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{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|social networking service|the 2010 movie|The Social Network}}{{other uses|Social network (disambiguation)}}
{{about|the theoretical concept as used in the social and behavioral sciences|social networking sites|social networking service|the 2010 movie|The Social Network}}{{other uses|Social network (disambiguation)}}
{{Sociology}}
{{Sociology}}
A '''social network''' is stupid [[social structure]] made up of a set of actors (such as individuals or organizations) and the dyadic ties between these actors (such as relationships, connections, or interactions). A social network perspective is employed to model the structure of a social group, how this structure influences other variables, or how structures change over time.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=978521382694 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1-27 |publisher=Cambridge University Press}}</ref> The study of these structures uses methods in [[social network analysis]] to identify influential nodes, local and global structures, and network dynamics. Social networks are distinct from [[information network|information]], [[biological network|biological]], or [[electrical network]]s, but theories and methods generalizing to all of these [[complex network]]s are studied in the field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892-895 |doi=10.1126/science.1165821 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca}}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1-20 |isbn=978-0-521-19533-1}}</ref>
A '''social network''' is a [[social structure]] made up of a set of actors (such as individuals or organizations) and the dyadic ties between these actors (such as relationships, connections, or interactions). A social network perspective is employed to model the structure of a social group, how this structure influences other variables, or how structures change over time.<ref name="WF94CH1">{{cite book|last1=Wasserman |first1=Stanley |last2=Faust |first2=Katherine |year=1994 |title=Social Network Analysis: Methods and Applications |isbn=978521382694 |chapter=Social Network Analysis in the Social and Behavioral Sciences |pages=1-27 |publisher=Cambridge University Press}}</ref> The study of these structures uses methods in [[social network analysis]] to identify influential nodes, local and global structures, and network dynamics. Social networks are distinct from [[information network|information]], [[biological network|biological]], or [[electrical network]]s, but theories and methods generalizing to all of these [[complex network]]s are studied in the field of [[network science]].<ref>{{cite journal|journal=Science |year=2009 |volume=323 |number=5916 |pages=892-895 |doi=10.1126/science.1165821 |title=Network Analysis in the Social Sciences |first1=Stephen P. |last1=Borgatti |first2=Ajay |last2=Mehra |first3=Daniel J. |last3=Brass |first4=Giuseppe |last4=Labianca}}</ref><ref name="EK">{{cite book|title=Networks, Crowds, and Markets: Reasoning about a Highly Connected World |first1=David |last1=Easley |first2=Jon |last2=Kleinberg |publisher=Cambridge University Press |year=2010 |chapter=Overview |pages=1-20 |isbn=978-0-521-19533-1}}</ref>


Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Jacob Moreno]] is credited with developing the first ''sociograms'' in the 1930s to study interpersonal relationships as structures in which people were points and the relationships between them were drawn as connecting lines. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=1-59457-714-5 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref>
Social networks and the analysis of them is an inherently [[Interdisciplinarity|interdisciplinary]] academic field which emerged from [[social psychology]], [[sociology]], [[statistics]], and [[graph theory]]. [[Jacob Moreno]] is credited with developing the first ''sociograms'' in the 1930s to study interpersonal relationships as structures in which people were points and the relationships between them were drawn as connecting lines. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s.<ref name="WF94CH1"/><ref name="Freeman History">{{cite book|last=Freeman |first=Linton |year=2004 |publisher=Empirical Press |isbn=1-59457-714-5 |title=The Development of Social Network Analysis: A Study in the Sociology of Science}}</ref>

Revision as of 11:49, 20 March 2012

A social network is a social structure made up of a set of actors (such as individuals or organizations) and the dyadic ties between these actors (such as relationships, connections, or interactions). A social network perspective is employed to model the structure of a social group, how this structure influences other variables, or how structures change over time.[1] The study of these structures uses methods in social network analysis to identify influential nodes, local and global structures, and network dynamics. Social networks are distinct from information, biological, or electrical networks, but theories and methods generalizing to all of these complex networks are studied in the field of network science.[2][3]

Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships as structures in which people were points and the relationships between them were drawn as connecting lines. These approaches were mathematically formalized in the 1950s and theories and methods of social networks became pervasive in the social and behavioral sciences by the 1980s.[1][4]

Overview

Evolution graph of a social network: Barabási model.

A Social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies (social units, see differentiation). The term is used to describe a social structure determined by such interactions. The ties (sometimes called edges, links, or connections) in the structure are called "nodes". The nodes through which any given social unit connects represent the convergence of the various social contacts of that unit. Many kinds of relationships may form the "network" between such nodes, but interpersonal "bridges" are a defining characteristic of social networks. Social network approaches are useful for modeling and explaining many social phenomena. The theoretical approach is, necessarily, relational. An axiom of the social network approach to understanding social interaction is that social phenomena should be primarily conceived and investigated through the properties of relations between and within units, instead of the properties of these units themselves. Thus, one common criticism of social network theory is that individual agency is essentially ignored,[5] although this is not the case in practice (see agent-based modeling). Precisely because many different types of relations, singular or in combination, form into a network configuration, network analytics are useful to a broad range of research enterprises. In social science, these fields of study include, but are not limited to anthropology, biology, communication studies, economics, geography, information science, organizational studies, social psychology, sociology, and sociolinguistics. Scholars in these and other areas have used the idea of "social network" loosely for almost a century to connote complex sets of relationships between members of social units across all scales of analysis, from the local to the global as well as the scale-free.

Background

Some of the ideas of social network theory are found in writings going back to the ancient Greeks. In the late 1800s, both Émile Durkheim and Ferdinand Tönnies foreshadow the idea of social networks in their theories and research of social groups. Tönnies argued that social groups can exist as personal and direct social ties that either link individuals who share values and belief (Gemeinschaft, German, commonly translated as "community") or impersonal, formal, and instrumental social links (Gesellschaft, German, commonly translated as "society").[6] Durkheim gave a non-individualistic explanation of social facts arguing that social phenomena arise when interacting individuals constitute a reality that can no longer be accounted for in terms of the properties of individual actors.[7] Georg Simmel, writing at the turn of the twentieth century, pointed to the nature of networks and the effect of network size on interaction and examined the likelihood of interaction in loosely-knit networks rather than groups.[8]

Major developments in the field can be seen in the 1930s by several groups in psychology, anthropology, and mathematics working independently.[5] In psychology, in the 1930s, Jacob L. Moreno began systematic recording and analysis of social interaction in small groups, especially classrooms and work groups (see sociometry). In anthropology, the foundation for social network theory is the theoretical and ethnographic work of Bronislaw Malinowski,[9] Alfred Radcliffe-Brown,[10] and Claude Lévi-Strauss.[11] A group of social anthropologists associated with Max Gluckman and the Manchester School, including John A. Barnes,[12] J. Clyde Mitchell and Elizabeth Bott Spillius,[13][14] often are credited with performing some of the first fieldwork from which network analyses were performed.[5] In sociology, the early (1930s) work of Talcott Parsons set the stage for taking a relational approach to understanding social structure.[15][16] Later, drawing upon Parsons' theory, the work of sociologist Peter Blau provides a strong impetus for analyzing the relational ties of social units with his work on social exchange theory.[17][18][19] By the latter 1900s, a growing number of scholars worked to combine the different tracks and traditions. One group consisted of sociologist Harrison White and his students at the Harvard University Department of Social Relations. Mark Granovetter[20] and Barry Wellman[21] are among the former students of White who elaborated and championed the analysis of social networks.

Levels of analysis

Self-organization of a network, based on Nagler, Levina, & Timme, (2011)[22]

In general, social networks are self-organizing, emergent, and complex, such that a globally coherent pattern appears from the local interaction of the elements that make up the system.[23][24] These patterns become more apparent as network size increases. However, a global network analysis of, for example, all interpersonal relationships in the world—or even one global region—is not feasible and is likely to contain so much information as to be uninformative. Thus, social networks are analyzed by the number and type of relationships relevant to the researcher's theoretical question. Such analyses can be delimited according to theory such that a specific set of persons whose relationships are to be analyzed fall within a specific scale or, again according to theory, may be targeted to analyzing specific types of relationships and be scale-free. Although levels of analysis are not necessarily mutually exclusive, there are three general levels into which networks may fall: micro-level, meso-level or middle-range, and macro-level.

Micro level

At the micro-level, social network research typically begins with an individual, snowballing as social relationships are traced, or may begin with a small group of individuals in a particular social context.

Social network diagram, micro-level.

Actor level

The smallest unit of analysis in a social network is an individual in their social setting, i.e., an "actor". Actor-centered network analysis often centers on network characteristics such as centrality, prestige and roles such as isolates, liaisons, and bridges. Such analyses, sometimes referred to as ego-centric or ego networks, are most commonly used in the fields of psychology or social pyschology, ethnographic kinship analysis or other genealogical studies of relationships between individuals.

Dyadic level

Simply put, a dyad is a social relationship between two individuals. Network research on dyads may concentrate on structure of the relationship, social equality, and tendencies toward reciprocity.

Triadic level

Add one individual to a dyad, and you have a triad. Research at this level may concentrate on factors such as balance and transitivity, as well as social equality and tendencies toward reciprocity.

Subset level

Subset levels of network research problems begin at the micro-level, but may crossover into the meso-level of analysis. Subset level research may focus on distance and reachability, cliques, cohesive subgroups, or other group action, group actions or behavior.

Meso level

In general, meso-level theories begin with a population size that falls between the micro- and macro-levels. However, meso-level may also refer to analyses that are specifically designed to reveal connections between micro- and macro-levels. Meso-level networks are low density and may exhibit causal processes distinct from interpersonal micro-level networks. [25]

Social network diagram, meso-level

Organizations

Formal organizations are social groups that distribute tasks for a collective goal. There are a variety of legal types of organizations, including: corporations, governments, non-governmental organizations, international organizations, armed forces, charities, not-for-profit corporations, partnerships, cooperatives, and universities. A hybrid organization is a body that operates in both the public sector and the private sector, simultaneously fulfilling public duties and developing commercial market activities. As a result the hybrid organization becomes a mixture of a government and a corporate organization. Network research on organizations may focus on either intra-organizational or inter-organizational ties in terms of formal or informal relationships.

Randomly-distributed networks

Exponential random graph models of social networks became state-of-the-art methods of social network analysis in the 1980s. This framework has the capacity to represent social-structural effects commonly observed in many human social networks, including general degree-based structural effects commonly observed in many human social networks as well as reciprocity and transitivity, and at the node-level, homophily and attribute-based activity and popularity effects, as derived from explicit hypotheses about dependencies among network ties. Parameters are given in terms of the prevalence of small subgraph configurations in the network and can be interpreted as describing the combinations of local social processes from which a given network emerges. These probability models for networks on a given set of actors allow generalization beyond the restrictive dyadic independence assumption of micro-networks, allowing models to be built from theoretical structural foundations of social behavior. [26]

Examples of a random network and a scale-free network. Each graph has 32 nodes and 32 links. Note the "hubs" in the scale-free diagram (on the right).

Scale-free networks

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. In network theory a scale-free ideal network is a random network with a degree distribution that unravels the size distribution of social groups. [27] Specific characteristics of scale-free networks vary with the theories and analytical tools used to create them, however, in general, scale-free networks have some common characteristics. One notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and may serve specific purposes in their networks, although this depends greatly on the social context. Another general characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law.[28]

The Barabási model of network evolution shown above is an example of a scale-free network.

Macro level

Rather than tracing interpersonal interactions, macro-level analyses generally trace the outcomes of interactions, such as economic or other resource transfer interactions over a large population.

Diagram: section of a large-scale social network

Large-scale networks

Large-scale network is a term somewhat synonymous with "macro-level" as used, primarily, in social and behavioral sciences, in economics. Originally, the term was used extensively in the computer sciences (see large-scale network mapping).

Complex networks

Most larger social networks display features of social complexity, which involves substantial non-trivial features of network topology, with patterns of complex connections between elements that are neither purely regular nor purely random (see, complexity science, dynamical system and chaos theory), as do biological, and technological networks. Such complex network features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of agency-directed networks these features also include reciprocity, triad significance profile (TSP, see network motif), and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.[29]

Theory clusters

Communications

Communication Studies are often considered a part of both the social sciences and the humanities, drawing heavily on fields such as sociology, psychology, anthropology, information science, biology, political science, and economics as well as rhetoric, literary studies, and semiotics. Many communications concepts describe the transfer of information from one source to another, and can thus be conceived of in terms of a network.

Community

In J.A. Barnes' day, a "community" referred to a specific geographic location and studies of community ties had to do with who talked, associated, traded, and attended church with whom. Today, however, there are extended "online" communities developed through telecommunications devices and social network services. Such devices and services require extensive and ongoing maintenance and analysis, often using network science methods. Community development studies, today, also make extensive use of such methods.

Complex Networks

Complex networks require methods specific to modelling and interpreting social complexity and complex adaptive systems, including techniques of dynamic network analysis.

Criminal networks

In criminology and urban sociology, much attention has been paid to the social networks among criminal actors. For example, Andrew Papachristos has studied gang murders as a series of exchanges between gangs. Murders can be seen to diffuse outwards from a single source, because weaker gangs cannot afford to kill members of stronger gangs in retaliation, but must commit other violent acts to maintain their reputation for strength.

Diffusion of innovations

Diffusion of ideas and innovations studies focus on the spread and use of ideas from one actor to another or one culture and another, and seek to explain why some become "early adopters" of ideas and innovations.

Demography

In demography, the study of social networks has led to new sampling methods for estimating and reaching populations that are hard to enumerate (for example, homeless people or intravenous drug users.) For example, respondent driven sampling is a network-based sampling technique that relies on respondents to a survey recommending further respondents.

Economic sociology

The field of sociology focuses almost entirely on networks of outcomes of social interactions. More narrowly, economic sociology considers behavioral interactions of individuals and groups through social capital and social "markets". Sociologists, such as Mark Granovetter, have developed core principles about the interactions of social structure, information, ability to punish or reward, and trust that frequently recur in their analyses of political, economic and other institutions. Granovetter examines how social structures and social networks can affect economic outcomes like hiring, price, productivity and innovation and describes sociologists’ contributions to analyzing the impact of social structure and networks on the economy.[30]

Health care

Analysis of social networks is increasingly incorporated into heath care analytics, not only in epidemological studies but also in models of patient communication and education, disease prevention, mental health diagnosis and treatment, and in the study of health care organizations and systems.[31]

Human ecology

Human ecology is an interdisciplinary and transdisciplinary study of the relationship between humans and their natural, social, and built environments. The scientific philosophy of human ecology has a diffuse history with connections to geography, sociology, psychology, anthropology, zoology, and natural ecology.[32][33]

Language/Linguistics

Studies of language and lingustics, particularly evolutionary linguistics, focus on the development of linguistic forms and transfer of changes, sounds or words, from one language system to another through networks of social interaction. Social networks are also important in language shift, as groups of people add and/or abandon languages to their repertoire.

Organizational Studies

Research studies of formal or informal organizational relationships, organizational communication, economics, economic sociology, and other resource transfers.

Social capital

Social capital is a sociological concept which refers to the value of social relations and the role of cooperation and confidence to achieve positive outcomes. The term refers to the value one can get from their social ties. For example, newly arrived immigrants can make use of their social ties to established migrants to acquire jobs they may otherwise have trouble getting (e.g., because of lack of knowledge of language).

Structural Holes

Structural holes refer to the absence of ties between two parts of a network. Finding and exploiting a structural hole can give an entrepreneur a competitive advantage. For example, a unique combination of business ties can allow them to combine expertise from two otherwise disconnected fields to create novel products. They can also act as brokers, reaping a reward from mediating trade between the communities. This concept was developed by sociologist Ronald Burt, and is sometimes referred to as an alternate conception of social capital (above).

See also

References

  1. ^ a b Wasserman, Stanley; Faust, Katherine (1994). "Social Network Analysis in the Social and Behavioral Sciences". Social Network Analysis: Methods and Applications. Cambridge University Press. pp. 1–27. ISBN 978521382694. {{cite book}}: Check |isbn= value: length (help)
  2. ^ Borgatti, Stephen P.; Mehra, Ajay; Brass, Daniel J.; Labianca, Giuseppe (2009). "Network Analysis in the Social Sciences". Science. 323 (5916): 892–895. doi:10.1126/science.1165821.
  3. ^ Easley, David; Kleinberg, Jon (2010). "Overview". Networks, Crowds, and Markets: Reasoning about a Highly Connected World. Cambridge University Press. pp. 1–20. ISBN 978-0-521-19533-1.
  4. ^ Freeman, Linton (2004). The Development of Social Network Analysis: A Study in the Sociology of Science. Empirical Press. ISBN 1-59457-714-5.
  5. ^ a b c Scott, John P. (2000). Social Network Analysis: A Handbook (2nd edition). Thousand Oaks, CA: Sage Publications.
  6. ^ Tönnies, Ferdinand (1887). Gemeinschaft und Gesellschaft, Leipzig: Fues's Verlag. (Translated, 1957 by Charles Price Loomis as Community and Society, East Lansing: Michigan State University Press.)
  7. ^ Durkheim, Emile (1893). De la division du travail social: étude sur l'organisation des sociétés supérieures, Paris: F. Alcan. (Translated, 1964, by Lewis A. Coser as The Division of Labor in Society, New York: Free Press.)
  8. ^ Simmel, Georg (1908). Soziologie, Leipzig: Duncker & Humblot.
  9. ^ Malinowski, Bronislaw (1913). The Family Among the Australian Aborigines: A Sociological Study. London: University of London Press.
  10. ^ Radcliffe-Brown, Alfred Reginald (1930) The social organization of Australian tribes. Sydney, Australia: University of Sydney Oceania monographs, No.1.
  11. ^ Lévi-Strauss, Claude ([1947]1967). Les structures élémentaires de la parenté. Paris: La Haye, Mouton et Co. (Translated, 1969 by J. H. Bell, J. R. von Sturmer, and R. Needham, 1969, as The Elementary Structures of Kinship, Boston: Beacon Press.)
  12. ^ Barnes, John (1954). "Class and Committees in a Norwegian Island Parish." Human Relations, (7): 39-58.
  13. ^ Freeman, Linton C. and Barry Wellman (1995). "A note on the ancestoral Toronto home of social network analysis." Connections, 18(2): 15-19.
  14. ^ Savage, Mike (2008). "Elizabeth Bott and the formation of modern British sociology." The Sociological Review, 56(4): 579–605.
  15. ^ Parsons, Talcott ([1937] 1949). The Structure of Social Action: A Study in Social Theory with Special Reference to a Group of European Writers. New York, NY: The Free Press.
  16. ^ Parsons, Talcott (1951). The Social System. New York, NY: The Free Press.
  17. ^ Blau, Peter (1956). Bureaucracy in Modern Society. New York: Random House, Inc.
  18. ^ Blau, Peter (1960). "A Theory of Social Integration." The American Journal of Sociology, (65)6: 545-556 , (May).
  19. ^ Blau, Peter (1964). Exchange and Power in Social Life.
  20. ^ Granovetter, Mark (2007). "Introduction for the French Reader," Sociologica 2: 1–8
  21. ^ Wellman, Barry (1988). "Structural analysis: From method and metaphor to theory and substance." Pp. 19-61 in B. Wellman and S. D. Berkowitz (eds.) Social Structures: A Network Approach, Cambridge, UK: Cambridge University Press.
  22. ^ Nagler, Jan, Anna Levina and Marc Timme (2011). "Impact of single links in competitive percolation." Nature Physics, 7: 265-270.
  23. ^ Newman, Mark, Albert-László Barabási and Duncan J. Watts (2006). The Structure and Dynamics of Networks (Princeton Studies in Complexity). Oxford: Princeton University Press.
  24. ^ Wellman, Barry (2008). "Review: The development of social network analysis: A study in the sociology of science." Contemporary Sociology, 37: 221-222.
  25. ^ Hedström,Peter, Rickard Sandell, and Charlotta Stern (2000).”Mesolevel Networks and the Diffusion of Social Movements: The Case of the Swedish Social Democratic Party.”American Journal of Sociology, 106(1): 145–72.
  26. ^ Cranmer, Skyler J. and Bruce A. Desmarais (2011). "Inferential Network Analysis with Exponential Random Graph Models." Political Analysis, 19(1): 66-86.
  27. ^ Moreira, André A., Demétrius R. Paula, Raimundo N. Costa Filho, José S. Andrade, Jr. (2006). "Competitive cluster growth in complex networks". Physical Review E. 73 (6). arXiv:cond-mat/0603272. doi:10.1103/PhysRevE.73.065101.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  28. ^ Barabási, Albert-László (2003). Linked: how everything is connected to everything else and what it means for business, science, and everyday life. New York, NY: Plum.
  29. ^ Strogatz, Steven H. (2001). "Exploring complex networks." Nature, 410: 268-276.
  30. ^ Granovetter, Mark (2005). "The Impact of Social Structure on Economic Outcomes." The Journal of Economic Perspectives, 19(1): 33-50
  31. ^ Levy, Judith and Bernice Pescosolido (2002). Social Networks and Health. Boston, MA: JAI Press.
  32. ^ Crona, Beatrice and Klaus Hubacek (eds.) (2010). "Special Issue: Social network analysis in natural resource governance." Ecology and Society, 48.
  33. ^ Ernstson, Henrich (2010). "Reading list: Using social network analysis (SNA) in social-ecological studies." Resilience Science

Further reading

  • Wellman, Barry; Berkowitz, S.D. (1988). Social Structures: A Network Approach. Structural Analysis in the Social Sciences. Cambridge University Press. ISBN 0-521-24441-2.
  • Scott, John (1991). Social Network Analysis: a handbook. SAGE. ISBN 978-0-7619-6338-7.
  • Wasserman, Stanley; Faust, Katherine (1994). Social Network Analysis: Methods and Applications. Structural Analysis in the Social Sciences. Cambridge University Press. ISBN 978-0-521-38269-4.
  • Barabási, Albert-László (2003). Linked: How everything is connected to everything else and what it means for business, science, and everyday life. Plum. ISBN 978-0-452-28439-5.
  • Freeman, Linton C. (2004). The Development of Social Network Analysis: A Study in the Sociology of Science. Empirical Press. ISBN 1-59457-714-5.
  • Barnett, George A. (2011). Encyclopedia of Social Networks. SAGE. ISBN 978-1-4129-7911-5.
  • Kadushin, Charles (2012). Understanding Social Networks: Theories, Concepts, and Findings. Oxford University Press. ISBN 978-0-19-537946-4.

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