In this article I discuss some insights about optimization of social networks. Basically I suggest that "trust is not preserved" along relationship paths of more than 3 hops. In other words, social networks should never forward messages beyond 3 hops. Doing so makes the communication of that message effectively arbitrary, adding noise to the system and degrading utility for users.
The Law of Density
The value of a social network, to each member, is inversely proportional to the density of the network, defined as the average number of relationships per member. As a network becomes denser, it becomes harder to manage -- each member experiences more "social overload" and there are an increasing number of routes between members. As density increases the likelihood that any two members of the network are connected by at least one path increases, and furthermore the distance of such paths decreases.
Social networks, if they are successful, asymptotically tend towards maximal density -- networks in which everyone is linked to everyone else by 1 hop, at which point there is no more social benefit of belonging to the network.
The Law of Sparsity
It appears from this cursory analysis that social networks are most beneficial when they are sparse, but not too sparse. If they are too sparse then the likelihood of getting a message from a randomly chosen member to another randomly chosen member becomes increasingly low -- essentially they break apart into disjoint subnetworks or islands. Sparsity is good, in moderation.
The Law of Balance
So under what conditions are social networks "socially beneficial" to their members. It appears that social benefit of social networks is directly related to striking a balance between sparsity and density. If the network is too sparse, it breaks into disjoint graphs or average length of the path between any two randomly chosen members is very long.
The Law of Intermediation
Let H be the average number of "hops" from member to member in a social network, where H greater than 1.
In networks where H = 1 there is no value because everyone is connected to everyone else by one hop. This is no better than sending direct e-mail messages -- there is no intermediation. In other words, the social network does not act as a filter on messages.
At H = 2 everyone is on average connected to anyone else by 2 hops. For example, to reach you, I go through one mutual friend. So that is 2 hops: one hop from me to our mutual friend, and one hop from them to you. In H = 2 networks there is the potential for a high degree of trust between participants because all interactions are intermediated by parties who know both members of the interaction. In other words the intermediary member has a direct relationship with the originator of an interaction and the target of that interaction. Because the intermediary knows both parties they can act as effective agent for both of them -- they know who the parties are, what their interests, qualifications, needs and priorities are, and can therefore filter messages between them in a manner that tailors to their individual interests.
But when H = 3 the benefits of the network start to break down: On average most interactions are now intermediated by parties who do not have a direct relationship with both parties to the interaction. Instead, intermediaries have relationships with only one of the parties -- thus interactions will on average be half as likely to be relevant to recipients. To put it simply, people start getting requests to forward messages that are either from people they don't know, or to people they don't know. Because they don't directly know both parties they cannot act as filters for both parties. But they can at least represent one of the parties (either the originator or the target of a given interaction), which is better than nothing.
When H = 4 the situation becomes dramatically worse -- now interactions are on average always intermediated by at least one party who does not know either party to the interaction. For that intermediary the choice of whether to forward the message or not becomes arbitrary since they have no knowledge of the sender or the recipient directly. This intermediary may trust that the intermediaries they get the message from have direct relationships to the parties to the interaction however, so at least there is still some modicum of trust and reliability that is preserved as the message travels.
Beyond H = 4, I would suggest the value of a social network to its participants (as a filter and discovery mechanism) falls off so dramatically that it is really not significantly greater than brute-force search. Another way of saying this is that the relevance of messages becomes similar to just sending unsolicited emails. Because now there is at least 1 hop in every interaction in which the intermediary really doesn't know anyone who directly knows either of the parties to the interaction.
What does this mean for social networking companies? I think it means they should not bother forwarding messages more than 3 hops. If a path is more than 3 hops it actually decreases the value of the network to members and should not be used.
Therefore it would seem that the optimal size of H (the average number of relationship hops to get a message from member x to member y) is in the range H = 2 to 3. E-mail has H=1 -- that is on average messages are direct -- they travel over direct, unmediated relationships from sender to recipient. It is too unmediated and is only good for direct personal relationships.
Listservers and discussion boards have H=2, where the server acts as an intermediary. Social networking apps are most useful when H=3 -- in which case every interaction on average goes across two intermediaries and every intermediary knows at least one of the parties to the interaction. H=3 provides a good balance -- just enough fuzziness to enable useful networking to take place, but not so much fuziness and indirection to be irrelevant. H = 4 is borderline -- anyone who is 4 hops from me is pretty unlikely to be relevant to my interests because the interaction must be mediated by someone who doesn't know either me or the them. Beyond H = 4 there is really no point in using a social network instead of a directory or brute-force search.
Dave Douglass has suggested some refinements and further ideas based on this article.