Axiomatic Ranking of Network Role Similarity
Abstract
A key task in social network and other complex network analysis is role analysis: describing and categorizing nodes according to how they interact with other nodes. Two nodes have the same role if they interact with equivalent sets of neighbors. The most fundamental role equivalence is automorphic equivalence. Unfortunately, the fastest algorithms known for graph automorphism are nonpolynomial. Moreover, since exact equivalence may be rare, a more meaningful task is to measure the role similarity between any two nodes. This task is closely related to the structural or linkbased similarity problem that SimRank attempts to solve. However, SimRank and most of its offshoots are not sufficient because they do not fully recognize automorphically or structurally equivalent nodes. In this paper we tackle two problems. First, what are the necessary properties for a role similarity measure or metric? Second, how can we derive a role similarity measure satisfying these properties? For the first problem, we justify several axiomatic properties necessary for a role similarity measure or metric: range, maximal similarity, automorphic equivalence, transitive similarity, and the triangle inequality. For the second problem, we present RoleSim, a new similarity metric with a simple iterative computational method. We rigorously prove that RoleSim satisfies all the axiomatic properties. We also introduce an iceberg RoleSim algorithm which can guarantee to discover all pairs with RoleSim score no less than a userdefined threshold without computing the RoleSim for every pair. We demonstrate the superior interpretative power of RoleSim on both both synthetic and real datasets.
Axiomatic Ranking of Network Role Similarity
Ruoming Jin Victor E. Lee Hui Hong 
Department of Computer Science 
Kent State University, Kent, OH, 44242, USA 
{jin,vlee,hhong}@cs.kent.edu 
In social science, it is wellestablished that individual agents tend to play roles or assume positions within their interaction network. For instance, in a university, each individual can be classified into the position of faculty member, administration, staff, or student. Each role may be further partitioned into subroles: faculty may be further classified into tenuretrack or nontenuretrack positions, etc. Indeed, role discovering is a major research subject in classical social science [?]. Interestingly, recent studies have found not only do roles appear in other types of networks, including food webs [?], world trade [?], and even software systems [?], but also roles can help predict node functionality within their domains. For instance, in a protein interaction network, proteins with similar roles tend to serve similar metabolic functions. Thus, if we know the function of one protein, we can predict that all other proteins having a similar role would also have similar function [?].
Role is complementary to network clustering, a major tool in analyzing network structures. Network clustering attempts to decompose a network into densely connected components. It produces a high level structural model consisting of a small number of “clusternodes” and the “superedges” between these clusternodes. Since its goal is to minimize the number of edges (interactions) between clusters, it will result in strong interactions between nodes within each cluster. Given this, the clustering scheme inevitably overlooks and oversimplifies the interaction patterns of each node. For instance, each node in a cluster may take very different “roles”: some of them may serve as the core of the clusters, some may be peripheral nodes, and some serve as the connectors to link between clusters. Indeed, those nodes with similar or same roles may not even directly link to each other as they may simply share similar interaction patterns. Furthermore, even when a network lacks modularity structure, for instance, a hierarchical structure, roles can still be applied for characterizing the interaction patterns of each node. To sum, “roles” provide an orthogonal abstraction for simplifying and highlighting the complex interactions among nodes.
A central question in studying the roles in a network system is how to define role similarity. In particular, how can we rank two nodes’ role similarity in terms of their interaction patterns? Despite its vital importance for network analysis and decades of work by social scientists, joined recently by computer scientists, no satisfactory metric for role similarity has yet emerged. A key issue is the encapsulation of graph automorphism (and its generalization) into a role similarity metric: if two nodes are automorphically equivalent, then they should share the same role and their role similarity should be maximal. From a network topology viewpoint, automorphic nodes have equivalent surroundings, so one can replace the other. Figure Axiomatic Ranking of Network Role Similarity illustrates a graph with nodes and being automorphically equivalent. Automorphism can be further generalized in terms of coloration: assuming each node is assigned a color, then two nodes are equivalent if their neighborhoods consist of the same color spectrum [?].
Traditionally, the social science community has approached role analysis by defining suitable mathematical equivalence relations so that nodes can be partitioned into equivalence classes (roles). An essential property of these equivalences is that they should positively confirm automorphic equivalence, i.e., if any two nodes are automorphic, then they are roleequivalent. (The converse is not necessarily true.) Automorphism confirmation is an instance of verifying a solution, which is often algorithmically less complex than discovering a solution. Therefore, even though there is no known polynomialtime algorithm for discovering graph automorphism ^{1}^{1}1The computational complexity of graph isomorphism and automorphism are still unproven to be either or ., role equivalence algorithms [?, ?, ?] can still guarantee to satisfy the aforementioned automorphism confirmation property. These equivalence rules also directly correspond to the aforementioned coloration.
However, by relying on strict equivalence rules, these role modeling schemes can produce only binary similarity metrics: two nodes are either equivalent (similarity ) or not (similarity ). In realworld networks, usually only a very small portion of the nodepairs would satisfy an equivalence criteria [?] and among those, many are simply trivially equivalent (such as singletons or children of the same parent). In addition, strict rulebased equivalence is not robust with respect to network noise, such as falsepositive or falsenegative interactions. Thus, it is desirable in many real world applications to rank nodepairs by their degree of similarity or provide a realvalued node similarity metric.
Several recent research works have proposed to measure realvalued structural similarity or to rank nodes’ similarity based on their interaction patterns [?, ?]. SimRank [?] is one of the bestknown such measures. It generates a node similarity measure based on the following principle: “two nodes are similar if they link to similar nodes”. Mathematically, for any two different nodes and , SimRank computes their similarity recursively according to the average similarity of all the neighbor pairs (a neighbor of paired with a neighbor of ). A single node has selfsimilarity value . This is equivalent to the probability that two simultanous random walkers, starting at and , will eventually meet. Most of the existing node structural similarity measures [?, ?, ?, ?, ?, ?] are variants of SimRank. Though SimRank seems to capture the intuition of the above recursive structural similarity, its random walk matching does not satisfy the basic graph automorphism condition. For example, in Figure Axiomatic Ranking of Network Role Similarity, though and are automorphically equivalent, SimRank assigns them a value of 0.226. We discuss this further in Section Axiomatic Ranking of Network Role Similarity. To our best knowledge, there is no available realvalued structural similarity measure satisfying the automorphic equivalence requirement. Since automorphic equivalence is a pivotal characteristic of the notion of role, its lack disqualifies these existing measures from serving as authentic role similarity measures. Here is a paradox: SimRank and its variants seem to implement the recursive structural similarity definition of automorphic equivalence (two nodes are similar if they link to similar nodes), yet they do not produce desired results (to assign value to those pairs).
Thus we have an open problem: Can we derive a realvalued role similarity measure or ranking which complies with the automorphic equivalence requirement? In this paper, we develop the first realvalued similarity measure to solve this problem. In addition, our measure is also a metric, i.e., it satisfies the triangle inequality. The key feature of our role similarity measure is a weighted generalization of the Jaccard coefficient to measure the neighborhood similarity between two nodes. Unlike SimRank, which considers the average similarity among all possible pairings of neighbors, our measure considers only those pairs in the optimal matching of their two neighbor sets which maximizes the targeted similarity function. We show this approach successfully resolves the aforementioned SimRank paradox.
In social network analysis, the traditional approach for formalizing roles and role groups is to define a equivalence relation and to partition the actors into equivalence classes. Actors who fulfill the same role are equivalent. Over the years, four definitions, offering different degrees of strictness, have stood out. These four, in decreasing strictness order, are structural equivalence, automorphic equivalence, equitable partition, and regular equivalence. Figure Axiomatic Ranking of Network Role Similarity shows how these different definitions generate different roles from the same network.
Let be a graph with vertex set and edge set .
For any node , let be the neighbors of and be the degree of .
Equivalence  Neigh. Rule  Nonsingleton Classes 

Structural  exactly same  {S3,S4},{J3,J4}, {L3,L4,L5} 
Automorphic, Exact Color.  same number per class  {S1,J1},{S2,J2}, {S3,S4,J3,J4}, {L3,L4,L5} 
Regular  same class  {S1,J1,L1},{S2,J2,L2}, {S3,S4,J3,J4,L3,L4,L5} 
Structural Equivalence: Two actors are structurally equivalent if they interact with the same set of others [?]. Mathematically, and are structurally equivalent if and only if . For example, consider the extended family shown in Figure Axiomatic Ranking of Network Role Similarity. , , and are siblings, , , and are spouses, and the remaining nodes are their children. Each family’s children, , , and form a nontrivial equivalence class. However, none of the parents can be grouped together via structural equivalence. This equivalence model is too strict to be useful for simplifying a large network and to discover meaningful roles.
Automorphic Equivalence: Two actors (nodes) and are automorphically equivalent if there is an automorphism of such that [?]. An automorphism of a graph is a permutation of vertex set such that for any two nodes and , iff . In social terms, and can swap names, along with possibly some other name swaps, while preserving all the actoractor relationships. Let be the group of all automorphisms of graph . For any two nodes and in , if for some . Note that is an equivalence relation on ; if we say that is automorphically equivalent to . The equivalence classes generated under (or ) are called orbits. The equivalence class for vertex is called the orbit of , and denoted as . Each orbit corresponds to a role in the automorphic equivalence. Understanding the importance of automorphic equivalence and applying it to role modeling was a major breakthrough in classical social network research. In our example Figure Axiomatic Ranking of Network Role Similarity, from the topology alone, we cannot distinguish between the Smith family and the Jones family. The Lee family is distinct, because it has three children instead of two. Therefore, the equivalence classes are , and . Interestingly, we can observe that automorphically equivalent classes must have equivalent indirect relations as well, such as equivalent inlaws and cousins. However, automorphic equivalence is hard to compute and still very strict.
Exact Coloration (Equitable Partition): An exact coloration of graph assigns a color to each node, such that any two nodes share the same color iff they have the same number of neighbors of each color [?]. Nodes of the same color form an equivalence class. An exact coloration is also referred to as equitable partition [?] and graph divisor [?] and is often applied in the vertex classification/refinement for canonical labeling of graph isomorphism test [?, ?]. A graph may have several exact colorations; in general we seek the fewest colors. In our running example, the structural equivalence partitioning and the automorphic partitioning offer two different exact colorations. Exact coloration relaxes automorphism by considering only immediate neighborhood equivalence. Two nodes with the same color under an exact coloration may not necessarily be automorphically equivalent, but the graph automorphic equivalence does introduce an exact coloration by assigning a unique color to each orbit. Like autmomorphic equivalence, exact coloration equivalence provides a recursive aspect to role modeling.
Regular Equivalence (Bisimulation): Two actors are regularly equivalent if they interact with the same variety of role classes, where class is recursively defined by regular equivalence [?]. Unlike automorphic equivalence and exact coloration, regular equivalence does not care about the cardinality of neighbor relationships, only whether they are nonzero. For example, using regular equivalence, all three families are now equivalent. There are only three equivalence classes: , , and . Note that under regular equivalence, any two automorphically equivalent nodes may be partitioned into the same regular equivalence class. In computer science, the regular equivalence is often referred to as the bisimulation, which is widely used in automata and modal logic [?].
An equivalence relation, however, tells us nothing about nonequivalent items. Using our example, the intuitive and realworld need is for a measure that not only recognizes automorphic equivalence, such as Smith child/spouse/parent to Jones child/spouse/parent, but also tell us that a Lee child/spouse/parent has strong similarity to either a Lee or Smith child/spouse/parent. Over the years, several methods have been developed for addressing various linkbased similarity problems (cocitation [?], coupling [?], SimRank [?]). Recently, several researchers have tried to apply these measurements to role modeling [?, ?]. However, none of these encompass the aforementioned automorphic equivalence property and thus are inadequate for measuring role similarity. To deal with this shortcoming and to clarify the problem, we first identify a list of axiomatic properties that all role similarity measures should obey.
Definition 1
(Axiomatic Role Similarity Properties) Given a graph , any that measures the neighborbased role similarity between vertices and in should satisfy properties P1 to P5:

P1) Range: , for all and .

P2) Symmetry: .

P3) Automorphism confirmation: If , .

P4) Transitive similarity: If , , then .

P5) Triangle inequality: , where distance is defined as .
Any node similarity measure satisfying the first four conditions (without triangle inequality) is called an admissible role similarity measure. Any node similarity measure satisfying all five conditions is an admissible role similarity metric. If the converse of the automorphic confirmation property is also true (if , then ), then the node similarity measure(metric) is an ideal role similarity measure(metric).
Property 1 describes the standard normalization where 1 means fully similar and 0 means completely dissimilar (i.e., the two neighborhoods have nothing in common). Property 2 indicates that similarity, like distance, must be symmetric. Property 3 expresses our idea that fully similar means automorphically equivalent. Property 4 claims that the similarity between two nodes is equal to the similarity between equivalent members of the first two node’s respective equivalence classes. In other words, we can simply define the similarity for the orbits, i.e., . This guarantees consistency of values at an orbitlevel. Property 5 assumes the measure is metriclike, i.e., satisfying the triangle inequality. This is much stronger than transitivity, enforcing an ordering of values. Indeed, the only condition which excludes from being a strict distance metric is the automorphic equivalence (it allows the distance between two different nodes to be ). In addition, note that Property 5 implies Property 4.
Lemma 1
(Transitive Similarity) For any and , if and , then .
Proof: From triangle inequality, we have and (). Thus, . Similarly, , , and . Put together, we have .
However, since most similarity measures do not necessarily satisfy the triangle inequality, we explicitly include Property 4 as one of the axiomatic properties. Further, Property 3 is an essential criterion which distinguishes the role similarity measure from other existing measures. As we discussed earlier, the automorphic equivalence can be relaxed to exact coloration or regular equivalence. In this case, we may replace Property 3 accordingly. Our work will focus on the automorphic equivalence though it can handle its generalization as well.
Theorem 1
(Generalized Transitive Similarity) For any two pairs of nodes , , if and , then, their cross similarities are all equal, i.e., .
Proof: From the triangle inequality, we have and (). Thus, . Similarly, , , and . Put together, we have .
Thus, if we partition the nodes into equivalence classes where similarity equals , we can simply record the similarity values between equivalent classes. Let and be the equivalence classes for node and , respectively. Then, we can define .
Theorem 2
(Binary Admissibility) Given any equivalence relation that also satisfies automorphism confirmation (P3), its binary indicator function is an admissible similarity metric.
Proof: Binary values satisfy the Range(P1). Any equivalence relation satisfies symmetry (P2) and transitivity (P4), by definition. For triangle inequality( P5), consider all possible cases:
Binary values satisfy the Range(P1). Any equivalence relation satisfies symmetry (P2) and transitivity (P4), by definition. For triangle inequality( P5), consider all possible cases:
Case 1: All in the same class:
Case 2: All in different classes:
Case 3: and in the same class:
Case 4: and one other in the same class:
Note that automorphic equivalence, regular equivalence, and exact coloration all satisfy P3, so they are admissible metrics. In addition, the binary similarity measure introduced by automorphic equivalence is an ideal role similarity metric. Though these binary similarity measures are admissible, they provide no meaningful information about crossclass similarities, because they set if . We would like a realvalued measure that ranks the degree of role similarity.
Before presenting our proposed realvalued role similarity metric for network roles, we first examine some similarity measures proposed in earlier works. We will see that these do not satisfy our required properties.
The SimRank [?] similarity between nodes and is the average similarity between ’s neighbors and ’s neighbors:
where is a decay factor, , so that the influence of neighbors decreases with distance. The original SimRank measure is for directed graphs. Here, we focus on its undirected version, though our comments also hold for the directed version. SimRank values can be computed iteratively, with successively iterations approaching a unique solution, much as PageRank [?] does.
Theorem 3
SimRank is not an admissible role similarity measure.
Proof: We give examples where property 3 (automorphic equivalence) does not hold. In Figure (a), and have the same neighbors. By even the strictest definition (structural equivalence), and have the same role. However, since SimRank’s initial assumption is that there is no similarity among , , and , when it computes the average similarity of and ’s neighbors, it will never discover their equivalence. Assuming the best case where ,, and are structurally equivalent and using the recommended , converges to only 0.667. If the neighbors are not equivalent, to should still be equivalent, but SimRank gives an even lower value. SimRank has an another problem (Figure (b)) when there is an odd distance between two nodes. Nodes and are automorphically equivalent, but because there are no nodes that are an equal distance from both and , !
We note that other variants of SimRank [?, ?, ?, ?, ?, ?] also do not meet the automorphic equivalence property for to similar reasons. More discussion of these variants can be found in the Appendix.
To produce an admissible realvalued role similarity measure, we face two key challenges: First, it is computationally difficult to satisfy the automorphic equivalence property. Though not proven to be NPcomplete, the graph automorphism problem has no known polynomial algorithm [?]. Second, all the existing realvalued role similarity measures have problems dealing with even simple conditions such as structural equivalence (Subsection Axiomatic Ranking of Network Role Similarity). To meet these challenges, we take the following approach: Given an initial simplistic but admissible role similarity measurement for any pair of nodes in a graph, refine the measurement by expressing the similarity in terms of neighboring values, while maintaining the automorphic and structural equivalence properties. In the following, we formally introduce RoleSim, the first admissible realvalued role similarity measure (metric) and its associated properties.
Given a graph , the RoleSim measure realizes the recursive node structural similarity principle “two nodes are similar if they relate to similar objects” as follows.
Definition 2
(RoleSim metric) Given two vertices and , where and denote their respective neighbohoods and and denote their respective degrees, then
(1) 
where , , and is a matching between and , i.e., . The parameter is a decay factor, .
The decay factor, similar to the one used in PageRank [?], both dampens the recursive effect and guarantees a minimal RoleSim score of . We will sometimes abbreviate as . refers to the entire matrix of values. Figure Axiomatic Ranking of Network Role Similarity illustrates the matching process. The grid is the subset of the RoleSim matrix of values corresponding to the pairings of neighbors of these two vertices. A matching selects one cell per row and column. If the number of rows differs from the number of columns, then the matching size is limited to . A maximal matching is a matching where the total value of selected cells is maximum. In contrast, SimRank computes the average of every cell in the neighbor grid.
RoleSim is built on top of a natural generalization of the Jaccard coefficient, which measures the similarity between two sets and as . The Jaccard coefficient has been used previously to measure nodenode similarity based on their neighborhood commonality [?]. In our generalization, however, sets and do not necessarily share any common element; instead, there is a matching between similar elements in and , i.e., . Let record the similarity between and .
Definition 3
(Generalized Jaccard Coefficient) The generalized Jaccard coefficient measures the similarity between two sets and under matching , defined as
(2) 
The original Jaccard coefficient is a special case which uses the following matching : Let if ; otherwise . Then define . Thus, the generalized Jaccard coefficient reduces to . Comparing Eq. (1) and (2), we see that the heart of is equivalent to the maximum of the generalized Jaccard coefficient between and , among all matchings . Then,
(3) 
The definition and significance of the RoleSim for any node pair is closely related to maximal weighted matching. For any nodes and in graph , define a weighted bipartite graph , with each edge having weight . Let the total weight of neighbor matching between and be . Let be the maximal weighted matching for . It is clear that
(4) 
Using this, we can represent in terms of maximal weighted matching . In Figure Axiomatic Ranking of Network Role Similarity, the shaded cells represent the maximal matching: .
Theorem 4
(Maximal Weighted Matching) The RoleSim between nodes and corresponds linearly to the maximal weighted matching for the bipartite graph , with each edge having the weight :
(5) 
Proof: We need to show that Equations (1) and (5) are equivalent. Without loss of generality, let . First, we show that the cardinality of the maximal weighted matching . It cannot be greater, because there are insufficient elements in . It cannot be smaller, because if it were, there must exist an available edge between an uncovered node in with one in . Adding this edge would increase the matching (every edge has weight ). If , it follows that . Thus, the denominators in Equations (1) and (5) are constant and identical. It is then a trivial observation that the numerators are in fact the same. Therefore, the maximal value for the entire Equation (1) is the same as the value in (5).
Theorem 4 not only shows the key equilibrium for the role similarities RoleSim between pairs of nodes in a graph , but shows that each iteration can be computed using existing maximal matching algorithms.
RoleSim values can be computed iteratively and are guaranteed to converge, just as in PageRank and SimRank. First we outline the procedure. In the next section, we prove that the calculated values comprise an admissible role similarity metric.
Step 1: Let the initial matrix of RoleSim scores be any set of admissible scores between any pair of nodes in .
Step 2: Compute the iteration scores from the iteration’s values, . Specifically, for any nodes and ,
(6) 
Based on Theorem 4, we compute Equation (6) by finding the maximal weighted matching in the weighted bipartite graph with each edge having weight .
Step 3: Repeat Step until values converge for each pair of nodes in .
Theorem 5
(Convergence) For any admissible set of RoleSim scores , the iterative computational procedure for RoleSim converges, i.e., for any pair,
(7) 
This can be proven by showing that the maximum absolute difference between any and is monotonically decreasing. The proof is in the Appendix.
Unlike PageRank and SimRank which converge to values independent of the initialization, the convergent RoleSim score is sensitive to the initialization. That is, different initial values may generate different final RoleSim values. Rather than being a disadvantage, this is actually the key to coping with the graph automorphism complexity, by allowing the ranking to utilize prior knowledge (the equivalence relationship) of the network topological structure.
Here, we present one of the key contributions of this paper: the axiomatic admissibility of RoleSim. If the initial computation is admissible, and because the iterative computation of Equation (5) maintains admissibility (i.e., is an invariant transform of the axiomatic properties), then the final measure is admissible.
Theorem 6
(Invariant Transformation) If the iteration is an admissible role similarity metric, then so is .
Properties (Range) and (Symmetry) are trivially invariant, so we will focus on Properties (Automorphic Equivalence), (Transitive Similarity), and (Triangle Inequality).
Lemma 2
(Automorphism Confirmation Invariance) If the iteration satisfies Axiom (Automorphism Confirmation), then so does .
Proof: For nodes , there is a permutation of vertex set , such that , and any edge iff . This indicates that provides a onetoone equivalence between nodes in and . Also, and have the same number of neighbors, i.e., . So, it is clear that the maximal weighted matching in the bipartite graph selects pairs of weight 1 each. Thus, .
Lemma 3
(Transitive Similarity Invariance) If the iteration satisfies Axiom (Transitive Similarity), then so does .
Proof: We know for any , , . Denote the maximal weighted matching between and as . Since there is a onetoone equivalence correspondence between and and a onetoone equivalence correspondence between and , we can construct a matching between and as follows: . Since the transitive similarity property holds for , we have . Thus, , and
Lemma 4
(Triangle Inequality Invariance) If the iteration satisfies Axiom (Triangle Inequality), then so does .
Proof: For iteration , for any nodes , , and , , where . We must prove that this inequality still holds for the next iteration: .
Observation: if there is a matching between and which satisfies , then . This is because , where is the maximal weighted matching between and , and thus, .
We break down the proof into three cases:
Case 1. (), Case 2. (), and
Case 3. ().
Case 1 (): Since is smallest, . Define matching between and as Then using our observation above:
Cases 2 and 3 can be proven by a similar technique; the details are in the Appendix.
By combining the admissible initial configurations given in Sec Axiomatic Ranking of Network Role Similarity with Theorem 6 on invariance, we have shown that the iterative RoleSim computation generates a realvalued, admissible role similarity measure.
Theorem 7
(Admissibility) If the initial is an admissible role similarity measure, then at each th iteration, is also admissible. When RoleSim computation converges, the final measure is admissible.
According to Theorem 7, an initial admissible RoleSim measurement is needed to generate the desired realvalued role similarity ranking. What initial admissible measures or prior knowledge should we use? We consider three schemes:

ALL1 : for all .

DegreeBinary (DB): If two nodes have the same degree (), then ; otherwise, .

DegreeRatio (DR): .
These schemes come from the following observation: nodes that are automorphically equivalent have the same degree. Basically, equal degree is a necessary but not sufficient condition for automorphism. This observation is key to RoleSim: degree affects both the size of a maximal matching set and the denominator of the Jaccard Coefficient.
Theorem 8
(Admissible Initialization) ALL1, DegreeBinary, and DegreeRatio are all admissible role similarity measures. Moreover, DegreeBinary and ALL1 are admissible role similarity metrics.
Proof: It is easy to see that ALL1 degenerately satisfies all the axioms of a role similarity metric. We focus on the two degreebased schemes. Clearly, they satisfy Range(P1) and Symmetry(P2). If , then , so they both satisfy Automorphism Confirmation (P3). For transitive similarity (P4), we only need to show that depends only on class membership (Theorem 1). For these schemes, class is defined by degree, and the measurement clearly depends only on degree. Finally, because DegreeBinary and ALL1 are binary indicators of equivalence, Theorem 2 states that they are metrics.
Note that SimRank’s initialization ( iff ) is NOT admissible, because it does exactly the wrong thing: setting the initial value of any potentially equivalent nodes to 0. SimRank iterations try to build up from zero. However, due to its problems with structural equivalence and oddlength paths that we noted, SimRank will never increase the value enough to discover equivalent pairs that were neglected at the start.
In addition, we make the following interesting observations on the different initialization schemes.
Lemma 5
Let be the matrix of RoleSim values at the first iteration after (All1 initialization). Let be the matrix of RoleSim initialized by the DegreeRatio (DR) scheme. Then, .
This lemma can be easily derived by following the definition of RoleSim formula. Basically, the DegreeRatio (DR) is exactly equal to the RoleSim state one iteration after ALL1 initialization. Thus, ALL1 and DR generate the same final results. The simple formula for DR is much faster than neighbor matching, so DR is essentially one iteration faster. On the other hand, we may consider the simple ALL1 scheme to be sufficient, since it works as well as the more sophisticated DR. Especially, after the simple initialization, RoleSim’s maximal matching process automatically discriminates between nodes of different degree and continues to learn differences among neighbors as it iterates. Also, both ALL1 and DR initialization have the following convergence property:
Theorem 9
(Monotone Convergence) If ALL1 initialization is used, each RoleSim value is monotonically decreasing (or nonincreasing): for all .
Proof: At any iteration, the RoleSim value for any is the maximal matching of its neighbors. The value can increase only if some neighbor matchings increase. If no value increased in the previous iteration, then no value can increase in the current iteration. In the first iteration after ALL1, clearly no value increases. Therefore, no value ever increases.
Indeed, this monotone convergence property can be generalized into the following format: if (for any () pair, ), then we have . Note that the DegreeBinary (DB) initialization scheme does not have this property. In our experiments, we will further empirically study these initialization schemes.
Given nodes, we have nodepair similarity values to update for each iteration. For each nodepair, we must perform a maximal weighted matching. For weighted bipartite graph , the fastest algorithm based on augmenting paths (Hungarian method) can compute the maximal weighted matching in , where and .
A fast greedy algorithm offers a approximation of the globally optimal matching in time [?]. If an equivalence matching exists (i.e., ), the greedy method will find it. This is important, because it means that a greedy RoleSim computation still generates an admissible measure. Using greedy neighbor matching, the overall time complexity of RoleSim is , where is the number of iterations and is the average of over all vertexpair bipartite graphs in . The space complexity is .
Node similarity ranking in general is computationally expensive because we need to compute the similarity for nodepairs. A graph with nodes needs about GB memory to simply maintain the similarity values, assuming bytes per value. Indeed, this is a major problem for almost all node similarity ranking algorithms. However, in most applications, we are interested only in the highest similarity pairs, which typically compose only a very small fraction of all pairs. Thus, in order to improve the scalability of RoleSim, we ask the following question: Can we identify the highsimilarity pairs without computing all pair similarities? Formally, we consider the following question:
Definition 4
(Iceberg RoleSim) Given a threshold , the Iceberg RoleSim problem is to discover all pairs for which and then approximate their RoleSim scores.
The goal is to identify and compute those highsimilarity pairs without materializing the majority of the low similarity pairs. To solve Iceberg RoleSim, we consider a twostep approach: 1) use pruning rules to rule out pairs whose similarity score must be less than ; and 2) apply RoleSim iterative computation to the remaining candidate pairs. Since RoleSim computation must match all neighborpairs () of a candidate pair (), we have to handle neighborpairs (such as ) which are not themselves candidate pairs. Here, we employ upper and lower bounds for estimating RoleSim values for the noncandidate pairs.
Upper and Lower Bound for RoleSim:
Lemma 6
Given nodes , and without loss of generality, , if , then similarity .
Proof:
Given this, assuming , since matching , then is in the range . Furthermore, to facilitate our discussion, we further define . Now, we introduce the following pruning rules to filter out those pairs whose RoleSim cannot be greater than or equal to threshold , without knowing their exact scores (Without loss of generality, let ):

If , then

If maximal matching weight , then

Assume neighbor lists and are sorted by degree, with and being the first items. The maximum possible similarity of this pair is . If the shorter list has the smaller degree (), and if , then .
Rule 1 is just a restatement of Lemma 6. Rule 2 is based on the upper bound of RoleSim value. Rule 3 requires more explanation: continuing from Rule 2, we begin to consider all the pairings of neighbors. Because is the shorter list, every member must contribute to the final matching. Either will be in the matching or not. If it is, then an upper bound for is if every remaining pair has weight 1, yielding . Additionally, because the lists are sorted, , for . So, if is too small to satisfy Rule 2, then all pairings using are too small. This rule allows us to shortcircuit the full neighbor matching
We now outline our approach, which is formalized in Algorithm 1. To generate the initial iceberg hash map, we sort nodes by degree (line 3) and sort each node’s list of neighbors, by degree (lines 4 to 6). The first sort allows us to consider only those nodepairs that are sufficiently similar in degree (line 8, pruning rule 1). We compute the estimated similarity for the first pair of neighbors. Note that this estimatation formula is the same as DegreeRatio initialization. If this weight is below the limit defined in Rule 3, we terminate this pair’s candidacy and move on (lines 9 to 12). Otherwise, compute the remainder of neighborpair initial similarities, and perform a maximal matching. If the matching weight exceeds the minimum bound (Rule 2), then this nodepair and its similarity are inserted into the hash table (lines 13 to 16). After iterating though all qualified nodepairs, we have our full hash table. We now perform RoleSim iterations, but only on members of the table, which is orders of magnitude smaller than a complete similarity matrix. When a noncandidate pair’s value is needed (as a neighborpair of a candidate pair), we apply the following estimate based on its lower and upper bound (assuming ):
In the experimental evaluation, we will empirical study the effect of on the estimation accuracy.
In this section we experimentally investigate the ranking ability and performance of the RoleSim algorithm for computing role similarity metric values. We compare RoleSim to several stateoftheart node similarity algorithms, analyze the effect of different initialization schemes, and measure the scalability of Iceberg RoleSim. Specifically, we focus on the following questions:

How do different initialization schemes perform in terms of their final RoleSim score and computational efficiency?

Do nodepairs with high RoleSim scores actually have similar network roles? For any two nodes known to have similar network roles, do they receive high role similarity scores?

How much less memory and time does Iceberg RoleSim use, and how closely does its rankings match standard RoleSim’s?
Clearly, the ideal validation study requires an explicit role model and role similarity measure, which often do not exist. In the following study, we utilize a wellknown rolerelated random graph model and external measures of real datasets which provide strong role indication for these evaluations.
We set for both RoleSim and SimRank, defining convergence to be when values change by less than 1% of their previous values. We ran several RoleSim tests with both exact matching and greedy matching. The results were nearly identical ( of cells have no difference; maximum difference was small), so we focus on greedy matching from here on. We implemented the algorithms in C++ and ran all large tests on a 2.0GHz Linux machine with dualcore Opteron CPU and 4.0GB RAM.
For our tests, we use three types of graphs:
BL: the probabilistic blockmodel [?], where each block is generally considered to be corresponding to a role [?]. Here, nodes are partitioned into blocks. Each node in block has probability of linking to each node in block . Thus, the underlying blockmodel may serve as the groundtruth for testing role similarity.
SF: Large ScaleFree random graphs ^{2}^{2}2http://pywebgraph.sourceforge.net/ are used for testing scalability of the Iceberg RoleSim computation.
Realworld networks, with a measureable feature similar to social role, are used for validating RoleSim performance.
In Section Axiomatic Ranking of Network Role Similarity we discussed that DegreeRatio initialization generates the same results as ALL1 by shortcutting the first iteration. This reduces the computation time by roughly . Now we ask: Does DegreeBinary initialization (DB, binary indicator which equals 1 when degrees ) give similar results, quickly?
We ran RoleSim using both ALL1 and DB on 12 graphs, some scalefree and some blockmodel, having 500 to 10,000 nodes, and edge densities from 1 to 10. We then converted values to percentile ranking, where means the highest value and is the median value. Test results are summarized in Table Axiomatic Ranking of Network Role Similarity. The high correlation coefficient means the rankings are virtually identical, so the rankings are not very sensitive to the initialization method. Moreover, DB took from less time to converge. Overall, seems to be the preferred initialization scheme in terms of computational efficiency. Thus, we adopt it for the rest of the experiments.
Relative to All1  DegreeBinary  Degree  
Initialization  Min  Avg.  Max  Ratio 
Diff. in percentile rank  0.14%  0.38%  11.17%  none 
Pearson correl. coeff.  0.9994  0.9998  0.9999  1 
Relative execution time  0.32  0.52  0.80  
Relative # iterations  0.38  0.58  0.88  1 fewer 
How well does RoleSim discover roles in complex graphs? Specifically, given a ground truth knowledge of roles, do nodes having similar roles have high scores? To answer this question, we generated probabilistic blockmodel graphs, where blocks behave like "noisy" roles, due to sampling variance. We generated graphs with nodes and either 3 or 5 blocks. We varied the edge density , with higher densities for graphs with more blocks. The size of each block and the values were randomized; we generated 3 random instances for each graph class. We compared RoleSim to the stateoftheart SimRank, SimRank++ [?], and PSimRank [?].
For each measure and trial, we ranked the similarity scores. This serves to normalize the scoring among the four measures. Then, for each graph, we computed the average ranking of all pairs of nodes within the same block. We then averaged the three trials for each graph class.
Our results (Figure Axiomatic Ranking of Network Role Similarity) show that RoleSim outperforms all other algorithms across all the tested conditions. None of the algorithms score perfectly, due to the inherent edge distribution variance of the probabilistic model. PSimRank is better than SimRank, perhaps because it uses Jaccard Coefficient weighting, a step towards our RoleSim approach. Accuracy takes time. SimRank and SimRank++ run at the same speed. PSimRank is about 1.5 to 2 times slower, and standard RoleSim is about twice as slow as SimRank.
We applied RoleSim and the best alternative measure, PSimRank, to a realworld network having an external role measure. Our first dataset [?] is a coauthor network of 2000 database researchers. Two authors are linked if they coauthored a paper from 2003 to 2008. We pruned the network to the largest connected component (1543 nodes, 15483 edges). An author’s role depends recursively on the number of connections to other authors, and the roles of those others. Hence, it measures collaboration. We use the Gindex as a proxy measure for coauthor role (Hindex provides similar results and thus is omitted here). The Gindex measures the influence of a scientific author’s publications, its value being the largest integer such that the most cited publications have at least citations. While Gindex and coauthor role are not precisely the same, Gindex score is influenced strongly by the underlying role. High impact authors tend to be highly connected, especially with other high impact authors. If a paper is highly cited, this boosts the score of every coauthor. Thus, we expect that if two authors have similar Gindex scores, their nodepair is likely to have a high role simlarity value. To normalize RoleSim, PSimRank, and Gindex values, we converted each raw value to a percentile rank.
Figure (a) addresses our second validation question (high rank similar roles?). For the top ranked 0.01% of authorpairs, their difference in Gindex ranking is about 20 points, for both RoleSim and PSimRank, well below the randompair value of 33. A belowaverage difference confirms that the authors are relatively similar. However, as we expand the search towards 10%, RoleSim continues to detect authors with similar authorship performance, while PSimRank converges to random scoring.
To validate performance, we binned the authors into 10 roles based on Gindex value (bottom 10%, next 10%, etc.). For every pair of authors within the same role decile, we looked up role similarity percentile rank and computed an average per bin. We also computed averages for pairs of authors not in the same bin (dissimilar roles). Figure Axiomatic Ranking of Network Role Similarity shows our results. The average withinbin RoleSim value is consistently between 55% and 60%, better than the randompair score of 50, and independent of whether the Gindex is high or low. It performs equally well for all roles. PSimRank withinbin scores (dashed line), however, are inconsistent. Performance of PSimRank is worse than random for low Gscores, perhaps due to low density of links in the network. For the crossbin data, the Xaxis is the difference in decile bins for the two authors in a pair. The falling line of RoleSim indicates that role similarity correctly decreases as Gindex scores become less similar. For PSimRank, however, the crossbin scores (dashed line) hover around 50, equivalent to random scoring.
Our second dataset is a snapshot of the Internet at the level of autonomous systems (22963 nodes and 48436 edges), as generated by [?]. Several studies have confirmed that the Internet is hierarchically organized, with a densely connected core and stubs (singlyconnected nodes) at the periphery [?, ?]. A node’s position within the network (proximity to the core) and its relation to others (such as density of connections) affects its efficiency for routing and its robustness. Inspired by [?], we use shells to delineate roles.
The core of a graph is the induced subgraph where every node connects to at least other nodes in the subgraph. If , then the core must be an induced subgraph of the core. The shell is defined as the ’ring’ of nodes that are included in a graph’s core but not its core. In other words, we can decompose a graph into a set of nested rings, becoming denser as we move inward.
Using Kshells as our roles, we perform tests and analyses similar to those of the coauthor network. In Figure (b) we see that both measures do well for the top 0.1%, but PSimRank’s falters significantly when the range is expanded to the top 1%.
Next, we treat shells the same way that we treated Gindex decile bins in the previous test. See Figure Axiomatic Ranking of Network Role Similarity. Unlike decile bins, the shells do not have equal sizes. Kshells 1, 2, and 3 together contain 92% of all nodes. To clarify how these three shells dominate, we also show horizontal lines representing the combined weighted average rank of all withinshell comparisons. RoleSim’s withinshell values are consistently high, averaging 70%. Conversely, PSimRank finds strong aboveaverage similarity for the small highK shells, but nearly random similarity for shells 1 to 3, pulling its overall performance down to 50%.
In crossshell analysis, RoleSim is able to distinguish different shells very well: RoleSim approaches zero as shell difference approaches maximum. On the other hand, PSimRank shows almost no correlation to shell difference. Many of its scores are aboveaverage when they should be belowaverage (dissimilar). On the whole, it seems that PSimRank is not detecting role, but something related to connectedness and density.
In all these experiments, we can see that RoleSim provides positive answer to the role similarity ranking: 1) nodepairs with similar roles have higher RoleSim ranking than nodepairs with dissimilar roles, and 2) high RoleSim ranking indicates that nodes have similar roles. PSimRank scores, however, do not correlate with network role similarity.
In this experiment, we study how Iceberg RoleSim performs in terms of reducing computational time and storage, and its accuracy at approximating the RoleSim score for high similar nodepairs. Here, we generated scalefree graphs with up to nodes and edge densities of , , and . We compared standard RoleSim to Iceberg RoleSim, with values of and . The parameter , which is the weighting for estimated nonstored values, is set to midpoint . For the scalefree graphs, the relative scale of the iceberg compared to the full similarity matrix depends on and edge density, but it is almost independent of the number of nodes. Table Axiomatic Ranking of Network Role Similarity shows that the icebergs’ hash tables are only to of the full similarity matrices. Higher density graphs tend to have more structural variation and thus fewer highly similar node pairs. In Figure Axiomatic Ranking of Network Role Similarity, we see that Iceberg RoleSim is an order of magnitude faster. To check that the ranking has not changed significantly, we computed the Pearson correlation coefficient for each graph’s Iceberg RoleSim’s rankings vs. the rankings from the corresponding portion of the full similarity matrix. For , the average coefficient is 0.823, and for , it is 0.880. Both show very strong correlation, indicating IcebergRoleSim’s very good accuracy at ranking rolesimilarity pairs.
Edge Density  Iceberg Size, as fraction of full matrix  

1  2.77%  1.47% 
2  2.47%  0.63% 
5  3.53%  0.15% 
Next we fixed at and varied from 0 to 1.0 to see how sensitive is the accuracy of Iceberg RoleSim with respect to . The results from scalefree grapha are shown in Figure Axiomatic Ranking of Network Role Similarity. The labels describe the number of nodes and edges of each graph. Most graphs prefer , but some prefer a midrange value. Any value in the lower half seems acceptable.
The role similarity problem is a distinct special case of the more general structural or link similarity problems, which find applications in cocitation and bibliographic networks [?], recommender systems, [?] and Web search [?]. Link similarity means that two objects accrue some amount of similarity if they have similar links.
Formal definitions of role, which enable a clear idea of what is being measured, arose from the social science community [?, ?, ?]. Block partitioning can be used directly to group nodes into roles [?]. However, block modeling does not produce individual nodepair similarities. Therefore, it is not useful as a ranking method.
SimRank [?] is the best known algorithm to implement a recursive definition of object similarity: two objects are similar if they relate to similar objects. SimRank has an elegant random walk interpretation: is the probability that two independent simultaneous random walkers, beginning at and , will eventually meet at some node. However, the more neighbors that and have in common, the less likely that they will both randomly choose the same neighbor. This then explains SimRank’s problem with structural equivalence. Recently, Zhao [?] has pointed out that inneighbor and outneighbor SimRank can be used as a univeral framework to describe cocitation (common inneighbors), bibliographic coupling (commnon outneighbors), or a weighted combination of the two. The number of iterations reflects the search radius for discovering similarity. As we note in Section Axiomatic Ranking of Network Role Similarity, SimRank has an undesirable trait: its values decrease when the number of common neighbors increases. Several works have tried to address this problem. SimRank++ [?] adds a socalled evidence weight which partially compensates for the neighbor matching cardinality problem. In [?], they execute Monte Carlo simulations of "intelligent" random walks, where they force the overall probability of meeting to be Jaccard coefficient . Recently, MatchSim [?] has also used maximal matching of neighbors to address problems with SimRank’s scoring. However, our formulations have small but important differences. Because they retained SimRank’s initialization, their work does not guarantee automorphic equivalence in the final results. Also, their work is intuitionbased, without a theory of correctness. They provide one specific formulation, while we define a theoretical framework for any admissible measure or metric. Because RoleSim satisfies the triangle inequality, it is a true metric.
We have developed RoleSim, the first realvalued role similarity measure that confirms automorphic equivalence. We have also presented a set of axioms which can test any future measure to see if it is an admissible measure or metric. Our experimental tests demonstrate RoleSim’s correctness and usefulness on real world data, opening up exciting possibilities for scientific and business applications. At the same time, we see that other wellknown measures, while suitable for other tasks, are not suitable for role similarity. This axiomatic approach may prove useful for developing and validating solutions to other related tasks.
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APPENDIX
Proof for Theorem 5 (RoleSim Convergence) Let the difference of scores between iterations and be . Also, let be the maximal absolute difference across all and in iteration . To prove converge, we will show that is monotonically decreasing, i.e., . For any node pair , let the maximal weighted matching between and computed at iteration be . Note that its weight is . Without loss of generality, assume , so that and . Given this, we observe that
Therefore, . Then,