Finding communities in directed networks

Estimating Mixed Memberships in Directed Networks by Spectral Clustering

Community detection is an important and powerful way to understand the latent structure of complex networks in social network analysis. This paper considers the problem of estimating community memberships of nodes in a directed network, where a node may belong to multiple communities. For such a directed network, existing models either assume that each node belongs solely to one community or ignore variation in node degree. Here, a directed degree corrected mixed membership (DiDCMM) model is proposed by considering degree heterogeneity. An efficient spectral clustering algorithm with a theoretical guarantee of consistent estimation is designed to fit DiDCMM. We apply our algorithm to a small scale of computer-generated directed networks and several real-world directed networks.

Analysis of risk propagation using the world trade network

An economic system is an exemplar of a complex system in which all agents interact simultaneously. Interactions between countries have generally been studied using the flow of resources across diverse trade networks, in which the degree of dependence between two countries is typically measured based on the trade volume. However, indirect influences may not be immediately apparent. Herein, we compared a direct trade network to a trade network constructed using the personalized PageRank (PPR) encompassing indirect influences. By analyzing the correlation of the gross domestic product (GDP) between countries, we discovered that the PPR trade network has greater explanatory power on the propagation of economic events than direct trade by analyzing the GDP correlation between countries. To further validate our observations, an agent-based model of the spreading economic crisis was implemented for the Russia-Ukraine war of 2022. The model also demonstrates that the PPR explains the actual impact more effectively than the direct trade network. Our research highlights the significance of indirect and long-range relationships, which have often been overlooked.

Network analysis of pig movements in Ecuador: Strengthening surveillance of classical swine fever

The analysis of domestic pig movements has become useful to understand the disease spread patterns and epidemiology, which facilitates the development of more effective animal diseases control strategies. The aim of this work was to analyse the static and spatial characteristics of the pig network, to identify its trading communities and to study the contribution of the network to the transmission of classical swine fever. In this regard, we used the pig movement records from the National veterinary service of Ecuador (2017-2019), using social network analysis and spatial analysis to construct a network with registered premises as nodes and their movements as edges. Furthermore, we also created a network of parishes as its nodes by aggregating their premises movements as edges. The annual network metrics showed an average diameter of 20.33, a number of neighbours of 2.61, a shortest path length of 4.39 and a clustering coefficient of 0.38 (small-world structure). The most frequent movements were to or from markets (55%). Backyard producers made up 89% of the network premises, and the top 2% of parishes (highest degree) contributed to 50% of the movements. The highest frequencies of movements between parishes were in the centre of the country, while the highest frequency of movements to abattoirs was in the south-west. Finally, the pattern of CSF disease outbreaks within the Ecuador network was likely the result of network transmission processes. In conclusion, our results represented the first exploratory analysis of domestic pig movements at premise and parish levels. The surveillance system could consider these results to improve its procedures and update the disease control and management policy, and allow the implementation of targeted or risk-based surveillance. This article is protected by copyright. All rights reserved.

Flow stability for dynamic community detection

Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. An important task in these systems is to extract a simplified view of their time-dependent network of interactions. Community detection in temporal networks usually relies on aggregation over time windows or consider sequences of different stationary epochs. For dynamics-based methods, attempts to generalize static-network methodologies also face the fundamental difficulty that a stationary state of the dynamics does not always exist. Here, we derive a method based on a dynamical process evolving on the temporal network. Our method allows dynamics that do not reach a steady state and uncovers two sets of communities for a given time interval that accounts for the ordering of edges in forward and backward time. We show that our method provides a natural way to disentangle the different dynamical scales present in a system with synthetic and real-world examples.

Consistency landscape of network communities

The concept of community detection has long been used as a key device for handling the mesoscale structures in networks. Suitably conducted community detection reveals various embedded informative substructures of network topology. However, regarding the practical usage of community detection, it has always been a tricky problem to assign a reasonable community resolution for networks of interest. Because of the absence of the unanimously accepted criterion, most of the previous studies utilized rather ad hoc heuristics to decide the community resolution. In this work, we harness the concept of consistency in community structures of networks to provide the overall community resolution landscape of networks, which we eventually take to quantify the reliability of detected communities for a given resolution parameter. More precisely, we exploit the ambiguity in the results of stochastic detection algorithms and suggest a method that denotes the relative validity of community structures in regard to their stability of global and local inconsistency measures using multiple detection processes. Applying our framework to synthetic and real networks, we confirm that it effectively displays insightful fundamental aspects of community structures.

Overlapping Community Detection in Directed and Undirected Attributed Networks Using a Multiobjective Evolutionary Algorithm

In many real-world networks, the structural connections of networks and the attributes about each node are always available. We typically call such graphs attributed networks, in which attributes always play the same important role in community detection as the topological structure. It is shown that the very existence of overlapping communities is one of the most important characteristics of various complex networks, while the majority of the existing community detection methods was designed for detecting separated communities in attributed networks. Therefore, it is quite challenging to detect meaningful overlapping structures with the combination of node attributes and topological structures. Therefore, in this article, we propose a multiobjective evolutionary algorithm based on the similarity attribute for overlapping community detection in attributed networks (MOEA-SA _OV ). In MOEA-SA _OV , a modified extended modularity EQ_OV , dealing with both directed and undirected networks, is well designed as the first objective. Another objective employed is the attribute similarity S_A . Then, a novel encoding and decoding strategy is designed to realize the goal of representing overlapping communities efficiently. MOEA-SA _OV runs under the framework of the nondominated sorting genetic algorithm II (NSGA-II) and can automatically determine the number of communities. In the experiments, the performance of MOEA-SA _OV is validated on both synthetic and real-world networks, and the experimental results demonstrate that our method can effectively find Pareto fronts about overlapping community structures with practical significance in both directed and undirected attributed networks.

Generalized Markov stability of network communities

We address the problem of community detection in networks by introducing a general definition of Markov stability, based on the difference between the probability fluxes of a Markov chain on the network at different timescales. The specific implementation of the quality function and the resulting optimal community structure thus become dependent both on the type of Markov process and on the specific Markov times considered. For instance, if we use a natural Markov chain dynamics and discount its stationary distribution (that is, we take as reference process the dynamics at infinite time) we obtain the standard formulation of the Markov stability. Notably, the possibility to use finite-time transition probabilities to define the reference process naturally allows detecting communities at different resolutions, without the need to consider a continuous-time Markov chain in the small time limit. The main advantage of our general formulation of Markov stability based on dynamical flows is that we work with lumped Markov chains on network partitions, having the same stationary distribution of the original process. In this way the form of the quality function becomes invariant under partitioning, leading to a self-consistent definition of community structures at different aggregation scales.

Cut Based Method for Comparing Complex Networks

Revealing the underlying similarity of various complex networks has become both a popular and interdisciplinary topic, with a plethora of relevant application domains. The essence of the similarity here is that network features of the same network type are highly similar, while the features of different kinds of networks present low similarity. In this paper, we introduce and explore a new method for comparing various complex networks based on the cut distance. We show correspondence between the cut distance and the similarity of two networks. This correspondence allows us to consider a broad range of complex networks and explicitly compare various networks with high accuracy. Various machine learning technologies such as genetic algorithms, nearest neighbor classification, and model selection are employed during the comparison process. Our cut method is shown to be suited for comparisons of undirected networks and directed networks, as well as weighted networks. In the model selection process, the results demonstrate that our approach outperforms other state-of-the-art methods with respect to accuracy.

Predicting species emergence in simulated complex pre-biotic networks

An intriguing question in evolution is what would happen if one could "replay" life's tape. Here, we explore the following hypothesis: when replaying the tape, the details ("decorations") of the outcomes would vary but certain "invariants" might emerge across different life-tapes sharing similar initial conditions. We use large-scale simulations of an in silico model of pre-biotic evolution called GARD (Graded Autocatalysis Replication Domain) to test this hypothesis. GARD models the temporal evolution of molecular assemblies, governed by a rates matrix (i.e. network) that biases different molecules' likelihood of joining or leaving a dynamically growing and splitting assembly. Previous studies have shown the emergence of so called compotypes, i.e., species capable of replication and selection response. Here, we apply networks' science to ascertain the degree to which invariants emerge across different life-tapes under GARD dynamics and whether one can predict these invariant from the chemistry specification alone (i.e. GARD's rates network representing initial conditions). We analysed the (complex) rates' network communities and asked whether communities are related (and how) to the emerging species under GARD's dynamic, and found that the communities correspond to the species emerging from the simulations. Importantly, we show how to use the set of communities detected to predict species emergence without performing any simulations. The analysis developed here may impact complex systems simulations in general.

Finding multiple core-periphery pairs in networks

With a core-periphery structure of networks, core nodes are densely interconnected, peripheral nodes are connected to core nodes to different extents, and peripheral nodes are sparsely interconnected. Core-periphery structure composed of a single core and periphery has been identified for various networks. However, analogous to the observation that many empirical networks are composed of densely interconnected groups of nodes, i.e., communities, a network may be better regarded as a collection of multiple cores and peripheries. We propose a scalable algorithm to detect multiple nonoverlapping groups of core-periphery structure in a network. We illustrate our algorithm using synthesized and empirical networks. For example, we find distinct core-periphery pairs with different political leanings in a network of political blogs and separation between international and domestic subnetworks of airports in some single countries in a worldwide airport network.

Recent Themes in Social Networking Service Research

The body of literature addressing the phenomenon related to social networking services (SNSs) has grown rather fast recently. Through a systematic and quantitative approach, this study identifies the recent SNS research themes, which are the issues discussed by a coherent and growing subset of this literature. A set of academic articles retrieved from the Web of Science database is used as the basis for uncovering the recent themes. We begin the analysis by constructing a citation network which is further separated into groups after applying a widely used clustering method. The resulting clusters all consist of articles coherent in citation relationships. This study suggests eight fast growing recent themes. They span widely encompassing politics, romantic relationships, public relations, journalism, and health. Among them, four focus their issues largely on Twitter, three on Facebook, and one generally on both. While discussions on traditional issues in SNSs such as personality, motivations, self-disclosure, narcissism, etc. continue to lead the pack, the proliferation of the highlighted recent themes in the near future is very likely to happen.

Cycle and flow trusses in directed networks

When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analysing directed networks. Although many notions of a directed module have been proposed, no consensus has been reached. This lack of consensus results partly because there might exist distinct types of modules in a single directed network, whereas most previous studies focused on an independent criterion for modules. To address this issue, we propose a generic notion of the so-called truss structures in directed networks. Our definition of truss is able to extract two distinct types of trusses, named the cycle truss and the flow truss, from a unified framework. By applying the method for finding trusses to empirical networks obtained from a wide range of research fields, we find that most real networks contain both cycle and flow trusses. In addition, the abundance of (and the overlap between) the two types of trusses may be useful to characterize module structures in a wide variety of empirical networks. Our findings shed light on the importance of simultaneously considering different types of modules in directed networks.

A Network of Networks Perspective on Global Trade

Mutually intertwined supply chains in contemporary economy result in a complex network of trade relationships with a highly non-trivial topology that varies with time. In order to understand the complex interrelationships among different countries and economic sectors, as well as their dynamics, a holistic view on the underlying structural properties of this network is necessary. This study employs multi-regional input-output data to decompose 186 national economies into 26 industry sectors and utilizes the approach of interdependent networks to analyze the substructure of the resulting international trade network for the years 1990-2011. The partition of the network into national economies is observed to be compatible with the notion of communities in the sense of complex network theory. By studying internal versus cross-subgraph contributions to established complex network metrics, new insights into the architecture of global trade are obtained, which allow to identify key elements of global economy. Specifically, financial services and business activities dominate domestic trade whereas electrical and machinery industries dominate foreign trade. In order to further specify each national sector's role individually, (cross-)clustering coefficients and cross-betweenness are obtained for different pairs of subgraphs. The corresponding analysis reveals that specific industrial sectors tend to favor distinct directionality patterns and that the cross-clustering coefficient for geographically close country pairs is remarkably high, indicating that spatial factors are still of paramount importance for the organization of trade patterns in modern economy. Regarding the evolution of the trade network's substructure, globalization is well-expressed by trends of several structural characteristics (e.g., link density and node strength) in the interacting network framework. Extreme events, such as the financial crisis 2008/2009, are manifested as anomalies superimposed to these trends. The marked reorganization of trade patterns, associated with this economic crisis in comparison to "normal" annual fluctuations in the network structure is traced and quantified by a new widely applicable generalization of the Hamming distance to weighted networks.

Tensor Spectral Clustering for Partitioning Higher-order Network Structures

Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.

Hyper-brain networks support romantic kissing in humans

Coordinated social interaction is associated with, and presumably dependent on, oscillatory couplings within and between brains, which, in turn, consist of an interplay across different frequencies. Here, we introduce a method of network construction based on the cross-frequency coupling (CFC) and examine whether coordinated social interaction is associated with CFC within and between brains. Specifically, we compare the electroencephalograms (EEG) of 15 heterosexual couples during romantic kissing to kissing one’s own hand, and to kissing one another while performing silent arithmetic. Using graph-theory methods, we identify theta–alpha hyper-brain networks, with alpha serving a cleaving or pacemaker function. Network strengths were higher and characteristic path lengths shorter when individuals were kissing each other than when they were kissing their own hand. In both partner-oriented kissing conditions, greater strength and shorter path length for 5-Hz oscillation nodes correlated reliably with greater partner-oriented kissing satisfaction. This correlation was especially strong for inter-brain connections in both partner-oriented kissing conditions but not during kissing one’s own hand. Kissing quality assessed after the kissing with silent arithmetic correlated reliably with intra-brain strength of 10-Hz oscillation nodes during both romantic kissing and kissing with silent arithmetic. We conclude that hyper-brain networks based on CFC may capture neural mechanisms that support interpersonally coordinated voluntary action and bonding behavior.

Community analysis in directed networks: in-, out-, and pseudocommunities

When analyzing important classes of complex interconnected systems, link directionality can hardly be neglected if a precise and effective picture of the structure and function of the system is needed. If community analysis is performed, the notion of "community" itself is called into question, since the property of having a comparatively looser external connectivity could refer to the inbound or outbound links only or to both categories. In this paper, we introduce the notions of in-, out-, and in-/out-community in order to correctly classify the directedness of the interaction of a subnetwork with the rest of the system. Furthermore, we extend the scope of community analysis by introducing the notions of in-, out-, and in-/out-pseudocommunity. They are subnetworks having strong internal connectivity but also important interactions with the rest of the system, the latter taking place by means of a minority of its nodes only. The various types of (pseudo-)communities are qualified and distinguished by a suitable set of indicators and, on a given network, they can be discovered by using a "local" searching algorithm. The application to a broad set of benchmark networks and real-world examples proves that the proposed approach is able to effectively disclose the different types of structures above defined and to usefully classify the directionality of their interactions with the rest of the system.

Epidemic threshold in directed networks

Epidemics have so far been mostly studied in undirected networks. However, many real-world networks, such as the online social network Twitter and the world wide web, on which information, emotion, or malware spreads, are directed networks, composed of both unidirectional links and bidirectional links. We define the directionality ξ as the percentage of unidirectional links. The epidemic threshold τ(c) for the susceptible-infected-susceptible (SIS) epidemic is lower bounded by 1/λ(1) in directed networks, where λ(1), also called the spectral radius, is the largest eigenvalue of the adjacency matrix. In this work, we propose two algorithms to generate directed networks with a given directionality ξ. The effect of ξ on the spectral radius λ(1), principal eigenvector x(1), spectral gap (λ(1)-|λ(2)|), and algebraic connectivity μ(N-1) is studied. Important findings are that the spectral radius λ(1) decreases with the directionality ξ, whereas the spectral gap and the algebraic connectivity increase with the directionality ξ. The extent of the decrease of the spectral radius depends on both the degree distribution and the degree-degree correlation ρ(D). Hence, in directed networks, the epidemic threshold is larger and a random walk converges to its steady state faster than that in undirected networks with the same degree distribution.

Flow motifs reveal limitations of the static framework to represent human interactions

Networks are commonly used to define underlying interaction structures where infections, information, or other quantities may spread. Although the standard approach has been to aggregate all links into a static structure, some studies have shown that the time order in which the links are established may alter the dynamics of spreading. In this paper, we study the impact of the time ordering in the limits of flow on various empirical temporal networks. By using a random walk dynamics, we estimate the flow on links and convert the original undirected network (temporal and static) into a directed flow network. We then introduce the concept of flow motifs and quantify the divergence in the representativity of motifs when using the temporal and static frameworks. We find that the regularity of contacts and persistence of vertices (common in email communication and face-to-face interactions) result on little differences in the limits of flow for both frameworks. On the other hand, in the case of communication within a dating site and of a sexual network, the flow between vertices changes significantly in the temporal framework such that the static approximation poorly represents the structure of contacts. We have also observed that cliques with 3 and 4 vertices containing only low-flow links are more represented than the same cliques with all high-flow links. The representativity of these low-flow cliques is higher in the temporal framework. Our results suggest that the flow between vertices connected in cliques depend on the topological context in which they are placed and in the time sequence in which the links are established. The structure of the clique alone does not completely characterize the potential of flow between the vertices.

Profiling core-periphery network structure by random walkers

Disclosing the main features of the structure of a network is crucial to understand a number of static and dynamic properties, such as robustness to failures, spreading dynamics, or collective behaviours. Among the possible characterizations, the core-periphery paradigm models the network as the union of a dense core with a sparsely connected periphery, highlighting the role of each node on the basis of its topological position. Here we show that the core-periphery structure can effectively be profiled by elaborating the behaviour of a random walker. A curve--the core-periphery profile--and a numerical indicator are derived, providing a global topological portrait. Simultaneously, a coreness value is attributed to each node, qualifying its position and role. The application to social, technological, economical, and biological networks reveals the power of this technique in disclosing the overall network structure and the peculiar role of some specific nodes.

Emergence of scale-free close-knit friendship structure in online social networks

Although the structural properties of online social networks have attracted much attention, the properties of the close-knit friendship structures remain an important question. Here, we mainly focus on how these mesoscale structures are affected by the local and global structural properties. Analyzing the data of four large-scale online social networks reveals several common structural properties. It is found that not only the local structures given by the indegree, outdegree, and reciprocal degree distributions follow a similar scaling behavior, the mesoscale structures represented by the distributions of close-knit friendship structures also exhibit a similar scaling law. The degree correlation is very weak over a wide range of the degrees. We propose a simple directed network model that captures the observed properties. The model incorporates two mechanisms: reciprocation and preferential attachment. Through rate equation analysis of our model, the local-scale and mesoscale structural properties are derived. In the local-scale, the same scaling behavior of indegree and outdegree distributions stems from indegree and outdegree of nodes both growing as the same function of the introduction time, and the reciprocal degree distribution also shows the same power-law due to the linear relationship between the reciprocal degree and in/outdegree of nodes. In the mesoscale, the distributions of four closed triples representing close-knit friendship structures are found to exhibit identical power-laws, a behavior attributed to the negligible degree correlations. Intriguingly, all the power-law exponents of the distributions in the local-scale and mesoscale depend only on one global parameter, the mean in/outdegree, while both the mean in/outdegree and the reciprocity together determine the ratio of the reciprocal degree of a node to its in/outdegree. Structural properties of numerical simulated networks are analyzed and compared with each of the four real networks. This work helps understand the interplay between structures on different scales in online social networks.

PageRank and rank-reversal dependence on the damping factor

PageRank (PR) is an algorithm originally developed by Google to evaluate the importance of web pages. Considering how deeply rooted Google's PR algorithm is to gathering relevant information or to the success of modern businesses, the question of rank stability and choice of the damping factor (a parameter in the algorithm) is clearly important. We investigate PR as a function of the damping factor d on a network obtained from a domain of the World Wide Web, finding that rank reversal happens frequently over a broad range of PR (and of d). We use three different correlation measures, Pearson, Spearman, and Kendall, to study rank reversal as d changes, and we show that the correlation of PR vectors drops rapidly as d changes from its frequently cited value, d_{0}=0.85. Rank reversal is also observed by measuring the Spearman and Kendall rank correlation, which evaluate relative ranks rather than absolute PR. Rank reversal happens not only in directed networks containing rank sinks but also in a single strongly connected component, which by definition does not contain any sinks. We relate rank reversals to rank pockets and bottlenecks in the directed network structure. For the network studied, the relative rank is more stable by our measures around d=0.65 than at d=d_{0}.

Existence and significance of communities in the World Trade Web

The World Trade Web (WTW), which models the international transactions among countries, is a fundamental tool for studying the economics of trade flows, their evolution over time, and their implications for a number of phenomena, including the propagation of economic shocks among countries. In this respect, the possible existence of communities is a key point, because it would imply that countries are organized in groups of preferential partners. In this paper, we use four approaches to analyze communities in the WTW between 1962 and 2008, based, respectively, on modularity optimization, cluster analysis, stability functions, and persistence probabilities. Overall, the four methods agree in finding no evidence of significant partitions. A few weak communities emerge from the analysis, but they do not represent secluded groups of countries, as intercommunity linkages are also strong, supporting the view of a truly globalized trading system.

Finding and testing network communities by lumped Markov chains

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition that is based on a quality threshold. By means of a lumped Markov chain model of a random walker, a quality measure called “persistence probability” is associated to a cluster, which is then defined as an “-community” if such a probability is not smaller than . Consistently, a partition composed of -communities is an “-partition.” These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired -level allows one to immediately select the -partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the quality of each single community. Given its ability in individually assessing each single cluster, this approach can also disclose single well-defined communities even in networks that overall do not possess a definite clusterized structure.

Disentangling categorical relationships through a graph of co-occurrences

The mesoscopic structure of complex networks has proven a powerful level of description to understand the linchpins of the system represented by the network. Nevertheless, the mapping of a series of relationships between elements, in terms of a graph, is sometimes not straightforward. Given that all the information we would extract using complex network tools depend on this initial graph, it is mandatory to preprocess the data to build it on in the most accurate manner. Here we propose a procedure to build a network, attending only to statistically significant relations between constituents. We use a paradigmatic example of word associations to show the development of our approach. Analyzing the modular structure of the obtained network we are able to disentangle categorical relations, disambiguating words with success that is comparable to the best algorithms designed to the same end.

Map equation for link communities

Community structure exists in many real-world networks and has been reported being related to several functional properties of the networks. The conventional approach was partitioning nodes into communities, while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently. We extended the map equation method, which was originally developed for node communities, to find link communities in networks. This method is tested on various kinds of networks and compared with the metadata of the networks, and the results show that our method can identify the overlapping role of nodes effectively. The advantage of this method is that the node community scheme and link community scheme can be compared quantitatively by measuring the unknown information left in the networks besides the community structure. It can be used to decide quantitatively whether or not the link community scheme should be used instead of the node community scheme. Furthermore, this method can be easily extended to the directed and weighted networks since it is based on the random walk.

Interarrival times of message propagation on directed networks

One of the challenges in fighting cybercrime is to understand the dynamics of message propagation on botnets, networks of infected computers used to send viruses, unsolicited commercial emails (SPAM) or denial of service attacks. We map this problem to the propagation of multiple random walkers on directed networks and we evaluate the interarrival time distribution between successive walkers arriving at a target. We show that the temporal organization of this process, which models information propagation on unstructured peer to peer networks, has the same features as SPAM reaching a single user. We study the behavior of the message interarrival time distribution on three different network topologies using two different rules for sending messages. In all networks the propagation is not a pure Poisson process. It shows universal features on Poissonian networks and a more complex behavior on scale free networks. Results open the possibility to indirectly learn about the process of sending messages on networks with unknown topologies, by studying interarrival times at any node of the network.

Coarse graining for synchronization in directed networks

Coarse-graining model is a promising way to analyze and visualize large-scale networks. The coarse-grained networks are required to preserve statistical properties as well as the dynamic behaviors of the initial networks. Some methods have been proposed and found effective in undirected networks, while the study on coarse-graining directed networks lacks of consideration. In this paper we proposed a path-based coarse-graining (PCG) method to coarse grain the directed networks. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model on four kinds of directed networks, including tree networks and variants of Barabási-Albert networks, Watts-Strogatz networks, and Erdös-Rényi networks, we find our method can effectively preserve the network synchronizability.

Trade communities and their spatial patterns in the German pork production network

The German trade network of pig holdings and related enterprises for the time period of 01 June 2006-31 December 2008 was analyzed. Available data comprised of the entire German trade information with about 121,287 pig premises and their links in the pork production chain (breeders to slaughter houses). During the study period, 330,000 trade connections between premises were recorded. Communities which are large scale structures comprising of premises which are in close trade contact with each other were detected by modularity maximization. Almost 97% of the pig holdings could be assigned to one of nine major communities. Contacts between communities are rare. Trade communities do not only form spatial clusters, but were also associated with specific regions within the territory of Germany. Communities are to some extent separated by 'trade borders', which may impede disease spread. Interestingly, the trade borders often failed to match with administrative borders of the German federal states. This finding is important, because it illustrates the need for new strategies of cross-border disease management strategies.

International population movements and regional Plasmodium falciparum malaria elimination strategies

Calls for the eradication of malaria require the development of global and regional strategies based on a strong and consistent evidence base. Evidence from the previous global malaria eradication program and more recent transborder control campaigns have shown the importance of accounting for human movement in introducing infections to areas targeted for elimination. Here, census-based migration data were analyzed with network analysis tools, Plasmodium falciparum malaria transmission maps, and global population databases to map globally communities of countries linked by relatively high levels of infection movements. The likely principal sources and destinations of imported cases in each region were also mapped. Results indicate that certain groups of countries, such as those in West Africa and central Asia are much more strongly connected by relatively high levels of population and infection movement than others. In contrast, countries such as Ethiopia and Myanmar display significantly greater isolation in terms of likely infection movements in and out. The mapping here of both communities of countries linked by likely higher levels of infection movement, and "natural" migration boundaries that display reduced movement of people and infections between regions has practical utility. These maps can inform the design of malaria elimination strategies by identifying regional communities of countries afforded protection from recolonization by surrounding regions of reduced migration. For more isolated countries, a nationally focused control or elimination program is likely to stand a better chance of success than those receiving high levels of visitors and migrants from high-transmission regions.

Edge direction and the structure of networks

Directed networks are ubiquitous and are necessary to represent complex systems with asymmetric interactions--from food webs to the World Wide Web. Despite the importance of edge direction for detecting local and community structure, it has been disregarded in studying a basic type of global diversity in networks: the tendency of nodes with similar numbers of edges to connect. This tendency, called assortativity, affects crucial structural and dynamic properties of real-world networks, such as error tolerance or epidemic spreading. Here we demonstrate that edge direction has profound effects on assortativity. We define a set of four directed assortativity measures and assign statistical significance by comparison to randomized networks. We apply these measures to three network classes--online/social networks, food webs, and word-adjacency networks. Our measures (i) reveal patterns common to each class, (ii) separate networks that have been previously classified together, and (iii) expose limitations of several existing theoretical models. We reject the standard classification of directed networks as purely assortative or disassortative. Many display a class-specific mixture, likely reflecting functional or historical constraints, contingencies, and forces guiding the system's evolution.