Topics In Social Computing (67810) Module 2 (Dynamics) Cascades, Memes, and Epidemics (Networks Crowds & Markets Ch. 19)

Spread of Ideas, Behaviors, Innovations How do things spread?

Spread of Ideas, Behaviors, Innovations Last time we talked about social contagion, and the spread of behaviors in a population Our basic assumptions were: Each individual’s choice is affected by the number of adopters it observes Each individual observes the rest of the population The second assumption is far from true, and we will relax it.

Structure Matters Exposure to an idea / adoption of a technology depends directly on your social contacts, not on arbitrary individuals in the population (We will see different models) The main point we’ll be seeing in all the models: Structure matters a great deal, and affects the dynamics in interesting ways.

Threshold Models for Diffusion of Technologies

Example Suppose all of your friends use skype, and none of them use whatsapp for instant messaging from their phone. Which app will you install? What if your friends split evenly?

Threshold Models for Diffusion of Technologies Our starting point in the first model will be reasoning about rational behavior on the part of the nodes. We will assume that there are two technologies A,B that compete with each other. Each person can choose only one technology. The benefit from choosing a technology depends on what your social contacts have picked. B ? A ?

We will assume one technology is better: 𝑎>𝑏 We summarize the utilities across each link in tabular form where (x,y) denote the utility of the row player (x), and of the column player (y) The utilities above imply the technologies are completely incompatible (no utility from an A-B link to either party). We will assume one technology is better: 𝑎>𝑏 B A (0,0) (a,a) (b,b)

Assumption: A node will choose the technology that maximizes utility. Since nodes have several contacts, their utility will be the sum of utilities from each link. Assumption: A node will choose the technology that maximizes utility. B A ? B A (0,0) (a,a) (b,b)

The node will choose technology A iff 𝑝⋅𝑑⋅𝑎> 1−𝑝 ⋅𝑑⋅𝑏 B A (0,0) (a,a) (b,b) B A ? Suppose a node (with degree 𝑑) has a fraction 𝑝 of neighbors that use tech. A, and (1−𝑝) neighbors that use tech. B The node will choose technology A iff 𝑝⋅𝑑⋅𝑎> 1−𝑝 ⋅𝑑⋅𝑏 𝑎 𝑏 > 1−𝑝 𝑝 𝑎 𝑏 +1> 1 𝑝 𝑝> 𝑏 𝑎+𝑏 =𝜃 Which implies there is a simple threshold (smaller than ½) for which a node will adopt the better technology.

The Dynamics The interaction proceeds in rounds. (we will quickly see that the notion of time is not as important in this model) In each round, we simultaneously ask all nodes if they want to switch to a different technology (given the current state).

Cascades and Invasion We will assume technology B has been adopted by the entire network (it was established earlier) and technology A now appears (a recent development) at some nodes (we will call these nodes the seed). Will the superior technology A take over the entire network? B A

Example 1 Suppose technology 𝐴 is a really good one: 𝑎=10,𝑏=1 From what we’ve seen before: 𝜃= 𝑏 𝑎+𝑏 = 1 11 B A

Example 1 𝑎=10,𝑏=1 ; 𝜃= 1 11 After one round: After a second round: (it’s easy to see eventually A takes over). B A B A

Example 2 B A 𝑎=6,𝑏=5; 𝜃= 5 11 After one round: But now no node will update! B A

Example 3 B A 𝑎=6,𝑏=5; 𝜃= 5 11 After one round: And technology A will oscillate on these nodes (To prevent this in the model we will assume from now on that the initial seed never changes back to B) B A

Adoption from a Stable Seed If the seed is stable, then every node that switches from B to A will never switch back. Why? (prove this by induction) We can therefore ask a simple question: Given a stable seed, will the entire graph ever adopt technology A, or will adoption be stopped? What are the barriers for adoption?

Clusters are a Barrier B A Every node in this cluster has more links inside than outside. In particular, at most 2/5 of the links lead outside. This implies that if the seed is outside of the cluster (even if the seed is the rest of the graph) a threshold below 2/5 is needed to get nodes inside to adopt A.

Clusters are a Barrier B A Breaking up this cluster would cause A to spread, e.g., by targeting some of the nodes in it (adding them to the seed)

Cluster Density We say that a cluster 𝐶⊆𝑉 has density 𝑝 if each node in 𝐶 has at least a fraction 𝑝 of its edges inside the cluster. (notice that a cluster of density p is also a cluster of lower densities) 𝑝= 1 2 𝑝= 3 5

Cluster Density Facts: A connected component is a cluster of density 1. The union of two clusters of density p, is a cluster of density p.

Cluster Density Characterizes the Threshold Theorem: Consider a graph 𝐺, A set of initial adopters 𝑆⊆𝑉 (that do not switch strategies), and an adoption threshold 𝜃. All nodes will adopt technology 𝐴 if and only if the rest of the graph does not contain a cluster of density ≥ 1−𝜃

Proof (density ≥1−𝜃 implies no full adoption): Let C be a cluster of density ≥1−𝜃 that does not contain seed nodes. No node in C will ever adopt tech A. By induction: If no node in C adopts technology A in round t, all nodes have no more than 𝜃 neighbors that adopted A, and so at round t+1 no node adopts A either.

Proof (no full adoption implies cluster with density ≥1−𝜃): Let C be the set of nodes that never adopt tech A. (C is not empty). Claim: C has a density of at least 1−𝜃 This is because by construction all nodes outside C adopted tech A, and if some node in C had more than 𝜃 external neighbors, it would adopt A too.

Cascades on infinite graphs To model the fact that seeds are usually smaller than the graph: Consider an infinite graph with some structure. Will a cascade spread in this graph if started from a finite seed?

Cascades on infinite graphs The simplest example: It is easy to see that if the threshold is under ½, then the technology spreads. If the threshold is above ½, it will not spread regardless of the seed B A

The Cascade Capacity Define the cascade capacity of an infinite graph as the largest value (supremum) of the threshold 𝜃 for which a finite set of adopters can get everyone to adopt technology A. We’ve just seen that the cascade capacity of the infinite chain is 1/2.

Example What is the cascade capacity of this graph?

Example Answer: The capacity is 1/3 How to prove: show that with a lower threshold some seed causes a cascade show that with a higher threshold no seed causes a cascade

Example A cascade capacity of 1/3 implies for example that 𝑎=2,𝑏=1 will not Spread. 𝜃= 𝑏 𝑎+𝑏 = 1 3 Tech A that spreads must be better than B by a factor of more than 2.

How about now?

The Capacity is 1/2 (proof?)

What is the largest possible capacity? Can we create a graph with capacity > ½? This is equivalent to finding a graph in which an inferior technology spreads. Theorem: All infinite graphs with finite degree have a cascade capacity ≤ 1 2 . Proof: Assume to the contrary, that some graph has a greater cascade capacity. Then there exists a threshold 𝜃> 1 2 for which this graph has a cascade from a finite seed.

Consider at round t, the set of nodes 𝑆 𝑡 that has adopted A Let 𝑁 𝑡 be the number of edges from 𝑆 𝑡 to the rest of the graph. B B A B A A B B

Claim 1: 𝑁 0 is positive (and finite). Claim 2: 𝑁 𝑡 strictly decreases with time A contradiction: the number of rounds must therefore be bounded, and the graph cannot fully adopt A. B B A B A A B B

Proof of Claim 2: ( 𝑁 𝑡 strictly decreases with time) Consider a node that adopts A at round t+1. Because 𝜃> 1 2 it must have more neighbors in 𝑆 𝑡 than outside of it. Therefore 𝑁 𝑡+1 < 𝑁 𝑡 B B A B A A B B

Compatibility and Cascades What if a participant could adopt both tech A and tech B at the same time? (You could potentially install two instant messaging programs) +

Compatibility and Cascades We assume that if you adopt both strategies, you get the best of both worlds. In this case won’t everyone just adopt both technologies? AB B A (a,a) (0,0) (b,b) (max(a,b), max(a,b))

Compatibility and Cascades Assume the utility of each node is the sum of utilities from interactions with each neighbor, Assume adopting tech AB has an additional cost of 𝑐 that is paid only once. AB B A (a,a) (0,0) (b,b) (max(a,b), max(a,b))

Will a cascade occur on an infinite chain? AB* B A (a,a) (0,0) (b,b) (max(a,b), max(a,b)) *(and a cost of 𝑐 for adopting AB) Will a cascade occur on an infinite chain? B A

AB* B A (a,a) (0,0) (b,b) (max(a,b), max(a,b)) *(and a cost of 𝑐 for adopting AB) Notice that behavior does not change if we multiply a,b,c by a constant. We shall therefore assume that b=1 for convenience.

It is enough to consider what happens to a boundary node in 2 cases: (a,a) (0,0) (b,b) (max(a,b), max(a,b)) *(and a cost of 𝑐 for adopting AB) It is enough to consider what happens to a boundary node in 2 cases: Case 1 A ? B Case 2 AB ? B

Case 1 𝑈 𝐴 >𝑈 𝐵 iff a>1 𝑈 𝐴𝐵 >𝑈(𝐴) iff 1>c ? B 𝑈 𝐴 =𝑎 𝑈 𝐵 =1 𝑈 𝐴𝐵 =𝑎+1−𝑐 𝑈 𝐴 >𝑈 𝐵 iff a>1 𝑈 𝐴𝐵 >𝑈(𝐴) iff 1>c 𝑈 𝐴𝐵 >𝑈(𝐵) iff a>c

Case 2 𝑈 𝐴 >𝑈 𝐵 iff a>2 𝑈 𝐴𝐵 >𝑈(𝐴) iff 1>c AB ? B 𝑈 𝐴 =𝑎 𝑈 𝐵 =2 𝑈 𝐴𝐵 =𝑎+1−𝑐 𝑈 𝐴 >𝑈 𝐵 iff a>2 𝑈 𝐴𝐵 >𝑈(𝐴) iff 1>c 𝑈 𝐴𝐵 >𝑈(𝐵) iff a>c+1

Combining the two cases

Combining the two cases Interesting implication: if a=1.5 high values of c imply cascade, low values of c imply cascade, Intermediate values block the cascade. The ability to adopt AB gets in the way!

Questions for further thought What happens when values off the diagonal are not 0? What if there are three competing strategies? B A (x,0) (a,a) (b,b) (0,x)

Adobe Writer and Adobe Reader Can you speculate why adobe would find it beneficial to give away adobe reader (for reading pdf files) for free?

Adobe Writer and Adobe Reader Assume you are a potential client, willing to pay money for adobe writer. What if no one can read documents that you create? What if everyone can? Adobe gave reader for free in order to create a market for its more sophisticated tools.

More Examples: Cellphone carriers (e.g., AT&T) offer unlimited free calls to other users of the same network. Why does google offer google docs + gmail + google plus for free? Are the costs really covered by advertising?

Questions for additional thought Can you explain why new versions of Microsoft Word allow you to save documents in formats readable by previous versions? How useful is the first telephone in the world? How did this affect the way phones were deployed? In contrast, is a cell phone useful in a world with no other cell phones? Why did Skype succeed as a voice-over-IP client where others failed?

Weak & Strong Ties Recall Granovetter’s view of weak links: These are links that are less likely to have triadic closure, often connect between dense clusters (rather than inside) Connect to remote location (form bridges) Lack of triadic closure implies that these links do not transmit the cascades well in the threshold model.

Weak & Strong Ties The threshold model is suited for things that spread well across strong links (but tight clusters also form barriers to the spread). If a close friend is “infected” other close friends are also likely to be. People may have a high threshold when engaged in high risk activities, like protesting against an oppressive regime. Weak links in this case do not help much. Weak links are much more important when spreading things that do not require a threshold