Project 2 22C:021 Computer Science II : Data Structures

Power-law Distribution The power-law degree distribution captures the phenomena that a small number of vertices in a graph have high degree, whereas most of the vertices have very small degree Captures phenomenon prevalent in real networks

Requirements Run 3 experiments Capture results from experiments and prepare a deliverable  Details on deliverables will be up soon.

Experiment 1 Run experiments to support the hypothesis:  “random graphs generated in Project 1 do not satisfy the power-law distribution” Write a method  int [] degreeDistribution ()  This method returns the distribution such that element i holds the number of vertices with degree i. Use results to determine if the hypothesis holds.

Experiment 2 Implement a graph similar to one from Project 1  Randomly generate graph  Instead of random rewiring use preferential rewiring. Project description explains these terms well Repeat tests from Experiment 1 on this new graph Determine if this new graph generation mechanism supports power-law distribution

How do I determine probabilities? Experiment 2 expects you to wire vertices depending on how “popular” they are. Popularity is defined as the degree of a vertex  Higher the degree, the more popular a vertex is You connect to vertices depending on their degree (preferential rewiring) This is different from Project 1, where a random non-neighbor was chosen to replace an edge (uniform rewiring)

How do I determine probabilities? Determine a set Z that contains all non- neighbors of v  v is the vertex you're trying to wire For each z in Z  compute probability of that vertex being picked as probZ [i] = (degree (Z[0]) + degree (Z[1])... +degree(Z[i])) / D  D is the sum of all degrees of vertices in Z

How do I determine probabilities? Now, generate a random real number in the range 0 to 1 Find the first i such that p ≤ probZ[i]  Note that probZ is a non-decreasing seires. Connect to Z[i]

Experiment 3 What are we trying to determine?  Can verticies in our models use “local information” to connect to a different vertex Run experiments on both  Uniform rewiring  Preferential rewiring This experiment is not difficult to figure  See project description on the course webpage.

Finding distances between vertices Use greedy algorithm to find distance between two vertices (in hops)  On a vertex v find a vertex w that is closer to destination than v is, and use w as the next hop “Closer”: Remember: coordinates of each vertex are available in graph Use breadth-first-search to determine shortest path hop distance Use Dijkstra's shortest path algorithm to determine the shortest path Euclidean-distance

Dijkstra's shortest path algorithm S <- emptyset d[v] <- infinity for all v in V d[s] <- 0 initialize the binary heap H to V while(H is non-empty)do { v <- deleteMin(H) S <- S union v for(each u, neighbor of v)do { if(d[u] > d[v] + w(u, v)) { d[u] <- d[v] + w(u, v) changePriority(H, u, d[u]) }

Dijkstra's shortest path algorithm changePriority() will run in O(logn)  But... Provided that we know where u is within H Since H is an array this might take O(n) time  O(n) is not good  One approach to fix this: Have each node store its index in H When changePriority() gets u, it can fetch u.heapPosition Make sure you update u.heapPosition whenever you perform a heap operation