Small-World Graphs for High Performance Networking

Reem AlshahraniKent State University

Main Types of Graphs

• Many real-world large scale networks e.g. social networks exhibit a set of properties that cannot be captured by traditional networks. • Regular Graph: each node has the same number of links• Random Graph: The number of links per node is random• Small-World Graph: Intermediary and more relevant to real-world

What is Small-World Graph?

• A Small-World graph is a Complex graph• Most nodes are not neighbors of one another, • most nodes can be reached from every other by a small number of

hops or steps.• In Small-World networks: The diameter of the network grows slowly

as the logarithm of the number of nodes. (How could that happen?)

Six Degrees of Separation

• Was first described by the Hungarian writer Frigyes Karinthy in 1929• In 1967, Milgram’s experiments [2] in the social network of the world

showed that any person turns out to be linked to any other person by roughly six connections. • In 1998, Watts and Strogatz [1] published the first Small-World

network mathematical model• In 2000 Kleinberg’s Model guarantees navigable graph with diameter

= O(logN)

Definition of the Problem

• In recent years, data centers or servers’ farms have been increasingly used in hospitals, universities and enterprise to run a variety of applications that meet consumers’ needs. • The main challenge is how to build a scalable DCN that delivers

significant aggregate bandwidth with low latency. • Network topology is one of the key determinants of application

performance in DCNs.

Graph Representation of the Problem Current Problems: • Hierarchical topologies bottlenecks & high cost• Random topologies high cost & complex• Structured topologies large diameter

Goals

• Main Goal: Improve the performance of data centers through improving the underlying structure and the connectivity.• Eliminates all hierarchical switches • Lower the number of links to decrease power consumption and cost • Easier maintenance. • Shorten the diameter relative to the number of nodes in the network

to decrease the latency while increasing the bandwidth. • More Fault tolerance

Construct a Graph for the DCNs

• A set of nodes represent the servers in the network. • The communication links are represented as edges

2D Grid Diameter = 2(N1/2-1) 3D Torus Diameter = 3/2(N1/3) Ring Diameter = N-1

Special properties that help to solve the graph problem more efficiently• The underlying nodes are connected in a regular pattern such as torus or cube• Such that every node can route efficiently to nodes in its immediate neighborhood

Interpret the Small-World Networks as Data Center NetworksRegular graph can be transformed into a Small-World in which the average number of edges between any two vertices is logarithmic.

• DCNs are modified by the adding a small number of long-range random links to link selected nodes throughout the datacenter based on Kleinberg’s Model• incorporates geographic distance in the distribution of long-range random

links• This model is proven to construct Navigable graph using greedy heuristic to

rout the messages with no more than logN steps where N is the number of nodes.

Kleinberg’s Model

• Based on an N X N, 2-D grid, where each node is locally connected to 4 immediate neighbors within one step (p=1)

Kleinberg’s Model

• Based on an N X N, 2-D grid, where each node is locally connected to 4 immediate neighbors within one step

q=2

• Add q directed random links per node

Kleinberg’s Model

• Based on an N X N, 2-D grid, where each node is locally connected to 4 immediate neighbors within one step

• Add q directed random links per node

q=2

• Define d(u,v): lattice distance between u & v

d(u,v)=2+5=7u

v

Kleinberg’s Model

• Based on an N X N, 2-D grid, where each node is locally connected to 4 immediate neighbors within one step

• Add q directed random links per node

q=2

• Define d(u,v): lattice distance between u & v

• Now, u has a link to v with probability proportional

to d -r(u,v).

• r is clustering exponent that determines the probability of a connection between two nodes as a function of their lattice distance

d(u,v)=2+5=7u

v

Kleinberg’s Model

• Based on an N X N, 2-D grid, where each node is locally connected to 4 immediate neighbors within one step

• Add q directed random links per node

q=2

• Define d(u,v): lattice distance between u & v

• Now, u has a link to v with probability proportional

to d -r(u,v).

• r = 2 (# dimentions) guarantees the distance between any randomly selected nodes is at most logN

d(u,v)=2+5=7u

v

Properties of Small-World Graphs

• The First Property: • Logarithmic diameter

• The Small-World graph has a log diameter which is considerably small comparing to the diameter of many data center networks. • By adapting this idea from the Small-World graph, it will be possible

for any well-structured data center network with a large diameter to have a logarithmic diameter.• Short path between almost all pairs

Properties of Small-World Graphs

• The Second Property:• Greedy routing

• Incorporating the geographic distance in the distribution of long-range random links enables a simple greedy algorithm to effectively route packets within the datacenter using at most O(logN) hops

Properties of Small-World Graphs

• The Third Property:• Clustering

• They are rich in structured short-range connections and have a few random long-range connections. • If there is a link (u,v) and (u,w) then it is more likely to be a link (v,w)• “The neighbors of u are more likely to be neighbors” • That increases the resilience of the network

Illustration of Small-World Networks[3]

The final graph will be a regular graph combined with a random subgraph

Is that graph problem hard on general graphs? • Most of the data center have well–structured topologies. • The SW phenomena can be implemented on regular graphs with any

dimensions. • Each node should be able to route efficiently to other nodes in its

immediate neighborhood • The dimension of the network determines the clustering exponent r

which determines the routing time.

References

• [1] D. J. Watts and S. H. Strogatz. Collective Dynamics of ‘Small-World’ Networks. In Nature, 393:440–442, June 1998.• [2] Shin, Ji-Yong, Bernard Wong, and Emin Gün Sirer. "Small-world

datacenters." Proceedings of the 2nd ACM Symposium on Cloud Computing. ACM, 2011.• [3] S. Milgram. The Small World Problem. In Psychology Today, 2:60–

67, 1967• [4] Wikipedia contributors. "Small-world routing." Wikipedia, The Free

Encyclopedia. Wikipedia, The Free Encyclopedia, 24 Aug. 2014. Web. 25 Nov. 2014.