This document discusses assortative mixing and partitions in multilayer networks. It defines graph partitions, ordering of partitions, self-similarity of partitions, and how partitions can be viewed as enumerative attribute assignments. It then defines the assortativity coefficient of a partition to measure how strongly vertices in the same group of a partition are connected. Finally, it discusses how to measure assortativity between two different partitions of a graph by focusing on their intersection partition.

1. Multilayerity Within Multilayerity?
Assortative Mixing of
Multiple Partitions in
Mulilayer Networks
Moses A. Boudourides1 & Sergios T. Lenis2
Department of Mathematics
University of Patras, Greece
1Moses.Boudourides@gmail.com
2sergioslenis@gmail.com
8 June 2015
Examples and Python code:
http://mboudour.github.io/2015/06/09/Multilayerity-Within-Multilayerity%3F.html
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

2. Graph Partitions
Let
G = (V , E)
be a (simple undirected) graph with (finite) set of vectors V
and (finite) set of edges E.
The terms “graph” and “(social) network” are used here interchangeably.
If S1, S2, . . . , Ss ⊂ V (for a positive integer s) are s nonempty
(sub)sets of vertices of G, the (finite) family of subsets
S = S(G) = {S1, S2, . . . , Ss},
forms a partition (or s–partition) in G, when, for all
i, j = 1, 2, · · · , s, i = j,
Si ∩ Sj = ∅ and
V =
s
i=1
Si .
S1, S2, . . . , Ss are called groups (of vertices) of partition S.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

3. For every graph, there exists (at least one) partition of its
vertices.
Two trivial graph partitions are:
the |V |–partition into singletons of vertices and
Spoint = Spoint(G) = {{v}: v ∈ V },
the 1–partition into the whole vertex set
Stotal = Stotal(G) = {V }.
If G is a bipartite graph with (vertex) parts U and V , the
bipartition of G is denoted as:
Sbipartition = Sbipartition(G) = {U, V }.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

4. Examples of Nontrivial Graph Partitions
1 Structural (endogenous) partitions:
Connected components
Communities
Chromatic partitions (bipartitions, multipartitions)
Degree partitions
Time slicing of temporal networks
Domination bipartitions (ego– and alter–nets)
Core–Periphery bipartitions
2 Ad hoc (exogenous) partitions:
Vertex attributes (or labels or colors)
Layers (or levels)
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

5. Multilayer Partitions and Multilayer Graphs
A multilayer partition of a graph G is a partition
L = L(G) = {L1, L2, . . . , L }
into ≥ 2 groups of vertices L1, L2, . . . , L , which are called
layers.
A graph with a multilayer partition is called a multilayer
graph.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

6. Ordering Partitions
Let P = {P1, P2, . . . , Pp} and Q = {Q1, Q2, . . . , Qq} be two
(different) partitions of vertices of graph G = (V , E).
Partition P is called thicker than partition Q (and Q is called
thinner than P), whenever, for every j = 1, 2, · · · , q, there
exists a i = 1, 2, · · · , p such that
Qj ⊂ Pi .
Partitions P and Q are called (enumeratively) equivalent,
whenever P is both thicker and thinner than partition Q.
Obviously, partitions P and Q are equivalent, whenever
|V | ≥ pq and,
for all i = 1, 2, · · · , p and j = 1, 2, · · · , q,
Pi ∩ Qj = ∅.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

7. Self–Similarity of Partitions
A partition P of vertices of graph G (such that |V | ≥ p2) is
called (enumeratively) self–similar, whenever, for any
i = 1, 2, · · · , p, there is a p–(sub)partition
Pi
= {Pi
1, Pi
2, . . . , Pi
p}
of the (induced) subgraph Pi .
In this case, graph G is called (enumeratively) self–similar
with respect to partition P.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

8. Graph Partitions as Enumerative Attribute Assignments
An assignment of enumerative attributes (or discrete
attribute assignment) to vertices of graph G is a mapping
A: V → {1, 2, · · · , α}, for some α ≤ |V |,
through which the vertices of G are classified according to the
values they take in the set {1, 2, · · · , α}.
Every p–partition P = {P1, P2, . . . , Pp} of vertices of G
corresponds to a p–assignment AP of enumerative attributes
to vertices of G distributing them in the groups of partition P.
Conversely, every assignment of enumerative attributes to
vertices of G taking values in the (finite) set {1, 2, · · · , α}
corresponds to a partition Pα = {Pα
1 , Pα
2 , . . . , Pα
p } of the
vertices of G such that, for any k = 1, 2, · · · , α,
Pα
k = {v ∈ V : A(v) = k} = A−1
(k).
Thus, vertex partitions and enumerative vertex attribute
assignments are coincident.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

9. Let P be a partition of G into p groups of vertices and A be
an assignment of α discrete attributes to vertices of G.
Partition P is called compatible with attribute assignment A,
whenever partition P is thinner than partition Pα, i.e.,
whenever, for every k = 1, 2, · · · , α, there exists (at least one)
i = 1, 2, · · · , p such that
Pi = A−1
(k).
Apparently, partition P is compatible with attribute
assignment A if anly if p ≥ α.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

10. Assortativity of a Partition
Let P = {P1, P2, . . . , Pp} be a vertex partition of graph G.
Identifying P to a p–assignment of enumerative attributes to
the vertices of G , one can define (cf. Mark Newman, 2003)
the (normalized) enumerative attribute assortativity (or
discrete assortativity) coefficient of partition P as follows:
rP =
tr MP − ||M2
P||
1 − ||M2
P||
,
where MP is the p × p (normalized) mixing matrix of
partition P.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

11. In general,
−1 ≤ rP ≤ 1,
where
rP = 0 signifies that there is no assortative mixing of graph
vertices with respect to their assignment to the p groups of
partition P, i.e., graph G is configured as a perfectly mixed
network (the random null model).
rP = 1 signifies that there is a perfect assortative mixing of
graph vertices with respect to their assignment to the p groups
of partition P, i.e., the connectivity pattern of these groups is
perfectly homophilic.
When rP atains a minimum value, which lies in general in the
range [−1, 0), this signifies that there is a perfect
disassortative mixing of graph vertices with respect to their
assignment to the p groups of partition P, i.e., the
connectivity pattern of these groups is perfectly heterophilic.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

12. Assortative Mixing Among Partitions
Let P = {P1, P2, . . . , Pp} and Q = {Q1, Q2, . . . , Qq} be two
(different) partitions of vertices of graph G.
Then, for any i = 1, 2, · · · , p, j = 1, 2, · · · , q, Pi ∩ Qj = ∅ and
the intersection of P and Q
P ∩ Q = {Pi ∩ Qj : i = 1, 2, · · · , p, j = 1, 2, · · · , q}
is also a vertex partition of G.
Notice that partition P ∩ Q is compatible with any one of the
following three discrete attribute assignments on G:
AP , in which, for any group Pi ∩ Qj of P ∩ Q, all vertices of
Pi ∩ Qj are assigned a value in the set {1, 2, · · · , p},
AQ, in which, for any group Pi ∩ Qj of Q ∩ Q, all vertices of
Pi ∩ Qj are assigned a value in the set {1, 2, · · · , q} and
AP∩Q, in which, for any group Pi ∩ Qj of P ∩ Q, all vertices
of Pi ∩ Qj are assigned a value in the set {1, 2, · · · , pq}.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

13. Thus, by focusing on partition P ∩ Q and the discrete
attribute assignment AP, one defines the discrete
assortativity coefficient of partition P with regards to
partition Q as
rPQ = rP∩Q
and the mixing matrix of partition P with regards to
partition Q as
MPQ = MP∩Q.
Similarly, by focusing on partition P ∩ Q and the discrete
attribute assignment AQ, one defines the discrete
assortativity coefficient of partition Q with regards to
partition P, rQP, and the mixing matrix of partition Q
with regards to partition P, MQP.
In general,
rPQ = rQP.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks

14. Special Cases
If Q = Spoint, then rPSpoint
is the attribute assortativity
coefficient rP of graph G equipped with the vertex attributes
corresponding to partition P, i.e.,
rPSpoint
= rP.
If P = Stotal, then, for any partition Q,
rStotalQ = 1.
If G is bipartite and P = Sbipartition, then, for any partition Q,
rSbipartitionQ = −1.
M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks