Machine Learning and Modeling for Social Networks

Transcription

1 Machine Learning and Modeling for Social Networks Olivia Woolley Meza, Izabela Moise, Nino Antulov-Fatulin, Lloyd Sanders 1

2 Introduction to Networks Computational Social Science D-GESS Olivia Woolley Meza 08/05/17 2

3 Motivation Basic concepts and definitions Adjacency matrix, paths, connected components Centrality Degree, closeness, Page Rank, betweenness Structural features (of social networks) Heterogeneity, assortativity, clustering, small world, communities Network models Random graphs, generative models 3

4 4

5 Multiple interconnected social media platforms source: H. Alani, M. Rowe, Mining and Comparing Engagement Dynamics Across Multiple Social Media Platforms, ACM Web Science Conference (WebSci)

6 Seven bridges of Königsberg In 1736 Euler posted the following problem: Is it possible to have a walk in the city of Königsberg, that crosses each of the seven bridges only once? 6

7 Networks: abstraction and representation of relations Source: Wikipedia Solution: No! Unless a node is a starting or endpoint, it must have an even number of edges if every edge is traversed only once. 7

8 Social networks Jacob L. Moreno introduced sociograms in his 1934 book Who Shall Survive? Understand the individual through it s relation to the group 8

9 Table 3.1 Basic statistics for a number of published networks. The properties measured are as follows: total number of vertices n; total number of edges m; mean degree z; mean vertex vertex distance l; type of graph, directed or undirected; exponent α of degree distribution if the distribution follows a power law (or if not; in/out-degree exponents are given for directed graphs); clustering coefficient C (1) from (3.3); clustering coefficient C (2) from (3.6); degree correlation coefficient r, section 3.6. The last column gives the citation for the network in the bibliography. Blank entries indicate unavailable data. Biological Technological Information Social Network Type n m z l α C (1) C (2) r Ref(s). film actors undirected [20, 415] company directors undirected [105, 322] math coauthorship undirected [107, 181] physics coauthorship undirected [310, 312] biology coauthorship undirected [310, 312] telephone call graph undirected [8, 9] messages directed / [136] address books directed [320] student relationships undirected [45] sexual contacts undirected [264, 265] WWW nd.edu directed / [14, 34] WWW Altavista directed /2.7 [74] citation network directed / [350] Roget s Thesaurus directed [243] word co-occurrence undirected [119, 157] Internet undirected [86, 148] power grid undirected [415] train routes undirected [365] software packages directed / [317] software classes directed [394] electronic circuits undirected [155] peer-to-peer network undirected [6, 353] metabolic network undirected [213] protein interactions undirected [211] marine food web directed [203] freshwater food web directed [271] neural network directed [415, 420] Source: Newman, M.E. (2003).The structure and function of complex networks. SIAM review, 45(2),

10 Basic definitions A graph G is defined as G(N,L) Set of nodes (vertices) N Nodes can have attributes Set of links (edges) L Directed (arcs) or undirected (edges) Unweighted or weighted (distance, traveling time, etc.) Links of different types can exist (multiplex networks) 10

11 Network Representations: Adjacency Matrix aij = existence of interaction between i and j wij = weights of interaction between i and j (e.g. number of communications per unit time) C A Adjacency Matrix A has entries aij a ij = ( 1, if w ij > 0 0, else 11

12 Network Representations: Edge and Adjacency Lists Edge list Adjacency list C A

13 Paths Path of length n = ordered collection of n+1 nodes and n links. Eg: (A,B,C,E), (A,D), (C,D),(D,E,C) in G =(N,L) Circuit = closed path (last node = first node) A B C Number of walks length k is given by powers or adjacency matrix D E 13

14 Geodesic paths The geodesic (shortest) path between i and j is minimum number of traversed edges A B A B C C D D E E Distance d(i,j) = shortest path between i and j Diameter D of the graph = max(d(i,j)) 14

15 Connected components A graph G=(N,L) is connected if and only if there exists a path connecting any two nodes in G Connected (Tree) Not Connected (Forest) Connected with loops 15

16 Centrality Measures 16

17 Degree, Strength, Closeness aij = existence of interaction between i and j wij = weight of interaction between i and j (e.g. number of communications per unit time) dij = distance between i and j Node degree k i = Â j a ij Node flux/strength F i = Â j w ij Node closeness D i = Â j d ij 17

18 Eigenvector centrality and PageRank Brin, Sergey, and Lawrence Page. "Reprint of: The anatomy of a large-scale hypertextual web search engine." Computer networks (2012): Eigenvector centrality x i is higher the more high-scoring others a node is connected to: x (t+1) i = nx j=1 A ij x (t) j Solution is dominated by the largest eigenvalue 1 as t!1 Ax = 1 x x i = 1 1 nx A ij x j j=1 PageRank x i downgrades common in-links and deals with directed links: x i = nx j=1 A ij x j k out j + x = AD 1 x + 1 D = max(k out, 1) 1 18

19 Betweenness centrality v Idea: Controlling network flows The number of shortest paths passing through a node v: σ st = number of shortest paths from s to t σ st (v) = number of shortest paths from s to t passing through v 19

20 Structural features (of social networks) 20

21 Giant Component A giant component is a connected component which size scales with the size of the network 21

22 Centrality heterogeneity P(Outdegree >= x) e-06 1e-08 1 P(Indegree >= x) e-06 1e Degree (a) Flickr Degree (b) LiveJournal Degree (c) Orkut Degree (d) YouTube source: Mislove et al. (2007) 22

23 Assortative mixing or homophily Birds of a feather flock together Can be any characteristic E.g. Degree assortativity: Average nearest-neighbor degree for vertices with degree k k nn Degree (a) Flickr (0.49) Degree (b) LiveJournal (0.34) Degree (c) Orkut (0.36) Degree (d) YouTube (0.19) source: Mislove et al. (2007) 23

24 Transitivity and clustering My friends tend to be friends Local clustering coefficient C(i): fraction of pairs of neighbors of a node that are also neighbors of each other. Equivalently, number of closed triples. Source Costa (2008) Question: What is the local clustering coefficient for the node i? Global clustering coefficient: network average 24

25 Small world property Empirical puzzle: Social worlds that are highly clustered but at the same time global distances are short e.g. there are at most 6 degrees of separation between any two randomly chosen individuals Randomly rewire a link with probability p Source: Watts, D. J., & Strogatz, S. H. (1998) A small-world network is a network where the typical distance L between two randomly chosen nodes grows logarithmically with total number of nodes N 25

26 Modularity and community structure Source: Newman, M. E. J. (2011) Computational Social Science D-GESS Olivia Woolley Meza 08/05/17 26

27 Community detection vs Graph partition Graph partitioning specifies the number of subgroups or number of nodes in each subgroup Hierarchical clustering, k means Community detection infers the subgroups from the network structure Divisive algorithms (recursively removing highest betweenness edges) Random walk algorithms (maximizing the time random walkers spend within a community) Modularity 27

28 Modularity m Q s 1 l s L d s 2L 2 is over the m modules of t observed fraction within group connections expected fraction within group connections m modules L links in total l s links within module s d s total module degree Basic idea: High fraction of links within group compared to chance (a null model) Community detection: Find partition with maximal modularity Q 28

29 1. Random graphs 2. Configuration models 3. Generative models Network Models 29

30 1. Random graphs (Erdos-Renyi) Start with a number of nodes n (not connected) Define probability of connection p For all the possible couples of nodes a link is created with probability p 30

31 Degree and clustering are easily computable The degree distribution is the Binomial distribution Pr(k) = n 1 k p k (1 p) n 1 k k! in the limit of large n Pr(k) ' The average degree is: <k> = p(n-1) (n 1)k k! p k e c = ck k! e c Clustering coefficient C is simply p (the probability of any pair existing) No heterogeneous degree distributions No small-world scaling with clustering 31

32 Percolation transition (a) The formation of the Giant Component is not a smooth process Emerges suddenly when <k>=1 This phenomenon is called 1st order phase-transition Size of the giant component S (b) all small, tree-like components giant component + small tree-like components mean Mean degree <k> c Source: A. Clauset Network lectures Figure 1: (a) Graphical solutions to Eq. (11), showing the curve y =1 e cs for three choices of c along with the curve y = S. The locations of their intersection gives the numerical solutions to Eq. (11). Any solution S>0impliesagiant component. (b) The solution to Eq. (11) as a function of c, showing the discontinuous emergence of a giant component at the critical point c = 1, along with some examples random graphs from di erent points on the c axis. 32

33 2. Configuration model Fix the degree sequence or degree distribution Find a network that samples uniformly over all other properties E.g. assign degrees to nodes and add stubs ` Uniformly at random sample stubs and connect them Problem : Creates self and duplicate edges (works better as network size grows) This process can be generalized to any property (see also Exponential Random Graph models ) 33

34 3. Generative models: Preferential attachment Algorithm: Start with a random connected graph At each time step create a new node and attach it to each node i with probability pi proportional to the node degree ki p i = k i P j k j P(K > k) Generates power-law tails (richer-get-richer) P (K) k 3 k 34

35 Network packages MATLAB: MatlabBGL (Boost Graph Library) matlabbgl or Python: NetworkX igraph (originally R, now also python and C/C++) 35

36 Representing and visualizing networks Gephi ( -> Easy and common Pajek ( -> Easy to use NWB ( -> Good for Analysis Visone ( JUNG ( -> library Net Draw ( Pegasus ( -> for huge data 36

37 References Handbook of graphs and networks: from the Genome to the Internet, edited by S. Bornholdt, H. G. Schuster. John Wiley and Sons, Watts,D.J.,& Strogatz, S.H. (1998).Collective dynamics of small- world networks. nature, 393(6684), Newman, M.E. (2003).The structure and function of complex networks. SIAM review, 45(2), Mislove, A., et al. (2007) "Measurement and analysis of online social networks." Proceedings of the 7th ACM SIGCOMM conference on Internet measurement. ACM. Newman, M. E. (2009). Networks: an introduction. Oxford University Press. Easley, D., & Kleinberg, J. (2010). Networks, crowds, and markets. Cambridge: Cambridge University Press. Newman, M. E. J. (2011). Communities, modules and large-scale structure in networks. Nature Physics, 8(1), Fortunato, S. "Community detection in graphs." Physics reports (2010): Laszlo Barabasi web site: 37