2.1 Preliminaries
2.1.2 Equivalence, Partition, and Role Assignment
Classification of vertices can be expressed by three mathematical notions:
by a vertex partition, by an equivalence relation on the vertex set, or by a mapping of vertices to some set of classes or positions (called a role assign-ment). Here we establish that these are just three different formulations for the same underlying concept.
LetV be a set and ∼⊆V ×V a (binary) relation on V. In this context we also write u ∼ v for (u, v) ∈∼. The relation ∼ is called an equivalence relation onV if it satisfies for all u, v, w∈V
• v ∼v (reflexive),
• u∼v implies v ∼u (symmetric),
• u∼v and v ∼w implies u∼w (transitive).
If∼is an equivalence relation onV andv ∈V then [v] ={u; u∼v}is called itsequivalence class. We can consider the elements of an equivalence relation as edges of a graph on V. Such a graph G has the following properties: all
edge weights of G are one, every vertex has a loop, G is undirected, and G is the disjoint union of unconnected cliques (the equivalence classes). See Fig. 2.1(left) for an example.
Figure 2.1: Left: graph of an equivalence relation on a set of five vertices, associated to the partition {{1,2,3},{4,5}}. Right: the associated role as-signment maps 1, 2, and 3 to the class A and 4 and 5 to the classB.
Apartition P ={C1, . . . , Ck}of V is a set of non-empty, disjoint subsets Ci ⊆ V, called classes, such that V = Sk
i=1Ci. That is, each vertex is in exactly one class.
The set of equivalence classes of an equivalence relation is a partition.
Conversely, a partition induces an equivalence relation by defining that two vertices are equivalent if and only if they are members of the same class.
These two mappings from equivalence relations to partitions and vice versa are mutually inverse.
A third formalization of this concept adopts the point of view that vertex classification is the assignment of classes to vertices.
Definition 2.1.1 A role assignment for V is a surjective mapping r: V → W onto a set W of positions, classes, or colors.
(Note that, in contrast to a frequent usage of the term “vertex coloring” in computer science, a role assignment can map adjacent vertices to the same class.)
A role assignment r: V → W defines a partition of V by taking the inverse-images r−1(w) ={v ∈V ; r(v) = w}, w ∈W as classes. Conversely an equivalence relation induces a role assignment forV by the class mapping v 7→[v].
We do not distinguish two role assignments that differ only in a renaming of the set of classes:
Remark 2.1.2 Let r: V → W and r0: V → W0 be two role assignments.
Then, the following two assertions are equivalent.
1. r and r0 define the same equivalence relation on V; 2. there is a bijection ϕ: W →W0 such that r0 =ϕ◦r.
In either case we do not distinguish between r and r0.
For instance, Fig. 2.1(right) shows a role assignment associated to the par-tition {{1,2,3},{4,5}}. If we labeled the two classes with (say) X and Y instead of A and B, we would obtain an equivalent role assignment.
Remark 2.1.3 Partitions, equivalence relations, and role assignments mu-tually stand in a canonical one-to-one correspondence. For the remainder of this chapter, definitions and theorems stated for one of these concepts trans-late to the other two.
If u ∼v then we say that u and v occupy the same position, play the same role, or are role-equivalent (according to the equivalence ∼).
Two specific and trivial role assignments are the identity partition and the complete partition. The identity partition is the partition in which each vertexv is in a singleton class{v}. It is associated to the equivalence relation in which every vertex is only equivalent to itself and to the identity role assignment v 7→ v which maps every vertex to a different position. The complete partition is the partition that has only one class (the entire vertex set). It is associated to the equivalence relation in which every pair of vertices is equivalent and to the role assignment that maps every vertex to the same position.
Lattice of Equivalence Relations
Obviously, there is in general more than one equivalence relation on a given vertex set. Here we define a partial order on this set that turns out to be a lattice (see e. g., [48]).
Equivalence relations on a set V are subsets of V ×V, thus they can be partially ordered by set-inclusion (∼1≤∼2 iff ∼1⊆∼2). (A partial order is a binary relation≤with the properties thatu≤vandv ≤wimpliesu≤wand u≤v andv ≤uimpliesu=v. In contrast to a linear ordering, two elements are not necessarily comparable in a partial ordering.) If ∼1≤∼2 then ∼1 is called finer than∼2 and ∼2 is called coarser than ∼1. An equivalence ∼1 is therefore finer than an equivalence∼2if, whenever two vertices are equivalent according to ∼1 then they are equivalent according to ∼2. Thus a finer equivalence relation makes (possibly) more distinctions between vertices.
We introduce formally the notion of a lattice. Let X be a set that is partially ordered by ≤ and Y ⊆ X. We call an element y0 of X an upper bound (a lower bound) for Y if for all y ∈ Y, y ≤ y0 (y0 ≤ y). We call an elementy0 ofX the supremum (infimum) ofY, if it is an upper bound (lower bound) and if each upper bound (lower bound) of Y is larger (smaller) than y0. The second condition implies that suprema and infima (if they exist) are unique. Instead of supremum we also say least upper bound and instead of infimumgreatest lower bound. The supremum ofY is denoted by sup(Y) the infimum by inf(Y). We also write sup(x, y) or inf(x, y) instead of sup({x, y}) or inf({x, y}), respectively. A lattice is a partially ordered set L, such that for all a, b∈ L, sup(a, b) and inf(a, b) exist. The supremum sup(a, b) is also called thejoin ofa and b and denoted bya∨b. The infimum inf(a, b) is also called the meet of a and b and denoted bya∧b. A lattice is called complete if suprema and infima exist for all subsets (not only two-element subsets). A subset X ⊆ Y of a lattice Y is called a sublattice of Y if the meet and join operation ofY restricted to elements of X, always yields an element ofX. It is easy to see that a sublattice is a lattice. Note however that a subset of a lattice that is itself a lattice is not necessarily a sublattice. This distinction becomes important later, since the property of being a sublattice implies the existence of certain hull and interior operations (see, e. g., Corollary. 2.3.15 and Theorem 3.7.4).
The set of all equivalence relations on a vertex set V is a lattice (but not a sublattice of the lattice of subsets of V ×V). If ∼1 and ∼2 are two equivalence relations on V, then their intersection (as sets) is the infimum of
∼1 and ∼2. That is, if ∼inf= inf(∼1,∼2), then
u∼inf v ⇐⇒ u∼1 v and u∼2 v .
Thus, the infimum of two role assignments distinguishes between vertices that play a different role in either one of the two role assignments. As an example see Figs. 2.2 and 2.3. Figure 2.2 shows the personal network of an Argentinian immigrant to Spain, together with two different partitions.
The partitionP1 according to country of origin and partitionP2 according to country of residence. Some pairs of actors (e. g., 1 and 3) are equivalent inP1 but not inP2. Conversely, other pairs of actors (e. g., 3 and 24) are equivalent in P2 but not in P1. Thus, none of the two equivalence relations identifies all pairs that the other does and therefore P1 and P2 are incomparable.
Figure 2.3 shows the partition P3 which is defined to be the intersection of P1 andP2 in Fig. 2.2. Whenever two vertices are equivalent in P3, then they are equivalent in P1 and in P2.
The supremum of two equivalence relations ∼1 and ∼2 is slightly more complicated. It must contain all pairs of vertices that are equivalent in either
Figure 2.2: Personal network of an Argentinian immigrant to Spain parti-tioned in two different ways. Left: Partition P1 according to country of origin. Black for Argentina, white for Spain, the grey vertex is from a Euro-pean country different from Spain. Right: PartitionP2 according to country of residence. Black for Argentina, white for Spain.
∼1or∼2, but also vertices that are related by a chain of such pairs. The union
∼1 ∪ ∼2 of two equivalence relations is in general not an equivalence relation since it is not transitive. The transitive closure of a relation R ⊆ V ×V is defined to be the relation S ⊆V ×V, where for allu, v ∈V
uSv ⇔ ∃k∈N, ∃w1, . . . , wk∈V such that
u=w1, v =wk, and ∀i= 1, . . . , k−1 it is wiRwi+1 . The transitive closure of a symmetric relation is symmetric, the transitive closure of a reflexive relation is reflexive and the transitive closure of any relation is transitive. It follows that, if ∼1 and ∼2 are two equivalence re-lations on V, then the transitive closure of their union is the supremum of
∼1 and ∼2. The supremum of two role assignments identifies vertices that play the same role in either of the two role assignments. The supremum of the two partitions P1 and P2 shown in Fig. 2.2, is the complete partition.
For instance, Vertices 37 and 38 are equivalent in the supremum since 37 is equivalent to 3 in P1 and 3 is equivalent to 38 in P2.
Theorem 2.1.4 ([48]) The set of equivalence relations is a lattice.
Figure 2.3: Partition P3 which is the intersection (infimum) of the two parti-tions from Fig. 2.2. Black vertices are from Argentina and live in Argentina.
Light-grey vertices are from Argentina and live in Spain. The dark-grey ver-tex (Number 38) is from a country in Europe and lives in Spain. White vertices are from Spain and live in Spain.
The identity partition is the minimum element in the lattice of equivalence relations, the complete partition is the maximum element.