Viral Fashion, Networks & Statistical Physics

One possible explanation is simple: Zetsche’s influence over what is appropriate is so great that his action single handedly changed the minds of the vast majority of German office workers.

An alternative explanation is more complex, and potentially more insightful.

The suddenness of the change in dress habits suggests that wearing a tie may not be everyones’ first preference (a point confirmed by almost anyone who wears a tie) and that instead it is done to conform to others.

Zetsche’s action broke that conformity in a public setting, influencing perhaps only a few other individuals to not wear a tie.

The actions of these individuals then influences others, and the effect expands through the population.

This takes an epidemiological approach, suggesting the decision to wear a tie in the workplace be modelled much in the same way as disease spreads through a population.

However simplified, exploring such a model has the potential to highlight how trends and norms spread through social groups.

How tie wearing evolves after an influencer (red) doesn't wear a tie for one day.

Light colours indicate that a tie is worn, dark indicate that no tie is worn.

This model assumes that each individual decides to wear a tie or not each day based on what the majority of the individuals on their network were wearing the previous day, with a slight preference for not wearing a tie.

The proportion of the population wearing a tie over time shows the same dramatic decrease as in Germany.

The influencer wears no tie on day 5.

The change to a new stable state of no ties is abrupt.

This process suggests that population all wearing a tie is a metastable state — if the influencer never forgoes a tie, no individual will either, and every individual will wear a tie indefinitely.

However, if the system is disturbed by somebody (who is referred to as the influencer, their identity being irrelevant), the action will spread and nobody will be left wearing a tie.

Such a state is comparable to superheated water, where pure water will remain liquid even at temperatures above 100.

All that is required to cause vaporisation through all the water is a small contamination to be introduced to provide a nucleation site.

Furthermore, this change described in both the model and the example is abrupt, there doesn’t seem to be place for a steady state where ties and non-tie wearers coexist, just as liquid and vaporised water don’t commonly exist under the same conditions.

When describing the states of water and other substances, phase diagrams are used which detail the phase (solid, liquid, or gas) of the substance under various pressure and temperature.

Source: MITThese similarities suggest that there may be a phase diagram for tie wearing habits, which describes how the stable state of the system varies as parameters of the model are tweaked.

This is the steady state of the system after the introduction of an influencer.

This allows us to explore how the German scenario could have changed if people didn’t really mind wearing a tie, or if Zetsche only influenced a small subset of the population.

Tie wearing phase diagram starting at state where everyone wears a tie.

Black indicates no-tie is stable, grey indicates tie wearing is stable.

Influence is measured as the proportion of individuals that are connected to the influencer’s node in the network.

Tie annoyance is the size of the penalty to utility from wearing a tie in relation to the utility gained from conforming, a measure of how much individuals would prefer to not wear a tie.

The above diagram shows that Zetsche’s influence isn’t extremely instrumental in determining the final state post meeting, as long as at least some others are influenced.

However, there is a minimum level of tie annoyance required to overcome the desire to conform.

Further parameters can be tweaked.

The stable states are robust to the number of individuals in the network, as long as the average number of connections is small compared to the number of individuals.

The connections between individuals can also be tweaked, in both their average number and variability.

The phase diagrams are produced for a range of connections and connection variability.

As both the average number of connections and variability of connections increases, the opportunities for an influencer to change the habits of the population decrease.

Further, across all connection configurations the influence of the influencer doesn’t have a large effect, and instead the dislike of ties seems to be the determining factor.

This is a logical conclusion, as the dislike of ties is what drives the initial tie wearing state to be metastable as opposed to inherently stable.

So, is this model useful?These simulations and complex resulting behaviour rests on a group of interacting individuals following a set of simple rules.

This reductionist approach ignores the nuances of office fashion, of which there are plenty.

Ignoring them undoubtedly makes the behaviour of each individual wrong.

However, this doesn’t mean that the statistical behaviour of the whole population is incorrect, as long as the errors in the model of each individual’s decisions roughly cancel each other out.

In reality some in the office pay little attention to the dress of others, while others ensure that their dress not only conforms but rises above average.

Through these population scale properties, the model can shed light on the process at hand and enhance our understanding of how norms can spread through society.

In other applications, models of this type have shown to be useful for simulating pedestrian movements and social policy.

Technical notes:Individual decision making:The decision to wear a tie or not is modelled as a utility maximisation problem, using the following utility function:Whichever decision (tie or no tie) yields the greater utility is taken for the day.

Network layout:The networks were generated as fully connected random graphs.

While small world networks would have been preferable to model social connections, the use of random graphs yielded similar characteristic path lengths and reduced run time for lengthly simulations.

Code:All code is available here on Github.

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