Localization and centrality in networks

Spectra of random networks with arbitrary degrees

We derive a message-passing method for computing the spectra of locally treelike networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees-the so-called configuration model. We find that the latter approximation works well for all but the sparsest of networks. We also derive bounds on the position of the band edges of the spectrum, which are important for identifying structural phase transitions in networks.

Analysis of Topological Parameters of Complex Disease Genes Reveals the Importance of Location in a Biomolecular Network

Network biology and medicine provide unprecedented opportunities and challenges for deciphering disease mechanisms from integrative viewpoints. The disease genes and their products perform their dysfunctions via physical and biochemical interactions in the form of a molecular network. The topological parameters of these disease genes in the interactome are of prominent interest to the understanding of their functionality from a systematic perspective. In this work, we provide a systems biology analysis of the topological features of complex disease genes in an integrated biomolecular network. Firstly, we identify the characteristics of four network parameters in the ten most frequently studied disease genes and identify several specific patterns of their topologies. Then, we confirm our findings in the other disease genes of three complex disorders (i.e., Alzheimer’s disease, diabetes mellitus, and hepatocellular carcinoma). The results reveal that the disease genes tend to have a higher betweenness centrality, a smaller average shortest path length, and a smaller clustering coefficient when compared to normal genes, whereas they have no significant degree prominence. The features highlight the importance of gene location in the integrated functional linkages.

Localization of eigenvector centrality in networks with a cut vertex

We show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set, the most extreme example being networks with a cut vertex. Three distinct types of localization are identified in these structures. One is related to the well-established hub node localization phenomenon and the other two are introduced and characterized here. We gain insights into these problems by deriving the relationship between eigenvector centrality and Katz centrality. This leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix.

Cycle Based Network Centrality

Paths and cycles are the two pivotal elements in a network. Here, we demonstrate that paths, particularly the shortest ones, are incomplete in information network. However, based on such paths, many network centrality measures are designed. While extensive explorations on paths have been made, modest studies focus on the cycles on measuring network centrality. We study the relationship between the shortest cycle and the shortest path from extensive real-world networks. The results illustrate the incompleteness of the shortest paths on measuring network centrality. Noticing that the shortest cycle is much more robust than the shortest path, we propose two novel cycle-based network centrality measures to address the incompleteness of paths: the shortest cycle closeness centrality (SCC) and the all cycle betweenness centrality (ACC). Notwithstanding we focus on the network centrality problem, our findings on cycles can be applied to explain the incompleteness of paths in applications and could improve the applicability into more scenarios where the paths are employed in network science.

Relevance of backtracking paths in recurrent-state epidemic spreading on networks

The understanding of epidemics on networks has greatly benefited from the recent application of message-passing approaches, which allow us to derive exact results for irreversible spreading (i.e., diseases with permanent acquired immunity) in locally treelike topologies. This success has suggested the application of the same approach to recurrent-state epidemics, for which an individual can contract the epidemic and recover repeatedly. The underlying assumption is that backtracking paths (i.e., an individual is reinfected by a neighbor he or she previously infected) do not play a relevant role. In this paper we show that this is not the case for recurrent-state epidemics since the neglect of backtracking paths leads to a formula for the epidemic threshold that is qualitatively incorrect in the large size limit. Moreover, we define a modified recurrent-state dynamics which explicitly forbids direct backtracking events and show that this modification completely upsets the phenomenology.

Disease networks identify specific conditions and pleiotropy influencing multimorbidity in the general population

Multimorbidity is an emerging topic in public health policy because of its increasing prevalence and socio-economic impact. However, the age- and gender-dependent trends of disease associations at fine resolution, and the underlying genetic factors, remain incompletely understood. Here, by analyzing disease networks from electronic medical records of primary health care, we identify key conditions and shared genetic factors influencing multimorbidity. Three types of diseases are outlined: "central", which include chronic and non-chronic conditions, have higher cumulative risks of disease associations; "community roots" have lower cumulative risks, but inform on continuing clustered disease associations with age; and "seeds of bursts", which most are chronic, reveal outbreaks of disease associations leading to multimorbidity. The diseases with a major impact on multimorbidity are caused by genes that occupy central positions in the network of human disease genes. Alteration of lipid metabolism connects breast cancer, diabetic neuropathy and nutritional anemia. Evaluation of key disease associations by a genome-wide association study identifies shared genetic factors and further supports causal commonalities between nervous system diseases and nutritional anemias. This study also reveals many shared genetic signals with other diseases. Collectively, our results depict novel population-based multimorbidity patterns, identify key diseases within them, and highlight pleiotropy influencing multimorbidity.

Improved collective influence of finding most influential nodes based on disjoint-set reinsertion

Identifying vital nodes in complex networks is a critical problem in the field of network theory. To this end, the Collective. Influence (CI) algorithm has been introduced and shows high efficiency and scalability in searching for the influential nodes in. the optimal percolation model. However, the crucial part of the CI algorithm, reinsertion, has not been significantly investigated. or improved upon. In this paper, the author improves the CI algorithm and proposes a new algorithm called Collective-Influence-Disjoint-Set-Reinsertion (CI_DR) based on disjoint-set reinsertion. Experimental results on 8 datasets with scales of a million nodes and 4 random graph networks demonstrate that the proposed CI_DR algorithm outperforms other algorithms, including Betweenness centrality, Closeness centrality, PageRank centrality, Degree centrality (HDA), Eigenvector centrality, Nonbacktracking centrality and Collective Influence with original reinsertion, in terms of the Robustness metric. Moreover, CI_DR is applied to an international competition on optimal percolation and ultimately ranks in 7th place.

Spreading Processes in Multiplex Metapopulations Containing Different Mobility Networks

We propose a theoretical framework for the study of spreading processes in structured metapopulations, with heterogeneous agents, subjected to different recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze classical epidemic models within this framework and obtain an excellent agreement with extensive Monte Carlo simulations. This agreement allows us to derive analytical expressions of the epidemic threshold and to face the challenge of characterizing a real multiplex metapopulation, the city of Medellín in Colombia, where different recurrent mobility patterns are observed depending on the socioeconomic class of the agents. Our framework allows us to unveil the geographical location of those patches that trigger the epidemic state at the critical point. A careful exploration reveals that social mixing between classes and mobility crucially determines these critical patches and, more importantly, it can produce abrupt changes of the critical properties of the epidemic onset.

The spread of disease is strongly driven by how humans move and segregate across cities, regions, and countries. Understanding how epidemics arise from the interplay of these behaviors is crucial for implementing efficient containment and prevention policies. In this work, we propose a theoretical framework for the spread of pathogens that incorporates the real patterns of mobility, demographics, and human diversity observed in urban environments.

Our framework represents various populations in a community as layers in a multiplex network, where nodes correspond to geographical areas and layers account for movement patterns. With this framework, we analyze a real urban system, the city of Medellin in Colombia, considering the interplay between six socioeconomic classes displaying disparate demographic segregation and mobility habits. Our framework can identify those neighborhoods and classes that trigger epidemic outbreaks.

Remarkably, we observe that small changes to both mobility and social mixing among these subpopulations can trigger abrupt changes. As a consequence, containment strategies targeting a certain neighborhood can quickly change from efficient to useless. Finally, we provide a physical interpretation of the abrupt changes of those patches driving the unfolding of epidemics based on the effective number of contacts of agents.

Our framework provides a reliable, time-saving platform for analyzing the spread of pathogens among various populations, allowing researchers to identify areas critical to the unfolding of diseases. With further improvements, our formalism could be extended to accommodate more sophisticated commuting patterns and epidemic models.

Generalized Erdős numbers for network analysis

The identification of relationships in complex networks is critical in a variety of scientific contexts. This includes the identification of globally central nodes and analysing the importance of pairwise relationships between nodes. In this paper, we consider the concept of topological proximity (or 'closeness') between nodes in a weighted network using the generalized Erdős numbers (GENs). This measure satisfies a number of desirable properties for networks with nodes that share a finite resource. These include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to new methods to measure centrality. We show that the square of the leading eigenvector of an importance matrix defined using the GENs is strongly correlated with well-known measures such as PageRank, and define a personalized measure of centrality that is also well correlated with other existing measures. The utility of this measure of topological proximity is demonstrated by showing the asymmetries in both the dynamics of random walks and the mean infection time in epidemic spreading are better predicted by the topological definition of closeness provided by the GENs than they are by other measures.

Finding influential nodes for integration in brain networks using optimal percolation theory

Global integration of information in the brain results from complex interactions of segregated brain networks. Identifying the most influential neuronal populations that efficiently bind these networks is a fundamental problem of systems neuroscience. Here, we apply optimal percolation theory and pharmacogenetic interventions in vivo to predict and subsequently target nodes that are essential for global integration of a memory network in rodents. The theory predicts that integration in the memory network is mediated by a set of low-degree nodes located in the nucleus accumbens. This result is confirmed with pharmacogenetic inactivation of the nucleus accumbens, which eliminates the formation of the memory network, while inactivations of other brain areas leave the network intact. Thus, optimal percolation theory predicts essential nodes in brain networks. This could be used to identify targets of interventions to modulate brain function.

Exact distribution of spacing ratios for random and localized states in quantum chaotic systems

Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra are modeled through an appropriate ensemble described by random matrix theory. However, a small subset of states violates this principle and displays eigenstate localization, a counterintuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and modify the spectral statistics. In this work, a 3×3 random matrix model is used to obtain exact results for the ratio of spacing between a generic and localized state. We consider time-reversal-invariant as well as noninvariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.

Analytical connection between thresholds and immunization strategies of SIS model in random networks

Devising effective strategies for hindering the propagation of viruses and protecting the population against epidemics is critical for public security and health. Despite a number of studies based on the susceptible-infected-susceptible (SIS) model devoted to this topic, we still lack a general framework to compare different immunization strategies in completely random networks. Here, we address this problem by suggesting a novel method based on heterogeneous mean-field theory for the SIS model. Our method builds the relationship between the thresholds and different immunization strategies in completely random networks. Besides, we provide an analytical argument that the targeted large-degree strategy achieves the best performance in random networks with arbitrary degree distribution. Moreover, the experimental results demonstrate the effectiveness of the proposed method in both artificial and real-world networks.

Network analytics approach towards identifying potential antivirulence drug targets within the Staphylococcus aureus staphyloxanthin biosynthetic network

Staphylococcus aureus is associated with several clinically significant infections among humans and infections associated with antibiotic-resistant strains are growing in frequency. Antivirulence strategies shift the target of drugs from bacterial growth to the infection process resulting to milder evolutionary pressure for the development of bacterial resistant strains. Staphyloxanthin (STX) is a yellowish-orange carotenoid pigment synthesized by S. aureus and this carotenoid functions as an important virulence factor for the bacteria. In this study, we elucidated whether network analytics can be used as a viable tool to identify significant components in the STX biosynthetic network which in-turn could serve as possible antivirulence drug targets. For confirmation, we correlated our results to known drugs that were able to inhibit STX biosynthesis. Throughout this study, we established that crtN(1) activity and 4,4'-diaponeurosporene amounts are significant components in the STX biosynthetic network and, moreover, network analytics can aid in identifying antivirulence drug targets within the STX biosynthetic network. Similarly, we found that network analytics is capable of identifying multiple potential targets simultaneously. Taken together, we propose that an effective antivirulence drug against S. aureus STX biosynthesis would involve targeting crtN(1) activity, 4,4'-diaponeurosporene levels, or both components.

Dynamic range maximization in excitable networks

We study the strategy to optimally maximize the dynamic range of excitable networks by removing the minimal number of links. A network of excitable elements can distinguish a broad range of stimulus intensities and has its dynamic range maximized at criticality. In this study, we formulate the activation propagation in excitable networks as a message passing process in which a critical state is reached when the largest eigenvalue of the weighted non-backtracking matrix is close to one. By considering the impact of single link removal on the largest eigenvalue, we develop an efficient algorithm that aims to identify the optimal set of links whose removal will drive the system to the critical state. Comparisons with other competing heuristics on both synthetic and real-world networks indicate that the proposed method can maximize the dynamic range by removing the smallest number of links, and at the same time maintaining the largest size of the giant connected component.

The effects of graded levels of calorie restriction: VII. Topological rearrangement of hypothalamic aging networks

Connectivity in a gene-gene network declines with age, typically within gene clusters. We explored the effect of short-term (3 months) graded calorie restriction (CR) (up to 40 %) on network structure of aging-associated genes in the murine hypothalamus by using conditional mutual information. The networks showed a topological rearrangement when exposed to graded CR with a higher relative within cluster connectivity at 40CR. We observed changes in gene centrality concordant with changes in CR level, with Ppargc1a, and Ppt1 having increased centrality and Etfdh, Traf3 and Abcc1 decreased centrality as CR increased. This change in gene centrality in a graded manner with CR, occurred in the absence of parallel changes in gene expression levels. This study emphasizes the importance of augmenting traditional differential gene expression analyses to better understand structural changes in the transcriptome. Overall our results suggested that CR induced changes in centrality of biological relevant genes that play an important role in preventing the age-associated loss of network integrity irrespective of their gene expression levels.

Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Accurate ranking of influential spreaders in networks based on dynamically asymmetric link weights

We propose an efficient and accurate measure for ranking spreaders and identifying the influential ones in spreading processes in networks. While the edges determine the connections among the nodes, their specific role in spreading should be considered explicitly. An edge connecting nodes i and j may differ in its importance for spreading from i to j and from j to i. The key issue is whether node j, after infected by i through the edge, would reach out to other nodes that i itself could not reach directly. It becomes necessary to invoke two unequal weights w_{ij} and w_{ji} characterizing the importance of an edge according to the neighborhoods of nodes i and j. The total asymmetric directional weights originating from a node leads to a novel measure s_{i}, which quantifies the impact of the node in spreading processes. An s-shell decomposition scheme further assigns an s-shell index or weighted coreness to the nodes. The effectiveness and accuracy of rankings based on s_{i} and the weighted coreness are demonstrated by applying them to nine real-world networks. Results show that they generally outperform rankings based on the nodes' degree and k-shell index while maintaining a low computational complexity. Our work represents a crucial step towards understanding and controlling the spread of diseases, rumors, information, trends, and innovations in networks.

Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment

This study highlights the relevance of network-guided controllability analysis as a precision oncology tool. Target controllability through networks is potentially relevant to cancer research for the identification of therapeutic targets. With reference to a recent study on multiple phenotypes from 22 osteosarcoma (OS) cell lines characterized both in vitro and in vivo, we found that a variety of critical proteins in OS regulation circuits were in part phenotype specific and in part shared. To generalize our inference approach and match cancer phenotypic heterogeneity, we employed multitype networks and identified targets in correspondence with protein sub-complexes. Therefore, we established the relevance for diagnostic and therapeutic purposes of inspecting interactive targets, namely those enriched by significant connectivity patterns in protein sub-complexes. Emerging targets appeared with reference to the OS microenvironment, and relatively to small leucine-rich proteoglycan members and D-type cyclins, among other collagen, laminin, and keratin proteins. These described were evidences shared across all phenotypes; instead, specific evidences were provided by critical proteins including IGFBP7 and PDGFRA in the invasive phenotype, and FGFR3 and THBS1 in the colony forming phenotype.

Eigenvector centrality for geometric and topological characterization of porous media

Solving flow and transport through complex geometries such as porous media is computationally difficult. Such calculations usually involve the solution of a system of discretized differential equations, which could lead to extreme computational cost depending on the size of the domain and the accuracy of the model. Geometric simplifications like pore networks, where the pores are represented by nodes and the pore throats by edges connecting pores, have been proposed. These models, despite their ability to preserve the connectivity of the medium, have difficulties capturing preferential paths (high velocity) and stagnation zones (low velocity), as they do not consider the specific relations between nodes. Nonetheless, network theory approaches, where a complex network is a graph, can help to simplify and better understand fluid dynamics and transport in porous media. Here we present an alternative method to address these issues based on eigenvector centrality, which has been corrected to overcome the centralization problem and modified to introduce a bias in the centrality distribution along a particular direction to address the flow and transport anisotropy in porous media. We compare the model predictions with millifluidic transport experiments, which shows that, albeit simple, this technique is computationally efficient and has potential for predicting preferential paths and stagnation zones for flow and transport in porous media. We propose to use the eigenvector centrality probability distribution to compute the entropy as an indicator of the "mixing capacity" of the system.

Super-Resolution Community Detection for Layer-Aggregated Multilayer Networks

Applied network science often involves preprocessing network data before applying a network-analysis method, and there is typically a theoretical disconnect between these steps. For example, it is common to aggregate time-varying network data into windows prior to analysis, and the trade-offs of this preprocessing are not well understood. Focusing on the problem of detecting small communities in multilayer networks, we study the effects of layer aggregation by developing random-matrix theory for modularity matrices associated with layer-aggregated networks with N nodes and L layers, which are drawn from an ensemble of Erdős-Rényi networks with communities planted in subsets of layers. We study phase transitions in which eigenvectors localize onto communities (allowing their detection) and which occur for a given community provided its size surpasses a detectability limit K* . When layers are aggregated via a summation, we obtain [Formula: see text], where T is the number of layers across which the community persists. Interestingly, if T is allowed to vary with L, then summation-based layer aggregation enhances small-community detection even if the community persists across a vanishing fraction of layers, provided that T/L decays more slowly than 𝒪(L-1/2). Moreover, we find that thresholding the summation can, in some cases, cause K* to decay exponentially, decreasing by orders of magnitude in a phenomenon we call super-resolution community detection. In other words, layer aggregation with thresholding is a nonlinear data filter enabling detection of communities that are otherwise too small to detect. Importantly, different thresholds generally enhance the detectability of communities having different properties, illustrating that community detection can be obscured if one analyzes network data using a single threshold.

Nonbacktracking expansion of finite graphs

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

Shortest Paths in Multiplex Networks

The shortest path problem is one of the most fundamental networks optimization problems. Nowadays, individuals interact in extraordinarily numerous ways through their offline and online life (e.g., co-authorship, co-workership, or retweet relation in Twitter). These interactions have two key features. First, they have a heterogeneous nature, and second, they have different strengths that are weighted based on their degree of intimacy, trustworthiness, service exchange or influence among individuals. These networks are known as multiplex networks. To our knowledge, none of the previous shortest path definitions on social interactions have properly reflected these features. In this work, we introduce a new distance measure in multiplex networks based on the concept of Pareto efficiency taking both heterogeneity and weighted nature of relations into account. We then model the problem of finding the whole set of paths as a form of multiple objective decision making and propose an exact algorithm for that. The method is evaluated on five real-world datasets to test the impact of considering weights and multiplexity in the resulting shortest paths. As an application to find the most influential nodes, we redefine the concept of betweenness centrality based on the proposed shortest paths and evaluate it on a real-world dataset from two-layer trade relation among countries between years 2000 and 2015.

Efficient collective influence maximization in cascading processes with first-order transitions

In many social and biological networks, the collective dynamics of the entire system can be shaped by a small set of influential units through a global cascading process, manifested by an abrupt first-order transition in dynamical behaviors. Despite its importance in applications, efficient identification of multiple influential spreaders in cascading processes still remains a challenging task for large-scale networks. Here we address this issue by exploring the collective influence in general threshold models of cascading process. Our analysis reveals that the importance of spreaders is fixed by the subcritical paths along which cascades propagate: the number of subcritical paths attached to each spreader determines its contribution to global cascades. The concept of subcritical path allows us to introduce a scalable algorithm for massively large-scale networks. Results in both synthetic random graphs and real networks show that the proposed method can achieve larger collective influence given the same number of seeds compared with other scalable heuristic approaches.

EIGENVECTOR-BASED CENTRALITY MEASURES FOR TEMPORAL NETWORKS. MULTISCALE MODELING & SIMULATION : A SIAM INTERDISCIPLINARY JOURNAL 2017;15:537-574. [PMID: 29046619 PMCID: PMC5643020 DOI: 10.1137/16m1066142]

Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network during different time windows, and we couple centrality matrices for the layers into a supra-centrality matrix of size NT × NT whose dominant eigenvector gives the centrality of each node i at each time t. We refer to this eigenvector and its components as a joint centrality, as it reflects the importances of both the node i and the time layer t. We also introduce the concepts of marginal and conditional centralities, which facilitate the study of centrality trajectories over time. We find that the strength of coupling between layers is important for determining multiscale properties of centrality, such as localization phenomena and the time scale of centrality changes. In the strong-coupling regime, we derive expressions for time-averaged centralities, which are given by the zeroth-order terms of a singular perturbation expansion. We also study first-order terms to obtain first-order-mover scores, which concisely describe the magnitude of nodes' centrality changes over time. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among top-billed actors during the Golden Age of Hollywood, and citations of decisions from the United States Supreme Court.

Unification of theoretical approaches for epidemic spreading on complex networks

Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical message-passing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into developing an accurate theoretical approach to spreading dynamics on complex networks.

Topological structure and the H index in complex networks

The generalized H(n) Hirsch index of order n has been recently introduced and shown to interpolate between the degree and the K-core centrality in networks. We provide a detailed analytic characterization of the properties of sets of nodes having the same H(n), within the annealed network approximation. The connection between the Hirsch indices and the degree is highlighted. Numerical tests in synthetic uncorrelated networks and real-world correlated ones validate the findings. We also test the use of the Hirsch index for the identification of influential spreaders in networks, finding that it is in general outperformed by the recently introduced nonbacktracking centrality.

Leveraging percolation theory to single out influential spreaders in networks

Among the consequences of the disordered interaction topology underlying many social, technological, and biological systems, a particularly important one is that some nodes, just because of their position in the network, may have a disproportionate effect on dynamical processes mediated by the complex interaction pattern. For example, the early adoption of a commercial product by an opinion leader in a social network may change its fate or just a few superspreaders may determine the virality of a meme in social media. Despite many recent efforts, the formulation of an accurate method to optimally identify influential nodes in complex network topologies remains an unsolved challenge. Here, we present the exact solution of the problem for the specific, but highly relevant, case of the susceptible-infected-removed (SIR) model for epidemic spreading at criticality. By exploiting the mapping between bond percolation and the static properties of the SIR model, we prove that the recently introduced nonbacktracking centrality is the optimal criterion for the identification of influential spreaders in locally tree-like networks at criticality. By means of simulations on synthetic networks and on a very extensive set of real-world networks, we show that the nonbacktracking centrality is a highly reliable metric to identify top influential spreaders also in generic graphs not embedded in space and for noncritical spreading.

Predicting the epidemic threshold of the susceptible-infected-recovered model

Researchers have developed several theoretical methods for predicting epidemic thresholds, including the mean-field like (MFL) method, the quenched mean-field (QMF) method, and the dynamical message passing (DMP) method. When these methods are applied to predict epidemic threshold they often produce differing results and their relative levels of accuracy are still unknown. We systematically analyze these two issues-relationships among differing results and levels of accuracy-by studying the susceptible-infected-recovered (SIR) model on uncorrelated configuration networks and a group of 56 real-world networks. In uncorrelated configuration networks the MFL and DMP methods yield identical predictions that are larger and more accurate than the prediction generated by the QMF method. As for the 56 real-world networks, the epidemic threshold obtained by the DMP method is more likely to reach the accurate epidemic threshold because it incorporates full network topology information and some dynamical correlations. We find that in most of the networks with positive degree-degree correlations, an eigenvector localized on the high k-core nodes, or a high level of clustering, the epidemic threshold predicted by the MFL method, which uses the degree distribution as the only input information, performs better than the other two methods.

Measuring the potential of individual airports for pandemic spread over the world airline network

Background

Massive growth in human mobility has dramatically increased the risk and rate of pandemic spread. Macro-level descriptors of the topology of the World Airline Network (WAN) explains middle and late stage dynamics of pandemic spread mediated by this network, but necessarily regard early stage variation as stochastic. We propose that much of this early stage variation can be explained by appropriately characterizing the local network topology surrounding an outbreak’s debut location.

Methods

Based on a model of the WAN derived from public data, we measure for each airport the expected force of infection (AEF) which a pandemic originating at that airport would generate, assuming an epidemic process which transmits from airport to airport via scheduled commercial flights. We observe, for a subset of world airports, the minimum transmission rate at which a disease becomes pandemically competent at each airport. We also observe, for a larger subset, the time until a pandemically competent outbreak achieves pandemic status given its debut location. Observations are generated using a highly sophisticated metapopulation reaction-diffusion simulator under a disease model known to well replicate the 2009 influenza pandemic. The robustness of the AEF measure to model misspecification is examined by degrading the underlying model WAN.

Results

AEF powerfully explains pandemic risk, showing correlation of 0.90 to the transmission level needed to give a disease pandemic competence, and correlation of 0.85 to the delay until an outbreak becomes a pandemic. The AEF is robust to model misspecification. For 97 % of airports, removing 15 % of airports from the model changes their AEF metric by less than 1 %.

Conclusions

Appropriately summarizing the size, shape, and diversity of an airport’s local neighborhood in the WAN accurately explains much of the macro-level stochasticity in pandemic outcomes.

Electronic supplementary material

The online version of this article (doi:10.1186/s12879-016-1350-4) contains supplementary material, which is available to authorized users.

Distinct types of eigenvector localization in networks

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes’ centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q^−γ, localization occurs on the largest hub if γ > 5/2; for γ < 5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

Urban Scaling of Cities in the Netherlands

We investigated the socioeconomic scaling behavior of all cities with more than 50,000 inhabitants in the Netherlands and found significant superlinear scaling of the gross urban product with population size. Of these cities, 22 major cities have urban agglomerations and urban areas defined by the Netherlands Central Bureau of Statistics. For these major cities we investigated the superlinear scaling for three separate modalities: the cities defined as municipalities, their urban agglomerations and their urban areas. We find superlinearity with power-law exponents of around 1.15. But remarkably, both types of agglomerations underperform if we compare for the same size of population an agglomeration with a city as a municipality. In other words, an urban system as one formal municipality performs better as compared to an urban agglomeration with the same population size. This effect is larger for the second type of agglomerations, the urban areas. We think this finding has important implications for urban policy, in particular municipal reorganizations. A residual analysis suggests that cities with a municipal reorganization recently and in the past decades have a higher probability to perform better than cities without municipal restructuring.

Message-passing approach for recurrent-state epidemic models on networks

Epidemic processes are common out-of-equilibrium phenomena of broad interdisciplinary interest. Recently, dynamic message-passing (DMP) has been proposed as an efficient algorithm for simulating epidemic models on networks, and in particular for estimating the probability that a given node will become infectious at a particular time. To date, DMP has been applied exclusively to models with one-way state changes, as opposed to models like SIS and SIRS where nodes can return to previously inhabited states. Because many real-world epidemics can exhibit such recurrent dynamics, we propose a DMP algorithm for complex, recurrent epidemic models on networks. Our approach takes correlations between neighboring nodes into account while preventing causal signals from backtracking to their immediate source, and thus avoids "echo chamber effects" where a pair of adjacent nodes each amplify the probability that the other is infectious. We demonstrate that this approach well approximates results obtained from Monte Carlo simulation and that its accuracy is often superior to the pair approximation (which also takes second-order correlations into account). Moreover, our approach is more computationally efficient than the pair approximation, especially for complex epidemic models: the number of variables in our DMP approach grows as 2mk where m is the number of edges and k is the number of states, as opposed to mk^{2} for the pair approximation. We suspect that the resulting reduction in computational effort, as well as the conceptual simplicity of DMP, will make it a useful tool in epidemic modeling, especially for high-dimensional inference tasks.

Universality in the spectral and eigenfunction properties of random networks

By the use of extensive numerical simulations, we show that the nearest-neighbor energy-level spacing distribution P(s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ≡αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that the Brody distribution characterizes well P(s) in the transition from α=0, when the vertices in the network are isolated, to α=1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

Predicting percolation thresholds in networks

We consider different methods, which do not rely on numerical simulations of the percolation process, to approximate percolation thresholds in networks. We perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools. Our study reveals that the inverse of the largest eigenvalue of the nonbacktracking matrix of the graph often provides a tight lower bound for true percolation threshold. However, in more than 40% of the cases, this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution. We find that the performance of all indicators becomes worse as the value of the true percolation threshold grows. Thus, none of them represents a good proxy for the robustness of extremely fragile networks.