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United States Patent Application 
20180322219

Kind Code

A1

Bekas; Konstantinos
; et al.

November 8, 2018

SIMPLIFICATION OF LARGE NETWORKS AND GRAPHS
Abstract
Embodiments relate to simplifying large and complex networks and graphs
using global connectivity information based on calculated node
centralities. An aspect includes calculating node centralities of a graph
until a designated number of central nodes are detected. A percentage of
the central nodes are then selected as pivot nodes. The neighboring nodes
to each of the pivot nodes are then collapsed until the graph shrinks to
a predefined threshold of total nodes. Responsive to the number of total
nodes reaching the predefined threshold, the simplified graph is
outputted.
Inventors: 
Bekas; Konstantinos; (Horgen, CH)
; Curioni; Alessandro; (Gattikon, CH)

Applicant:  Name  City  State  Country  Type  INTERNATIONAL BUSINESS MACHINES CORPORATION  Armonk  NY  US   
Family ID:

1000003446855

Appl. No.:

16/033361

Filed:

July 12, 2018 
Related U.S. Patent Documents
       
 Application Number  Filing Date  Patent Number 

 13900024  May 22, 2013  10083250 
 16033361   

Current U.S. Class: 
1/1 
Current CPC Class: 
G06F 9/3001 20130101; G06F 17/30958 20130101; G06F 17/30961 20130101 
International Class: 
G06F 17/30 20060101 G06F017/30; G06F 9/30 20060101 G06F009/30 
Claims
1. A computerimplemented method for reducing a memory footprint of a
network, the method comprising: utilizing an original network to model
dynamics, wherein the original network has an original memory footprint
in a computer system; calculating node centralities of the original
network until a designated number of central nodes are detected, wherein
the calculating of the node centralities of the original network further
comprises: approximating a product of a matrix exponential and a random
probe vector of an adjacency matrix, the adjacency matrix representing
the original network; and computing a diagonal of the adjacency matrix
based on the product of the matrix exponential and the random probe
vector; selecting a percentage of the central nodes as pivot nodes;
collapsing neighboring nodes to each pivot node of the original network
to shrink the original network to generate a reduced network until the
reduced network reaches a predefined threshold of total nodes; and
utilizing the reduced network to model the dynamics, wherein the reduced
network has a reduced memory footprint in the computer system as compared
to the original footprint.
2. The computerimplemented method of claim 1, wherein the collapsing
further comprises: acquiring a first set of neighboring nodes for each
pivot node; acquiring a second set of neighboring nodes for each node in
the first set of neighboring nodes; deleting the first set of neighboring
nodes; and establishing the second set of neighboring nodes as neighbors
for a current pivot node.
3. The computerimplemented method of claim 2, wherein the first set of
neighboring nodes and the second set of neighboring nodes are not pivot
nodes.
4. The computerimplemented method of claim 1, wherein the approximating
of the product of the matrix and the random probe vector further
comprises: computing an orthogonal Krylov basis and tridiagonal matrix
using a Lanczos algorithm; computing a matrix exponential of the
tridiagonal matrix; and computing a current approximation of the product
of the matrix exponential and the random probe vector.
5. The computerimplemented method of claim 1, wherein the computing of
the diagonal further comprises calculating the diagonal based on a
formula D.sub.s=SUM.sub.1.sup.s (v.sub.i.x F(A)v.sub.i) ./
SUM.sub.1.sup.s (v.sub.i.x v.sub.i), where D is a diagonal, v.sub.i is
the random probe vector, s is the total number of required vectors, A is
the adjacency matrix of size N, F(A) is the matrix exponential, .x
symbolizes elementwise multiplication, and ./ symbolizes elementwise
division.
6. The computerimplemented method of claim 5, wherein the computing of
the diagonal further comprises: a) initializing vectors Q, W, and D of
length N to zero; b) initializing the random probe vector v.sub.i; c)
computing the product of the matrix and the random probe vector; d)
updating vector Q by calculating Q=Q+v.sub.i.x Z, where Z is the product
of the matrix and the random probe vector; e) updating vector W by
calculating W=W+v.sub.i.x v.sub.i; f) updating vector D by calculating
D=D+Q ./ W; and g) repeating operations bf until a designated end
condition is reached.
7. The computerimplemented method of claim 6, wherein the designated end
condition comprises a selected one of a condition where the difference of
a previously estimated diagonal and the estimated diagonal is smaller
than a designated diagonal tolerance, a condition where the maximum
number of steps s has been reached, and a condition where the percentage
of nodes with highest centrality has converged.
8. A computer system comprising: a memory having computerreadable
instructions; and one or more processors for executing the
computerreadable instructions, the computerreadable instructions
comprising: utilizing an original network to model dynamics, wherein the
original network has an original memory footprint in a computer system;
calculating node centralities of the original network until a designated
number of central nodes are detected, wherein the calculating of the node
centralities of the original network further comprises: approximating a
product of a matrix exponential and a random probe vector of an adjacency
matrix, the adjacency matrix representing the original network; and
computing a diagonal of the adjacency matrix based on the product of the
matrix exponential and the random probe vector; selecting a percentage of
the central nodes as pivot nodes; collapsing neighboring nodes to each
pivot node of the original network to shrink the original network to
generate a reduced network until the reduced network reaches a predefined
threshold of total nodes; and utilizing the reduced network to model the
dynamics, wherein the reduced network has a reduced memory footprint in
the computer system as compared to the original footprint.
9. The computer system of claim 8, wherein the collapsing further
comprises: acquiring a first set of neighboring nodes for each pivot
node; acquiring a second set of neighboring nodes for each node in the
first set of neighboring nodes; deleting the first set of neighboring
nodes; and establishing the second set of neighboring nodes as neighbors
for a current pivot node.
10. The computer system of claim 9, wherein the first set of neighboring
nodes and the second set of neighboring nodes are not pivot nodes.
11. The computer system of claim 8, wherein the approximating of the
product of the matrix and the random probe vector further comprises:
computing an orthogonal Krylov basis and tridiagonal matrix using a
Lanczos algorithm; computing a matrix exponential of the tridiagonal
matrix; and computing a current approximation of the product of the
matrix exponential and the random probe vector.
12. The computer system of claim 8, wherein the computing of the diagonal
further comprises calculating the diagonal based on a formula
D.sub.s=SUM.sub.1.sup.s (v.sub.i .x F(A)v.sub.i) .times./ SUM.sub.1.sup.s
(v.sub.i .x v.sub.i), where D is a diagonal, v.sub.i is the random probe
vector, s is the total number of required vectors, A is the adjacency
matrix of size N, F(A) is the matrix exponential, .x symbolizes
elementwise multiplication, and ./ symbolizes elementwise division.
13. The computer system of claim 12, wherein the computing of the
diagonal further comprises: a) initializing vectors Q, W, and D of length
N to zero; b) initializing the random probe vector v.sub.i; c) computing
the product of the matrix and the random probe vector; d) updating vector
Q by calculating Q=Q+v.sub.i.x Z, where Z is the product of the matrix
and the random probe vector; e) updating vector W by calculating
W=W+v.sub.i.x v.sub.i; f) updating vector D by calculating D=D+Q ./ W;
and g) repeating operations bf until a designated end condition is
reached.
14. A computer program product for reducing a memory footprint of a
network, the computerprogram product comprising a computerreadable
storage medium having program instructions embodied therewith, the
program instructions executable by a processor to cause the processor to
perform a method comprising: utilizing an original network to model
dynamics, wherein the original network has an original memory footprint
in a computer system; calculating node centralities of the original
network until a designated number of central nodes are detected, wherein
the calculating of the node centralities of the original network further
comprises: approximating a product of a matrix exponential and a random
probe vector of an adjacency matrix, the adjacency matrix representing
the original network; and computing a diagonal of the adjacency matrix
based on the product of the matrix exponential and the random probe
vector; selecting a percentage of the central nodes as pivot nodes;
collapsing neighboring nodes to each pivot node of the original network
to shrink the original network to generate a reduced network until the
reduced network reaches a predefined threshold of total nodes; and
utilizing the reduced network to model the dynamics, wherein the reduced
network has a reduced memory footprint in the computer system as compared
to the original footprint.
15. The computer program product of claim 14, wherein the collapsing
further comprises: acquiring a first set of neighboring nodes for each
pivot node; acquiring a second set of neighboring nodes for each node in
the first set of neighboring nodes; deleting the first set of neighboring
nodes; and establishing the second set of neighboring nodes as neighbors
for a current pivot node.
16. The computer program product of claim 15, wherein the first set of
neighboring nodes and the second set of neighboring nodes are not pivot
nodes.
17. The computer program product of claim 14, wherein the approximating
of the product of the matrix and the random probe vector further
comprises: computing an orthogonal Krylov basis and tridiagonal matrix
using a Lanczos algorithm; computing a matrix exponential of the
tridiagonal matrix; and computing a current approximation of the product
of the matrix exponential and the random probe vector.
18. The computer program product of claim 14, wherein the computing of
the diagonal further comprises calculating the diagonal based on a
formula D.sub.s=SUM.sub.1.sup.s (v.sub.i.x F(A)v.sub.i) ./
SUM.sub.1.sup.s (v.sub.i.x v.sub.i), where D is a diagonal, v.sub.i is
the random probe vector, s is the total number of required vectors, A is
the adjacency matrix of size N, F(A) is the matrix exponential, .x
symbolizes elementwise multiplication, and ./ symbolizes elementwise
division.
19. The computer program product of claim 18, wherein the computing of
the diagonal further comprises: a) initializing vectors Q, W, and D of
length N to zero; b) initializing the random probe vector v.sub.i; c)
computing the product of the matrix and the random probe vector; d)
updating vector Q by calculating Q=Q+v.sub.i.x Z, where Z is the product
of the matrix and the random probe vector; e) updating vector W by
calculating W=W+v.sub.i.x v.sub.i; f) updating vector D by calculating
D=D+Q ./ W; and g) repeating operations bf until a designated end
condition is reached.
20. The computer program product of claim 19, wherein the designated end
condition comprises a selected one of a condition where the difference of
a previously estimated diagonal and the estimated diagonal is smaller
than a designated diagonal tolerance, a condition where the maximum
number of steps s has been reached, and a condition where the percentage
of nodes with highest centrality has converged.
Description
CROSSREFERENCE TO RELATED APPLICATION
[0001] This application is a continuation of U.S. patent application Ser.
No. 13/900,024, filed May 22, 2013, the disclosure of which is
incorporated by reference herein in its entirety.
BACKGROUND
[0002] The present invention relates generally to graph theory, and more
specifically, to simplifying large and complex networks and graphs using
global connectivity information based on calculated node centralities.
[0003] Graph theory is the study of graphs, which are mathematical
structures used to model pairwise relations between objects. A graph in
this context is made up of vertices or nodes and lines called edges that
connect them. Graphs are widely used in applications to model many types
of relations and process dynamics in physical, biological, social and
information systems. Accordingly, many practical problems in modern
technological, scientific and business applications are typically
represented by graphs.
[0004] The centrality of a node is a widely used measure to determine the
relative importance a node within a full network or graph. Node
centralities may be used to determine which nodes are important in a
complex network, to understand influencers, or to find hot spot links.
For example, node centralities are typically used to determine how
influential a person is within a social network, or, in the theory of
space syntax, how important a room is within a building or how wellused
a road is within an urban network.
BRIEF SUMMARY
[0005] According to an embodiment of the present invention, a method for
simplifying large and complex networks and graphs using global
connectivity information based on calculated node centralities is
provided. The method includes calculating node centralities of a graph
until a designated number of central nodes are detected. A percentage of
the central nodes are then selected as pivot nodes. The neighboring nodes
to each of the pivot nodes are then collapsed until the graph shrinks to
a predefined threshold of total nodes. Responsive to the number of total
nodes reaching the predefined threshold, the simplified graph is
outputted.
[0006] According to another embodiment of the present invention, a system
for simplifying large and complex networks and graphs using global
connectivity information based on calculated node centralities is
provided. The system includes a computer processor and logic executable
by the computer processor. The logic is configured to implement a method.
The method includes calculating node centralities of a graph until a
designated number of central nodes are detected. A percentage of the
central nodes are then selected as pivot nodes. The neighboring nodes to
each of the pivot nodes are then collapsed until the graph shrinks to a
predefined threshold of total nodes. Responsive to the number of total
nodes reaching the predefined threshold, the simplified graph is
outputted.
[0007] According to a further embodiment of the present invention, a
computer program product for simplifying large and complex networks and
graphs using global connectivity information based on calculated node
centralities is provided. The computer program product includes a storage
medium having computerreadable program code embodied thereon, which when
executed by a computer processor, causes the computer processor to
implement a method. The method includes calculating node centralities of
a graph until a designated number of central nodes are detected. A
percentage of the central nodes are then selected as pivot nodes. The
neighboring nodes to each of the pivot nodes are then collapsed until the
graph shrinks to a predefined threshold of total nodes. Responsive to the
number of total nodes reaching the predefined threshold, the simplified
graph is outputted.
[0008] Additional features and advantages are realized through the
techniques of the present invention. Other embodiments and aspects of the
invention are described in detail herein and are considered a part of the
claimed invention. For a better understanding of the invention with the
advantages and the features, refer to the description and to the
drawings.
BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS
[0009] The subject matter which is regarded as the invention is
particularly pointed out and distinctly claimed in the claims at the
conclusion of the specification. The forgoing and other features, and
advantages of the invention are apparent from the following detailed
description taken in conjunction with the accompanying drawings in which:
[0010] FIG. 1 depicts a block diagram of a computer system according to an
embodiment;
[0011] FIG. 2 depicts a flow diagram of a process for calculating the most
important nodes in large and complex networks and graphs according to an
embodiment;
[0012] FIG. 3 depicts a PF process for approximating the product of the
matrix exponential and the random probe vector according to an
embodiment;
[0013] FIG. 4 depicts a Lanczos process for calculating a Krylov basis and
a tridiagonal matrix according to an embodiment; and
[0014] FIG. 5 depicts a graph coarsening process for collapsing a large
graph using calculated central nodes as pivot nodes according to an
embodiment.
DETAILED DESCRIPTION
[0015] Embodiments disclosed herein are directed to simplifying large and
complex networks and graphs using global connectivity information based
on calculated node centralities. Aspects of embodiments include
calculating node centralities of a graph until a designated number of
central nodes are detected. A percentage of the central nodes are then
selected as pivot nodes. The neighboring nodes to each of the pivot nodes
are then collapsed until the graph shrinks to a predefined threshold of
total nodes. Responsive to the number of total nodes reaching the
predefined threshold, the simplified graph is outputted.
[0016] Embodiments include a recursive method, system, computerprogram
product for collapsing a complex graph or network to a simpler graph or
network by using a percentage of the most central nodes as pivots and
agglomerating the relevant edges and neighboring nodes. The recursive
method, system, computerprogram product for collapsing a complex graph
may be implemented using the disclosed embodiment for computing node
centralities, which overcomes the very high cost of computing node
centralities with conventional techniques.
[0017] There typically exists a very high cost for computing node
centralities. One example of a method for computing node centralities
involves the matrix exponential. Consider the adjacency matrix A of an
undirected graph (e.g., network). That is, the (i,j) element of matrix A
is nonzero only if node i and node j are adjacent. Typically, this
nonzero value is set to either 1 or to a certain weight according to
some user specified weighting scheme. Consider the matrix exponential
E=exp(A)=I+A+A2/2!+A3/3!+A4/4!+. . . The diagonal element E(i,i) of the
matrix exponential is then the node centrality of node i.
[0018] Accordingly, a calculation of the node centralities is accomplished
by: (i) calculating the spectral decomposition of the adjacency matrix A:
A=QDQ.sup.T, where D is a diagonal matrix, Q is the matrix of orthogonal
eigenvectors, and Q.sup.T stands for the transpose of this matrix (i.e.,
every element (i,j) of Q becomes element (j,i) of its transpose), (ii)
calculating the matrix exponential exp(D), which amounts to calculating
the scalar exponentials exp(D(i,i)), i=1, . . . N, where N is the size of
the matrix (i.e., the number of nodes in the graph), and (iii)computing
the diagonal entries of the matrix product E=Q exp(D)Q.sup.T.
[0019] An alternative of method for calculating node centralities replaces
the matrix exponential with a resolvent function (AzI).sup.1, where I
is the identity matrix and z is a suitably selected scalar number. This
calculation is accomplished by: (i) computing the Cholesky decomposition
of matrix (AzI): AzI=R.sup.TR, (ii) solving the linear systems
(R.sup.1R)z.sub.i=e.sub.i, i=1, . . . ,N, where the vectors e.sub.i have
1 at the ith element and zero elsewhere, and (iii) computing the
diagonal entries as d.sub.i=e.sub.i.sup.Tz.sub.i.
[0020] The methods discussed above are simple, elegant and can use
standard linear algebra packages such as Linear Algebra Package (LAPACK).
However, the major caveat is their cost that increases as the cube of the
number of nodes of the graph. That is, computing the node centralities of
a graph (e.g., network) with 10,000 nodes already requires 1 tera
floating point operations per second (FLOP) of computations, while
interesting graphs can easily reach sizes of tens of millions of nodes. A
graph with 50 million nodes would require at least 125 zeta FLOP of
computations (125.times.10.sup.21). This would take the most powerful
contemporary supercomputer more than 90 days of computations.
[0021] Alternative conventional methods for computing node centralities in
large networks and graphs may sample the nodes of the graph in an attempt
to reduce the number of the nodes, and thus the computational complexity.
These methods, however, generate huge biases and offer very limited ways
of regulating and understanding the error.
[0022] Embodiments disclosed herein identify only the most central nodes
of a large and complex graph. In other words, embodiments quickly home in
on the most important (i.e., central) nodes to drastically reduce cost
and memory footprint of calculating node centralities of a large and
complex graph. The disclosed embodiments combine a stochastic estimator
for the diagonal of a matrix with methods for approximating the product
of a matrix exponential times a vector and mixed precision low complexity
methods to solve linear systems of equations. Embodiments may exploit the
stochastic nature of a stochastic estimator for the diagonal of a matrix
and can stop at any point during the iteration when a user specified
number of central nodes has been detected.
[0023] Accordingly, embodiments may implement an exemplary recursive
method, system, computerprogram product for collapsing a complex graph
or network to a simpler graph by using a percentage of the most central
nodes as pivots and agglomerating the relevant edges and neighboring
nodes.
[0024] Referring now to FIG. 1, a block diagram of a computer system 10
suitable for calculating node centralities in large and complex networks
and graphs according to exemplary embodiments is shown. Computer system
10 is only one example of a computer system and is not intended to
suggest any limitation as to the scope of use or functionality of
embodiments described herein. Regardless, computer system 10 is capable
of being implemented and/or performing any of the functionality set forth
hereinabove.
[0025] Computer system 10 is operational with numerous other general
purpose or special purpose computing system environments or
configurations. Examples of wellknown computing systems, environments,
and/or configurations that may be suitable for use with computer system
10 include, but are not limited to, personal computer systems, server
computer systems, thin clients, thick clients, cellular telephones,
handheld or laptop devices, multiprocessor systems, microprocessorbased
systems, set top boxes, programmable consumer electronics, network PCs,
minicomputer systems, mainframe computer systems, and distributed cloud
computing environments that include any of the above systems or devices,
and the like.
[0026] Computer system 10 may be described in the general context of
computer systemexecutable instructions, such as program modules, being
executed by the computer system 10. Generally, program modules may
include routines, programs, objects, components, logic, data structures,
and so on that perform particular tasks or implement particular abstract
data types. Computer system 10 may be practiced in distributed cloud
computing environments where tasks are performed by remote processing
devices that are linked through a communications network. In a
distributed computing environment, program modules may be located in both
local and remote computer system storage media including memory storage
devices.
[0027] As shown in FIG. 1, computer system 10 is shown in the form of a
generalpurpose computing device, also referred to as a processing
device. The components of computer system may include, but are not
limited to, one or more processors or processing units 16, a system
memory 28, and a bus 18 that couples various system components including
system memory 28 to processor 16.
[0028] Bus 18 represents one or more of any of several types of bus
structures, including a memory bus or memory controller, a peripheral
bus, an accelerated graphics port, and a processor or local bus using any
of a variety of bus architectures. By way of example, and not limitation,
such architectures include Industry Standard Architecture (ISA) bus,
Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA) bus, Video
Electronics Standards Association (VESA) local bus, and Peripheral
Component Interconnects (PCI) bus.
[0029] Computer system 10 may include a variety of computer system
readable media. Such media may be any available media that is accessible
by computer system/server 10, and it includes both volatile and
nonvolatile media, removable and nonremovable media.
[0030] System memory 28 can include computer system readable media in the
form of volatile memory, such as random access memory (RAM) 30 and/or
cache memory 32. Computer system 10 may further include other
removable/nonremovable, volatile/nonvolatile computer system storage
media. By way of example only, storage system 34 can be provided for
reading from and writing to a nonremovable, nonvolatile magnetic media
(not shown and typically called a "hard drive"). Although not shown, a
magnetic disk drive for reading from and writing to a removable,
nonvolatile magnetic disk (e.g., a "floppy disk"), and an optical disk
drive for reading from or writing to a removable, nonvolatile optical
disk such as a CDROM, DVDROM or other optical media can be provided. In
such instances, each can be connected to bus 18 by one or more data media
interfaces. As will be further depicted and described below, memory 28
may include at least one program product having a set (e.g., at least
one) of program modules that are configured to carry out the functions of
embodiments of the disclosure.
[0031] Program/utility 40, having a set (at least one) of program modules
42, may be stored in memory 28 by way of example, and not limitation, as
well as an operating system, one or more application programs, other
program modules, and program data. Each of the operating system, one or
more application programs, other program modules, and program data or
some combination thereof, may include an implementation of a networking
environment. Program modules 42 generally carry out the functions and/or
methodologies of embodiments of the invention as described herein.
[0032] Computer system 10 may also communicate with one or more external
devices 14 such as a keyboard, a pointing device, a display 24, etc.; one
or more devices that enable a user to interact with computer
system/server 10; and/or any devices (e.g., network card, modem, etc.)
that enable computer system/server 10 to communicate with one or more
other computing devices. Such communication can occur via Input/Output
(I/O) interfaces 22. Still yet, computer system 10 can communicate with
one or more networks such as a local area network (LAN), a general wide
area network (WAN), and/or a public network (e.g., the Internet) via
network adapter 20. As depicted, network adapter 20 communicates with the
other components of computer system 10 via bus 18. It should be
understood that although not shown, other hardware and/or software
components could be used in conjunction with computer system 10. Examples
include, but are not limited to: microcode, device drivers, redundant
processing units, external disk drive arrays, RAID systems, tape drives,
and data archival storage systems, etc.
[0033] An embodiment combines a stochastic estimator for the diagonal of a
matrix with methods to approximate the product of a matrix exponential
times a vector and mixed precision low complexity methods to solve linear
systems of equations.
[0034] The stochastic diagonal estimator of an embodiment uses the formula
D.sub.s=SUM.sub.1.sup.s (v.sub.i .x F(A)v.sub.i) ./ SUM.sub.i.sup.s
(v.sub.i .x v.sub.i), where D is a diagonal, v.sub.i is a selected
vector, s is the total number of required vectors, A is the adjacency
matrix, F(A) is the matrix exponential, and .x and ./ symbolize
elementwise multiplication and division, respectively. Elementwise
multiplication for two vectors x, y is a process that includes setting a
variable u=1, calculating a result res(u)=x(u)*y(u), and increasing a
counter u=u+1 until u<=n, where n is the size (i.e., length) of the
vectors x, y, and res. In this particular case, n is the number of nodes
of the graph. Similarly, elementwise division and addition of two
vectors x, y is processed simply by substituting the calculation of
res(u) described above with res(u)=x(u)/y(u) and res(u)=x(u)+y(u),
respectively.
[0035] The stochastic diagonal estimator of an embodiment requires access
to approximations of the products F(A)v.sub.i for carefully selected
vectors v.sub.i. According to an embodiment, the total number of required
vectors s is set to be much smaller than the size N of the adjacency
matrix. An embodiment for approximating the matrix vector product
F(A)v.sub.i requires a small number k of matrix vector products with the
adjacency matrix A itself (i.e., k<<N). Thus, since the adjacency
matrices of graphs are typically quite sparse by definition, this means
that each matrixvector product with the adjacency matrix costs O(N).
Therefore, the total cost of an embodiment is: O(Nks). The numbers k and
s are selected to be at most a few hundred according to an embodiment.
Thus, the graph with 50 million nodes would now require only 12.5 terra
FLOP (12.5.times.10.sup.12), which is approximately 8 orders of magnitude
less than contemporary methods. Accordingly, it would take about
1/100.sup.th of a second to calculate node centralities on the best
contemporary supercomputer.
[0036] With reference to FIG. 2, a process 200 performed by an embodiment
of the processing device 16 of computer system 10 is generally shown. As
shown in FIG. 2, the process 200 calculates the most important (i.e.,
central) nodes in large and complex networks and graphs according to an
embodiment.
[0037] Process 200 may receive the following data as input: graph G and
its adjacency matrix A of size N, user tolerances tol_diagonal and
tol_function, maximum number of steps s (i.e., a maximum required number
of vectors) and k (i.e., a maximum number of Lanczos steps), and a
required percentage of nodes with highest centrality. The user tolerances
tol_diagonal and tol_function are user defined and may change with regard
to particular applications. Based on these input values, the process 200
will output estimated node centralities for graph G.
[0038] According to an embodiment, vectors Q, W, D of length N are
initialized to zero. At block 210, a random probe vector vi is generated.
Vector Z is then computed (i.e., Z:=F(A)v.sub.i) by calling a process PF
(i.e., PF(A, v.sub.i)) of an embodiment to approximate the product of the
matrix exponential F(A) and the random probe vector v.sub.i, as shown in
block 220. One embodiment of process PF is described in further detail
below with respect to FIG. 3.
[0039] At block 230, vectors Q and W are updated according to an
embodiment. Vector Q is updated by calculating Q=Q+v.sub.i .x Z. Vector W
is updated by calculating W=W+v.sub.i .x v.sub.i. The diagonal of
adjacency matrix A is then updated according to an embodiment, as shown
in block 240. Vector D is updated by calculating D=D 30 Q ./ W.
[0040] At block 250, an embodiment of the process 200 determines whether a
designated end condition has been reached. Responsive to an end condition
at block 250, the process 200 is completed, as shown in block 260.
However, if an end condition has not been reached at block 250, the
process 200 restarts at block 210 and generates another random probe
vector v.sub.i. According to an embodiment, a designated end condition
may include a condition where the difference of a previously estimated
diagonal and the estimated diagonal is smaller than tol_diagonal, a
condition where the maximum number of steps s has been reached, or a
condition where the percentage of nodes with highest centrality has
converged. According to an embodiment, responsive to the condition where
the maximum number of steps s has been reached, the process 200 may
consider increasing the number of maximum number of steps s prior to
restarting at block 210.
[0041] With reference to FIG. 3, an embodiment of a PF process 300 for
approximating the product of the matrix exponential F(A) and the random
probe vector vi according to an embodiment is shown. Process 300 may
receive the following data as input: adjacency matrix A of size N, input
vector v.sub.i, maximum number of steps k, user tolerance tol_function,
e.sub.1, which is the vector of length N with 1 as its first position and
zero elsewhere, and a chunk size m<k. The process 300 of an embodiment
outputs an approximation to vector Z :=F(A)v.sub.i.
[0042] At block 310, beta b is set to be the Euclidean norm of vector vi
according to an embodiment. A start_step variable is initially set to 1,
as shown in block 320, and a stop_step variable is set to m, where i=1,
as shown in block 330. At block 340, a Lanczos method of an embodiment is
performed from start_step until stop_step to compute an orthogonal Krylov
basis Vstop_step and tridiagonal matrix Tstop_step. According to an
embodiment, matrix Vstop_step has size N rows and i*m columns and Matrix
Tstop_step has i*m rows and columns. The Lanczos method of an embodiment
is described in further detail in FIG. 4 below.
[0043] At block 350, a matrix exponential of tridiagonal matrix Tstop_step
(i.e., exp(Tstop_step) is computed according to an embodiment. At block
360, a current approximation Z:=beta
V.sub.stop.sub._.sub.stepexp(T.sub.stop.sub._.sub.step) e.sub.1 is
computed according to an embodiment. At block 370, an embodiment
determines whether vector Z has converged (i.e., current approximation of
Z and the previous approximation Z differ in norm in less than user
tolerance tol_function). If it is determined that Z has converged at
block 370, then the process 300 stops, as shown in block 380.
[0044] If, however, it is determined that Z has not yet converged at block
370, then the start_step is then incremented to stop_step+1 and the
stop_step is then incremented to stop_step+m, as shown in block 390. If
the maximum number of steps k has not been exceeded (i.e., i*m >k)
then the process 300 is restarted at block 340. Otherwise, if the maximum
number of steps k has been exceeded, the process 300 stops according to
an embodiment.
[0045] With reference to FIG. 4, a Lanczos process 400 for calculating a
Krylov basis and a tridiagonal matrix according to an embodiment is
shown. Process 400 may receive the following data as input: adjacency
matrix A of size N, input vector V and maximum number of steps k (i.e., a
maximum number of Lanczos steps). The process 400 of an embodiment
outputs a Krylov basis V and a tridiagonal matrix T. According to this
embodiment, W is a vector of length n and a.sub.i and b.sub.i are scalar
values.
[0046] At block 405, beta b is set to be the Euclidean norm of input
vector v according to an embodiment. The first column of basis matrix V
is initialized to V(:,1)=v/b, as shown in block 410. At block 415, V(:,0)
is set to 0, b.sub.1 is set to 0 and, at block 420, variable i is
initially set to 1.
[0047] At block 420, W is computed using the formula A*V(:,i)b.sub.i*
V(:,i1) according to an embodiment. According to an embodiment, the
matrix vector multiplication in this formula is implemented by setting a
variable i=1, j=1, res(i)=0, computing res(i)=res(i)+A(i,j) * x(j), and
incrementing j. If j is less than or equal to n, then
res(i)=res(i)+A(i,j) * x(j) is recomputed and j is incremented again. If
j is greater than n, then i is incremented. If i greater than n, then the
multiplication of the vector stops. However, if i is less than or equal
to n, then res(i) is reset to 0, res(i)=res(i)+A(i,j) * x(j) is
recomputed and j is incremented again.
[0048] At block 425, a.sub.i is computed using the formula w.sup.T*
V(:,i), where T(i,i):=a.sub.i. W is then computed using the formula
Wa.sub.i * V(:,i), as shown in block 430. At block 435, b.sub.i+1 is set
to equal .parallel.w.parallel. and T(i+1,i)=T(i, i+1) :=b.sub.i+1. At
block 440, i is incremented. If i is less than or equal to k at block
445, then the process 400 is stopped, as shown in block 450. If i is
greater than k, then the process 400 is restarted at block 420.
Accordingly, the symmetric tridiagonal matrix T is given as main
diagonal: [a1 a2 a3 . . . ] and the super and subdiagonal [b2 b3 b4].
[0049] According to an alternative embodiment, the PF process of block 220
for approximating vector Z may be calculated using a resolvent function
(i.e., (AzI).sup.1v.sub.i), where I is the identity matrix and z is a
suitably selected scalar number. According to this embodiment, the PF
process may receive the following data as input: adjacency matrix A of
size N, input vector v.sub.i, maximum number of steps k, and user
tolerance tol_function. The process of this embodiment outputs an
approximation to vector Z. The process for calculating
Z:=(AzI).sup.1v.sub.i), using k number of steps for the internal
conjugate gradient method and user tolerance tol_function, is disclosed
in U.S. Patent Application Publication No. 20120005247, titled
"Processing of Linear Systems of Equations" and filed on Aug. 18, 2011,
the entire contents of which are hereby incorporated by reference.
[0050] With reference to FIG. 5, a graph coarsening process 500 performed
by an embodiment of the processing device 16 of computer system 10 is
generally shown. As shown in FIG. 5, the graph coarsening process 500
collapses a large graph using the most central nodes as pivot nodes
according to an embodiment.
[0051] The graph coarsening process 500 may receive the following data as
input: an original graph O, a percentage of nodes to keep p, and a
designated number of nodes I of simplified graph B. Based on these input
values, the graph coarsening process 500 will output a simplified graph B
according to an embodiment.
[0052] The node centralities of the original graph O are calculated using
the exemplary process 200 of an embodiment, as shown in FIG. 2. At block
505, a simplified graph B is set to be equal to the original graph O
according to an embodiment. At block 510, a percentage of the most
central nodes p in simplified graph B is selected and placed in a set C
according to an embodiment.
[0053] For each node in set C, the graph coarsening process 500 of an
embodiment then finds all neighboring nodes that are not central nodes
(i.e., nodes in set C) and places them in a set S1, as shown in block
515. Further, for each node in S1 at block 520, the graph coarsening
process 500 of an embodiment finds all neighboring nodes that are not
central nodes (i.e., nodes in set C) and places them in set S2, as shown
in block 525. At block 530, it is determined whether all nodes in set S1
have been visited. If not, then then next node in S1 is selected at block
520 and all its neighboring nodes that are not central nodes (i.e., nodes
in set C) are placed in set S2, as shown in block 525. If all the nodes
in set S1 have been visited at block 530, then all the nodes in set S1
are deleted, as shown in block 535. All nodes in set S2 are then made to
be neighbors of the current central node according to an embodiment, as
shown in block 540. At block 545, the current simplified graph B is
updated with node deletions and new neighbor edges according to an
embodiment.
[0054] At block 550, it is determined whether all the current central
nodes in set C have been examined according to an embodiment. If not,
then for the next node in set C, the graph coarsening process 500 of an
embodiment finds all neighboring nodes that are not central nodes and
places them in a set S1, as shown in block 515. However, if all the
current central nodes in set C have been examined at block 550, then the
graph coarsening process 500 of an embodiment determines whether the
simplified graph B is smaller than the designated number of nodes I, as
shown in block 555. If the simplified graph B is not smaller than the
designated number of nodes I, then the node centralities of simplified
graph B are calculated, as shown in block 560, and the graph coarsening
process 500 of an embodiment restarts at block 510 until the simplified
graph B is smaller than the designated number of nodes I. The node
centralities of the simplified graph B are calculated using the exemplary
process 200 of an embodiment, as shown in FIG. 2. If the simplified graph
B is smaller than the designated number of nodes I, then the graph
coarsening process 500 of an embodiment is complete, as shown in block
565.
[0055] Embodiments disclosed herein are directed to simplifying large and
complex networks and graphs using global connectivity information based
on calculated node centralities. Aspects of embodiments include
calculating node centralities of a graph until a designated number of
central nodes are detected. A percentage of the central nodes are then
selected as pivot nodes. The neighboring nodes to each of the pivot nodes
are then collapsed until the graph shrinks to a predefined threshold of
total nodes. Responsive to the number of total nodes reaching the
predefined threshold, the simplified graph is outputted.
[0056] Technical effects and benefits include an extremely reduced cost
because geometric and connectivity information within the original graph
is preserved in the remaining central nodes. Embodiments reduce memory
footprint and traffic to memory. Conventional methods need to compute the
eigenvector matrix Q. In contrast to the adjacency matrix A which is
sparse, matrix Q is dense. Thus, memory requirements run at O(N.sup.2).
The memory requirements of the disclosed embodiments remain O(N). In
addition, only a few vectors need to remain in the cache system, thus
traffic to the memory subsystem is kept to O(N) words, while conventional
methods need at least O(N.sup.2) words. Additionally, it is very often
the case that only a few nodes of the graph are interesting with respect
to having a high centrality. The disclosed embodiments exploit its
stochastic nature and can stop at any point during the iteration when a
user specified number of central nodes has been detected. The
conventional method, on the other hand, needs to pay the full cost before
any results are available. Moreover, the disclosed embodiments are based
on matrix vector products. This means that the only operation necessary
is the application of the adjacency matrix on a vector. This can easily
be accomplished on distributed data. That means, there is not a need to
assemble the adjacency matrix, but rather, embodiments may work directly
with the raw data collections.
[0057] Embodiments may also be applied to distributed memory machines
without the need for large scale shared memory systems that are very
expensive and extremely difficult to scale. Embodiments require no access
to a global system memory, but rather may be implemented by message
parsing, thus facilitating the use of distributed machines and cloud
infrastructures. Thus, the combination of a distributed memory system
with embodiments disclosed herein leads to simplifying complex graphs and
networks.
[0058] The terminology used herein is for the purpose of describing
particular embodiments only and is not intended to be limiting of the
disclosure. As used herein, the singular forms "a", "an" and "the" are
intended to include the plural forms as well, unless the context clearly
indicates otherwise. It will be further understood that the terms
"comprises" and/or "comprising," when used in this specification, specify
the presence of stated features, integers, steps, operations, elements,
and/or components, but do not preclude the presence or addition of one or
more other features, integers, steps, operations, elements, components,
and/or groups thereof.
[0059] The corresponding structures, materials, acts, and equivalents of
all means or step plus function elements in the claims below are intended
to include any structure, material, or act for performing the function in
combination with other claimed elements as specifically claimed. The
description of the present disclosure has been presented for purposes of
illustration and description, but is not intended to be exhaustive or
limited to the disclosure in the form disclosed. Many modifications and
variations will be apparent to those of ordinary skill in the art without
departing from the scope and spirit of the disclosure. The embodiments
were chosen and described in order to best explain the principles of the
disclosure and the practical application, and to enable others of
ordinary skill in the art to understand the disclosure for various
embodiments with various modifications as are suited to the particular
use contemplated.
[0060] Further, as will be appreciated by one skilled in the art, aspects
of the present disclosure may be embodied as a system, method, or
computer program product. Accordingly, aspects of the present disclosure
may take the form of an entirely hardware embodiment, an entirely
software embodiment (including firmware, resident software, microcode,
etc.) or an embodiment combining software and hardware aspects that may
all generally be referred to herein as a "circuit," "module" or "system."
Furthermore, aspects of the present disclosure may take the form of a
computer program product embodied in one or more computer readable
medium(s) having computer readable program code embodied thereon.
[0061] Any combination of one or more computer readable medium(s) may be
utilized. The computer readable medium may be a computer readable signal
medium or a computer readable storage medium. A computer readable storage
medium may be, for example, but not limited to, an electronic, magnetic,
optical, electromagnetic, infrared, or semiconductor system, apparatus,
or device, or any suitable combination of the foregoing. More specific
examples (a nonexhaustive list) of the computer readable storage medium
would include the following: an electrical connection having one or more
wires, a portable computer diskette, a hard disk, a random access memory
(RAM), a readonly memory (ROM), an erasable programmable readonly
memory (EPROM or Flash memory), an optical fiber, a portable compact disc
readonly memory (CDROM), an optical storage device, a magnetic storage
device, or any suitable combination of the foregoing. In the context of
this document, a computer readable storage medium may be any tangible
medium that can contain, or store a program for use by or in connection
with an instruction execution system, apparatus, or device.
[0062] A computer readable signal medium may include a propagated data
signal with computer readable program code embodied therein, for example,
in baseband or as part of a carrier wave. Such a propagated signal may
take any of a variety of forms, including, but not limited to,
electromagnetic, optical, or any suitable combination thereof. A
computer readable signal medium may be any computer readable medium that
is not a computer readable storage medium and that can communicate,
propagate, or transport a program for use by or in connection with an
instruction execution system, apparatus, or device.
[0063] Program code embodied on a computer readable medium may be
transmitted using any appropriate medium, including but not limited to
wireless, wireline, optical fiber cable, RF, etc., or any suitable
combination of the foregoing.
[0064] Computer program code for carrying out operations for aspects of
the present disclosure may be written in any combination of one or more
programming languages, including an object oriented programming language
such as Java, Smalltalk, C++ or the like and conventional procedural
programming languages, such as the "C" programming language or similar
programming languages. The program code may execute entirely on the
user's computer, partly on the user's computer, as a standalone software
package, partly on the user's computer and partly on a remote computer or
entirely on the remote computer or server. In the latter scenario, the
remote computer may be connected to the user's computer through any type
of network, including a local area network (LAN) or a wide area network
(WAN), or the connection may be made to an external computer (for
example, through the Internet using an Internet Service Provider).
[0065] Aspects of the present disclosure are described above with
reference to flowchart illustrations and/or block diagrams of methods,
apparatus (systems) and computer program products according to
embodiments of the disclosure. It will be understood that each block of
the flowchart illustrations and/or block diagrams, and combinations of
blocks in the flowchart illustrations and/or block diagrams, can be
implemented by computer program instructions. These computer program
instructions may be provided to a processor of a general purpose
computer, special purpose computer, or other programmable data processing
apparatus to produce a machine, such that the instructions, which execute
via the processor of the computer or other programmable data processing
apparatus, create means for implementing the functions/acts specified in
the flowchart and/or block diagram block or blocks.
[0066] These computer program instructions may also be stored in a
computer readable medium that can direct a computer, other programmable
data processing apparatus, or other devices to function in a particular
manner, such that the instructions stored in the computer readable medium
produce an article of manufacture including instructions which implement
the function/act specified in the flowchart and/or block diagram block or
blocks.
[0067] The computer program instructions may also be loaded onto a
computer, other programmable data processing apparatus, or other devices
to cause a series of operational steps to be performed on the computer,
other programmable apparatus or other devices to produce a computer
implemented process such that the instructions which execute on the
computer or other programmable apparatus provide processes for
implementing the functions/acts specified in the flowchart and/or block
diagram block or blocks.
[0068] The flowchart and block diagrams in the Figures illustrate the
architecture, functionality, and operation of possible implementations of
systems, methods, and computer program products according to various
embodiments of the present disclosure. In this regard, each block in the
flowchart or block diagrams may represent a module, segment, or portion
of code, which comprises one or more executable instructions for
implementing the specified logical function(s). It should also be noted
that, in some alternative implementations, the functions noted in the
block may occur out of the order noted in the figures. For example, two
blocks shown in succession may, in fact, be executed substantially
concurrently, or the blocks may sometimes be executed in the reverse
order, depending upon the functionality involved. It will also be noted
that each block of the block diagrams and/or flowchart illustration, and
combinations of blocks in the block diagrams and/or flowchart
illustration, can be implemented by special purpose hardwarebased
systems that perform the specified functions or acts, or combinations of
special purpose hardware and computer instructions.
* * * * *