Pedro Manuel Pinto Ribeiro

g-tries: an efficient data structure for discovering network motifs

In this paper we propose a novel specialized data structure that we call g-trie, designed to deal with collections of subgraphs. The main conceptual idea is akin to a prefix tree in the sense that we take advantage of common topology by constructing a multiway tree where the descendants of a node share a common substructure. We give algorithms to construct a g-trie, to list all stored subgraphs, and to find occurrences on another graph of the subgraphs stored in the g-trie. We evaluate the implementation of this structure and its associated algorithms on a set of representative benchmark biological networks in order to find network motifs. To assess the efficiency of our algorithms we compare their performance with other known network motif algorithms also implemented in the same common platform. Our results show that indeed, g-tries are a feasible, adequate and very efficient data structure for network motifs discovery, clearly outperforming previous algorithms and data structures.

G-Tries: a data structure for storing and finding subgraphs

The ability to find and count subgraphs of a given network is an important non trivial task with multidisciplinary applicability. Discovering network motifs or computing graphlet signatures are two examples of methodologies that at their core rely precisely on the subgraph counting problem. Here we present the g-trie, a data-structure specifically designed for discovering subgraph frequencies. We produce a tree that encapsulates the structure of the entire graph set, taking advantage of common topologies in the same way a prefix tree takes advantage of common prefixes. This avoids redundancy in the representation of the graphs, thus allowing for both memory and computation time savings. We introduce a specialized canonical labeling designed to highlight common substructures and annotate the g-trie with a set of conditional rules that break symmetries, avoiding repetitions in the computation. We introduce a novel algorithm that takes as input a set of small graphs and is able to efficiently find and count them as induced subgraphs of a larger network. We perform an extensive empirical evaluation of our algorithms, focusing on efficiency and scalability on a set of diversified complex networks. Results show that g-tries are able to clearly outperform previously existing algorithms by at least one order of magnitude.

Comparison of Co-authorship Networks across Scientific Fields Using Motifs

Comparing scientific production across different fields of knowledge is commonly controversial and subject to disagreement. Such comparisons are often based on quantitative indicators, such as papers per researcher, and data normalization is very difficult to accomplish. Different approaches can provide new insight and in this paper we focus on the comparison of different scientific fields based on their research collaboration networks. We use co-authorship networks where nodes are researchers and the edges show the existing co-authorship relations between them. Our comparison methodology is based on network motifs, which are over represented patterns, or sub graphs. We derive motif fingerprints for 22 scientific fields based on 29 different small motifs found in the corresponding co-authorship networks. These fingerprints provide a metric for assessing similarity among scientific fields, and our analysis shows that the discrimination power of the 29 motif types is not identical. We use a co-authorship dataset built from over 15,361 publications inducing a co-authorship network with over 32,842 researchers. Our results also show that we can group different fields according to their fingerprints, supporting the notion that some fields present higher similarity and can be more easily compared.

Parallel discovery of network motifs

Many natural structures can be naturally represented by complex networks. Discovering network motifs, which are overrepresented patterns of inter-connections, is a computationally hard task related to graph isomorphism. Sequential methods are hindered by an exponential execution time growth when we increase the size of motifs and networks. In this article we study the opportunities for parallelism in existing methods and propose new parallel strategies that adapt and extend one of the most efficient serial methods known from the Fanmod tool. We propose both a master-worker strategy and one with distributed control, in which we employ a randomized receiver initiated methodology capable of providing dynamic load balancing during the whole computation process. Our strategies are capable of dealing both with exact and approximate network motif discovery. We implement and apply our algorithms to a set of representative networks and examine their scalability up to 128 processing cores. We obtain almost linear speedups, showcasing the efficiency of our proposed approach and are able to reach motif sizes that were not previously achievable using conventional serial algorithms.

Efficient subgraph frequency estimation with g-tries

Many biological networks contain recurring overrepresented elements, called network motifs. Finding these substructures is a computationally hard task related to graph isomorphism. G-Tries are an efficient data structure, based on multiway trees, capable of efficiently identifying common substructures in a set of subgraphs. They are highly successful in constraining the search space when finding the occurrences of those subgraphs in a larger original graph. This leads to speedups up to 100 times faster than previous methods that aim for exact and complete results. In this paper we present a new efficient sampling algorithm for subgraph frequency estimation based on g-tries. It is able to uniformly traverse a fraction of the search space, providing an accurate unbiased estimation of subgraph frequencies. Our results show that in the same amount of time our algorithm achieves better precision than previous methods, as it is able to sustain higher sampling speeds.

Efficient Parallel Subgraph Counting Using G-Tries

Finding and counting the occurrences of a collection of subgraphs within another larger network is a computationally hard problem, closely related to graph isomorphism. The subgraph count is by itself a very powerful characterization of a network and it is crucial for other important network measurements. G-tries are a specialized data-structure designed to store and search for subgraphs. By taking advantage of subgraph common substructure, g-tries can provide considerable speedups over previously used methods. In this paper we present a parallel algorithm based precisely on g-tries that is able to efficiently find and count subgraphs. The algorithm relies on randomized receiver-initiated dynamic load balancing and is able to stop its computation at any given time, efficiently store its search position, divide what is left to compute in two halfs, and resume from where it left. We apply our algorithm to several representative real complex networks from various domains and examine its scalability. We obtain an almost linear speedup up to 128 processors, thus allowing us to reach previously unfeasible limits. We showcase the multidisciplinary potential of the algorithm by also applying it to network motif discovery.

Towards a faster network-centric subgraph census

Determining the frequency of small subgraphs is an important computational task lying at the core of several graph mining methodologies, such as network motifs discovery or graphlet based measurements. In this paper we try to improve a class of algorithms available for this purpose, namely network-centric algorithms, which are based upon the enumeration of all sets of k connected nodes. Past approaches would essentially delay isomorphism tests until they had a finalized set of k nodes. In this paper we show how isomorphism testing can be done during the actual enumeration. We use a customized g-trie, a tree data structure, in order to encapsulate the topological information of the embedded subgraphs, identifying already known node permutations of the same subgraph type. With this we avoid redundancy and the need of an isomorphism test for each subgraph occurrence. We tested our algorithm, which we called FaSE, on a set of different real complex networks, both directed and undirected, showcasing that we indeed achieve significant speedups of at least one order of magnitude against past algorithms, paving the way for a faster network-centric approach.

Parallel Subgraph Counting for Multicore Architectures

Computing the frequency of small subgraphs on a large network is a computationally hard task. This is, however, an important graph mining primitive, with several applications, and here we present a novel multicore parallel algorithm for this task. At the core of our methodology lies a state-of-the-art data structure, the g-trie, which represents a collection of subgraphs and allows for a very efficient sequential search. Our implementation was done using Pthreads and can run on any multicore personal computer. We employ a diagonal work sharing strategy to dynamically and effectively divide work among threads during the execution. We assess the performance of our Pthreads implementation on a set of representative networks from various domains and with diverse topological features. For most networks, we obtain a speedup of over 50 for 64 cores and an almost linear speedup up to 32 cores, showcasing the flexibility and scalability of our algorithm. This paves the way for the usage of such counting algorithms on larger subgraph and network sizes without the obligatory access to a cluster.

Dynamic inference of social roles in information cascades

Nodes in complex networks inherently represent different kinds of functional or organizational roles. In the dynamic process of an information cascade, users play different roles in spreading the information: some act as seeds to initiate the process, some limit the propagation and others are in-between. Understanding the roles of users is crucial in modeling the cascades. Previous research mainly focuses on modeling users behavior based upon the dynamic exchange of information with neighbors. We argue however that the structural patterns in the neighborhood of nodes may already contain enough information to infer users’ roles, independently from the information flow in itself. To approach this possibility, we examine how network characteristics of users affect their actions in the cascade. We also advocate that temporal information is very important. With this in mind, we propose an unsupervised methodology based on ensemble clustering to classify users into their social roles in a network, using not only their current topological positions, but also considering their history over time. Our experiments on two social networks, Flickr and Digg, show that topological metrics indeed possess discriminatory power and that different structural patterns correspond to different parts in the process. We observe that user commitment in the neighborhood affects considerably the influence score of users. In addition, we discover that the cohesion of neighborhood is important in the blocking behavior of users. With this we can construct topological fingerprints that can help us in identifying social roles, based solely on structural social ties, and independently from nodes activity and how information flows.

Motif Mining in Weighted Networks

Unexpectedly frequent subgraphs, known as motifs, can help in characterizing the structure of complex networks. Most of the existing methods for finding motifs are designed for unweighted networks, where only the existence of connection between nodes is considered, and not their strength or capacity. However, in many real world networks, edges contain more information than just simple node connectivity. In this paper, we propose a new method to incorporate edge weight information in motif mining. We think of a motif as a subgraph that contains unexpected information, and we define a new significance measurement to assess this subgraph exceptionality. The proposed metric embeds the weight distribution in subgraphs and it is based on weight entropy. We use the g-trie data structure to find instances of k-sized subgraphs and to calculate its significance score. Following a statistical approach, the random entropy of subgraphs is then calculated, avoiding the time consuming step of random network generation. The discrimination power of the derived motif profile by the proposed method is assessed against the results of the traditional unweighted motifs through a graph classification problem. We use a set of labeled ego networks of co-authorship in the biology and mathematics fields. The new proposed method is shown to be feasible, achieving even slightly better accuracy. Since it does not require the generation of random networks, it is also computationally faster, and because we are able to use the weight information in computing the motif importance, we can avoid converting weighted networks into unweighted ones.