IEEE Systems, Man and Cybernetics Magazine - April 2022 - 34

systems, community Mining, and so on, relevant knowledge
services are carried out based on the reasoning
results provided by the analytic plane.
Tensor-Based Knowledge
Representation and Fusion
The knowledge graph is often multisource and dynamic
due to the high complexity and heterogeneity of the CPSS.
It is necessary and meaningful to fuse the knowledge
acquired from different scenarios and devices as well as
that generated continuously over time. Specifically, the
knowledge generated by common things in different scenarios
and periods is very relevant. If the inference results
depend only on any separate knowledge, it will be one-sided
and inadequate. Therefore, reasoning over integrated
knowledge will be beneficial to intelligent decision making.
We first represent knowledge graphs as Boolean tensors,
and then we propose a series of graph tensor operations for
knowledge graph fusion. Graph tensor operations and definitions
are summarized in Table 1.
Tensor Representation
We first give a general definition of knowledge graph
tensor representation. Given a knowledge graph
GA ,, ,, ,AA T
^hf
12 N
" ^h,
N
aa aA12
An
fg1 ## # N
12
, In
relations, and T denotes the facts of the structure
,, ,.AA
that represents G can be defined as X BII
where B 01= " ,,
Xa ,, ,aa 1
12
12 N
## #g I
,
G^hf N = indicates the fact aaa12
exists; otherwise, it does not exist.
denote the length of set A .n
^hf ,, ,
N
Intragraph Operations
The intragraph operations are used to update and
modify vertexes or edges in a knowledge graph, including
deleting, contracting, and extracting subgraphs,
Table 1. The graph tensor operations
and definitions.
Type
Intragraph
Graph Tensor Operation
Extract subgraph
Delete edge
Delete vertex
Contract edge
Contract vertex
Intergraph
Definition
X sub
XGrXGtXrr
Xtt
,
+
ij+
and
so on. More formal ized definitions are given
as follows:
◆ Delete edge: Given
rA ,n where An
!
the graph tensor XGrfrom
^hf
GA ,, ,, .AA T
12 N
XGr◆
Delete vertex: Given tA ,m where Am
the graph tensor XG t^hf
deleting
the r-th slice of tensor X .G
!
from GA ,, ,,
12 N
Xr Rt! ^h
G
is a relations set,
denotes removing the relation r
can be obtained by
is a entity set,
denotes removing the entity t
dent on t, which are denoted by Rt .^h XG t◆
Contract edge: Given r ,i rA ,
tions set, the graph tensor Xrr
ij+
j ! n where An
ij+
on
AA T and the edges that are incican
be
obtained by deleting the t-th and r-th, slices of tensor
,.
is a reladenotes
the contraction
of the relations ri and rj ^hf
Xrr can be derived by adding the r -thi
slices of tensor X .G
and j
◆ Contract vertex: Given t ,i tAj
with t ,j where Am
ij+
! m and ti
Xtt denotes contracting the entities ti
GA ,, ,, .AA T
12 N
r -th
has relations
is an entity set. The graph tensor
and tj
removing all edges between the two vertexes. Xtt
can be calculated by adding the t -thi
and j
denotes the set of entities or
The graph tensor
G !
and setting the corresponding entries of XG to 0.
◆ Extract subgraph: The graph tensor Xsub
extracting a subgraph of ^hf
entities belong to entities set Am
Xsub
GA ,, ,, ,AA T
12 N
to relations set A .n
is a subtensor of X .G
Intergraph Operations
The intergraph operations are used to integrate two or
more graphs, including union, intersection, and product,
and so on. More formalized definitions are given
as follows:
◆ Graph tensor union: Given two knowledge graphs
GA ,, ,,
1 AA T
12
^hf
N
1 ^hf ,, ,, ,
N
representation tensors are XG1
andGT whose
and X ,G2
2 BB B12
2
respectively.
The union of the two graph can be obtained by graph
tensor union operation, i.e.,
XX ,X
G t GGU12
,
=
where t
,
denotes implementing union operation on two entries
at the same index of the two graph tensor. XGU
is the
representation tensor of the union graph with size
II ,I2
1
CC C
N
## and IC
n
and ,G2
is the length of set CA .Bnn n
= ,
◆ Graph tensor intersection: Given two knowledge
graphs G1
XG1
and X ,G2
XX ,X
G t GGI12
+
=
where t
their representation tensors are
as described earlier. The intersection of
the two graphs can be calculated by the graph tensor
intersection, i.e.,
+ denotes
executing intersection operation on two entries at the
same index of the two graph tensor. XGI
ij+
Graph tensor union XXG t G12
Graph tensor intersection XXG t G12
Graph tensor product XXG t G12
#
Graph tensor decomposition XXG t G12
+
34 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE April 2022
sentation tensor of the intersected graph with size
II ,I2
1
CC C
N
## and IC
n
is the length of set CA .Bnn n
= +
◆ Graph tensor product: Given two graphs G1
XX ,X
G t GGP12
# =
where t
filling 1s to XGP
G .2
XGP
and ,G2
the product of them can be obtained by graph tensor
product, i.e.,
according to the indexes in G1
is the representation tensor of the product
# stands for the
and
is the repreand
ij+
t
-th slices
denotes
whose
and relations belong
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