scholarly journals Automorphic Equivalence of Many-Sorted Algebras

2015 ◽  
Vol 24 (3) ◽  
pp. 209-240 ◽  
Author(s):  
A. Tsurkov
Keyword(s):  
Automorphic Equivalence

scholarly journals Learning Stochastic Equivalence based on Discrete Ricci Curvature

2021 ◽  
Author(s):  
Xuan Guo ◽  
Qiang Tian ◽  
Wang Zhang ◽  
Wenjun Wang ◽  
Pengfei Jiao
Keyword(s):  
Ricci Curvature ◽  
Structure Identification ◽  
Network Embedding ◽  
Discrimination Ability ◽  
Role Identification ◽  
Stochastic Equivalence ◽  
Regular Equivalence ◽  
Role Based ◽  
Low Dimensional ◽  
Automorphic Equivalence

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.

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

2021 ◽  
Author(s):  
Weiren Yu ◽  
Sima Iranmanesh ◽  
Aparajita Haldar ◽  
Maoyin Zhang ◽  
Hakan Ferhatosmanoglu
Keyword(s):  
Web Search ◽  
Similarity Measures ◽  
Computational Time ◽  
Pairwise Similarity ◽  
Large Graphs ◽  
Triangular Inequality ◽  
Graph Theoretic ◽  
Role Based ◽  
Automorphic Equivalence ◽  
Similarity Information

AbstractRoleSim 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.

scholarly journals Automorphic equivalence in the classical varieties of linear algebras

2017 ◽  
Vol 27 (08) ◽  
pp. 973-999 ◽  
Author(s):  
A. Tsurkov
Keyword(s):  
Quotient Group ◽  
Free Algebras ◽  
Inner Automorphisms ◽  
Automorphic Equivalence

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.

scholarly journals Automorphic equivalence within gapped phases in the bulk

2020 ◽  
Vol 278 (8) ◽  
pp. 108422 ◽  
Author(s):  
Alvin Moon ◽  
Yoshiko Ogata
Keyword(s):  
Automorphic Equivalence

scholarly journals AUTOMORPHIC EQUIVALENCE PROBLEM FOR FREE ASSOCIATIVE ALGEBRAS OF RANK TWO

2007 ◽  
Vol 17 (02) ◽  
pp. 221-234 ◽  
Author(s):  
VESSELIN DRENSKY ◽  
JIE-TAI YU
Keyword(s):  
Associative Algebra ◽  
Simple Proof ◽  
Equivalence Problem ◽  
Free Associative Algebra ◽  
Associative Algebras ◽  
Nontrivial Automorphism ◽  
Simpler Algorithm ◽  
Unpublished Thesis ◽  
Rank 2 ◽  
Automorphic Equivalence

Let K 〈x,y〉 be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K 〈x,y〉 are equivalent under an automorphism of K 〈x,y〉. A modification of our algorithm solves the problem whether or not an element in K 〈x,y〉 is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K 〈x,y〉. An algorithm for equivalence of polynomials under automorphisms of ℂ[x,y] was presented by Wightwick. Another, much simpler algorithm for automorphic equivalence of two polynomials in K[x,y] for any algebraically closed constructive field K was given by Makar-Limanov, Shpilrain, and Yu. In our approach we combine an idea of the latter three authors with an idea from the unpublished thesis of Lane used to describe automorphisms which stabilize elements of K 〈x,y〉. This also allows us to give a simple proof of the corresponding result for K[x,y] obtained by Makar-Limanov, Shpilrain, and Yu.

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

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Víctor Martínez ◽  
Fernando Berzal ◽  
Juan-Carlos Cubero
Keyword(s):  
Equivalence Class ◽  
State Of The Art ◽  
Equivalence Classes ◽  
Network Node ◽  
Distance Metric ◽  
Role Mining ◽  
And Function ◽  
Vector Representations ◽  
Automorphic Equivalence ◽  
Node Embeddings

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.

Sign in / Sign up

Export Citation Format

Share Document