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 Spreading and Influence on social networks Computational Social Science D-GESS Olivia Woolley Meza 09/05/17 2

3 Outline Measuring online and offline networks Network sampling Inferring networks through social sensing Simple contagion and disease spread Complex contagion Threshold model Meme content Limited attention and competition 3

4 Measuring networks 4

5 The challenge of sampling networks Why sample? It can be extremely costly or impossible to collect an entire network Computations are too expensive (time or memory) on a full network The difficulty is in preserving properties of the network Sampling protocol induces structure No general procedures for inverting a sample to obtain true distribution 5

6 Standard sampling procedures Probabilistic sampling (knowledge of full network is assumed) Uniform sampling of vertices (and neighbors) Uniform sampling of edges Degree or attribute proportional sampling sampled Seed-based sampling (e.g. crawlers) Snowball sampling: unsampled halo BFS/DFS: pick a seed i and include all vertices for an l+1 breadth or depth -first search tree rooted at i Random walk: pick a seed and jump to neighbors uniformly at random 6

7 Biases emerging from sampling Uniform node sampling only preserves degree distribution of random graphs: e.g. power-law degree distributions are not preserved (Stumpf et al., 2005) Snowball sampling is biased towards higher degree nodes and can induce powerlaw degree distributions in random networks (Clauset, A. and Moore, C., 2005) 7

8 Biases emerging from sampling TABLE II. The changes of quantities in networks by each sampling method. As the sampling fraction gets lower at the very right of each sampling method indicates this, stands for increase, for decrease, for the same, and for depending on networks. Degree Exponent BC Exponent Assortativity r Clustering Coefficient C Node Link Snowball Source: Lee, Sang Hoon, Pan-Jun Kim, and Hawoong Jeong. "Statistical properties of sampled networks." Physical Review E 73.1 (2006):

9 Corrected random walk sampling Metropolis-Hastings Random Walk Let k i denote the degree of node i. At node v a walker chooses a neighbor w uniformly at random. Then: If k v k w the walker moves to w If kv < k w the walker moves to w with probability k v /k w Effectively, we are more likely to transition to low degree nodes, correcting to bias towards higher degree nodes. P(kv = k) P(kv = k) P(kv = k) node degree k node degree k node degree k Degree distribution (pdf) estimated by the sampling techniques and the ground truth (uniform sampler). MHRW and RWRW agree almo source: Gjoka, Minas, et al. "Walking in Facebook: A case study of unbiased sampling of OSNs." Infocom, 2010 Proceedings IEEE. 9

10 Social sensing of offline networks 10

11 Bluetooth sensors to infer social ties Source: Stopczynski, Arkadiusz, et al. "Measuring large-scale social networks with high resolution." PloS one 9.4 (2014): e

12 Structure of face to face interactions How do we infer social ties from proximity patterns? Null model Temporal variation Additional data sources? Source: Stopczynski, Arkadiusz, et al. "Measuring large-scale social networks with high resolution." PloS one 9.4 (2014): e

13 Online and Offline networks Can we infer online social activity form offline activity and vice-versa? Source: Stopczynski, Arkadiusz, et al. "Measuring large-scale social networks with high resolution." PloS one 9.4 (2014): e

14 Spreading and Influence Computational Social Science D-GESS Olivia Woolley Meza 09/05/17 14

15 Modeling spreading processes contact rate recovery rate S I R usceptible nfected ecovered Kermack-McKendrick or SIR model 15

16 Long range connections Short-range clustered connections S I R usceptible nfected ecovered 16

17 Equation approximation Well-mixed assumption Infinite population size N Total number in a state: S I R Total fraction in a state: s j r t s = bsj t j = bsj t r = gj gj Base reproductive number R 0 = R 0 < 1 =) Infection dies out pop. contact rate recovery rate β γ 17

18 Approximating spreading on a network Assume that each of the k neighbors is equally likely to be of type Infected. Probability that a node with k neighbors becomes infected in time interval dt: Expected number of infected neighbors 1 (1 dt) kj ' kj dt (Leading order of Taylor expansion when βdt << 1) Probability contact occurs with a single infected neighbor 18

19 Mean-field network approximation Assume that the mean degree approximates the dynamics every node experiences (mean-field assumption) t s = hkibsj t j = hkibsj t r = gj gj R 0 = β k γ Easier for disease to invade the population for larger <k> 19

20 Spreading threshold and the adjacency spectrum We can write the dynamics on terms of the Adjacency matrix A and find a condition for the epidemic threshold in terms of the largest eigenvalue of A, λ1 dj k dt = Then s k X dj k A kl j l j k when s k 1 dt = X l l dj dt = Mj where M = A I Conveniently, the eigenvectors vr of M and A are the same, and the eigenvalues of M are a translation of the eigenvalues of A: Mv r = Av r Iv r =( )v r ( A kl j l kl )j k Writing the solution as a linear combination of eigenvectors vr nx j(t) = C r v r e ( r )t which is dominated by the largest eigenvalue as t r=1 Thus we are above the epidemic threshold if > 1 1 > 0 equivalently 1 (This derivation follows (Newman 2009), where you can find more details) 20

21 Complex Contagion Network Heterogeneity Meso structure Temporal structure Transmission Mechanis)c independent dependent Strategic choice Exogenous influence Individual Agent Heterogeneity Limita)ons - - Memory En)ty Salience Interac)on Evolu)on 21

22 Threshold model k neighbors m infected neighbors Probability of adop)on given a fixed threshold Φ Φ adoption threshold S usceptible F (m, k) = ( 0 for m apple k 1 for m> k I nfected Watts, Duncan J. "A simple model of global cascades on random networks." Proceedings of the National Academy of Sciences 99.9 (2002): Nodes with more connec)ons are less suscep)ble Captures social reinforcement 22

23 Conditions for a global outbreak Outcome depends on: Threshold distribution Topology (Degree distribution) Initial seed Global cascade requires a percolating component of nodes that adopt if one neighbor is active (connected to seed) On a uniform random graph: hki 1 < hki apple1/ Watts, Duncan J. "A simple model of global cascades on random networks." Proceedings of the National Academy of Sciences 99.9 (2002):

24 Conditions for a global outbreak Mixed effect of degree: Degree must he high enough to ensure connectivity Larger degree can inhibit spread Mixed effect of clustering: For adoption, a density Φ of adopting neighbors is necessary Clusters of connection density 1-Φ outside of the initial adopters won t adopt Contrary to simple contagion where larger degree and less clustering facilitate spread 24

25 Does meme content affect virality? Compute exposure curves for hashtags on Twitter P(K) is the probability a user adopts when they are exposed exactly K times P Estimate P(K) using the fraction of all individuals with K exposures, who have not adopted, and who adopt before K+1 exposures K Figure 3: Sample exposure curves for hashtags #cantlivewithout (blue) and #hcr (red). Source: Romero, Daniel M., Brendan Meeder, and Jon Kleinberg. "Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter." Proceedings of the 20th international conference on World wide web. ACM,

26 Does meme content affect virality? Manual classification into different topics suggests systematic differences in persistence F(P) P 0.01 Persistence F(P) = Political Idioms Music Technology Movies Sports Games Celebrity K Figure 2: F (P ) for the different types of hashtags.the black dots are the average F (P ) among all hashtags, the red x is the average for the specific category, and the green dots indicate the 90% expected interval where the average for the specific set of hashtags would be if the set was chosen at random. Each point is the average of a set of at least 10 hashtags Figure 3: Sample exposure curves for hashtags #cantlivewithout (blue) and #hcr (red). Source: Romero, Daniel M., Brendan Meeder, and Jon Kleinberg. "Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter." Proceedings of the 20th international conference on World wide web. ACM,

27 Does meme content affect virality? Compute exposure curves for hashtags on Twitter P(K) is the probability a user adopts when they are exposed exactly K times P Estimate P(K) using the fraction of all individuals with K exposures, who have not adopted, and who adopt before K+1 exposures Claim: P(K) estimation for higher K selects higher degree nodes, which experience more information overload and are less susceptible Figure 3: Sample exposure curves for hashtags #cantlivewithout (blue) and #hcr (red). K Source: Romero, Daniel M., Brendan Meeder, and Jon Kleinberg. "Differences in the mechanics of information diffusion across topics: idioms, political hashtags, and complex contagion on twitter." Proceedings of the 20th international conference on World wide web. ACM,

28 Limited human attention and meme competition Source: Weng, Lilian, et al. "Competition among memes in a world with limited attention." Scientific reports 2 (2012). Figure 5 Illustration of the meme diffusion model. Each user has a memory and a screen, both with limited size. (a) Memes are propagated along follower links. (b) The memes received by a user appear on the screen. With probability p n, the user posts a new meme, which is stored in memory. (c) Otherwise, with probability 1 p n, the user scans the screen. Each meme x in the screen catches the user s attention with probability p r. Then with probability p m a random meme from memory is triggered, or x is retweeted with probability 1 p m. (d) All memes posted by the user are also stored in memory. Variation in success across memes can be explained purely through limited attention and network heterogeneity Deep influence of News Feeds and other information filters 28

29 References Stumpf, Michael PH, Carsten Wiuf, and Robert M. May. "Subnets of scale-free networks are not scale-free: sampling properties of networks." Proceedings of the National Academy of Sciences of the United States of America (2005): Clauset, Aaron, and Cristopher Moore. "Accuracy and scaling phenomena in Internet mapping." Physical Review Letters 94.1 (2005): Newman, M. E. (2009). Networks: an introduction. Oxford University Press. Barrat, A., Barthelemy, M., & Vespignani, A. Dynamical processes on complex networks. Cambridge: Cambridge University Press.(2008) Porter, Mason A., and James P. Gleeson. "Dynamical systems on networks." Frontiers in Applied Dynamical Systems: Reviews and Tutorials 4 (2016). Good source for all network related (including data links): 29