One way of characterizing the topological and structural properties of vertices and edges in a graph is by using structural similarity measures. Measures like Cosine, Jaccard and Dice compute the similarities restricted to the immediate neighborhood of the vertices, bypassing important structural properties beyond the locality. Others measures, such as the generalized edge clustering coefficient, go beyond the locality but with high computational complexity, making them impractical in large-scale scenarios. In this paper we propose a novel similarity measure that determines the structural similarity by dynamically diffusing and capturing information beyond the locality. This new similarity is modeled as an iterated function that can be solved by fixed point iteration in super-linear time and memory complexity, so it is able to analyze large-scale graphs. In order to show the advantages of the proposed similarity in the community detection task, we replace the local structural similarity used in the SCAN algorithm with the proposed similarity measure, improving the quality of the detected community structure and also reducing the sensitivity to the parameter @math of the SCAN algorithm.
By definition, the structural similarity is a local function because only requires information about the inmediate neighborhood of vertices @math and @math . In fact, given two vertices @math and @math , the structural similairy @math can be computed in @math time. Furthermore, @math time is required to compute the structural similarity for each pair of vertices in a graph @math , such that the term @math corresponds to the arboricity of @math @cite_19 . The structural similarity is just an extension to the context of graphs of the Cosine Similarity. Another two popular similarity measures extended to the context of graphs are Jaccard and Dice, defined as @math and @math respectively.
@cite_19
2064723547 2146607065 2607500032 2104812688
The human perception of stylistic similarity transcends structure and function: for instance, a bed and a dresser may share a common style. An algorithmically computed style similarity measure that mimics human perception can benefit a range of computer graphics applications. Previous work in style analysis focused on shapes within the same class, and leveraged structural similarity between these shapes to facilitate analysis. In contrast, we introduce the first structure-transcending style similarity measure and validate it to be well aligned with human perception of stylistic similarity. Our measure is inspired by observations about style similarity in art history literature, which point to the presence of similarly shaped, salient, geometric elements as one of the key indicators of stylistic similarity. We translate these observations into an algorithmic measure by first quantifying the geometric properties that make humans perceive geometric elements as similarly shaped and salient in the context of style, then employing this quantification to detect pairs of matching style related elements on the analyzed models, and finally collating the element-level geometric similarity measurements into an object-level style measure consistent with human perception. To achieve this consistency we employ crowdsourcing to quantify the different components of our measure; we learn the relative perceptual importance of a range of elementary shape distances and other parameters used in our measurement from 50K responses to cross-structure style similarity queries provided by over 2500 participants.We train and validate our method on this dataset, showing it to successfully predict relative style similarity with near 90 accuracy based on 10-fold cross-validation. Comparing graphs to determine the level of underlying structural similarity between them is a widely encountered problem in computer science. It is particularly relevant to the study of Internet topologies, such as the generation of synthetic topologies to represent the Internet's AS topology. We derive a new metric that enables exactly such a structural comparison: the weighted spectral distribution. We then apply this metric to three aspects of the study of the Internet's AS topology. i) We use it to quantify the effect of changing the mixing properties of a simple synthetic network generator. ii) We use this quantitative understanding to examine the evolution of the Internet's AS topology over approximately seven years, finding that the distinction between the Internet core and periphery has blurred over time. iii) We use the metric to derive optimal parameterizations of several widely used AS topology generators with respect to a large-scale measurement of the real AS topology. Structural identity is a concept of symmetry in which network nodes are identified according to the network structure and their relationship to other nodes. Structural identity has been studied in theory and practice over the past decades, but only recently has it been addressed with representational learning techniques. This work presents struc2vec, a novel and flexible framework for learning latent representations for the structural identity of nodes. struc2vec uses a hierarchy to measure node similarity at different scales, and constructs a multilayer graph to encode structural similarities and generate structural context for nodes. Numerical experiments indicate that state-of-the-art techniques for learning node representations fail in capturing stronger notions of structural identity, while struc2vec exhibits much superior performance in this task, as it overcomes limitations of prior approaches. As a consequence, numerical experiments indicate that struc2vec improves performance on classification tasks that depend more on structural identity. Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu