An Automorphic Distance Metric and Its Application to Node Embedding for Role Mining

Role is a fundamental concept in the analysis of the behavior and function of interacting entities in complex networks. Role discovery is the task of uncovering the hidden roles of nodes within a network. Node roles are commonly defined in terms of equivalence classes. Two nodes have the same role if they fall within the same equivalence class. Automorphic equivalence, where two nodes are equivalent when they can swap their labels to form an isomorphic graph, captures this notion of role. The binary concept of equivalence is too restrictive, and nodes in real-world networks rarely belong to the same equivalence class. Instead, a relaxed definition in terms of similarity or distance is commonly used to compute the degree to which two nodes are equivalent. In this paper, we propose a novel distance metric called automorphic distance, which measures how far two nodes are from being automorphically equivalent. We also study its application to node embedding, showing how our metric can be used to generate role-preserving vector representations of nodes. Our experiments confirm that the proposed automorphic distance metric outperforms a state-of-the-art automorphic equivalence-based metric and different state-of-the-art techniques for the generation of node embeddings in different role-related tasks.

RoleSim*: Scaling axiomatic role-based similarity ranking on large graphs

RoleSim and SimRank are among the popular graph-theoretic similarity measures with many applications in, e.g., web search, collaborative filtering, and sociometry. While RoleSim addresses the automorphic (role) equivalence of pairwise similarity which SimRank lacks, it ignores the neighboring similarity information out of the automorphically equivalent set. Consequently, two pairs of nodes, which are not automorphically equivalent by nature, cannot be well distinguished by RoleSim if the averages of their neighboring similarities over the automorphically equivalent set are the same. To alleviate this problem: 1) We propose a novel similarity model, namely RoleSim*, which accurately evaluates pairwise role similarities in a more comprehensive manner. RoleSim* not only guarantees the automorphic equivalence that SimRank lacks, but also takes into account the neighboring similarity information outside the automorphically equivalent sets that are overlooked by RoleSim. 2) We prove the existence and uniqueness of the RoleSim* solution, and show its three axiomatic properties (i.e., symmetry, boundedness, and non-increasing monotonicity). 3) We provide a concise bound for iteratively computing RoleSim* formula, and estimate the number of iterations required to attain a desired accuracy. 4) We induce a distance metric based on RoleSim* similarity, and show that the RoleSim* metric fulfills the triangular inequality, which implies the sum-transitivity of its similarity scores. 5) We present a threshold-based RoleSim* model that reduces the computational time further with provable accuracy guarantee. 6) We propose a single-source RoleSim* model, which scales well for sizable graphs. 7) We also devise methods to scale RoleSim* based search by incorporating its triangular inequality property with partitioning techniques. Our experimental results on real datasets demonstrate that RoleSim* achieves higher accuracy than its competitors while scaling well on sizable graphs with billions of edges.

Learning Stochastic Equivalence based on Discrete Ricci Curvature

Role-based network embedding methods aim to preserve node-centric connectivity patterns, which are expressions of node roles, into low-dimensional vectors. However, almost all the existing methods are designed for capturing a relaxation of automorphic equivalence or regular equivalence. They may be good at structure identification but could show poorer performance on role identification. Because automorphic equivalence and regular equivalence strictly tie the role of a node to the identities of all its neighbors. To mitigate this problem, we construct a framework called Curvature-based Network Embedding with Stochastic Equivalence (CNESE) to embed stochastic equivalence. More specifically, we estimate the role distribution of nodes based on discrete Ricci curvature for its excellent ability to concisely representing local topology. We use a Variational Auto-Encoder to generate embeddings while a degree-guided regularizer and a contrastive learning regularizer are leveraged to improving both its robustness and discrimination ability. The effectiveness of our proposed CNESE is demonstrated by extensive experiments on real-world networks.

Automorphic equivalence in the classical varieties of linear algebras

This research is a continuation of [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi:10.1142/S0219498814500261]. In this paper, we consider some classical varieties of linear algebras over the field [Formula: see text] such that [Formula: see text]. We study the relation between the geometric equivalence and automorphic equivalence of the algebras of these varieties. If we denote by [Formula: see text] one of these varieties, then [Formula: see text] is a category of the finite generated free algebras of the variety [Formula: see text]. In this paper, we calculate for the considered varieties the quotient group [Formula: see text], where [Formula: see text] is a group of all the automorphisms of the category [Formula: see text] and [Formula: see text] is a subgroup of all inner automorphisms of this category. The quotient group [Formula: see text] measures the possible difference between the geometric equivalence and automorphic equivalence of algebras from the variety [Formula: see text]. The results of this paper and [Tsurkov, Automorphic equivalence of linear algebras, J. Algebra Appl. 13(7) (2014), doi: 10.1142/S0219498814500261] are summarized in the table at the end of Sec. 5. We can see from this table that in all considered varieties of the linear algebras the group [Formula: see text] is generated by cosets which are presented by no more than two types of the strongly stable automorphisms of the category [Formula: see text]. The first type of automorphisms is connected with the changing of the multiplication by scalar and a the second type is connected with the changing of the multiplication of the elements of the algebras. In Sec. 6, we present some examples of the pairs of linear algebras such that the considered strongly stable automorphisms provide the automorphic equivalence of these algebras, but these algebras are not geometrically equivalent.

Automorphic equivalence of linear algebras

This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H1 and H2 are called geometrically equivalent if and only if the H1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H1 and H2 are called automorphically equivalent if and only if the H1-closed sets coincide with the H2-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ0 of the finitely generated free algebras of the variety Θ. By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group 𝔄/𝔜 is big. Here 𝔄 is the group of all automorphisms of the category Θ0, 𝔜 is a normal subgroup of all inner automorphisms of the category Θ0. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group 𝔄/𝔜 in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group 𝔄/𝔜 is isomorphic to the group (U(kS2)/U(k{e}))λ Aut k, where S2 is the symmetric group of the set which has 2 elements, U(kS2) is the group of all invertible elements of the group algebra kS2, e ∈ S2, U(k{e}) is a group of all invertible elements of the subalgebra k{e}, Aut k is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group 𝔄/𝔜 is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.

Knowledge Bases Over Algebraic Models

This paper discusses the informational equivalence problem for knowledge bases. The authors show that using some mathematical approach it is possible to attack this problem and end up with an implementable algorithm. An essential part of the paper is devoted to the explanation of the mathematical idea which stands behind this algorithm. The authors try to do that in common terms or, at least, in less formal terms. In the second part of the paper mathematical methods are applied to study the properties of automorphic equivalence of knowledge bases (multi-models) and show that this notion is much wider than the total isomorphism (identity) of knowledge bases. In order to make the paper self-contained, the reader is provided with the formal definition of a knowledge base. Further development of the theoretical approach presented in the paper can lead to practical applications. For example, it can be used for preventing duplication of information in knowledge bases and in other tasks of improving knowledge management.