Small-World Graphs for High Performance Networking Reem Alshahrani Kent 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(N 1/2 -1) 3D Torus Diameter = 3/2(N 1/3 ) 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 Networks Regular 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=7 u 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=7 u 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=7 u 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 [2] Shin, Ji-Yong, Bernard Wong, and Emin Gün Sirer. "Small-world datacenters." Proceedings of the 2nd ACM Symposium on Cloud Computing. ACM, [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 Web. 25 Nov