Evolutional Codes: Novel Efficient Graph Data Representation for Mobile Edge Computing

To transmit big graph data emerging from prevalent social networks and biological networks over mobile devices has become challenging nowadays. Since graphs are usually represented by the associated adjacency matrices or graph data structures, dynamic node (vertex) re-ordering to reflect the nodes' real relations (for example, spatial or structural relations) is desirable in practice but impossible for these two existing graph-representation methods. Lately, we proposed a novel evolutional coding technique, which can restructure graph data dynamically, flexibly, and efficiently for transmission among mobile devices. Memory efficiency, in terms of number of memory storage units, is a crucial performance measure for mobile edge computing. To select the optimal graph-representation strategy for arbitrary graph data, we define a new graph-taxonomy metric dependent on the edge density (edges per node) in this work. According to our newly defined graph-taxonomy metric, one can choose the optimal graph-representation solution among the three aforementioned methods in terms of required memory-storage space for any application. Pertinent theorems and proofs for the underlying comparative studies on graph representation are presented in this article.

I. Introduction

A graph is often employed to interpret the relationships among a set of objects. These objects are called nodes or vertices and the relationship of any pair of nodes is referred to as an edge or a link. Many complex networks, including social networks [1], [2], [3], [4], biological networks [5], [6], finite state machines [7], [8], and molecular structures [9], [10] can be modeled as graphs. The most common way to represent and store graphs is the adjacency matrix formation [11], [12], [13], [14], [15], [16]. Adjacency matrix is a convenient mathematical tool for representing the relationship (as an edge) between each pair of nodes. However, it would incur redundant memory storage containing many zero entries if a graph has only few edges [17]. In other words, such an adjacency matrix is a sparse matrix (i.e., it contains few nontrivial entries). Moreover, when a sparse adjacency matrix (graph) is involved in a linear system of equations (as appearing in many attribute association problems), intermediate redundant computations related to the zero entries could be avoided. Another way to represent and store graphs is the graph data structure (or adjacency list) [18], [19], [20]. In contrast to the adjacency matrix formation, the graph data structure only stores nontrivial entries. However, if a graph contains numerous edges, i.e., it is a dense graph, it is computationally cumbersome to partition a subgraph (to extract a node-centric subgraph) through the graph data structure [21], [22], [23].