††footnotetext: This document is the results of a research project funded by
the Cyber-Defence Campus, armasuisse Science and Technology.

# TechRank

Anita Mezzetti  Credit Suisse Loïc Maréchal  Cyber-Defence Campus, armasuisse
Science and Technology  University of Lausanne (HEC Lausanne)  Dimitri Percia
David  University of Applied Science (HES-SO Valais-Wallis)  William Lacube
Cyber-Defence Campus, armasuisse Science and Technology  Sébastien Gillard
GSEM, University of Geneva  Military Academy, ETH Zurich  Michael Tsesmelis
Cyber-Defence Campus, armasuisse Science and Technology  Thomas Maillart
GSEM, University of Geneva  Alain Mermoud  Cyber-Defence Campus, armasuisse
Science and Technology

###### Abstract

We introduce TechRank, a recursive algorithm based on a bi-partite graph with
weighted nodes. We develop TechRank to link companies and technologies based
on the method of reflection. We allow the algorithm to incorporate exogenous
variables that reflect an investor’s preferences. We calibrate the algorithm
in the cybersecurity sector. First, our results help estimate each entity’s
influence and explain companies’ and technologies’ ranking. Second, they
provide investors with a quantitative optimal ranking of technologies and
thus, help them design their optimal portfolio. We propose this method as an
alternative to traditional portfolio management and, in the case of private
equity investments, as a new way to price assets for which cash flows are not
observable.

JEL classification: C14, C69, G17, G24

Keywords: private equity, bipartite networks, technology monitoring, portfolio
optimization

##  1 Introduction

This work investigates the innovation structure and the dynamics underlying
the life cycle of technologies. We fill two research gaps. The first concerns
the identification of future benefits and risks of emerging technologies for
the society. The second regards the valuation or early-stage companies and
optimal investment decisions. To fill these gaps, we introduce the TechRank
algorithm. Our methodology assigns a score to each entities, i.e.,
technologies and firms, based upon their contribution to the technological
ecosystem. We expect this method to help stakeholders in forming optimal
decisions for investment, procurement, and technology monitoring.

We calibrate our model in the cybersecurity sector, although TechRank could
apply to any sector. The cybersecurity technological landscape represents a
particular challenge for this calibration, given the important share of start-
ups and fast innovations it yields.[16]. Moreover, the important number of
cyber-attacks and the increasing costs they incur has boosted cybersecurity
investments.111 The New York Times: “As Cyberattacks Surge, Security Start-Ups
Reap the Rewards” by Erin Woo (July 26, 2021).
Yahoo Finance: “Microsoft Securing its Position with Cybersecurity
Investments” by TipRanks (July 20, 2021). According to Bloomberg, “the global
cybersecurity market size is expected to reach USD 326.4 billion by 2027,
registering a compound annual growth rate of 10.0% from 2020 to
2027”.222Bloomberg: “Global Cybersecurity Market Could Exceed $320 Billion in
Revenues by 2027” (July 29, 2020).

To develop the TechRank algorithm, we first model and map the ecosystem of
companies and technologies from the Crunchbase dataset using a bi-partite
network. The bi-partite network structure accurately describes this complex
and heterogeneous system. We evaluate the relative influence of the network
nodes in the ecosystem by adapting a recursive algorithm that estimates
network-centrality.

This methodology should help decision-makers and investors to assess the
influence of entities in the cybersecurity ecosystem, reducing investment
uncertainties. In fact, around 90% of startups fail and in 42% of the cases
this is due to incorrect evaluation of the market demand. The second reason
(29%) is because they run out of funding and personal money.333Findstack: “The
Ultimate List of Startup Statistics for 2021” by Jack Steward. Christensen
(1997) highlights that well-managed companies also break down, because they
over-invest in new technologies[8]. Thus, by selecting the right technologies
to invest in goes along with the optimal investment strategy.

Our research takes inspiration from Google’s PageRank algorithm, that ranks
web pages according to readers’ interest[25]. We use a similar approach with
bi-partite networks to assign a score to companies and technologies. Our
method is flexible and permits to incorporate investors’ preferences such as
location or previous funding rounds. TechRank let the investor select
entities’ features that reflect their interests. The algorithm uses their
choices as input, which tweaks the entities’ score to reflect them. This
enables investors to select a personalized portfolio strategy using a
quantitative methodology. The evaluation of companies and new technologies
largely depends on investors’ personal choices, which may lead to misread
market demand. This work aims to lead to more methodical decision making for
investors.

The remainder of this article proceeds as follows. Section 2 presents the
literature review and hypotheses. Section 3 details the data and the
methodology. Section 4 presents the results. Section 5 concludes.

##  2 Literature review and hypotheses development

###  2.1 Centrality measures

In network analysis, the centrality estimates the importance of nodes through
ranking. The most simple centrality estimate is the “degree”, which counts the
number of neighbours of a node. One of its drawbacks is that it does not show
which one stands in the center of the network. Two nodes may share the same
degree, while being more or less peripheral. Thus, the degree is a local
centrality measure, which does not capture the influence across nodes within
the graph.

Fig. 1: Central and peripheral nodes

This figure depicts the difference between central (red) and peripheral
(brown) nodes in a graph.

Another important centrality measure is the “closeness”, which measures how
long it takes for information to spread from one node to the next.
Specifically, closeness is defined as the reciprocal of the “farness”, i.e.
the sum of distances of one node with respect to all other nodes. The
“betweenness centrality” of a node measures how often a node stands in the
shortest path between a pair of other nodes (see, e.g., Bavelas; 1948, Saxena
and Iyengar; 2020, and Freeman; 1978)2, 28, 14).

Another strand of research focuses on the top-K shortest path identification
in a complex network, a topic less tackled by the literature than centrality.
To rank nodes, one must compute the centrality of all nodes and compare them
to extract the rank, which is not always feasible due to the size of the
network. To overcome this problem Saxena and Iyengar (2017) attempt to
estimate the global centrality of a node without analyzing the whole
network[29]. Similarly, Bavelas (1948) develops a structural centrality
measure in the context of social graphs[2]. Other centrality concepts include
the eigenvector, Katz, or PageRank centralities[5, 25, 20]. Finally, Freeman
(1978) creates a formal mathematical framework for centrality, which includes
degree, closeness, and betweenness and advocates for the combination of
different kinds of centrality measures[14].

###  2.2 Page Rank

Page, Brin, Motwan, and Winograd (1999) develop the PageRank algorithm[25].
Its primary goal is to rank web pages objectively, a challenge with the fast-
growing web. PageRank assigns a score to each web page based on its relations
with other web pages in the graph. Other fields have benefited PageRank
providing modifications and improvements. Xing and Ghorbani (2014) extend the
algorithm and propose the weighted PageRank (WPR)[32]. This algorithm assigns
larger rank values to more important pages, instead of dividing the rank
evenly among its outlink pages.444Given a web page W, an inlink of W is a link
of another web page that includes a link pointing to W. An outlink of W is a
link appearing in W, which points to another web page. Each outlink page gets
a value proportional to its popularity, taking into account the links weights.
On caveat of PageRank and its variants is that they do not consider n-partite
structures, yet, web pages can all be linked to one another. Bi-partite
networks address this issue and capture this complexity, among other
structures.

###  2.3 Bi-partite networks

Networks are a fundamental tool to capture the relations between entities.
Graphs ($G$) are composed by vertices ($V$) and edges ($E$), and we denote
$G=(V,E)$. We build links and mathematically analyze many properties of the
whole system and of singular entities. To graphically represent a real system,
we synthesize its information into a simple graph framework. This
simplification generates an information loss in the modelling process. Simple
network structures might discard important information about the structure and
function of the original system [23]. As a consequence, the failure of a very
small fraction of nodes in one network may lead to the complete fragmentation
of a system[6]. To solve the problem, extensions to the simple structure
$G=(V,E)$ are added and yield graphs with more powerful features. For
instance, in case vertices are connected by relationships of different kinds,
Battiston, Nicosia, and Latora (2014) advocate to work with multiplex
networks, i.e. networks where each node appears in a set of different layers,
and each layer describes all the edges of a given type [1]. When it is
possible to distinguish the nature of the edges, multiplex networks are an
effective approach, which starts from embedding the edges in different layers
according to their type. However, even if we have two kinds of nodes, the
nature of the edges is unique. Therefore, a more suitable approach is bi-
partite networks. Bi-partite networks are for instance, a good way to describe
the technological and business landscape. In Figure 2, we depict two sets of
nodes, companies and technologies, which are interconnected but do not present
edges within the same set.

Fig. 2: Bi-partite structure of companies and technologies

The left panel depicts a typical bi-partite structure. The right panel
provides an illustration of this structure with companies (layer 1) and
technologies (layer 2).

There are multiple adaptations of the PageRank algorithm to bi-partite
structures[3, 12, 31, 21]. In particular, Benzi, Estrada, and Klymko (2015),
Donato, Laura, Leonardi, and Millozzi (2004), and Tu, Jiang, Song, and Zhang
(2018) extend the PageRank algorithm to multiplex networks. They assume that
only some clusters of the graph are multiplex networks and extend the PageRank
algorithm only to analyze the sub-graph centrality. Bi-partite networks are
used to transform directed into undirected networks with twice the number of
vertices.

Klein, Maillart, and Chuang (2015) extend PageRank in the Wikipedia editors
and articles context [21]. The application of this algorithm to the case of
interactions between companies and technologies is straightforward. A major
benefit of this approach is that it starts from an unweighted graph, linking
authors and articles. They develop a recursive algorithm in which the two
entities contribute to the quality (for articles) or the expertise (for
authors) of each other. They develop a bi-partied random walker by building
the adjacency matrix $M_{e,a}$ that takes value 1 if editor $e$ has edited
article $a$ and 0 otherwise, which tracks all the editors’ contributions. They
obtain $M_{e,a}\in\mathbb{R}^{n_{e},n_{a}}$, where $n_{e}$ and $n_{a}$ are the
number of editors and articles. They sort editors by the number of articles’
contributions and assign a contribution (quality) value to each editor
(article) based on their degree. The expertise $w_{e}^{0}$ (quality
$w_{a}^{0}$) is given by the number of articles (editors) they have worked on
(have received modifications).

The second part of the algorithm follows a Markov process in its iterations.
The step $w^{n}$ ($w^{n}=w^{n}(\alpha,\beta)$) only depends on information
available at $w^{n-1}$. At each step, the algorithm incorporates information
about the expertise of editors and the quality of articles, within the bi-
partite network structure. The process is a random walker with jumps, whose
transition probability is zero in the case $M_{e,a}=0$. Next, the authors
define two variables for the transition probability, $G_{e,a}(\beta)$ and
$G_{e,a}(\alpha))$. $G_{e,a}(\beta)$ represents the probability of jumping
from article $a$ to editor $e$ and $G_{a,e}(\alpha)$ represents the
probability to jump from an editor to an article. Both parameters depend on
initial conditions and the selection of optimal parameters is done through a
grid search that maximizes the Spearman rank-correlation between the rank
given by the model and a ground-truth metrics obtained independently. Finally
Klein et al. (2015) observe a “less-is-more” situation since the presence of
too many editors working on an article is detrimental to its quality. Studying
different categories of Wikipedia articles they find $\alpha$ to remain
constant, while $\beta$ varies significantly across categories.

Estimating the global rank of a node starting from local information and
centrality measures is still an open research question in many sectors [28].
In particular, no research to our knowledge use this approach for investment’s
decision and portfolio optimization. Yet, this approach could help overcome
the limitations of standards financial models in private equity, in which the
network structure is easily obtainable, whereas the cash flow process is not.

###  2.4 Private equity valuation

Private firms are not required to publicly disclose their financial
statements, which makes it difficult to measure their past performance and
estimate their expected returns, without insider information. Moreover, since
they are not listed on exchanges we do not observe expectations of market
participants. Thus, standard asset pricing methods fall short in this context.
Similarly, private equity analysts must rely on insider or private information
to value private firms. These valuations generally occur around a financing
round and impose to take into account the capital dilution to compute realized
returns on the firm[17].

One approach that attempts to overcome these limitations and estimate the
expected returns and risk in venture capital is Cochrane (2005)[9]. He uses a
maximum likelihood estimation method to obtain these values at the market and
sector levels, such as healthcare, biotechnology, technology companies, and
retail services. He finds a mean arithmetic return of 59%, an alpha of 32%, a
beta of 1.9 and a volatility of 86% (equivalent to a 4.7% daily volatility).
Given that the distribution of returns is heavily positively skewed in venture
capital, he adopts a logarithmic model that also accounts for the inherent
selection bias of this asset class.555Most of venture capital data is private
and available data are more often related to successful firms than under-
performing ones. Ewens (2009) updates this method on returns computed from
financing round to the next[13]. He adopts a three-regime mixture model
(failure, medium returns, and “home-runs”). He also corrects for the selection
bias and obtains an alpha of 27% and a beta of 2.4. He finds that 60% of all
venture capital investments have a negative log return. Altogether the results
are similar, venture capital investments exhibit positive alpha, large beta,
and a high volatility. Other attempts to evaluate the market parameters of the
venture capital asset class includes Korteweg and Nagel (2016) and Moskowitz
and Vissing-Jorgensen (2010) with results in line with the aforementioned
studies[22, 24].

Another strand of research attempts to index and benchmark the private equity
market. Peng (2001) builds a venture capital index from 1987 to 1999 from
about 13,000 financing rounds targeting over 5,600 firms[26]. He addresses the
problems of missing data, censored data, and sample selection using a re-
weighting procedure and method of moment regressions. From the index
perspective, the results are qualitatively the same, that is, high and
volatile returns to venture capital (average return of 55.18% per year). He
finds his index to display a much higher volatility than the S&P 500 and
NASDAQ indices and high exposure to these indices (betas of 2.4 and 4.7,
respectively). Other venture capital indices construction includes Hwang,
Quigley, and Woodward (2005), Schmidt (2006), and Cuming, Has and Schweizer
(2013), all obtaining results on par with the aforementioned studies.

One limitation of the above studies is that they only estimate these
parameters at an aggregate level. An investor could form her investment’s
decisions and portfolio choices segregating among sectors, but not obtain the
actual firms’ parameters. One exception is Schwartz and Moon (2000), who
provide an approach based upon real-options theory to price individual
firms[30]. However, this method requires the observations of cash flows and
they only provide one calibration example with Amazon. Thus, there remains a
caveat in methodology to help investors’ forming optimal decisions using all
the information available. Given the recent venture capital boom, Zhong,
Chuanren, Zhong, and Xiong (2018) advocates for the use of quantitative
methodologies of screening and evaluation[33]. However, there is a clear
research gap on methodologies enabling to value early stage companies and form
optimal portfolios. These methodologies either only enable to value a sector,
instead of a specific company, or require the use of cash flows that are
unobservable. Non-financial features and relations between companies and
technologies are instead numerous and easily observable[10]. We thus formulate
our hypotheses as follows,

$H_{1}$ Using a bi-partite network structure allows to create an algorithm
that rank companies based on their links with technologies.

$H_{2}$ This algorithm and its ranking may be improved and tilted towards
investors’ preferences.

$H_{3}$ The performance of the ranking is independent from the sector
considered.

##  3 Data and methods

###  3.1 Data

We use Crunchbase data.666Crunchbase website: https://www.crunchbase.com/
Crunchbase is a commercial database that provides access to financial and
managerial data on private and public companies globally. It was created in
2007 by TechCrunch, which is a source of information about start-up activities
and their financing within and across countries. This database has been
largely adopted by both academics and industry practitioners[4]. It is also
used by international organizations such as the OECD[11].

Crunchbase is made of data collected with a multifaceted approach that
combines crowd-sourcing (through venture programs or direct contributions),
machine learning, in-house processing, and aggregation of third-party
providers data. Crunchbase updates and revises data on a daily basis, which is
organized into several entities such as organizations, people, events,
acquisitions, or IPOs. The primary focus of Crunchbase is the technology
industry, although it also includes data on other sectors.

Data can be accessed in two ways: using an API or downloading a *.csv file
directly from the Crunchbase website. Data is split in several databases
depending on their type. We provide a non-exhaustive list in Table
1.777Crunchbase daily CSV export from https://data.crunchbase.com/docs/daily-
csv-export. Data downloaded on April 28, 2021.

Field name | Description
---|---
organizations | Organisation profiles
organization_desc | Long descriptions of organisation profiles
acquisitions | List of all acquisitions
org_parents | Map between parent organisations and subsidiaries
ipos | Detail for each IPO
people | People profiles
people_desc | Long descriptions for people profiles
degrees | Detail of people’s education background
jobs | List of all jobs and advisory roles
investors | Active investors (organisations and people)
investments | All investments
investment_partners | Partners responsible for their firm’s investments
funds | Details of investments funds
funding_rounds | Details for each funding round in the dataset
events | Event details
event_appearances | Event participation details

Table 1: Crunchbase files description

This table reports the main fields available from Crunchbase.

We first analyze the Crunchbase dataset dedicated to investors. With a total
of 185,784 investors divided into 78,001 (41.98%) organisations and 107,783
(58.02%) persons, there are more investors than target companies. Figure 3
shows that the majority of investors are pure investors (87.11%). Some
organisations are both investee and investor (12.65%). The remainder of the
sample are typically universities. Crunchbase ranks the top 1,000 investors
through its proprietary algorithm. Figure 3 indicates that the majority of
investors are located in the USA (29.62%). In particular, there is a wide gap
between the first and second country, China, where 7.04% of investors are
located.

Fig. 3: Summary statistics of investors in Crunchbase data

The left panel depicts the distribution of investors according to their types.
The right panel depicts the number of investors per country.

###  3.2 Methodology

####  3.2.1 Adaptation of the work by Klein et al. (2015).

In this research we use a bi-partite network that describes the relations
among companies and the technologies they are involved in. Figure 2 describes
the typical bi-partite network structure. This structure benefits from
advances in fields such as network theory, Markov chains, and machine
learning. We adapt the recursive algorithm with the method of Hidalgo,
Hausmann, and Dasgupta (2009)[18]. We expect the resulting rank to incorporate
the positive influence of well-established companies on technologies and, at
the same time, the positive impact of new companies that explore new fields.
We build the adjacency matrix $M^{CT}_{c,t}\in\mathbb{R}^{n^{c},n^{t}}$, which
takes value 1 if a company $c$ works on a technology $t$ and 0 otherwise.
$n^{c}$ and $n^{t}$ represent the number of companies and technologies. We
assume that well-established companies have more means to diversify their
expertise and therefore, that an entity has a relatively high number of
neighbours[7, 15]. Thus, we initialize the algorithm with the degree, i.e.
counting the neighbours, of each entity,

$\begin{cases}w_{c}^{0}&=\sum_{t=1}^{n^{t}}M^{CT}_{c,t}=k_{c}\\\
w_{t}^{0}&=\sum_{c=1}^{n^{c}}M^{CT}_{c,t}=k_{t}\end{cases}$ (1)

The algorithm is a “random walker” that incorporates information about company
expertise and technology relevance at each step. The transition probabilities,
$G_{c,t}$ and $G_{t,c}$, describe the extent to which the entities weights
change over the iterations. If the relation between $c$ and $t$ increases
(decreases) the value, the entity weight increases (decreases) in proportion
with the transition probabilities. We define $G_{c,t}$ and $G_{t,c}$,

$\begin{cases}G_{c,t}(\beta)&=\frac{M^{CT}_{c,t}k_{c}^{-\beta}}{\sum_{c^{\prime}=1}^{n^{c}}M^{CT}_{c^{\prime},t}k_{c}^{\prime-\beta}}\\\
G_{t,c}(\alpha)&=\frac{M^{CT}_{c,t}k_{t}^{-\alpha}}{\sum_{t^{\prime}=1}^{n^{t}}M^{CT}_{c,t^{\prime}}k_{t}^{\prime-\alpha}},\end{cases}$
(2)

where $\alpha$ and $\beta$ inform how coordination generates value. Next, we
define the recursive step,

$\begin{cases}w_{c}^{n+1}&=\sum_{t=1}^{n^{t}}G_{c,t}(\beta)w_{t}^{n}\\\
w_{t}^{n+1}&=\sum_{c=1}^{n^{c}}G_{t,c}(\alpha)w_{c}^{n}\end{cases}$ (3)

As in PageRank, the recursion ends when the algorithm converges. Our algorithm
allows to consider the market complexity and feedback loops (investments’
impact on companies and on technologies). We discuss this feature and the
optimization of $\alpha$ and $\beta$ after the addition of exogenous factors.

####  3.2.2 Inclusion of exogenous factors

We include exogenous factors as ground truth in the parameters’ calibration
step. This allows to keep the algorithm tractable, while letting it capture
the technological structure. We use this ground truth to compute the Spearman
correlation, $\rho_{c}$ for companies and $\rho_{t}$ for technologies. Because
$\rho_{c}$ and $\rho_{t}$ depend on $\alpha$ and $\beta$ [see Eq. (2)], we
find the parameters that maximize these correlations,

$\begin{cases}(\alpha^{*},\beta^{*})=\operatorname*{arg\,max}_{\alpha,\beta}\rho_{c}(\alpha,\beta)\\\
(\alpha^{*},\beta^{*})=\operatorname*{arg\,max}_{\alpha,\beta}\rho_{t}(\alpha,\beta),\end{cases}$
(4)

and we solve this optimization problem with a grid search. Eq. (4) shows that
parameters depend on both companies and technologies. This dependence enables
to create the structure of the bi-partite graph. To obtain the correlation
between the TechRank score, which assigns a weight $w_{c}$ ($w_{t}$) to each
company (technology) and the ground truth evaluation, which assigns
$\hat{w_{c}}$ ($\hat{w_{t}}$) to each company (technology), we normalize both
TechRank results and the exogenous measure in the same range $[0,1]$.

Investors use the entities’ features to select companies and the investment
amount they want to allocate. We suppose that an investor has $n^{(C)}$
features to pick from, denoted as $f^{(C)}_{1},\ldots,f^{(C)}_{n^{(C)}}$,
where $C$ ($T$) represents the association with the companies (technologies).
Each feature $f^{(C)}_{i}$ is associated with a percentage of interest
$p^{(C)}_{i}$ and $\sum_{i=0}^{n^{(C)}}p^{(C)}_{i}=1$. For instance, if a
company’s features are the amount of previous investments and its geographical
proximity to the investor, $n^{(C)}=2$. An investor may then decide to be
interested at 80% in the first feature and at 20% in the second, by selecting
$p^{(C)}_{1}=0.8$ and $p^{(C)}_{2}=0.2$. Investors may also be pushed back by
a feature, in which case we multiply it by -1. We define all notations in
Table 2.

Variable | $\in$ | Description
---|---|---
$n^{(C)}$ | $\mathbb{N}$ | Number of external features available for companies.
$n^{c}$ | $\mathbb{N}$ | Number of companies.
$n^{(T)}$ | $\mathbb{N}$ | Number of external features available for technologies.
$n^{t}$ | $\mathbb{N}$ | Number of technologies.
$p^{(C)}_{i}$ | $[0,1]$ |  Percentage of interest in the company preference number $i$.
$p^{(T)}_{j}$ | $[0,1]$ |  Percentage of interest in the technology preference number $j$.
$f^{(C)}_{i}$ | $\mathbb{R}^{n^{c}}$ |  Vector of factors associated to the company preference number $i$.
$f^{(T)}_{j}$ | $\mathbb{R}^{n^{t}}$ |  Vector of factors associated to the technology preference number $j$.
$n^{i}$ | $\mathbb{N}$ | Number of investors.
$M^{CT}$ | $\mathbb{R}^{n^{c}\cdot n^{t}}$ | Adjacency matrix of the C-T bipartite network
$M^{IC}$ | $\mathbb{R}^{n^{i}\cdot n^{c}}$ | Adjacency matrix of the I-C bipartite network
$\gamma^{i,c}_{t}$ | $\mathbb{R}$ |  Amount in funding round between $c$ and $i$ at time $t$
$e^{IC}$ | $\mathbb{R}^{n^{i}\cdot n^{c}}$ | Total amount of investment between each investor to each company
$e^{C}$ | $\mathbb{R}^{n^{c}}$ | Total amount of investments toward each company
$e^{T}$ | $\mathbb{R}^{n^{t}}$ | Total amount of investments toward each technology
$e^{C}_{\text{max}}$ | $\mathbb{R}$ | Maximum amount of total investments among all the companies
$e^{T}_{\text{max}}$ | $\mathbb{R}$ | Maximum amount of total investments among all the technologies
$f^{C}_{c}$ | $[0,1]$ |  Factor related to previous investments into the company number $c$
$f^{T}_{t}$ | $[0,1]$ |  Factor related to previous investments into the technology number $t$

Table 2: Variable definitions

This table presents the variable definitions used throughout the article.

We convert quantitative and qualitative properties into a number
$f^{(C)}_{i}\in[0,1]$. Once we have created all the factors
$f^{(C)}=f^{(C)}_{1},\ldots,f^{(C)}_{n^{(C)}}$, the exogenous evaluation
$\hat{w_{c}}$ is given by,

$\hat{w_{c}}=\sum_{i=1}^{n^{(C)}}p^{(C)}_{i}f^{(C)}_{i}=p^{(C)}\cdot f^{(C)}.$
(5)

Considering $\sum_{i=0}^{n^{(C)}}p^{(C)}_{i}=1$ and that $f^{(C)}_{i}\in[0,1]$
for each company $i$, we have $\hat{w_{c}}\in[0,1]$. The same holds for
$\hat{w_{t}}$. Finally, we have

$\begin{cases}\hat{w_{c}}=p^{(C)}\cdot f^{(C)}\\\ \hat{w_{t}}=p^{(T)}\cdot
f^{(T)}\\\ \sum_{i=0}^{n^{(C)}}p^{(C)}_{i}=1\\\
\sum_{i=0}^{n^{(T)}}p^{(T)}_{i}=1,\end{cases}$ (6)

where $n^{(C)}$ ($n^{(T)}$) is the number of the company- (technology-)related
features and $f^{(C)}=(f^{(C)}_{1},\ldots,f^{(C)}_{n^{(C)}})$. To select the
features, we use Crunchbase data about companies and investors (see in Table
1).

###  3.3 Previous Investments

We assume that previous investments is an essential factor to evaluate
companies. Investors may be willing to invest in companies which have already
received capital or look for higher returns, targeting newer firms.

To compute this factor, we use the Crunchbase field “funding_rounds”, which
reports the amount of all funding rounds from an investor $i$ to a company
$c$. We capture this structure with another bi-partite network that describes
the links among investors (I) and companies (C). In this case, we weight the
edges by the sum of all previous investments from investor $i$ to company $c$,
until the current period ($\mathcal{T}$), and compute the adjacency matrix
$M^{IC}$. We define the amount of a single investment from $i$ to $c$ at time
$t$ by $\gamma^{i,c}_{t}$. The weight of the edge $i-c$ is given by
$e_{i,c}=\sum_{t=0}^{\mathcal{T}}\gamma^{i,c}_{t}$ (see Table 2). We then sum
the contribution of all investors to find the attribute $f^{C}_{c}\in[0,1]$
for a company $c$.888Note that here, $f_{c}^{C}$ represents the factor related
to a company. Next, we normalize and divide all investments by the maximum
investment.

Fig. 4: Investors-companies bi-partite network

This figure depicts a stylized bi-partite network between investors and
companies.

Figure 4 depicts the investment structure as an example. We consider two
investors $i_{1}$ and $i_{2}$ and three companies $c_{1}$, $c_{2}$, and
$c_{3}$. We compute the maximum $e_{\text{max}}$ as
$\max\\{e_{11}+e_{21},e_{22},e_{23}\\}$ and the features related to the
investments for each company as,

$\begin{cases}f^{C}_{1}=\frac{e_{11}+e_{21}}{e_{\text{max}}}\\\
f^{C}_{2}=\frac{e_{22}}{e_{\text{max}}}\\\
f^{C}_{3}=\frac{e_{23}}{e_{\text{max}}},\end{cases}$ (7)

where $n^{i}$ ($n^{c}$) is the total number of investors (companies).
Generalizing, we get,

$\begin{cases}e^{IC}_{i,c}=\sum_{t=0}^{\mathcal{T}}\gamma^{i,c}_{t}\hskip
39.83368pt\forall i,c\\\ e^{C}_{c}=\sum_{i=1}^{n^{i}}e_{i,c}M^{IC}_{i,c}\hskip
25.6073pt\forall c\\\ e_{\text{max}}=\underset{c}{\max{}}e^{C}_{c}\\\
f^{(C)}_{c}=e^{C}_{c}/e_{\text{max}},\end{cases}$ (8)

for each $c\in{1,\ldots,n^{c}}$. We present the corresponding algorithm in the
Appendix 1. With Eq. (8), for each company we have a factor between 0 and 1
that summarizes the amount of previous investments.

Fig. 5: Tripartite structure of investors, companies, and technologies

The left hand side figure depicts a typical tri-partite structure. The right-
hand side provides an illustration of this structure with investments (layer
1), companies (layer 2), and technologies (layer 3).

We link the two bi-partite structures investment-companies and companies-
technoéogies to obtain an I-C-T tri-partite structure depicted in Figure 5.
This structure allows to assign some features to technologies from companies
(direct link) or investors (indirect link). Thus, we can find the amount of
previous investments on a technology through companies’ funding rounds. The
previous investments’ factor for technology is given by,

$\begin{cases}e^{(I,C)}_{i,c}=\sum_{t=0}^{(T)}\gamma^{t}_{i,c}\hskip
17.07182pt\forall i,c\\\ e^{C}_{c}=\sum_{i=1}^{n^{i}}e_{i,c}\hskip
31.2982pt\forall c\\\ e^{T}_{t}=\sum_{c=1}^{n^{c}}e_{c}M^{CT}_{c,t}\\\
e_{\text{max}}=\underset{t}{\max{}}e^{T}_{t}\\\
f^{(T)}_{t}=e^{T}_{t}/e_{\text{max}}\end{cases}.$ (9)

We provide the algorithm of this methodology in the Appendix 2.

###  3.4 Location

The second feature we consider is the distance between investors and
companies’ locations. We retrieve the addresses of firms and investors from
Crunchbase ($c\\_address$) and map them to geographic coordinates. We compute
the Haversine approximation to measure the distance. We detail the Haversine
approximation in the Appendix Distance computation. Investors may prefer
short-distance investments or places with high potential. If they face some
investment’s restrictions, we filter the companies based on the criteria
before applying the algorithm. Otherwise, we add a distance factor to the
algorithm.

We use the Haversine distance $h$ to obtain a factor $f_{c}^{(C)}\in[0,1]$ for
each company. We consider the distance $h_{i,c}$ between the company $c$ and
the investor $i$. We assume that the factor is the proximity so that
$f_{c}^{(C)}$ tends to one as the distance decreases,

$f_{c}^{(C)}\rightarrow 1\hskip 14.22636pt\text{when}\hskip
14.22636pth_{i,c}\rightarrow 0.$ (10)

To compute $f_{c}^{(C)}$, we first find $h_{i,c}$ for each company and
identify the maximum distance $h_{\text{max}}$ among all companies. We
normalize by the maximum to obtain a distance that lies in the $[0,1]$ range
with, $f_{c}^{(C)}=1-h_{i,c}/h_{\text{max}}$, so that a distance of zero
corresponds to a value of $f_{c}^{(C)}=1$. We report the algorithm in the
Appendix 3. We implement the algorithm and run the experiments using Python
and the NumPy, Pandas, NetworkX, Matplotlib, and Seaborn libraries.

##  4 Results

###  4.1 Cybersecurity field

We select all the companies whose description contains at least two
cybersecurity-related terms and obtain 2,429 companies and 477
technologies.999The word list is in the Appendix List of words used to
identify companies’ sectors Figure 6 display the structure of the bi-partite
network between technologies and companies.

Fig. 6: Bi-partite network of cybersecurity companies

This figure describes the bi-partite network of a subset of cybersecurity
companies (red nodes) and the technologies they are involved in (blue nodes).
The nodes size represents the number of neighbours.

We assume that investors are only interested in previous investments, both for
technologies and companies. We examine how the parameters’ calibration step
changes when we change the investors’ preferences using a smaller sample of
companies. Figure 7 shows the optimization in which the correlations
$\rho_{c}$ and $\rho_{t}$ change according to $\alpha$ and $\beta$. In Table 3
We identify the optimal $\alpha^{*}$ and $\beta^{*}$ to be of 0.04 and -1.88
for companies and 0.48 and -2.00 for technologies, respectively. Next, we plug
these values in the recursive algorithm.

Fig. 7: Grid search of parameters $\alpha$ and $\beta$

This figure displays the results of the grid search for parameters $\alpha$
and $\beta$ for 2,429 companies and 477 technologies in cybersecurity when
investors preferences are fully set in previous investments.

Companies | Technologies
---|---
Number | $\alpha^{*}$ | $\beta^{*}$ | Number | $\alpha^{*}$ | $\beta^{*}$
10 | -0.36 | 1.92 | 26 | -2.00 | 0.00
100 | -0.04 | 0.92 | 134 | 0.52 | -1.04
499 | -0.08 | 0.88 | 306 | 0.68 | -1.36
997 | -0.12 | 0.80 | 371 | -2.00 | 0.00
1,494 | -0.12 | 0.80 | 416 | 0.92 | -0.12
1,990 | -0.04 | 0.92 | 449 | 0.56 | -2.00
2,429 | 0.04 | -1.88 | 477 | 0.48 | -2.00

Table 3: Optimal parameters in cybersecurity

This table reports the optimal parameters $\alpha$ and $\beta$ for companies
and technologies in cybersecurity, depending on the number of companies and
linked technologies considered as input.

We illustrate the evolution of the TechRank random walker in Figure 8. While
the entities’ positions significantly change over the first steps, they
gradually stabilize. With the 2,429 companies and 477 technologies, the
algorithm requires 723 (1,120) iterations for companies (technologies) to
converge. Entities starting with a high score (the initialisation is the
degree of the node) do not significantly change rank and remain among the best
ones. Thus, the algorithm assigns good scores to entities with many
neighbours. Instead, entities starting with a low degree may significantly
change their score, especially in the case of technologies. TechRank does not
only recognize the importance of the most established entities, it also
enables to identify emerging technologies. Figure 11 shows the first
classified entities in cybersecurity.

Fig. 8: TechRank scores evolution in cybersecurity

This figure displays the TechRank scores evolution over the iterations for
2,429 companies and 477 technologies in cybersecurity.

We check how TechRank performs when we change the number of companies and
technologies. We fix the number of companies $n^{c}$, which yields a resulting
number of technologies $n^{t}$. For instance, in the cybersecurity field, by
selecting 10 companies randomly, we get 26 technologies. Considering that
there are 2,429 cybesecurity-related companies on Crunchbase, we study the
runtime running the algorithm for 10, 100, 499, 997, 1494, 1990, and 2,429
companies and 26, 134, 306, 372, 431, 456, and 477 technologies respectively.

Figure 9 displays the results of TechRank applied on a subset of 10
cybersecurity companies. We note that “AppOmni”’s position does not change
over the iteration, while two of its technologies, “Software as a Service”
(Saas) and “cloud management” increase their scores. In Figure 10 we display
this restricted network of 10 companies, that shows that SaaS and cloud
management do not have other links. Hence, the strength of this company
depends on its ability to combine important technologies (software, cyber
security, and cloud security) with more exotic fields. Similarly, “Integrity
Market Group” is the single involved in some fields (marketing, digital
marketing, and advertising). This company does not use more established
technologies and thus does not improve its score. Again, in Figure 10 we
observe that these technologies lie out of the main network. Conversely,
“Lacework” and “Acronis” follow an opposite trend. Lacework (Acronis)
significantly increases (decreases) its score. One explanation for this
behaviour is the fact that Acronis is involved in a lot of technologies, most
of which are not explored by other companies. On the other hand, Lacework
relies on recognized technologies (security, cloud security, and software).
Interestingly, the compliance technology, benefits from its connections,
increasing its rank by three positions.

Fig. 9: TechRank scores evolution of 10 companies in cybersecurity

This figure displays the TechRank scores evolution over the iterations on a
subset of 10 companies and 26 technologies in cybersecurity.

Fig. 10: Circular network representation of 10 companies in cybersecurity

This figure displays a circular network representation of a subset of 10
companies and 26 technologies in cybersecurity.

Fig. 11: TechRank top five scores in cybersecurity

This left (right) panel displays the top five companies (technologies)
according to the TechRank score when run on a subset of 10 companies in
cybersecurity.

Table 4 reports the number of algorithm iterations before reaching
convergence. The number of iterations needed appears to be independent from
the number of entities. Technologies need more iterations than companies,
which we explain by the fact that there are many more companies than
technologies. Since each company has at least one edge, the technology nodes
have a higher degree than the companies on average. Thus, we expect the
structure and the dynamics related to technologies to be more complex. The
algorithm complexity does not only depend on the number of entities, but also
on the network structure.

Companies | Technologies
---|---
Number | Iterations C | Number | Iterations T
10 | 32 | 26 | 18
100 | 100 | 134 | 155
499 | 134 | 306 | 2,469
997 | 196 | 371 | 194
1,494 | 180 | 416 | 871
1,990 | 240 | 449 | 5,000
2,429 | 723 | 477 | 1,120

Table 4: TechRank convergence

This table reports the number of TechRank iterations before convergence for
companies and technologies in cybersecurity.

###  4.2 Investment strategy

We investigate how investors can select the strategy that reflects their
preferences. If investors prefer to focus on technologies, they should choose
companies working on the best technologies as selected by the (highest)
TechRank score. This decision implies many criteria such as the number of
technologies they want to be invested in, the capital allocation for each
company, and the diversification. We sketch the procedure to solve this
decision process quantitatively in the Appendix A1.

###  4.3 Comparison with the Crunchbase rank

Crunchbase assigns a rank to the top companies of each industry, that takes
into account the entity’s strength of relationships, funding events, news
articles, and
acquisitions.101010https://about.crunchbase.com/blog/influential-companies/ We
compare our results in the cybersecurity sector with the Crunchbase rank and
investigate the strength of the association between the two scores using
Spearman’s correlation.

To make the ranks comparable, we convert our algorithm’s output into a
ranking. The resulting Spearman’s correlation of 1.4% indicates that the two
ranks are uncorrelated. We explain these differences by the fact that the
Crunchbase rank is fixed, while TechRank is customizable according to
investors’ preferences. Moreover, the Crunchbase rank focuses on the company’s
level of activity and not on its market influence. Furthermore, the Crunchbase
rank results from an algorithm that involves all the companies, while we focus
only on a subset. We attempt to change the investors’ preferences and never
obtain correlation coefficients above 2%. Other explanations we propose for
this divergence includes the fact we assign a weight which identifies the
distance between entities in the ranking. In the same line, TechRank allows
decision-makers to set a threshold as starting parameter before running the
algorithm. Finally, the Crunchbase algorithm is not open source and we do not
know its mechanism, which makes the identification of the divergence’s source
difficult.

###  4.4 Runtime

We detail the code related to the TechRank algorithm in the Appendix 1. We run
it on a machine with a 16-cores Intel Xeon CPU E5-2620 v4 @ 2.10GHz and with
126GB of memory. We investigate the variations in runtime given changes in the
number of companies and technologies.

The runtime is a positive function of the number of entities. For technologies
the curve is much steeper than that for companies. However, considering that
the number of technologies is a direct link to the number of companies, we
repeat the experiment treating companies and technologies together. The random
walk phase lines represent the runtime to convergence. There is a strong
similarity between the runtime for companies and technologies, which is
surprising given their different numbers. This also shows how strongly they
are correlated and supports the capability of TechRank to capture the
complexity of the cybersecurity technological landscape. Table 4 reports all
the runtimes and we report the corresponding runtime comparisons for
technologies and companies of both cybersecurity and medical field in the
Appendix A2.

Companies | Technologies
---|---
Number | Parameters’ calibration | Convergence | Number | Parameters’ calibration | Convergence
10 | 10.21 | 0.56 | 26 | 11.75 | 0.57
100 | 28.69 | 13.24 | 134 | 35.37 | 13.72
499 | 189.03 | 470.10 | 306 | 154.79 | 483.25
997 | 730.43 | 2,023.18 | 371 | 312.65 | 2,392.46
1,494 | 1,372.17 | 4,514.11 | 416 | 482.18 | 4,404.48
1,990 | 2,057.42 | 8,396.26 | 449 | 656.95 | 8,096.69
2,429 | 3,230.99 | 16,890.26 | 477 | 1,071.84 | 12,779.62

Table 5: TechRank runtime

This table reports the TechRank runtime for companies and technologies in
cybersecurity.

###  4.5 Exogenous factors

We conduct a sensitivity analysis based on investors’ preferences. We restrict
the analysis to 1,000 companies only, given the to long runtime required. We
assume investors to be interested in firm location only and consider the case
of an investor based in New York City and in San Francisco, in turn. In Table
6 we report the outcome in terms of location for the five top ranked companies
in both cases. We uncover a location change in the company ranking, with the
first one being in the state of New York (investor based in New York City) and
in the state of California (investor based in San Francisco), respectively.
The companies with lower rank also reflects these geographical preferences,
albeit with some exceptions (Singapore and Beijing). This implies that, even
if remote companies are disadvantaged, their other attributes overcome this
flaw.

Company rank/ Investor location | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
New York City |  New York City (USA) |  Massachusetts (USA) |  Quebec (Canada) |  California (USA) |  Singapore (Singapore)
San Francisco |  California (USA) |  Illinois (USA) |  California (USA) |  Beijing (China) |  Arizona (USA)

Table 6: Companies TechRank scores with location

This table reports the location of the top five TechRank companies’ scores
when the geographical preference is fully set in the location of the investors
(New York City and San Francisco).

###  4.6 Robustness tests

To test the robustness of our algorithm and benchmark the cybersecurity
sector, we apply TechRank for companies in the medical sector. We choose this
sector given the important number of companies (twice the number of companies
working in cybersecurity). We select the companies with the same methodology,
which yields a total of 4,996 companies and 437 technologies. Figure 12 shows
the results of TechRank in the medical sector. The runtime for these
companies, reported in the Appendix A2, is on par with those of the
cybersecurity sector. To make the two fields comparable, we set as x-label the
number of entities, for both companies and technologies. The results reveal
that the runtime of the two fields, for both the parameter calibration and the
random walker steps, follow the same behaviour, for both companies and
technologies. Increasing the number of entities does not yield any significant
change both in terms of convergence and runtime behaviour. Finally, unlike
Klein et al. (2015), for which the $\alpha$ remains constant and $\beta$
changes significantly, we observe that all of our parameters significantly
change across sectors[21].

Fig. 12: TechRank scores evolution in the medical field

This figure displays the TechRank scores evolution over the iterations for
2,429 companies and 477 technologies in the medical field.

##  5 Conclusion

###  5.1 Limitations

We choose technologies related to cybersecurity according to Crunchbase
description, e.g., security, privacy, or confidentiality. Since these words
may overlap other fields, we require the description to contain at least two
of them to classify a company as cybersecurity. This naive strategy could be
improved with more sophisticated techniques, such as natural language
processing NLP. We also face limitation given the lack of information about
companies’ resources allocation towards technologies. We only have a list of
technologies for each company without more information. Instead, it would be
helpful to observe the amount of expenditures towards each technologies. Our
algorithm is static as we do not have access to time series and it would be
interesting to study how the bi-partite network changes. Finally, we are well
aware that introducing exogenous variables may induce a bias given potential
outliers. Our normalization procedure divides the factors by their maximum,
which may lead to unproportionate results if the maximum is an outlier.
However, we do not believe that removing outliers is a viable solution, since
this would lead to overlook potentially profitable opportunities.

###  5.2 Further research

This research can be expanded with time series, to investigate, for instance,
the outcome of a company divesting from a technology, or investing in a new
one. This would also enables to assess which new technologies are successful.
This would give more insights about investment choices towards too recent
ideas. A focus on percolation theory could help assess the effects of a node
disappearance in the network [27]. For this purpose, machine learning methods
could also be employed. Further research should be devoted to investigate
additional exogenous factors in TechRank, enabling investors the widest range
of features available possible. Alternative exogenous factors include the
inception date of the company, number of employees, social networks activity,
and even the Crunchbase rank, which is itself based upon entity’s strength of
relations, funding events, news articles, or acquisitions. By the same token,
further investigation should confirm the pertinence of the TechRank algorithm
in fields that include more entities, as an increase in nodes could lead to
coordination problems. Finally, it would be crucial to assess the long-term
effects of the TechRank algorithm on investments returns and technologies’
development through back-testing, that once again, requires time series.

###  5.3 Conclusion

We introduce TechRank, an algorithm that assigns a score to companies and
technologies in complex systems. This methodology constitutes the first step
towards a new data-driven investment strategy, which enables investors to
follow their preferences while benefiting from a quantitative approach. We
include investors’ preference based upon a case-by-case study. Our algorithm
convergence depends on the number of entities and the complexity of the
relationships within the bi-partite network. Using a restricted number of
companies in cybersecurity, we analyze the TechRank scores and we explain the
score variations of entities over iterations. We also explore how results
change depending on the company’s location. Finally, we conduct robustness
tests in the medical field, for which our results are qualitatively similar.

We believe that our approach brings value to help investors’ form their
decisions. Moreover, our algorithm’s flexibility allows to include exogenous
factors and preferences, which is impossible in alternative existing company
ranks, such as that of Crunchbase. Given our algorithm performance for
cybersecurity, a highly complex market, as a case study, we believe that our
algorithm would perform well in all markets. TechRank is a complementary, if
not alternative, way to look at portfolio choices.

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## Appendix

Fig. A1: Flowchart of the investment process

This flowchart sketches a potential investment process that uses TeckRank and
investment preferences (exogenous factors and investment styles) before
reaching an optimal investment and portfolio choice.

### List of words used to identify companies’ sectors

List of words related to cybersecurity: cybersecurity, confidentiality,
integrity, availability, secure, security, safe, reliability, dependability,
confidential, confidentiality, integrity, availability, defence, defensive,
privacy.

List of words related to the medical field: cure, medicine, surgery, doctors,
nurses, hospital, medication, prescription, pill, health, cancer, antibiotic,
HIV, cancers, disease, resonance, rays, CAT, blood, blood transfusion,
accident, injuries, emergency, poison, transplant, biotechnology, health care,
healthcare, health-tech, genetics, DNA, RNA, lab, heart, lung, lungs, kidneys,
brain, gynaecologist, cholesterol, diabetes, stroke, infections, infection,
ECG, sonogram.

Fig. A2: Runtime comparisons

The left panel displays the grid search runtime for cybersecurity and medical
fields. The right panel displays the parameters’ calibration runtime for the
cybersecurity and medical fields. The ordinate axis uses a logarithmic scale.

1:$e^{C}\leftarrow[0]\cdot len(c\\_names)$

2:for $c\in range(c\\_names)$ do

3: for $i\in range(i\\_names)$ do

4: for $c\in range(i\\_names)$ do

5: $e^{IC}_{i,c}\leftarrow\sum_{t=0}^{\mathcal{T}}\gamma^{i,c}_{t}$
$\triangleright$ $\gamma^{t}_{i,c}$ is the amount of the investment from $i$
to $c$ at time $t$

6: $e^{C}[c]\leftarrow e^{C}[c]+e^{IC}_{i,c}$

7: end for

8: end for

9:end for

10:$e^{C}_{max}\leftarrow\max{(e^{C})}$

11:$f^{C}\leftarrow e^{C}/e_{max}$ $\triangleright$ $f^{C}$: list of previous
investments for each technology

12:return $f^{C}$

Algorithm 1 Previous investments factor for companies

1:$e^{C}\leftarrow[0]\cdot len(c\\_names)$

2:for $c\in range(c\\_names)$ do

3: for $i\in range(i\\_names)$ do

4: $e^{IC}_{i,c}\leftarrow\sum_{t=0}^{(T)}\gamma^{IC}_{t}$ $\triangleright$
$\gamma^{t}_{i,c}$ is the amount of the investment from $i$ to $c$ at time $t$

5: $e^{C}[c]\leftarrow e^{C}[c]+e^{IC}_{i,c}$

6: end for

7:end for

8:$e^{T}\leftarrow e^{C}\cdot M^{CT}$ $\triangleright$ Matrix multiplication

9:$e_{max}\leftarrow\max{(e^{T})}$

10:$f^{T}\leftarrow e^{T}/e_{max}$ $\triangleright$ $f^{T}$: list of previous
investments for each technology

11:return $f^{T}$

Algorithm 2 Previous investments factor for technologies

1:$h\\_dict\leftarrow\\{\\}$

2:for $c\\_name,c\\_address\in c\\_locations$ do

3: $lat\leftarrow c\\_address.latitude$

4: $lon\leftarrow c\\_address.longitude$

5: $h\leftarrow haver\\_dist(lat,lon,lat\\_inv,lon\\_in)$ $\triangleright$
haver_dist is a function we have created

6: $h\\_dict[c\\_name]\leftarrow 1/h$

7:end for

8:$h\\_max\leftarrow\max{(h\\_dict)}$

9:for $c\\_name,h\in h\\_dict$ do

10: $h\\_dict[c\\_name]\leftarrow 1-h/h\\_max$

11:end for

12:return $h\\_dict$

Algorithm 3 Geographic coordinates factor

### Distance computation

We obtain the distance between two points on earth with the Haversine
approximation ($\operatorname*{hav}(\theta)$), using latitude and longitude of
the locations[19].

Let $(\lambda_{1},\phi_{1})$ and $(\lambda_{2},\phi_{2})$ be the longitude and
latitude in radiance of two points on a sphere and $\theta$ the central angle
given by the spherical law of cosines, the Haversine distance writes,

$h=\operatorname*{hav}(\theta)=\operatorname*{hav}(\phi_{2}-\phi_{1})+\cos{\phi_{1}}\cos{\phi_{2}}\operatorname*{hav}(\lambda_{2}-\lambda_{1}).$
(11)