2.3 Requiring Neighborhood Equivalence
2.3.3 Regular Equivalence
.
Summary for equitable partitions. Equitable partitions are powerful tools in algebraic and spectral graph theory. While some problems around equitable partitions are N P-complete, there are efficient algorithms to com-pute the maximal equitable partition of a graph, or to comcom-pute the coarsest equitable refinement of an a priori partition. These algorithms could be used to compute role assignments, but, due to irregularities, the results contain in most cases too many classes and miss the underlying (possibly perturbed) structure. In empirical networks the maximal equitable partition is often close to the identity partition and hence trivial. Structural similarities, which will be introduced in Sect. 3.2 are a relaxation of equitable partition which is tolerant against irregularities.
2.3.3 Regular Equivalence
Regular equivalence goes back to the idea ofstructural relatedness of Sailer [78], who proposed that actors play the same role if they are connected to role-equivalent actors—in contrast to structural equivalence, where they have to be connected to identical actors. A formalization of this idea is given by the definition of bisimulation (compare [70]) and [66]). The term regular equivalence has been introduced by White and Reitz [87]. Borgatti and Ev-erett [34] gave an equivalent definition in terms of colorings. A coloring is regular if vertices that are colored the same, have the same colors in their neighborhoods. In contrast to equitable partitions multiple occurrence of a color makes no difference for regular equivalence. Therefore it is sufficient to consider regular equivalence on graphs whose edges all have weight one which we adopt as the graph model of this section. If r: V → W is a role assignment and U ⊆V then r(U) ={r(u) ; u∈U} ⊆W is the set of classes
or colors that members of U have. Note that r(U) is a set, i. e., multiple occurrences of the same element (color) are ignored.
Definition 2.3.8 Let G= (V, E)be a graph. A role assignment r: V →W is called
1. out-regular if r(u) =r(v)⇒r(N+(u)) =r(N+(v));
2. in-regular if r(u) =r(v)⇒r(N−(u)) =r(N−(v));
3. regular if both conditions hold;
for all u, v ∈V.
An example of a regular equivalence can be seen in Fig. 2.9. Note that, in contrast to equitable partitions, regularly equivalent vertices can have different degrees.
Figure 2.9: Graph with a non-trivial regular equivalence indicated by the vertex coloring. Note that the maximal regular equivalence of this graph is the complete partition.
There are many more equivalent definitions for regular equivalence (see e. g., [87, 17]). We recall one of these that uses Boolean matrix multiplication.
The Boolean characteristic matrix of an equivalence relation∼ on a set ofn vertices is the n×n matrix S defined by
Suv =
(1 if u∼v ; 0 else .
IfA and B are two n×n matrices with entries in{0,1} the Boolean product AB is defined to be then×n matrix whose ij’th entry is
(AB)ij =
n
_
k=1
Aik∧Bkj .
Theorem 2.3.9 ([17]) Let G = (V, E) be a graph with adjacency matrix A and ∼ an equivalence relation on V with Boolean characteristic matrix S.
Then, ∼is a regular equivalence if and only if AS =SAholds under Boolean matrix multiplication.
The use of Boolean matrix multiplication ignores multiple occurrences of a color in the neighborhood of a vertex, which is consistent with the definition of regular equivalence. In contrast, the corresponding theorems for structural equivalence and equitable partition (Theorems 2.2.3 and 2.3.3) are formu-lated using real matrix multiplication, since for those types of equivalences the number of occurrences of a color matters.
Elementary Properties of Regular Equivalence
The identity partition is regular for all graphs. More generally, every equi-table partition (hence every orbit partition, and every structural equivalence) is regular. It is easy to see that equitable partitions are a proper subset of regular equivalences.
The next proposition characterizes when the complete partition is regular.
A sink is a vertex with empty out-neighborhood, asource is one with empty in-neighborhood.
Proposition 2.3.10 ([14]) The complete partition of a graph G = (V, E) is regular if and only if G contains neither sinks nor sources or E =∅.
This implies the simple observation that for undirected graphs the maximal regular equivalence is the division into isolates and non-isolates (and therefore does not provide useful information). For instance the complete partition is regular for the graph in Fig. 2.9, since this graph has no isolates.
We remember that the identity and the complete partition are called trivial role assignments. A graph with at least 3 vertices whose only regular role assignments are trivial is called role primitive in [33]. Constructing directed role primitive graphs is trivial. For every directed path only the identity partition is regular. Directed graphs which have exactly the identity and the complete partition as regular partitions are, e. g., directed cycles of prime length, since every non-trivial regular equivalence induces a non-trivial
divisor of the cycle length. Undirected role primitive graphs are not that easy to find but exist as well.
Theorem 2.3.11 ([33]) The graph in Figure 2.10 is role primitive.
Figure 2.10: A role-primitive undirected graph
It is easy to see that in the complete graph every role assignment is regular (if all vertices in the complete graph have loops than every role assignment is even structural).
Lattice Structure and Regular Interior
For a given graph there might be many regular equivalences and the maximal regular equivalence is often trivial. Thus, in order to make regular equiva-lence a useful tool for role assignment one has to clarify whether it is possible to characterize specific non-trivial elements of the set of regular equivalences.
One step in this direction is the observation that this set has a lattice struc-ture, although it is not a sublattice of the lattice of all equivalence relations.
See the definitions for lattice and sublattice in Sect. 2.1.2.
Theorem 2.3.12 ([14]) The set of all regular equivalences of a graph G forms a lattice, where the supremum is a restriction of the supremum in the lattice of all equivalences. (For the infimum see Proposition 2.3.13 and Corollary 2.3.16.)
Although the supremum in the lattice of regular equivalences is a re-striction of the supremum in the lattice of all equivalences, the infimum is not.
Proposition 2.3.13 ([14]) The lattice of regular equivalences is not a sub-lattice of the sub-lattice of all equivalences.
Proof We show that the infimum is not a restriction of the infimum in the lattice of all equivalences (which is simply intersection). Consider the graph in Figure 2.11 where the intersection of the two regular partitions {{A, C, E},{B, D}}and{{A, C},{B, D, E}}is{{A, C},{B, D},{E}}, which
is not regular.
Figure 2.11: The intersection of the two regular equivalences {{A, C, E},{B, D}} and {{A, C},{B, D, E}} is not regular for this graph.
The fact that the supremum in the lattice of regular equivalences is a restriction of the supremum in the lattice of all equivalences implies the exis-tence of a maximum regular equivalence which lies below a given (arbitrary) equivalence.
Definition 2.3.14 ([17]) Let G be a graph and ∼ an equivalence relation on its vertex set. An equivalence relation ∼1 is called the regular interior of
∼ if it satisfies the following three conditions. The equivalence ∼1 is regular, it is smaller than or equal to ∼, and for all ∼2 satisfying the former two conditions it holds ∼2≤∼1.
Another name for regular interior is coarsest regular refinement.
Corollary 2.3.15 Let G be a graph and ∼ an equivalence relation on its vertex set. Then the regular interior of ∼ exists. On the other hand there is in general no minimum regular equivalence above a given equivalence (called regular closure or regular hull).
Proof The first part has been shown in [17]. For the second part recall the example in the proof of Prop. 2.3.13 shown in Figure 2.11). It is easy to verify that the regular partitions {{A, C, E},{B, D}}and {{A, C},{B, D, E}}are both above the (non-regular) partition {{A, C},{B, D},{E}} and are both
minimal with this property.
Corollary 2.3.16 ([17]) The infimum (in the lattice of regular equivalence relations) of two regular equivalence relations ∼1 and ∼2 is given by the regular interior of the intersection of ∼1 and ∼2.
Regular Equivalence, Bisimulation, and Dynamic Logic
Marx and Masuch [66] pointed out the close relationship between regular equivalence, bisimulation (see, e. g., [70]), and dynamic logic (see [66] and
references therein). Bisimulations are used, e. g., to prove equivalence of finite automata: A finite automaton is a directed graph G = (V, E), whose vertices are called states and whose edges are labeled with the letters of a finite alphabet. Furthermore, an automaton has aninitial state s∈V and a set of terminal states F ⊆V. The labels on a directed path from the initial state to any terminal state form a word that is said to be accepted by the automaton. The language defined by an automaton is the set of accepted words. Two automata are equivalent if they accept the same language. A partition of the vertex set of a graph is called stable if it is out-regular in the sense of Def. 2.3.8. Consider the partitionP0 ={F, V \F}into terminal and non-terminal states and let P be the coarsest stable refinement (out-regular interior) of P0. The role graph with respect to P defines an (in many cases smaller) equivalent automaton. See Fig. 2.12 for an example.
Figure 2.12: Left: Finite automaton with initial stateS and terminal states 2 and 6 (in black). All edges are assumed to have the label a. Middle: Coloring indicates classes of equivalent states (corresponding to the coarsest stable refinement of the partition into terminal and non-terminal states).
Right: Minimal equivalent automaton with initial state {S,3} and terminal state {2,6}. The automaton defines the language{a2a3i; i∈N}.
There are quite efficient solutions for algorithmic problems around bisim-ulation (compare, e. g., [72]). Of course, these results carry over to regular equivalence—even if the original articles do not use this term.
Computation of Regular Interior
The regular interior (see Definition 2.3.14) of an equivalence relation ∼ is the coarsest regular refinement of ∼. It can be computed, starting with ∼, by a number of refinement steps in each of which currently equivalent
ver-tices with non-equivalent neighborhoods are split, until all equivalent verver-tices have equivalent neighborhoods. For an example of such a computation see Figure 2.13. Note that the maximal regular equivalence for this graph is the complete partition. So, in this example the regular interior of a certain input partition provides more useful information than the maximal regular equivalence.
Figure 2.13: Computation of the regular interior for the graph shown in Fig. 2.9. Left: initial partition (some white vertices have black neighbors some do not); right: first step (white vertices with black neighbors got a black-white color gradient (bw-vertices), some of these have white neighbors some do not); the second and final step is shown in Fig. 2.9 (the class of bw-vertices has been split into grey and bw-vertices, the partition is now regular).
The running time for computing the regular interior depends heavily on how the refinement steps are organized. catrege [15] is the most well-known algorithm in the social network literature and runs in O(n3) time for a graph on n vertices. catrege implements the refinement steps in a rather straightforward manner: the algorithm maintains a partition P which is initially set to the input partition and will be the regular interior at the end. In each refinement step, catrege tests for each pair of vertices that are equivalent with respect toP whether their neighborhoods are equivalent.
If so, they remain equivalent, otherwise they are separated in this refinement step. The algorithm terminates if no changes happen. The number of refine-ment steps is bounded by n, since in each refinement step (except the last) the number of equivalence classes grows by at least one. The running time of one refinement step is in O(n2).
Tarjan and Paige [72] present a sophisticated algorithm for therelational coarsest partition problem. Their algorithm runs in O(mlogn) time on a
graph with n vertices and m edges and is well-known in the bisimulation literature. (See [66] for the relationship between bisimulation and regular equivalence.) The algorithm presented in [72] computes the coarsest stable refinement of an input partition, corresponding to the out-regular interior (compare Def. 2.3.8). However, it is possible to compute the coarsest refine-ment that is stable with respect to a bounded number of relations in the same asymptotical running time. Since stable with respect to the edge rela-tion and its inverse is equivalent to regular, a O(mlogn) time bound for the regular interior follows.
The algorithm from [72] maintains a partition P that is initially set to the input partition and will be the coarsest stable refinement at the end. In each step the partition is refined from the point of view of only one class C:
all classes are split into those vertices that have outgoing edges terminating in C and those that do not have outgoing edges terminating in C. Such a refinement step can be implemented such that its running time is proportional to the sum of the degrees of all vertices in C. Furthermore, the splitting classes can be chosen in such a way that if a vertex is used a second time as a splitting vertex then the class containing this vertex has at most half the size of the previous time. It follows that each vertex is used at most logn+ 1 times as a splitting vertex, so that the overall running time is asymptotically logn times the sum over all degrees, i. e., in O(mlogn).
Since the algorithm from Paige and Tarjan is much faster thancatrege, it is preferable to compute the regular interior—if regularity is required in the strict sense. However, it should be noted that catrege can be used to compute a degree of regular equivalence (compare, e. g., [65]) of two vertices that are not strictly regular equivalent. Such a relaxation is certainly nec-essary for empirical data (compare Sect. 2.6) and seems to be infeasible for the algorithm from Paige and Tarjan.
The Role Assignment Problem
The maximal regular equivalence (MRE) can be computed quite efficiently, however, it is often trivial: for undirected graphs the MRE is simply the division into isolates and non-isolates and for directed graphs the MRE is the complete partition if the graph contains neither sinks nor sources. While the regular interior of a carefully chosen input partition might be more useful in some cases, we do not know of any guidelines to chose a good input partition.
To get non-trivial regular equivalences, one might want to compute only those that have a number of equivalence classes different from one and n (in most cases a small number different from one). However, as will be noted below, it is N P-complete to decide whether a graph admits a regular equivalence
with exactly k classes for all k ≥ 2. Similarly, the more specific problem of determining a regular equivalence that yields a given role graph is N P-hard as well. Thus, computing non-trivial regular equivalences can probably not be solved efficiently. See Sect. 3.4 for further reasons why efficient solutions of the role assignment problem would be desirable.
The results below are from Fiala and Paulusma [37]. Let k ∈ N and R be an undirected graph, possibly with loops.
Problem 2.3.17 (Regular k-Role Assignment) Given an undirected graph G. Is there a regular equivalence for G with exactly k equivalence classes?
Problem 2.3.18 (RegularR-Role Assignment)Given an undirected graph G. Is there a regular role assignment r :V(G)→V(R) with role graph R?
Note that we require role assignments to be surjective and that we consider for regular equivalences the role graph without edge-weights (see Def. 2.1.5).
Theorem 2.3.19 ([37]) Regular k-Role Assignment is polynomially solv-able for k = 1 and it is N P-complete for all k ≥2.
Theorem 2.3.20 ([37]) Regular R-Role Assignment is polynomially solv-able if each connected component ofR consists of a single vertex (with or with-out a loop), or consists of two vertices withwith-out loops and it is N P-complete otherwise.
Although k-role assignability can not be tested efficiently, there is an easy-to-verify sufficient condition that guarantees the existence of regular k-role assignments. Briefly, the condition is that the minimal degree is not too far from the maximal degree.
Theorem 2.3.21 ([76]) For allk ∈Nthere is a constant ck ∈Rsuch that for all graphsGwith minimal degreeδ =δ(G)and maximal degree∆ = ∆(G) satisfying
δ≥cklog(∆) ,
there is a regular equivalence for G with exactly k equivalence classes.
The proof in [76] makes use of the so-called probabilistic method [5].
Actually, it shows that for each graph satisfying the conditions of the theorem there is a vertex partition into exactly k classes such that every vertex is adjacent to at least one vertex from every class. In particular this partition is regular and its role graph is the complete graph on k vertices including all loops.
It is not completely clear whether Theorem 2.3.21 favors the use of regular equivalences for defining role assignments: on one hand it is nice to know that under certain conditions non-trivial role assignments exist. On the other hand these conditions are typically not satisfied in empirical networks as these are often reported to have many vertices of very low degree (like degree one or two) and other vertices of very high degree.
Perfect Equivalence
Results so far indicate that the maximal regular equivalence is in many cases (almost) trivial and that non-trivial regular equivalences are computation-ally intractable. Perfect equivalence is a restriction of regular equivalence which requires additionally that if two vertices are non-equivalent then their neighborhoods must be non-equivalent. Thus, perfect equivalence requires that there must be a reason if two vertices are not considered as equiva-lent. In our definition of perfect equivalence we do not include out-perfect or in-perfect (compare Def. 2.3.8).
Definition 2.3.22 ([34]) A role assignmentrdefines a perfect equivalence if for all u, v ∈V
r(u) = r(v) ⇐⇒ r(N+(u)) =r(N+(v)) and r(N−(u)) =r(N−(v)).
A regular equivalence is perfect if and only if the induced role graph has no structurally equivalent vertices. The idea that there must be a reason that two vertices are non-equivalent could also be formulated for structural equivalence (requiring that two vertices are equivalent if and only if their neighborhoods are identical). However, in this case we would only specify the maximal structural equivalence. On the other hand, perfect equivalence defines a non-trivial subset of all regular equivalences.
The set of perfect equivalence relations of a graph is a lattice [34], which is neither a sublattice of all equivalence relations (Sect. 2.1.2) nor of the lattice of regular equivalence relations. A perfect interior of an equivalence relation
∼ is a coarsest perfect refinement of∼ (compare Def. 2.3.14). In contrast to the regular interior, the perfect interior does not exist in general.
Theorem 2.3.23 In general, the transitive closure (see Sect. 2.1.2) of the union of two perfect equivalence relations is not perfect. In particular, for some equivalences there is no perfect interior.
Proof For the first statement see Fig. 2.14. For the second, note that P1 and P2 are both perfect refinements ofP and are both maximal with respect
to this property.
Figure 2.14: Graph for the proof of Theorem 2.3.23. Transitive clo-sure of the two perfect equivalences P1 = {{1,5},{2,6}{3,4}} and P2 = {{1,2},{5,6}{3},{4}} is the (automorphic) equivalence P = {{1,2,5,6},{3,4}} (indicated by the coloring) which is not perfect.
The second statement has a more trivial proof: For a graph with two strong structurally equivalent vertices, the identity partition has no perfect refine-ment. The graph in Fig. 2.14 also shows that an automorphic equivalence is not necessarily perfect, which is certainly a drawback of perfect equivalence.
Many decision problems concerning perfect equivalence areN P-complete as well. This can be seen by Theorem 2.3.20 restricted to role graphs without strong structurally equivalent vertices.
Summary for perfect equivalence. Perfect equivalence is a restriction of regular equivalence, but it does not seem to yield better role assignments.
Some mathematical properties of regular equivalences get lost and there are examples where the condition on perfect equivalence rules out meaningful regular role assignments, or automorphic equivalences.
Relative Regular Equivalence
Relative regular equivalence expresses the idea that equivalent vertices have equivalent neighborhoods in a coarser, predefined measure. Again, we do not formulate the definitions for relative out-regular and relative in-regular (compare Def. 2.3.8).
Definition 2.3.24 ([17]) Let G = (V, E) be a graph and r: V → W and
Definition 2.3.24 ([17]) Let G = (V, E) be a graph and r: V → W and