Networks are represented graphically. The topology of a network has implications for the distributed system sitting on top if it.

✏️ Graphs 101

• Graphs are either man-made (internet, WWW, social media) or natural (biological, ecosystems, DNA)
• When deployed at-scale, graphs have unexpected dynamic behaviors (ex: election results can never be polled accurately at-scale)
• 6 (or 5) is recurring magic number for graphs.
  • 6 degrees of Kevin Bacon - graph of all actors where edges connect actors that work together shows that any actor can reach Kevin within 6 hops
  • Same can be observed for many naturally occurring graphs, including a study on Facebook social graph

🔧 Network Properties

(1) Clustering Coefficient

• CC = P(A ↔️ B | A ↔️ C && B ↔️ C)
• the probability of A connected to B given that A connected to C and B connected to C
• usually high in human social network

(2) Path Length

Given these properties, we have three types of graphs

🎓 Graph Degree Distribution

Degree distribution is the probability of a given node having k edges. Some variants of this are:

• Regular graph = all nodes same degree, extremely uncommon
• Degree distribution follows a Gaussian distribution
• Random graph = probability that given node has k neighbors follows an exponential distribution related = $e^{-k*c}$ where $c$ is some constant
• Power law = probability that given node has k neighbors follows $k^{-\alpha}$
  • These are scale-free, meaning the degree distribution is mostly unchanged by scale or # of nodes (Gnutella has 3.4 edges per vertex)
  • WWW is a small-world + power-law graph with $\alpha = [2.1, 2.4]$

⭐ Small-world Graphs that follow Power Law

Extended ring graph can be transformed into a random graph by replacing some edges, but in this process, there is a happy medium where you get the best of both worlds.

PRO: Highly resilient to random attacks, killing large number of randomly chosen nodes cannot disconnect the graph

CON: Weakness is that if the few important nodes (high-degree nodes which are less than 5% of all nodes) are targeted and chosen, the graph will become disconnect

Naturally, to get the shortest path, the high-degree vertices will have heavy overload, BUT in the real world this is mitigated with randomization. For example, ISPs in a network have contracts that make certain paths more expensive, and as a result the high-degree vertex isn’t always taken, thus creating a natural load balancing effort, never causing congestion by overloading the high-degree switch/router.

**💡 Trivia!**

Erdos was a mathematician that wrote many papers with various people. 
Erdos himself has Erdos number = 0
Anyone that worked with Erdos has Erdos number = 1 
Anyone that worked with someone that worked with Erdos has Erdos number = 2 
It is proven that any researcher is within 5-6 hops of hops away from Erdos, i.e. the magic number! 

Note not all small-world networks follow the power law, this is an example of such a network.