Top-Ups 2.0

1 Introduction

“Top-Ups and Telephone” discovered primary mechanisms for how legal innovations diffuse throughout the legal advisory network. We now test the theory of how diffusion happens using a family history pattern. How do firms pass down the knowledge firm by firm?

2 Methods

We assume that firms gain knowledge of how a top-up option works by doing a deal that includes a top-up option with another firm or a deal with a prior adopter of the top-up option. We classify a firm as a “parent,” meaning they pass on the knowledge of the top-up innovation to another firm if:

The cosigner of deal that includes a top-up option is a prior Adopter
The proximate Adopter to a firm: the firm who was a prior Adopter and most recently signed a deal with the firm before it became an Adopter
If the cosigner of the deal is not a prior adopter, then we use the proximate adopter as the parent firm.
All prior Adopters who a firm interacted with the firm before it became an Adopter.

We construct each of these definitions of parent. The following results will use definitions (3) and (4) where specified. Thus, the parent is classified as the cosigner of the deal who is a prior adopter or if the cosigner is not a prior adopter, then we use the most proximate adopter as the parent firm. Similar robustness results could use the other two definitions of parents with fewer firms to analyze as fewer parent relationships exist.

Then, we look at the parents of each firm’s parents and so on to construct a “genealogical” relationship of how firms are connected through the adoption of the top-up option adoption.

We also construct the Cosine Similarity between the texts of each deal to test the theory that as firms become farther removed away from each other through genealogic distance, so does the text of their deal.

3 Results

Table 1: Number of Adopters and their Parents by Type
	Number
Adopter Firms	76
Cosigner Adopter	65
Cosigner Prior Adopter	42
Proximate Adopter	47
Cosign Prior Adopter or Proximate Adopter	63
Multiple Proximate Adopters	37

Figure 1: Histogram of the Number of Interactions with Prior Adopters

Figure 1 shows the frequency distribution of the number of interactions each firm has with a prior Adopter. These are also the number of multiple parent Adopters for each Adopter. The average number of interactions with a prior adopter is 4.72. Though the distribution is right skewed with the majority of firms having only one interaction with a prior Adopter - this is the most proximate Adopter to the top-up option adoption decision.

3.1 Descriptive Statistics

This section shows descriptive statistics about the hierarchy and number of parent and children Adopter firms. These relationships are created by observing the parents of each firm’s parents and so on.

Table 2: Count of Total and Unique Previous Parents of Firms
	Total	Unique
Adopter	76	76
Parent	63	34
Grandparent	46	17
GGP	32	9
GGGP	15	5
GGGGP	7	3
GGGGGP	3	2
GGGGGGP	1	1

Table 2 shows the distribution of the number of parent firm types. This table captures that there are multiple firms who have multiple children. There farthest line that can be traced is that of the fifth great-grandparent. One of these firms exists. The number of farther back connections decreases over time.

Table 3: Number and Mean of Children Firms per Parent Firm
	Sum	Mean
Children	63	1.85
Grandchildren	46	2.71
Great Grandchildren	32	3.56
GG GC	15	3.00
GGG GC	7	2.33
GGGG GC	3	1.50
GGGGG GC	1	1.00

Table 3 shows the number of each type of child firm. It also captures the average number of children that each firm ancestor has. The largest number of firms in a category is the children category as not all firms share in the hierarchal structure. The average number of grandchildren that each grandparent has is 2.71.

These relationships are explored further in the next section.

3.2 Firm Trees

This section shows the relationships between successive parents in a firm family tree structure.

Figure 2: Relationships of firms who share Heller Ehrman as a common ancestor.

flowchart LR
  
  130[Heller Ehrman] --> 61[Wilson Sonsini] 
  61[Wilson Sonsini] --> 44[Skadden Arps]
  61[Wilson Sonsini] --> 50[Paul Weiss]
  61[Wilson Sonsini] --> 58[Fenwick & West]
  61[Wilson Sonsini] --> 86[Perkins Cole]
  61[Wilson Sonsini] --> 285[Crowell & Moring]
  
  44[Skadden Arps] --> 2[Sullivan Cromwell]
  44[Skadden Arps] --> 3[Kirkland & Ellis]
  44[Skadden Arps] --> 9[Cooley LLP]
  44[Skadden Arps] --> 15[Dechert]
  44[Skadden Arps] --> 32[Davis Polk]
  44[Skadden Arps] --> 92[Dorsey & Whitney]
  44[Skadden Arps] --> 94[Husch Blackwell]
  44[Skadden Arps] --> 198[Akin-Gump]
  
  50[Paul Weiss] --> 141[Bryan Cave LLP]
  58[Fenwick & West] --> 104[Pillsbury Winthrop]
  285[Crowell & Moring] --> 84[O'Melveny]
  285[Crowell & Moring] --> 125[Ballard Spahr]

Figure 3: Flowchart for Heller Ehrman family

Figure 2 shows the parent and child relationships of firms whose common ancestor is firm 130: Heller Ehrman.

Figure 4: Relationships of all Adopters

Figure 4 shows the relationships of all Adopters.

Figure 5: Family Relationships Using Multiple Parents

Figure 5 shows the relationships of the firms when using multiple parent relationships. This graph is shown at only four generational levels of the family 130, Heller Ehrman. It reproduces Figure 2 in the multiple parent version.

3.3 Textual Similarity between Adopters

Next, we calculate the cosine similarity between the Adopter firms and their parents at multiple generational levels.

3.3.1 Histograms

Figure 6: Correlation plot of the cosine similarity between Adopter and its parent (p1) and grandparent (p2), great-grandparent (p3), great-great-grandparent (p4) and third great grandparent (p5).

Figure 6 shows the lower triangular Spearman correlation coefficient of the cosine similarity between an Adopter firm and its parent (p1), grandparent (p2), great-grandparent (p3), great-great-grandparent (p4), and third great grandparent (p5). The fourth and fifth generations are negatively correlated with each other. The cosine similarity of the parent is positively correlated to other generations.

Figure 7 shows the distribution of the cosine similarities between an Adopter and its parent, grandparent, great-grandparent, and great-grandparent. As the generations become farther removed from the Adopter, the cosine similarities approach one, which shows the texts become less related.

Figure 8: Histogram of Firm and Parent Cosine-Similarity by Generation Removed

Figure 8 shows the same distributions as Figure 7 but in a stacked version.

3.3.2 Heatmaps

First, Figure 9 shows the heatmap of the cosine-similarities of all deals including a top-up option.

Figure 9: Heatmap: Cosine Similarity of all deals that include a top-up option

Next, we show the Heatmap of the adopters, ordered by adoption date in Figure 10.

We break down the adopters into subfamily groups to show if the generation level or the subfamily group within the family matters for the text similarity of the deal in the adoption patterns.

Figure 11 shows a heatmap of family 130 - firm Heller Ehrman, by family subgroups and by generation. We do not see any clear difference grouping by subfamily or generational level. The heatmaps are relatively similar.

Figure 12 shows a heatmap of family 10 of firm Gibson Dunn Crut by generation and subfamily level. Again, notice no clear difference in the ordering structure of how the firms are situated in relation to each other by generation level or subfamily groups.

3.3.3 DNA test for multiple adopters

Finally, we return to the case of using multiple parents for each adopters and analyzing their textual similarity.

Figure 13 shows for two firms that have the most deals with a prior adopter: 43 (Simpson Thatcher) and 44 (Skadden Arps) the relationship between the deal date and the cosine similarity with the prior Adopter who they signed with. There seems to be no linear trend of deal date and the cosine similarity.

Figure 14 shows the relationship between the cosine similarity of the Adopter and its multiple parent Adopters and the date on which the Prior Adopter became an Adopter. While no linear trend appears for Simpson Thacher, Skaaden Arps shows a negative relationship between the prior Adopter’s adoption date and the cosine similarity of the prior Adopter.