  Given a set of people and a set of events they attend, we address the problem
of measuring connectedness or tie strength between each pair of persons given
that attendance at mutual events gives an implicit social network between
people. We take an axiomatic approach to this problem. Starting from a list of
axioms that a measure of tie strength must satisfy, we characterize functions
that satisfy all the axioms and show that there is a range of measures that
satisfy this characterization. A measure of tie strength induces a ranking on
the edges (and on the set of neighbors for every person). We show that for
applications where the ranking, and not the absolute value of the tie strength,
is the important thing about the measure, the axioms are equivalent to a
natural partial order. Also, to settle on a particular measure, we must make a
non-obvious decision about extending this partial order to a total order, and
that this decision is best left to particular applications. We classify
measures found in prior literature according to the axioms that they satisfy.
In our experiments, we measure tie strength and the coverage of our axioms in
several datasets. Also, for each dataset, we bound the maximum Kendall's Tau
divergence (which measures the number of pairwise disagreements between two
lists) between all measures that satisfy the axioms using the partial order.
This informs us if particular datasets are well behaved where we do not have to
worry about which measure to choose, or we have to be careful about the exact
choice of measure we make.
