# What is loyalty?

Loyalty is a measure of a person's attachment to one or more parties. We believe simple party membership to be an insufficient measure of loyalty, for example:

- If a person joins party
*A*today, but was a member of party*B*for the last three years, that person's loyalty to*B*does not vanish immediately. - If a husband and wife are members in parties
*A*and*B*respectively, it is unlikely that either spouse is 100% loyal to their party's vision. - If
*X*is not a member of any party, but*X*'s father is a member of party*A*, then we consider*X*loyal to*A*to a certain degree. - If a person is an associate in a company that has numerous deals with a party, then we consider the person loyal to that party to a certain degree.

To account for these possibilities, we define loyalty in a more complex way. Specifically, **a person's loyalty to a party is a fraction between 0% and 100% such that the sum of a person's loyalties is 100%**.

#### The weights of various relationships

Party membership is a good indication of a person's loyalty, but it is not the only relevant relationship. To deal with cases 2 and 3 above, we give weights to various relationships:

- Party membership has a weight of 1.
- Close kinship (husband/wife, son/daughter, brother/sister, parent, brother/sister-in-law, mother/father-in-law, son/daughter-in-law) has a weight of 0.5.
- Distant kinship has a weight of 0.25

We plan to take other relationships into account in the future, such as partnership in companies with political clients (case 4 above).

#### Time coefficients

An active relationship (party membership, marriage, association in a company) is an indication of loyalty, but past relationships are also relevant. To deal with case 1 above we define the following time coefficients:

- An active relationship is worth 365 points.
- A relationship during the previous year is worth one point for each day it was active.
- A relationship during the two years before the first is worth 0.5 points for each day it was active.

We intend for relationships to start from a coefficient of 33.33% on the day of their inception and to increase to 100% over three years.

For example, if *X* was a member of party *A* in 2017 and 2018, of party *B* in 2019, then joins party *C* on January 1, 2020, then each relationship is worth 365 points, so their coefficients will be equal. If *X* remains in party *C* throughout 2020, then on January 1, 2021:

- Membership in
*C*is worth 365 + 365 points (active relationship on that day, plus every day in 2020). - Memberships in
*A*and*B*are worth 182.5 points each (365 days × 0.5 points each, for the years 2018 and 2019 respectively). - Therefore, membership in
*C*has a coefficient of 730 / 1,095 = 66.66%, while memberships in*A*and*B*have coefficients of 182.5 / 1,095 = 16.66% each.

#### How we calculate loyalty

With the above elements we can proceed to calculate the loyalties of all persons to all parties known to Dignitas. Intuitively, to determine the loyalty of *X* we do the following experiment: we start from *X* and follow one of its relationships at random, giving greater probability to higher-weight relationships (party membership) and to recent or long-term relationships. By following the relationship we reach another entity *Y*, which could be a party of which *X* is or was a member, a company with which *X* is or was associated or a person to whom *X* is or was related. If *Y* is a party, the experiment ends. Otherwise, starting from*Y* we follow a new relationship at random, as many times as needed until we reach a party.

If we repeated this experiment millions of times, we would likely follow many different paths. All these paths end at one party or another. Finally, we count the frequencies of ending at parties *A*, *B*, *C*, ... These frequencies are precisely the loyalty of *X*. If *X* has been a member of a party for at least three years and has no relatives in other parties, *X*'s loyalty to that party will be 100%. In practice, however, more complex distributions arise.

In the interest of transparency, we also give a formal mathematical definition. Let *G* = (*V*, *E*) be the Dignitas graph in which the nodes *V* are entities (persons, parties, companies), and the edges *E* are relations between entities. The edges have values which are computed according to the weights and time coefficients above. The outgoing edges of each node are normalized (the sum of their values is 1). Then *G* is a Markov chain in which the nodes corresponding to parties are absorbing states. We define the loyalty of an entity *u* to be the distribution of absorption probabilities of *u* in nodes corresponding to parties.