Queryable Compression on Time-evolving Web and Social Networks with Streaming

Time-evolving web and social network graphs are modeled as a set of pages/individuals (nodes) and their arcs (links/relationships) that change over time. Due to their popularity, they have become increasingly massive in terms of their number of nodes, arcs, and lifetimes. However, these graphs are extremely sparse throughout their lifetimes. For example, it is estimated that Facebook has over a billion vertices, yet at any point in time, it has far less than 0.001% of all possible relationships. The space required to store these large sparse graphs may not fit in most main memories using underlying representations such as a series of adjacency matrices or adjacency lists. We propose building a compressed data structure that has a compressed binary tree corresponding to each row of each adjacency matrix of the time-evolving graph. We do not explicitly construct the adjacency matrix, and our algorithms take the time-evolving arc list representation as input for its construction. Our compressed structure allows for directed and undirected graphs, faster arc and neighborhood queries, as well as the ability for arcs and frames to be added and removed directly from the compressed structure (streaming operations). We use publicly available network data sets such as Flickr, Yahoo!, and Wikipedia in our experiments and show that our new technique performs as well or better than our benchmarks on all datasets in terms of compression size and other vital metrics.

Fast Algorithm for Calculating Transitive Closures of Binary Relations in the Structure of a Network Object

The article proposes a fast algorithm for constructing the transitive closures between all pairs of nodes in the structure of a network object, which can have both directional and non-directional links. The algorithm is based on the disjunctive addition of the elements of certain rows of the adjacency matrix, which models (describe) the structure of the original network object. The article formulates and proves a theorem that using such a procedure, the matrix of transitive closures of a network object can be obtained from the adjacency matrix in two iterations (traversal) on such an array. An estimate of the asymptotic computational complexity of the proposed algorithm is substantiated. The article presents the results of an experimental study of the execution time of such an algorithm on network structures of different dimensions and with different connection densities. For this indicator, the developed algorithm is compared with the well-known approaches of Bellman, Warshall-Floyd, Shimbel, which can also be used to determine the transitive closures of binary relations of network objects. The corresponding graphs of the obtained dependences are given. The proposed algorithm (the logic embedded in it) can become the basis for solving problems of monitoring the connectivity of various subscribers in data transmission networks in real time when managing the load in such networks, where the time spent on routing information flows directly depends on the execution time of control algorithms, as well as when solving other problems on the network structures.

Relationship between adjacency and distance matrix of graph of diameter two

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0. In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is <em>D=2(J-I)-A</em>. From this relationship, we also determine the value of the determinant matrix <em>A+D</em> and the upper bound of determinant of matrix <em>D</em>.

Integral Mixed Cayley Graphs over Abelian Groups

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The mixed Cayley graph $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and edge set $\left\{ (a,b): b-a\in S \right\}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(\Gamma,S)$ over an abelian group $\Gamma$ in terms of its connection set $S$.

Extending Proximity Measures to Attributed Networks for Community Detection

Proximity measures on graphs are extensively used for solving various problems in network analysis, including community detection. Previous studies have considered proximity measures mainly for networks without attributes. However, attribute information, node attributes in particular, allows a more in-depth exploration of the network structure. This paper extends the definition of a number of proximity measures to the case of attributed networks. To take node attributes into account, attribute similarity is embedded into the adjacency matrix. Obtained attribute-aware proximity measures are numerically studied in the context of community detection in real-world networks.

Laplacian integral graphs with a given degree sequence constraint

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Structural synthesis of plane kinematic chain inversions without detecting isomorphism

A plane kinematic chain inversion refers to a plane kinematic chain with one link fixed (assigned as the ground link). In the creative design of mechanisms, it is important to select proper ground links. The structural synthesis of plane kinematic chain inversions is helpful for improving the efficiency of mechanism design. However, the existing structural synthesis methods involve isomorphism detection, which is cumbersome. This paper proposes a simple and efficient structural synthesis method for plane kinematic chain inversions without detecting isomorphism. The fifth power of the adjacency matrix is applied to recognize similar vertices, and non-isomorphic kinematic chain inversions are directly derived according to non-similar vertices. This method is used to automatically synthesize 6-link 1-degree-of-freedom (DOF), 8-link 1-DOF, 8-link 3-DOF, 9-link 2-DOF, 9-link 4-DOF, 10-link 1-DOF, 10-link 3-DOF and 10-link 5-DOF plane kinematic chain inversions. All the synthesis results are consistent with those reported in literature. Our method is also suitable for other kinds of kinematic chains.

{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.