# The Six Handshake Theory: How Mathematics Explains Our Connectedness

Once upon a time in the early twentieth century, the Hungarian writer Fridyes Karinthy, in his story “Chain Links,” introduced an amazing game in which participants had to connect with any person of the 1.8 billion inhabiting the Earth, using only personal acquaintances. According to his hypothesis, the chain of connections consisted of no more than five people, i.e. six levels of connection. This idea attracted attention and became known as the six handshake theory.

For a long time it remained a mystery why exactly six handshakes was the minimum number to establish a connection between people. However, a team of mathematicians from MIPT and King Juan Carlos University in Madrid found the answer to this question. They have developed a computer mathematical model that proves the theory of the six handshakes to be true. Their results were published in the journal Physycal Review X.

The basic idea of the model is that connections between people are formed through horizontal ties in societies, rather than through hierarchical structures. Scientists modeled the evolution of the social system and found that the six-handshake effect occurs when participants seek to establish useful connections at minimal cost.

According to the researchers, people often strive to acquire valuable connections or become the right person, but this requires some effort and cost. Only those connections for which the benefits exceed the costs will realistically emerge. In the model, participants in the network compete for centrality and try to become the most important intermediaries for as many people as possible with as little effort as possible. At the same time, agents influence each other, securing profitable connections and severing unprofitable ones. As a result, an optimal balance is established, in which the participants no longer seek to change anything. This equilibrium leads to a network with a diameter of six, regardless of its size.

Thus, according to scientists’ calculations, in all possible networks all participants are connected with each other through six virtual handshakes. This model confirms the old theory and explains the mechanism of social phenomenon through mathematical rules.

The historical roots of the theory of the six handshakes go back to the beginning of the 20th century, when Fridyesh Carinty in his story “Links in the Chain” introduced the game with the establishment of connections through acquaintances. However, the idea of these six handshakes permeated people’s minds and became popular only after experimental testing and scientific research.

According to the standard explanation through hierarchical structures, the chain of connections could contain more than seven or eight steps, but the rule of six handshakes worked even without such leaders. This suggests that horizontal ties in societies play a major role in the formation of ties.

The theory of the six handshakes has not only practical value, but also scientific significance. It helps us understand how people organize their connections and why the shortest path is limited to six handshakes. This theory also demonstrates that we live in a “tight world,” where all people are connected through a limited number of connections.

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