caution in interpreting graph-theoretic diagnostics

Transcription

1 April 17, 2013

2 What is a network [1, 2, 3]

3 What is a network [1, 2, 3]

4 What is a network [1, 2, 3]

5 What is a network [1, 2, 3]

6 What is a network a collection of more or less identical agents or objects, each pair of which are either tied or not information (World Wide Web, Wikipedia) affiliation (collaboration, boards of directors) a substrate of locations and pathways on which structures or processes are sustained communication (neural network, power grid, Internet) social (simple / complex contagion) a system of relations among an open-ended collection of concepts phraseological (keyword, narrative) biological (ecosystem, gene regulation)

7 What is a network a collection of more or less identical agents or objects, each pair of which are either tied or not information (World Wide Web, Wikipedia) affiliation (collaboration, boards of directors) a substrate of locations and pathways on which structures or processes are sustained communication (neural network, power grid, Internet) social (simple / complex contagion) a system of relations among an open-ended collection of concepts phraseological (keyword, narrative) biological (ecosystem, gene regulation)

8 What is a network a collection of more or less identical agents or objects, each pair of which are either tied or not information (World Wide Web, Wikipedia) affiliation (collaboration, boards of directors) a substrate of locations and pathways on which structures or processes are sustained communication (neural network, power grid, Internet) social (simple / complex contagion) a system of relations among an open-ended collection of concepts phraseological (keyword, narrative) biological (ecosystem, gene regulation)

9 graphs: a universal framework? Many structural properties are ubiquitous to real-world networks: fat-tailed degree distributions Erős Rényi graphs (asymptotically) produce Poisson degree distributions Real-world graphs have more high-degree nodes and more low-degree nodes; often follow a Pareto (power law) distribution homophily Birds of a feather flock together Nodes of similar degree tend to be connected transitivity The friend of my friend is my friend Graphs have lots (like, several orders of magnitude) more triangles than uniform chance predicts

10 graphs: a universal framework? Many structural properties are ubiquitous to real-world networks: fat-tailed degree distributions Erős Rényi graphs (asymptotically) produce Poisson degree distributions Real-world graphs have more high-degree nodes and more low-degree nodes; often follow a Pareto (power law) distribution homophily Birds of a feather flock together Nodes of similar degree tend to be connected transitivity The friend of my friend is my friend Graphs have lots (like, several orders of magnitude) more triangles than uniform chance predicts

11 graphs: a universal framework? Many structural properties are ubiquitous to real-world networks: fat-tailed degree distributions Erős Rényi graphs (asymptotically) produce Poisson degree distributions Real-world graphs have more high-degree nodes and more low-degree nodes; often follow a Pareto (power law) distribution homophily Birds of a feather flock together Nodes of similar degree tend to be connected transitivity The friend of my friend is my friend Graphs have lots (like, several orders of magnitude) more triangles than uniform chance predicts

12 graphs: a universal framework? Many structural properties are ubiquitous to real-world networks: fat-tailed degree distributions Erős Rényi graphs (asymptotically) produce Poisson degree distributions Real-world graphs have more high-degree nodes and more low-degree nodes; often follow a Pareto (power law) distribution homophily Birds of a feather flock together Nodes of similar degree tend to be connected transitivity The friend of my friend is my friend Graphs have lots (like, several orders of magnitude) more triangles than uniform chance predicts

13 graphs: a universal framework? motivations in tension: 1. depth combinatorial granularity (node types, edge direction) qualitative attributes (tie strength, reference class) 2. generalizability cross-disciplinary comparisons (homophily, clustering) general theory (small world, preferential attachment)

14 graphs: a universal framework? motivations in tension: 1. depth combinatorial granularity (node types, edge direction) qualitative attributes (tie strength, reference class) 2. generalizability cross-disciplinary comparisons (homophily, clustering) general theory (small world, preferential attachment)

15 graphs: a universal framework? motivations in tension: 1. depth combinatorial granularity (node types, edge direction) qualitative attributes (tie strength, reference class) 2. generalizability cross-disciplinary comparisons (homophily, clustering) general theory (small world, preferential attachment)

16 preferential attachment growth models Take an evolving graph G(t) = (N(t), E(t)) over discrete time t Z 0. Each time step new nodes and links may appear while existing nodes and links may disappear. A new node i links to an existing node i with probability P(i). The graph exhibits (degree) preferential attachment if P(i) deg(i).

17 preferential attachment growth models Take an evolving graph G(t) = (N(t), E(t)) over discrete time t Z 0. Each time step new nodes and links may appear while existing nodes and links may disappear. A new node i links to an existing node i with probability P(i). The graph exhibits (degree) preferential attachment if P(i) deg(i).

18 preferential attachment growth models Take an evolving graph G(t) = (N(t), E(t)) over discrete time t Z 0. Each time step new nodes and links may appear while existing nodes and links may disappear. A new node i links to an existing node i with probability P(i). The graph exhibits (degree) preferential attachment if P(i) deg(i).

19 preferential attachment growth models Take an evolving graph G(t) = (N(t), E(t)) over discrete time t Z 0. Each time step new nodes and links may appear while existing nodes and links may disappear. A new node i links to an existing node i with probability P(i). The graph exhibits (degree) preferential attachment if P(i) deg(i). A B F D C E

20 preferential attachment growth models random (but same node degrees) preferential attachment

21 preferential attachment growth models preferential attachment random (but same node degrees)

22 preferential attachment by degree Attachment preference , , , Collaborators

23 preferential attachment by degree Attachment preference , , , Collaborators (logarithmic binning)

24 geodesic preferential linking The geodesic distance l(i, i ) between i and i in G is the minimum length of a path (i, i 1, i 2,..., i ) where each adjacent pair are tied. Two existing nodes i and i form a link with probability P(i, i ). Say that G(t) exhibits geodesic preferential linking if P(i, i ) l(i, i ).

25 geodesic preferential linking The geodesic distance l(i, i ) between i and i in G is the minimum length of a path (i, i 1, i 2,..., i ) where each adjacent pair are tied. Two existing nodes i and i form a link with probability P(i, i ). Say that G(t) exhibits geodesic preferential linking if P(i, i ) l(i, i ).

26 geodesic preferential linking The geodesic distance l(i, i ) between i and i in G is the minimum length of a path (i, i 1, i 2,..., i ) where each adjacent pair are tied. Two existing nodes i and i form a link with probability P(i, i ). Say that G(t) exhibits geodesic preferential linking if P(i, i ) l(i, i ).

27 geodesic preferential linking The geodesic distance l(i, i ) between i and i in G is the minimum length of a path (i, i 1, i 2,..., i ) where each adjacent pair are tied. Two existing nodes i and i form a link with probability P(i, i ). E A B Say that G(t) exhibits geodesic preferential linking if C D P(i, i ) l(i, i ). F

28 geodesic preferential linking Proportion of pairs linked 1e 07 1e 05 1e , , , 2006 slope = 5.61 slope = 6.39 slope = Separation

29 affiliation networks G B = (N P, E B ) G U = (N, E U )

30 affiliation networks A 1 B E 5 2 C 4 D 3 G U = (N, E U ) G B = (N P, E B )

31 affiliation networks A 1 B A E 5 2 C 4 D 3 D C E B G B = (N P, E B ) G U = (N, E U )

32 affiliation networks A 1 B E 5 2 C 4 D 3 D C E B A N = {A, B, C, D, E} P = {1, 2, 3, 4, 5} E = {(A, 1), (B, 1), (B, 2), (C, 2), (C, 3), (C, 4), (C, 5), (D, 2), (D, 3), (E, 5)} N = {A, B, C, D, E} E = {(A, B), (B, C), (B, D), (C, D), (C, E)}

33 affiliation networks Coming up: Time series plots for the Mathematical Reviews range of values long-term trends short-term shifts contrast between pure and applied subnetworks

34 assortative mixing Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The (degree) assortativity of G U [5] is r conn = E(rs) E(r)E(s) Var(r)Var(s), where r, s range over the remaining degrees of pairs of neighbors. Let the assortativity by productivity of G B be r prod = E(pq) E(p)E(q) Var(p)Var(q), where p, q range over the remaining productivities of pairs of coauthors.

35 assortative mixing Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The (degree) assortativity of G U [5] is r conn = E(rs) E(r)E(s) Var(r)Var(s), where r, s range over the remaining degrees of pairs of neighbors. Let the assortativity by productivity of G B be r prod = E(pq) E(p)E(q) Var(p)Var(q), where p, q range over the remaining productivities of pairs of coauthors.

36 assortative mixing Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The (degree) assortativity of G U [5] is r conn = E(rs) E(r)E(s) Var(r)Var(s), where r, s range over the remaining degrees of pairs of neighbors. Let the assortativity by productivity of G B be r prod = E(pq) E(p)E(q) Var(p)Var(q), where p, q range over the remaining productivities of pairs of coauthors.

37 assortative mixing Assortativity by connectivity r conn Terminal year (5 year windows)

38 assortative mixing Assortativity by productivity r prod Terminal year (5 year windows)

39 clustering coefficients Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The global clustering coefficient of G U [6] is C = #{closed 2-paths}. #{2-paths} The coauthorship-exclusive clustering coefficient of G B is C = #{closed chordless author-centered 4-paths}. #{chordless author-centered 4-paths}

40 clustering coefficients Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The global clustering coefficient of G U [6] is C = #{closed 2-paths}. #{2-paths} The coauthorship-exclusive clustering coefficient of G B is C = #{closed chordless author-centered 4-paths}. #{chordless author-centered 4-paths}

41 clustering coefficients Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The global clustering coefficient of G U [6] is C = #{closed 2-paths}. #{2-paths} The coauthorship-exclusive clustering coefficient of G B is C = #{closed chordless author-centered 4-paths}. #{chordless author-centered 4-paths}

42 clustering coefficients Take the attribution graph G B = (N P, E B N P) with coauthorship graph projection G U = (N, E U N N). The global clustering coefficient of G U [6] is C = #{closed 2-paths}. #{2-paths} The coauthorship-exclusive clustering coefficient of G B is E F 2 C B 1 A D 3 C = #{closed chordless author-centered 4-paths}. #{chordless author-centered 4-paths}

43 clustering coefficients Global clustering coefficient C Terminal year (5 year window)

44 clustering coefficients Exclusive clustering coefficient C X Terminal year (5 year window)

45 summary Make provisional conclusions. Beware of unificationism.

46 references Adamic L A, Glance N (2005) The political blogosphere and the 2004 U.S. election. Proceedings of the 3rd international workshop on Link discovery. ACM, New York, NY, USA, Goh K et al (2007) The human disease network. PNAS 104 (21): TutorVista Retrieved 1 April Glänzel W (2002) Coauthorship patterns and trends in the sciences ( ). Open Access publications from Katholieke Universiteit Leuven. Newman M E J (2003) Mixing patterns in networks. Physical Review E 67: Barrat and Weigt (2000) On the properties of small-world network models. The European Physical Journal B, 13(3):