Contagion. Complex Networks CSYS/MATH 303, Spring, Prof. Peter Dodds

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1 Complex Networks CSYS/MATH 33, Spring, 211 Basic Social Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3. License. 1 of 58

2 Outline Basic Basic Social Social 2 of 58

3 models Some large questions concerning network contagion: 1. For a given spreading mechanism on a given network, what s the probability that there will be global spreading? 2. If spreading does take off, how far will it go? 3. How do the details of the network affect the outcome? 4. How do the details of the spreading mechanism affect the outcome? 5. What if the seed is one or many nodes? Basic Social Next up: We ll look at some fundamental kinds of spreading on generalized random networks. 3 of 58

4 Spreading mechanisms uninfected infected General spreading mechanism: State of node i depends on history of i and i s neighbors states. Doses of entity may be stochastic and history-dependent. May have multiple, interacting entities spreading at once. Basic Social 4 of 58

5 Spreading on Random Networks For random networks, we know local structure is pure branching. Successful spreading is contingent on single edges infecting nodes. Success Failure: Basic Social Focus on binary case with edges and nodes either infected or not. 5 of 58

6 condition We need to find: r = the average # of infected edges that one random infected edge brings about. Define b k as the probability that a node of degree k is infected by a single infected edge. r = k= + kp k k }{{} prob. of connecting to a degree k node k= {}}{ kp k k (1 } {{ b k) } Prob. of no infection b k }{{} Prob. of infection }{{} # outgoing infected edges (k 1) }{{} # outgoing infected edges Basic Social 6 of 58

7 condition Our contagion condition is then: r = (k 1) kp k k b k > 1. k= Basic Social Case 1: If b k = 1 then R = (k 1) kp k k k= = k(k 1) k > 1. Good: This is just our giant component condition again. 7 of 58

8 condition Case 2: If b k = b < 1 then R = (k 1) kp k k b > 1. k= Basic Social A fraction (1-b) of edges do not transmit infection. Analogous phase transition to giant component case but critical value of k is increased. Aka bond percolation ( ). Resulting degree distribution P k : P k = bk i=k ( ) i (1 b) i k P i. k Insert question from assignment 7 ( ) We can show F P (x) = F P (bx + 1 b). 8 of 58

9 condition Cases 3, 4, 5,...: Now allow b k to depend on k Asymmetry: Transmission along an edge depends on node s degree at other end. Possibility: b k increases with k... unlikely. Possibility: b k is not monotonic in k... unlikely. Possibility: b k decreases with k... hmmm. b k is a plausible representation of a simple kind of social contagion. The story: More well connected people are harder to influence. Basic Social 9 of 58

10 condition Example: b k = 1/k. r = k=1 = (k 1)kP k b k = k k=1 (k 1)P k k k=1 (k 1)kP k k k = 1 1 P k Basic Social Since r is always less than 1, no spreading can occur for this mechanism. Decay of b k is too fast. Result is independent of degree distribution. 1 of 58

11 condition Example: b k = H( 1 k φ) where < φ 1 is a threshold and H is the Heaviside function ( ). Infection only occurs for nodes with low degree. Call these nodes vulnerables: they flip when only one of their friends flips. Basic Social r = k=1 (k 1)kP k b k = k k=1 (k 1)kP k H( 1 k k φ) = 1 φ k=1 (k 1)kP k k where means floor. 11 of 58

12 condition The contagion condition: r = 1 φ k=1 (k 1)kP k k > 1. Basic Social As φ 1, all nodes become resilient and r. As φ, all nodes become vulnerable and the contagion condition matches up with the giant component condition. Key: If we fix φ and then vary k, we may see two phase transitions. Added to our standard giant component transition, we will see a cut off in spreading as nodes become more connected. 12 of 58

13 Social Some important models (recap from CSYS 3) Tipping models Schelling (1971) [8, 9, 1] Simulation on checker boards. Idea of thresholds. Threshold models Granovetter (1978) [7] Herding models Bikhchandani et al. (1992) [1, 2] Social learning theory, Informational cascades,... Basic Social 14 of 58

14 Threshold model on a network Basic Social Original work: A simple model of global cascades on random networks D. J. Watts. Proc. Natl. Acad. Sci., 22 [12] Mean field Granovetter model network model Individuals now have a limited view of the world 15 of 58

15 Threshold model on a network Interactions between individuals now represented by a network Network is sparse Individual i has k i contacts Influence on each link is reciprocal and of unit weight Basic Social Each individual i has a fixed threshold φ i Individuals repeatedly poll contacts on network Synchronous, discrete time updating Individual i becomes active when number of active contacts a i φ i k i Activation is permanent (SI) 16 of 58

16 Threshold model on a network Basic Social t=1 t=2 t=3 e e e a a a d d d b c b c b c All nodes have threshold φ = of 58

17 The most gullible Vulnerables: Recall definition: individuals who can be activated by just one contact being active are vulnerables. The vulnerability condition for node i: 1/k i φ i. Means # contacts k i 1/φ i. Key: For global cascades on random networks, must have a global component of vulnerables [12] For a uniform threshold φ, our contagion condition tells us when such a component exists: Basic Social r = 1 φ k=1 (k 1)kP k k > of 58

18 Cascades on random networks ( n.b., z = k ) Cross section of the cascade window from Fig. 1, at.18. (a) The * e time required for a cascade to terminate diverges at both the lower and boundaries of the cascade window, indicating two phase transitions. (b) rison between connected components of the network and the properties bal cascades. The frequency of global cascades in the numerical model circles) is well approximated by the fractional size of theextended vulneruster (short dashes). subcomponent For comparison, the size of the vulnerable >. cluster is own, both the exact solution derived in the text (long dashes) and the e over 1, realizations of a random graph (crosses). The exact and ical solutions agree everywhere except at the upper phase transition, the finite size of the network (n 1,) affects the numerical results., the average sizeboundary [3, 4, 11] of global cascades is shown (solid circles) and compared e exact solution for the largest connected component (solid line). e average size of global cascades (solid circles) is clearly not ned either by the size of the vulnerable cluster Sv, or by Se, but the connectivity of the network as a whole. This is a surprising APPLIED Top curve: final fraction infected if successful. Middle curve: chance of starting a global spreading event (cascade). Bottom curve: fractional size of vulnerable subcomponent. [12] Cascades occur only if size of vulnerable System is robust-yet-fragile just below upper Ignorance facilitates spreading. MATHEMATICS Basic Social 19 of 58

19 Cascades on random networks Basic Social Time taken for cascade to spread through network. [12] Two phase transitions. ( n.b., z = k ) Largest vulnerable component = critical mass. Now have endogenous mechanism for spreading from an individual to the critical mass and then beyond. 2 of 58

20 Cascade window for random networks Basic Social ( n.b., z = k ) Outline of cascade window for random networks. 21 of 58

21 Cascade window for random networks Basic Social 3 z no cascades S Fraction of Vulnerables Final cascade size.2 No Cascades No Cascades Possible Cascades Low influence z High influence influence 1 5 cascades Example networks φ = uniform individual threshold 22 of 58

22 Social Prob(activation) Granovetter s Threshold model recap φ Assumes deterministic response functions φ = threshold of an individual. f (φ ) = distribution of thresholds in a population. F (φ ) = cumulative distribution = φ φ = f (φ )dφ φ t = fraction of people rioting at time step t. Basic Social 24 of 58

23 Social Sciences Threshold models Basic Social At time t + 1, fraction rioting = fraction with φ φ t. φ t+1 = φt f (φ )dφ = F (φ ) φ t = F (φ t) Iterative maps of the unit interval [, 1]. 25 of 58

24 Social Sciences Threshold models Action based on perceived behavior of others. Pr(a i,t+1 =1) 1 A f (φ ) 2.5 B φ t+1 = F (φ t ) 1 C Basic Social φ 1 i φ i,t.5 1 φ.5 1 φ t Two states: S and I Recover now possible (SIS) φ = fraction of contacts on (e.g., rioting) Discrete time, synchronous update (strong assumption!) This is a Critical mass model 26 of 58

25 Social Sciences Threshold models Basic f(γ) φ t Social γ φ t Example of single stable state model 27 of 58

26 Social Sciences Threshold models Implications for collective action theory: 1. Collective uniformity individual uniformity 2. Small individual changes large global changes Basic Social Next: Connect mean-field model to network model. Single seed for network model: 1/N. Comparison between network and mean-field model sensible for vanishing seed size for the latter. 28 of 58

27 All-to-all versus random networks all to all networks random networks 1 1 A B.8.8 Basic Social F (a t+1 ) f (φ ) S φ a 1 C.8 a t 1 D.8 k F (a t+1 ) f (φ ) S φ.2 a.4.6 at.8 a k 29 of 58

28 Threshold contagion on random networks Three key pieces to describe analytically: Basic Social 1. The fractional size of the largest subcomponent of vulnerable nodes, S vuln. 2. The chance of starting a global spreading event, P trig = S trig. 3. The expected final size of any successful spread, S. n.b., the distribution of S is almost always bimodal. 31 of 58

29 Threshold contagion on random networks First goal: Find the largest component of vulnerable nodes. Recall that for finding the giant component s size, we had to solve: Basic Social F π (x) = xf P (F ρ (x)) and F ρ (x) = xf R (F ρ (x)) We ll find a similar result for the subset of nodes that are vulnerable. This is a node-based percolation problem. For a general monotonic threshold distribution f (φ), a degree k node is vulnerable with probability b k = 1/k f (φ)dφ. 32 of 58

30 Threshold contagion on random networks Everything now revolves around the modified generating function: F (vuln) P (x) = b k P k x k. k= Basic Social Generating function for friends-of-friends distribution is related in same way as before: F (vuln) R (x) = d dx F (vuln) P (x) d dx F (vuln). P (x) x=1 33 of 58

31 Threshold contagion on random networks Functional relations for component size g.f. s are almost the same... ( ) F π (vuln) (x) = 1 F (vuln) P (1) +xf (vuln) P F ρ (vuln) (x) }{{} central node is not vulnerable F (vuln) ρ (x) = 1 F (vuln) R (1) }{{} first node is not vulnerable ( +xf (vuln) R F (vuln) ρ ) (x) Basic Social Can now solve as before to find S vuln = 1 F (vuln) π (1). 34 of 58

32 Threshold contagion on random networks Second goal: Find probability of triggering largest vulnerable component. Assumption is first node is randomly chosen. Same set up as for vulnerable component except now we don t care if the initial node is vulnerable or not: ( ) F π (trig) (x) = xf P F ρ (vuln) (x) ( ) F ρ (vuln) (x) = 1 F v) (vuln) (1) + xf F ρ (vuln) (x) R R Basic Social Solve as before to find P trig = S trig = 1 F (trig) π (1). 35 of 58

33 Threshold contagion on random networks Third goal: Find expected fractional size of spread. Not obvious even for uniform threshold problem. Difficulty is in figuring out if and when nodes that need 2 hits switch on. Problem solved for infinite seed case by Gleeson and Cahalane: Seed size strongly affects cascades on random networks, Phys. Rev. E, 27. [6] Developed further by Gleeson in Cascades on correlated and modular random networks, Phys. Rev. E, 28. [5] Basic Social 36 of 58

34 Expected size of spread Idea: Randomly turn on a fraction φ of nodes at time t = Capitalize on local branching network structure of random networks (again) Now think about what must happen for a specific node i to become active at time t: t = : i is one of the seeds (prob = φ ) t = 1: i was not a seed but enough of i s friends switched on at time t = so that i s threshold is now exceeded. t = 2: enough of i s friends and friends-of-friends switched on at time t = so that i s threshold is now exceeded. t = n: enough nodes within n hops of i switched on at t = and their effects have propagated to reach i. Basic Social 37 of 58

35 Expected size of spread = active, t= ϕ = 1/3 Basic Social i 38 of 58

36 Expected size of spread t=4 = active at t= = active at t=1 = active at t=2 = active at t=3 = active at t=4 Basic Social i ϕ = 1/3 39 of 58

37 Expected size of spread Notes: Calculations are possible nodes do not become inactive. Not just for threshold model works for a wide range of contagion processes. We can analytically determine the entire time evolution, not just the final size. We can in fact determine Pr(node of degree k switching on at time t). Asynchronous updating can be handled too. Basic Social 4 of 58

38 Expected size of spread Basic Social Pleasantness: Taking off from a single seed story is about expansion away from a node. Extent of spreading story is about contraction at a node. 41 of 58

39 Expected size of spread Notation: φ k,t = Pr(a degree k node is active at time t). Notation: b kj = Pr (a degree k node becomes active if j neighbors are active). Our ( starting point: φ k, = φ. ) k j φ j (1 φ ) k j = Pr (j of a degree k node s neighbors were seeded at time t = ). Basic Social Probability a degree k node was a seed at t = is φ (as above). Probability a degree k node was not a seed at t = is (1 φ ). Combining everything, we have: φ k,1 = φ + (1 φ ) k j= ( ) k φ j j (1 φ ) k j b kj. 42 of 58

40 Expected size of spread For general t, we need to know the probability an edge coming into a degree k node at time t is active. Notation: call this probability θ t. We already know θ = φ. Story analogous to t = 1 case: Basic Social k i ( ) ki φ i,t+1 = φ + (1 φ ) θ j t j (1 θ t) k i j b ki j. j= Average over all nodes to obtain expression for φ t+1 : φ t+1 = φ + (1 φ ) k= P k k j= ( ) k θ j t j (1 θ t) k j b kj. So we need to compute θ t... massive excitement of 58

41 Expected size of spread First connect θ to θ 1 : θ 1 = φ + (1 φ ) k=1 kp k k k 1 ( k 1 j j= ) θ j (1 θ ) k 1 j b kj Basic Social kp k k = R k = Pr (edge connects to a degree k node). piece gives Pr(degree node k activates) of its neighbors k 1 incoming neighbors are active. k 1 j= φ and (1 φ ) terms account for state of node at time t =. See this all generalizes to give θ t+1 in terms of θ t of 58

42 Expected size of spread Two pieces: edges first, and then nodes 1. θ t+1 = φ }{{} exogenous +(1 φ ) with θ = φ. 2. φ t+1 = φ }{{} exogenous k 1 kp k ( ) k 1 θ j t k j (1 θ t) k 1 j b kj k=1 j= }{{} social effects +(1 φ ) k ( ) k P k θ j t j (1 θ t) k j b kj. k= j= }{{} social effects Basic Social 45 of 58

43 Comparison between theory and simulations ES P. GLEESON AND DIARMUID J. CAHALANE z ρ 15 (a) R 1 (b) z FIG. 1. Color online Average density of active nodes in a sson random graph of mean degree z and uniform threshold e R. a Color-coded values of from Eq. 1 on the R,z plane seed fraction =1 2. Lines show approximations to the glocascade boundaries: the solid line encloses the region where 5 atisfied; the dashed line represents the extended cascade bound- From Gleeson and Cahalane [6] z 1 Basic PHYSICAL REVIEW E 75, Social 1 Pure 15 random networks (a) with simple threshold responses R = uniform threshold (our5 φ ); z = average degree; ρ = φ; q = θ; N = 1 5. ρ R φ = 1 3,.5 1 2, 1 and 1 (b) 2. Cascade.5 window is for φ = 1 2 case. FIG. 2. Color online As Fig. 1, but threshold distribution is Gaussian increases. with mean R and standard deviation.2, and seed fraction =. a Cascade boundaries as in Fig. 1; the arrow marks the critical point R c,z c described in the text; b theory and numerical simulation results N=1 5, average over 1 realizations at 46 R of 58 =.2 solid,.362 dashed, and.38 dash-dotted z Sensible expansion of cascade window as φ

44 Notes: Retrieve cascade condition for spreading from a single seed in limit φ. Depends on map θ t+1 = G(θ t ; φ ). First: if self-starters are present, some activation is assured: Basic Social G(; φ ) = k=1 kp k k b k >. meaning b k > for at least one value of k 1. If θ = is a fixed point of G (i.e., G(; φ ) = ) then spreading occurs if G (; φ ) = 1 k (k 1)kP k b k1 > 1. k= Insert question from assignment 8 ( ) 47 of 58

45 Notes: In words: If G(; φ ) >, spreading must occur because some nodes turn on for free. If G has an unstable fixed point at θ =, then cascades are also always possible. Basic Social Non-vanishing seed case: Cascade condition is more complicated for φ >. If G has a stable fixed point at θ =, and an unstable fixed point for some < θ < 1, then for θ > θ, spreading takes off. Tricky point: G depends on φ, so as we change φ, we also change G. 48 of 58

46 General fixed point story: Basic Social θt+1 = G(θt; φ) θt+1 = G(θt; φ) θt+1 = G(θt; φ) θt 1 Given θ (= φ ), θ will be the nearest stable fixed point, either above or below. n.b., adjacent fixed points must have opposite stability types. Important: Actual form of G depends on φ. So choice of φ dictates both G and starting point can t start anywhere for a given G. θt 1 θt 1 49 of 58

47 Comparison between theory and simulations z ρ 15 (a) R 1 (b).5 PHYSICAL REVIEW E 75, z FIG. 2. Color online As Fig. 1, but threshold distribution is aussian From with mean Gleeson R and standard andeviation Cahalane.2, and seed [6] fraction =. a Cascade boundaries as in Fig. 1; the arrow marks the itical point R c,z c described in the text; b theory and numerical mulation results N=1 5, average over 1 realizations at R.2 solid,.362 dashed, and.38 dash-dotted. Now allow thresholds to be distributed according to a Gaussian with mean R. R =.2,.362, and.38; σ =.2. φ = but some nodes have thresholds so effectively φ >. Now see a (nasty) discontinuous phase transition for low k. Basic Social 5 of 58

48 Comparison between theory and simulations ED SIZE STRONGLY AFFECTS CASCADES ON RANDOM q q q 1 (a) z 1 (b) z 1 (c) z From Gleeson and Cahalane [6] FIG. 3. Color online Bifurcation diagrams as described in text r dependence of q on z for =.2 and = at R= a.35, b 71, and c.375. PHYSICAL REVIEW E 75, Basic root finding for the system of three equations Social q= + 1 G q, 1 G q =1, and G q =. For =.2 this yields R c,z c =.3543,3.136 ; this point is Network marked version with an arrow in Fig. 2 a. We remark that the discontinuous transition for from θ t+1 q = 1 G(θ to q t ; φ which ). induces a similar transi- tion in through Eq. 1 occurs due to a saddle-node bifurcation 24. This behavior is quite generic, occurring for a wide variety of parameters, with the exception of the special case point studied here: by Watts. θ For = = and F = as in 13, the coefficient C is zero and the fixed-point equation always has always takes off. a root at q=, with transcritical bifurcations on the q= line giving rise to the observed transitions. However, any nonzero seed size replaces the transcritical bifurcations with saddlenode.35, bifurcations.371, as described and.375. above. We have confirmed the accuracy of these results and Eq. 1 against numerical simulations on other configuration model network topologies 1, including power-law degree distributions with exponential for cutoff : (b) pand k k (c) exp k/ lead 17. to We turn now to the derivation Eqs Our analytical higher approach fixed is based points on methods of G. introduced by Dhar et al. to study the zero-temperature random-field Ising model on Bethe lattices 22. The RFIM is a spin-based model of magnetic materials, and its zero-temperature limit has been bifurcations appear and extensively studied as a model for systems exhibiting hysteresis merge and Barkhausen (b and noise c). 21. A Bethe lattice of coordination number z for integer z is an infinite tree where every node has exactly z neighbors. Dhar et al. derive analytical results valid on Bethe lattices, but their numerical simulations show that the theory also applies very accurately 51 of to58 Plots of stability points n.b.: is not a fixed Top to bottom: R = n.b.: higher values of θ Saddle node

49 Spreadarama Bridging to single seed case: Consider largest vulnerable component as initial set of seeds. Not quite right as spreading must move through vulnerables. But we can usefully think of the vulnerable component as activating at time t = because order doesn t matter. Rebuild φ t and θ t expressions... Basic Social 52 of 58

50 Spreadarama Two pieces modified for single seed: 1. θ t+1 = θ vuln + (1 θ vuln ) k=1 kp k k k 1 ( ) k 1 θ j t j (1 θ t) k 1 j b kj j= Basic Social with θ = θ vuln = Pr an edge leads to the giant vulnerable component (if it exists). 2. φ t+1 = S vuln + (1 S vuln ) k P k k= j= ( ) k θ j t j (1 θ t) k j b kj. 53 of 58

51 Time-dependent solutions Synchronous update Done: Evolution of φ t and θ t given exactly by the maps we have derived. Basic Social Asynchronous updates Update nodes with probability α. As α, updates become effectively independent. Now can talk about φ(t) and θ(t). More on this later of 58

52 I [1] S. Bikhchandani, D. Hirshleifer, and I. Welch. A theory of fads, fashion, custom, and cultural change as informational cascades. J. Polit. Econ., 1: , [2] S. Bikhchandani, D. Hirshleifer, and I. Welch. Learning from the behavior of others: Conformity, fads, and informational cascades. J. Econ. Perspect., 12(3):151 17, pdf ( ) [3] J. M. Carlson and J. Doyle. Highly optimized tolerance: A mechanism for power laws in design systems. Phys. Rev. E, 6(2): , pdf ( ) Basic Social 55 of 58

53 II [4] J. M. Carlson and J. Doyle. Highly optimized tolerance: Robustness and design in complex systems. Phys. Rev. Lett., 84(11): , 2. pdf ( ) [5] J. P. Gleeson. Cascades on correlated and modular random networks. Phys. Rev. E, 77:46117, 28. pdf ( ) [6] J. P. Gleeson and D. J. Cahalane. Seed size strongly affects cascades on random networks. Phys. Rev. E, 75:5613, 27. pdf ( ) [7] M. Granovetter. Threshold models of collective behavior. Am. J. Sociol., 83(6): , pdf ( ) Basic Social 56 of 58

54 III [8] T. C. Schelling. Dynamic models of segregation. J. Math. Sociol., 1: , Basic Social [9] T. C. Schelling. Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities. J. Conflict Resolut., 17: , pdf ( ) [1] T. C. Schelling. Micromotives and Macrobehavior. Norton, New York, [11] D. Sornette. Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin, 2nd edition, of 58

55 IV Basic Social [12] D. J. Watts. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci., 99(9): , 22. pdf ( ) 58 of 58