The Establishment Game. Motivation

Transcription

1 Motivation

2 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc.

3 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems.

4 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems. On the other hand the traits of the nodes might affect the structure and properties of the network.

5 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems. On the other hand the traits of the nodes might affect the structure and properties of the network. Questions:

6 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems. On the other hand the traits of the nodes might affect the structure and properties of the network. Questions: What are the properties of a trait-based network?

7 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems. On the other hand the traits of the nodes might affect the structure and properties of the network. Questions: What are the properties of a trait-based network? Are growing trait-based networks different from static ones?

8 Motivation The network models so far neglect the attributes, traits of the nodes. A node can represent anything, people, web pages, computers, etc. Because of this, these models are very general, they can characterize the behavior of a broad range of systems. On the other hand the traits of the nodes might affect the structure and properties of the network. Questions: What are the properties of a trait-based network? Are growing trait-based networks different from static ones? What about the real world? Are there real trait-based networks?

9 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution.

10 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step.

11 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step. In each time step: A new node with random trait vector is added.

12 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step. In each time step: A new node with random trait vector is added. The new node chooses randomly k putative partners from the other nodes.

13 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step. In each time step: A new node with random trait vector is added. The new node chooses randomly k putative partners from the other nodes. The trait similarity between the new node and one of its putative partners is calculated. If the similarity exceeds a given threshold (θ) an edge between the two nodes will be added.

14 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step. In each time step: A new node with random trait vector is added. The new node chooses randomly k putative partners from the other nodes. The trait similarity between the new node and one of its putative partners is calculated. If the similarity exceeds a given threshold (θ) an edge between the two nodes will be added. This procedure is repeated for each putative partner of the new node.

15 The Algorithm Each node has a trait vector of length d. The traits can have continuous or discrete values and are drawn from a given probability distribution. Begin with one node with random traits in the first time step. In each time step: A new node with random trait vector is added. The new node chooses randomly k putative partners from the other nodes. The trait similarity between the new node and one of its putative partners is calculated. If the similarity exceeds a given threshold (θ) an edge between the two nodes will be added. This procedure is repeated for each putative partner of the new node. In our simulations we used: binary trait-vectors drawn from a uniform distribution; number of matching bits as the trait similarity.

16 Parameters: k = 2, d = 3, θ =

17 Parameters: k = 2, d = 3, θ =

18 Parameters: k = 2, d = 3, θ =

19 Parameters: k = 2, d = 3, θ =

20 Parameters: k = 2, d = 3, θ =

21 Parameters: k = 2, d = 3, θ =

22 Parameters: k = 2, d = 3, θ =

23 Parameters: k = 2, d = 3, θ =

24 Parameters: k = 2, d = 3, θ =

25 Parameters: k = 2, d = 3, θ =

26 Parameters: k = 2, d = 3, θ =

27 Parameters: k = 2, d = 3, θ =

28 Parameters: k = 2, d = 3, θ =

29 The Degree Distribution Calculating the degree distribution: p : Probability that two randomly generated trait-vectors are compatible.

30 The Degree Distribution Calculating the degree distribution: p : Probability that two randomly generated trait-vectors are compatible. For binary trait-vectors and the common bits similarity function: d i p = 2 d i=0 ( ) d i

31 The Degree Distribution II d m (t) : expected number of nodes with degree m at time t.

32 The Degree Distribution II d m (t) : expected number of nodes with degree m at time t. The k = 1 case is simple: d 0 (t + 1) =d 0 (t) d 0(t) p + 1 p (1) t

33 The Degree Distribution II d m (t) : expected number of nodes with degree m at time t. The k = 1 case is simple: d 0 (t + 1) =d 0 (t) d 0(t) p + 1 p t (1) d 1 (t + 1) =d 1 (t) + 2 d (t) t p + (1 d 0(t) + d 1 (t) )p t (2)

34 The Degree Distribution II d m (t) : expected number of nodes with degree m at time t. The k = 1 case is simple: d 0 (t + 1) =d 0 (t) d 0(t) p + 1 p t (1) d 1 (t + 1) =d 1 (t) + 2 d (t) t p + (1 d 0(t) + d 1 (t) )p t (2) d m (t + 1) =d m (t) d m(t) t p + d m 1(t) p (m > 1) (3) t

35 The Degree Distribution III From our simulations we found that d m (t) grows linearly with time, ie. d m (t) = p m t.

36 The Degree Distribution III From our simulations we found that d m (t) grows linearly with time, ie. d m (t) = p m t. Searching solutions of this form, we can solve the recursion above easily: p 0 = 1 p 1 + p, p m = p ( ) m p (4) 1 + p

37 The Degree Distribution III From our simulations we found that d m (t) grows linearly with time, ie. d m (t) = p m t. Searching solutions of this form, we can solve the recursion above easily: p 0 = 1 p 1 + p, p m = p ( ) m p (4) 1 + p Clearly, p m gives the probability of a randomly chosen node having degree m.

38 The Degree Distribution IV The general case is more difficult: d 0 (t + 1) =d 0 (t) k d 0(t) p + (1 p) k (5) t

39 The Degree Distribution IV The general case is more difficult: d 0 (t + 1) =d 0 (t) k d 0(t) t d i (t + 1) =d i (t) k d 1(t) t p + (1 p) k (5) p + k d ( ) i 1(t) k p + p i (1 p) k i t i (1 i k) (6)

40 The Degree Distribution IV The general case is more difficult: d 0 (t + 1) =d 0 (t) k d 0(t) t d i (t + 1) =d i (t) k d 1(t) t d m (t + 1) =d m (t) k d m(t) t p + (1 p) k (5) p + k d ( ) i 1(t) k p + p i (1 p) k i t i (1 i k) (6) p + k d m 1(t) p (m > k) (7) t

41 The Degree Distribution V. Again, we found that d m (t) p m t with the appropriate p m factors.

42 The Degree Distribution V. Again, we found that d m (t) p m t with the appropriate p m factors. The solutions for these equations: p i = p i i j=0 ( k ) i j (1 p) k j 1 + kp 1 + kp ( ) k j (0 i k) (8)

43 The Degree Distribution V. Again, we found that d m (t) p m t with the appropriate p m factors. The solutions for these equations: p i = p i i j=0 ( k ) i j (1 p) k j 1 + kp 1 + kp ( ) k j (0 i k) (8) ( ) m k kp p m = p k (m > k) (9) 1 + kp

44 The Degree Distribution VI. log(freqency) 1 Simulation Analytical result e Degree Parameters: d = 9, k = 1, θ = 5, n = 1000, averaged over 100 runs.

45 The Degree Distribution VI. log(freqency) 1 Simulation Analytical result e Degree Parameters: d = 9, k = 1, θ = 5, n = 1000, averaged over 100 runs. Conclusion: the degree distribution is exponential.

46 The Degree Distribution VI. log(freqency) 1 Simulation Analytical result e Degree Parameters: d = 9, k = 1, θ = 5, n = 1000, averaged over 100 runs. Conclusion: the degree distribution is exponential. Conclusion for real networks: in the trait-based world it is very unlikely that nodes with very many edges exist. One can t have very many friends, co-authors, co-actors, etc. because of natural limits.

47 The Cluster Distribution Cluster: set of connected nodes, with no connection to the outside world.

48 The Cluster Distribution Cluster: set of connected nodes, with no connection to the outside world. N m (t): expected number of clusters with m nodes, at time t.

49 The Cluster Distribution Cluster: set of connected nodes, with no connection to the outside world. N m (t): expected number of clusters with m nodes, at time t. Again, the k = 1 case is simple: N 1 (t) =d 0 (t), by definition (10)

50 The Cluster Distribution Cluster: set of connected nodes, with no connection to the outside world. N m (t): expected number of clusters with m nodes, at time t. Again, the k = 1 case is simple: N 1 (t) =d 0 (t), by definition (10) N m (t + 1) =N m (t) + p (m 1)N m 1(t) t p mn m(t) t (11)

51 And since N m (t) c m t, we get: The Cluster Distribution c m = (1 p)(m 1)! p m 1 m j= jp (12)

52 And since N m (t) c m t, we get: The Cluster Distribution c m = (1 p)(m 1)! p m 1 m j= jp (12) We can calculate the c m -s for different p values p=0.2 p=0.4 p=0.5 p= e 05 1e 06 1e 07 1e 08 1e 09 1e

53 The Cluster Distribution The log-log plots for the cluster distribution show straight lines, meaning that it follows a power-law. The cluster size follows a scale-free power-law distribution.

54 The Cluster Distribution The log-log plots for the cluster distribution show straight lines, meaning that it follows a power-law. The cluster size follows a scale-free power-law distribution. Conclusion for real networks: in the trait-based world, there are clusters with very different sizes, from very small to very large.

55 Giant clusters The clustering behavior of the model is dual: for some parameters a giant cluster (=contains almost all the nodes) exists, for others not.

56 Giant clusters The clustering behavior of the model is dual: for some parameters a giant cluster (=contains almost all the nodes) exists, for others not e Examples for the dual clustering behaviors. Parameters d = 256, k = 1, n = 1000, θ 1 = 125, θ 2 = 110. Both figures are averaged over 100 runs.

57 Giant clusters The clustering behavior of the model is dual: for some parameters a giant cluster (=contains almost all the nodes) exists, for others not e Examples for the dual clustering behaviors. Parameters d = 256, k = 1, n = 1000, θ 1 = 125, θ 2 = 110. Both figures are averaged over 100 runs. It seems that there is a phase transition in the system, with increasing k or with decreasing θ (the dimension of the trait vector is constant).

58 e e e e e e e e Parameters: d = 256, k = 1, n = 1000, θ = , averaged over 100 runs.

59 The Cluster Distribution III. Even if there is a giant cluster in the network, separated clusters of a broad range of sizes exist.

60 The Cluster Distribution III. Even if there is a giant cluster in the network, separated clusters of a broad range of sizes exist

61 The Cluster Distribution III. Even if there is a giant cluster in the network, separated clusters of a broad range of sizes exist Conclusion for real networks: in the trait-based world, even if most of the nodes are in the same group, there exist many small groups of different sizes.

62 Bond percolation: Connection to bond percolation

63 Connection to bond percolation Bond percolation: Nodes on a lattice, many different types of lattices possible.

64 Bond percolation: Connection to bond percolation Nodes on a lattice, many different types of lattices possible. Edges are added with a given constant probability (p b ) between the neighboring nodes.

65 Bond percolation: Connection to bond percolation Nodes on a lattice, many different types of lattices possible. Edges are added with a given constant probability (p b ) between the neighboring nodes.

66 Bond percolation: Connection to bond percolation Nodes on a lattice, many different types of lattices possible. Edges are added with a given constant probability (p b ) between the neighboring nodes.

67 Bond percolation: Connection to bond percolation Nodes on a lattice, many different types of lattices possible. Edges are added with a given constant probability (p b ) between the neighboring nodes. Bond percolation is well studied, we know that a phase transition exists, for certain lattices we know the transition threshold.

68 Connection to bond percolation II Unfortunately in our model the connection probability is not constant, we cannot apply these results immediately.

69 Connection to bond percolation II Unfortunately in our model the connection probability is not constant, we cannot apply these results immediately. Expected value of p b in our model: p b = 2pk N (13)

70 Growing networks vs. static ones The fact that the network is grown, rather than created as a complete entity, leaves an identifiable signature in the network topology.

71 Growing networks vs. static ones The fact that the network is grown, rather than created as a complete entity, leaves an identifiable signature in the network topology. Older nodes have higher degree in a grown graph.

72 Growing networks vs. static ones The fact that the network is grown, rather than created as a complete entity, leaves an identifiable signature in the network topology. Older nodes have higher degree in a grown graph. Because of this, the grown network is more connected than the static one.

73 Growing networks vs. static ones The cluster distribution does not always follow a power-law in case of static networks: 1 1 Parameters: d = 256, θ = 125, k = 1, n = 1000, averaged over 100 runs e e

74 Growing networks vs. static ones The cluster distribution does not always follow a power-law in case of static networks: 1 1 Parameters: d = 256, θ = 125, k = 1, n = 1000, averaged over 100 runs e e Conclusion for real networks: growing networks are more connected than static ones.

75 Collecting Data For evaluating the relevance of our model, we have to collect real trait-based network data.

76 Collecting Data For evaluating the relevance of our model, we have to collect real trait-based network data. Freshmen friendship network in K College.

77 Collecting Data For evaluating the relevance of our model, we have to collect real trait-based network data. Freshmen friendship network in K College. What to collect:

78 Collecting Data For evaluating the relevance of our model, we have to collect real trait-based network data. Freshmen friendship network in K College. What to collect: Friendship data for building the network itself.

79 Collecting Data For evaluating the relevance of our model, we have to collect real trait-based network data. Freshmen friendship network in K College. What to collect: Friendship data for building the network itself. Personality traits. How to describe people s personality using a finite set of numbers?

80 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality:

81 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression.

82 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression. Extraversion. The extent to which an individual is outgoing, active, high-spirited.

83 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression. Extraversion. The extent to which an individual is outgoing, active, high-spirited. Openness to experience. People with high levels of openness to experience: imagination, curiosity, originality, open-mindedness. People with low level of it are practical, traditional and set in their ways.

84 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression. Extraversion. The extent to which an individual is outgoing, active, high-spirited. Openness to experience. People with high levels of openness to experience: imagination, curiosity, originality, open-mindedness. People with low level of it are practical, traditional and set in their ways. Agreeableness. Flexible, trusting, good-natured, cooperative, forgiving vs. hardheaded, direct, skeptical, competitive.

85 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression. Extraversion. The extent to which an individual is outgoing, active, high-spirited. Openness to experience. People with high levels of openness to experience: imagination, curiosity, originality, open-mindedness. People with low level of it are practical, traditional and set in their ways. Agreeableness. Flexible, trusting, good-natured, cooperative, forgiving vs. hardheaded, direct, skeptical, competitive. Conscientiousness. Defendable, careful, responsible, organized, good at planning, high will to achieve vs. easygoing, not very well organized, sometimes careless.

86 Collecting Data The big five. Five traits, proven to be general and enough for describing people s personality: Neuroticism. The extent to which an individual experiences and displays negative effects: anxiety, vulnerability, depression. Extraversion. The extent to which an individual is outgoing, active, high-spirited. Openness to experience. People with high levels of openness to experience: imagination, curiosity, originality, open-mindedness. People with low level of it are practical, traditional and set in their ways. Agreeableness. Flexible, trusting, good-natured, cooperative, forgiving vs. hardheaded, direct, skeptical, competitive. Conscientiousness. Defendable, careful, responsible, organized, good at planning, high will to achieve vs. easygoing, not very well organized, sometimes careless. We designed a WWW-based survey for collecting the data.

87 Collecting Data

88 Future Work Other measures to calculate. Connectedness, betweenness, closeness, etc.

89 Future Work Other measures to calculate. Connectedness, betweenness, closeness, etc. Comparison with real world data. It is not an easy task to find structure data and traits data for the same network.

90 Future Work Other measures to calculate. Connectedness, betweenness, closeness, etc. Comparison with real world data. It is not an easy task to find structure data and traits data for the same network. Other games to play:

91 Future Work Other measures to calculate. Connectedness, betweenness, closeness, etc. Comparison with real world data. It is not an easy task to find structure data and traits data for the same network. Other games to play: Similarity game vs. diversity game.

92 Future Work Other measures to calculate. Connectedness, betweenness, closeness, etc. Comparison with real world data. It is not an easy task to find structure data and traits data for the same network. Other games to play: Similarity game vs. diversity game. Local game vs. global game.

93 Summary Graph theory is shown to be a useful and effective method to study networks of social systems.

94 Summary Graph theory is shown to be a useful and effective method to study networks of social systems. Some recently developed network models are able to capture properties of acquaintanceship structures.

95 Summary Graph theory is shown to be a useful and effective method to study networks of social systems. Some recently developed network models are able to capture properties of acquaintanceship structures. Traits have important role in network formation.

96 Summary Graph theory is shown to be a useful and effective method to study networks of social systems. Some recently developed network models are able to capture properties of acquaintanceship structures. Traits have important role in network formation. Growing trait-based networks are different from static ones.

97 Summary Graph theory is shown to be a useful and effective method to study networks of social systems. Some recently developed network models are able to capture properties of acquaintanceship structures. Traits have important role in network formation. Growing trait-based networks are different from static ones. Research still in progress.

98 Collaborators Máté Lengyel Jan Tobochnik Péter Érdi Becky Warner Acknowledgement K College Psychology Department

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