Tag Archives: friends

Actuarial science

Networks to reinvent insurance?

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The theory of networks, or graphs, was born in 1735, following the work of Leonard Euler, who tried to find a walk – starting from a given point – that would bring us back to that point by passing once and only once through each of the seven bridges in the city of Königsberg. These networks can be compared to metro networks, consisting of stations (nodes), linked between two by rails, or not, or more generally a road network, which can give rise to congestion studies, for example. But today, networks are mainly social, connecting people through friendships, professional, family, or monetary ties. Network analysis makes it possible to create relatively homogeneous communities, accepting to share a risk, recreating a mutualisation.

Network and credit

In genealogy, we will have hierarchical networks, a child being linked to his parents, who are themselves linked to their parents. In sociology, social networks make it possible to analyze the links between individuals (or organizations) within a group. Friendships can be studied in a schoolyard (a link that could be an invitation to a birthday party) or e-mail exchanges in a company (the Enron e-mail database has been widely used, with over 180,000 messages exchanged between 36,000 employees). Figure 1 shows two networks of 20 individuals (A, B, …, T).

 

Figure 1: Random networks, 20 nodes (Watts-Strogatz and Barbasi)

In a Facebook or Linkedin type vision, we will say that E and F are linked, in the sense of “friends”, if there is a segment linking points E and F. A network can be directed, for example if we study the exchange of messages (E wrote to F), or money loans (E lent money to F). If historically only adjacency was studied (existence or not of links), we can now add weights, for example the amount of a financial loan. Babutsidze (2012) thus studies the positions of French and German banks in interbank lending within the European zone (the nodes are then the banks). The study of networks within village communities in developing countries has led to a better understanding of informal finance mechanisms. Banerjee et al (2013) study the dissemination of information in a network, and more particularly microfinance loans.

While networks are useful for better organizing microcredit, CNN noted in 2015 that Facebook allowed credit organizations to use a borrower’s social network to determine whether or not it represents a good credit risk. In particular, if friends’ credit scores were too low, a person could be denied credit. This situation is dangerous because of the particular properties of networks, and more particularly the paradox of friends.

From the very small world to the paradox of friends

In 1929, Frigyes Karinthy hypothesized that any person on earth could be connected to any other person by a succession of individual relationships involving at most 6 links. “We should select anyone from the world’s 1.5 billion people, anyone, anywhere. It seems that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the chosen individuals using nothing other than the network of personal acquaintances. This theory of six handshakes originated in a new literary novel. It will be necessary to wait for the work of Michael Gurevich in the 1960s, then Stanley Milgram ten years later, to see the first attempts to quantify these relationships appear, under the name “Small World Problem”.

While Leskovec & Horvitz (2008) confirmed this order of magnitude, by analyzing several billion messages exchanged using the Windows Live Messenger platform, more recently, Baghat et al (2016) estimated that any two people on Facebook were connected by an average of three and a half people. On the random network on the left, a person has, on average, 2 friends, while a random friend has, on average, 2.25 friends. On the right-hand network, the gap is even greater, because if there too a person has, on average, 2 friends, a random friend will have on average more than 4 friends.

 

Figure 2: Random networks, 500 nodes (Watts-Strogatz and Barbasi)

This paradox, observed in 1991 by sociologist Scott Feld, is very easily demonstrated. Heuristically, we can see a link with the probabilistic property \frac{\mathbb{E}[X^2]}{\mathbb{E}[X]}=\mathbb{E}[X]+\frac{\text{Var}[X]}{\mathbb{E}[X]}>\mathbb{E}[X] where the term on the left is the number of friends of my friends, divided by my number of friends. The difference is all the greater the greater the dispersion of the number of friends. If the left-hand network is very dense, the right-hand network on the other hand has a power law property: the distribution of the number of friends follows a power law (or Zipf law, or Pareto’s law). Figure 3 shows the distribution of the number of friends on a network, in a double logarithmic scale: linearity indicates a distribution according to power. This type of distribution can be found in a very large number of networks, particularly Facebook, as shown by Wohlgemuth & Matache (2014).

 

Figure 3: Distribution of the number of friends on simulated random networks (Watts-Strogatz and Barbasi in red)

The classic interpretation is that some people are central in the network, with a very large number of connections. This property is well known in marketing (we will then speak of a “peer effect“) but it also has impacts in risk management or public health. Chrisakis & Fowler (2010) have shown that influenza epidemics can be detected almost two weeks in advance, by monitoring the infection in a social network. In particular, the analysis of the health of central people in a network is “an ideal way to predict outbreaks, but detailed information doesn’t exist for most groups, and to produce it would be time-consuming and costly”. To return to the example of the credit score, if it is found to be correlated to the number of friends, the friends paradox makes it dangerous to use the friends’ score as an indicator of an individual’s risk!

The importance of homophilia

Another important feature of networks is the notion of homophilia, introduced in 2001 in sociology by two important articles, corresponding to the tendency to be connected to one’s peers. McPherson et al (2001) assumed that similarity generates connection, and therefore people’s personal networks are homogeneous across many socio-demographic, behavioural and intrapersonal characteristics. Moody (2001) studied friendships in elementary school playgrounds in the United States, with a focus on interracial friendships. Easley & Kleinberg (2010) thus presents a number of consequences of homophilia, ranging from the creation of tables at business meals to the granting of credit in the United States. The measurement of homophilia is the same as asking, taking into account pre-existing groups (according to gender, age, socio-professional category, etc.) how the links are distributed, between groups, or within groups.

 

Figure 4: Low homophilia (left) and high homophilia (right)

In an insurance context, an actuary seeks to create tariff classes, groups that are homogeneous in terms of risks, according to explanatory variables (the so-called tariff variables). People who live in the same place, drive the same types of vehicles, and have the same characteristics, are likely to be in the same class. But if homophilia exists in a population, a tariff group could perhaps be observed on a network of friends. Why not then consider creating groups within a network?

Using insurance networks

In this spirit, in 2010, Friendsurance was launched in Germany and has more than 100,000 insured in 2018. In France, a short collaborative insurance experiment was launched in 2015, with Inspeer, offering to share damage insurance deductibles (in car or home insurance) with friends. These types of collaborative insurance, sometimes called peer-to-peer insurance, are based on the formation of small groups by a broker. A portion of the insurance premiums paid is paid to a group fund, the other portion to a third party insurance company. Minor damage suffered by the policyholder is first covered by this group fund. For claims exceeding the deductible, the usual insurer is used. A group can be formed by the insured, forming a social network a bit like Facebook. In this model, the only requirement is that all group members must have the same type of insurance (e. g. liability insurance with legal expenses insurance).

As Schiller (2013) noted, this type of mechanism has many virtues, the first being to reduce costs and the risk of fraud. There is no tendency to cheat on the cost of a claim when the risk is borne by family members or friends. The anonymity of mutuality that exists in the law of large numbers is disappearing. But aren’t we reinventing version 2.0. of the tontine associations, with the strong return of risk sharing within close-knit communities?

References

Joshua Angrist. The perils of peer effects. Labor Economics, 30, 98-108, 2014

Zakaria Babutsidze. Positions of French and German Banks in European interbank lending network. OFCE, Mars 2012.

Abhijit Banerjee, Arun Chandrasekhar, Esther Duflo & Matthew Jackson. Diffusion of Microfinance. Science, 341,

Smriti Bhagat, Moira Burke, Carlos Diuk, Ismail Onur Filiz & Sergey Edunov. Three and a half degrees of separation. Facebook Research, 2016.

Ananya Bhattacharya. Facebook patent: Your friends could help you get a loan – or not. 4 août 2015, CNN,

Nicholas Christakis & James Fowler Social Network Sensors for Early Detection of Contagious Outbreaks. PLoS ONE. 5 (9): e12948. arXiv:1004.4792 2015

David Easley & Jon Kleinberg. Networks, Crowds, and Markets. Cambridge University Press. 2010.

Scott Feld. Why your friends have more friends than you do, American Journal of Sociology, 96 (6): 1464–1477, 1991.

Matthew Jackson. Social and Economic Networks. Princeton University Press, 2010.

Jure Leskovec & Eric Horvitz. Planetary-Scale Views on a Large Instant-Messaging Network. Microsoft Research, 2008.

Miller McPherson, Lynn Smith-Lovin & James Cook. Birds of a Feather: Homophily in Social Networks. Annual Review of Sociology. 27: 415–444, 2001.

James Moody. Race, School Integration, and Friendship Segregation in America. American Journal of Sociology, 107 (3): 679-716, 2001.

Wesley Perkins, Michael Haines & Richard Rice. Misperceiving the college drinking norm and related problems: a nationwide study of exposure to prevention information, perceived norms and student alcohol misuse. Journal of Studies on Alcohol 66 (4) : 470-478, 2005.

Ben Schiller. A Social Network For Insurance That Cuts Costs And Reduces Fraud. Fast Company, October 2013,

Brad Walker. How Peer-to-Peer Companies Are Transforming the Insurance Sector. The Street, Avril 2016,

Jason Wohlgemuth & Mihaela Matache. Small-World Properties of Facebook Group

Networks. Complex Systems, 23 (2014).

[i] Complete data can be downloaded from https://snap.stanford.edu/data/email-Enron.html

[ii] https://www.friendsurance.com/ and https://www.inspeer.me/ respectively

2017/2018, networks

Game of Friendship Paradox

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In the introduction of my course next week, I will (briefly) mention networks, and I wanted to provide some illustration of the Friendship Paradox. On network of thrones (discussed in Beveridge and Shan (2016)), there is a dataset with the network of characters in Game of Thrones. The word “friend” might be abusive here, but let’s continue to call connected nodes “friends”. The friendship paradox states that

People on average have fewer friends than their friends

This was discussed in Feld (1991) for instance, or Zuckerman & Jost (2001). Let’s try to see what it means here. First, let us get a copy of the dataset

download.file("https://www.macalester.edu/~abeverid/data/stormofswords.csv","got.csv")
GoT=read.csv("got.csv")
library(networkD3)
simpleNetwork(GoT[,1:2])

Because it is difficult for me to incorporate some d3js script in the blog, I will illustrate with a more basic graph,

Consider a vertex v\in V in the undirected graph G=(V,E) (with classical graph notations), and let d(v) denote the number of edges touching it (i.e. v has d(v) friends). The average number of friends of a random person in the graph is \mu = \frac{1}{n_V}\sum_{v\in V} d(v)=\frac{2 n_E}{n_V} The average number of friends that a typical friend has is
\frac{1}{n_V}\sum_{v\in V} \left(\frac{1}{d(v)}\sum_{v'\in E_v} d(v')\right)But
\sum_{v\in V} \left(\frac{1}{d(v)}\sum_{v'\in E_v} d(v')\right)=\sum_{v,v' \in G} \left(<br /> \frac{d(v')}{d(v)}+\frac{d(v)}{d(v')}\right)=\sum_{v,v' \in G}\left(\frac{d(v')^2+d(v)^2}{d(v)d(v')}\right)=\sum_{v,v' \in G} \left(\frac{(d(v')-d(v))^2}{d(v)d(v')}+2\right){\color{red}{\succ}}\sum_{v,v' \in G} \left(2\right)=\sum_{v\in V} d(v)
Thus,\frac{1}{n_V}\sum_{v\in V} \left(\frac{1}{d(v)}\sum_{v'\in E_v} d(v')\right)\succ \frac{1}{n_V}\sum_{v\in V} d(v)
Note that this can be related to the variance decomposition \text{Var}[X]=\mathbb{E}[X^2]-\mathbb{E}[X]^2i.e.\frac{\mathbb{E}[X^2]}{\mathbb{E}[X]} =\mathbb{E}[X]+\frac{\text{Var}[X]}{\mathbb{E}[X]}\succ\mathbb{E}[X](Jensen inequality). But let us get back to our network. The list of nodes is

M=(rbind(as.matrix(GoT[,1:2]),as.matrix(GoT[,2:1])))
nodes=unique(M[,1])

and we each of them, we can get the list of friends, and the number of friends

friends = function(x) as.character(M[which(M[,1]==x),2])
nb_friends = Vectorize(function(x) length(friends(x)))

as well as the number of friends friends have, and the average number of friends

friends_of_friends = function(y) (Vectorize(function(x) length(friends(x)))(friends(y)))
nb_friends_of_friends = Vectorize(function(x) mean(friends_of_friends(x)))

We can look at the density of the number of friends, for a random node,

Nb  = nb_friends(nodes)
Nb2 = nb_friends_of_friends(nodes)
hist(Nb,breaks=0:40,col=rgb(1,0,0,.2),border="white",probability = TRUE)
hist(Nb2,breaks=0:40,col=rgb(0,0,1,.2),border="white",probability = TRUE,add=TRUE)
lines(density(Nb),col="red",lwd=2)
lines(density(Nb2),col="blue",lwd=2)


and we can also compute the averages, just to check

mean(Nb)
[1] 6.579439
mean(Nb2)
[1] 13.94243

So, indeed, people on average have fewer friends than their friends.