class: center, middle, inverse, title-slide .title[ # Structural Holes and Communities ] .author[ ### David Garcia <br><br> <em>ETH Zurich, Chair of Systems Design</em> ] .date[ ### Social Data Science ] --- layout: true <div class="my-footer"><span>David Garcia - Social Data Science - ETH Zurich, Chair of Systems Design</span></div> --- # Triadic closure <div style="float:right"> <img src="closedTriad.png" alt="A strong tie." width=500pt/> </div> > **Triadic closure:** The tendency of individuals in social networks to create links to neighbors at distance 2 **Local clustering coefficient**: `$$C_i = \frac{2*t(i)}{d_i *(d_i−1)}$$` - `\(d_{out}(i)\)` is the outdegree of `\(i\)` (>1) - `\(t(i)\)` is the number of pairs of neighbors of `\(i\)` that are connected - can be averaged or computed globally --- # The strength of weak ties <div style="float:right"> <img src="strongTie.png" alt="A strong tie." width=400pt/> </div> **Strong ties:** People who are very close to you (family, friends, etc.) - Provide reciprocal emotional support - Long-lasting, require energy and time **Weak ties:** People you know but not very close to you (colleagues from work, school, etc.) - Provide information, for example about employment and housing - Are also faster to disappear > **The strength of weak ties:** Weak ties are weaker than strong ties, but are also many more than strong ties. (Mark Granovetter) --- # Structural holes and communities .center[] --- ### Measuring modularity The measure of [**modularity**](https://www.pnas.org/content/103/23/8577.short) quantifies the tendency of a network to be divided in densely connected subgroups that are sparsely connected across groups. **Q-modularity** measures this tendency: `$$Q = \frac{1}{2m} \sum_{u,v} [ A_{u,v} - \frac{k_u k_v}{2m} ]\delta(c_u, c_v)$$` - `\(u\)` and `\(v\)` are two nodes of the network. The sum goes over all pairs of nodes. - `\(m\)` is the number of edges - `\(A\)` is the adjacency matrix ($A_{u,v} = 1$ if and only if u and v are connected) - `\(k_u\)` and `\(k_v\)` are the degrees of `\(u\)` and `\(v\)` - `\(c_u\)` is the community of `\(u\)` and `\(c_v\)` is the community of `\(v\)` - `\(δ(c_u,c_v)\)` takes value `\(1\)` if and only if u and v are in the same community, otherwise it's `\(0\)` --- ## Finding community structures .center[] .center[Example: Louvain algorithm] --- ## When holes are bridged: The small world concept Small world networks have the following three properties: - Most nodes are not neighbors (sparse network) - The mean distance between two nodes is small - They have high global clustering coefficient <div style="float:right"> <img src="map_us.png" alt="Milgram's experiment." width=400pt/> </div> **Stanley Milgram's experiment:** - 160 people in Omaha try to reach one person in Boston by mail to their acquaintances - 44 letters reached the target with six steps on average - Short path length as evidence of small-world - **Guess: what is the average path length on Facebook?** --- # The Bacon number .center[]