# Social Capital Within Social Health Networks (part 2)

In my last blog I introduced the concept of triadic closure_ and the fundamental effects triadic closure has on both a micro and macro level for network phenomena.  One can understand from an intuitive perspective how triadic closure operates and everyone usually can find direct affective examples of this phenomena. Social Capital is both altruistic and opportunistic in its function. Triadic closure can operate in the same fashion. In most cases if friend B and friend C are more likely to become acquaintances it's usually because of opportunity, incentive, or trust. Some of the trust and incentive aspects can have far reaching effects such as latent stress. For example, when A (for whatever reason) has a need to bring B and C together even though B and C are not friends yet both trust A.

This brings us to how important those ties are within a triadic closure. Within the three aspects of opportunity, incentive, and trust we have the notional functions of strong and weak ties within the closure. Strong ties denote strong relationships. A weak tie has a lesser direct effect (although not negative). Therefore we can say:

If node A edges to nodes B and C, then the B-C edge has a high probability to form if A's edges to B and C are strong ties.

This makes sense from the basic concerns of what happens within most physical relationships. Obviously networks at scale (e.g. very large scale social graph architectures) cannot have strong ties at every edge. To this end most aspects of "strength" within a social graph, which is probably linear in most cases, are a combination of the interaction time, sentiment, and confiding in one another as well as reciprocity. Reciprocity is also a large factor in computing the strength of ties.  There are many theories of what actually constitute a strong and weak tie and how this benefits the person via the social capital within a network. We will continually revisit this fundamental but important concept.

So how do we measure the closure and extended graph of several closures?

"Network intransitivity" measures the structure of the extended social networks in which persons are embedded. In terms of network topology, transitivity means the presence of a heightened number of triangles in the network—sets of three vertices each of which is connected to each of the others.

Let's consider three acquaintances: if Anames Bas a friend, and Bnames C as a friend, then A and Care 2 friendship steps apart. If A also names C as a friend, the resultant triad (A, B, C) is transitive. If does not nominate C, despite B's friendship with A and C, then the triad is intransitive. Transitive relations reflect closed, dense friendship groups, in which an individual's friends are friends with one another. Intransitivity indicates dissonant relations, where an individual's friendship circle spans multiple disconnected members. The intransitivity index can measure the proportion of an individual's friends' friends who were not also the individual's friends.  This is also a reflection of the clustering network coefficient we previously discussed. Given this behavior we can surmise that time and similarity play a major roll in the reasons behind weaker A-B and A-C ties, making a C-B tie less likely than if the A-B and A-C ties are strong ones and rendering C and B much less likely to have interactions and less "compatible" when actually interacting on a social network.

The basis for this interaction and interest model lies in the similarity of these networks. As mentioned in the previous blog there are inherent value systems in creating Social Capital. At PokitDok, we are specifically interested in calculating a measurement for Social Capital in Social Health Networks (#sharethehealth). It's this emphasis that creates value in the network. But, how do you calculate a triadic closure?

Input:

An undirected graph G=(V,E) with |V|>0 and |E|>0

Output (a loop of the following steps):

Steps:

1. Look for two nodes x, y that are non-adjacent and share a common neighbor
2. If no such pair of nodes x, y exists, then we're done
3. Add element {x,y} to set E
4. Repeat step 1

The output of the triadic closure algorithm has some notable properties:

• Consider the closing of a single triad (the closure part): Nodes x and y are non-adjacent and share a common neighbor; then x and y are joined by edge {x,y}.
• Adding edge {x,y} does not change the number of connected components in G. Nodes x and y were already connected before we joined them with {x,y}.
• Adding edge {x,y} increases the density of the connected component that contains nodes x and y.

How this graph density increases over time is extremely important to similar relationships and predicting future relations.  This nodal density is the important aspect of the clustering components and results in the similarity function we mentioned earlier.  This "sameness" or "likeness" is called "homophily".  There are two types of homophily in networks: status and value.

As for reference concerning homophily a familiar proverb has been in use since at least the mid 16th century. In 1545, William Turner used a version of it in his papist satire "The Rescuing of Romish Fox":

"Byrdes of on kynde and color flok and flye all wayes together."

Also known as:

"Birds of a feather flock together."

In their original formulation of homophily, Lazarsfeld and Merton (1954) distinguished between status homophily and value homophily. Recently sociologist McPherson, Smith-Lovin, and Cook (2001) cite over one hundred studies that have observed homophily in some form or another. These include age, gender, class, organizational role, and so forth.

Status homophily means that individuals with similar social status characteristics are more likely to associate with each other than by chance. Value homophily refers to people who "think" in similar ways or like similar things regardless of class or status.  (Note: the reason I placed "think" in quotes is my personal belief that humans react differently online that in person, thus thinking is a value perception which is another blog entirely.)

Major differences in the two are the speed at which they can change. Social status and class status are very slow phase changing systems compared to values, which if we use the current terminology of "Like", can actually occur immediately – albeit by accident via simply hitting a "like" or "share."

In the now classic paper by Robin Dunbar, Human Nature 14 (2003) he shows that that average size of a human social network is approximately 150. Note: this assumes an average degree of centrality based on this observation. He further shows that this is actually hard limited by our brains via the pre-frontal cortex. This has direct limits on our ability to reason within relationships and people. However given the world of social networks online this number would, at first glance, appear small. That said online presence takes much less effort to maintain than a close physical personal relationship (albeit, at the moment, we do not have virtual reality constructs that truly mimic our perceived reality – never say never).

So every person actually has different preferences for personal contact. From the misanthrope to the gadfly, we call these preferences "levels of engagement." So where does all this lead?

In our next post we will discuss how small-scale networks, clustering and homophily work together to classify what you may like or dislike from a programmatic standpoint, as we continue to delve into the true essence of #sharethehealth.

Until then,

Ted Tanner, CTO PokitDok (@tctjr)