Optimizing sentinel surveillance in static and temporal networksPetter Holme
The document discusses objective measures for comparing different approaches to sentinel surveillance in networks. It runs simulations of disease spread using the SIR model on empirical temporal and static networks. It finds that nodes identified as important by different objective measures like time to detection, time to detection or extinction, and frequency of detection do not always coincide. The correlations between measures depend on network structure, with stronger correlations in temporal networks. Centrality metrics are also correlated with objective measures, but not perfectly. The best objective measure depends on the aspect of sentinel surveillance that is most important.
The community detection in complex networks has attracted a growing interest and is the subject of several
researches that have been proposed to understand the network structure and analyze the network
properties. In this paper, we give a thorough overview of different community discovery strategies, we
propose taxonomy of these methods, and we specify the differences between the suggested classes which
helping designers to compare and choose the most suitable strategy for the various types of network
encountered in the real world.
How the information content of your contact pattern representation affects pr...Petter Holme
This document summarizes a presentation about how the structure of temporal networks, which model patterns of human contact over time, can affect the predictability of disease outbreaks. The presentation discusses how different levels of information contained in representations of contact patterns, from fully mixed to temporal network models, influence the size and uncertainty of epidemics simulated using an SIR compartmental model. It analyzes several real-world temporal network datasets and examines how metrics that characterize the network structure correlate with the shape of the relationship between outbreak size and the basic reproduction number R0.
Temporal Networks of Human InteractionPetter Holme
Temporal networks provide a framework for modeling systems of interactions that occur between nodes over time. These networks capture both the topological structure of connections as well as the timing of interactions. Three key aspects of temporal networks discussed in the document are:
1) Temporal networks can be represented using contact sequences that capture when interactions occur between nodes, unlike static networks which only represent connections.
2) The temporal structure of interactions, such as patterns in the timing of contacts, can impact dynamical processes unfolding on the network like information or disease spreading.
3) Randomizing the timing of contacts in empirical temporal network data can alter dynamical processes, highlighting the importance of temporal structure beyond just topology.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This literature survey discusses papers on the topological structure of social networks and information propagation. Regarding network structure, papers found that social networks like Facebook exhibit scale-free and small-world properties with high clustering. Different networks may have different structures depending on factors like symmetry. Regarding information spread, factors like sender involvement, tie strength, and communication influence forwarding. Messages tend to spread through closely connected friendship networks rather than broadly. Key influencers and the structure of interaction graphs also impact propagation patterns.
Optimizing sentinel surveillance in static and temporal networksPetter Holme
The document discusses objective measures for comparing different approaches to sentinel surveillance in networks. It runs simulations of disease spread using the SIR model on empirical temporal and static networks. It finds that nodes identified as important by different objective measures like time to detection, time to detection or extinction, and frequency of detection do not always coincide. The correlations between measures depend on network structure, with stronger correlations in temporal networks. Centrality metrics are also correlated with objective measures, but not perfectly. The best objective measure depends on the aspect of sentinel surveillance that is most important.
The community detection in complex networks has attracted a growing interest and is the subject of several
researches that have been proposed to understand the network structure and analyze the network
properties. In this paper, we give a thorough overview of different community discovery strategies, we
propose taxonomy of these methods, and we specify the differences between the suggested classes which
helping designers to compare and choose the most suitable strategy for the various types of network
encountered in the real world.
How the information content of your contact pattern representation affects pr...Petter Holme
This document summarizes a presentation about how the structure of temporal networks, which model patterns of human contact over time, can affect the predictability of disease outbreaks. The presentation discusses how different levels of information contained in representations of contact patterns, from fully mixed to temporal network models, influence the size and uncertainty of epidemics simulated using an SIR compartmental model. It analyzes several real-world temporal network datasets and examines how metrics that characterize the network structure correlate with the shape of the relationship between outbreak size and the basic reproduction number R0.
Temporal Networks of Human InteractionPetter Holme
Temporal networks provide a framework for modeling systems of interactions that occur between nodes over time. These networks capture both the topological structure of connections as well as the timing of interactions. Three key aspects of temporal networks discussed in the document are:
1) Temporal networks can be represented using contact sequences that capture when interactions occur between nodes, unlike static networks which only represent connections.
2) The temporal structure of interactions, such as patterns in the timing of contacts, can impact dynamical processes unfolding on the network like information or disease spreading.
3) Randomizing the timing of contacts in empirical temporal network data can alter dynamical processes, highlighting the importance of temporal structure beyond just topology.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This literature survey discusses papers on the topological structure of social networks and information propagation. Regarding network structure, papers found that social networks like Facebook exhibit scale-free and small-world properties with high clustering. Different networks may have different structures depending on factors like symmetry. Regarding information spread, factors like sender involvement, tie strength, and communication influence forwarding. Messages tend to spread through closely connected friendship networks rather than broadly. Key influencers and the structure of interaction graphs also impact propagation patterns.
The document summarizes a research paper that proposes a new algorithm called the leader-follower algorithm for detecting community structure in networks. The algorithm is based on identifying the internal structure of communities rather than external connectivity properties used by traditional methods like spectral clustering. It first distinguishes between leader nodes that connect different communities and loyal follower nodes that only have neighbors within their community. It then assigns loyal followers to leaders to form the communities. The algorithm is able to detect communities of any size and learn their number without requiring this as input, offering an improvement over spectral clustering especially for densely connected networks. Experiments on synthetic and real-world networks demonstrate its effectiveness.
Spreading processes on temporal networksPetter Holme
This document discusses temporal networks and how temporal structures can impact dynamical processes on networks. It begins by describing different types of temporal networks including person-to-person communication, information dissemination, physical proximity, and cellular biology networks. It then discusses methods for analyzing temporal network structures like inter-event times and how bursty or heavy-tailed distributions can slow spreading compared to memory-less processes. The document also presents examples of how neutralizing temporal structures like inter-event times or beginning/end times can impact spreading simulations. Finally, it discusses how different temporal network datasets exhibit diverse temporal structures.
This document discusses how rumors spread quickly through social networks. It simulates a simple rumor spreading process on real-world social networks like Twitter and Orkut as well as theoretical network models. The results show that rumors spread much faster in the structures of actual social networks and preferential attachment networks than in random or complete networks. Specifically, a rumor reaching 45.6 million Twitter users within 8 rounds of communication.
The purpose of the present scientific contribution is to investigate from the business economics standpoing the emerging phenomenon of company networks. In particular, through the analysis of the theory of networks will be proposed the principal categories of business networks, and even before this the concept of the network will be defined. The proposed research, qualitatively, represents the point of departure for the study of the network phenomenon in light of the current economic phase termed “economy of knowledge”. Moreover, the research questions are the following: From where does the theory of networks arise? Do company networks consider themselves equal to knowledge networks?
Subscriber Churn Prediction Model using Social Network Analysis In Telecommun...BAINIDA
Subscriber Churn Prediction Model using Social Network Analysis In Telecommunication Industry โดย เชษฐพงศ์ ปัญญาชนกุล อาจารย์ ดร. อานนท์ ศักดิ์วรวิชญ์
ในงาน THE FIRST NIDA BUSINESS ANALYTICS AND DATA SCIENCES CONTEST/CONFERENCE จัดโดย คณะสถิติประยุกต์และ DATA SCIENCES THAILAND
Scott Gomer presented on social network analysis (SNA). He reviewed literature on SNA and its use as a tool to analyze social structures and influence. He discussed SNA's capabilities in identifying key relationships and influencers through visual sociograms. However, SNA also has limitations such as complexity with large networks. Gomer collected binary data on relationships within a network and analyzed it using sociograms to illustrate examples of social link platforms. He concluded that SNA is a qualitative tool that can provide useful insights for marketing research by studying relationships.
Eugenio S Final Presentation Material Results Of Annotated Bibliography Apr12shirin0809
This annotated bibliography summarizes 18 sources related to computer-mediated communication (CMC) and its impact on organizational behavior. The sources cover a range of literature, including theoretical papers, reviews of empirical studies, and conceptual frameworks. Key findings include that CMC enables the maintenance of "weak ties" across organizations, new media is enabling richer forms of communication beyond just text, and emerging communication technologies and practices are giving rise to a new concept of "connected presence." The implications drawn are that CMC contributes value through both formal and informal communication, new theories of CMC need to consider multimedia aspects, and changes in personal media usage may influence workplace behaviors.
The science of networks is becoming an increasingly important and intriguing area of study that reveals many a patterns and relationships often hidden. This presentation is about the use of SNA to study the network of the Digital Library Community
Complex Networks Analysis @ Universita Roma TreMatteo Moci
This document discusses complex networks and their analysis. It provides a brief history of network analysis starting in the 18th century with Euler's work on the Seven Bridges of Königsberg problem. It then covers key topics like different types of networks, graph modeling approaches, measures to analyze networks, and applications of network analysis to domains like the web, social networks, and disease spreading. The document emphasizes that understanding network structure and interactions is important for studying complex systems and influences within networks.
Network Data Collection
The document discusses collecting social network data. It covers three main topics:
1) Introduces social network analysis and why networks are important in social science. Networks matter because of connections that allow diffusion and because positions in networks influence roles and behavior.
2) Discusses research design considerations for collecting network data, including specifying relations of interest based on theoretical mechanisms, boundary selection, and sampling approaches.
3) Addresses accuracy of network survey data and how to handle inaccurate or missing data. The goal is to systematically understand connections between actors using empirical network data and analysis methods.
This document discusses considerations for collecting social network data through surveys. It addresses research design elements like defining the relevant population boundaries and sampling approaches. For surveys specifically, it covers informed consent, name generator questions to identify social ties, response formats, and balancing depth of network detail collected versus sample size. The key challenges are defining the theoretical population of interest, collecting a sufficiently large and representative network sample, and designing survey questions that accurately capture social ties within time and resource constraints.
This document discusses the use of social network graphs and analytics. It provides an overview of key concepts in social network analysis (SNA) including representing social networks, identifying strong and weak ties, central nodes, and overall network structure. Examples are given of how SNA is used in business, law enforcement, social media sites, and more. Key measures discussed include degree, betweenness, closeness, eigenvector centrality, density, and clustering coefficient. The small-world phenomenon and preferential attachment are also covered.
This document discusses community detection in networks. Community detection aims to identify tightly knit groups within networks. The document outlines popular community detection algorithms like modularity maximization and stochastic block models. It also discusses applications of community detection to multilayer networks and examples like congressional voting networks and Facebook networks. Community detection is a useful tool for exploring network structure and identifying essential features in data.
This document discusses the history and challenges of network visualization. It outlines James Moody's presentation on the topic, which traces the evolution of network visualization from Euler's early work to modern approaches. Key challenges discussed include determining which social space to represent, how to handle multidimensional data, and dealing with issues of scale and density in large networks. The document argues that visualization allows researchers to gain insights that metrics alone cannot provide, by making the invisible visible and communicating complex features effectively.
This document summarizes key concepts for describing networks, including centrality measures, connectivity, cohesion, and roles. It discusses measuring the importance of individual nodes through degrees, closeness, betweenness, and power centrality. It also covers sociocentric measures like degree distributions, centralization, and density. Additionally, it explores local connectivity through triads, transitivity, and clustering coefficients as well as structural cohesion through components and cut points.
DOMINANT FEATURES IDENTIFICATION FOR COVERT NODES IN 9/11 ATTACK USING THEIR ...IJNSA Journal
The document presents a framework called SoNMine that identifies key players in the 9/11 covert network using node behavioral profiles. It generates profiles by analyzing node behaviors based on path types extracted from the network's multi-relational structure. The framework identifies outlier nodes with dense connections or high communication as influential players. It also determines dominant features that help classify normal and outlier nodes more accurately.
This document summarizes open problems and future directions in the field of social networks and health. It identifies key areas for methodological development including dynamic diffusion models, improved community detection techniques, and understanding triadic network structures. Important theoretical advances involve modeling multiplex and evolving networks over time as well as better understanding social mechanisms linking networks to health. Future data collection should incorporate electronic traces, return to community-based studies, and develop national samples capturing full network contexts.
This chapter will introduce you to the field of science known as Network Theory and tell you about the major researches that took place since its conceptualization. Since the course in question is social computing the chapter is written in a way to give examples and illustrations which mostly relate to social computing. It also contains theories and information which are mostly related to network theory and have some or no relation to social computing. But the basic purpose of this chapter is to explain Network theory and its applications in the field of social computing.
This document discusses Network Canvas, a software for collecting social network data developed by Michelle Birkett and others. It is being used in the RADAR study to collect longitudinal network data from over 1000 young men who have sex with men in Chicago. The software was designed to be intuitive for participants to select, position and manipulate nodes representing their social connections. It aims to capture complex network and attribute data across multiple time points. The document discusses the project workflow, comparisons to other network data collection methods, evaluation plans and sustainability efforts through workshops and community involvement.
10 More than a Pretty Picture: Visual Thinking in Network Studiesdnac
Visualization has been important in network science since its beginnings to make invisible structures visible. While metrics can describe networks, visualizations allow researchers to see relationships and patterns across multiple dimensions that numbers alone cannot reveal. Effective network visualizations communicate insights that would be difficult to understand otherwise, by depicting global patterns and local details simultaneously in a way that builds intuition about the network's structure and generating processes. However, challenges include lack of consistent display frameworks, integrating too much multidimensional information, and issues of scale for large and dynamic networks.
1) The document describes a model for evolving social networks where individuals are connected based on their social affinity for one another. 2) The model assigns each individual an affinity ranking of others, representing their personal interest or tension in being connected to that person. Individuals try to minimize their personal or global network energy by rewiring connections to those they have higher affinity for. 3) The document explores two approaches for initializing affinity rankings: random distributions and neighbor-correlated distributions where closer individuals start with higher affinity. It analyzes how the parameter q in the latter affects the structure of the final stable network state.
The problem of matchmaking in electronic social networks is formulated as an optimization problem.
In particular, a function measuring the matching degree of fields of interest of a search profile with
those of an advertising profile is proposed.
The document summarizes a research paper that proposes a new algorithm called the leader-follower algorithm for detecting community structure in networks. The algorithm is based on identifying the internal structure of communities rather than external connectivity properties used by traditional methods like spectral clustering. It first distinguishes between leader nodes that connect different communities and loyal follower nodes that only have neighbors within their community. It then assigns loyal followers to leaders to form the communities. The algorithm is able to detect communities of any size and learn their number without requiring this as input, offering an improvement over spectral clustering especially for densely connected networks. Experiments on synthetic and real-world networks demonstrate its effectiveness.
Spreading processes on temporal networksPetter Holme
This document discusses temporal networks and how temporal structures can impact dynamical processes on networks. It begins by describing different types of temporal networks including person-to-person communication, information dissemination, physical proximity, and cellular biology networks. It then discusses methods for analyzing temporal network structures like inter-event times and how bursty or heavy-tailed distributions can slow spreading compared to memory-less processes. The document also presents examples of how neutralizing temporal structures like inter-event times or beginning/end times can impact spreading simulations. Finally, it discusses how different temporal network datasets exhibit diverse temporal structures.
This document discusses how rumors spread quickly through social networks. It simulates a simple rumor spreading process on real-world social networks like Twitter and Orkut as well as theoretical network models. The results show that rumors spread much faster in the structures of actual social networks and preferential attachment networks than in random or complete networks. Specifically, a rumor reaching 45.6 million Twitter users within 8 rounds of communication.
The purpose of the present scientific contribution is to investigate from the business economics standpoing the emerging phenomenon of company networks. In particular, through the analysis of the theory of networks will be proposed the principal categories of business networks, and even before this the concept of the network will be defined. The proposed research, qualitatively, represents the point of departure for the study of the network phenomenon in light of the current economic phase termed “economy of knowledge”. Moreover, the research questions are the following: From where does the theory of networks arise? Do company networks consider themselves equal to knowledge networks?
Subscriber Churn Prediction Model using Social Network Analysis In Telecommun...BAINIDA
Subscriber Churn Prediction Model using Social Network Analysis In Telecommunication Industry โดย เชษฐพงศ์ ปัญญาชนกุล อาจารย์ ดร. อานนท์ ศักดิ์วรวิชญ์
ในงาน THE FIRST NIDA BUSINESS ANALYTICS AND DATA SCIENCES CONTEST/CONFERENCE จัดโดย คณะสถิติประยุกต์และ DATA SCIENCES THAILAND
Scott Gomer presented on social network analysis (SNA). He reviewed literature on SNA and its use as a tool to analyze social structures and influence. He discussed SNA's capabilities in identifying key relationships and influencers through visual sociograms. However, SNA also has limitations such as complexity with large networks. Gomer collected binary data on relationships within a network and analyzed it using sociograms to illustrate examples of social link platforms. He concluded that SNA is a qualitative tool that can provide useful insights for marketing research by studying relationships.
Eugenio S Final Presentation Material Results Of Annotated Bibliography Apr12shirin0809
This annotated bibliography summarizes 18 sources related to computer-mediated communication (CMC) and its impact on organizational behavior. The sources cover a range of literature, including theoretical papers, reviews of empirical studies, and conceptual frameworks. Key findings include that CMC enables the maintenance of "weak ties" across organizations, new media is enabling richer forms of communication beyond just text, and emerging communication technologies and practices are giving rise to a new concept of "connected presence." The implications drawn are that CMC contributes value through both formal and informal communication, new theories of CMC need to consider multimedia aspects, and changes in personal media usage may influence workplace behaviors.
The science of networks is becoming an increasingly important and intriguing area of study that reveals many a patterns and relationships often hidden. This presentation is about the use of SNA to study the network of the Digital Library Community
Complex Networks Analysis @ Universita Roma TreMatteo Moci
This document discusses complex networks and their analysis. It provides a brief history of network analysis starting in the 18th century with Euler's work on the Seven Bridges of Königsberg problem. It then covers key topics like different types of networks, graph modeling approaches, measures to analyze networks, and applications of network analysis to domains like the web, social networks, and disease spreading. The document emphasizes that understanding network structure and interactions is important for studying complex systems and influences within networks.
Network Data Collection
The document discusses collecting social network data. It covers three main topics:
1) Introduces social network analysis and why networks are important in social science. Networks matter because of connections that allow diffusion and because positions in networks influence roles and behavior.
2) Discusses research design considerations for collecting network data, including specifying relations of interest based on theoretical mechanisms, boundary selection, and sampling approaches.
3) Addresses accuracy of network survey data and how to handle inaccurate or missing data. The goal is to systematically understand connections between actors using empirical network data and analysis methods.
This document discusses considerations for collecting social network data through surveys. It addresses research design elements like defining the relevant population boundaries and sampling approaches. For surveys specifically, it covers informed consent, name generator questions to identify social ties, response formats, and balancing depth of network detail collected versus sample size. The key challenges are defining the theoretical population of interest, collecting a sufficiently large and representative network sample, and designing survey questions that accurately capture social ties within time and resource constraints.
This document discusses the use of social network graphs and analytics. It provides an overview of key concepts in social network analysis (SNA) including representing social networks, identifying strong and weak ties, central nodes, and overall network structure. Examples are given of how SNA is used in business, law enforcement, social media sites, and more. Key measures discussed include degree, betweenness, closeness, eigenvector centrality, density, and clustering coefficient. The small-world phenomenon and preferential attachment are also covered.
This document discusses community detection in networks. Community detection aims to identify tightly knit groups within networks. The document outlines popular community detection algorithms like modularity maximization and stochastic block models. It also discusses applications of community detection to multilayer networks and examples like congressional voting networks and Facebook networks. Community detection is a useful tool for exploring network structure and identifying essential features in data.
This document discusses the history and challenges of network visualization. It outlines James Moody's presentation on the topic, which traces the evolution of network visualization from Euler's early work to modern approaches. Key challenges discussed include determining which social space to represent, how to handle multidimensional data, and dealing with issues of scale and density in large networks. The document argues that visualization allows researchers to gain insights that metrics alone cannot provide, by making the invisible visible and communicating complex features effectively.
This document summarizes key concepts for describing networks, including centrality measures, connectivity, cohesion, and roles. It discusses measuring the importance of individual nodes through degrees, closeness, betweenness, and power centrality. It also covers sociocentric measures like degree distributions, centralization, and density. Additionally, it explores local connectivity through triads, transitivity, and clustering coefficients as well as structural cohesion through components and cut points.
DOMINANT FEATURES IDENTIFICATION FOR COVERT NODES IN 9/11 ATTACK USING THEIR ...IJNSA Journal
The document presents a framework called SoNMine that identifies key players in the 9/11 covert network using node behavioral profiles. It generates profiles by analyzing node behaviors based on path types extracted from the network's multi-relational structure. The framework identifies outlier nodes with dense connections or high communication as influential players. It also determines dominant features that help classify normal and outlier nodes more accurately.
This document summarizes open problems and future directions in the field of social networks and health. It identifies key areas for methodological development including dynamic diffusion models, improved community detection techniques, and understanding triadic network structures. Important theoretical advances involve modeling multiplex and evolving networks over time as well as better understanding social mechanisms linking networks to health. Future data collection should incorporate electronic traces, return to community-based studies, and develop national samples capturing full network contexts.
This chapter will introduce you to the field of science known as Network Theory and tell you about the major researches that took place since its conceptualization. Since the course in question is social computing the chapter is written in a way to give examples and illustrations which mostly relate to social computing. It also contains theories and information which are mostly related to network theory and have some or no relation to social computing. But the basic purpose of this chapter is to explain Network theory and its applications in the field of social computing.
This document discusses Network Canvas, a software for collecting social network data developed by Michelle Birkett and others. It is being used in the RADAR study to collect longitudinal network data from over 1000 young men who have sex with men in Chicago. The software was designed to be intuitive for participants to select, position and manipulate nodes representing their social connections. It aims to capture complex network and attribute data across multiple time points. The document discusses the project workflow, comparisons to other network data collection methods, evaluation plans and sustainability efforts through workshops and community involvement.
10 More than a Pretty Picture: Visual Thinking in Network Studiesdnac
Visualization has been important in network science since its beginnings to make invisible structures visible. While metrics can describe networks, visualizations allow researchers to see relationships and patterns across multiple dimensions that numbers alone cannot reveal. Effective network visualizations communicate insights that would be difficult to understand otherwise, by depicting global patterns and local details simultaneously in a way that builds intuition about the network's structure and generating processes. However, challenges include lack of consistent display frameworks, integrating too much multidimensional information, and issues of scale for large and dynamic networks.
1) The document describes a model for evolving social networks where individuals are connected based on their social affinity for one another. 2) The model assigns each individual an affinity ranking of others, representing their personal interest or tension in being connected to that person. Individuals try to minimize their personal or global network energy by rewiring connections to those they have higher affinity for. 3) The document explores two approaches for initializing affinity rankings: random distributions and neighbor-correlated distributions where closer individuals start with higher affinity. It analyzes how the parameter q in the latter affects the structure of the final stable network state.
The problem of matchmaking in electronic social networks is formulated as an optimization problem.
In particular, a function measuring the matching degree of fields of interest of a search profile with
those of an advertising profile is proposed.
The document provides biographical information about Master Chang Dsu Yao, founder of Taiji Chang Style martial arts. It describes how he studied for 20 years under the renowned Master Liu Pao Chün in northern China. It then outlines his military career and involvement in wars against Japan and in the Chinese civil war, where he was severely wounded multiple times. The document concludes with details of Master Chang founding the FEIK martial arts federation in Italy and spreading his art in Europe before his sudden death in Taiwan in 1992.
The Majority Rule is applied to a topology that consists of two coupled
random networks, thereby mimicking the modular structure observed in social
networks. We calculate analytically the asymptotic behaviour of the model and derive a
phase diagram that depends on the frequency of random opinion flips and on the inter-
connectivity between the two communities. It is shown that three regimes may take
place: a disordered regime, where no collective phenomena takes place; a symmetric
regime, where the nodes in both communities reach the same average opinion; an
asymmetric regime, where the nodes in each community reach an opposite average
opinion. The transition from the asymmetric regime to the symmetric regime is shown
to be discontinuous.
The document summarizes key findings from a research study on social media and wine consumers. Some of the main findings include:
- Integrated social media marketing strategies are gaining importance for wine brands, particularly for targeting younger, high-involvement consumers.
- Opinion leaders on social media can influence larger audiences, so engaging these influential users is important.
- Traditional print advertising is being used less and American consumers highly value online word-of-mouth and information searching.
The study analyzed survey responses from over 500 wine consumers, mostly American males aged 36-45, to understand their social media usage, purchasing behaviors, and how social influences affect their wine choices.
The document discusses graph matching and distance measures between graphs. It defines the graph distance di(a,b) between two graphs a and b as a function f(a,b). It then presents the Euclidean distance measure which calculates the distance as the square root of the sum of the squared differences between corresponding vertex attributes in a and b. The document also discusses finding an optimal assignment w between the vertices of two graphs to minimize the assignment cost Sp(w).
The document describes the Glasgow Reversing Club, which focuses on the topic of reverse engineering (RE). RE is the process of understanding an existing product like software or hardware by analyzing its components and functionality. The club aims to teach experienced and inexperienced individuals about RE through online tutorials, seminars from security experts, and reversing challenges. It also promotes social activities for members like games and showcasing hacking projects. The document provides examples of RE applications like malware analysis and outlines legal and illegal uses of the practice.
This document describes a new method for detecting community structure in complex networks based on node similarity. The method works as follows:
1. It calculates the similarity between all node pairs using a local node similarity metric.
2. It treats each node as its own community initially. Then it iteratively incorporates the community of the current node with the communities containing its most similar nodes.
3. It selects the most similar uncovered node as the next current node, and repeats the process until all nodes have been incorporated into communities.
The method requires only local network information and has a computational complexity of O(nk) for a network with n nodes and average degree k. It is evaluated on real and computer-generated networks, demonstrating
Networks & Health
This document provides an introduction and overview of social network analysis and its relevance to health research. It discusses key concepts such as what networks are, different types of network data including one-mode and two-mode data, and different levels of analysis including ego networks, partial networks, and complete networks. The document also discusses why networks matter for health through connectionist mechanisms like diffusion and positional mechanisms like social roles. Overall, the document serves as a high-level introduction to social network concepts and their application to health research.
01 Introduction to Networks Methods and Measuresdnac
This document provides an introduction to social network analysis. It discusses how networks matter through two fundamental mechanisms: connections and positions. Connections refer to the flow of things through networks, viewing networks as pipes. Positions refer to relational patterns and networks capturing role behavior, viewing networks as roles. The document also covers basic network data structures including nodes, edges, directed/undirected ties, binary/valued ties, and different levels of analysis such as ego networks and complete networks. It provides examples of one-mode and two-mode network data.
This document provides an introduction to social network analysis. It discusses how network analysis allows us to understand social connections and positions. There are two key mechanisms through which networks can impact outcomes: connections, where networks matter because of what flows through them, and positions, where networks capture roles and social exchange. Network analysis provides tools to empirically study patterns of social structure by mapping relationships between actors.
An information-theoretic, all-scales approach to comparing networksJim Bagrow
My presentation at NetSci 2018 on Portrait Divergence, a new approach to comparing networks that is simple, general-purpose, and easy to interpret.
The preprint: https://arxiv.org/abs/1804.03665
The code: https://github.com/bagrow/portrait-divergence
This document discusses the history and development of network analysis in the social sciences. It covers early concepts from the 19th century that laid the foundations, work in the mid-20th century applying mathematical and graph-theoretic approaches to social networks, and the proliferation of network analysis across many fields in the 1990s. It also examines key concepts of network analysis like centrality, structural holes, strength of weak ties, and the use of network models to understand macro-level phenomena emerging from micro-level actor interactions.
This document provides a high-level overview of protein-protein interaction networks and graph-theoretic modeling approaches. It discusses how experimental data on protein interactions is incomplete and noisy but still offers opportunities for biological insight. It describes how graph theory can be used to model these networks and compare them, such as by analyzing network properties and frequencies of small subgraph structures. Different random network models are also discussed as ways to understand properties of real biological networks. The goal is to provide a concise summary of this area and comment on possible future challenges through a graph-theoretic lens.
National Resource for Networks Biology's TR&D Theme 3: Although networks have been very useful for representing molecular interactions and mechanisms, network diagrams do not visually resemble the contents of cells. Rather, the cell involves a multi-scale hierarchy of components – proteins are subunits of protein complexes which, in turn, are parts of pathways, biological processes, organelles, cells, tissues, and so on. In this technology research project, we will pursue methods that move Network Biology towards such hierarchical, multi-scale views of cell structure and function.
IRJET- Predicting Social Network Communities Structure Changes and Detection ...IRJET Journal
This document discusses predicting changes in community structures of social networks and detecting spam bots. It proposes using digital DNA behavioral modeling to predict crucial events in how social network communities expand, shrink, or combine over time. Digital DNA reflects a user's unique pattern of interactions and can be used for social fingerprinting to efficiently distinguish real users from spam bots. The document reviews several related works and approaches for tracking community changes, predicting critical events, and modeling community evolution over time. It concludes that critical community changes can be predicted using digital DNA and that the proposed approach using social fingerprinting may effectively detect spam bots.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Networks provide connections and positions that influence health outcomes. Social network analysis examines relationships between actors to understand how networks impact behavior. Networks matter through both connectionist mechanisms like diffusion, and positional mechanisms like social roles. Network data can be analyzed at different levels from individual ego networks to global networks, and can involve one or multiple types of relationships between nodes. Social network data is commonly represented through matrices and lists to encode network structure and allow computational analysis.
Subgraph Frequencies: Mapping the Empirical and Extremal Geography of Large G...Gabriela Agustini
A growing set of on-line applications are generating data that can be viewed as very large collections of small, dense social graphs — these range from sets of social groups, events, or collabora- tion projects to the vast collection of graph neighborhoods in large social networks. A natural question is how to usefully define a domain-independent ‘coordinate system’ for such a collection of graphs, so that the set of possible structures can be compactly rep- resented and understood within a common space. In this work, we draw on the theory of graph homomorphisms to formulate and an- alyze such a representation, based on computing the frequencies of small induced subgraphs within each graph.
Six Degrees of Separation to Improve Routing in Opportunistic Networksijujournal
This document discusses using small-world network concepts for routing in opportunistic networks. It analyzes three real-world datasets representing contact graphs and finds they exhibit small-world properties with high clustering and short path lengths. The document proposes a simple routing algorithm that applies these findings and concludes it outperforms other algorithms in simulations by taking temporal contact factors into account.
SIX DEGREES OF SEPARATION TO IMPROVE ROUTING IN OPPORTUNISTIC NETWORKSijujournal
Opportunistic Networks are able to exploit social behavior to create connectivity opportunities. This
paradigm uses pair-wise contacts for routing messages between nodes. In this context we investigated if the
“six degrees of separation” conjecture of small-world networks can be used as a basis to route messages in
Opportunistic Networks. We propose a simple approach for routing that outperforms some popular
protocols in simulations that are carried out with real world traces using ONE simulator. We conclude that
static graph models are not suitable for underlay routing approaches in highly dynamic networks like
Opportunistic Networks without taking account of temporal factors such as time, duration and frequency of
previous encounters.
The International Journal of Engineering and Science (IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document provides an introduction to social network analysis (SNA), including its origins in social science and mathematics. SNA focuses on relationships between individuals, groups, or institutions rather than attributes of individuals. It represents social structures as networks and uses graph theory concepts to analyze them. SNA is useful for understanding information flow, identifying influential individuals, and improving network effectiveness. The document discusses key SNA concepts like networks, tie strength, central players, and network cohesion.
Online social network mining current trends and research issueseSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Improving Knowledge Handling by building intellegent social systemsnazeeh
This document discusses improving knowledge handling by building intelligent systems using social agent modelling. It proposes capturing knowledge from social environments by developing new features in social network analysis systems and using this knowledge to model multi-agent systems. The approach involves extending social network analysis to cover more qualitative factors like emotions, relationships and trust to better represent knowledge and simulate agent behavior. Capturing these social aspects from real networks can provide criteria to analyze and design intelligent multi-agent systems.
This document discusses predicting new friendships in social networks using temporal information. It describes research on predicting new links in social networks over time using supervised learning models trained on temporal features from past network interactions. The researchers used anonymized Facebook data over 28 months to train decision tree and neural network classifiers to predict new relationships, finding models using temporal information performed better than those without it.
Incremental Community Mining in Location-based Social NetworkIJAEMSJORNAL
A social network can be defined as a set of social entities connected by a set of social relations. These relations often change and differ in time. Thus, the fundamental structure of these networks is dynamic and increasingly developing. Investigating how the structure of these networks evolves over the observation time affords visions into their evolution structure, elements that initiate the changes, and finally foresee the future structure of these networks. One of the most relevant properties of networks is their community structure – set of vertices highly connected between each other and loosely connected with the rest of the network. Subsequently networks are dynamic, their underlying community structure changes over time as well, i.e they have social entities that appear and disappear which make their communities shrinking and growing over time. The goal of this paper is to study community detection in dynamic social network in the context of location-based social network. In this respect, we extend the static Louvain method to incrementally detect communities in a dynamic scenario following the direct method and considering both overlapping and non-overlapping setting. Finally, extensive experiments on real datasets and comparison with two previous methods demonstrate the effectiveness and potential of our suggested method.
Incremental Community Mining in Location-based Social Network
New Approaches Social Network
1. New approaches to model and study social networks
arXiv:physics/0701107v1 [physics.soc-ph] 9 Jan 2007
P.G. Lind
ICP, Universit¨ t Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany
a
CFTC, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisbon, Portugal
H.J. Herrmann
ICP, Universit¨ t Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart, Germany
a
IfB, HIF E12, ETH H¨ nggerberg, CH-8093 Z¨ rich, Switzerland
o u
Departamento de F´sica, Universidade Federal do Cear´ , 60451-970 Fortaleza, Brazil
ı a
Abstract. We describe and develop three recent novelties in network research which are
particularly useful for studying social systems. The first one concerns the discovery of some
basic dynamical laws that enable the emergence of the fundamental features observed in
social networks, namely the nontrivial clustering properties, the existence of positive degree
correlations and the subdivision into communities. To reproduce all these features we describe
a simple model of mobile colliding agents, whose collisions define the connections between
the agents which are the nodes in the underlying network, and develop some analytical
considerations. The second point addresses the particular feature of clustering and its
relationship with global network measures, namely with the distribution of the size of cycles
in the network. Since in social bipartite networks it is not possible to measure the clustering
from standard procedures, we propose an alternative clustering coefficient that can be used
to extract an improved normalized cycle distribution in any network. Finally, the third point
addresses dynamical processes occurring on networks, namely when studying the propagation
of information in them. In particular, we focus on the particular features of gossip propagation
which impose some restrictions in the propagation rules. To this end we introduce a quantity,
the spread factor, which measures the average maximal fraction of nearest neighbors which
get in contact with the gossip, and find the striking result that there is an optimal non-trivial
number of friends for which the spread factor is minimized, decreasing the danger of being
gossiped.
PACS numbers: 89.65.-s, 89.75.Fb, 89.75.Hc, 89.75.Da
2. Social networks: models and measures 2
1. Introduction
Contrary to what may be perceived at a first glance, social and physical models were brought
together several times, during the last four centuries. In fact, not only Maxwell and Boltzmann
were inspired by the statistical approaches in social sciences to develop the kinetic theory of
gases, but one can even cite the English philosopher Thomas Hobbes, who already in the
seventeenth century, using a mechanical approach, tried to explain how people acquaintances
and behaviors may contribute to the evolution towards a stable absolute monarchy[1, 2]. More
than making a historical perspective if these approaches were successful and correct or not, it
is almost unquestionable that, at a certain level, there are social phenomena that could be more
deeply understood by using approaches of statistical and physical models. Recently[3, 4, 5],
this perspective gained considerable strength from the increased interest on - and in several
senses well-succeed - network approach, where one describes complex systems by mapping
them on a graph (network) of nodes and links and studies their structure and dynamics with the
help of some statistical and topological tools from statistical physics and graph theory[6, 7].
When addressing the specific case of a social system, nodes represent individuals and
the connections between them represent social relations and acquaintances of a certain kind.
Social networks were studied in different contexts[8, 9, 10, 11, 12, 13, 14], ranging from
epidemics spreading and sexual contacts to language evolution and vote elections. However,
although they are ubiquitous, social networks differ from most other networks, yielding a still
broad spectrum of unanswered questions and improvements to be done when studying their
statistical and topological properties. In this paper, we will address three fundamental open
questions related to the typical structure and dynamics associated to social networks.
The first open question has to do with the modeling of social networks. The recent
broad study of empirical social networks has shown that they have three fundamental
features common to all of them[15]. First, they present the small-world effect[16] with
small average path lengths between nodes and high clustering coefficients meaning that
neighbors tend to be connected with each other. Second, they have positive correlations:
the highly (poorly) connected nodes tend to connected to other highly (poorly) connected
nodes. Third and last, invariably one observes an organization of the network into some
subsets of nodes (communities) more densely connected between each other. Although there
are arguments pointing out that all these features could be consequence from one another[15],
the modeling of specific social networks reproducing quantitatively all these features has not
been successful. Using a recent approach to construct networks, based on a system of mobile
agents, it is possible to reproduce all these features. In Section 2 we will further show that the
degree distributions characterizing social networks typically follow a specific one-parameter
distribution, so-called Brody distribution.
The second question is related to the intrinsic nature of the nodes. For certain social
networks there are intrinsic features of the individuals which must be considered in the
analysis. For instance, the gender in networks of sexual contacts[14] or the hierarchical
position in a network of social contacts inside some enterprise. From the network point of
view this distinction means to introduce multipartivity in the network, biasing the preferential
3. Social networks: models and measures 3
attachment between nodes that tend to connect with nodes of a certain type. When there
are two types of nodes, e.g. men and women, and the connections between them is strongly
related to this type, e.g. men can only match women and vice-versa, the standard measures to
analyze network structure fails. In particular, the standard clustering coefficient[16], is unable
to quantify the connectedness of broader neighborhoods that typically appear in multipartite
networks. In Sec. 3 we will revisit some of the clustering coefficients used to study clustering
in bipartite networks, and show how the combination of both clustering coefficients can
yield good estimates of normalized cycle distributions. Moreover, we will discuss a general
theoretical picture of a global measure of increasing order of clustering coefficients according
to some suitable expansion.
The third open question has to do with the heterogeneity of nodes in what concerns their
influence in the connections and therefore in the propagation phenomena on social networks.
In rumor propagation[17], for instance, one usually treats all connections equally in the spread
of some signal (opinion, rumor, etc). This is a suitable assumption for situations like the
spread of an opinion which is equally interesting to all nodes in the network, for example
political opinions to some vote election. However, there are also several social situations
where the signal is not equally interesting to all nodes, as the case of spreading of some
gossip about some common friend. In these cases there are connections which will be more
probably used to spread the signal than others, since not all our friends are also friends of the
particular person which is being gossiped about and therefore, either we tend to not tell the
gossip to them or they tend to not spread it even if they hear it. In Sec. 4 we will present a
simple model for gossip propagation and describe some striking features. Namely, that there
is an optimal number of friends, depending on the degree distribution and degree correlations
of the entire network, for which the danger to be gossiped is minimized.
Finally, in Sec. 5 we make final conclusions, giving an overview of future questions
which could be studied in social networks arising from the topics studied throughout the
paper.
2. Modeling social networks: an approach based on mobile agents
Since the study of social networks is mainly concerned with topological and statistical features
of people’s acquaintances[8, 9], the modeling of such networks has been done within the
framework of graph theory using suitable probabilistic laws for the distribution of connections
between individuals[3, 4, 5, 7]. This approach proved to be successful in several contexts, for
instance to describe community formation[18, 19] and their growth[20].
However, they present two major drawbacks. First, the graph approach may be suited to
describe the structure of social contacts and acquaintances, but lacks to give insight into the
social dynamical laws underlying it. Second, these models seem to be unable to reproduce all
the main features characteristic of social networks, at least at the fundamental level. In this
context, it was pointed out that[21, 22, 23] dynamical processes based on local information
should be also considered when modeling the network. Our recent proposal to overcome these
shortcomings was to construct networks, from a system of mobile agents following a simple
4. Social networks: models and measures 4
1
0 11
00
1
0
00
11 000000
111111
00
11
00
11 0
1 111111
000000 11
00
11 1
00 0
00
11 0
111 11
00 00
1
0
11
00 0
1
00
11
111111
000000
111111
000000 00
11
00 P
11
00
11
00 1
11
00
11 00
11
111111
000000 11
00 11
00
0
1 00
11 00
11
00 00
11 11 11
00 1 11 111111
0 00 000000 00
11
00
11 00
11
00
11
11
00
11 11
00 00 1 11
0 00 00 00 0
11 11 1 00
11 11
00
0
1 00
11 0
111
00 11
00
1 111111
0 000000 00
11
111
000 P
00
11 00
11
000
111
11
00
0
1 000000
111111 3
00
11 1
0 0 00
1 11
111111
000000 P2
1 11
0 00
00
00
11
11
1 11 11
0 00 00
111111
000000
11
00
000000
111111
1
0 00
11
11
00 00 0
11 1
0
1 00 00
11 11 11 1
00 0 1 11
0 00
000000
111111
00
11
000000
111111
0 00
1 11
00
11
0
1 111111
000000
111111
000000 00000
11111
00
11 000000
111111 11111
00000
0 00 0
1 11 1 111 0
000 0 1 000000
111111
00
11 t=0 11111
00000
11
00
0 00
1 11 1
0 11 1 11
00 0 00
1 000000
111111 00000 P
11111 4
00
11
11 111 11 111111
00 000 00 000000
00
111 11
0 00
0
1 000000
111111
11
00 11
00
111
000 000000
111111
0
1
111111
000000
0
1 11
00
0
1 1 11
0 00
0
1 1 1
0 0 0
1
000 P
111 1
11
00
P1
11111
00000 111
000 000 000
111 111
000
111
0000000 00
1111111 11
00
11
111 111
000 000
11111
00000 111
000 1111111 11
0000000 00
000
111
00000
11111
11
00
11111
00000 111
000 111
000
0000000 00
1111111 11
1111111
0000000
00
11
00000
11111
11
0000
11
11
00 00
11
000
111 0000000
1111111
00
11
00
11 00
11 0000000 P
1111111 3
0
1
00
11
1
0 11
00P3 1111111
0000000
0P
1 2 1111111
0000000
1111111
0000000
0
1
0
1
0
1
1111111
P20000000
1111111
0000000
0000000
1111111
00000
11111 000 0000000
111 1111111
11
00
0000000
1111111
00
11
00
11
00000
11111 11
0011
00
11
00
0
1
11111
00000
00
11 11
00
0 P
1 4
11
00
00000
11111
11
00 0
1
004
11
P t=1 0
1
1
0
0
1
t=2
Figure 1. Illustration of the two-dimensional mobile agents system. Initially there are no
connections between nodes and nodes move with some initial velocity v0 in a randomly chosen
direction (arrows). At t = 1 two nodes, P1 and P2 collide and a connection between them is
introduced (solid line), velocities are updated increasing their magnitude and choosing a new
random direction. At t = 2 two other collisions occur, between nodes P2 and P4 and between
nodes P1 and P3 . In this way a network of nodes and connections between them emerges as a
straightforward consequence of their motion (see text).
motion law[13, 14]. Here, we briefly review this model and further present the analytical
expression that fits the obtained degree distributions. In particular we show that the degree
distribution typically follows a Brody distribution[24].
2.1. The model
The model is given by a system of particles (agents) that move and collide with each other,
forming through those collisions the acquaintances between individuals. Consequently, the
network results directly from the time evolution of the system and is parameterized by two
single parameters, the density ρ of agents characterizing the system composition and the
maximal residence time Tℓ controlling its evolution. Each agent i is characterized by its
number ki of links and by its age Ai . When initialized each agent has a randomly chosen
age, position and moving direction with velocity v0 and one sets ki = 0. While moving,
the individuals follow ballistic trajectories till they collide. As a first approximation we
assume that social contacts do not determine which social contact will occur next. Therefore,
after collisions, the total momentum should not be conserved, with the two agents choosing
completely random new moving directions. Figure 1 sketches consecutive stages of the
evolution of such a system of mobile agents.
Assuming that large number of acquaintances tend to favor the occurrence of new
contacts, the velocity should increase with degree k, namely
v α
v(ki ) = (¯ki + v0 )ω, (1)
5. Social networks: models and measures 5
9 200
8
7
(b)
150
6
Tl /t 5 100 λ
4
3
2 50
1
(a)
0 0
10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
<k> <k>
Figure 2. Bridging between real social networks with average degree k and the system of
mobile agents that reproduce their topological and statistical features. In (a) the normalized
maximal residence time of agents is plotted as a function of the average degree, while in (b)
one plots the collision rate λ which is a unique function of the residence time, and scales with
k .
where v = 1 m/s is a constant to assure dimensions of velocity, ω = (ex cos θ + ey sin θ)
¯
with θ a random angle and ex and ey are unit vectors. The exponent α in Eq. (1) controls
the velocity update after each collision. Here, we consider α = 1. Further, the removal of
agents considered here is simply imposed by some threshold Tℓ in the age of the agents: when
Ai = Tℓ agent i leaves the system and a new agent j replaces it with kj = 0, vj = v0 and
randomly chosen moving direction. The selected values for Tℓ must be of the order of several
times the characteristic time τ between collisions, in order to avoid either premature death
of nodes. Too large values of Tℓ are also inappropriate since in that case each node may on
average collide with all other nodes yielding a fully connected network.
Similarly to other systems[25, 26], this finite Tℓ enables the entire system to reach a
non-trivial quasi-stationary state[13]. In fact, only by tuning Tℓ within an acceptable range
of small density values, one reproduces networks of social contacts. In Fig. 2a one sees
the normalized residence time Tℓ /τ as a strictly monotonic function of the average degree
k . From the residence time it is also possible to define a collision rate, as the fraction
between the average residence time Tℓ − A(0) = Tℓ /2 and the characteristic time τ , namely
λ = Tℓ /(2τ ) = v Tℓ /(2v0 τ0 ), where τ0 is the characteristic time of the system at the
beginning when all agents have velocity v0 . Figure 2b shows clearly that λ = 2 k .
By looking at Fig. 2 one now understands the main strength of the mobile agent model
here described: when taking a real network of social contacts and measuring the average
degree k the correspondence sketched in Fig. 2 straightforwardly returns the suitable value
of Tℓ that reproduces the topological and statistical features.
It was already reported[13, 27] that empirical networks extracted from a survey among
84 American schools are easily reproduced with this mobile agent model, in what concerns the
degree distribution, second-order correlations, community structure, average path length and
clustering coefficient. As an illustration, Fig. 3 shows the degree distribution P (k) of nine of
such schools (symbols). Such distributions are well fitted by Brody distributions (solid lines)
6. Social networks: models and measures 6
0.1
P(k) 0.01
N=1405 N=2152 N=1974
0.001 <k>=7.02 <k>=7.42 <k>=4.9
0.1
P(k) 0.01
N=1638 N=1877 N=2539
0.001 <k>=6.40 <k>=7.49 <k>=8.24
0.1
P(k) 0.01
N=1545 N=1710 N=1630
0.001 <k>=5.18 <k>=5.15 <k>=8.36
0.1 1 0.1 1 0.1 1
k/<k> k/<k> k/<k>
Figure 3. Degree distributions of nine different schools (symbols) from an in-school
questionnaire involving a total of 90118 students which responded to it in a survey between
1994 and 1995. Each school comprehends a number N of interviewed students and from
their questionnaires an average number k of acquaintances is extracted. With solid lines
we represent the fit obtained with a Brody distribution, Eq. (2), whose parameter value is
computed in Fig. 4.
defined as[24]:
¯ 1 ¯ ¯
PB (k) = (β + 1)η k β exp (−η k β+1 ), (2)
B
¯
with k = k/ k and
β+1
β+2
η=Γ . (3)
β+1
and B a normalization constant. Roughly, the Brody distribution in Eq. (2) is, apart some
special constants, the product of a power of k with an exponential with a negative exponent
proportional to a higher power of k. For the particular case β = 0, the Brody distribution
reduces to the exponential distribution having always a non-positive derivative.
The distributions in Fig. 3 were obtained with values of β slightly above zero, namely
between zero and one as shown in Fig. 4. In this case one is able to obtain the non trivial
positive slope which is typically observed for small k values in the degree distribution of
such social networks. Interestingly, Fig. 4 also shows a linear trend between the average
degree k in the network and the corresponding value of β which fits the degree distribution.
This guarantees that distribution in Eq. (2) has indeed one single parameter. How such a
distribution can be obtained from an analytical approach to the model of mobile agents is still
an open question and will be discussed in detail elsewhere.
7. Social networks: models and measures 7
1
0.9
0.8
β 0.7
0.6
0.5
0.4
4 5 6 7 8 9
<k>
Figure 4. The linear dependence between the parameter β of the Brody distribution in Eq. (2)
with the average number k of connections. Each bullet corresponds to one of the schools
whose degree distribution is plotted in Fig. 3. The solid line yields the fit β = 0.094 k +0.078.
3. Particular measures for social networks
To measure “the cliquishness of a typical neighborhood” in a network, Watts and Strogatz [16]
introduced a simple coefficient, called the clustering coefficient, which counts the number
of pairs of neighbors of a certain node which are connected with each other, forming a
cycle of size s = 3. While such tool enables one to access the structure of complex
networks arising in many systems [4, 7], helping to characterize small-world networks [16],
to understand synchronization in scale-free networks of oscillators [28] and to characterize
chemical reactions [29] and networks of social relationships [30, 31], there are other situations
where this measure does not suit. Namely, when the network presents a multipartite structure.
For instance, when there are two different kinds of nodes and connections link only nodes of
different type, the network is bipartite[30, 31, 32] and the bipartite structure does not allow
the occurrence of cycles with odd size, in particular with s = 3.
Bipartite networks are quite common for social systems[32, 33] where the two different
kinds of nodes represent e.g. the two genders. While the standard clustering coefficient in such
networks is always zero, they have in general non vanishing clustering properties[31] and
therefore more appropriate quantities to access such networks have been proposed, namely
coefficients counting larger cycles. In this Section, we will discuss how these different
clustering coefficients are related to each other and how one can use them to improve the
knowledge of the network structure.
The standard clustering coefficient C3 is usually defined[16] as the fraction between the
number of cycles of size s = 3 (triangles) observed in the network out of the total number of
possible triangles which may appear, namely
2ti
C3 (i) = . (4)
ki (ki − 1)
where ti is the number of existing triangles containing node i and ki is the number of neighbors
of node i, yielding a maximal number ki (ki − 1)/2 of triangles.
To access the cliquishness in bipartite networks one has proposed[21, 31, 32, 34] a
clustering coefficient C4 (i), sometimes called the grid coefficient[34], defined as the quotient
8. Social networks: models and measures 8
between the number of cycles of size s = 4 (squares) and the total number of possible squares.
Explicitly, for a given node i with two neighbors, say m and n, this coefficient yields[21]
qimn
C4,mn (i) = , (5)
(km − ηimn )(kn − ηimn ) + qimn
where qimn is the number of common neighbors between m and n (not counting i) and
ηimn = 1 + qimn + θmn with θmn = 1 if neighbors m and n are connected with each other and
0 otherwise.
After averaging over the nodes, the coefficients C3 and C4 characterize the contribution
of the first and second neighbors, respectively, for the network cliquishness. In order to
be a suitable quantity to measure the cliquishness of bipartite networks compared to their
monopartite counterparts, C4 must behave the same way as C3 when the network parameters
are changed, as it is indeed the case for C4 computed from Eq. (5). See Ref. [21] for details.
One should notice that in most m-partite networks, it is always possible to have cycles
of size s = 4, indicating that C4 is in some sense a more general clustering measure than C3 .
However, it could be the case that for a larger number of partitions forming the network, the
contribution of larger cycles increases. This is the case, for instance, of trophic relations in
an ecological network of different individuals from different species, where large cycles tend
to be abundant, namely the ones ranging from the higher predators to the plants at the lowest
trophic level. In such cases, a general clustering coefficient counting the fraction of possible
cycles of arbitrary size n may be needed. The generalization is straightforward yielding a
clustering coefficient Cn = En /Ln , where En is the number of existing cycles with size n,
Ln the maximal number of such cycles that is possible to be attained and n = 3, . . . , N for a
network of N nodes.
Having Cn for the required values of n, one is able to introduce a general clustering
measure of the network, given by the sum of all these contributions, namely
N N
En
C= αn Cn = αn , (6)
n=3 n=3
Ln
where αn is a coefficient that weights the contribution of each different clustering order n
and obeys the normalization condition N αn = 1. In general one can write En and Ln in
n=3
Eq. (6) as
En = NP (k1 )q(k1 , k2 )NP (k2 )q(k2 , k3) . . . NP (kn−1 )q(kn−1, kn ) (7)
k1 ,...,kn
N N!
Ln = Bn n! = (8)
(N − n)!
N
where Bn are the total combinations of n elements out of N, P (k) is the fraction of nodes with
k neighbors and q(k1 , k2) is the correlation degree distribution, i.e. the fraction of connections
linking a node with k1 neighbors to a node with k2 neighbors.
From Eq. (7) one can assume approximately that En ∼ ( P q N)n with P and q the
average fractions of P (k) and q(k1 , k2 ) respectively. Since Ln increases also as N n , a possible
suitable choice for α would be a constant, namely α = 1/(N − 2) obeying the normalization
9. Social networks: models and measures 9
(a) (b) (c)
Figure 5. Illustrative examples of cycles (size s = 6) where the most connected node (◦) is
connected to (a) all the other nodes composing the cycle, forming four adjacent triangles. In
(b) the most connected node is connected to all other nodes except one, forming two triangles
and one sub-cycle of size s = 4, while in (c) the same cycle s = 6 encloses two sub-cycles of
size s = 4 and no triangles (see text).
condition above. Having presented this general scenario, we now concentrate on the two first
clustering coefficients, C3 and C4 , to address the cycle size distribution.
We first show an estimate introduced in Ref. [35], which considers only the degree
distribution P (k) and the distribution of the standard clustering coefficient C3 (k). One starts
by considering the set of cycles with a central node, i.e. cycles with one node connected to
all other nodes composing the cycle, as illustrated in Fig. 5a. The central node composes
one triangle with each pair of connected neighbors. Due to this fact, the number of cycles
with size s can be easily estimated, since the number of different possible cycles to occur is
n0 (s, k) = Bs−1 (s−1)! , for a central node with k neighbors and the corresponding fraction of
k
2
these cycles which is expected to occur is p0 (s, k) = C3 (k)s−2 , yielding a total number of
s-cycles given by
kmax
Ns = Ngs P (k)n0 (s, k)p0 (s, k), (9)
k=s−1
where gs is a factor which takes into account the number of cycles counted more than once.
The estimate in Eq. (9) is a lower bound for the total number of cycles since it considers
only cycles with a central node. Further, this estimate only accounts for cycles up to size
s ≤ kmax + 1, with kmax the maximal degree and is not suited for bipartite networks where
C3 (k) = 0 for all k. Bipartite networks are typically composed of a set of nodes as those
illustrated in Fig. 5c, where no central node exists.
By using additionally the coefficient C4 (k) in a similar estimate, one is now able to take
into account several cycles without central nodes. One first considers the set of cycles of size
s with one node connected to all the others except one, as illustrated in Fig. 5b. Assuming
that this node has k neighbors, s − 2 of them belonging to the cycle one is counting for, one
k
has n1 (s, k) = Bs−2 (s − 2)!/2 different possible cycles of size s. The corresponding fraction
of such cycles which is expected to occur is given by p1 (s, k) = C3 (k)s−4 C4 (k)(1 − C3 (k)).
Writing an equation similar to Eq. (9), where instead of n0 (s, k) and p0 (s, k) one has n1 (s, k)
and p1 (s, k) respectively and the sum starts at s − 2 instead of s − 1, one has an additional
number Ns of estimated cycles which is not considered in estimate (9).
′
10. Social networks: models and measures 10
To improve the estimate further one repeats the same approach, taking out each time
one connection to the initial central node, increasing by one the number of elementary cycles
of size s = 4. Figure 5c illustrates a cycle of size s = 6 composed by two elementary
cycles of size 4. In general, for cycles composed by q sub-cycles of size 4 one finds
nq (s, k) = (s−q−1)! Bs−q−1 possible cycles of size s looking from a node with k neighbors
2
k
and a fraction pq (s, k) = C3 (k)s−2q−2 C4 (k)q (1 − C3 (k))q of them which are expected to be
observed.
Summing up over k and q yields our final expression
[s/2]−1 kmax
Ns = Ngs P (k)nq (s, k)pq (s, k). (10)
q=0 k=s−q−1
where [x] denotes the integer part of x. In particular, the first term (q = 0) is the sum in Eq. (9)
and the upper limit [s/2] − 1 of the first sum is obtained by imposing the exponent of C3 (k)
in pq (s, k) to be non-negative.
The estimate in Eq. (10) not only improves the estimated number computed from Eq. (9),
but also enables the estimate of cycles up to a larger maximal size[21], namely up to
s = 2kmax where kmax is the maximal number of neighbors in the network.
The estimate in Eq. (10) has also the advantage of being able to estimate cycles in
bipartite networks. Since for bipartite networks C3 (k) = 0, all terms in Eq. (10) vanish
except those for which the exponent of C3 (k) is zero, i.e. for s = 2(q + 1) with q an integer,
which naturally shows the absence of cycles of odd size in such networks.
For highly connected networks, both estimates should nevertheless yield similar results,
since in that case there is a very large number of both triangles and squares. For instance,
the so-called pseudo-fractal network[36] is a deterministic scale-free network, constructed
6
10 0.8
20
(a) (b)
10
16
(c)
5 10
10 12
10
10
8 0.6
4 4
10 10
0
10
0 20 40 60 80 100
N/(Ngs)
N/(Ngs)
3
10 0.4
2
10
0.2
1
10
0
10 0
0 20 40 60 80 20 40 60 80
s s
Figure 6. (a) The fraction Ns /N gs of the number of cycles estimated from Eqs. (9), dashed
lines, and (10), solid lines, compared with (b) the exact number of cycles as a function of
the size for the pseudo-fractal network [36]. From small to large curves one has pseudo-
fractal networks with m = 2, 3, 4, 5 generations (see text). In (c) one sees the comparison
between both estimates in a scale-free network with degree distribution P (k) = P0 k −γ with
(0) (0)
P0 = 0.737 and γ = 2.5, and coefficient distributions C3,4 (k) = C3,4 k −α with C3 = 2,
(0)
C4 = 0.33 and α = 0.9.
11. Social networks: models and measures 11
from three initial nodes connected with each other (generation m = 0), and iteratively adding
new generations of nodes such that in generation m + 1 one new node is added to each
edge of generation m and is connected to the two nodes joined by that edge. For these
networks, the exact number of cycles with size s can be written iteratively [37] and can be
directly compared to the one obtained with the two estimates above. Figure 6a shows the
two estimates, while in Fig. 6b the exact number is computed. We notice that both the real
number Ns of cycles and the normalized value Ns /(Ngs ), though different, yield the same
shape. Thus, although the estimates above are not able to explicit the geometrical factor gs ,
the corresponding normalized distributions agree very well with the real one. However, while
in this simple situation both estimates are similar, in general they can deviate significantly, as
illustrated in Fig. 6c. In such cases, the estimate (10) is closer to the real distribution of cycle
sizes[21].
4. Spreading phenomena in social networks
In the previous Section we show how the study of network structure can be addressed by
using tools as the clustering coefficient and first and second degree distributions. However,
although the ability to communicate within a network of contacts is favored by the network
topology[38], to study dynamical phenomena occurring on the network other measures are
necessary. Here, we focus on novel properties that help to ascertain the broadness and speed
of propagating phenomena through the network. We will describe two helpful quantities to
study propagation in a network. As we will see these tools are particularly suited for a simple
model of gossip propagation, that yields a striking result: in real social systems it is possible
to minimize the risk of being gossiped, by only choosing an optimal number of friendship
acquaintances.
We start by introducing the additional quantities in the context of gossip propagation. As
opposed to rumors, a gossip always targets the details about the behavior or private life of
a specific person. Some information of a specific gossip is created at time t = 0 about the
victim by one of its neighbors. Since typically the gossip tends to be of interest to only those
who know the victim personally, we consider first that it only spreads at each time step from
the vertices that know the gossip to all vertices that are connected to the victim and do not
yet know the gossip. Our dynamics is therefore like a burning algorithm [39], starting at the
originator and limited to sites that are neighbors of the victim. The gossip will spread until all
reachable neighbors of the victim know it, yielding a spreading time τ .
To measure how effectively the gossip or more generally the amount of information
attains the neighbors of the starting node (victim), we define the spreading factor f given
by
f = nf /k (11)
where nf is the total number of the k neighbors who eventually hear the gossip in a network
with N vertices (individuals). Although similar in particular cases, the spreading factor f and
the clustering coefficient are, in general, different because the later one only measures the
12. Social networks: models and measures 12
Figure 7. Semi-logarithmic plot of the spreading time τ as a function of the degree k for (a)
the Apollonian (n = 9 generations) and (b) the Barab´ si-Albert network with N = 104 nodes
a
for m = 3 (circles), 5 (squares) and 7 (triangles), where m is the number of edges of a new
site, and averaged over 100 realizations. In the inset of (a) we show a schematic design of the
Apollonian lattice for n = 3 generations. Fitting Eq. (12) to these data we have B = 1.1 in
(a) and B = 5.6 for large k in (b).
number of bonds between neighbors giving no insight about how they are connected.
In Fig. 7 one sees how the spreading time τ depends on the degree k of the starting
node. The Apollonian network[40] is illustrated in Fig. 7a, while the case of Barab´ si-Albert
a
networks is given in Fig. 7b. In both cases τ clearly grows logarithmically,
τ = A + Blogk, (12)
for large k. In the case of the Apollonian network, one can even derive this behavior
analytically as follows. In order to communicate between two vertices of the n-th generation,
one needs up to n steps, which leads to τ ∝ n. Since for the Apollonian network one has[40]
k = 3 × 2n−1 , one immediately obtains that τ ∝ logk.
For the Apollonian network all neighbors of a given victim are connected in a closed path
surrounding the victim, as can be seen from the inset of Fig. 7a, yielding f = 1. This stresses
the fact that the spread factor f is rather different from the clustering coefficient which in this
case is C = 0.828 [40].
Next, we will show that for these two features to appear one needs the existence of
degree correlations between connected nodes, as usually observed in real empirical networks.
In Fig. 8 we plot the results of gossip spreading on an empirical set of networks extracted
from survey data[41] in 84 U.S. schools. Here, the logarithmic growth of τ with k, shown in
Fig. 8a, follows the same dependence of the average degree knn of the nearest neighbors[42],
as illustrated in the inset. As in the case of the BA networks, we also find for the schools a
characteristic degree k0 for which f and therefore the gossip spreading is smallest. The inset
of Fig. 8b, however, gives clear evidence that the school networks are not scale-free. Since
the same optimal degree appears in Barab´ si-Albert networks, one argues that the existence
a
of this optimal number is not necessarily related to the degree distribution of the network, but
13. Social networks: models and measures 13
1
14
0.1
5
12 P(k) 0.01 0.9
knn
10 0.001
4 0.8
8 0.0001
τ 1 10
6 10 20 30
0.7 f
3 k
k
0.6
2
0.5
(a) (b)
1 0.4
1 10 1 k0 10
k k
Figure 8. Gossip propagation on a real friendship network of American students [41] averaged
over 84 schools. In (a) we show the spreading time τ and, in the inset, the average degree of
neighbors of nodes with degree k. In (b) the spread factor f , both as a function of degree k. In
the inset of (b) we see the degree distribution P (k).
rather to the degree correlations.
However, the relation between degree correlations, measured by knn , and the logarithmic
behavior of the spreading time is not straightforward. While in the empirical network we
found the same distribution for both knn and τ , in BA and APL networks knn follows a
power-law with k (not shown). As for the spread factor f , a mean field approach can
be derived, yielding an f -rate equation which depends in general on P (k) and two and
three-point correlations of the degree. In the case of uncorrelated networks, two and three-
point correlations reduce to simple expressions of the moments of the degree distribution.
Therefore, f is independent of the degree, similarly to what is observed for the density of
particles as derived by Catanzaro et al[43] in diffusion-annihilation processes on complex
networks. For correlated networks, as the empirical network here studied, the analytical
approach is not straightforward and will be presented elsewhere.
Another quantity of interest is the distribution P (τ ) of spreading times, which clearly
decays exponentially for the Apollonian network, as illustrated in Fig. 9a. This behavior
can be also obtained analytically by considering that P (τ )dτ = P (k)dk and using Eq. (12)
together with the degree distribution, P (k) ∝ k −γ , to obtain
P (τ ) ∝ eτ (1−γ)/B , (13)
for large k. The slope in Fig. 9a is precisely (1 − γ)/B = −0.17 using B from Fig. 7a and
γ = 2.58 from Ref. [40]. For the school network, P (τ ) follows also an exponential decay for
large τ , but with a 3.5 times smaller characteristic decay time, and has a maximum for small
τ , as seen in Fig. 9b (circles). Compared to the P (τ ) of the Barab´ si-Albert network with
a
m = 9 (solid line), the shapes are similar but the Barab´ si-Albert case is slightly shifted to
a
the right, due to the larger minimal number of connections.
Many other regimes of gossip and of propagation phenomena can be also addressed with
these two quantities. Namely, a more realistic scenario could be addressed by enabling each
14. Social networks: models and measures 14
1 1
(a)
P(τ) 0.1
0.1 0.01
0.001
0.0001
0.01
(b)
1e-05
1 3 5 7 9 1 3 5 7 9 11
τ τ
Figure 9. Distribution P (τ ) of spreading times τ for (a) the APL network of 8 generations,
and (b) the real school network (circles) and the BA network with m = 9 and N = 1000 (solid
line).
node to transfer information with a probability 0 ≤ p ≤ 1. Further, the assumption that
the person to which a gossip did not spread at the first attempt, will never get it, yields a
regime similar to percolation conditional to the neighborhood of the victim. Differently, if at
each time-step the neighbors which already know the gossip repeatedly try to spread it to the
common friends, one observes the same value of f measured for q = 1, and the spreading
time scales as τ ′ ∼ τ /q, where τ is measured for q = 1. Finally, other possible regimes
comprehend the situation where the gossip spreading over strangers, i.e. over nodes which are
not directly connected to the victim. Such cases are being studied in detail and results will be
presented elsewhere[44].
5. Discussion and conclusions
In this paper we presented and developed recent achievements in social network research,
concerning the modeling of empirical networks, and specific mathematical tools to address
their structure and dynamical processes on them.
Concerning the modeling of empirical networks, we described briefly a recent approach
based on a system of mobile agents. Further developments were given, namely in what
concerns the analytical expression which fits the typical degree distributions observed in
empirical social networks. We gave evidence that such distributions follow a Brody
distribution which depends on a single parameter that scales with the average degree of the
network. A question which now remains to be answered is how to derive such distribution
from an analytical and meaningful approach.
Showing that the usual clustering coefficient is, in general, inappropriate when
addressing the clustering properties of social networks, we described a suitable measure to
access these properties and presented its additional applications for estimating the distribution
of cycles of higher order. This additional clustering coefficient was also put in a general
framework with different other higher-order coefficients, that could be useful for particular
15. Social networks: models and measures 15
situations of multipartite networks. An expansion combining all possible coefficients was
also proposed, motivated by previous works[4], which depends only on the degree distribution
and degree-degree correlations. However, computational effort to compute such coefficients
increases exponentially with their order and therefore it is not yet clear how useful such an
expansion may be.
Finally, to study dynamical processes in social networks, in particular the propagation of
information, two simple measures were introduced. Namely, a spread factor, which measures
the maximal relative size of the neighborhood reached, when the information starts from a
local source (node), and a spreading time, which gives the number of sufficient steps to reach
such maximal size. This two measures gave rise to introduce a minimal model for gossip
propagation, which can be seen as a particular model of opinions. Within this specific model,
the spread factor was found to be minimized by a particular non-trivial degree of the source,
which is related to the degree-degree correlations arising in the network. If such possibility
of minimizing the danger of being gossiped can be tested in a real situation and which other
implications these findings have in other situations - e.g. in internet virus propagation - remain
open questions for forthcoming studies.
Acknowledgments
The authors thank M.C. Gonz´ lez, J.S. Andrade Jr., L. da Silva and O. Dur´ n for useful
a a
discussions. We thank the Deutsche Forschungsgemeinschaft, the Max Planck Price and the
Volkswagenstiftung.
References
[1] P. Ball, Physica A 314 1-14 (2002).
[2] P. Ball, Complexus 1 190-206 (2003).
[3] R. Albert and A.-L. Barab´ si, Rev. Mod. Phys. 74 47-97 (2002).
a
[4] M.E.J. Newman, SIAM Rev. 45 167-256 (2003).
[5] S.N. Dorogovtsev and J.F.F. Mendes, Adv. Phys. 51 1079-1187 (2002).
[6] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwanga, Physics Reports 424(4-5), 175-308
(2006).
[7] B. Bollob´ s, Modern Graph Theory (Springer, New York, 1998).
a
[8] M.E.J. Newman, Proc. Natl. Acad. Sci. 98, 404 (2001).
[9] M.E.J. Newman, D.J. Watts and S.H. Strogatz, PNAS 99(Supp. 1) 2566-2572 (2002).
[10] P.L. Krapivsky and S. Redner, Phys. Rev. Lett. 90, 238701 (2003).
[11] A. Rogers, Phys. Rev. Lett. 90, 158103 (2003).
[12] L.C. Freeman, The Development of Social Network Analysis (Vancouver, Canada, 2004).
[13] M.C. Gonz´ lez, P.G. Lind and H.J. Herrmann, Phys. Rev. Lett. 96 088702 (2006).
a
[14] M.C. Gonz´ lez, P.G. Lind and H.J. Herrmann, Eur. Phys. J. B 49, 371-376 (2006).
a
[15] M.E.J. Newman and J. Park, Phys. Rev. E 68 036122 (2003).
[16] D.J. Watts and S.H. Strogatz, Nature 393, 440-442 (1998).
[17] D.J. Daley and D.G. Kendall, Nature 204, 1118 (1964).
[18] A. Gr¨ nlund, P. Holme, Phys. Rev. E 70 036108 (2004).
o
[19] R. Toivonen, J.-P. Onnela, J. Saram¨ ki, J. Hyv¨ nen, K. Kaski, physics/0601114 (2006).
a o
[20] E.M. Jin, M. Girvan, M.E.J. Newman, Phys. Rev. E 64, 046132 (2001).
16. Social networks: models and measures 16
[21] P.G. Lind, M.C. Gonz´ lez and H.J. Herrmann, Phys. Rev. E 72, 056127 (2005); cond-mat/0504241.
a
[22] J. Davidsen, H. Ebel, and S. Bornholdt, Phys. Rev. Lett. 88, 128701 (2002).
[23] E. Eisenberg and E.Y. Levanon, Phys. Rev. Lett. 91, 138701 (2003).
[24] T.A. Brody, Lett. Nuovo Cimento 7 482 (1973).
[25] L.A.N. Amaral, A. Scala, M. Barth´ l´ my, H.E. Stanley, Proc. Natl. Acad. Sci. 21, 11149 (2000).
ee
[26] S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 62, 1842-1845 (2000).
[27] M.C. Gonz´ lez, P.G. Lind and H.J. Herrmann, Physica D 224 137 (2006).
a
[28] P.N. McGraw and M. Menzinger, cond-mat/0501663 (2005).
[29] P.F. Stadler, A. Wagner and D.A. Fell, Adv. Complex Sys. 4, 207-226 (2001).
[30] M.E.J. Newman, Social Networks 25, 83-95 (2003).
[31] P. Holme, C.R. Edling and F. Liljeros, Social Networks 26, 155 (2004).
[32] P. Holme, F. Liljeros, C.R. Edling and B.J. Kim, Phys. Rev. E 68, 056107 (2003).
[33] R. Guimer` , X. Guardiola, A. Arenas, A. D´az-Guilera, D. Streib and L.A.N. Amaral, “Quantifying the
a ı
creation of social capital in a digital community”, private report.
[34] G. Caldarelli, R. Pastor-Satorras and A. Vespignani, Eur. Phys. J. B 38, 183-186 (2004).
[35] A. V´ zquez, J.G. Oliveira and A.-L. Barab´ si, Phys. Rev. E 71, 025103(R) (2005).
a a
[36] S.N. Dorogovtsev, A.V. Goltsev and J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002).
[37] K. Klemm and P.F. Stadler, Phys. Rev. E 73, 025101(R) (2006); condmat/0506493.
[38] A. Trusina, M. Rosvall and K. Sneppen, Phys. Rev. Lett. 94, 238701 (2005).
[39] H.J. Herrmann, D.C. Hong and H.E. Stanley, J.Phys.A 17, L261 (1984).
[40] J.S. Andrade Jr., H.J. Herrmann, R.F.S. ANDRADE, L.R. da Silva, Phys. Rev. Lett. 94, 018702 (2005).
[41] Add Health program designed by J.R. Udry, P.S. Bearman and K.M. Harris funded by National Institute of
Child and Human Development (PO1-HD31921).
[42] M. Cantazaro, M. Bogu˜ a and R. Pastor-Satorras, Phys. Rev. E 71 027103 (2005).
n
[43] M. Cantazaro, M. Bogu˜ a and R. Pastor-Satorras, Phys. Rev. E 71, 056104 (2005).
n
[44] P.G. Lind, J.S. Andrade Jr., L.R. da Silva, H.J. Herrmann, in preparation (2007).