US20100306372A1

US20100306372A1 - System and method for analyzing the structure of logical networks

Info

Publication number: US20100306372A1
Application number: US12/791,320
Authority: US
Inventors: Sean P. Gorman; Rajendra G. I. Kulkani; Laurie Schintler; Roger R. Stough
Original assignee: George Mason University
Current assignee: George Mason University
Priority date: 2003-07-30
Filing date: 2010-06-01
Publication date: 2010-12-02
Also published as: US7765319B1

Abstract

Systems and methods for analyzing the structure of logical networks. Embodiments of the invention include ranking critical nodes according to regional hierarchies, distance hierarchies, global hierarchies, and relay hierarchies. Embodiments of the present invention are capable of testing the effectiveness of such hierarchies. In addition, critical nodes may be used to define critical regions.

Description

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No. 10/902,416, filed Jul. 30, 2004, which claims priority of U.S. Provisional Application No. 60/490,910, filed Jul. 30, 2003. These applications are incorporated by reference in their entireties.

FEDERALLY SPONSORED DEVELOPMENT

This invention was made with U.S. Government support under grant number 60NANB2D0108, awarded by the National Institute of Standards and Technology (NIST). The U.S. Government may have certain rights in this invention.

FIELD OF THE INVENTION

The invention relates to systems and methods for analyzing the structure of logical networks.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates a system, according to one embodiment of the present invention.

FIG. 2 illustrates the method of a regional hierarchy, according to one embodiment of the invention.

FIGS. 3-5 illustrate an example of the method of a regional hierarchy, according to one embodiment of the invention.

FIG. 6 illustrates the method of a distance hierarchy, according to one embodiment of the invention.

FIGS. 7-8 illustrate the method of a distance hierarchy, according to one embodiment of the invention.

FIG. 9 illustrates the method of the global hierarchy, according to one embodiment of the invention.

FIG. 10 illustrates the method of the relay hierarchy, according to one embodiment of the invention.

FIG. 11 illustrates the method of testing the effectiveness of the node criticality ranking hierarchies, according to one embodiment of the invention.

FIGS. 12-13 illustrate an example of the method of testing the effectiveness of the node criticality ranking hierarchies, according to one embodiment of the invention.

FIG. 14 illustrates the method of defining regions by node connectivity, according to one embodiment of the invention.

FIG. 15 illustrates an example of the method of defining regions by node connectivity, according to one embodiment of the invention.

DESCRIPTION OF SEVERAL EMBODIMENTS OF THE INVENTION

Embodiments of the present invention relate to systems and methods for analyzing the structure of logical networks. The embodiments outlined can be used in spatial and non-spatial contexts for a variety of logical network structures.

System

FIG. 1 illustrates a system, according to one embodiment of the present invention. The system includes a storage database 105, which stores the data utilized in the present invention (e.g., network data) and a user interface 175. The network data comprises, for example, but not limited to: satellite imagery data; digitized map data; topological map data; photo data; satellite geo-spatial data; telecommunication data; marketing data; demographic data; business data; North American Industrial Classification (NAIC) code location data; right-of-way routing layers data; metropolitan area fiber geo-spatial data; long haul fiber geo-spatial data; co-location facilities geo-spatial data; internet exchanges geo-spatial data; wireless towers geo-spatial data; wire centers geo-spatial data; undersea cables geo-spatial data; undersea cable landings geo-spatial data; data centers geo-spatial data; static network data; or dynamic network data; or any combination of the above. The right-of-way routing layers data comprises, for example, but not limited to: gas pipeline data; oil pipeline data; highway data; rail data; or electric power transmission lines data; or any combination of the above. The logical network data comprises, for example, but not limited to: static network data; or dynamic network data; or any combination of the above. The static network data comprises, for example, but not limited to: ip network data; or network topology data; or any combination of the above. The dynamic network data comprises, for example, but not limited to, network traffic data. The regional analysis comprises, for example, but not limited to: continent information; nation information; state information; county information; zip code information; census block information; census track information; time information; metropolitan information; or functional information; or any combination of the above. The function information comprises, for example, but not limited to: a formula; a federal reserve bank region; a trade zone; a census region; or a monetary region; or any combination of the above.
Data can be obtained by performing, for example, but not limited to: purchasing data; manually constructing data; mining data from external sources; probing networks; tracing networks; accessing proprietary data; or digitizing hard copy data; or any combination of the above.
The system also includes a ranking system 130, which can include: a region program 155, a distance program 165, a global program 161, or a relay program 170, or any combination thereof. The region program 155 is a node criticality ranking approach which defines global connections as links that connect two different regions and local connections as links within a region. The definition of region is fluid including geographic regions, topological regions, industrial sectors, markets, etc. The distance program 165 is a node criticality ranking approach which defines global connections as links over a certain distance threshold and local connections as links under a certain distance threshold. The definition of distance is fluid including Euclidean distance, Manhattan distance, latency, bandwidth, flow measurements etc. The global program 161 is a node criticality ranking approach which looks only at the number of global connections utilizing either the region program 155 or the distance program 165. The relay program 170 is a node criticality ranking approach which takes the ratio of the total capacity connected to a node (i.e., supply) and the demand for that capacity to identify nodes that are acting as relays between large demand areas.

Regional Hierarchy

In many networks one or more nodes can be identified in a specific region that are most critical to the operation of that region. The region could be geographic, non-geographic, or both. For example, in a geographic region, the most critical nodes for Internet connectivity or airline traffic in a specified geographic area could be identified. As another example, the network (an autonomous system) that is the most critical to the connectivity of financial institutions connected to the Internet could be determined. In addition, the region could be a fusion of both geographic and non-geographic areas where the region is an individual network (autonomous systems) and the interconnection of different networks happens in specific geographic locations. In this case, the most critical interconnection points (i.e., nodes) of several networks could be determined. Embodiments of the invention could be used in a variety of network scenarios, including supply chains, social networks, or any other logical network structure.
FIG. 2 illustrates the method of a regional hierarchy, according to one embodiment of the invention.
In step 205, the network data is loaded into the system as one or more nodes. For example, the sample city-to-city long haul data network illustrated in FIG. 3 could be loaded into the system. Each of the nodes in a network has a location indicated by an identifier. For example, in a geographic region, the location could be tied to a city name. In a non-geographic networks, locations can be indicated by other identifiers.
In step 210, each node in the network is assigned to a region based on the node's location. The regions can be defined in a fluid manner, depending on the desires of the user. In the city-to-city long haul data network example, the nodes could be allocated to census regions illustrated in FIG. 4.
In step 215; once each node in the network has been assigned to a region, links (i.e., connections) between nodes are designated as global or local. Links that occur within a region are designated as local links, and links that connect nodes located in different regions are designated as global links. In the city-to-city long haul data network example, a connection between Atlanta, Ga. and Jacksonville, Fla. would be designated as a local link because both nodes are located in the South Atlantic Region.
In step 220, once all links have been designated as global or local links, a ratio of global links to local links is taken for each node in the network, and then weighted by the total number of links to the node. Thus, in the city-to-city long haul data network example, a ratio of one city's (i.e., node's) global links to local links is computed, and then the ratio is weighted by the total connectivity of the network (i.e., the total number of nodes in the network). This would provide an indicator of how well the city acts as a regional connector in the network.
In one embodiment, this process is expressed mathematically as follows: Consider a large network of nodes n, spanning an area A consisting of regions r, with a variable number of nodes inside each region that have a variable number of connections from each region to other regions. For a region r with p number of nodes n, a p×p contiguity matrix represents connections between these nodes. As illustrated in FIG. 5, a contiguity or adjacency matrix M for the entire network of m number of regions r can be constructed as a block diagonal matrix, where matrices along the main diagonal (indicated in the boxes where there is no grid pattern) refer to the contiguity matrices for each of the regions. Interregional connections are represented as the off-block-diagonal elements (indicated in the boxes with a grid pattern).
If a node i in region r is connected to another node j in the same region, then that connection is considered as a local link and is denoted by q_i(r)j(r). If node i in region r is connected to node k in region s then that connection is considered as a global connection and is denoted by g_i(r)k(s). Thus, one may associate each node i(r) with a global connectivity index as a ratio between its global and local connections, weighted by the total number of global and local connections for the entire network.
The total number of global connections G is computed from the elements of the upper triangular block of matrix M, of m regions, each with a variable number of nodes:
$\begin{matrix} G = \sum_{i (1)} \sum_{s > 1}^{m} \sum_{k (s)} g_{i (1) k (s)} + \sum_{i (2)} \sum_{s > 2}^{m} \sum_{k (s)} g_{i (2) k (s)} + \dots + \sum_{i (m - 1)} \sum_{s > m - 1}^{m} \sum_{k (s)} g_{i (m - 1) k (s)} & (1) \end{matrix}$
Note that, because in is the last region in the block diagonal matrix, its global connections have already been computed in the previous m−1 blocks.
The total number of local connections L is a sum over all the local connections over m regions and is given by:
$\begin{matrix} L = \sum_{i (1)} \sum_{j (1) > i (1)} q_{i (1) j (1)} + \sum_{i (2)} \sum_{j (2) > i (2)} q_{i (2) j (2)} + \dots + \sum_{i (m)} \sum_{j (m) > i (m)} q_{i (m) j (m)} & (2) \end{matrix}$
Thus, for example, if Jacksonville, Fla. was located in the Southeast region and had local connections to other region in the Southeast, including Orlando, Fla., Atlanta, Ga., Tallahassee, Fla., and Charlotte, N.C., but also a connection outside of the Southeast to Washington, D.C. in the Mid-Atlantic region it would have one local connections (G) and four local connections (L). In a non-spatial context an example would be identifying a critical autonomous system in the financial sector. The Bank of New York could have local connections to other autonomous systems in the financial region such as Morgan Stanley and Goldman Sachs, and also have connections to autonomous systems outside of the financial region such as the Federal Reserve (Govt.), MCI (Telecom) Sprint (Telecom), and General Electric (Tech/Manufacturing). In this case the Bank of New York would have two local connections and four global connections.
The global connectivity index for a node i in region r is then given by:
$\begin{matrix} C_{i (r)} = (\frac{\sum_{s \neq r}^{m} \sum_{k (s)} g_{i (r) k (s)}}{1 + \sum_{j (r), j \neq 1} q_{i (r) j (r)}}) \times (G + L) & (3) \end{matrix}$
Note that the numeral of 1 in the denominator indicates a self-loop of a node.
Using the example of Jacksonville above the equation would then be plugged with G=1 and L=4 resulting in C_i(r)=[(1/(1+4))×(1+4)]=1 indicating a relatively low level of criticality in the network. Using the Bank of New York examples the equation would then be plugged with G=4 and L=2 resulting in C_i(r)=[(4/(1+2))×(4+2)]=8 indicating a relatively high level of criticality in the network.
When the hierarchies above are set for the city-to-city long haul data network example, the following node criticality ranking is produced:

Top Sixteen Nodes

	CMSA	Region Score

	New York	135.7567108
	Chicago	120.3182127
	San Francisco	111.5303899
	Washington	98.90846075
	Boston	93.70275229
	Dallas	92.40582839
	Denver	81.42618849
	St. Louis	56.1399932
	Cleveland	43.84487073
	Louisville	41.33944954
	Kansas City	39.37090433
	Seattle	34.70472307
	Phoenix	34.70472307
	Los Angeles	33.95740498
	Atlanta	33.68399592

Thus, the most critical nodes in the network, ranked beginning with the most critical node, are: New York, Chicago, San Francisco, Washington, etc.

Distance Hierarchy

FIG. 6 illustrates the method of the distance hierarchy, according to one embodiment of the invention. In step 605, the network data is loaded into the system as one or more nodes.
In step 610, the distances between the nodes are defined and calculated. Distance is defined according to the desire of the user (e.g., Euclidean distance, latency, capacity, flow data). In this example, distance is defined as Euclidean distance.
In step 615, the link between nodes is designated as global or local. The designation can be determined by automating the node criticality-ranking equation with an incremental set of test distances. The test distances are used to calculate the ratio of global to local links, weighted by the total number of links connected for each individual node in the network. In one embodiment, this process is expressed mathematically as follows:
$\begin{matrix} R = (\frac{\sum_{j} g_{ij} > D}{1 + \sum_{j} l_{ij} \leq D}) (\sum_{j} g_{ij} + \sum_{j} l_{ij}) & (4) \end{matrix}$
where Σg_ijrepresents the numbered links between node i and nodes having a distance greater than a threshold value D; and Σl_ijrepresents the number of links between node i and nodes having a distance less than or equal to the threshold D. Using the Jacksonville example again, the distance between Jacksonville and its five connecting cities would be calculated as follows: Jacksonville-Atlanta=287 miles, Jacksonville-Orlando=127 miles, Jacksonville-Tallahassee=157 miles, Jacksonville-Charlotte=339 miles, and Jacksonville-DC=647 miles. Using a threshold of 300 miles, there would be three local connections (Jacksonville-Atlanta, Jacksonville-Orlando, and Jacksonville-Tallahassee) and two global connections (Jacksonville-Charlotte and Jacksonville-DC). When these numbers are plugged into the equation, the result is R=[(2/(1+3))×(2+3)]=2.5, raising the relative criticality ranking of the city from the regional hierarchy. Distance could also be calculated by other functions such as the flow between two nodes. If the same example looked at the tonnage of goods shipped between Jacksonville and its connections, the calculation would be: Jacksonville-Atlanta=6000 tons, Jacksonville-Orlando=8000 tons, Jacksonville-Tallahassee=500 tons, Jacksonville-Charlotte=1500 tons, and Jacksonville-DC=250 tons. Using a threshold of 1000 tons as the break between global and local, there would be two local connections (Jacksonville-Tallahassee and Jacksonville-DC) and three global connections (Jacksonville-Atlanta, Jacksonville-Orlando, and Jacksonville-Charlotte). When these numbers are in turn plugged into the equation, the result is R=[(3/(1+2))×(3+2)]=5, raising the relative criticality ranking of the city from the previous definition of distance. The same calculation could be done using many other definitions of local and global to determine other relationships, such as bandwidth capacity between nodes or the number of passengers using an airline route.
The test distances are then loaded in the equation and the output is graphed for the various test distances. The inflection point of the graphed curved is used as the distance threshold to run the hierarchy. An example of this is illustrated below using the city-to-city data network utilized previously. A series of alternative distances for distance D (e.g., 100 miles, 200 miles) are selected and used to simulate global/local ratios utilizing the city-to-city data network:
where Dε[100, 200, 300 . . . 2700]
The simulations produce the graph presented in the FIG. 7, where the x-axis are the increments of the global/local ratio produced by different values of D, and the y-axis are the percentage of nodes with a global to local ratio greater than one. FIG. 7 shows a sharp shift at about 300 miles and a second shift at about 700 miles.
To find the exact point of inflection, the rate of change (i.e., derivative) in the global to local ratio is calculated, as illustrated in FIG. 8.
The rate of change illustrated in FIG. 8 clearly points to 300 miles being the primary point of inflection. Under such an assumptions, all links shorter than 300 miles are considered local and all links over 300 miles are considered global.
In step 620, the hierarchy of step 615 is utilized for each node in the network to produce a criticality ranking, which ranks each node according to its global/local ratio. A sample of the out put for the hierarchy is displayed below.

Top Sixteen Nodes

	CMSA	Global/Local Ratio

	Salt Lake City	342
	Denver	312
	San Francisco	159
	Dallas	94
	Seattle	79
	Chicago	71
	Los Angeles	65
	Atlanta	64
	Washington	62
	New York	59
	Phoenix	55
	Houston	48
	Miami	41
	Boston	41
	Kansas City	34

Global Hierarchy

FIG. 9 illustrates the method of the global hierarchy, according to one embodiment of the invention. This hierarchy is based on the number of global connections per node. The nodes are ranked based only on this count. In step 905, the network data is loaded into the system as one or more nodes.
In step 910, the distances between each node are defined and calculated. Distance is defined according to the desire of the user (e.g., Euclidean distance, latency, capacity, flow data). In this example, distance is defined as Euclidean distance.
In step 915, the links are ranked according to the following equation
$\begin{matrix} R_{L} = \sum_{j} g_{ij} > D & (5) \end{matrix}$
where R_L=the ranking of the link, and g_ijis the distance between nodes i and j and D is a threshold distance.
This ranking provides an indicator of how many long haul global connections a node has, dictated by connections longer than D. (E.g., D was 300 miles in the sample case presented in step 215). In the Jacksonville example, there were two global links in the distance hierarchy example thus R_L=2, or using the regional hierarchy's definition of global R_L=1. In the financial example R_L=4, or in the distance tonnage example R_L=3.
In step 920, the nodes are ranked based on the ranking of the links connected to each node.

Relay Hierarchy

FIG. 10 illustrates the method of the relay hierarchy, according to one embodiment of the invention. This hierarchy identifies relay nodes and their effect on the survivability of the network. Relay nodes are locations that are neither the ultimate origin nor destination of an interaction across a network. The primary purpose of a relay node is to receive flows in order to transmit them to another node with minimum delay and cost. Nodes that act as structural links to relay information to large markets could serve as critical junctures. The following method determines which nodes are disproportionately acting as relay nodes.
In step 1005, the network data is loaded into the system as one or more nodes. In step 1010, the total capacity and demand for each node in the network is determined. For the city-to-city long haul data network example, the total capacity and demand is the total amount of bandwidth connected to the node (i.e., city) and the total bandwidth demand for the node (i.e., city).
In step 1015, the ratio of capacity to demand is determined for each node in the network. Mathematically, this can be expressed as follows:
$\begin{matrix} R = \frac{\sum_{i = 1}^{n} c_{ij}}{\sum_{i = 1}^{n} b_{ij}} & (6) \end{matrix}$
where R=ratio of capacity to demand, c_ij=capacity, and b_ij=business demand.
The relay hierarchy could be another means used to access Jacksonville's criticality. Jacksonville's total connected capacity equals 15000 megabytes, but its demand for capacity is only 5000 megabytes, thus its relay ratio would R=(15,000/5000)=3. The same could be done with an airline network, where capacity is the total number of passengers landing at the airport and demand are the number of passengers for which Jacksonville is their destination.
In step 1020, the nodes in the network are ranked based on their ratio R of capacity to demand. The greater the ratio, the higher the rank. This approach provides a rough indicator of how much built capacity exceeds the consumption of capacity dictated by demand. A sample of the out put for the hierarchy of step 1015 is displayed below.

Top Sixteen Nodes

	MSA	Relay Ratio

	Kansas City	7.511627907
	Salt Lake City	3.395759717
	Indianapolis	3.208191126
	Seattle	2.962616822
	Portland	2.753665689
	Sacramento	2.679577465
	St. Louis	2.2382134
	Denver	1.951584507
	Atlanta	1.882087099
	Washington-Baltimore	1.795747423
	Chicago	1.712831503
	Philadelphia	1.695364238
	Orlando	1.485314685
	Jacksonville	1.45785877
	Phoenix	1.201257862

Testing Node Criticality Ranking Hierarchies

The above hierarchies may be compared to determine which hierarchies are most correct. In order to test the effect of the above hierarchies on a network, each hierarchy is subjected to simulations.
Accessibility Index. The most commonly used indicator of node criticality is the number of connections a node has, often called the degree or the accessibility index. To provide a comparison to the new hierarchies outlined in this application, the accessibility index will be calculated and plotted to provide a baseline. This allows a demonstration if the new hierarchies are doing better or worse than current methods when the hierarchies are tested in the following section.
FIG. 11 illustrates the method of testing the effectiveness of the node criticality ranking hierarchies, according to one embodiment of the invention. In step 1105, the network data is loaded into the system as one or more nodes. In step 1110, the criticality rankings produced by each hierarchy are loaded into the system.
In step 1115, the diameter and S-I index of each node in each hierarchy is measured. Each node is successively removed according to its rank and the diameter of the network and the S-I is measured for each removed node.
The diameter of the network is the minimum number of hops it takes to get from the two furthest nodes on the network. Mathematically this is expressed as:
Diameter=maximum D_ij
where D_ij=shortest path (in links) between the ith and jth node.
Thus, for example, the longest shortest path in the city-to-city network is Eugene, Oreg. to Ft. Myers Fla., which uses the following route: Eugene, Oreg. to Portland, Oreg. to Seattle, Wash. to Denver, Colo. to St. Louis, Mo. to Atlanta, Ga. to Orlando, Fla. to Tampa, Fla. to Ft. Myer, Fla. The longest shortest path has seven hops, and thus the diameter of the network is seven.
The S-I index of a graph is based on the frequency distribution of the shortest path lengths s_ijin the graph. Mathematically, it is defined as the pair (S,I), where:
$\begin{matrix} S = \frac{μ_{3}}{μ_{2}} and I = \frac{μ_{2}}{μ_{1}} & (7) \end{matrix}$
In the above equation, μ₁is the first moment (i.e., mean) of the frequency of all shortest paths in the network, μ₂is the second moment (i.e., variance) of the frequency of all shortest paths in the network, and μ₃is the third moment (i.e., kurtosis) of the frequency of all shortest paths in the network. Once each moment for the network has been calculated the S index is calculated by dividing the third moment by the second moment, and the I index is calculated by dividing the second moment by the first moment. For example in the city-to-city data network μ₁=2.8274, μ₂=0.8324, and μ₃=0.0444. Thus S=0.0534 and I=0.2944 both providing a measure of connectivity for the network. As nodes are removed from the network the connectivity decreases and the S and I index captures the loss quantitatively.
By examining the S-I index of the US IP network infrastructure as nodes are removed, one can obtain a quantitative indication of how disconnected the network becomes.
The results of both the diameter and S-I index analysis can be found in the example below.

Output of Diameter and S-I Index Analysis on Hierarchies


Diameter	CMSA	I = u2/u1	S = u3/u2

Binary Hierarchy

7		0.2937	0.0499
8	Atlanta	0.3416	0.0927
8	Chicago	0.3445	0.0449
8	San Francisco	0.3466	0.0424
10	Dallas	0.4415	0.4056
10	Washington	0.4441	0.3019
10	New York	0.4463	0.3133
10	Denver	0.4602	0.3656
10	Houston	0.5313	0.4742
10	Kansas City	0.5410	0.3871
10	Los Angeles	0.5085	0.2671
10	Cleveland	0.5037	0.2268
10	St. Louis	0.5096	0.1999
10	Salt Lake City	0.5069	0.1805
10	Boston 2	0.5145	0.1185
10	Phoenix	0.5374	0.1309

Regional Hierarchy

7		0.2937	0.0499
8	New York	0.3029	0.0454
8	Chicago	0.3063	−0.0125
8	San Francisco	0.3155	−0.0468
8	Washington	0.3318	−0.0793
9	Boston	0.3938	0.2081
10	Dallas	0.4804	0.4802
10	Denver	0.4962	0.5025
10	St. Louis	0.4982	0.4890
11	Cleveland	0.5915	0.6812
11	Louisville	0.5933	0.6759
11	Kansas City	0.6600	0.5959
12	Seattle	0.7778	0.9118
12	Phoenix	0.7752	0.8822
12	Los Angeles	0.7622	0.8810
12	Atlanta	0.7656	0.4362


	Distance Hierarchy

Diameter	CMSA	I = u2/u1	S = u3/u2

7		0.2937	0.0499
8	Salt Lake City	0.2935	0.0399
8	Denver	0.3003	0.0573
8	San Francisco	0.3061	0.0246
9	Dallas	0.4081	0.5258
9	Seattle	0.4072	0.5149
9	Chicago	0.4194	0.4465
9	Los Angeles	0.3841	0.2800
10	Atlanta	0.4205	0.1839
10	Washington	0.4420	0.0249
10	New York	0.4394	−0.1134
10	Phoenix	0.4583	−0.0784
11	Houston	0.5412	0.1520
13	Miami	0.7341	0.6719
14	Boston	0.9572	0.8135
16	Kansas City	1.3219	1.1954


	Global Hierarchy

Diameter	CMSA	I = u2/u1	S = u3/u2

7		0.2937	0.0499
8	San Francisco	0.2981	0.0258
8	Atlanta	0.3489	0.0779
8	Chicago	0.3518	0.0169
10	Dallas	0.4384	0.3208
10	Denver	0.4503	0.3691
10	Washington	0.4717	0.1825
10	New York	0.4672	0.0570
10	Salt Lake City	0.4649	0.0189
10	Los Angeles	0.4427	−0.0806
10	Houston	0.4932	−0.0264
11	Kansas City	0.5306	−0.0190
11	Seattle	0.5317	−0.0705
12	Phoenix	0.6464	0.2665
13	Boston	0.8097	0.3425
16	Miami	1.3219	1.1954


	Relay Node Hierarchy

Diameter	MSA	I = u2/u1	S = u3/u2

7		0.2937	0.0499
8	Kansas City	0.2958	0.0405
8	Salt Lake City	0.2956	0.0302
8	Indianapolis	0.2949	0.0227
8	Seattle	0.2942	0.0137
10	Portland	0.3654	0.6527
10	Sacramento	0.3834	0.7821
	St. Louis	0.3866	0.7927
10	Denver	0.4063	0.7470
10	Atlanta	0.4248	0.5493
10	Washington-	0.4254	0.4537
	Baltimore
10	Chicago	0.4285	0.3020
10	Philadelphia	0.4291	0.2970
10	Orlando	0.4412	0.2912
12	Jacksonville	0.5249	0.6122
12	Phoenix	0.5237	0.6021

The diameter results are the easiest to interpret and reveal some interesting findings. The hierarchies with the largest effect on the diameter of the network were the distance hierarchy and the global hierarchy, both of which ended in a diameter of 16 when the top 15 nodes (roughly 10%) were removed. The superior performance of the distance hierarchy confirmed that the best performing hierarchy would be one based on Euclidean distance. The global hierarchy was based on the presence of a large number of long distance links between two different regions. While it did not directly use Euclidean distance there is an obvious correlation between global links between different regions and a longer physical length.
The starting diameter of the network in the case of both the distance and global hierarchy was 7, and the end result of 16 was more than a doubling of the diameter. Thus, it took more than twice the number of hops to reach the two furthest places on the network. This results in a ripple effect across the network where it will take a minimum of twice the time to get from any point to another. This does not take into account the capacity of the links removed and how traffic will be redistributed across the network. While both hierarchies end up at 16 the global hierarchy accelerates more rapidly in the beginning while the distance hierarchy accelerates the diameter more quickly at the end of the nodal hierarchy. The next group of nodal hierarchies was the relay node and regional hierarchy which both end with a diameter of 12. Finally, the binary and bandwidth capacity hierarchy had the least impact each ending in a diameter of 10.
In step 1120, the results of step 1115 are plotted in a graph form. The graph format allows a visual indication of which node ranking hierarchy does a better job of identifying critical nodes in a network. The graph format also gives an indication of when the network experiences a catastrophic failure, breaking apart into disconnected components. The diameter relationship of the hierarchies is seen more clearly when all the nodal hierarchies are plotted with their diameters at each successive node removal, as illustrated in FIG. 12.
The graph illustrates two aspects of network resiliency, the diameter of the network, and the point at which the network Balkanizes, indicative of a catastrophic failure. The diameter of the network after each successive node removal is indicated by the number on the x axis. As the diameter increases, it is taking more hops to connect nodes in the network indicating a decrease in efficiency and an increase in latency. Balkanization is indicated at the point that the diameter of network stops increasing and drops of rapidly. At this point, the network has broken into two or more segments and the hierarchy takes the diameter of the largest remaining subgraph. Since the network has segmented into smaller parts, the diameter decreases to match the network's now smaller size. Since the network has now fractured into segments that can no longer communicate with each other, a catastrophic failure has occurred. When the hierarchies were compared using the above indicators, all the hierarchies outperformed the existing standard, the accessibility index. The global hierarchy reached the highest diameter, followed by the distance hierarchy, and regional hierarchy. While the global hierarchy reached the highest diameter the distance hierarchy's case is catastrophic Balkanization in the network first, closely followed by the global hierarchy and then the regional hierarchy. An examination of the S-I index confirms the findings of the diameter analysis.
FIG. 13 illustrates the S and I measure of the network as nodes are removed from the network, using the global hierarchy approach. The graph format clearly shows the similar effect S and I have with diameter as nodes are removed and the extreme sensitivity of S to network changes. The graphical approach is different from the typical plotting of the S and I onto the S-I plane as (X,Y) coordinates, but works well in this case to one demonstrate the connection between diameter and the S-I measures, and two show how increases in the S-I index are indicators of a disconnecting network.

Using Nodes to Define Regions

In the examples outlined above, a variety of hierarchies are used to determine what nodes in a network are most critical. Many times, the most critical nodes in a network are already known and may not involve connectivity or the measure outlined above. In this case it is useful to know what regions are impacted by these critical nodes. As before, the regions defined by the hierarchy can be geographic (e.g., a critical hub located in Atlanta) or non-geographic (e.g., a market or industrial sector).
FIG. 14 illustrates the method of defining regions by node connectivity, according to one embodiment of the invention. In step 1405, the network data is loaded into the system as one or more nodes. In step 1410, for a network N of nodes n, an adjacency matrix A and a distance matrix W are generated, based on the connectivity of the loaded data. Adjacency matrix A is the connectivity matrix of the network being analyzed. In the city-to-city data network, this would be cities and the connections between them. If there were a connection between Jacksonville and Atlanta, one would be entered in matrix A. If there were no connection, then a zero would be entered.
The distance matrix W indicates the distance between any two directly connected nodes in matrix A. In the case of the city-to-city data network, it is the number of miles between any two directly connected cities. For example, a connection between Jacksonville and Atlanta, the distance matrix, would have a value of 281 miles in the cell of the matrix representing the connection between Jacksonville and Atlanta. The members of matrix W represent the distance (e.g., physical distance, latency, or any other appropriate variable) between any two nodes of N.
In step 1415, the shortest path for each node in N is computed using adjacency matrix A. This is done by calculating the shortest number of hops to connect a single node individually with every other node in the network. This process is repeated for every node in the network, thus providing the shortest paths for each node in the network N.
In step 1420, the number of connections for each node in N is determined, and the nodes are ranked in descending order. Thus, assuming that the adjacency matrix A is symmetric, either egress or ingress connections c(i) are computed for each node i of N. These nodes are then ranked in descending order by ingress (egress) connections c.
In step 1425, a set m<n of an arbitrary number of top ranked nodes (e.g., such as, but not limited to, the nodes ranked in step 1420) is created. Thus, for example, the set of nodes could be n={New York, Washington, San Francisco, Seattle, Atlanta}, and the set m of top ranked nodes could be m={New York, Washington, San Francisco}. Selecting the number of hubs in the network is left to the user's discretion. The user can use one of the ranking hierarchies outlined above, or their own qualitative measures based on insider knowledge of a network. Thus, the number is arbitrary to the demands of the user, and which nodes in the network they determine to be critical.
In step 1430, for each member in the m set of nodes (e.g., hubs), a list of nodes that are one hop, two hops, three hops, etc. away from each other, is generated. Thus, for node j in the set m, lists L_r(j) (e.g., of nodes that are 1, 2, . . . s hop distant from node j) and rε[1, 2, . . . s] are created.
In step 1435, each node in the network follows its available shortest path until a node j in the m set of nodes is reached. This can be calculated by setting
$R_{j} = \sum_{r} L_{r} (j),$
where R_jrepresents a region around node j, which is included in the set m (i.e., jεm). In the city-to-city example, Atlanta, Ga., Washington, D.C., St. Louis, Mo., and San Francisco, Calif. all contain critical data warehouses designated as critical hubs by a firm. The region impacted by the loss of a data warehouse could then be ascertained using this hierarchy, by determining which nodes fall under a particular data warehouses region of connectivity. When Jacksonville's shortest path is calculated to all the hubs in the list, it is two hops from Atlanta, three hops from Washington, four hops from St. Louis and six hops from San Francisco. Thus the hierarchy would place Jacksonville as belonging to Atlanta's region.
Starting with the highest ranked node of set m, the list of nodes that are s hops away from node j (i.e., L_r(j)) is compared to the list of nodes that are s hops away from node k (i.e., L_r(k)), where k is not one of the highest ranking nodes included in the set m (i.e., k≠j). One-hop connections (if there are any) between the top nodes in the set of m nodes are not included.
In step 1440, if there are two or more nodes in the set reachable from equal shortest paths, this tie is broken by determining which node is more proximate. Proximity can be defined by distance, capacity, latency, or any other appropriate metric. Thus, if there is a common node q that is r hops away from both j and k, then the physical distances d_jqand d_kq. between nodes j to q and k to q from the distance matrix W are compared. If d_jq<=d_kqthen node q belongs to the list L_r(j) or region R_j, whose members are exactly r hops away from node jεm. If d_jq>d_kq, then q belongs to the list L_r(k) region R_k, whose members are exactly r hops away from node kεm.
Building on the data warehouse example, let Charlotte be assigned to a region and be two hops from Washington, two hops from Atlanta, three hops from St. Louis, and five hops from San Francisco. Because there is a tie between Washington and Atlanta, the tiebreaker would be done based on which city is closer to Charlotte. In the case of Euclidean distance, matrix W would be referenced, and the lower value would be selected (i.e., Washington is 350 miles from Charlotte, but Atlanta is only 200 miles from Charlotte.) Thus Charlotte would be placed in Atlanta's region. As with the distance hierarchy, different values can be used to indicate distance between two nodes (e.g., capacity, flow, etc.).
In step 1445, each node is placed in a set under its designated hub and attached to an attribute indicating how many hops the node is from its designated hub. In the data-warehousing example, both Charlotte and Jacksonville would have a two attributed to them, because they were both two hops away from Atlanta. Each of these lists comprises a region that can be mapped, as illustrated in FIG. 15.
In this example, nodes that are one hop from the regional hub are given the hubs abbreviated name (i.e., ATL=Atlanta) and cities that are more than one hop away are designated by the abbreviated name followed by the number of hops (i.e., ATL2=two hops away from Atlanta). It should be noted that the distance variable could be substituted with a bandwidth capacity variable, or other variable of the user's choice, as best fits the hierarchy's application. In this case, distance was used because network design most often incorporates a distance cost variable when selecting link build outs.

CONCLUSION

While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example, and not limitation. It will be apparent to persons skilled in the relevant art(s) that various changes in form and detail can be made therein without departing from the spirit and scope of the present invention. Thus, the present invention should not be limited by any of the above-described exemplary embodiments.
In addition, it should be understood that the Figures described above, which highlight the functionality and advantages of the present invention, are presented for example purposes only. The architecture of the present invention is sufficiently flexible and configurable, such that it may be utilized in ways other than that shown in the Figures.
Further, the purpose of the Abstract is to enable the U.S. Patent and Trademark Office and the public generally, and especially the scientists, engineers and practitioners in the art who are not familiar with patent or legal terms or phraseology, to determine quickly from a cursory inspection the nature and essence of the technical disclosure of the application. The Abstract is not intended to be limiting as to the scope of the present invention in any way.

Claims

1. A method for analyzing a structure of a network, comprising:

entering data form the network into a system as one or more nodes;

designating links between nodes as global or local; and

ranking the nodes utilizing link designation as global or local.