Spreading and Opinion Dynamics in Social Networks

Transcription

1 Spreading and Opinion Dynamics in Social Networks Gyorgy Korniss Rensselaer Polytechnic Institute 05/27/2013 1

2 Simple Models for Epidemiological and Social Contagion Susceptible-Infected-Susceptible (SIS) model [Contact Process (CP)] Threshold model (threshold contact process) Binary agreement model(s) (influencing with committed minorities) 2

3 The SIS model Harris, Ann. Probab, 2, 969 (1974); Liggett (1999); Kephart and White (1991). Infection (SI): node i becomes infected at rate if at least one of its neighbors is infected 0: Susceptible 1: Infected

4 The SIS model Cure (IS): at rate (independent of the neighborhood) 4

5 The SIS model local mean-field (MF) approximation: homogeneous MF (reasonable approx. for ER, SW networks):, for all nodes: 5

6 Homogeneous MF:, stationary (or equilibrium) state: if epidemic threshold: if 6

7 Homogeneous MF:, stationary (or equilibrium) state: absorbing phase: virus extinction if if active phase: non-zero prevalence epidemic threshold: 7

8 Heterogeneous (degree-lumped) MF (HMF): : relative fraction of infected nodes with degree Pastor-Satorras and Vespignani, PRL 86, 3200 (2001); PRE 63, (2001). 8

9 or uncorrelated random graphs: Pastor-Satorras and Vespignani, PRL 86, 3200 (2001); PRE 63, (2001). 9

10 Self-consistent solution for the steady-state: Pastor-Satorras and Vespignani, PRL 86, 3200 (2001); PRE 63, (2001). 10

11 Epidemic Threshold (HMF) (when 0 solution emerges, infection spreads and becomes endemic) for for if if Pastor-Satorras and Vespignani, PRE 65, (2002) 11

12 Epidemic Threshold (HMF) Implications for uncorr. scale-free (usf) nets. for for for for Pastor-Satorras and Vespignani, PRE 65, (2002) 12

13 Prevalence (HMF) (stationary density of infected individuals) / / ( for ) Pastor-Satorras and Vespignani, PRL 86, 3200 (2001); PRE 63, (2001). 13

14 Prevalence (stationary density of infected individuals) SW BA - - / Pastor-Satorras and Vespignani, Phys. Rev. E 63, (2001). 14

15 Exact results for arbitrary graphs Wang et al., Proc. SRDS IEEE (2003) p. 25. Ganesh et al., Proc. IEEE INFOCOM (2005) p Chakrabarti et al., ACM Trans. Inf. Syst. Secur. 10, 13 (2008). Chatterjee and Durrett, Ann. Probab. 37, 2332 (2009). Van Mieghem et al. IEEE/ACM Trans. Networking 17, 1 (2009). R. Durrett, PNAS 107,4491 (2010). Prakash et al., arxiv: (2010). Van Mieghem, EPL 97, (2012). 15

16 Exact results for arbitrary graphs SF: for for Chung, Lu, and Vu, PNAS 100, 6313 (2003). Chung, Lu, and Vu, Annals of Combinatorics 7, 21 (2003). for for Castellano and Pastor-Satorras et al., PRL 105, (2010). 16

17 Exact results for arbitrary graphs / for for Castellano and Pastor-Satorras et al., PRL 105, (2010). 17

18 Some Recommended Reading T.E. Harris, Ann. Probab. 2, (1974); J.O. Kephart and S.R. White, Proc IEEE Comp. Soc. Symp. on Research in Security and Privacy p ; T.M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Springer, New York, 1999). R. Pastor-Satorras and A. Vespignani, PRL 86, 3200 (2001); R. Pastor-Satorras and A. Vespignani, Phys. Rev. E 63, (2001); R. Pastor-Satorras and A. Vespignani, PRE 65, (2002); Y. Wang et al., Proc. SRDS IEEE (2003) p. 25; A. Ganesh et al., IEEE INFOCOM (2005), p. 1455; D. Chakrabarti et al., ACM Trans. Inf. Syst. Secur. 10, 13 (2008); S. Chatterjee and R. Durrett, Ann. Probab. 37, 2332 (2009); P. Van Mieghem et al. IEEE/ACM Trans. Networking 17, 1 (2009); R. Durrett, PNAS 107,4491 (2010); C. Castellano and R. Pastor-Satorras, Phys. Rev. Lett. 105, (2010); Prakash et al., arxiv: (2010); 18 Van Mieghem, EPL 97, (2012);

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20 Threshold model Rules and model description if a node is inactive, it becomes active if at least a fraction of its neighbors are active (could include deactivation at rate, won t discuss it here) 5 neighbors Threshold φ=0.5 Granovetter, AJS 83, (1978). Watts, PNAS 99, (2002). Watts and Dodds, J. Consumer Res. 34, (2007). 20

21 Threshold model Rules and model description if a node is inactive, it becomes active if at least a fraction of its neighbors are active (could include deactivation at rate, won t discuss it here) 5 neighbors Threshold φ=0.5 Fraction of active neighbors Granovetter, AJS 83, (1978). Watts, PNAS 99, (2002). Watts and Dodds, J. Consumer Res. 34, (2007). 21

22 Threshold model Rules and model description if a node is inactive, it becomes active if at least a fraction of its neighbors are active (could include deactivation at rate, won t discuss it here) 5 neighbors Threshold φ=0.5 Fraction of active neighbors Granovetter, AJS 83, (1978). Watts, PNAS 99, (2002). Watts and Dodds, J. Consumer Res. 34, (2007). 22

23 Threshold Model ER 0.18 size of giant component Prob. of global cascades Prob. of Global Cascades RGG 1-initial seed 3-initial seeds Threshold = graph percolation k ~1/ R ( k / ) Radius of Communication 1/ 2 (2d RGG : k R 2 ) Watts, PNAS 99, (2002). Lu et al., Proc. ICNSS 2006; arxiv:cs/

24 Threshold Model ER ~1/ ER (SF) (ER) ϕ Watts, PNAS 99, (2002). 24

25 Threshold model on 2d RGG Cascade window for single node and three node initial seeds Probability of Global Cascades (Cascade Window) 50 (a) 50 (b) Average Degree () k c2 ~ 1/ Threshold () k c1 ~ 4.5 (percolation transition) Threshold () single node initial seed 3 (adjacent) node initial seed Lu et al., Proc. ICNSS 2006; arxiv:cs/

26 Threshold model on RGG Cascade for a single node seed: 10,000, 8,

27 Threshold Model with multiple initiators For no global cascades for initial seed size (initiators) What is the density of initiators sufficient to trigger global cascades? Gleeson and Cahalane, PRE 75, (2007); Chen et al., IEEE ICDM (2010); Kempe et al., ACM SIGKDD (2003) Singh et al., arxiv: (2013). [see talk by Pramesh Singh, Wed 4pm] What is the impact of clustering on cascades (e.g., in empirical networks)? Centola et al., Physica A 374, (2006). Ikeda et al., J. Phys: Conf. Ser. 21, (2010). Singh et al., arxiv: (2013). 27

28 Some Recommended Reading Granovetter, Am. J. Soc. 83, (1978); Watts, PNAS 99, (2002); Kempe et al., ACM SIGKDD (2003); Centola et al., Physica A 374, (2006); Lu et al., Proc. ICNSS 2006; arxiv:cs/ ; Watts and Dodds, J. Consumer Res. 34, (2007); Gleeson and Cahalane, PRE 75, (2007); Chen et al., IEEE ICDM (2010); Ikeda et al., J. Phys: Conf. Ser. 221, (2010); Singh et al., arxiv: (2013); 28

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30 A Binary Agreement Model: The Naming Game Tipping points and the influence of committed minorities in social networks Galam and Jacobs, Physica A (2007). Lu et al., JEIC (2009); Xie et al., PRE (2010). Motivation: historical evidence for minority influence suffragette movement in the early 20th century American civil-rights movement

31 The impact of committed agents that never change their opinions Never doubt that a small group of thoughtful, committed, citizens can change the world. Indeed, it is the only thing that ever has. Margaret Mead Village Health Workers (VHW) - individuals within the social network committed to dispensing basic health advise. Often the most disenfranchised individuals who are picked to be VHWs Quantitative Model?

32 The impact of committed agents that never change their opinions Initially dominating opinion cluster (opinion B) exerts significant resistance just by their sheer network presence. Promoted ideology A (by committed individuals) does not spread into ideological vacuum, but instead, is inhibited by the existing opinion cluster. How much does it take to tip the system (fraction of committed individuals)? How long does it take to tip the system (time to reach a new consensus)?

33 Influencing with committed agents in social/opinion dynamics A model for negotiation/opinion dynamics: the Naming Game (Binary Agreement Model) Baronchelli et al. (2006) 1. At each step a speaker and a listener (neighbor of speaker) are chosen randomly 2. Speaker sends an opinion randomly selected from his list (Speaker Listener) if the sent opinion presents in listener s list, both retain only this opinion Case 1: the listener does not have this opinion; it adds the sent opinion to his list. A,B B A new info A,B A,B Case 2: the listener already has this opinion; both retain only this opinion A,B B A,B B B agree Committed agents: individuals who do not change states [Lu et al., JEIC (2009); Xie et al., PRE (2011)]

34 Tipping point in Social Networks: Influencing with Committed Minorities ER network (sparse random graph, N=200, k=5) p: fraction of agents committed to opinion A p p c p pc 34

35 p p c

36 p p c

37 density of existing opposing opinion Tipping point in Social Networks: Influencing with Committed Minorities fraction of committed agents FC density of existing opposing opinion tipping point ER 10 fraction of committed agents Xie et al., Phys. Rev. E 84, (2011) Zhang et al.,chaos 21, (2011) Xie et al., PLoS One (2012) SF 10 37

38 Tipping point in Social Networks: Influencing with Committed Minorities The meaning of never in finite networks with N >>1 nodes Time to reach consensus (T c ): p p : T ~ c c e cn p p c : T ~ log( N) c Xie et al., Phys. Rev. E 84, (2011); Zhang et al.,chaos 21, (2011); Singh et al., Phys. Rev. E 85, (2012). 38

39 Tipping Points in Real Networks High school network, N=1127, k=9 p c =4.8% network, N=1133, k=9.6 p c =5% Facebook, N=1893, k=15 p c =7.8% Tipping point in empirical social networks is weakly impacted by the average degree k and is typically bounded by 10% [Xie (2012)]. Theory/prediction: Xie et al., PLoS One (2012); Zhang, et al., PRE

40 Tipping Point: Mean-Field Approximation each individual can interact with all others ( complete graph ) the number of individuals, N, is large (N>>1) a small fraction p of individuals is committed to the initially minority opinion A (committed individuals can never change their opinions) p is the density of committed individuals with opinion A, n A is the density of individuals with opinion A n B is the density of individuals with opinion B n AB is the density of individuals with mixed opinion (AB) n A + n B + n AB = 1 -p dn dt dn dt A B n n A A n n B B n n AB AB n n AB AB n n AB AB n n A B 3 2 pn pn B AB Baronchelli et al., PRE (2007) Castelló et al., EPJB (2009) Baronchelli, PRE (2011) Xie et al., PRE (2011) Zhang et al., Chaos (2011) 40

41 Tipping point: flows in opinion space p: fraction of agents committed to opinion A p p c p p c A non-absorbing (B-dominated, mixed) stable fixed point exists; All trajectories starting from initial condition flow to the non-absorbing fixed point Only all-a consensus fixed point exists All trajectories flow to consensus fixed point. 41

42 NG in Social Networks with Strong Community Structure Lu et al., JEIC (2009) Add Health (high-school friendship networks, 2005)

43 The Effects of Competing Committed Groups: Mean-Field Approximation p A : density of committed individuals with opinion A p B : density of committed individuals with opinion B n A : density of individuals with opinion A, n A (0)=0; n B : density of individuals with opinion B, n B (0)=1-p A -p B ; n AB : density of individuals with mixed opinion (AB), p A +p B +n A +n B +n AB = 1 dn dt dn dt A B n n A A n n B B n n AB AB n n AB AB n n AB AB n n A B 3 2 p A p n A B n AB 3 2 p p B B n n A AB Xie et al., PLoS One (2012). 43

44 The Effects of Competing Committed Groups: Mean-Field Approximation p A : density of committed individuals with opinion A p B : density of committed individuals with opinion B B dominates Monostable Bistable A dominates Region I: Two stable steady states separated by a saddle point. Region II: One stable steady state. Xie et al., PLoS One (2012).

45 Characterize the system using the order parameter: (analogous to magnetization in spin systems) => opinion B dominates => opinion A dominates We start with the initial condition: All influencable nodes have opinion B

46 Traversing an off diagonal trajectory in parameter space Fraction committed to A is twice the fraction committed to B finite complete graphs B dominates A dominates Discontinuous transitions occur along curves bounding the bistable region

47 Traversing the diagonal trajectory in parameter space Fraction committed to A and B are equal finite complete graphs B dominates A dominates At the tip of the beak ( ) : System undergoes a continuous transition to a state where neither opinion dominates

48 In Region I, switches occur between the A-dominant and a B-dominant states. m Q. How does switching time scale with network size?

49 Qualitatively similar results for sparse networks Erdos-Renyi Networks Scale-free Networks

50 Some Recommended Reading Baronchelli et al., J. Stat. Mech.: Theory Exp. P06014 (2006); Baronchelli et al., Phys Rev E (2007); Castelló et al., EPJB 71, 557 (2009); Lu et al., J. Econ. Interact. Coord. 4, 221 (2009); S. Galam and F. Jacobs, Physica A 381, 366 (2007); Xie et al., Phys. Rev. E 84, (2011); Xie et al., PLoS One 7(3): e33215 (2012); Zhang et al., Chaos 21, (2011); Zhang et al., PRE 86, (2013); 50

51 Thanks: Q. Lu (Fermilab), J. Xie (Oracle), P. Singh (RPI), W. Zhang (RPI), J. Emenheiser (Naval Res.), M. Kirby (Arizona), D. Galehouse (RPI), S. Sreenivasan (RPI), B. Szymanski (RPI), C. Lim (RPI)