Contagion. Basic Contagion Models. Social Contagion Models. Network version All-to-all networks Theory. References. 1 of 58.

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1 Complex Networks CSYS/MATH 33, Spring, 2 Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Basic Social 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 Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3. License. of 58 4 of 58 Outline Basic Spreading on andom Networks Basic Basic Social Social For random networks, we know local structure is pure branching. Successful spreading is contingent on single edges infecting nodes. Success Failure: Social Focus on binary case with edges and nodes either infected or not. 2 of 58 5 of 58 models Some large questions concerning network contagion:. 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? Next up: We ll look at some fundamental kinds of spreading on generalied random networks. Basic Social condition We need to find: 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. k= + kp k prob. of connecting to a degree k node {}}{ kp k ( } {{ b k } Prob. of no infection k= b k Prob. of infection # outgoing infected edges (k # outgoing infected edges Basic Social 3 of 58 6 of 58

2 condition Basic condition Basic Our contagion condition is then: Social Example: b k = /k. Social (k kp k b k >. k= k= (k kp k b k = k= (k kp k k Case : If b k = = then (k kp k k= = k(k >. Good: This is ust our giant component condition again. = k= (k P k = P Since r is always less than, no spreading can occur for this mechanism. Decay of b k is too fast. esult is independent of degree distribution. 7 of 58 of 58 condition Case 2: If b k = b < = then (k kp k b >. k= Basic Social condition Example: b k = H( k φ where < φ is a threshold and H is the Heaviside function (. Infection only occurs for nodes with low degree. Basic Social A fraction (-b of edges do not transmit infection. Analogous phase transition to giant component case but critical value of is increased. Aka bond percolation (. esulting degree distribution P k : P k = bk i=k ( i ( b i k P i. k Insert question from assignment 7 ( We can show F P (x = F P (bx + b. 8 of 58 Call these nodes vulnerables: they flip when only one of their friends flips. k= φ = k= (k kp k (k kp k b k = k= (k kp k H( k φ where means floor. of 58 condition Basic condition Basic 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. Social The contagion condition: φ k= (k kp k >. As φ, 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, 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. Social 9 of 58 2 of 58

3 Social Basic Threshold model on a network Basic Social Social Some important models (recap from CSYS 3 Tipping models Schelling (97 [8, 9, ] Simulation on checker boards. Idea of thresholds. Threshold models Granovetter (978 [7] Herding models Bikhchandani et al. (992 [, 2] Social learning theory, Informational cascades,... t= t=2 t=3 e e a a d d b b b c c All nodes have threshold φ =.2. a c e d 4 of 58 7 of 58 Threshold model on a network Basic The most gullible Basic Original work: A simple model of global cascades on random networks D. J. Watts. Proc. Natl. Acad. Sci., 22 [2] Mean field Granovetter model network model Individuals now have a limited view of the world Social Vulnerables: ecall definition: individuals who can be activated by ust one contact being active are vulnerables. The vulnerability condition for node i: /k i φ i. Means # contacts k i /φ i. Key: For global cascades on random networks, must have a global component of vulnerables [2] For a uniform threshold φ, our contagion condition tells us when such a component exists: Social φ k= (k kp k >. 5 of 58 8 of 58 Threshold model on a network Basic Fig.. Cascade windows for the threshold model. TheSocial dashed line encloses the region of the (, plane in which the cascade condition (Eq. 5 is satisfied * for a uniform random graph with a homogenous threshold distribution f( (. The solid circles outline the region in which global Network cascades version occur for * the same parameter settings in the full dynamical model All-to-all fornetworks n, (averaged over random single-node perturbations. Interactions between individuals now represented by a network Network is sparse Individual i has k i contacts explicitly excluding the percolating cluster (when it exists from the sum nqnx n. Using Eq. 3b, it follows that Sv H( P G(H(, where H( satisfies Eq. 3a. Outside the cascade window, the only solution to Eq. 3a is H(, which yields Sv (and therefore no cascades as expected. But inside the cascade window, where the percolating vulnerable cluster exists, Eq.3a has an additional solution that corresponds to a non-ero value of Sv. In the special case of a uniform random graph with homogeneous thresholds, we obtain Sv Q(K, e (H Q(K, H, * * in which H satisfies H Q(K, e (H Q(K, H. We * * contrast this expression with that for the sie of the entire connected component of the graph, S e S (32, which is equivalent to allowing K 3 (or 3. In Fig. 2b we show the exact solutions * * for both Sv (long-dashed line and S (solid line for the case of *.8, and compare these quantities with the frequency and sie of global cascades observed in the full dynamical simulation of, nodes averaged over, random realiations of the network and the initial condition. (The corresponding numerical valuesforsv andsareindistinguishablefromtheanalyticalcurves, except near the upper boundary of the window. The frequency of global cascades (open circles that is, cascades that are successful is obviously related to the sie of the vulnerable component: the larger is Sv, the more likely a randomly chosen initial site is to be a part of it. In particular, if Sv does not percolate, then global cascades are impossible. Fig. 2b clearly supports this intuition, but it is equally clear that, within the cascade window, Sv seriously underestimates the likelihood of 6 a global of 58 cascade. The reason is that, according to our original decision rule, an individual s choice of state depends only on the states of its neighbors; hence, even stable vertices, although they do not participate in the initial stages of a global cascade, can still trigger them as long as they are directly adacent to the vulnerable cluster. The true likelihood of a global cascade is therefore determined by the sie of what we call the extended vulnerable cluster Se, consisting of the vulnerable cluster itself, and any stable vertices immediately adacent to it. We have not solved forse exactly (although this may be possible, but it is relatively simple to determine numerically, and as the corresponding (dotted curve in Fig. 2b demonstrates, the average value of Se is an excellent approximation to the observed frequency of global cascades. Influence on each link is reciprocal and of unit weight 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 Cascades on random networks ( n.b., = Fig. 2. Cross section of the cascade window from Fig., at.8. (a The * average time required for a cascade to terminate diverges at both the lower and upper boundaries of the cascade window, indicating two phase transitions. (b Comparison between connected components of the network and the properties of global cascades. The frequency of global cascades in the numerical model (open circles is well approximated by the fractional sie of theextended vulnerable cluster (short dashes. subcomponent For comparison, the sie of the vulnerable >. cluster is also shown, both the exact solution derived in the text (long dashes and the average over, realiations of a random graph (crosses. The exact and numerical solutions agree everywhere except at the upper phase transition, where the finite sie of the network (n, affects the numerical results. Finally, the average sieboundary [3, 4, ] of global cascades is shown (solid circles and compared with the exact solution for the largest connected component (solid line. The average sie of global cascades (solid circles is clearly not governed either by the sie of the vulnerable cluster Sv, or by Se, but by S, the connectivity of the network as a whole. This is a surprising result, the reason for which is not entirely clear, but a plausible explanation is as follows. If a global cascade is triggered by an initially small seed striking the extended vulnerable cluster, it is guaranteed to occupy the entire vulnerable cluster, and therefore a finite fraction of even an infinite network. At this stage, the small-seed condition no longer holds, and so nodes that are still in the off state can now have multiple (early-adopting neighbors in the on state. Hence, even individuals that were originally classified as stable (the early and late maority can now be toppled, allowing the cascade to occupy not ust the vulnerable component that allowed the cascade to spread initially, but the entire connected component of the graph. That the activation of a percolating APPLIED Top curve: final fraction infected if successful. Middle curve: chance of starting a global spreading event (cascade. Bottom curve: fractional sie of vulnerable subcomponent. [2] Cascades occur only if sie of vulnerable System is robust-yet-fragile ust below upper Ignorance facilitates spreading. MATHEMATICS Basic Social 9 of 58 Watts PNAS April 3, 22 vol. 99 no

4 No Cascades Low influence Example networks Cascades Possible No Cascades High influence Cascades on random networks Basic Social Basic d line encloses q. 5 is satisfied ribution f( cades occur for or n, sts from the ( P the cascade h yields Sv the cascade ts, Eq.3a has value of Sv. omogeneous, H, *, H. We * re connected quivalent to act solutions e case of * ncy and sie imulation of tions of the g numerical ytical curves, t is, cascades sie of the y a randomly f Sv does not. 2b clearly n the cascade of a global ecision rule, states of its do not partrigger them cluster. The ined by the consisting of immediately ugh this may erically, and nstrates, the the observed ( n.b., = Fig. 2. Cross section of the cascade window from Fig., at.8. (a The * average time required for a cascade to terminate diverges at both the lower and upper boundaries of the cascade window, indicating two phase transitions. (b Comparison between connected components of the network and the properties of global cascades. The frequency of global cascades in the numerical model (open circles is well approximated by the fractional sie of theextended vulnerable cluster (short dashes. For comparison, the sie of the vulnerable cluster is also shown, both the exact solution derived in the text (long dashes and the average over, realiations of a random graph (crosses. The exact and numerical solutions agree everywhere except at the upper phase transition, where the finite sie of the network (n, affects the numerical results. Finally, the average sie of global cascades is shown (solid circles and compared with the exact solution for the largest connected component (solid line. The average sie of global cascades (solid circles is clearly not governed either by the sie of the vulnerable cluster Sv, or by Se, but by S, the connectivity of the network as a whole. This is a surprising result, the reason for which is not entirely clear, but a plausible explanation is as follows. If a global cascade is triggered by an initially small seed striking the extended vulnerable cluster, it is guaranteed to occupy the entire vulnerable cluster, and therefore a finite fraction of even an infinite network. At this stage, the small-seed condition no longer holds, and so nodes that are still in the off state can now have multiple (early-adopting neighbors in the on state. Hence, even individuals that were originally classified as stable (the early and late maority can now be toppled, allowing the cascade to occupy not ust the vulnerable component that allowed the cascade to spread initially, but the entire connected component of the graph. That the activation of a percolating PNAS April 3, 22 vol. 99 no APPLIED Time taken for cascade to spread through network. [2] Two phase transitions. Largest vulnerable component = critical mass. Now have endogenous mechanism for spreading from an individual to the critical mass and then beyond. Cascade window for random networks MATHEMATICS Social 2 of 58 Basic 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. Social Sciences Threshold models At time t +, fraction rioting = fraction with φ φ t. φt φ t+ = f (φ dφ = F (φ φ t = F (φ t Iterative maps of the unit interval [, ]. Social 24 of 58 Basic Social ( n.b., = Outline of cascade window for random networks. 2 of of 58 Cascade window for random networks Basic Social Sciences Threshold models Basic no cascades S Fraction of Vulnerables Final cascade sie Social Action based on perceived behavior of others. Pr(a i,t+ = A φ i φ i,t f (φ 2.5 B φ φ t+ = F (φ t C φ t Social influence 5 cascades φ = uniform individual threshold Two states: S and I ecover now possible (SIS φ = fraction of contacts on (e.g., rioting Discrete time, synchronous update (strong assumption! This is a Critical mass model 22 of of 58

5 Social Sciences Threshold models Basic Threshold contagion on random networks Basic 3 Social Social f(γ γ φ t φ t Three key pieces to describe analytically:. The fractional sie 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 sie of any successful spread, S. n.b., the distribution of S is almost always bimodal. Example of single stable state model 27 of 58 3 of 58 Social Sciences Threshold models Implications for collective action theory:. Collective uniformity individual uniformity 2. Small individual changes large global changes Basic Social Threshold contagion on random networks First goal: Find the largest component of vulnerable nodes. ecall that for finding the giant component s sie, we had to solve: F π (x = xf P (F (x and F (x = xf (F (x Basic Social Next: Connect mean-field model to network model. Single seed for network model: /N. Comparison between network and mean-field model sensible for vanishing seed sie for the latter. 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 = /k f (φdφ. 28 of of 58 All-to-all versus random networks F (a t+ all to all networks A C.8 a f (φ a t 5.5 φ S random networks B D.8 k Basic Social Threshold contagion on random networks Everything now revolves around the modified generating function: F (vuln P (x = b k P k x k. k= Generating function for friends-of-friends distribution is related in same way as before: Basic Social F (a t f (φ φ S F (vuln (x = d dx F (vuln P (x d dx F (vuln. P (x x=.2 a a at k 29 of of 58

6 Threshold contagion on random networks Functional relations for component sie g.f. s are almost the same... ( F π (vuln (x = F (vuln P ( +xf (vuln P F (vuln (x central node is not vulnerable F (vuln (x = F (vuln ( first node is not vulnerable ( +xf (vuln F (vuln (x Can now solve as before to find S vuln = F (vuln π (. Basic Social 34 of 58 Expected sie of spread Idea: andomly turn on a fraction φ of nodes at time t = Capitalie 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 = : 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 Threshold contagion on random networks Basic Expected sie of spread Basic Second goal: Find probability of triggering largest vulnerable component. Assumption is first node is randomly chosen. Social = active, t= ϕ = /3 Social 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 (vuln (x ( F (vuln (x = F v (vuln ( + xf F (vuln (x i Solve as before to find P trig = S trig = F (trig π (. 35 of of 58 Threshold contagion on random networks Basic Expected sie of spread Basic Third goal: Find expected fractional sie 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 sie strongly affects cascades on random networks, Phys. ev. E, 27. [6] Developed further by Gleeson in Cascades on correlated and modular random networks, Phys. ev. E, 28. [5] Social t=4 i = active at t= = active at t= = active at t=2 = active at t=3 = active at t=4 Social ϕ = /3 36 of of 58

7 Expected sie of spread Notes: Calculations are possible nodes do not become inactive. Not ust for threshold model works for a wide range of contagion processes. We can analytically determine the entire time evolution, not ust the final sie. We can in fact determine Pr(node of degree k switching on at time t. Asynchronous updating can be handled too. Basic Social Expected sie 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 = case: k i ( ki φ i,t+ = φ + ( φ θ t ( θ t k i b ki. = Average over all nodes to obtain expression for φ t+ : φ t+ = φ + ( φ k P k k= = ( k θ t ( θ t k b k. Basic Social So we need to compute θ t... massive excitement... 4 of of 58 Expected sie of spread Basic Expected sie of spread Basic Social First connect θ to θ : 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. θ = φ + k kp k ( k ( φ θ ( θ k b k k= = kp k = k = Pr (edge connects to a degree k node. piece gives Pr(degree node k activates of its neighbors k incoming neighbors are active. k = φ and ( φ terms account for state of node at time t =. See this all generalies to give θ t+ in terms of θ t... 4 of of 58 Expected sie of spread Notation: φ k,t = Pr(a degree k node is active at time t. Notation: b k = Pr (a degree k node becomes active if neighbors are active. Our ( starting point: φ k, = φ. k φ ( φ k = Pr ( of a degree k node s neighbors were seeded at time t =. 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 ( φ. Combining everything, we have: φ k, = φ + ( φ k = ( k φ ( φ k b k. Basic Social Expected sie of spread Two pieces: edges first, and then nodes. θ t+ = φ exogenous k kp k ( k +( φ with θ = φ. 2. φ t+ = φ exogenous k= = θ t ( θ t k b k } {{ } social effects +( φ k ( k P k θ t ( θ t k b k. k= = social effects Basic Social 42 of of 58

8 Comparison between theory and simulations JAMES P. GLEESON AND DIAMUID J. CAHALANE 5 (a (b Basic PHYSICAL EVIEW E 75, Social FIG.. Color online Average density of active nodes in a FIG. 2. Color online As Fig., but threshold distribution is Poisson random graph of mean degree and uniform threshold Gaussian increases. with mean and standard deviation.2, and seed fraction value. a Color-coded values of from Eq. on the, plane =. a Cascade boundaries as in Fig. ; the arrow marks the with seed fraction = 2. Lines show approximations to the global cascade boundaries: the solid line encloses the region where 5 simulation results N= 5, average over realiations at critical point c, c described in the text; b theory and numerical 46 of 58 is satisfied; the dashed line represents the extended cascade boundary from Eq. 6. b Values of at =.8 from Eq. lines and =.2 solid,.362 dashed, and.38 dash-dotted. numerical simulations symbols, averaged over realiations = 3 and 2 close to the value marked by the arrow, with N= 5. Seed fractions are = 3 solid, 5 3 dashed, but the simulations and full theory show that the respective and 2 dot-dashed. The arrow marks the cascade boundary transitions are actually near 6.4 and 9.2. Condition 5 given by Notes: Eq. 5 in the limit of ero seed fraction. clearly does not accurately represent the effects ofbasic finite seed sie norjames of nonero P. GLEESON F ; see AND below, DIAMUID because J. CAHALANE the function. The theory shows etrieve excellent cascade agreement withcondition simulations. for G is spreading not well approximated from by aastraight line near Social the critical Note the smooth from to nea, and parameters for cascade transitions. the discontinuous single transitionseed back to in limit at larger φ values.. 5 (a We seek an improved cascade condition by extending the The location of the upper transition clearly depends sensitively on the initial Depends activated fraction on map. θ t+ = G(θ series for G q to order q 2. This approximation results in a quadratic t ; φ. equation for the fixed point q A global cascade occurs with high probability when a which we represent as aq 2 small seed results in a large value of. Writing G q as +bq +c=. Under the approximation of small q values, we interpret the existence of a positive root of this qua-.5 First: if self-starters are present, some activation is = C q with coefficients assured: dratic equation to imply q, and hence the impossibility C = k kf k= + n= n +nk p n of global cascades. 5 Note that the first-order approximation 4 5 requires b for global cascades. However, when the k, kp nonlinear k behavior of G is important, it is still possible for G(; φ global cascades to occur when b is negative, provided that and lineariing Eq. 2 near q= gives a first-order = condition for global cascades to occur: C the full solution of the quadratic equation precludes positive b k >., since this..2.3 k= real roots. We therefore extend thecascade boundary to include regions where either b as in 5 or b guarantees that q n increases with n, at least initially. This cascade condition may also be written as 2 4ac, the latter giving to meaning b first order in k > for at least one value of k k p k= k F k F (b k.. 5 If θ = is a fixed C point 2 4C C 2 +2 C C 2 2C 2 +4C C 2. of G (i.e., G(;.5 φ = then 6 In the limit spreading and with F =, occurs this reduces if to the condition derived by Watts using percolation arguments 3. It is straightforward to plot this extended cascade condition: On the, plane of Fig. a the condition 5 is satisfied the dashed lines in Figs. a and 2 a clearly give much inside the solid line; fo.8 in Fig. b, 5 predicts improved approximations to the actual boundaries for global cascade transitions at values G between (; φ5.7 and = for both (kcascades. kp FIG. k. b k Color > online. Average density of active nodes in a Poisson random graph of mean degree and uniform threshold k= value. a Color-coded values of from Eq. on the, plane with seed fraction = 2. Lines show approximations to the global cascade boundaries: the solid line encloses the region where 5 Insert question from assignment 8 ( is satisfied; the dashed line represents the extended cascade 47 of boundary from Eq. 6. b Values of at =.8 from Eq. lines and 58 numerical simulations symbols, averaged over realiations with N= 5. Seed fractions are = 3 solid, 5 3 dashed, and Notes: 2 dot-dashed. The arrow marks the cascade boundary given by Eq. 5 in the limit of ero seed fraction. Basic. The theory shows excellent agreement with simulations. Note the smooth transition from to Social near =, and In words: the discontinuous transition back to at larger values. The location of the upper transition clearly Network depends version sensitively on the initial activated fraction If G(; φ >, spreading must occur because some. A global cascade occurs with high probability when a small seed nodes turn on for free. results in a large value of. Writing G q as From Gleeson and Cahalane [6] Pure 5 random networks (a with simple threshold responses = uniform threshold (our5 φ ; = average degree; = φ; q = θ; N = 5. φ = 3,.5 2, and (b 2. Cascade.5 window is for φ = 2 case Sensible expansion of cascade window as φ If G has an unstable fixed point at θ =, then C cascades are also always possible. = k= + n= Non-vanishing seed case: = C q with coefficients Cascade condition is more complicated for k k φ >. If G has a stable fixed point at θ =, and an unstable fixed point for some < θ <, then for θ > θ, spreading takes off. Tricky point: G depends on φ, so as we change φ, we also change G. k n +nk p kf n k, 4 and lineariing Eq. 2 near q= gives a first-order condition for global cascades to occur: C, since this guarantees that q n increases with n, at least initially. This cascade condition may also be written as k= p k F k F. 5 In the limit and with F =, this reduces to the condition derived by Watts using percolation arguments 3. On the, plane of Fig. a the condition 5 is satisfied inside the solid line; fo.8 in Fig. b, 5 predicts cascade transitions at values between 5.7 and 5.8 for both 48 of 58 General fixed point story: θt+ = G(θt; φ θt θt+ = G(θt; φ Given θ (= φ, θ will be the nearest stable fixed point, either above or below. n.b., adacent 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. Comparison between theory and simulations 5 (a (b.5 PHYSICAL EVIEW E 75, = 3 and 2 close to the value marked by the arrow, but the simulations and full theory show that the respective transitions are actually near 6.4 and 9.2. Condition 5 clearly does not accurately represent the effects of finite seed SEED sie nor SIZE of STONGLY nonero F ; AFFECTS see below, CASCADES because ON the ANDOM function G is not well approximated by a straight line near the critical parameters for cascade transitions. (a We seek an improved cascade condition by extending the series for G q to order q 2. This approximation results in a quadratic.5 equation for the fixed point q which we represent as aq 2 +bq +c=. Under the approximation of small q values, we interpret the existence of a positive root of this quadratic equation to imply 2 q 4, and6 hence the 8 impossibility of global cascades. Note that the first-order approximation 5 requires b for global cascades. However, when the nonlinear behavior (b of G is important, it is still possible for global cascades to occur when b is negative, provided that the full solution of the quadratic equation precludes positive.5 real roots. We therefore extend the cascade boundary to include regions where either b as in 5 or b 2 4ac, the latter giving to first order in C 2 4C C 2 +2 C C 2 2C 2 +4C C 2. 6 (c It is straightforward to plot this extended cascade condition: the dashed.5 lines in Figs. a and 2 a clearly give much improved approximations to the actual boundaries for global cascades FIG. 2. Color online As Fig., but threshold distribution is Gaussian From with mean Gleeson and standard andeviation Cahalane.2, and seed [6] fraction =. a Cascade boundaries as in Fig. ; the arrow marks the critical point c, c described in the text; b theory and numerical simulation results N= 5, average over realiations at =.2 solid,.362 dashed, and.38 dash-dotted. θt θt+ = G(θt; φ Now allow thresholds to be distributed according to a Gaussian with mean. =.2,.362, and.38; σ =.2. φ = but some nodes have thresholds so effectively φ >. Now see a (nasty discontinuous phase transition for low. Comparison between theory and simulations q q q From Gleeson and Cahalane [6] FIG. 3. Color online Bifurcation diagrams as described in text for dependence of q on for =.2 and = at = a.35, b.37, and c.375. Figure 2 a shows on Poisson random graphs with thresholds drawn from a Gaussian distribution of mean and standard deviation =.2. Unlike the assumption in 3 that F =, a Gaussian distribution necessarily implies the presence of negative-valued thresholds among the population, so F. Negative-threshold agents act as a natural seed, since they activate regardless of the states of their neighbors. The presence of these innovators 3 allows us to set = in this case. The extended cascade condition 6 again gives a good approximation to the discontinuous transition at high values. Figure 2 b focuses on the low- transition θt Basic Social 49 of 58 Basic Social 5 of 58 PHYSICAL EVIEW E 75, Basic root finding for the system of three equations Social q= + G q, G q =, and G q =. For =.2 this yields c, c =.3543,3.36 ; this point is Network marked version with an arrow in Fig. 2 a. We remark that the discontinuous transition for from θ t+ q = G(θ to q t ; φ which. induces a similar transi- tion in through Eq. 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 Fo = and F = as in 3, the coefficient C is ero 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 nonero seed sie replaces the transcritical bifurcations with saddlenode.35, bifurcations.37, as described and.375. above. We have confirmed the accuracy of these results and Eq. against numerical simulations on other configuration model network topologies, including power-law degree distributions with exponential for cutoff : (b pand k k (c exp k/ lead 7. to We turn now to the derivation Eqs. 3. Our analytical higher approach fixed is based points on methods of G. introduced by Dhar et al. to study the ero-temperature random-field Ising model on Bethe lattices 22. The FIM is a spin-based model of magnetic materials, and its ero-temperature limit has been bifurcations appear and extensively studied as a model for systems exhibiting hysteresis merge and Barkhausen (b and noise c. 2. A Bethe lattice of coordination number for integer is an infinite tree where every node has exactly neighbors. Dhar et al. derive analytical results valid on Bethe lattices, but their numerical simulations show that the theory also applies very accurately 5 of to58 random graphs where every node has exactly neighbors, provided that short-distance loops are rare. To analye Watts model we extend the approach of 22 in two ways. First, we consider treelike random graphs with arbitrary degree distributions, rather than the Bethe lattices of 22. Second, we account for the PAP, which is the essential difference between Watts update rule and standard FIM dynamics. This difference between the update rules is crucial to our derivation of the dependence of the activated fraction. We begin by replacing the given random graph with degree distribution p k by a tree structure. The top level of the tree is a single node with degree k, and this is connected to Plots of stability points n.b.: is not a fixed Top to bottom: = n.b.: higher values of θ Saddle node

9 Spreadarama Basic I Basic 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. ebuild φ t and θ t expressions... Social [] S. Bikhchandani, D. Hirshleifer, and I. Welch. A theory of fads, fashion, custom, and cultural change as informational cascades. J. Polit. Econ., :992 26, 992. [2] S. Bikhchandani, D. Hirshleifer, and I. Welch. Learning from the behavior of others: Conformity, fads, and informational cascades. J. Econ. Perspect., 2(3:5 7, 998. pdf ( [3] J. M. Carlson and J. Doyle. Highly optimied tolerance: A mechanism for power laws in design systems. Phys. ev. E, 6(2:42 427, 999. pdf ( Social 52 of of 58 Spreadarama Two pieces modified for single seed:. θ t+ = θ vuln + k kp k ( k ( θ vuln θ t ( θ t k b k k= = with θ = θ vuln = Pr an edge leads to the giant vulnerable component (if it exists. 2. φ t+ = S vuln + ( S vuln k P k k= = ( k θ t ( θ t k b k. Basic Social 53 of 58 II [4] J. M. Carlson and J. Doyle. Highly optimied tolerance: obustness and design in complex systems. Phys. ev. Lett., 84(: , 2. pdf ( [5] J. P. Gleeson. Cascades on correlated and modular random networks. Phys. ev. E, 77:467, 28. pdf ( [6] J. P. Gleeson and D. J. Cahalane. Seed sie strongly affects cascades on random networks. Phys. ev. E, 75:563, 27. pdf ( [7] M. Granovetter. Threshold models of collective behavior. Am. J. Sociol., 83(6:42 443, 978. pdf ( Basic Social 56 of 58 Time-dependent solutions Basic III Basic Synchronous update Done: Evolution of φ t and θ t given exactly by the maps we have derived. Asynchronous updates Update nodes with probability α. As α, updates become effectively independent. Now can talk about φ(t and θ(t. Social [8] T. C. Schelling. Dynamic models of segregation. J. Math. Sociol., :43 86, 97. [9] T. C. Schelling. Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities. J. Conflict esolut., 7:38 428, 973. pdf ( [] T. C. Schelling. Micromotives and Macrobehavior. Norton, New York, 978. Social More on this later... [] D. Sornette. Critical Phenomena in Natural Sciences. Springer-Verlag, Berlin, 2nd edition, of of 58

10 IV Basic Social [2] D. J. Watts. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci., 99(9: , 22. pdf ( 58 of 58