1. Network Topologies William M. Jones Assistant Professor
Computer Science Department
Coastal Carolina University

2. Network Classifications
Networks have two broad classifications
Static
Dynamic
Static Networks
Static networks consist of point-to-point communication links among nodes and are also referred to as direct networks
Dynamic Networks
Dynamic networks are built using switches and communication links. Dynamic networks are also referred to as indirect networks.
For example ….

3. The only non-PC, networking devices are adjacent to the PCs themselves
The main idea here is that in strictly static networks, there are no networking devices that facilitate routing or switching that are not directly adjacent to processing nodes, in this case, PCs. In the more common type nowadays, switches and routers are present in the middle, with the PCs at the edges of the network infrastructure. This is only one of many ways to attempt to classify networks into two broad categories.
The only non-PC, networking devices are adjacent to the PCs themselves
Networking devices are potentially non-adjacent to the PCs.

4. Network Topologies
A variety of network topologies have been proposed and implemented
These topologies tradeoff performance for cost; a full analysis goes beyond the scope of this course
Commercial machines often implement hybrids of multiple topologies for reasons of packaging, cost, and available components
Any analysis of a network will include the engineering tradeoff questions encompassing redundancy, fault tolerance, cost, scalability, and this whole topic gets complicated very quickly. This idea in this presentation is simply to intro some simple topologies, and give a rudimentary sense of the costs / performances involved. And to quote the dean from the meeting the other day, EE302 should be about teaching them to think critically, and step through problems logically (or course, this is largely paraphrased  )
Let’s take a look at some common static network topologies …

5. Linear Arrays (1D) Bus Ring What are some of the drawbacks here?
What happens if link is cut?
Susceptible to congestion
Bus
Ring
The drawbacks largely depend on how the bus / ring is implemented. For example, if only one node can transmit at a given time (which is typically the notion of a bus) then you have a whole congestion issue at larger scales. Of course, there are optical techniques that use WDM that can render this topology less restrictive. Also, if only one link is cut, the whole system is cut into two pieces. If that happens, can the system continue to operate? Good question. It depends.
What are some of the drawbacks here?

6. Meshes (2D, 3D)
2 links cut
4 links cut
More connectivity  better fault tolerance  more cost
Up from the 1D bus, to the 2D mesh. (Keep in mind, all these topologies are mentioned in the book, just no meat is provided). What kind of failures can happen now? What happens as you want to make the network larger? Wireless sensor networks tend to be ad-hoc meshes due to the physical location of the sensors. Some of the original supercomputers were connected this way. What are the degree of the nodes here? 4 in the case of 2D, and 6 in the case of 3D. This would imply that strictly speaking, the PCs would need 4 and 6 NICs (or 6 ports at the very least) to implement these strategies.
Two and three dimensional meshes: (a) 2-D mesh with no wraparound; (b) 2-D mesh with wraparound link (2-D torus); and (c) a 3-D mesh with no wraparound.
In terms of graph theory, networks can be described by a set of vertices (nodes) and a set of edges (links)
What’s up with these topologies? 4+ NW wires / PC, I’ve never seen a computer like that before!

7. Hypercubes (4D)
To just add one more computer to network requires doubling the number of computers
Adding only one additional node severely
disrupts symmetry
I introduced this here because it involves binary numbers, which we’ve done in class, and because of Hamming distances are relevant (see next slide). One point to mention here is that if you want to increase the size of the network, what happens? Where do you add more nodes? You have to add new dimension, which involves doubling the size of the network, just to add one node. This is a limitation. You can’t just add one because the symmetry that makes all these static networks work efficiently, including routing, it the fact that there is symmetry.

8. Self explanatory.
Note that the vertices can be numbered so that the Hamming distance between any two adjacent vertices is 1. This plays an important role in determining a path from sender to receiver (i.e. routing). For example …

9. Source
Dest.
You can start with the MSB or the LSB, it doesn’t matter. This provides a succinct way of routing a message from sender to receiver. When I node receives a message, it only has to compare the destination address to it’s own address, and make a choice among 4 links (in this case) to forward it. This technique will provide the shortest path from sender to receiver. There may be more than one such shortest path, but there exists none shorter.
Suppose node 0101 wants to send a message to node Starting with the LSB of the numbers, compare the destination address bit to the sending address bit and if they are different, take the adjacent link that makes them the same. This is referred to as dimension order routing.

10. Step 1 -- 0101 to 1111: These are the same  do nothing
Source
Dest.
Here starts a sequence of slides showing the routing process. The 0th bits are the same, so just goto the next bit.
Step to 1111: These are the same  do nothing

11. Step 1 -- 0101 to 1111: These are the same  do nothing
Source
Dest.
The next bits are different. So The message must take a link that would change that 0 in the sender’s address to the 1 that is in the destination’s address.
Step to 1111: These are the same  do nothing
Step t0 1111: These are different  follow the edge that takes you to a vertex that is 0111 – (a given node only has to know the address of his neighbors to make this work, which is a nice simplification)

12. Source
Dest.
Step to 1111: These are the same  do nothing, go to next bit
Step to 1111: These are different  follow the edge that takes you to a vertex that is 0111
Step to 1111: These are the same  do nothing, go to next bit

13. Step 4 -- 0101 to 1111: These are different  take the adjacent edge
Source
Dest.
Step to 1111: These are the same  do nothing, go to next bit
Step to 1111: These are different  take the adjacent edge
From this we can see that the number of “hops” from source to destination is in fact the Hamming distance between their “addresses”

14. Example 0: Message from 0000 to 1111 route shown above
Source
Dest.
If a link goes down in the path of the deterministic route, this static routing approach would fail. A dynamic system would route around this failure. Doing this however means the system has to ensure the message does not flow indefinitely around the network, which requires additional intelligence. Another issue worth mentioning is the difference between cut through routing and store and forward (a common concept in digital communications). If the faster cut through routing is used, then deadlocks can occur in the network. This would need to be addressed.
Example 0: Message from 0000 to 1111 route shown above
Dimension ordered routing is deterministic (same route between a pair of node, each time a message is sent). What are the drawbacks of this approach? ANS: Assumes all links are up.

15. Example 1
Since the min # of hops from sender to receiver is t he Hamming distance between their addresses, it would depend on the number of bits in the address. In a system with 16 node, like above, that is 4, because log base 2 of 16 = 4. at most, all the bits would be different, yielding the largest Hamming distance, and therefore the greatest number of hops.
Given a network with “p” nodes (computers), what is the maximum number of hops (diameter) to get from source to destination?

16. Example Solution
Given a network with “p” nodes, what is the maximum number of hops to get from source to destination?
Given “p” nodes, each node address will be log2(p) bits wide. In the worst case, the max Hamming distance between source and destination address would be when every bit position is different (e.g to 0101); therefore, the max number of hops would be log2(p).
Ahh …. Didn’t realize I already had this answer here!!!!

17. Trees Another variation
Static because point to point
Dynamic because switching in hierarchy
Trees are popular, both physically and logically. Often, a logical tree pattern is overlaid on top of a non-tree physical network in order to simplify a system. See HW solution I sent out to learn more about trees.
Links higher up the tree potentially carry more traffic than those at the lower levels; therefore, we could ….
Note these PCs must have 3 NICs

18. Fat Trees
Increasing link capacity
It is indirect, aka, dynamic network because there are switching elements present at the upper levels. Here we see PCs only at the leaf nodes. See HW solution for a reasoning behind increasing the bandwidth as you go up the tree.
Increase the available bandwidth hierarchically. This is a VERY popular approach.
Direct (static) or indirect (dynamic) network topology?

19. BLOCK DIAGRAM - CVN BS1 BS2 ES1 What topology is this? Tree
BS1 is at the root, and the clients (at the leaf level) are connected to the edge switches
UHF
EHF
Elint
SHF
KGR
USC38
SSR1 KG
GALE
Lite
TRE
WSC - 3
WSC6
OTCIXS V6 KG
STREDs
TEAMS
Modem
NECC KG
DSCS
EHF-MDR
DAMA KG
CWSP KG
Compatible
DAMA
TADIXS
WSC - 3 V6
ADNS
GFCP
Unclas
STUIII
MDS/
Nav
Unclas
Router/
APS MDS
PLT
LAN
INE
Link
TAC - X
Rad Mer
Mux CP
TAC -
X (2)
TAC -
X (2)
TAC -
X (5)
ITS/APPS
ISDS/Profile
NAVMACS
TBMCS
CVIC
CVIC
CVIC
CVIC
Secret
TAC -
X (2)
METOC
Router
Supplot
BS1
BS2
SHF/CA
TAC - X
Voice
TAC - X
Flag Plot
Services
Alt CP
TFCC
ACDS
TAC - X NT
Srvrs
PDC/BDC
ES1
ES2
ES3
ES4
ES5
ES6
ES7
ES8
ASUW NT
Srvrs
TAC -
X (4)
NTWorkstations
TAC - X
TAC - X
TAC - X
EXC/SQL
TFCC
USW
ASUW
JAOC
ACDS
NTWorkstations
ES1
SCI
TAC -
X (5)
INE
Router
sci
NT Server

20. Completely Connected & Star Networks
Any PC can directly communication with any other PC at the same time (theoretically)
Note, each PC must have multiple network ports (NICs)
Single point of failure
The completely connected graph is the one most similar to the internal structure of a modern day crossbar switch. In (a), any computer can send to any other computer at the same time as any other computer is sending.
(a) A completely-connected network of eight nodes;
(b) a star connected network of nine nodes.
How do we evaluate these static networks?
What are the costs, tradeoffs, etc?
Let’s define some metrics …

21. Some Useful Metrics
Diameter: The distance between the farthest two nodes in the network. Specifies max number of hops. (smaller  better)
Bisection Width: The minimum number of wires you must cut to divide the network into two equal parts. (larger  better)
Cost: The number of links is a meaningful measure of the cost. However, a number of other factors, such as the ability to layout the network, the length of wires, etc., also factor in to the cost.
Arc Connectivity: The minimum number of links that must be removed to break the network into two disconnected networks, not necessarily equal in size (larger  better)
Let’s take a look at the previous networks ….

22. Linear Arrays Imagine any two arbitrary nodes trying to communicate.
Diameter: p-1
Bisection width: 1
Arc connectivity: 1
Cost: p-1
Diameter: p/2
Bisection width: 2
Arc connectivity: 2
Cost: p
Note “cost” here is the taken to be the number of links
Imagine any two arbitrary nodes trying to communicate.
Which topology is “better”? ANS: Ring
The bus costs less, but the ring is more fault tolerant
What is fault tolerance? Qualitatively, the ability to continue to function properly (albeit in a potentially degraded mode) in the presence of faults, e.g. link failures.

23. Meshes Which is “better”? Why? Arc connectivity Bisection width
2D mesh
2D torus
Diameter:
Bisection width:
Arc connectivity: 4
Cost:
Diameter:
Bisection width:
Arc connectivity: 2
Cost:
Which is “better”? Why?
Floor function

24. Trees (leaf nodes) Diameter: (log is base 2) Bisection width: 1
Arc connectivity: 1
Cost: p-1
(log is base 2)

25. Completely Connected and Star
Diameter: 2
Bisection width: 1
Arc connectivity: 1
Cost: p-1
Diameter: 1
Bisection width:
Arc connectivity: p-1
Cost:
Note p2 links
Note the tradeoff here between cost and the other metrics.

26. Summary Static Interconnection Networks
Logs are base 2
Network
Diameter
BisectionWidth
Arc Connectivity
Cost
(No. of links)
Completely-connected
Star
Complete binary tree
Linear array
2-D mesh, no wraparound
2-D wraparound mesh
Hypercube
Wraparound k-ary d-cube
Scalability: How a metric changes as a function of increasing “p”, for example:

27. Number of links
Put y axis lable, change the cost part
Note completely-connected is more costly (in terms of the number of links) than the others

28. Max number of hops (diameter)
Multi-objective optimization

29. The final decision will most likely be a compromise between performance and cost.

30. Example 2:
Given the choice between a binary tree and a 2D mesh, which is better, and why?
Justify your answer!

31. Example 2: ANS Justify your answer!
Given the choice between a binary tree and a 2D mesh, which is better, and why?
The binary tree b/c at scale, has a smaller diameter and also costs less (fewer links) than 2D mesh.
Justify your answer!

32. Example 3
Given a complete binary tree interconnection network where the two farthest nodes are separated by 6 hops, how much would you have to spend on the network links, assuming $10 per link?
Assuming only the leaf nodes are PCs, at $500 per PC, what would the total PC cost be?

33. Example 3 Answer
Given a complete binary tree interconnection network where the two farthest nodes are separated by 6 hops, how much would you have to spend on the network links, assuming $10 per link?
6 hops  diameter is 6  using the equation from table, p = 15 Then, given p = 15, number of links is p – 1 (from table) = 15-1 = 14 so at $10/link that is $140
Assuming only the leaf nodes are PCs, at $500 per PC, what would the total PC cost be? If you draw the tree out, you see that it has 8 leaf nodes so $8*500

34. Example 4
Which topology provides the best fault tolerance? Why?

35. Example 4 Answer
Which topology provides the best fault tolerance? Why? This is a tough question. The completely connected one seems to provide the best tolerance to link failures; of course, it has this highest link and NIC/node cost too.

36. Common Network Hardware: The Switch

37. Ports of a Switch

38. Internal Structure of Switch
Internal switching elements can be configured to allow any two mutually exclusive pairs of nodes to communicate at the same time.
As the number of switch ports increases, what happens to the number of internal switching elements?
It increases, but at what rate? p2!
Scalability: How many switching elements are there?
Which static network topology is this similar to?
What are the differences?
Looks like mesh, but is closer to completely connected, except that only 1 cable would extend from the switch port to the PC.

39. “Cost”
Rows = p
p2 elements
Cols = p

40. Comparison of a network topology with a common network device: the switch
Fully connected network
p2 links! (meaning a total of roughly p2 ports)
Any two pairs of nodes can communicate at the same time! (theoretically because PCs typically only have 1 network port)
p2 switching elements!
Any two mutually exclusive pairs of nodes can communicate at the same time!
Somewhat more restrictive; however the fully connected graph is unrealistic not only because of the number of links, but also because each node would have to have p-1 NICs ..  switch is a good compromise

41. Why is a switch a good compromise?
Allows a direct (1 hop) connection between any pair of computers
Reduces the number of cables from p2  p
Reduces the number of NICs per computer from p-1  1
However, increasing network size typically increases the switch cost quadratically (p2)
Due to it’s internal structure
Especially as “p” gets very large.