Spreading gossip in social networks

The Big Bang of an epidemic: a metapopulation approach to identify the spatiotemporal origin of contagious diseases and their universal spreading pattern

In this paper, we propose a mathematical framework that governs the evolution of epidemic dynamics, encompassing both intra-population dynamics and inter-population mobility within a meta-population network. By linearizing this dynamical system, we can identify the spatial starting point(s), namely the source(s) and the initiation time of the epidemic, which we refer to as the "Big Bang" of the epidemic. Furthermore, we introduce a novel concept of effective distance to track disease spread within the network. Our analysis reveals that the contagion geometry can be represented as a line with a universal slope, for any disease type (R0) or mobility network configuration. The mathematical derivations presented in this framework are corroborated by empirical data, including observations from the COVID-19 pandemic in Iran and the US and the H1N1 outbreak worldwide. Within this framework, to detect the Big Bang of an epidemic we require two types of data: (1) A snapshot of the active infected cases in each subpopulation during the linear phase. (2) A coarse-grained representation of inter-population mobility. Also even with access to only the first type of data, we can still demonstrate the universal contagion geometric pattern. Additionally, we can estimate errors and assess the precision of the estimations. This comprehensive approach enhances our understanding of when and where epidemics began and how they spread. It equips us with valuable insights for developing effective public health policies and mitigating the impact of infectious diseases on populations worldwide.

Trust based attachment

In social systems subject to indirect reciprocity, a positive reputation is key for increasing one's likelihood of future positive interactions [1-13]. The flow of gossip can amplify the impact of a person's actions on their reputation depending on how widely it spreads across the social network, which leads to a percolation problem [14]. To quantify this notion, we calculate the expected number of individuals, the "audience", who find out about a particular interaction. For a potential donor, a larger audience constitutes higher reputational stakes, and thus a higher incentive, to perform "good" actions in line with current social norms [7, 15]. For a receiver, a larger audience therefore increases the trust that the partner will be cooperative. This idea can be used for an algorithm that generates social networks, which we call trust based attachment (TBA). TBA produces graphs that share crucial quantitative properties with real-world networks, such as high clustering, small-world behavior, and powerlaw degree distributions [16-21]. We also show that TBA can be approximated by simple friend-of-friend routines based on triadic closure, which are known to be highly effective at generating realistic social network structures [19, 22-25]. Therefore, our work provides a new justification for triadic closure in social contexts based on notions of trust, gossip, and social information spread. These factors are thus identified as potential significant influences on how humans form social ties.

Random Walks on Networks with Centrality-Based Stochastic Resetting

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.

Network diffusion of gender diversity on boards: A process of two-speed opposing forces

Network diffusion processes or how information spreads through networks have been widely examined in numerous disciplines such as epidemiology, physics, sociology, politics, or computer science. In this paper, we extend previous developments by considering a generalization of the diffusion by considering the possibility of differences in the speed of diffusion and reduction depending on the forces’ directions. In this situation, the differential speed of diffusion produces deviations from the standard solution around the average of the initial conditions in the network. In fact, this asymmetry gives rise to non-linear dynamics in which, contrary to the symmetric case, the final solution depends on the topology of the graph as well as on the distribution of the initial values. Counter-intuitively, less central nodes in the network are able to exert a higher influence on the final solution. This behavior applies also for different simulated networks such as random, small-world, and scale-free. We show an example of this kind of asymmetric diffusion process in a real case. To do so, we use a network of US Boards of Directors, where boards are the nodes and the directors working for more than one board, are the links. Changes in the proportion of women serving on each board are influenced by the gradient between adjacent boards. We also show that there is an asymmetry: the gradient is reduced at a slower (faster) rhythm if the board has less (more) women than neighboring boards. We are able to quantify the accumulated effect of this asymmetry from 2000 to 2015 in the overall proportion of women on boards, in a 4.7 percentage points (the proportion should have been an 14.61% instead of the observed 9.93% in 2015).

A novel understanding of the nature of epistemic vice

My aim in this paper is to present and discuss a novel understanding of the nature of epistemic vice. I highlight that epistemic vice such as excessive curiosity, gossip and excessive inquisitiveness do not obstruct the acquisition, transmission and retention of knowledge and are not characterized by a deficiency of epistemic desires or vicious epistemic motivations. However, I argue that such traits ought to be classified as epistemic vices because the agent who possesses them causes epistemic harm to other agents through those traits' characteristic activities. To remedy obstructivism's inability to account for vices that cause epistemic harm in other ways besides blocking effective epistemic inquiry, I propose an amended version of this theory. I argue that epistemic vices are character traits, attitudes, and ways of thinking that obstruct the acquisition, transmission, and retention of knowledge and/or cause other kinds of epistemic harm. In addition, I propose a modified version of motivationalism that cashes out non-obstructing, excess-motivation vices in terms of motivation simply by acknowledging, and incorporating into theory, excessive epistemic drives and the negative epistemic (and non-epistemic) consequences stemming from them.

A novel understanding of the nature of epistemic vice

My aim in this paper is to present and discuss a novel understanding of the nature of epistemic vice. I highlight that epistemic vice such as excessive curiosity, gossip and excessive inquisitiveness do not obstruct the acquisition, transmission and retention of knowledge and are not characterized by a deficiency of epistemic desires or vicious epistemic motivations. However, I argue that such traits ought to be classified as epistemic vices because the agent who possesses them causes epistemic harm to other agents through those traits' characteristic activities. To remedy obstructivism's inability to account for vices that cause epistemic harm in other ways besides blocking effective epistemic inquiry, I propose an amended version of this theory. I argue that epistemic vices are character traits, attitudes, and ways of thinking that obstruct the acquisition, transmission, and retention of knowledge and/or cause other kinds of epistemic harm. In addition, I propose a modified version of motivationalism that cashes out non-obstructing, excess-motivation vices in terms of motivation simply by acknowledging, and incorporating into theory, excessive epistemic drives and the negative epistemic (and non-epistemic) consequences stemming from them.

Locating the source of large-scale outbreaks of foodborne disease

In today's globally interconnected food system, outbreaks of foodborne disease can spread widely and cause considerable impact on public health. We study the problem of identifying the source of emerging large-scale outbreaks of foodborne disease; a crucial step in mitigating their proliferation. To solve the source identification problem, we formulate a probabilistic model of the contamination diffusion process as a random walk on a network and derive the maximum-likelihood estimator for the source location. By modelling the transmission process as a random walk, we are able to develop a novel, computationally tractable solution that accounts for all possible paths of travel through the network. This is in contrast to existing approaches to network source identification, which assume that the contamination travels along either the shortest or highest probability paths. We demonstrate the benefits of the multiple-paths approach through application to different network topologies, including stylized models of food supply network structure and real data from the 2011 Shiga toxin-producing Escherichia coli outbreak in Germany. We show significant improvements in accuracy and reliability compared with the relevant state-of-the-art approach to source identification. Beyond foodborne disease, these methods should find application in identifying the source of spread in network-based diffusion processes more generally, including in networks not well approximated by tree-like structure.

A trust model for spreading gossip in social networks: a multi-type bootstrap percolation model

We introduce here a multi-type bootstrap percolation model, which we call T -Bootstrap Percolation ( T -BP), and apply it to study information propagation in social networks. In this model, a social network is represented by a graph G whose vertices have different labels corresponding to the type of role the person plays in the network (e.g. a student, an educator etc.). Once an initial set of vertices of G is randomly selected to be carrying a gossip (e.g. to be infected), the gossip propagates to a new vertex provided it is transmitted by a minimum threshold of vertices with different labels. By considering random graphs, which have been shown to closely represent social networks, we study different properties of the T -BP model through numerical simulations, and describe its implications when applied to rumour spread, fake news and marketing strategies.

Time evolution of the behaviour of Brazilian legislative Representatives using a complex network approach

The follow up of Representative behavior after elections is imperative for a democratic Representative system, at the very least to punish betrayal with no re-election. Our goal was to show how to follow Representatives’ and how to show behavior in real situations and observe trends in political crises including the onset of game changing political instabilities. We used correlation and correlation distance matrices of Brazilian Representative votes during four presidential terms. Re-ordering these matrices with Minimal Spanning Trees displays the dynamical formation of clusters for the sixteen year period, which includes one Presidential impeachment. The reordered matrices, colored by correlation strength and by the parties clearly show the origin of observed clusters and their evolution over time. When large clusters provide government support cluster breaks, political instability arises, which could lead to an impeachment, a trend we observed three years before the Brazilian President was impeached. We believe this method could be applied to foresee other political storms.

EpiRank: Modeling Bidirectional Disease Spread in Asymmetric Commuting Networks

Commuting network flows are generally asymmetrical, with commuting behaviors bi-directionally balanced between home and work locations, and with weekday commutes providing many opportunities for the spread of infectious diseases via direct and indirect physical contact. The authors use a Markov chain model and PageRank-like algorithm to construct a novel algorithm called EpiRank to measure infection risk in a spatially confined commuting network on Taiwan island. Data from the country's 2000 census were used to map epidemic risk distribution as a commuting network function. A daytime parameter was used to integrate forward and backward movement in order to analyze daily commuting patterns. EpiRank algorithm results were tested by comparing calculations with actual disease distributions for the 2009 H1N1 influenza outbreak and enterovirus cases between 2000 and 2008. Results suggest that the bidirectional movement model outperformed models that considered forward or backward direction only in terms of capturing spatial epidemic risk distribution. EpiRank also outperformed models based on network indexes such as PageRank and HITS. According to a sensitivity analysis of the daytime parameter, the backward movement effect is more important than the forward movement effect for understanding a commuting network's disease diffusion structure. Our evidence supports the use of EpiRank as an alternative network measure for analyzing disease diffusion in a commuting network.

Scaling properties of d-dimensional complex networks

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located d-dimensional networks. In this paper, we study the scaling properties of a wide class of d-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through r_{ij}^{-α_{A}} (α_{A}≥0). We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient for d=1,2,3,4 and typical values of α_{A}. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable α_{A}/d. These observations confirm the exist- ence of three regimes. The first one occurs in the interval α_{A}/d∈[0,1]; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a q exponential with q constant and above unity. The critical value α_{A}/d=1 that emerges in many of these properties is replaced by α_{A}/d=1/2 for the β exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately-long-range interactions, and reflects in an index q monotonically decreasing with α_{A}/d increasing from its critical value to a characteristic value α_{A}/d≃5. Finally, the third regime is Boltzmannian-like (with q≃1) and corresponds to short-range interactions.

The Network Source Location Problem in the Context of Foodborne Disease Outbreaks

In today’s globally interconnected food system, outbreaks of foodborne disease can spread widely and cause considerable impact on public health. Food distribution is a complex system that can be seen as a network of trade flows connecting supply chain actors. Identifying the source of an outbreak of foodborne disease distributed across this network can be solved by considering this network structure and the dimensions of information it contains. The literature on the network source identification problem has grown widely in recent years covering problems in many different contexts, from contagious disease infecting a human population, to computer viruses spreading through the Internet, to rumors or trends diffusing through a social network. Much of this work has focused on studying this problem in analytically tractable frameworks, designing approaches to work on trees and extending to general network structures in an ad hoc manner. These simplified frameworks lack many features of real-world networks and problem contexts that can dramatically impact transmission dynamics, and therefore, backwards inference of the transmission process. Moreover, the features that distinguish foodborne disease in the context of source identification have not previously been studied or identified. In this article we identify these features, then provide a review of existing work on the network source identification problem, categorizing approaches according to these features. We conclude that much of the existing work cannot be implemented in the foodborne disease problem because it makes assumptions about the transmission process that are unrealistic in the context of food supply networks—that is, identifying the source of an epidemic contagion whereas foodborne contamination spreads through a transport network-mediated diffusion process, or because it requires data that is not available—complete observations of the contamination status of all nodes in the network.

Fast and accurate detection of spread source in large complex networks

Spread over complex networks is a ubiquitous process with increasingly wide applications. Locating spread sources is often important, e.g. finding the patient one in epidemics, or source of rumor spreading in social network. Pinto, Thiran and Vetterli introduced an algorithm (PTVA) to solve the important case of this problem in which a limited set of nodes act as observers and report times at which the spread reached them. PTVA uses all observers to find a solution. Here we propose a new approach in which observers with low quality information (i.e. with large spread encounter times) are ignored and potential sources are selected based on the likelihood gradient from high quality observers. The original complexity of PTVA is O(N α ), where α ∈ (3,4) depends on the network topology and number of observers (N denotes the number of nodes in the network). Our Gradient Maximum Likelihood Algorithm (GMLA) reduces this complexity to O (N2log (N)). Extensive numerical tests performed on synthetic networks and real Gnutella network with limitation that id's of spreaders are unknown to observers demonstrate that for scale-free networks with such limitation GMLA yields higher quality localization results than PTVA does.

Influence of trust in the spreading of information

The understanding and prediction of information diffusion processes on networks is a major challenge in network theory with many implications in social sciences. Many theoretical advances occurred due to stochastic spreading models. Nevertheless, these stochastic models overlooked the influence of rational decisions on the outcome of the process. For instance, different levels of trust in acquaintances do play a role in information spreading, and actors may change their spreading decisions during the information diffusion process accordingly. Here, we study an information-spreading model in which the decision to transmit or not is based on trust. We explore the interplay between the propagation of information and the trust dynamics happening on a two-layer multiplex network. Actors' trustable or untrustable states are defined as accumulated cooperation or defection behaviors, respectively, in a Prisoner's Dilemma setup, and they are controlled by a memory span. The propagation of information is abstracted as a threshold model on the information-spreading layer, where the threshold depends on the trustability of agents. The analysis of the model is performed using a tree approximation and validated on homogeneous and heterogeneous networks. The results show that the memory of previous actions has a significant effect on the spreading of information. For example, the less memory that is considered, the higher is the diffusion. Information is highly promoted by the emergence of trustable acquaintances. These results provide insight into the effect of plausible biases on spreading dynamics in a multilevel networked system.

How Behaviour and the Environment Influence Transmission in Mobile Groups

The movement of individuals living in groups leads to the formation of physical interaction networks over which signals such as information or disease can be transmitted. Direct contacts represent the most obvious opportunities for a signal to be transmitted. However, because signals that persist after being deposited into the environment may later be acquired by other group members, indirect environmentally-mediated transmission is also possible. To date, studies of signal transmission within groups have focused on direct physical interactions and ignored the role of indirect pathways. Here, we use an agent-based model to study how the movement of individuals and characteristics of the signal being transmitted modulate transmission. By analysing the dynamic interaction networks generated from these simulations, we show that the addition of indirect pathways speeds up signal transmission, while the addition of physically-realistic collisions between individuals in densely packed environments hampers it. Furthermore, the inclusion of spatial biases that induce the formation of individual territories, reveals the existence of a trade-off such that optimal signal transmission at the group level is only achieved when territories are of intermediate sizes. Our findings provide insight into the selective pressures guiding the evolution of behavioural traits in natural groups, and offer a means by which multi-agent systems can be engineered to achieve desired transmission capabilities.

Effective information spreading based on local information in correlated networks

Using network-based information to facilitate information spreading is an essential task for spreading dynamics in complex networks. Focusing on degree correlated networks, we propose a preferential contact strategy based on the local network structure and local informed density to promote the information spreading. During the spreading process, an informed node will preferentially select a contact target among its neighbors, basing on their degrees or local informed densities. By extensively implementing numerical simulations in synthetic and empirical networks, we find that when only consider the local structure information, the convergence time of information spreading will be remarkably reduced if low-degree neighbors are favored as contact targets. Meanwhile, the minimum convergence time depends non-monotonically on degree-degree correlation, and a moderate correlation coefficient results in the most efficient information spreading. Incorporating the local informed density information into contact strategy, the convergence time of information spreading can be further reduced, and be minimized by an moderately preferential selection.

Role of dimensionality in complex networks

Deep connections are known to exist between scale-free networks and non-Gibbsian statistics. For example, typical degree distributions at the thermodynamical limit are of the form , where the q-exponential form optimizes the nonadditive entropy S_q (which, for q → 1, recovers the Boltzmann-Gibbs entropy). We introduce and study here d-dimensional geographically-located networks which grow with preferential attachment involving Euclidean distances through . Revealing the connection with q-statistics, we numerically verify (for d = 1, 2, 3 and 4) that the q-exponential degree distributions exhibit, for both q and k, universal dependences on the ratio α_A/d. Moreover, the q = 1 limit is rapidly achieved by increasing α_A/d to infinity.

How multiple social networks affect user awareness: The information diffusion process in multiplex networks

The information diffusion process in single complex networks has been extensively studied, especially for modeling the spreading activities in online social networks. However, individuals usually use multiple social networks at the same time, and can share the information they have learned from one social network to another. This phenomenon gives rise to a new diffusion process on multiplex networks with more than one network layer. In this paper we account for this multiplex network spreading by proposing a model of information diffusion in two-layer multiplex networks. We develop a theoretical framework using bond percolation and cascading failure to describe the intralayer and interlayer diffusion. This allows us to obtain analytical solutions for the fraction of informed individuals as a function of transmissibility T and the interlayer transmission rate θ. Simulation results show that interaction between layers can greatly enhance the information diffusion process. And explosive diffusion can occur even if the transmissibility of the focal layer is under the critical threshold, due to interlayer transmission.

Robust-yet-fragile nature of interdependent networks

Interdependent networks have been shown to be extremely vulnerable based on the percolation model. Parshani et al. [Europhys. Lett. 92, 68002 (2010)] further indicated that the more intersimilar networks are, the more robust they are to random failures. When traffic load is considered, how do the coupling patterns impact cascading failures in interdependent networks? This question has been largely unexplored until now. In this paper, we address this question by investigating the robustness of interdependent Erdös-Rényi random graphs and Barabási-Albert scale-free networks under either random failures or intentional attacks. It is found that interdependent Erdös-Rényi random graphs are robust yet fragile under either random failures or intentional attacks. Interdependent Barabási-Albert scale-free networks, however, are only robust yet fragile under random failures but fragile under intentional attacks. We further analyze the interdependent communication network and power grid and achieve similar results. These results advance our understanding of how interdependency shapes network robustness.

Traffic congestion in interconnected complex networks

Traffic congestion in isolated complex networks has been investigated extensively over the last decade. Coupled network models have recently been developed to facilitate further understanding of real complex systems. Analysis of traffic congestion in coupled complex networks, however, is still relatively unexplored. In this paper, we try to explore the effect of interconnections on traffic congestion in interconnected Barabási-Albert scale-free networks. We find that assortative coupling can alleviate traffic congestion more readily than disassortative and random coupling when the node processing capacity is allocated based on node usage probability. Furthermore, the optimal coupling probability can be found for assortative coupling. However, three types of coupling preferences achieve similar traffic performance if all nodes share the same processing capacity. We analyze interconnected Internet autonomous-system-level graphs of South Korea and Japan and obtain similar results. Some practical suggestions are presented to optimize such real-world interconnected networks accordingly.

Collaboration networks from a large CV database: dynamics, topology and bonus impact

Understanding the dynamics of research production and collaboration may reveal better strategies for scientific careers, academic institutions, and funding agencies. Here we propose the use of a large and multidisciplinary database of scientific curricula in Brazil, namely, the Lattes Platform, to study patterns of scientific production and collaboration. Detailed information about publications and researchers is available in this database. Individual curricula are submitted by the researchers themselves so that coauthorship is unambiguous. Researchers can be evaluated by scientific productivity, geographical location and field of expertise. Our results show that the collaboration network is growing exponentially for the last three decades, with a distribution of number of collaborators per researcher that approaches a power-law as the network gets older. Moreover, both the distributions of number of collaborators and production per researcher obey power-law behaviors, regardless of the geographical location or field, suggesting that the same universal mechanism might be responsible for network growth and productivity. We also show that the collaboration network under investigation displays a typical assortative mixing behavior, where teeming researchers (i.e., with high degree) tend to collaborate with others alike.

Critical behavior of the ideal-gas Bose-Einstein condensation in the Apollonian network

We show that the ideal Boson gas displays a finite-temperature Bose-Einstein condensation transition in the complex Apollonian network exhibiting scale-free, small-world, and hierarchical properties. The single-particle tight-binding Hamiltonian with properly rescaled hopping amplitudes has a fractal-like energy spectrum. The energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation. A finite-size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise estimates for the exponents characterizing the condensed fraction, correlation size, and specific heat. The critical exponents, as well as the power-law behavior of the density of states at the bottom of the band, are similar to those of the ideal Boson gas in lattices with spectral dimension d(s)=2ln(3)/ln(9/5)~/=3.74.

Stability and responsiveness in a self-organized living architecture

Robustness and adaptability are central to the functioning of biological systems, from gene networks to animal societies. Yet the mechanisms by which living organisms achieve both stability to perturbations and sensitivity to input are poorly understood. Here, we present an integrated study of a living architecture in which army ants interconnect their bodies to span gaps. We demonstrate that these self-assembled bridges are a highly effective means of maintaining traffic flow over unpredictable terrain. The individual-level rules responsible depend only on locally-estimated traffic intensity and the number of neighbours to which ants are attached within the structure. We employ a parameterized computational model to reveal that bridges are tuned to be maximally stable in the face of regular, periodic fluctuations in traffic. However analysis of the model also suggests that interactions among ants give rise to feedback processes that result in bridges being highly responsive to sudden interruptions in traffic. Subsequent field experiments confirm this prediction and thus the dual nature of stability and flexibility in living bridges. Our study demonstrates the importance of robust and adaptive modular architecture to efficient traffic organisation and reveals general principles regarding the regulation of form in biological self-assemblies.

While migrating, the nomadic army ant Eciton burchellii forms long trails of workers that can extend over hundreds of meters in the rain forest. To facilitate the movement of sometimes millions of individuals on uneven and unpredictable terrains, part of the ant workers link together their legs and bodies to form temporary bridges over gaps along the trails. In this work we showed that these bridges were formed readily when the flow of ants hit an unspanned gap and were dismantled very quickly after traffic has ceased on the trail. However, we also observed that the bridges were formed and remained stable under a large spectrum of the traffic intensities on the trail. Using field experiments and computer simulations we discovered the construction rules used by the ants to create these living structures that are capable of enduring variations of the traffic while remaining highly responsive to its interruption. These results offer important insights about the mechanisms that regulate biological self-assemblies and they have potential applications in swarm robotics and swarm intelligence.

Sampling properties of directed networks

For many real-world networks only a small "sampled" version of the original network may be investigated; those results are then used to draw conclusions about the actual system. Variants of breadth-first search (BFS) sampling, which are based on epidemic processes, are widely used. Although it is well established that BFS sampling fails, in most cases, to capture the IN component(s) of directed networks, a description of the effects of BFS sampling on other topological properties is all but absent from the literature. To systematically study the effects of sampling biases on directed networks, we compare BFS sampling to random sampling on complete large-scale directed networks. We present new results and a thorough analysis of the topological properties of seven complete directed networks (prior to sampling), including three versions of Wikipedia, three different sources of sampled World Wide Web data, and an Internet-based social network. We detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks. Most notably, we find that sampling method and coverage affect both the bow-tie structure and the number and structure of strongly connected components in sampled networks. In addition, at a low sampling coverage (i.e., less than 40%), the values of average degree, variance of out-degree, degree autocorrelation, and link reciprocity are overestimated by 30% or more in BFS-sampled networks and only attain values within 10% of the corresponding values in the complete networks when sampling coverage is in excess of 65%. These results may cause us to rethink what we know about the structure, function, and evolution of real-world directed networks.

Social influence and spread dynamics in social networks

Social networks often serve as a critical medium for information dissemination, diffusion of epidemics, and spread of behavior, by shared activities or similarities between individuals. Recently, we have witnessed an explosion of interest in studying social influence and spread dynamics in social networks. To date, relatively little material has been provided on a comprehensive review in this field. This brief survey addresses this issue. We present the current significant empirical studies on real social systems, including network construction methods, measures of network, and newly empirical results. We then provide a concise description of some related social models from both macro- and micro-level perspectives. Due to the difficulties in combining real data and simulation data for verifying and validating real social systems, we further emphasize the current research results of computational experiments. We hope this paper can provide researchers significant insights into better understanding the characteristics of personal influence and spread patterns in large-scale social systems.

Locating the source of diffusion in large-scale networks

How can we localize the source of diffusion in a complex network? Because of the tremendous size of many real networks-such as the internet or the human social graph-it is usually unfeasible to observe the state of all nodes in a network. We show that it is fundamentally possible to estimate the location of the source from measurements collected by sparsely placed observers. We present a strategy that is optimal for arbitrary trees, achieving maximum probability of correct localization. We describe efficient implementations with complexity O(N(α)), where α=1 for arbitrary trees and α=3 for arbitrary graphs. In the context of several case studies, we determine how localization accuracy is affected by various system parameters, including the structure of the network, the density of observers, and the number of observed cascades.

Propagation on networks: an exact alternative perspective

By generating the specifics of a network structure only when needed (on-the-fly), we derive a simple stochastic process that exactly models the time evolution of susceptible-infectious dynamics on finite-size networks. The small number of dynamical variables of this birth-death Markov process greatly simplifies analytical calculations. We show how a dual analytical description, treating large scale epidemics with a Gaussian approximation and small outbreaks with a branching process, provides an accurate approximation of the distribution even for rather small networks. The approach also offers important computational advantages and generalizes to a vast class of systems.

A universal model for mobility and migration patterns

Introduced in its contemporary form in 1946 (ref. 1), but with roots that go back to the eighteenth century, the gravity law is the prevailing framework with which to predict population movement, cargo shipping volume and inter-city phone calls, as well as bilateral trade flows between nations. Despite its widespread use, it relies on adjustable parameters that vary from region to region and suffers from known analytic inconsistencies. Here we introduce a stochastic process capturing local mobility decisions that helps us analytically derive commuting and mobility fluxes that require as input only information on the population distribution. The resulting radiation model predicts mobility patterns in good agreement with mobility and transport patterns observed in a wide range of phenomena, from long-term migration patterns to communication volume between different regions. Given its parameter-free nature, the model can be applied in areas where we lack previous mobility measurements, significantly improving the predictive accuracy of most of the phenomena affected by mobility and transport processes.

Quantifying the connectivity of a network: the network correlation function method

Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small-world networks), and a power-law degree distribution (scale-free networks). The topological features of a network are commonly related to the network's functionality. However, the topology alone does not account for the nature of the interactions in the network and their strength. Here, we present a method for evaluating the correlations between pairs of nodes in the network. These correlations depend both on the topology and on the functionality of the network. A network with high connectivity displays strong correlations between its interacting nodes and thus features small-world functionality. We quantify the correlations between all pairs of nodes in the network, and express them as matrix elements in the correlation matrix. From this information, one can plot the correlation function for the network and to extract the correlation length. The connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network. Using this method, we distinguish between a topological small world and a functional small world, where the latter is characterized by long-range correlations and high connectivity. Clearly, networks that share the same topology may have different connectivities, based on the nature and strength of their interactions. The method is demonstrated on metabolic networks, but can be readily generalized to other types of networks.

Efficient network reconstruction from dynamical cascades identifies small-world topology of neuronal avalanches

Cascading activity is commonly found in complex systems with directed interactions such as metabolic networks, neuronal networks, or disease spreading in social networks. Substantial insight into a system's organization can be obtained by reconstructing the underlying functional network architecture from the observed activity cascades. Here we focus on Bayesian approaches and reduce their computational demands by introducing the Iterative Bayesian (IB) and Posterior Weighted Averaging (PWA) methods. We introduce a special case of PWA, cast in nonparametric form, which we call the normalized count (NC) algorithm. NC efficiently reconstructs random and small-world functional network topologies and architectures from subcritical, critical, and supercritical cascading dynamics and yields significant improvements over commonly used correlation methods. With experimental data, NC identified a functional and structural small-world topology and its corresponding traffic in cortical networks with neuronal avalanche dynamics.

In many complex systems found across disciplines, such as biological cells and organisms, social networks, economic systems, and the Internet, individual elements interact with each other, thereby forming large networks whose structure is often not known. In these complex networks, local events can easily propagate, resulting in diverse spatio-temporal activity cascades, or avalanches. Examples of such cascading activity are the propagation of diseases in social networks, cascades of chemical reactions inside a cell, the propagation of neuronal activity in the brain, and e-mail forwarding on the Internet. Although the observation of a single cascade provides limited insight into the organization of a complex network, the observation of many cascades allows for the reconstruction of very robust features of network organization, providing valuable insight into network function as well as network failure. The current work develops new algorithms for an efficient reconstruction of relatively large networks in the context of cascading activity. When applied to the brain, these algorithms uncover the structural and functional features of gray matter networks that display activity cascades in the form of neuronal avalanches.

Temporal correlations of local network losses

We introduce a continuum model describing data losses in a single node of a packet-switched network (like the Internet) which preserves the discrete nature of the data loss process. By construction, the model has critical behavior with a sharp transition from exponentially small to finite losses with increasing data arrival rate. We show that such a model exhibits strong fluctuations in the loss rate at the critical point and non-Markovian power-law correlations in time, in spite of the Markovian character of the data arrival process. The continuum model allows for rather general incoming data packet distributions and can be naturally generalized to consider the buffer server idleness statistics.