Method and apparatus of machine learning using a network with software agents at the network nodes and then ranking network nodes
An apparatus and method are provided for rapidly ranking network nodes according to input ranking criteria. The links (i.e., first-order paths) between nodes are expressed in a first-order path matrix, which is used to generate nth-order path matrices as nth powers of the first-order path matrix and summed as a power series to generate a surrogate ranking operator (SRO) representing as a single matrix operation a sum over paths of all orders. Thus, in contrast to conventional ranking methods that require multiple recursive steps to account for the interrelatedness of linked nodes, a ranking is produced by multiplying the SRO by a state vector representing the input ranking criteria.
1. A website ranking apparatus, comprising:
a network communications interface connected to an internet including a plurality of websites, and configured to receive data elements from the websites;
a memory storing a database of the websites, the database of the websites including a plurality of nodes representing the respective websites and data elements received therefrom, and the plurality of nodes being respectively connected by a plurality of links representing connections between the websites; and
processing circuitry configured to
receive, via a user interface of the apparatus, an input query from a user, the input query including one or more keywords of a website search;
determine a first-order path matrix including values representing one-link paths between pairs of the plurality of nodes, which are connected by a respective one of the plurality of links;
generate a surrogate ranking operator (SRO) using a power series of the first-order path matrix;
rank the websites using a matrix product between a state vector of the input query, including the one or more keywords, and the generated SRO to determine a ranking of the websites; and
display, via the user interface, the ranked websites in order of the ranking of the websites, as a recommendation of best websites satisfying the received input query including the one or more keywords of the website search,
wherein the processing circuitry is further configured to generate the SRO by summing the power series such that, in the power series, each nth-order path matrix in the power series is divided by 2 raised to an nth power, when the plurality of links are ordered, and each nth-order path matrix in the power series is divided by a factorial of n, when the plurality of links are not ordered.
2. A recommender apparatus, comprising:
a memory storing a database of a network, wherein the network is a network of websites, the database of the network including a plurality of nodes, each representing a respective website and containing respective data elements, and the plurality of nodes being respectively connected by a plurality of links; and
processing circuitry configured to:
determine a first-order path matrix including values representing first-order adjacencies from the plurality of links that connect the plurality of nodes of the network database;
generate a surrogate ranking operator (SRO) using a power series of the first-order path matrix;
receive, via a user interface of the apparatus, one or more input data elements including one or more keywords of a website search, representing ranking criteria, and generate an input state vector of the input data elements including the one or more keywords;
rank the websites of the network using a matrix product of the generated SRO and the input state vector to determine a ranking of the websites, wherein the processing circuitry is further configured to rank the websites of the network using a personalized preference matrix together with the SRO to personalize the rankings according to personal preferences of the user; and
display, via the user interface, the ranked websites in order of the ranking of the websites, as recommendations of best websites corresponding to the one or more keywords of the website search,
wherein the processing circuitry is further configured to generate the SRO by summing the power series such that, in the power series, each nth-order path matrix in the power series is divided by 2 raised to an nth power, when the plurality of links are ordered, and each nth-order path matrix in the power series is divided by a factorial of n, when the plurality of links are not ordered.
3. The recommender apparatus of claim 2 , wherein the processing circuitry is configured to generate the SRO by:
decomposing the first-order path matrix into eigenvalues and eigenvectors to generate a diagonal matrix of the eigenvalues, a unitary matrix having the eigenvectors as column vectors, and a Hermitian conjugate of the unitary matrix;
generating nth-order path matrices as an nth power of the first-order path matrix, by taking the nth power of the diagonal matrix of the eigenvalues and multiplying on the left by the unitary matrix and on the right by the Hermitian conjugate of the unitary matrix; and
summing the nth-order path matrices using a power series.
4. The recommender apparatus of claim 3 , wherein the processing circuitry is further configured to generate the SRO by decomposing the first-order path matrix into the eigenvalues, wherein the eigenvalues represent a spectrum of the first-order path matrix.
5. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to modify the first-order path matrix prior to generating the SRO by:
normalizing the first-order path matrix using a determinant of the first-order path matrix; and
ensuring that the first-order path matrix satisfies a bistochasticity condition,
wherein the SRO is generated using the modified first-order path matrix to calculate the power series of the first-order path matrix.
6. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to determine the first-order path matrix by using a discrete approximation to diffusion operator, wherein a rate of diffusion is greater across links of the plurality of links that represent a greater connection between pairs of nodes corresponding to the respective links.
7. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to determine the first-order path matrix using
elements of the first-order path matrix that are real numbers when the plurality of links express are undirected,
the elements of the first-order path matrix that are one of bivectors and complex numbers when the plurality of links are unidirectional,
the elements of the first-order path matrix that are one of tessarines, bicomplex numbers, and split complex numbers when the plurality of links are bidirectional, and
the elements of the first-order path matrix that are multivectors when the plurality of links are mixed.
8. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to determine the first-order path matrix using
elements of the first-order path matrix that are real numbers when the plurality of links express correlations between pairs of nodes of the network,
the elements of the first-order path matrix that are one of bivectors and complex numbers when the plurality of links express associations between pairs of nodes of the network,
the elements of the first-order path matrix that are one of tessarines, bicomplex numbers, and split complex numbers when the plurality of links express associations between pairs of nodes of the network, and
the elements of the first-order path matrix that are multivectors when the plurality of links express attitudinal relationships between pairs of nodes of the network.
9. The recommender apparatus of claim 3 , wherein the processing circuitry is further configured to generate the SRO by eliminating degeneracy in the eigenvalues before generating the diagonal matrix of the eigenvalues.
10. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to generate the SRO, which is path adjusted by removing path redundancies in the nth-order path matrices prior to summing the nth-order path matrices using the power series.
11. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to rank the nodes of the network using metadata to augment the SRO.
12. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to receive the one or more input data elements, wherein upon the one or more input data elements including at least one data element that is absent from the data elements contained by plurality of nodes, and the at least one data element is mapped onto the input state vector using a measure that is one of a conceptual relativity measure, a semantic distance measure, a Jaccard measure, a mutual information measure, and a positive pointwise mutual information measure.
13. The recommender apparatus of claim 2 , wherein the processing circuitry is further configured to determine the first-order path matrix, wherein the first-order path matrix is based on an operator of quantum graph theory that is one of a Hamiltonian operator, a Laplacian operator, and a Lagrangian operator.
14. A method of ranking a plurality of websites, the method comprising:
storing, in a non-transitory computer readable medium, a database of the websites, the database of the websites including a plurality of nodes representing the respective websites, and the plurality of nodes being respectively connected by a plurality of links representing connections between the websites;
determining, using processing circuitry, a first-order path matrix including values representing first-order adjacencies from the plurality of links that connect the plurality of nodes of the database of the websites;
generating, using the processing circuitry, a surrogate ranking operator (SRO) using a power series of the first-order path matrix;
receiving, via a user interface, one or more input data elements, including one or more keywords of a website search, representing ranking criteria, and generate an input state vector of the input data elements including the one or more keywords;
ranking, using the processing circuitry, the websites using a matrix product of the generated SRO and the input state vector to determine a ranking of the websites; and
displaying, via the user interface, the ranked websites user in order of the ranking of the websites, as recommendations of best websites corresponding to the one or more keywords of the website search,
wherein the generating step further comprises generating the SRO by summing the power series such that, in the power series, each nth-order path matrix in the power series is divided by 2 raised to an nth power, when the plurality of links are ordered, and each nth-order path matrix in the power series is divided by a factorial of n, when the plurality of links are not ordered.
15. The method of claim 14 , wherein generating the SRO further includes:
decomposing the first-order path matrix into eigenvalues and eigenvectors to generate a diagonal matrix of the eigenvalues, a unitary matrix having the eigenvectors as column vectors, and a Hermitian conjugate of the unitary matrix;
generating nth-order path matrices as an nth power of the first-order path matrix, by taking the nth power of the diagonal matrix of the eigenvalues and multiplying on the left by the unitary matrix and on the right by the Hermitian conjugate of the unitary matrix; and
summing the nth-order path matrices using a power series.
16. A non-transitory computer-readable medium storing executable instructions, wherein the instructions, when executed by processing circuitry, cause the processing circuitry to perform the method according to claim 14 of ranking the websites.