## Social Hierarchy Dynamics

We have discovered that groups of twelve male mice living in a large environment form linear social hierarchies. Here are data from five different groups over a three-week period. These are a subset from a larger dataset that we present in our recent paper - Williamson CM, Lee W & Curley JP, 2016, Animal Behaviour. The first row of matrices below shows the wins and losses accrued by each individual. Rows represent winners and columns represents losers. Therefore in Cohort A, individual A beat individual B 100 times, but individual B beat individual A 0 times. These matrices can be transformed into binarized matrices - shown in the second row for each cohort. A "1" represents individuals in rows that win more often than they lose against individuals in columns.

## Cohort A

## Cohort B

## Cohort C

## Cohort D

## Cohort E

The matrices are ranked A-L (1-12) according to the I&SI ranking algorithm.
This attempts to move all 1's above the diagonal of the matrix. If all 1's were in the upper triangle it would make a perfect hierarchy.
The number of 1s underneath the diagonal reprsents the ** number of inconsistencies (I)**.
The algorithm then attempts to move those 1s that are beneath the triangle as close as possible to the diagonal.
The total distance of all 1s from the digaonal is called the

**. These social hierarchies are highly linear with only occasional rank inconsistency.**

*strength of inconsistencies (SI)*## Inequality in Wins

## Inequality in Losses

We have also discovered that alpha males vary in their degree of despotism. Although all alpha males win >95% of their fights in stable hierarchies, there are some alpha males who monopolize almost all of the fights that occur. In other hierarchies, the number of fights made by other sub-dominant and mid-ranking animals are proportionally much higher.

The graphs on the left show the ** Lorenz curves** for the cumulative proportion of wins and losses by individuals by cohort.
By calculating the

**of each curve, we are able to determine those cohorts that have a high coefficient indicating highly despotic hierarchy. Much more detail regarding dynamic changes in the inequality of wins and losses is described in the paper.**

*Gini Coefficient*## Temporal changes in social dominance

Changes in the relative social status of individuals can be tracked over time using a
pairwise contest model - ** the Glicko rating system**. All individuals start with the same 2200±300 rating.
Individuals gain points for winning fights and lose points for losing fights.
The number of points won and lost are relative to the degree of difference between the two individuals in each contest.
The charts below represent three weeks of data from each cohort. Hover over each line to identify its rank.

## Cohort A

## Cohort B

## Cohort C

## Cohort D

## Cohort E

We have found that stable individual differences in ranks emerge between 3-5 days after group formation (after about 25% of time above). Alpha males typically emerge first with sub-dominant and middle ranking animals establishing their ranks afterwards. Interestingly, the initial alpha males to emerge are not necessarily those that eventually ascertain the top rank. There are a number of other methods for quantitatively evaluating how hierarchies emerge and how stable they are over the long-term. For instance, in the paper we discuss our findings using various social network measures of dynamic hierarchical organization over time.

Williamson C, Lee W & Curley JP, 2016, Temporal Dynamics of Social Hierarchy Formation and Maintenance in Male Mice, Animal Behaviour 115: 259-272.

We have also studied network structure dynamics in groups of 30 male mice. See here for more info.

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