Random Graph Generator University of CS 8910 – Final Research Project Presentation Professor: Dr. Zhu Presented: December 8, 2010 By: Hanh Tran

Overview Why RG Generator? Basic RG Concepts Implementation Features Randomness Verification Degree Distribution Example Convergence Example Results & Analysis Summary

Why Choose RG Generator Internet, WWW, social networks, scientific- collaboration networks Random Graphs are used as models Few RG generators available (commercial product/open source) Very specific (test new algorithm), complex, unfriendly interface

Basic RG Concepts Directed vs. Undirected Weighted vs. Non-weighted Minimum Edge Connected Graph

Basic RG Concepts (2) Complete Graph

Basic RG Concepts (3) Minimum < Sparse < Dense < Complete Degree Distribution - # nodes w/ certain # of edges

Implementation Features Minimum Graph = (n – 1) edges Sparse Graph = has at least as many as (n – 1) edges, but no more than (n 2 +n-2)/4 edges Dense Graph = has at least as many as (n 2 +n-2)/4 edges, but no more than n (n – 1)/2 edges Complete Graph = n (n – 1)/2 edges

Randomness Verification Random network – node linkages follow a Poisson distribution in a bell shape.

Randomness Verification (2)

Degree Distribution Degree Distribution = P(k) = n k /n Where n k = # nodes w/ degree k, n = # nodes in network Example 1. Input: n = 10, type = minimum, degree = 5, distribution = 70% Output: Minimum Graph contains 9 max edges. Complete Graph contains 45 max edges. Sparse Graph contains greater than 9 but less than 27 edges. Dense Graph contains greater or equal 27 but less than 45 edges. ********************************************************* We will create 9 edges for this Minimum edge graph. *********************************************************

Degree Distribution (2) Initial Frequency Array Node 0 has 2 edge(s). Node 1 has 3 edge(s). Node 2 has 2 edge(s). Node 3 has 3 edge(s). Node 4 has 1 edge(s). Node 5 has 2 edge(s). Node 6 has 2 edge(s). Node 7 has 1 edge(s). Node 8 has 1 edge(s). Node 9 has 1 edge(s). Currently 0 nodes have 5 or more number of edges. That represents 0 %... Adding more edges.

Degree Distribution (3) After adding an edge between nodes 0 and 9, the results are: Frequency Array Node 0 has 3 edge(s). Node 1 has 3 edge(s). Node 2 has 2 edge(s). Node 3 has 3 edge(s). Node 4 has 1 edge(s). Node 5 has 2 edge(s). Node 6 has 2 edge(s). Node 7 has 1 edge(s). Node 8 has 1 edge(s). Node 9 has 2 edge(s).

Degree Distribution (4) After randomly selecting nodes 0 and 3, which already have an edge, the results are unchanged: Frequency Array Node 0 has 3 edge(s). Node 1 has 3 edge(s). Node 2 has 2 edge(s). Node 3 has 3 edge(s). Node 4 has 1 edge(s). Node 5 has 2 edge(s). Node 6 has 2 edge(s). Node 7 has 1 edge(s). Node 8 has 1 edge(s). Node 9 has 2 edge(s).

Degree Distribution (5) After adding an edge between nodes 8 and 9, the results are: Frequency Array Node 0 has 3 edge(s). Node 1 has 3 edge(s). Node 2 has 2 edge(s). Node 3 has 3 edge(s). Node 4 has 1 edge(s). Node 5 has 2 edge(s). Node 6 has 2 edge(s). Node 7 has 1 edge(s). Node 8 has 2 edge(s). Node 9 has 3 edge(s).

Degree Distribution (6)... After 29 more iterations, the results are: Frequency Array Node 0 has 5 edge(s). Node 1 has 5 edge(s). Node 2 has 5 edge(s). Node 3 has 5 edge(s). Node 4 has 2 edge(s). Node 5 has 4 edge(s). Node 6 has 4 edge(s). Node 7 has 5 edge(s). Node 8 has 6 edge(s). Node 9 has 5 edge(s). 70% Achieved. We have 23 edges in this Sparse Graph.

Distributed Agreement Distributed agreement has been used in numerous areas: Network communication Computer architecture operating systems software engineering

Distributed Agreement Some examples: Byzantine agreement in flight control systems Multi-value consensus in critical applications commit protocols in database transactions Leader election from a number of hosts Load balancing among a number of hosts Clock synchronization among hosts

Distributed Agreement General approach to reach agreement: Broadcast: Each node broadcasts its value to neighbors Collect: Each node forms a multiset of values Average: Average the values received. It is hoped that agreement is reached when the above round is repeated a sufficient number of times.

Conditions for agreement Convergence: All hosts eventually halt with values that are within a predefined tolerance of each other Validity: The agreed upon values are within the range of initial values before the agreement process started.

Application of RG in Distributed Agreement Generation of RG can be used as a test-bed for: Simulating the distributed agreement process. Validating the process of distributed alogorithms. Better intuitive understanding of agreement process Comparing the simulation results with mathematical results. How fast convergence takes place in a realistic environment

Example of distributed agreement How fast do the node values come to a consensus? Example 2. Input: n = 6, type = minimum, convergence = on Output: We will create 5 edges for this Minimum edge graph. Adjacency Matrix: n

Convergence (2) Minimum Graph of 6 nodes, with random values

Convergence (3) INITIAL value diameter: 98 – 31 = 67 After ROUND 1, diameter of values becomes 44  converged by: (67-44)/67 = 34% After ROUND 2, diameter of values becomes 32  converged by : (44-32)/44 = 27% After ROUND 3, diameter of values becomes 24 After ROUND 4, diameter of values becomes 18 After ROUND 5, diameter of values becomes 14 After ROUND 6, diameter of values becomes 11 After ROUND 7, diameter of values becomes 09 Overall after 7 rounds the values converged by : (67-09)/67 = 87% If one expects to have 100% convergence, this process should continue until the diameter values becomes 0.

Some general results noticed Randomness = larger networks with higher connectivity produce a clearer bell-shape distribution. Degree Distribution = no clear impact based on network size. Network diameter seems to have more impact on the number of rounds to reach the desired convergence rate. Dense connectivity implies faster convergence in terms of number of rounds needed. Convergence rate and number of rounds to reach a desired level of convergence highly depends on the initial distribution of values.

Summary Basic framework for generating random networks with some key features, DD and Convergence. Future implementation of directed and weighted edges will provide additional data pertaining to convergence speed. Test the distribution agreement in presence of failures (such as link failures).