Random graph models of social networks

Some features of the spread of epidemics and information on a random graph

Random graphs are useful models of social and technological networks. To date, most of the research in this area has concerned geometric properties of the graphs. Here we focus on processes taking place on the network. In particular we are interested in how their behavior on networks differs from that in homogeneously mixing populations or on regular lattices of the type commonly used in ecological models.

Exploring biological network structure with clustered random networks

BACKGROUND

Complex biological systems are often modeled as networks of interacting units. Networks of biochemical interactions among proteins, epidemiological contacts among hosts, and trophic interactions in ecosystems, to name a few, have provided useful insights into the dynamical processes that shape and traverse these systems. The degrees of nodes (numbers of interactions) and the extent of clustering (the tendency for a set of three nodes to be interconnected) are two of many well-studied network properties that can fundamentally shape a system. Disentangling the interdependent effects of the various network properties, however, can be difficult. Simple network models can help us quantify the structure of empirical networked systems and understand the impact of various topological properties on dynamics.

RESULTS

Here we develop and implement a new Markov chain simulation algorithm to generate simple, connected random graphs that have a specified degree sequence and level of clustering, but are random in all other respects. The implementation of the algorithm (ClustRNet: Clustered Random Networks) provides the generation of random graphs optimized according to a local or global, and relative or absolute measure of clustering. We compare our algorithm to other similar methods and show that ours more successfully produces desired network characteristics.Finding appropriate null models is crucial in bioinformatics research, and is often difficult, particularly for biological networks. As we demonstrate, the networks generated by ClustRNet can serve as random controls when investigating the impacts of complex network features beyond the byproduct of degree and clustering in empirical networks.

CONCLUSION

ClustRNet generates ensembles of graphs of specified edge structure and clustering. These graphs allow for systematic study of the impacts of connectivity and redundancies on network function and dynamics. This process is a key step in unraveling the functional consequences of the structural properties of empirical biological systems and uncovering the mechanisms that drive these systems.

Connectivity, coverage and placement in wireless sensor networks

Wireless communication between sensors allows the formation of flexible sensor networks, which can be deployed rapidly over wide or inaccessible areas. However, the need to gather data from all sensors in the network imposes constraints on the distances between sensors. This survey describes the state of the art in techniques for determining the minimum density and optimal locations of relay nodes and ordinary sensors to ensure connectivity, subject to various degrees of uncertainty in the locations of the nodes.

The development, design, testing, refinement, simulation and application of an evaluation framework for communities of practice and social-professional networks

BACKGROUND

Communities of practice and social-professional networks are generally considered to enhance workplace experience and enable organizational success. However, despite the remarkable growth in interest in the role of collaborating structures in a range of industries, there is a paucity of empirical research to support this view. Nor is there a convincing model for their systematic evaluation, despite the significant potential benefits in answering the core question: how well do groups of professionals work together and how could they be organised to work together more effectively? This research project will produce a rigorous evaluation methodology and deliver supporting tools for the benefit of researchers, policymakers, practitioners and consumers within the health system and other sectors. Given the prevalence and importance of communities of practice and social networks, and the extent of investments in them, this project represents a scientific innovation of national and international significance.

METHODS AND DESIGN

Working in four conceptual phases the project will employ a combination of qualitative and quantitative methods to develop, design, field-test, refine and finalise an evaluation framework. Once available the framework will be used to evaluate simulated, and then later existing, health care communities of practice and social-professional networks to assess their effectiveness in achieving desired outcomes. Peak stakeholder groups have agreed to involve a wide range of members and participant organisations, and will facilitate access to various policy, managerial and clinical networks.

DISCUSSION

Given its scope and size, the project represents a valuable opportunity to achieve breakthroughs at two levels; firstly, by introducing novel and innovative aims and methods into the social research process and, secondly, through the resulting evaluation framework and tools. We anticipate valuable outcomes in the improved understanding of organisational performance and delivery of care. The project's wider appeal lies in transferring this understanding to other health jurisdictions and to other industries and sectors, both nationally and internationally. This means not merely publishing the results, but contextually interpreting them, and translating them to advance the knowledge base and enable widespread institutional and organisational application.

Backbone of complex networks of corporations: the flow of control

We present a methodology to extract the backbone of complex networks based on the weight and direction of links, as well as on nontopological properties of nodes. We show how the methodology can be applied in general to networks in which mass or energy is flowing along the links. In particular, the procedure enables us to address important questions in economics, namely, how control and wealth are structured and concentrated across national markets. We report on the first cross-country investigation of ownership networks, focusing on the stock markets of 48 countries around the world. On the one hand, our analysis confirms results expected on the basis of the literature on corporate control, namely, that in Anglo-Saxon countries control tends to be dispersed among numerous shareholders. On the other hand, it also reveals that in the same countries, control is found to be highly concentrated at the global level, namely, lying in the hands of very few important shareholders. Interestingly, the exact opposite is observed for European countries. These results have previously not been reported as they are not observable without the kind of network analysis developed here.

Deciphering the connectivity structure of biological networks using MixNet

Background

As biological networks often show complex topological features, mathematical methods are required to extract meaningful information. Clustering methods are useful in this setting, as they allow the summary of the network's topology into a small number of relevant classes. Different strategies are possible for clustering, and in this article we focus on a model-based strategy that aims at clustering nodes based on their connectivity profiles.

Results

We present MixNet, the first publicly available computer software that analyzes biological networks using mixture models. We apply this method to various networks such as the E. coli transcriptional regulatory network, the macaque cortex network, a foodweb network and the Buchnera aphidicola metabolic network. This method is also compared with other approaches such as module identification or hierarchical clustering.

Conclusion

We show how MixNet can be used to extract meaningful biological information, and to give a summary of the networks topology that highlights important biological features. This approach is powerful as MixNet is adaptive to the network under study, and finds structural information without any a priori on the structure that is investigated. This makes MixNet a very powerful tool to summarize and decipher the connectivity structure of biological networks.

Generic aspects of complexity in brain imaging data and other biological systems

A key challenge for systems neuroscience is the question of how to understand the complex network organization of the brain on the basis of neuroimaging data. Similar challenges exist in other specialist areas of systems biology because complex networks emerging from the interactions between multiple non-trivially interacting agents are found quite ubiquitously in nature, from protein interactomes to ecosystems. We suggest that one way forward for analysis of brain networks will be to quantify aspects of their organization which are likely to be generic properties of a broader class of biological systems. In this introductory review article we will highlight four important aspects of complex systems in general: fractality or scale-invariance; criticality; small-world and related topological attributes; and modularity. For each concept we will provide an accessible introduction, an illustrative data-based example of how it can be used to investigate aspects of brain organization in neuroimaging experiments, and a brief review of how this concept has been applied and developed in other fields of biomedical and physical science. The aim is to provide a didactic, focussed and user-friendly introduction to the concepts of complexity science for neuroscientists and neuroimagers.

Epidemic thresholds in dynamic contact networks

The reproductive ratio, R0, is a fundamental quantity in epidemiology, which determines the initial increase in an infectious disease in a susceptible host population. In most epidemic models, there is a specific value of R0, the epidemic threshold, above which epidemics are possible, but below which epidemics cannot occur. As the complexity of an epidemic model increases, so too does the difficulty of calculating epidemic thresholds. Here we derive the reproductive ratio and epidemic thresholds for susceptible-infected-recovered (SIR) epidemics in a simple class of dynamic random networks. As in most epidemiological models, R0 depends on two basic epidemic parameters, the transmission and recovery rates. We find that R0 also depends on social parameters, namely the degree distribution that describes heterogeneity in the numbers of concurrent contacts and the mixing parameter that gives the rate at which contacts are initiated and terminated. We show that social mixing fundamentally changes the epidemiological landscape and, consequently, that static network approximations of dynamic networks can be inadequate.

Explosive percolation in random networks

Networks in which the formation of connections is governed by a random process often undergo a percolation transition, wherein around a critical point, the addition of a small number of connections causes a sizable fraction of the network to suddenly become linked together. Typically such transitions are continuous, so that the percentage of the network linked together tends to zero right above the transition point. Whether percolation transitions could be discontinuous has been an open question. Here, we show that incorporating a limited amount of choice in the classic Erdös-Rényi network formation model causes its percolation transition to become discontinuous.

A Core/Periphery Perspective on Individual Creative Performance: Social Networks and Cinematic Achievements in the Hollywood Film Industry

The paper advances a relational perspective to studying creativity at the individual level. Building on social network theory and techniques, we examine the role of social networks in shaping individuals' ability to generate a creative outcome. More specifically, we argue that individuals who occupy an intermediate position between the core and the periphery of their social system are in a favorable position to achieve creative results. In addition, the benefits accrued through an individual's intermediate core/periphery position can also be observed at the team level, when the same individual works in a team whose members come from both ends of the core/periphery continuum. We situate the analysis and test our hypotheses within the context of the Hollywood motion picture industry, which we trace over the period 1992–2003. The theoretical implications of the results are discussed.

This work is licensed under a Creative Commons Attribution 4.0 International License. You are free to copy, distribute, transmit and adapt this work, but you must attribute this work as “Organization Science. Copyright © 2017 INFORMS. https://doi.org/10.1287/orsc.1070.0350 , used under a Creative Commons Attribution License: http://creativecommons.org/licenses/by/4.0/ .”

Scale-free networks as entropy competition

Complex networks describe several and different real-world systems consisting of a number of interacting elements. A very important characteristic of such networks is the degree distribution that strongly controls their behavior. Based on statistical mechanics, three classes of uncorrelated complex networks are identified here, depending on the role played by the connectivities amongst elements. In particular, by identifying the connectivities of a node with the number of its nearest neighbors, we show that the power law is the most probable degree distribution that both nodes and neighbors, in a reciprocal competition, assume when the respective entropy functions reach their maxima, under mutual constraint. As a result, we obtain scaling exponent values as a function of the structural characteristics of the whole network. Moreover, our approach sheds light on the exponential and Poissonian degree distributions, derived, respectively, when connectivities are thought of as degenerated connections or as half-edges. Thus, all three classes of degree distributions are derived, starting from a common principle and leading to a general and unified framework for investigating the network structure.

Mobility and social network effects on extremist opinions

Understanding the emergence of extreme opinions and in what kind of environment they might become less extreme is a central theme in our modern globalized society. A model combining continuous opinions and observed discrete actions (CODA) capable of addressing the important issue of measuring how extreme opinions might be has been recently proposed. In this paper I show extreme opinions to arise in a ubiquitous manner in the CODA model for a multitude of social network structures. Depending on network details reducing extremism seems to be possible. However, a large number of agents with extreme opinions is always observed. A significant decrease in the number of extremists can be observed by allowing agents to change their positions in the network.

Socioeconomic networks with long-range interactions

We study a modified version of a model previously proposed by Jackson and Wolinsky to account for communication of information and allocation of goods in socioeconomic networks. In the model, the utility function of each node is given by a weighted sum of contributions from all accessible nodes. The weights, parametrized by the variable delta , decrease with distance. We introduce a growth mechanism where new nodes attach to the existing network preferentially by utility. By increasing delta , the network structure evolves from a power-law to an exponential degree distribution, passing through a regime characterized by shorter average path length, lower degree assortativity, and higher central point dominance. In the second part of the paper we compare different network structures in terms of the average utility received by each node. We show that power-law networks provide higher average utility than Poisson random networks. This provides a possible justification for the ubiquitousness of scale-free networks in the real world.

Looking for robust properties in the growth of an academic network: the case of the Uruguayan biological research community

Graph-theoretical methods have recently been used to analyze certain properties of natural and social networks. In this work, we have investigated the early stages in the growth of a Uruguayan academic network, the Biology Area of the Programme for the Development of Basic Science (PEDECIBA). This transparent social network is a territory for the exploration of the reliability of clustering methods that can potentially be used when we are confronted with opaque natural systems that provide us with a limited spectrum of observables (happens in research on the relations between brain, thought and language). From our social net, we constructed two different graph representations based on the relationships among researchers revealed by their co-participation in Master's thesis committees. We studied these networks at different times and found that they achieve connectedness early in their evolution and exhibit the small-world property (i.e. high clustering with short path lengths). The data seem compatible with power law distributions of connectivity, clustering coefficients and betweenness centrality. Evidence of preferential attachment of new nodes and of new links between old nodes was also found in both representations. These results suggest that there are topological properties observed throughout the growth of the network that do not depend on the representations we have chosen but reflect intrinsic properties of the academic collective under study. Researchers in PEDECIBA are classified according to their specialties. We analysed the community structure detected by a standard algorithm in both representations. We found that much of the pre-specified structure is recovered and part of the mismatches can be attributed to convergent interests between scientists from different sub-disciplines. This result shows the potentiality of some clustering methods for the analysis of partially known natural systems.

Susceptible-infected-recovered epidemics in dynamic contact networks

Contact patterns in populations fundamentally influence the spread of infectious diseases. Current mathematical methods for epidemiological forecasting on networks largely assume that contacts between individuals are fixed, at least for the duration of an outbreak. In reality, contact patterns may be quite fluid, with individuals frequently making and breaking social or sexual relationships. Here, we develop a mathematical approach to predicting disease transmission on dynamic networks in which each individual has a characteristic behaviour (typical contact number), but the identities of their contacts change in time. We show that dynamic contact patterns shape epidemiological dynamics in ways that cannot be adequately captured in static network models or mass-action models. Our new model interpolates smoothly between static network models and mass-action models using a mixing parameter, thereby providing a bridge between disparate classes of epidemiological models. Using epidemiological and sexual contact data from an Atlanta high school, we demonstrate the application of this method for forecasting and controlling sexually transmitted disease outbreaks.

SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE's. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.

Properties of the organization of memory for people: Evidence from dream reports

Steyvers and Tenenbaum (2005) showed that semantic networks for words have three organizational properties: short average path lengths, high clustering, and power law degree distribution. If these are general properties of memory organization, they would apply to memory for other complex material, including people and relations between them. In addition, if during dreaming, characters are generated via knowledge in the dreamer's memory, the three properties would be found in a relational network of characters in dreams. In dream reports from three individuals, two characters in the same dream were considered affiliated. Resulting social networks have the three properties, with the power law holding when low degrees are omitted. One network with a tree-like outline is different from the other two. Results suggest associative memory has the three properties, and demonstrate that dream reports are a potentially valuable source for information about social networks.

Do scale-free regulatory networks allow more expression than random ones?

In this paper, we compile the network of software packages with regulatory interactions (dependences and conflicts) from Debian GNU/Linux operating system and use it as an analogy for a gene regulatory network. Using a trace-back algorithm we assemble networks from the pool of packages with both scale-free (real data) and exponential (null model) topologies. We record the maximum number of packages that can be functionally installed in the system (i.e., the active network size). We show that scale-free regulatory networks allow a larger active network size than random ones. This result might have implications for the number of expressed genes at steady state. Small genomes with scale-free regulatory topologies could allow much more expression than large genomes with exponential topologies. This may have implications for the dynamics, robustness and evolution of genomes.

Network frailty and the geometry of herd immunity

The spread of infectious disease through communities depends fundamentally on the underlying patterns of contacts between individuals. Generally, the more contacts one individual has, the more vulnerable they are to infection during an epidemic. Thus, outbreaks disproportionately impact the most highly connected demographics. Epidemics can then lead, through immunization or removal of individuals, to sparser networks that are more resistant to future transmission of a given disease. Using several classes of contact networks-Poisson, scale-free and small-world-we characterize the structural evolution of a network due to an epidemic in terms of frailty (the degree to which highly connected individuals are more vulnerable to infection) and interference (the extent to which the epidemic cuts off connectivity among the susceptible population that remains following an epidemic). The evolution of the susceptible network over the course of an epidemic differs among the classes of networks; frailty, relative to interference, accounts for an increasing component of network evolution on networks with greater variance in contacts. The result is that immunization due to prior epidemics can provide greater community protection than random vaccination on networks with heterogeneous contact patterns, while the reverse is true for highly structured populations.

Generating graphs for visual analytics through interactive sketching

We introduce an interactive graph generator, GreenSketch, designed to facilitate the creation of descriptive graphs required for different visual analytics tasks. The human-centric design approach of GreenSketch enables users to master the creation process without specific training or prior knowledge of graph model theory. The customized user interface encourages users to gain insight into the connection between the compact matrix representation and the topology of a graph layout when they sketch their graphs. Both the human-enforced and machine-generated randomnesses supported by GreenSketch provide the flexibility needed to address the uncertainty factor in many analytical tasks. This paper describes more than two dozen examples that cover a wide variety of graph creations from a single line of nodes to a real-life small-world network that describes a snapshot of telephone connections. While the discussion focuses mainly on the design of GreenSketch, we include a case study that applies the technology in a visual analytics environment and a usability study that evaluates the strengths and weaknesses of our design approach.

Graph theoretic properties of networks formed by the Delaunay tessellation of protein structures

The Delaunay tessellation of several sets of real and simplified model protein structures has been used to explore graph theoretic properties of residue contact networks. The system of contacts defined by residues joined by edges in the Delaunay simplices can be thought of as a graph or network and analyzed using techniques from elementary graph theory and the theory of complex networks. Such analysis indicates that protein contact networks have small world character, but technically are not small world networks. This approach also indicates that networks formed by native structures and by most misfolded decoys can be differentiated by their respective graph properties. The characteristic features of residue contact networks can be used for the detection of structural elements in proteins, such as the ubiquitous closed loops consisting of 22-32 consecutive residues, where terminal residues are Delaunay neighbors.

Social inertia in collaboration networks

This work is a study of the properties of collaboration networks employing the formalism of weighted graphs to represent their one-mode projection. The weight of the edges is directly the number of times that a partnership has been repeated. This representation allows us to define the concept of social inertia that measures the tendency of authors to keep on collaborating with previous partners. We use a collection of empirical datasets to analyze several aspects of the social inertia: (1) its probability distribution, (2) its correlation with other properties, and (3) the correlations of the inertia between neighbors in the network. We also contrast these empirical results with the predictions of a recently proposed theoretical model for the growth of collaboration networks.

Gene sequence signatures revealed by mining the UniGene affiliation network

BACKGROUND

In the post-genomic era, developing tools to decode biological information from genomic sequences is important. Inspired by affiliation network theory, we investigated gene sequences of two kinds of UniGene clusters (UCs): narrowly expressed transcripts (NETs), whose expression is confined to a few tissues; and prevalently expressed transcripts (PETs) that are expressed in many tissues.

RESULTS

We explored the human and the mouse UniGene databases to compare NETs and PETs from different perspectives. We found that NETs were associated with smaller cluster size, shorter sequence length, a lower likelihood of having LocusLink annotations, and lower and more sporadic levels of expression. Significantly, the dinucleotide frequencies of NETs are similar to those of intergenic sequences in the genome, and they differ from those of PETs. We used these differences in dinucleotide frequencies to develop a discriminant analysis model to distinguish PETs from intergenic sequences.

CONCLUSIONS

Our results show that most NETs resemble intergenic sequences, casting doubts on the quality of such UniGene clusters. However, we also noted that a fraction of NETs resemble PETs in terms of dinucleotide frequencies and other features. Such NETs may have fewer quality problems. This work may be helpful in the studies of non-coding RNAs and in the validation of gene sequence databases.

N-body decomposition of bipartite author networks

In this paper, we present a method to project co-authorship networks, that accounts in detail for the geometrical structure of scientists' collaborations. By restricting the scope to three-body interactions, we focus on the number of triangles in the system, and show the importance of multi-scientist (more than two) collaborations in the social network. This motivates the introduction of generalized networks, where basic connections are not binary, but involve arbitrary number of components. We focus on the three-body case and study numerically the percolation transition.

Small-world properties emerge in highly compartmentalized networks with intermediate group sizes and numbers

Many recent studies have focused on two statistical properties observed in diverse real-world networks: the small-world property and compartmentalization [D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998); M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. 99, 7821 (2002)]. Models that include group affiliations have been shown to produce networks with high clustering coefficients, a necessary condition for small-world properties [M. E. J. Newman, Phys. Rev. E, 68, 026121 (2003); M. E. J. Newman and J. Park, Phys. Rev. E 68, 036122 (2003)]. However, the consequences of varying the number and size of groups in a network are not well understood. In order to investigate the consequences of group organization, we examined sets of networks that varied simultaneously in the size and number of groups, while maintaining the same overall size and average degree. Here we show that the small-world property arises in maximally compartmentalized and clustered networks that occur in the intermediate region between few, very large groups and many, very small groups.

Infection in social networks: using network analysis to identify high-risk individuals

Simulation studies using susceptible-infectious-recovered models were conducted to estimate individuals' risk of infection and time to infection in small-world and randomly mixing networks. Infection transmitted more rapidly but ultimately resulted in fewer infected individuals in the small-world, compared with the random, network. The ability of measures of network centrality to identify high-risk individuals was also assessed. "Centrality" describes an individual's position in a population; numerous parameters are available to assess this attribute. Here, the authors use the centrality measures degree (number of contacts), random-walk betweenness (a measure of the proportion of times an individual lies on the path between other individuals), shortest-path betweenness (the proportion of times an individual lies on the shortest path between other individuals), and farness (the sum of the number of steps between an individual and all other individuals). Each was associated with time to infection and risk of infection in the simulated outbreaks. In the networks examined, degree (which is the most readily measured) was at least as good as other network parameters in predicting risk of infection. Identification of more central individuals in populations may be used to inform surveillance and infection control strategies.

Transition from local to global phase synchrony in small world neural network and its possible implications for epilepsy

Temporal correlations in the brain are thought to have very dichotomous roles. On one hand they are ubiquitously present in the healthy brain and are thought to underlie feature binding during information processing. On the other hand, large-scale synchronization is an underlying mechanism of epileptic seizures. In this paper we show a potential mechanism for the transition to pathological coherence underlying seizure generation. We show that properties of phase synchronization in a two-dimensional lattice of nonidentical coupled Hindmarsh-Rose neurons change radically depending on the connectivity structure of the network. We modify the connectivity using the small world network paradigm and measure properties of phase synchronization using a previously developed measure based on assessment of the distributions of relative interspike intervals. We show that the temporal ordering undergoes a dramatic change as a function of topology of the network from local coherence strongly dependent on the distance between two neurons, to global coherence exhibiting a larger degree of ordering and spanning the whole network.

A tool for filtering information in complex systems

We introduce a technique to filter out complex data sets by extracting a subgraph of representative links. Such a filtering can be tuned up to any desired level by controlling the genus of the resulting graph. We show that this technique is especially suitable for correlation-based graphs, giving filtered graphs that preserve the hierarchical organization of the minimum spanning tree but containing a larger amount of information in their internal structure. In particular in the case of planar filtered graphs (genus equal to 0), triangular loops and four-element cliques are formed. The application of this filtering procedure to 100 stocks in the U.S. equity markets shows that such loops and cliques have important and significant relationships with the market structure and properties.

Random initial condition in small Barabasi-Albert networks and deviations from the scale-free behavior

Barabasi-Albert networks are constructed by adding nodes via preferential attachment to an initial core of nodes. We study the topology of small scale-free networks as a function of the size and average connectivity of their initial random core. We show that these two parameters may strongly affect the tail of the degree distribution, by consistently leading to broad-scale or single-scale networks. In particular, we argue that the size of the initial network core and its density of connections may be the main responsible for the exponential truncation of the power-law behavior observed in some small scale-free networks.

Random networks with tunable degree distribution and clustering

We present an algorithm for generating random networks with arbitrary degree distribution and clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and Poisson degree distributions with variable levels of clustering. Such networks may be used as models of social networks and as a testable null hypothesis about network structure. Finally, we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network, and on the size of the giant component. Some analysis of these effects is presented.

Self-organization of collaboration networks

We study collaboration networks in terms of evolving, self-organizing bipartite graph models. We propose a model of a growing network, which combines preferential edge attachment with the bipartite structure, generic for collaboration networks. The model depends exclusively on basic properties of the network, such as the total number of collaborators and acts of collaboration, the mean size of collaborations, etc. The simplest model defined within this framework already allows us to describe many of the main topological characteristics (degree distribution, clustering coefficient, etc.) of one-mode projections of several real collaboration networks, without parameter fitting. We explain the observed dependence of the local clustering on degree and the degree-degree correlations in terms of the "aging" of collaborators and their physical impossibility to participate in an unlimited number of collaborations.