Quantum Social Networks (I love science)

Social networks (SNs) are a traditional subject of study in social sciences [1–3] and may be tackled from many perspectives, including complexity and dynamics [4–6]. Recently, they have attracted much attention after their tremendous growth through the internet. A SN is a set of people, ‘actors’, with a pattern of interactions between them. In principle, there is no restriction on the nature of these interactions. In practice, in actual SNs, these interactions are based on relationships or mutual acquaintances (common interests, friendship, kinship, etc). However, to our knowledge, SNs have never been discussed on the basis of general interactions which can give rise to them.

This is precisely the aim of a paper by Adan Cabello, Lars Danielsen, Antonio Lopez-Tarrida and Jose Portillo published by IOP Science.  The authors introduce a physical approach to social networks (SNs) in which each actor is characterized by a yes–no test on a physical system. This allows them to consider SNs beyond those originated by interactions based on pre-existing properties, as in a classical SN (CSN). As an example of SNs beyond CSNs, they introduce quantum SNs (QSNs) in which actor i is characterized by a test of whether or not the system is in a quantum state |ψi. They show that QSNs outperform CSNs for a certain task and some graphs. They identify the simplest of these graphs and show that graphs in which QSNs outperform CSNs are increasingly frequent as the number of vertices increases. They also discuss more general SNs and identify the simplest graphs in which QSNs cannot be outperformed.

Whenever I write science articles, I am usually asked how this anything to do with marketing.  Well, this one does: Any actual SN through the internet, like Facebook or Twitter, is complex enough to potentially benefit from assigning quantum tests to the actors. An example is the following: suppose that a company wants to sell a product to as many Facebook users as possible. Under the (correct) assumption that Facebook is a CSN, the optimal strategy would be to identify the biggest subgroup of mutually linked actors, single out their common interest, and then design a commercial targeting this common interest. However, if Facebook were a QSN with exactly the same links as the actual Facebook, then the company would have a larger positive feedback by linking its commercial to the results of the quantum tests.

As in a CSN, the vertices of a QSN can be organized in communities or clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Given a graph G, community detection might be simpler if the graph represents a QSN rather than a CSN. The reason is that a QSN with a given G requires a (quantum) physical system of dimension (i.e. number of perfectly distinguishable states) dQ = ξ ( £˛G), with ξ ( £˛G) the orthogonal rank of the complement of G, defined as the minimum d such that there exists an orthogonal representation of £˛G in d dimensions. dQ dC and, in most cases, dQ < dC.  Once a community is detected, the study of its induced subgraph will tell us whether or not it has a quantum advantage. Note that QSNs with no global quantum advantage can contain induced subgraphs (e.g., representing communities) with quantum advantage.

On the experimental side, constructing a simple QSN with advantage over its classical counterpart is within actual experimental capabilities. The simplest example is a pentagon in which each actor has a device for testing the appropriate quantum state.

The authors do an outstanding job of outlining the construct (with graphic examples) of the benefits of the process, especially how Classical Social Networks fail to measure to Quantum Social Networks.  SN is typically described by a graph in which vertices represent actors and edges represent the result of their mutual interactions, the graph does not capture the nature of the interactions or explain why actor i is linked or not to other actors.  From this perspective, the graph gives an incomplete description.  With the ever expanding importance of social media among advertising agencies, I highly recommend downloading their paper.

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