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Six Degrees: The Science of a Connected Age: Chapter Eight: Thresholds, Cascades and Predictability (Watts, 2003)

• How do information cascades occur?
• What is important in thinking about this?

The key to information cascades (as opposed to disease outbreaks etc) is the idea of decisions with externalities (people are influenced by their experiences/characteristics etc, and also other people).

Threshold Model of Decision-Making – Disease contagion occurs independently of the past BUT social contagion is highly contingent on others’ opinions.

Threshold rule: We change alternatives once we reach a critical threshold. The threshold may vary by individual, and also by the certainty of the situation

Threshold rules are also determined by decision externalities:

1. Market Externalities (e.g. fax machine).
2. Coordination Externalities in social dilemmas: long/short-term payoff preferences and perception of individual influence.

Capturing Differences – Wide variability increases the probability of cascades:

People are different in terms of:

1. Situation specific information and expertise.
2. Strength of conviction.

The amount of others we listen to varies (depending on time, importance, etc):

1. ↑ volume of opinions = ↓ influence of individual opinions
2. ↑ volume of opinions = ↑ chance that relevant information gets lost.

Cascades in Social Networks – Variability of degree (number of
neighbours/opinions) is central:

• What is the basis for the individual adoption of an innovation?
• Low individual threshold or high density of adopting neighbours?

In order to set the problem out mathematically, define:

1. Innovator: Node activated randomly at start of innovation cycle.
2. Early Adopter (Vulnerable Node): Node which switches to active if it has a single active neighbour.
3. Stable Node: Nodes which are not vulnerable.
4. Critical Upper Degree: Max number of neighbours a node can have and still be activated by one neighbour.

Cascades, Percolation and Phase Transitions – What is the condition for a global cascade in a social system? Recap: Each node is either stable or vulnerable (i.e. each node does not have the same probability of being infected).

1. ↑Early adopters, ↑probability of spread.
2. ↑Size of connected cluster of early adopters, ↑distance of spread.

Global cascades are dependent on presence of a percolating cluster: Well connected nodes are stable (w.r.t individual influence) AND stable nodes cannot be part of a vulnerable cluster. THUS, effective percolation of vulnerable clusters is generally only possible with relatively un-connected nodes.

There are three characteristics of a network where cascades CANNOT occur (phase diagram, Fig. 8.5):

1. High average threshold: Nodes are too resistant to change.
2. Low connectivity: Nodes are vulnerable, but innovation cannot spread. In this context, well connected nodes play a large role in social contagion.
3. High connectivity (this is surprising): All nodes are paying attention to a great number of other nodes, thus the influence of individual (innovator) nodes is low (i.e. all nodes are locally stable) and innovations die out.

Crossing the Chasm – Successful cascades are not as dependent on the characteristics of the innovation (price, quality, design etc) as is commonly thought. Further, situations characterised by high connectivity may appear stable over long periods, but are vulnerable to discontinuous phase transitions (either there are no cascades, or the entire network is affected).

1. At the lower boundary (rel.low connectivity), cascades can only go so far (these are relatively common and small.)
2. At the upper boundary (rel.high connectivity), as soon as a vulnerable cluster activates, stable nodes quickly become exposed to great numbers of early adopters thus triggering a cascade through the whole network (these are relatively rare and large.)
3. Low connectivity implies that some special individuals are important for linking clusters, BUT where connectivity is not a problem, being connected to people who are easily influenced is more important that being connected to a large number or people.

A Nonlinear View of History/Power to the People – Outcomes are hard to predict
without understanding the underlying structure of the network:

1. ‘Success’ and ‘quality’ are in most cases not as strongly linked as we think.
2. History is the result of a contingent mechanism – the consensus of large numbers of individuals (decisions with externalities) – “The seed alone is not enough” (e.g. Napster, architecture, human genome).

Robustness Revisited – Build robust, prepare for fragility:

1. ‘Normal Accidents’ – some disasters are caused by the unexpected interaction of normal behaviour, and are thus almost inevitable.
2. ‘HOT’ – Fixing weak points in robust systems as they present themselves increases the robustness of the system, but doesn’t fix its underlying fragility.

Original Text: in Watts, D. (2003) Six Degrees – The Science of a Connected Age
Six Degrees: The Science of a Connected Age (Open Market Edition)