemics.

Role of masks in mitigating viral spread on networks

Masks have remained an important mitigation strategy in the fight against COVID-19 due to their ability to prevent the transmission of respiratory droplets between individuals. In this work, we provide a comprehensive quantitative analysis of the impact of mask-wearing. To this end, we propose a novel agent-based model of viral spread on networks where agents may either wear no mask or wear one of several types of masks with different properties (e.g., cloth or surgical). We derive analytical expressions for three key epidemiological quantities: The probability of emergence, the epidemic threshold, and the expected epidemic size. In particular, we show how the aforementioned quantities depend on the structure of the contact network, viral transmission dynamics, and the distribution of the different types of masks within the population. Through extensive simulations, we then investigate the impact of different allocations of masks within the population and tradeoffs between the outward efficiency and inward efficiency of the masks. Interestingly, we find that masks with high outward efficiency and low inward efficiency are most useful for controlling the spread in the early stages of an epidemic, while masks with high inward efficiency but low outward efficiency are most useful in reducing the size of an already large spread. Last, we study whether degree-based mask allocation is more effective in reducing the probability of epidemic as well as epidemic size compared to random allocation. The result echoes the previous findings that mitigation strategies should differ based on the stage of the spreading process, focusing on source control before the epidemic emerges and on self-protection after the emergence.

Generic properties of the oligodendrocyte - axon network

The role of oligodendrocytes (OLs) extends beyond saltatory conduction to a modulatory role in neural information processing. Given this exalted role, we take first steps to frame the OL - axon interaction as a network of cells. We find that the OL - axon network has a natural encoding as a bipartite network, allowing us to determine key network properties, estimate the number of OLs or axons in various brain regions and determine the robustness of the network to random removal of cell nodes.

Topological data analysis of human brain networks through order statistics

Understanding the common topological characteristics of the human brain network across a population is central to understanding brain functions. The abstraction of human connectome as a graph has been pivotal in gaining insights on the topological properties of the brain network. The development of group-level statistical inference procedures in brain graphs while accounting for the heterogeneity and randomness still remains a difficult task. In this study, we develop a robust statistical framework based on persistent homology using the order statistics for analyzing brain networks. The use of order statistics greatly simplifies the computation of the persistent barcodes. We validate the proposed methods using comprehensive simulation studies and subsequently apply to the resting-state functional magnetic resonance images. We found a statistically significant topological difference between the male and female brain networks.

Identifying Taxonomic Units in Metagenomic DNA Streams on Mobile Devices

With the emergence of portable DNA sequencers, such as Oxford Nanopore Technology MinION, metagenomic DNA sequencing can be performed in real-time and directly in the field. However, because metagenomic DNA analysis tasks, e.g., classification, taxonomic units assignment, etc., are compute and memory intensive, and the available methods are designed for batch processing, the current metagenomic tools are not well suited for mobile devices. In this work, we propose a new memory-efficient approach to identify Operational Taxonomic Units (OTUs) in metagenomic DNA streams on mobile devices. Our method is based on finding connected components in overlap graphs constructed over a real-time stream of long DNA reads as produced by the MinION platform. We propose an efficient algorithm to maintain connected components when an overlap graph is streamed and show how redundant information can be removed from the stream by transitive closures. We also propose how our algorithms can be integrated into a larger DNA analysis pipeline tailored for mobile computing. Through experiments on simulated and real-world metagenomic data, executed on the actual mobile device, we demonstrate that our resulting solution is able to recover OTUs with high precision. Our experiments also demonstrate the compounding benefits of introducing feedback loops in the DNA analysis pipeline.

Nonequilibrium dynamics of the Ising model on heterogeneous networks with an arbitrary distribution of threshold noise

The Ising model on networks plays a fundamental role as a testing ground for understanding cooperative phenomena in complex systems. Here we solve the synchronous dynamics of the Ising model on random graphs with an arbitrary degree distribution in the high-connectivity limit. Depending on the distribution of the threshold noise that governs the microscopic dynamics, the model evolves to nonequilibrium stationary states. We obtain an exact dynamical equation for the distribution of local magnetizations, from which we find the critical line that separates the paramagnetic from the ferromagnetic phase. For random graphs with a negative binomial degree distribution, we demonstrate that the stationary critical behavior as well as the long-time critical dynamics of the first two moments of the local magnetizations depend on the distribution of the threshold noise. In particular, for an algebraic threshold noise, these critical properties are determined by the power-law tails of the distribution of thresholds. We further show that the relaxation time of the average magnetization inside each phase exhibits the standard mean-field critical scaling. The values of all critical exponents considered here are independent of the variance of the negative binomial degree distribution. Our work highlights the importance of certain details of the microscopic dynamics for the critical behavior of nonequilibrium spin systems.

Distribution of the number of cycles in directed and undirected random regular graphs of degree 2

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with lnN, while the length of the longest cycle scales with N. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles 〈S〉 scales with lnN. Here we present exact analytical results for the distribution P_{N}(S=s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large N limit. The moments and cumulants of P_{N}(S=s) are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of N objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

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.

Aging in binary-state models: The Threshold model for complex contagion

We study the non-Markovian effects associated with aging for binary-state dynamics in complex networks. Aging is considered as the property of the agents to be less prone to change their state the longer they have been in the current state, which gives rise to heterogeneous activity patterns. In particular, we analyze aging in the Threshold model, which has been proposed to explain the process of adoption of new technologies. Our analytical approximations give a good description of extensive Monte Carlo simulations in Erdős-Rényi, random-regular and Barabási-Albert networks. While aging does not modify the cascade condition, it slows down the cascade dynamics towards the full-adoption state: the exponential increase of adopters in time from the original model is replaced by a stretched exponential or power law, depending on the aging mechanism. Under several approximations, we give analytical expressions for the cascade condition and for the exponents of the adopters' density growth laws. Beyond random networks, we also describe by Monte Carlo simulations the effects of aging for the Threshold model in a two-dimensional lattice.

Higher-order rich-club phenomenon in collaborative research grant networks

Modern scientific work, including writing papers and submitting research grant proposals, increasingly involves researchers from different institutions. In grant collaborations, it is known that institutions involved in many collaborations tend to densely collaborate with each other, forming rich clubs. Here we investigate higher-order rich-club phenomena in networks of collaborative research grants among institutions and their associations with research impact. Using publicly available data from the National Science Foundation in the US, we construct a bipartite network of institutions and collaborative grants, which distinguishes among the collaboration with different numbers of institutions. By extending the concept and algorithms of the rich club for dyadic networks to the case of bipartite networks, we find rich clubs both in the entire bipartite network and the bipartite subnetwork induced by the collaborative grants involving a given number of institutions up to five. We also find that the collaborative grants within rich clubs tend to be more impactful in a per-dollar sense than the control. Our results highlight advantages of collaborative grants among the institutions in the rich clubs.

Predicting molecule size distribution in hydrocarbon pyrolysis using random graph theory

Hydrocarbon pyrolysis is a complex process involving large numbers of chemical species and types of chemical reactions. Its quantitative description is important for planetary sciences, in particular, for understanding the processes occurring in the interior of icy planets, such as Uranus and Neptune, where small hydrocarbons are subjected to high temperature and pressure. We propose a computationally cheap methodology based on an originally developed ten-reaction model and the configurational model from random graph theory. This methodology generates accurate predictions for molecule size distributions for a variety of initial chemical compositions and temperatures ranging from 3200 to 5000 K. Specifically, we show that the size distribution of small molecules is particularly well predicted, and the size of the largest molecule can be accurately predicted provided that this molecule is not too large.

Fragility Induced by Interdependency of Complex Networks and Their Higher-Order Networks

The higher-order structure of networks is a hot research topic in complex networks. It has received much attention because it is closely related to the functionality of networks, such as network transportation and propagation. For instance, recent studies have revealed that studying higher-order networks can explore hub structures in transportation networks and information dissemination units in neuronal networks. Therefore, the destruction of the connectivity of higher-order networks will cause significant damage to network functionalities. Meanwhile, previous works pointed out that the function of a complex network depends on the giant component of the original(low-order) network. Therefore, the network functionality will be influenced by both the low-order and its corresponding higher-order network. To study this issue, we build a network model of the interdependence of low-order and higher-order networks (we call it ILH). When some low-order network nodes fail, the low-order network's giant component shrinks, leading to changes in the structure of the higher-order network, which further affects the low-order network. This process occurs iteratively; the propagation of the failure can lead to an eventual network crash. We conducted experiments on different networks based on the percolation theory, and our network percolation results demonstrated a first-order phase transition feature. In particular, we found that an ILH is more fragile than the low-order network alone, and an ILH is more likely to be corrupted in the event of a random node failure.

Condensates formed by prion-like low-complexity domains have small-world network structures and interfaces defined by expanded conformations

Biomolecular condensates form via coupled associative and segregative phase transitions of multivalent associative macromolecules. Phase separation coupled to percolation is one example of such transitions. Here, we characterize molecular and mesoscale structural descriptions of condensates formed by intrinsically disordered prion-like low complexity domains (PLCDs). These systems conform to sticker-and-spacers architectures. Stickers are cohesive motifs that drive associative interactions through reversible crosslinking and spacers affect the cooperativity of crosslinking and overall macromolecular solubility. Our computations reproduce experimentally measured sequence-specific phase behaviors of PLCDs. Within simulated condensates, networks of reversible inter-sticker crosslinks organize PLCDs into small-world topologies. The overall dimensions of PLCDs vary with spatial location, being most expanded at and preferring to be oriented perpendicular to the interface. Our results demonstrate that even simple condensates with one type of macromolecule feature inhomogeneous spatial organizations of molecules and interfacial features that likely prime them for biochemical activity.

Sharpest possible clustering bounds using robust random graph analysis

Complex network theory crucially depends on the assumptions made about the degree distribution, while fitting degree distributions to network data is challenging, in particular for scale-free networks with power-law degrees. We present a robust assessment of complex networks that does not depend on the entire degree distribution, but only on its mean, range, and dispersion: summary statistics that are easy to obtain for most real-world networks. By solving several semi-infinite linear programs, we obtain tight (the sharpest possible) bounds for correlation and clustering measures, for all networks with degree distributions that share the same summary statistics. We identify various extremal random graphs that attain these tight bounds as the graphs with specific three-point degree distributions. We leverage the tight bounds to obtain robust laws that explain how degree-degree correlations and local clustering evolve as a function of node degrees and network size. These robust laws indicate that power-law networks with diverging variance are among the most extreme networks in terms of correlation and clustering, building a further theoretical foundation for the widely reported scale-free network phenomena such as correlation and clustering decay.

Degree distributions under general node removal: Power-law or Poisson?

Perturbations made to networked systems may result in partial structural loss, such as a blackout in a power-grid system. Investigating the resulting disturbance in network properties is quintessential to understand real networks in action. The removal of nodes is a representative disturbance, but previous studies are seemingly contrasting about its effect on arguably the most fundamental network statistic, the degree distribution. The key question is about the functional form of the degree distributions that can be altered during node removal or sampling. The functional form is decisive in the remaining subnetwork's static and dynamical properties. In this work, we clarify the situation by utilizing the relative entropies with respect to the reference distributions in the Poisson and power-law form, to quantify the distance between the subnetwork's degree distribution and either of the reference distributions. Introducing general sequential node removal processes with continuously different levels of hub protection to encompass a series of scenarios including uniform random removal and preferred or protective (i.e., biased random) removal of the hub, we classify the altered degree distributions starting from various power-law forms by comparing two relative entropy values. From the extensive investigation in various scenarios based on direct node-removal simulations and by solving the rate equation of degree distributions, we discover in the parameter space two distinct regimes, one where the degree distribution is closer to the power-law reference distribution and the other closer to the Poisson distribution.

Structure of university licensing networks

We study the structure and evolution of networks of inventors involved in university licensing and patenting. In particular, we focus on networks of inventors that have successfully licensed a university patent (i.e., licensing networks), and investigate levels of their fragmentation, cliquishness, and whether they exhibit the small world phenomenon. We find that these licensing networks are more fragmented and cliquish than the networks of inventors engaged in all (not necessarily licensed) patents (i.e., patenting networks), and that they are not small worlds. Additionally, by comparing the created licensing networks to random subnetworks of the patenting networks, we find that concerns in regard to the potential effects of opportunistic behavior are, to some degree, justified. We detect an interesting collaboration behavior of inventors who license, which we designate as dualistic opportunistic behavior.