Path Criticality Routing
3.2 Robustness Quantification
This first step towards designing a robust routing algorithm is to clarify the meaning of robustness. In this thesis we provide a quantitative definition of robustness.
3.2.1 Definition of Robustness
From a control-theoretical point of view, robustness is the capability of a network to keep itself in a stable mode when changes take place in different parameters of the network in unpredictable fashion. In order to fix our notion of robustness we begin with our definition of robustness. There are three major types of changes that may affect the performance of the network:
1. Changes in network topology including capacity.
2. Changes in community of interest (CoI), the set of active source-destination pairs.
3. Changes in Traffic demand.
Throughout this thesis, we call a ”network topology”, ”network control strategy”
or a ”traffic engineering method” robust if its performance is not sensitive to changes in topology, traffic or community of interest. We aim to develop robust methods by studying the interaction between flow assignment and network structure with the help of graph-theoretical concepts.
In order to have a robust routing plan we need to recognize the effect of link and node changes on network connectivity. Connectivity is a well studied subject in graph theory [52, 53] which allows us to define some useful metrics to measure the sensitivity of the network to node or link failures.
Capacity of a network is another key issue in flow assignment problem. Clearly the paths with more capacity are desired since the low capacity paths are prone to
congestion. Hence an intelligent routing plan should avoid routing the flows onto the low capacity paths and should request for capacity increases for those paths if possible.
Finally traffic demand directly affects the routing plan. The traffic demand profile may change from time to time (e.g. week-day traffic profile). We need to find routing schemes that are robust to the predicted traffic patterns and unpredicted ones to the extent possible.
We will next introduce two metrics to estimate the effect of these characteristics:
Link Criticality Index(LCI) and Path Criticality Index (PCI) which are based on the theory of graphs [83, 84]. We will subsequently propose our first heuristic routing algorithm based on PCI. This algorithm provides the ”general topology manager”
block of AutoNet.
3.2.2 Link Criticality Index (LCI)
Freeman [83] introduced a useful measure in graph theory called betweenness cen-trality. Suppose that we are measuring the centrality of node k. The betweenness centrality of the node is defined as the share of times a node i traverses a node k in order to reach a node j via the shortest path(s). A similar definition of betweenness centrality is also valid for the links of a network graph. Suppose n(sp)sd is the number of shortest paths between source-destination pair s−d and n(sp)sd (l) is the number of shortest paths between source s and destination d containing the specific link l . According to the original definition of betweenness, the betweenness of link l for source-destination pair s − d would be n
(sp) sd (l)
n(sp)sd . The total betweenness of link l is the sum of shortest path betweennesses of link l for all possible source-destination pairs: bl = Ps,dn(sp)sd (l)
n(sp)sd .
Example 3.2.1 Fig. 3.3 shows a high-level view of a network with 3 different active source destination pairs s1−d1, s2−d2, s3−d3. There are 3 paths between s1 and d1(P1, P2, P3), 2 paths between s2 and d2 (P4, P5), and 3 paths between s3 and d3 (P6, P7, P8). Among these
paths P1, P3, P5, P6, P7, P8are shortest paths. Further, paths P1, P2, P5, P7, P8include link l. These information are summarized in table 3.1. Based on table 3.1 shortest-path betweenness of link l can be obtained for different source-destination pairs as follows:
b(sp)s
1d1(l) = 1
2 b(sp)s
2d2(l) = 1
1 b(sp)s
3d3(l) = 2
3
The shortest-path link betweenness is then:
b(sp)(l) = 1 2 + 1
1+ 2 3 = 13
6
Figure 3.3: Test Network to Study Link Be-tweenness
Path Src Dest SP? l in Path?
P1 s1 d1 Y Y
P2 s1 d1 N Y
P3 s1 d1 Y N
P4 s2 d2 N N
P5 s2 d2 Y Y
P6 s3 d3 Y N
P7 s3 d3 Y Y
P8 s3 d3 Y Y
Table 3.1: Path Characteristics for the Test Network
In order to provide robustness, the shortest path is not necessarily the best path because it may overload some paths and underload other possible paths. For this
reason we modified the definition of link/node betweenness (we call it deterministic betweenness) as follows:
Definition 3.2.2 Let nsd be the total number of simple (loop-free) paths between source-destination pair s − d and nsd(l) be the total number of simple paths between source s and destination d containing the specific link l. Now one can define deterministic betweenness of link l for source destination pair s − d as the fraction nnsd(l)
sd . The total deterministic betweenness of link l is the sum of all these fractions for active source-destination pairs.
b(l) = X
(s,d)∈CoI(G)
nsd(l) nsd
(3.1)
By active source-destination pairs in definition 3.2.2 we mean those nodes which are actively sending and/or receiving traffic,or in brief community of interest for network G (CoI(G)). One can easily see two major differences between our modified definition of betweenness and the original shortest path one:
• In deterministic betweenness all the paths are involved, whereas in shortest-path betweenness only shortest paths are considered.
• In deterministic betweenness only the active path set (CoI) is involved in defini-tion of betweenness, whereas in original shortest path betweenness, all possible node pairs are considered.
Example 3.2.3 For the network of example 3.2.1, one can find the deterministic betweenness of link l for different source-destination pairs as follows:
bs1ld1 = 2 3 bs2ld2 = 1 2 bs3ld3 = 2 3 b(l) = 2
3 + 1 2 +2
3 = 11 6
Figure 3.4: Weighted Trap Network
Example 3.2.4 Consider network of Fig. 3.4 which is called ”Trap Network”. Details of information about deterministic betweenness is given in Table 3.2. In this table, psx(d) shows the deterministic betweenness of link x for source-destination pair s − d.
The trap topology is well-known in the context of survivable routing. Suppose there is a demand from node 1 for node 6. The min-hop path from node 1 to 6 is the straight line 1 → 3 → 4 → 6. It appears that this path is the best choice to run the demand, but in survivable routing we need to assign backup paths to each primary route. In trap network there is no link-disjoint backup path for 1 → 3 → 4 → 6.
Therefore it would be beneficial to choose path 1 → 2 → 4 → 6 (1 → 3 → 5 → 6) as the primary route for demands from 1 to 6. Then the link-disjoint backup path will be 1 → 3 → 5 → 6 (1 → 2 → 4 → 6). We will return to the trap topology in chapter 5 and will find its optimal weight assignment to maximize network robustness.
Deterministic link/node betweenness can characterize the topological load of a link/node regardless of the nature of the traffic. We use the available capacity of link l to characterize its load carrying capacity. Now we define our metric to quantify the robustness of a given link.