Metaphorical meaning dynamics: Identifying patterns in the metaphorical evolution of English words using mathematical modeling techniques

Random growth

The randomly growing hypergraph evolves according to the following principle: Initially, there are only the 414 vertices and no hyperedges. At each step of the growth process there are three possibilities:

1. With probability p₁ a new hyperedge with one vertex in the tail and one vertex in the head is created h_і = (1, 1). This happens whenever the processed line in the data contains a metaphorical transfer of a word which did not appear in the previously processed data lines.

2. With probability p₂ one vertex is added to the tail of one existing hyperedge h_і = (t_і + 1, h_і). This corresponds to a word which already took part in one or more mappings also mapping from a previously unrelated source to a previously related target. The probability is equal for all existing hyperedges (words) at one step.

3. With probability p₃ one vertex is added to the head of one existing hyperedge h_і = (t_і, h_і + 1). This corresponds to a word which already took part in one or more mappings also mapping from an established source of this word to a new target. The probability is equal for all existing hyperedges (words) at one step.

One of the above processes will occur at each step, meaning p₁ + p₂ + p₃ = 1.

To derive the predicted distribution of tail numbers at each step, we describe the change in the hyperedge size distributions between steps:

where N (c, t) stands for the number of hyperedges with a tail number of c at step t. The first term represents the possibility of a new hyperedge with one tail being created. For c > 1, the predicted value of N (c, t) decreases if a hyperedge of tail number c grows by one additional vertex, which occurs with probability p2〈N(t) . ­Similarly, the predicted value increases if a hyperedge of tail number c – 1 grows by one vertex, which is represented by the last term.

Considering the probability distribution rather than the predicted value using P(c, t)=〈 N(c, t) 〉〈 N(t) 〉 (2) gives:

As ⟨N (t)⟩ = p₁ . t, this transforms to:

Assuming that the cardinality distribution does not change at t → ∞, we insert the ­following approximations:

and set the case c = 1 aside. This turns 4 into

The smallest variation step of c is 1, so the left side corresponds to the probability change in c:

For any c ≠ 1 this is solved by an exponential function and accordingly the ­probability distribution is

The derivation of the probability distributions of the head sizes is exactly analogous to p₃ instead of p₂. Taking c to be the number of head vertices, the probability distribution of hyperedge head sizes is then described by

Growth with preferential attachment

For the preferential attachment model, there are also three options at each step. However, in contrast to random growth, the probability of an additional tail or head vertex is not evenly distributed across the hyperedges. Instead, it is proportional to the tail size for an additional tail and to the head size for an additional head. Because of this, the rate of change of the predicted number of hyperedges of tail or head size c will be proportional to ktc¯ , with c̄ the mean expected size either of heads or tails of all hyperedges.

Explicitly, the three options at each step are;

1. Adding a new hyperedge with both tail and head number 1 with probability p₁.

2. With probability p₂ one existing hyperedge is selected. The selection occurs with a probability proportional to each hyperedge’s tail number. An additional vertex is added to this hyperedge’s tail.

3. With probability p₃ one existing hyperedge is selected. The selection occurs with a probability proportional to each hyperedge’s head number. An additional vertex is added to this hyperedge’s head.

The mean tail number

As such, the expected tail number distribution changes at each step according to

Considering probabilities ⟨N (c, t)⟩ = P (c, t) · ⟨N⟩ = P (c, t) · p₁ · t results in

Assuming also here that for t → ∞ the cardinality distribution becomes stable and thus time independent, we again insert 5 and omit the case of c = 1:

where the latter reformulation results from the fact that the right-hand side contains the rate of change in c. This gives the differential equation:

the solution to which is spanned by

Interchanging p₂ and p₃, the probability distribution for the head sizes is analogously