Social Networks: an Overview Jan Treur Agent Systems Research Group, VU University Amsterdam De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands Email:
[email protected] URL: http://www.few.vu.nl/~treur
Abstract This document provides a brief and mostly nontechnical overview of a number of concepts from the literature on social networks. It is used as one of the introductory texts in course Behaviour Dynamics in Social Networks in the Master programme on Artificial Intelligence, specialisation Human Ambience.
1 Introduction In this document a brief overview is given of different types of networks that often are distinguished. When persons function in a social context they interact with a number of other persons they know. These persons themselves also interact with a number of persons. And so on and on. When each person is depicted by a node and for each of these interactions arcs between te nodes are drawn, this results in a social network structure. An example of this based on Facebook connections is shown in Fig. 1. Note that as such arcs indicate that interaction takes place, and interaction in principle means that persons affect each other, from a dynamical perspective they can also be considered relations that represent mutual causal effects on certain agent states. Another example, based on telephone calls is shown in Fig. 2.
Fig. 1. Example social network based on Facebook connections
1
Fig. 2. Example social network based on telephone calls
The area of Social Networks has already a longer tradition of more than 40 years, starting in Social Science, but more recently has gradually developed stronger and moved into other disciplines as well, such as Biology, Neuroscience, Mathematics, Physics, Economics, Informatics, Artificial Intelligence, and Web Science; see, for
2
example (Boccalettia, Latorab, Morenod, Chavez, Hwanga, 2006; Valente, 2010). As in this widened scope not always a social context is involved, nowadays the subject is often just named Networks. Matrix representation of a network A network can be represented by a square table or matrix for the pairs of nodes, with a value 1 for connected nodes and 0 for nodes not connected, as shown in Table 1 (note that for convenience the 0’s have been left out). Table 1 Matrix representation of a network A A B C D E F G H I J K L
B 1
1 1 1 1
1 1
C 1
D 1
E 1 1 1
1
F
H
I
1 1 1
1
1
1 1
1 1 1
J
K
1 1
1
L
1
1
1
G
1 1
1 1 1
1 1
1 1
1
1 1
1
1 1
1
1
1 1
degree 4 3 3 4 7 4 5 4 2 4 5 3
This specific matrix represents the example network shown in Figure 3. The last column shows for each node the sum of the values in the row. This represents the number of connections of the node, also called the degree of the node. Note that in this specific example the connections are assumed bidirectional; therefore the matrix is symmetric in its diagonal. Networks with non-bidirectional connections show non-symmetric matrices. B L
F
A E C
J G
K
D H I
Fig. 3. Example network represented by the matrix in Table 1
To characterise different types of networks, over the years some criteria have been identified:
The length of the shortest path between two given nodes in the network The number of connections of a given node (called the degree), and how that varies over the network The occurrence of clusters of tightly connected groups of nodes (subcommunities) The variation of strengths of the connections over the network
Each of these criteria will be briefly introduced.
3
2 Shortest Paths and Small World Networks Examples of the paths in a network are the following. In the World Wide Web consider a Web page is connected to another one when a link to this other Webpage occurs. Then a path from one Webpage to another one is based on a number of mouseclicks bringing you from the one to the other. The shortest path from one Webpage to another one is the minimal number of mouseclicks needed to get from the one to the other. For another example, consider the networks of persons who are connected when they know each other: they are acquaintances or more (e.g., at least once they have spoken with each other). Suppose a k-th order contact is a contact with shortest path of length k. You can wonder, for example, which order of contact you have with the queen, or with Barack Obama. In general it may be expected that the shortest paths between two arbitrary nodes in a larger network may still be very long, unless the nodes in the network have many connections. For example, it may be expected that the number of steps needed to connect you to an arbitrary person anywhere can be very large, as every person knows only a limited number of other persons. However, in the social networks in the real world shortest paths seem much shorter than one would expect. The notion ‘small world’ is sometimes used in relation to the experience when you find out that a person you meet for the first time knows somebody you know as well; this would make k = 2. Usually this is felt as a surprise and not seldom expressed by something like ‘What a small world!’. Small World Networks The term small world network has been adopted to indicate a network in which
the average number of (direct) connections per node is low but still the shortest paths are short
Of course, when practically all nodes in a network are directly connected, and therefore the number of connections per node is high, then automatically the shortest paths will be very short, and even 1 for a fully connected network (everybody knows everybody). It has been found that many networks in the real world that have been analysed fall in the class of small world networks. This does not only concern social networks, but also, for example, in Biology metabolic networks, in Neuroscience networks in the brain, and in Web Science networks on the World Wide Web. For the human society the conjecture has even been put forward that any two persons on earth are in a k-th order contact with k at most 6. In Fig. 4 an example of a small world network is shown.
Fig. 4. Example of a small world network on Facebook
4
Random networks Random networks satisfy the criterion that there is a fixed number p in the interval [0, 1] such that for every pair of nodes they are connected with probability p. Fig. 5 shows examples of random networks for p = 0.1, 0.25, and 0.5, respectively. Usually this leads to a high number of connections per node, especially in large networks (see also Fig. 7). Therefore in principle they do not satisfy the criteria for small world networks.
Fig. 5. Examples of random networks for different link probabilities
Fully Connected Network A random network with p = 1 is called a fully connected network: all pairs of nodes are connected. For an example, see Fig. 6.
Fig. 6. Example of a fully connected network
3 Distribution of the Number of Connections per Node and Scale-Free Networks In the case of a random network with n nodes the number of connections per node, also called degree, has a distribution with top around p (n-1). More precisely, suppose the fraction of nodes with degree k is denoted as P(k). The graph of P(k) as a function of k given in Fig. 7 shows the (frequency or probability) distribution of the degrees over a network. This particular case shows the distribution for a random network of n = 20 nodes with p = 0.5, in which case p (n-1) = 9.5. Note that for p = 1 (fully connected network) there are no nodes with degree lower than n1: in that case all nodes have the maximal degree n-1.
5
Fig. 7. Distribution of node degrees for a random network with n = 20 and p = 0.5
Networks with distributions that have more dominance in the lower degrees, as is the case for small world networks that often occur in reality have been further explored. This has led to the notion of scale-free networks.
Scale-Free Networks One specific type of small world network is called a scale-free network; this satisfies a distribution of node degrees according to a power law: the fraction of nodes with degree k, denoted as P(k) is approximated by a function f(k) = ck- for some which typically (but not always) may be between 2 and 3, and c a constant. The pattern of such a distribution is shown in Fig. 8.
Fig. 8. Power law distribution of node degrees for a scale-free network
The name ‘scale-free’ was derived from a property of the function f representing the power law. This function f satisfies the property that for any a exists such that for all x it holds: f(x) = f(x) Therefore scaling up the degree k by any factor still provides the same type of distribution. Fig. 8 shows a scalefree network structure for a section of the Internet.
6
Fig. 9. Example of a scale-free network for a section of the Internet
The conjecture is that many networks in the real world are scale-free, for example World Wide Web link networks, biological networks, and social networks. In (Barabasi and Albert, 1999, p. 510) the results shown in Table 2 are put forward. For a large network of WWW pages it turned out that it satisfies a power law with about 2.1. Also a social network of actors with connections when they were playing at least once in the same movies had a power law structure, with about 2.3. Similarly it was found that the network of citations between scientific papers satisfies a power law with about 3 and an electrical power grid (in the western US) with about 4. Such empirical results have led to the hypothesis that networks in practice usually scale-free with a distribution of degrees according to a power law with between 2 and 4. Further empirical evidence for this hypothesis is still being acquired. Table 2 Some empirical results showing for which exponent the power law holds for networks in practice network
WWW 2.1
Actor network 2.3
Citation network 3
Power grid 4
As a simple example, suppose the network shown in Fig. 10 is given. The question may be posed whether this network is scale-free, and if so, which applies.
Fig. 10. Another example of a scale-free network
7
By counting, the degree distribution can be found as shown in the middle column of Table 3. degree
empirical
power law
1 2 3 4 5 6 7
18 5 2 2 0 1 2
18.00 4.50 2.00 1.13 0.72 0.50 0.37
Table 3 Empirical data on degrees for the network in Fig. 10 compared with a power function with = 2 and c = 18.
This distribution has been approximated by a power law distribution function f(k) = ck- by choosing = 2 and c = 18; see the column at the right hand side in Table 2. In Fig. 11 for both (empirical and power law) distributions the graph is shown. This shows that the network depicted in Fig. 10 is scale-free.
20 18 16 14 12 10 8
empirical power law
6 4 2 0 1
2
3
4
5
6
7
Fig. 11. Graph for empirical degree data and power law for the network in Fig. 10
A useful approach to estimate which exponent fits well to a set of empirical data is by applying the logarithm ln to the power law, thus obtaining: ln f(k) = ln c - ln k This is a linear relation between ln f(k) and ln k, which will show as a straight line in a graph. When this logarithm is also applied to the empirical data, the data shown Table 4 are obtained by applying the logarithm to the cells of Table 3.
8
log degree 0 0.693147 1.098612 1.386294 1.609438 1.791759 1.94591
log empirical 2.89037176 1.60943791 0.69314718 0.69314718 - 0 0.69314718
log power law 2.890372 1.088189 0.03398 -0.71399 -1.29417 -1.7682 -2.16899
Table 4 Logarithm of empirical data on degrees for the network in Fig. 10 compared with a logarithm of a power function (with = 2 and c = 18)
This can be depicted in a (double logarithmic) graph as shown in Fig. 12. Note that the value - resulting from applying ln to 0 is depicted here as 0. 3.5 3 2.5 2 1.5
log empirical
1
log power law
0.5 0 -0.5
0
0.5
1
1.5
2
2.5
-1 -1.5
Fig. 12. Double logarithmic graph for empirical degree data and power law for the network in Fig. 10
In this form for any given set of empirical data the parameters c and can be estimated by minimizing the sum of the distances, or squares thereof as is done by a least square approximation method; see also (Clauset, Shalizi, and Newman, 2009). Some other examples of scale-free networks are shown in Fig. 13 and Fig. 14.
Fig. 13. Yet another example of a scale-free network
9
Fig. 14. Example of a scale-free network concerning ownership for stocks traded in 2001 on the New York Stock Exchange.
In the graph shown in Fig. 15 the different types of distributions of node degrees are shown in a comparative manner: for a random network, a fully connected network and a scale-free network..
Fully connected
Fig. 15. Comparison of different distributions of degrees: scale-free, random and fully connected networks
10
4 Clusters, Bridges, and Fractal Networks In small world networks and especially scale-free networks, often a form of clustering structure can be found. Sometimes also clusters themselves again have their own clustering structure, which gives the network a repetitive, fractal structure. Clusters and Bridges As is shown in many examples, in a scale-free network often certain clusters can be identified of subgroups of nodes that have a relatively high number of mutual connections but practically no connection with nodes outside the cluster; for example, see Fig. 16.
Fig. 16. Example of clustering in a language network
If there are no such connections between a cluster and other clusters at all, the whole cluster would be isolated. However, usually there are at least some connections from one cluster to another; they are called bridge connections or simply bridges. Some examples are shown in Fig. 17 and Fig. 18. It is possible that the clusters themselves are fully connected (sometimes called a clique), as in the case of the examples of Fig. 17 and Fig. 18.
Cluster 2
Cluster 1
bridges
bridges
bridges
Fig. 17. Fully connected clusters (cliques) and bridges between them
11
Cluster 3
Fig. 18. Another example showing fully connected clusters (cliques) and bridges between them
Fractal Networks In the example shown in Fig. 18 seems to have the remarkable property that the overall structure of the clusters and bridges has a similar structure as the internal structure in the clusters themselves. This is made even more clear in Fig. 18, where the clusters have been highlighted. The overall structure consists of five fully connected clusters, just like each cluster consists of five fully connected nodes. The is called a repetitive or fractal network structure.
Fig. 19. Repetitive structure in the network shown in Fig. 17
Fractal networks can have more than two levels of repetition. Examples are shown in Fig. 20 (which is similar to the two-level structure shown in Fig. 19), Fig. 21 (three level fractal structure), and Fig. 22 (four-level fractal structure; best visible when zooming in). In these examples it is assumed that bridge connections between two clusters are connected to one node with each of the clusters, although for the sake of simplicity this is not explicitly depicted.
12
Fig. 20. Two-level fractal network (as in Fig. 18)
Fig. 21. A three-level fractal network
13
Fig. 22. A four-level fractal network
For these examples it can be analysed what the numbers of nodes, the degrees and the shortest paths are. The twolevel fractal network shown in Fig. 20 has 52 = 25 nodes. The degrees of the nodes are 4 or 5 if it is assumed that different bridges are connected to different nodes in the clusters. Shortest paths vary from 1 to at most 3. The same can be done for the three-level and four-level fractal networks shown in Fig. 21 and Fig. 22. They have 53 = 125 and 54 = 625 nodes, respectively. The node degrees are still 4 or 5, under the same assumption that different bridges are never connected to the same node. Shortest paths vary from 1 to at most 5 in the three-level case and 1 to at most 7 in the four-level case. As a generalisation, for an n-level fractal network of this type, the number of nodes is 5n, the node degrees are 4 or 5, and the shortest paths vary from 1 to at most 2n-1. Table 5 summarises this. Note that if m is the number of nodes, then the length s of the maximal shortest paths has a logarithmic relation to m: s = 2n-1 = 2 ln(5n)/ ln(5) -1 = 2 ln(m)/ ln(5) -1 Such a logarithmic relation between number m of nodes and average shortest path is sometimes considered a criterion for a network to be a small world network.
14
levels 1 2 3 4 .. n
nodes 5 25 125 625 .. 5n
node degree 4 4 or 5 4 or 5 4 or 5 .. 4 or 5
max shortest path 1 3 5 7 .. 2n - 1
Table 5. Overview of numbers and degrees of nodes and maximal shortest path lengths in a fractal network structure
6 Variation in Connection or Tie Strength: Weighted and Multidimensional Networks Above it was assumed that nodes in a network are connected or not connected. However, it has since long been recognized that connections can have different gradual strengths and types. Some can be very weak, like being acquaintances, others can be very strong like between best friends or in a partner or family relation. Weighted networks: nonbinary connection strengths, weak ties and strong ties Weak connections are often called weak ties, and strong connections strong ties. In general, a weighted network is a network in which for each pair of nodes a connection or tie weight (or strength) is defined as a number in the interval [0, 1]. The weak ties are those with lower weight and strong ties those with higher weight.
Fig. 23. Weak and strong ties based on telephone calls
15
A claim made by Rapoport (1957) is the weak tie hypothesis: if A has strong ties to both B and C, then there is a high probability that B and C at least have a weak tie. As an example, your two best friends probably at least know each other, or, as another example, your parents and partner probably know each other.
Fig. 24. Weak tie hypothesis
The idea is (Granovetter, 1973, 1983) that strong ties usually occur within the clusters and lead to weak ties making the cluster more tightly connected. Moreover, the bridges between clusters are usually formed by weak ties, and are essential for contagion from one cluster to another; this principle is sometimes indicated by the strength of weak ties. By Krackhardt (1992) the strong tie hypothesis was formulated. This hypothesis claims that sometimes for more fundamental types of contagion weak ties are not sufficient. Especially for contagion that comes with uncertainty a connection is needed that makes a person feel secure, which usually is a characteristic of a strong tie but not of a weak tie. Such more fundamental types of contagion would not make it over bridges when they are only weak ties. For weighted networks the matrix representation can contain nonnegative real numbers as (weight) values, usually taken from the interval [0, 1]. An example of this is shown in Table 6 (for the network shown in Fig. 3; again the 0’s have been left out). Note that this example network is not bidirectional. Table 6 Matrix representation of a weighted network to from A B C D E F G H I J K L
A
B
C
D
0.6 0.8 1 0.7 0.5 0.6 0.3 0.5 0.4 0.6 1 0.2 0.5 0.9
E
F
G
H
I
0.5 0.4 0.8 1 0.9 0.4 0.8 0.7 0.3 0.1 0.3 0.3 0.1 0.4 0.8 1 0.4 0.7 0.3 0.2 0.3 0.9
J
K
L
0.7 0.3 0.7 0.2 0.2 0.5 0.9 0.7 0.1 0.2 0.8 0.9
Multidimensional Networks In many cases the same individual functions as a node in different networks. Sometimes it can be useful to model these multiple networks as a multidimensional network: a network where between nodes connections of a number of different types can occur. For example, individuals may interact by direct contact, by phone, by email, or by Twitter. To cover these four different types of connections would make a 4-dimensional network. Some persons are connected in all four dimensions, others only in some of the four. For some papers on this topic, see (Berlingerio, Coscia, Giannotti, Monreale, Pedreschi, 2011; Kazienko, Musial, Kukla, Kajdanowicz, and Bródka, 2011). In the area of customer relationship management the area of multi-channel management focuses on ways in which companies can interact with their customers using different communication channels such as the four mentioned above; see, for example, (Pieterson and Ebbers, 2008; Slack, Rowley, Coles, 2008).
16
7
Dynamics in Networks
Above the focus was on the structure of networks, without taking into account that they may shown some form of dynamics. Two types of dynamics are often considered:
dynamics of node states by mutual contagion through the links connecting them dynamics of the connections
The first type of dynamics is usually indicated by words such as diffusion, spread or contagion. The node states that are spread can be of different types, such as:
Information or beliefs Examples: o Did you know that A is dating B? o There is a new restaurant nearby Opinions or valuations Examples: o Do you also like this movie? o This phone is not good Emotions Examples: o Showing a happy face o Talking with a sad voice, o Saying or writing that you feel bad o Spreading your sweat and pheromones (e.g., for fear or disgust, or for sexual arousal) Diseases Examples: o Contagion of your flu to somebody else o Tuberculosis epidemic Behaviour, habits or lifestyle Examples: o Contagion of sport activities among friends o Smoking behaviour o Use of alcohol or drugs
The second type of dynamics concerns changes in the connectivity structure of the network. This is sometimes indicated by evolving networks or network growth. Also such changes can be of different types, such as:
Link addition Examples: o Meeting somebody, o Becoming friends with somebody o Citing a paper
Link removal Examples:
17
o Unfriend somebody o Breaking up a relationship Strengthening a link Examples: o After more and more activities together a weak tie becomes a strong tie o Discovering that you have much in common with somebody Weakening a link Examples: o Weakening of a friendship due to moving to an area far away o Neglecting the contact with somebody during a very busy period
A number of general principles are used in modelling of networks in order to conduct simulation experiments and mathematical analysis. Examples of such principles are (e.g., Newman, 2003):
Random link addition When it is assumed that persons meet each other by chance, this can be modelled by link addition in a network between randomly chosen pairs of nodes. Triadic closure Examples of this principle are transitive addition (if A is connected to B and B to C, then A will get a connection with C: ‘your friends are my friends’), and the weak tie hypothesis discussed in Section 6. More becomes more This principle expresses that nodes that already have more connections get more extra connections than nodes with less connections (the rich become more rich and the poor remain poor). Analyses have been made showing that applying this principle usually leads to scalefree networks; cf. (Price, 1976; Barabási and Albert, 1999; Newman, 2003). Homophily This principle indicates that the more similar the states of two connected nodes are, the stronger their connection will become: ‘birds of a feather flock together’ (‘soort zoekt soort’); see, for example, (Byrne, 1986; McPherson, Smith-Lovin, and Cook, 2001; Mislove, Viswanath, Gummadi, Druschel, 2010). When also the states are assumed dynamic, this principle can be combined with contagion of states into a circular causal relation: state ⇄ link When in a practical situation a network is found in which similar states and strong connections occur together, due to such a circular causal relation it is difficult to tell which of the two principles (or both?) was causing this situation; see also, for example (Aral, Muchnik, and Sundararajan, 2009; Shalizi and Thomas, 2011; Steglich, Snijders, Pearson, 2010; Mundt, Mercken, Zakletskaia, 2012).
8
Network Analysis and Network Interventions
An important role of networks is that they form a basis for diffusion or contagion processes for various matters, for example, diseases, information, innovations, opinions, emotions, behaviours, lifestyles. Monitoring and analysing the dynamics of given diffusion or contagion processes is one thing. Many dynamic models have been put forward to support such analysis, varying from agent-based to population-based models. However, having such analysis means available, they can be used for prediction as well, and also for what-if simulation: predicting what will happen if some action is undertaken. More generally, methods can be developed to determine what types of network
18
intervention actions can be undertaken under which conditions in order to achieve some specific goal. Examples of such goals are:
avoiding that an epidemic will develop achieving that many persons will know about a new product that you bring out achieving that more people will adopt a healthy lifestyle achieving that elderly persons are not in a situation in which they have almost no social contacts
As an example, persons with large numbers of connections may be identified (for example, with thousands or even millions of followers on Twitter), as they may play a crucial role to reach many nodes. As another example, in a strongly clustered kind of network mutual influence or contagion can take place according to a kind of repeated sequence of waterfalls, where at each step some time is passing to get a cluster affected, after which a next cluster is affected. In such a process the bridge connections play an important role, so network interventions may focus on them. In (Valente, 2010; 2012) more can be found on this area of network interventions.
References Aral, S., Muchnik, L., and Sundararajan, A. (2009). Distinguishing Influence Based Contagion From Homophily Driven Diffusion in Dynamic Networks. Proceedings of the National Academy of Sciences (USA) 106:21544-1549. Barabási, A.L., and Albert, R. (1999). Emergence of Scaling in Random Networks. Science, vol 286, pp. 509-512. Barabási, A.L., and Oltvai, Z.N. (2004). Network biology: understanding the cell's functional organization. Nature Reviews Genetics 5, 101-113 (February 2004) Berlingerio, M., Coscia, M., Giannotti, F., Monreale, A., Pedreschi, D., (2011). Foundations of Multidimensional Network Analysis. In: International Conference on Advances in Social Networks Analysis and Mining, 2011, pp. 485-489. Boccalettia, S., Latorab, V., Morenod, Y., Chavez, M., Hwanga, D.-U. (2006). Complex networks: Structure and dynamics. Physics Reports 424 (2006) 175 – 308. Byrne, D. (1986). The attraction hypothesis: Do similar attitudes affect anything? Journal of Personality and Social Psychology, 51(6), 1986, 1167-1170. Giles, J., Computational Social Science: Making the Links. Nature, vol 488, 2012, pp. 448-450. Clauset, A., C.R. Shalizi, and M.E.J. Newman (2009). Powerlaw distributions in empirical data. SIAM Review 51(4): 661–703. Granovetter, M.S. (1973). The Strength of Weak Ties, Amer. J. of Sociology, Vol. 78, Issue 6, May 1360-80. Granovetter, M.S. (1983). The Strength of the Weak Tie: Revisited. Sociological Theory, vol. 1, 201-33. Groot, J.H.B. de, Smeets, M.A.M, Kaldewaij, A., Duijndam, M.J.A., and R. Semin, G.R. (2012). Chemosignals Communicate Human Emotions. Psychological Science, 2012. Online First, doi:10.1177/0956797612445317. Kazienko, P., Musial, K., Kukla, E., Kajdanowicz, T., and Bródka, P., (2011). Multidimensional Social Network: Model and Analysis. P. Jędrzejowicz et al. (eds.), Proc. of the Third International Conference on Computational Collective Intelligence, ICCCI 2011, Part I, LNCS 6922, pp. 378–387, 2011. Krackhardt, D. (1992). The Strength of Strong Ties: The Importance of Philos in Organizations. In N. Nohria & R. Eccles (eds.), Networks and Organizations: Structure, Form, and Action: 216-239. Boston, MA: Harvard Business School Press. Liben-Nowell, D., Kleinberg, J. (2007). The Link-Prediction Problem for Social Networks. Journal of the American Society for Information Science and Technology, vol. 58, pp. 1019–1031 Marsden, P.V., Campbell, K.E. (1990). Measuring Tie Strength. Social Forces, 63, 1990, 482–501. McPherson, M., Smith-Lovin, L., Cook, J.M. (2001). Birds of a feather: homophily in social networks. Annu. Rev. Sociol. 27, 415–444 (2001). Mislove, A., Viswanath, B., Gummadi, K.P., Druschel, P. (2010). You Are Who You Know: Inferring User Profiles in Online Social Networks. Proc. WSDM’10, February 4–6, 2010, New York City, New York, USA, pp. 251-260 Mundt, M.P., Mercken, L, Zakletskaia, L.I. (2012). Peer selection and influence effects on adolescent alcohol use: a stochastic actor-based model. BMC pediatrics, 12, 115. Pappalardo, L., Rossetti, G., Pedreschi, D., (2012). How well do we know each other?: Detecting tie strength in multidimensional social networks, Proc. Of the ASONAM’12 Complex Social Network Analysis Workshops, ASONAM – CSNA’12 (16) Pieterson, P., and Ebbers, W. (2008). The use of service channels by citizens in the Netherlands: implications for multi-channel management. International Review of Administrative Sciences vol. 74, 2008, pp. 95-110
19
Price, D.J. de S. (1976). A general theory of bibliometric and other cumulative advantage processes. J. Amer. Soc. Inform. Sci., 27, p. 292-306. Rapoport, A., (1957). Contribution to the Theory of Random and Biased Nets. Bulletin of Mathematical Biology 19:257-77. Shalizi, C.R., and Thomas, A.C. (2011). Homophily and Contagion are Generically Confounded in Observational Social Network Studies. Sociological Methods & Research 40(2) 211–239 Slack, F., Rowley, J., Coles, S. (2008). Consumer behaviour in multi-channel contexts: the case of a theatre festival. Internet Research, vol. 18, 2008, pp. 46-59. Steglich, C.E.G., Snijders, T.A.B., Pearson, M. (2010). Dynamic networks and behavior: separating selection from influence. Sociol Methodol 40, 329–393. Valente, T.W. (2010). Social Networks and Health: Models, Methods, and Applications. New York, Oxford University Press, 2010. Valente, T.W. (2012). Network Interventions. Science 337, 49-53, 2012 Watts, D.J., and Strogatz, S.H. (1998). Collective dynamics of ‘small-world’ networks. Nature 393: 440–442. Watts, D.J. (1999). Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press.
Figure references Fig. 1 Fig. 2 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9
Fig. 10 Fig. 13 Fig. 14 Fig. 16 Fig. 18 Fig. 23 Fig. 24
http://www.digitaltrainingacademy.com/socialmedia/2009/06/social_networking_map.php http://www.nature.com/news/computational-social-science-making-the-links-1.11243 Giles, J., Computational Social Science: Making the Links. Nature, vol 488, 2012, pp. 448-450. http://polysingularity.com/social/ Perseguers, S., Lewenstein, M., Acín, A. & Cirac, J. I., Quantum random networks. Nature Physics 6, 539–543 (2010) doi:10.1038/nphys1665 Graham, B., Patterns – The Art, Soul, and Science of Beholding Nature. http://www.patternsinnature.org/Book/RandomNetworks.html Graham, B., Patterns – The Art, Soul, and Science of Beholding Nature. http://www.patternsinnature.org/Book/RandomNetworks.html Networks; The Power of Hubs http://idm09.wordpress.com/2009/11/03/networks-the-power-of-hubs/ Dovrolis, C., Network Science: Methods and Applications Course http://www.cc.gatech.edu/~dovrolis/Courses/cs8803ns-F09.html Graham, B., Patterns – The Art, Soul, and Science of Beholding Nature. http://www.patternsinnature.org/Book/RandomNetworks.html Leslie-Pelecky, D., It really is a small world http://twistedphysics.typepad.com/cocktail_party_physics/2009/01/it-really-is-a-small-world.html Sabroe, I., Parker, L.C., Calverley, P.M.A., Dower, S.K., Whyte, M.K.B., Pathological networking: a new approach to understanding COPD. Postgrad Med J 2008;84:259-264 doi:10.1136/thx.2007.077768 Caldarelli, G., The Structure of Biological and Social Systems. SIAM News, Volume 37, Number 3, April 2004 http://www.mathaware.org/mam/04/essays/systemsstructure.html http://www.news.ku.edu/2008/february/5/language.shtml Church, C., To View or Not to View: ‘Why’ is the Question http://www.streamfoundations.com/index.php/2010/11/30/to-view-or-not-to-view-why-is-the-question/ http://www.nature.com/news/computational-social-science-making-the-links-1.11243 Giles, J., Computational Social Science: Making the Links. Nature, vol 488, 2012, pp. 448-450. Interpersonal ties http://en.wikipedia.org/wiki/Interpersonal_ties
20
