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adjacensmatriser

Adjacensmatriser is a term used in theoretical graph theory to denote a family of matrices that encodes adjacency relations across multiple steps or layers. It is a coined concept intended to generalize single-step adjacency information to multi-step connectivity in networks.

Formally, for a finite simple graph G = (V,E) with n vertices, an adjacensmatriser A is a sequence

The collection {A_k} thus generalizes the single-step adjacency information to multi-step connectivity, and A_1 determines the

Properties include symmetry for undirected graphs, the potential inclusion or exclusion of diagonal entries to reflect

Applications lie in the study of multi-step reachability, diffusion and contagion processes on networks, and the

Example: in a simple chain on three vertices 1—2—3, A_1 has ones at (1,2) and (2,3); A_2

See also: adjacency matrix, path counting, powers of a matrix, multi-layer networks.

{A_k}_{k≥1}
of
n×n
matrices
over
nonnegative
integers
or
booleans,
depending
on
the
chosen
convention.
A_1
is
the
ordinary
adjacency
matrix
A,
where
A(i,j)
=
1
if
{vi,vj}
is
an
edge
and
0
otherwise.
For
k≥2,
A_k
encodes
connections
via
walks
of
length
k
between
vertices:
in
the
integer
version,
(A_k)_{i,j}
equals
the
number
of
walks
of
length
k
from
vi
to
vj;
in
the
boolean
version,
(A_k)_{i,j}
=
1
if
there
exists
at
least
one
such
walk,
and
0
otherwise.
others
via
matrix
multiplication
(A^k
in
the
standard
algebra
gives
the
number
of
walks
of
length
k;
boolean
powers
give
reachability).
loops,
and
compatibility
with
edge
weights
in
a
weighted
graph.
analysis
of
layered
or
time-evolving
graphs
where
different
A_k
capture
behavior
at
different
horizons.
Extensions
include
adaptation
to
directed
graphs,
hypergraphs,
or
time-varying
graphs.
has
a
one
at
(1,3),
indicating
a
two-step
connection.