Titre : Title : Graphlet characteristics in directed networks Auteurs

Graphlet analysis is part of network theory that does not depend on the choice of the network null model and can provide comprehensive description of the local network structure. Here, we propose a novel method for graphlet-based analysis of directed networks by computing first the signature vector for every vertex in the network and then the graphlet correlation matrix of the network. This analysis has been applied to brain effective connectivity networks by considering both direction and sign (inhibitory or excitatory) of the underlying directed (effective) connectivity. In particular, the signature vectors for brain regions and the graphlet correlation matrices of the brain effective network are computed for 40 healthy subjects and common dependencies are revealed. We found that the signature vectors (node, wedge, and triangle degrees) are dominant for the excitatory effective brain networks. Moreover, by considering only those correlations (or anti correlations) in the correlation matrix that are significant (>0.7 or <−0.7) and are presented in more than 60% of the subjects, we found that excitatory effective brain networks show stronger causal (measured with Granger causality) patterns (G-causes and G-effects) than inhibitory effective brain networks. The complexity of systems is frequently the result of non-trivial local connectivity and interaction of its constituents parts. A number of network structural characteristics have recently been the subject of particularly intense research, including degree distributions 1 , community structure 2,3 , and various measures of vertex cen-trality 4,5 , to mention only a few. Vertices may have attributes associated with them; for example, properties of proteins in protein-protein interaction networks 6 , users' social network profiles 7 , or authors' publication histories in co-authorship networks 8. Two approaches that focus on the local connectivity of subgraphs within a network are Motifs and Graphlets. Motifs are defined as sub-graphs that repeat frequently in the networks i.e they repeat at frequency higher than in the random graphs 9,10 , and they depend on the choice of the network's null model. In contrast, graphlets are induced sub-graphs of a network that appear at any frequency and hence are independent of a null model. They have been introduced recently 11 and they have found numerous applications as building blocks of network analysis in various disciplines ranging from social science 12,13 to biology 14,15. In social science, graphlet analysis (known as sub-graph census) is widely adopted in sociometric studies 12. Much of the work in this vein focused on analyzing triadic tendencies as important structural features of social networks (e.g., transi-tivity or triadic closure) as well as analyzing triadic configurations as the basis for various social network theories (e.g., social balance, strength of weak ties, stability of ties, or trust 16). In biology graphlets were used to infer protein structure 17 , to compare biological networks 14,15 , and to characterize the relationship between disease and structure of networks 18. Many of the real-world networks are directed, but until now no method has been proposed based on graphlets that can provide information about local structure of directed networks. Here, we offer a graphlet-based approach for analysis of the local structure of a directed network. In the method proposed in this manuscript, we compute for each vertex, a vector of structural features, called signature vector, based on the number of graphlets associated with the vertex, and for the network its graphlet correlation matrix, measuring graphlet dependencies which reveal unknown organizational principles of the network. We applied the technique to brain effective networks of 40 healthy subjects, and we found that many of the subjects share similar patterns in their network's local structure. In brain networks a node is associated with different types of elements, depending on the level of interest in the brain, and an edge represents the connection or interaction between two elements 19. If the brain is studied on

Scientific Reports

The characterization of topology is crucial in understanding network evolution and behavior. This paper presents an innovative approach, the GHuST framework to describe complex-network topology from graphlet decomposition. This new framework exploits the local information provided by graphlets to give a global explanation of network topology. The GHuST framework is comprised of 12 metrics that analyze how 2- and 3-node graphlets shape the structure of networks. The main strengths of the GHuST framework are enhanced topological description, size independence, and computational simplicity. It allows for straight comparison among different networks disregarding their size. It also reduces the complexity of graphlet counting, since it does not use 4- and 5-node graphlets. The application of the novel framework to a large set of networks shows that it can classify networks of distinct nature based on their topological properties. To ease network classification and enhance the graphical r...

Scientific Reports, 2016

We are flooded with large-scale, dynamic, directed, networked data. Analyses requiring exact comparisons between networks are computationally intractable, so new methodologies are sought. To analyse directed networks, we extend graphlets (small induced sub-graphs) and their degrees to directed data. Using these directed graphlets, we generalise state-of-the-art network distance measures (RGF, GDDA and GCD) to directed networks and show their superiority for comparing directed networks. Also, we extend the canonical correlation analysis framework that enables uncovering the relationships between the wiring patterns around nodes in a directed network and their expert annotations. On directed World Trade Networks (WTNs), our methodology allows uncovering the core-broker-periphery structure of the WTN, predicting the economic attributes of a country, such as its gross domestic product, from its wiring patterns in the WTN for up-to ten years in the future. It does so by enabling us to tr...

Bioinformatics, 2007

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&#39;s local structure ...

Bioinformatics, 2006

Motivation: Algorithmic and modeling advances in the area of protein–protein interaction (PPI) network analysis could contribute to the understanding of biological processes. Local structure of networks can be measured by the frequency distribution of graphlets, small connected non-isomorphic induced subgraphs. This measure of local structure has been used to show that high-confidence PPI networks have local structure of geometric random graphs. Finding graphlets exhaustively in a large network is computationally intensive. More complete PPI networks, as well as PPI networks of higher organisms, will thus require efficient heuristic approaches. Results: We propose two efficient and scalable heuristics for finding graphlets in high-confidence PPI networks. We show that both PPI and their model geometric random networks, have defined boundaries that are sparser than the ‘inner parts’ of the networks. In addition, these networks exhibit ‘uniformity’ of local structure inside the networ...

2020

Networks are fundamental to the study of complex systems, ranging from social contacts, message transactions, to biological regulations and economical networks. In many realistic applications, these networks may vary over time. Modeling and analyzing such temporal properties is of additional interest as it can provide a richer characterization of relations between nodes in networks. In this paper, we explore the role of graphlets in network classification for both static and temporal networks. Graphlets are small non-isomorphic induced subgraphs representing connected patterns in a network and their frequency can be used to assess network structures. We show that graphlet features, which are not captured by state-of-the-art methods, play a significant role in enhancing the performance of network classification. To that end, we propose two novel graphlet-based techniques, gl2vec for network embedding, and gl-DCNN for diffusion-convolutional neural networks. We demonstrate the efficac...

Applied Network Science, 2021

Complex networks that arise in nature, such as those from biochemistry, neurobiology, ecology, and engineering, exhibit some of the same, simple structures that occur with greater than expected frequency, known as "network motifs" (Milo et al. 2002). Network motifs refer to recurring, significant patterns of interaction between sets of nodes, or actors, and they represent the basic building blocks of graphs, typically through the overrepresentation of a particular substructure. Identifying such local network patterns among small numbers of graph nodes, or "graphlets, " has been particularly useful in providing insight into the functioning of basic, biological networks, including those in bacteria and yeasts (Alon 2007; Shenn-Orr et al. 2002) and food-web structures (Stouffer and Bascompte 2010).

The European Physical Journal B, 2010

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with interactions between at most k links. In this framework we investigate the dependencies between several observables commonly used to quantify structure in networks, such as the degree distribution, cluster and assortativity coefficients.

2012

We introduce the graphlet decomposition of a weighted network, which encodes a notion of social information based on social structure. We develop a scalable inference algorithm, which combines EM with Bron-Kerbosch in a novel fashion, for estimating the parameters of the model underlying graphlets using one network sample. We explore some theoretical properties of the graphlet decomposition, including computational complexity, redundancy and expected accuracy. We demonstrate graphlets on synthetic and real data. We analyze messaging patterns on Facebook and criminal associations in the 19th century.

2020

We develop graphlet analysis for multiplex networks and discuss how this analysis can be extended to multilayer and multilevel networks as well as to graphs with node and/or link categorical attributes. The analysis has been adapted for two typical examples of multiplexes: economic trade data represented as a 957-plex network and 75 social networks each represented as a 12-plex network. We show that wedges (open triads) occur more often in economic trade networks than in social networks, indicating the tendency of a country to produce/trade of a product in local structure of triads which are not closed. Moreover, our analysis provides evidence that the countries with small diversity tend to form correlated triangles. Wedges also appear in the social networks, however the dominant graphlets in social networks are triangles (closed triads). If a multiplex structure indicates a strong tie, the graphlet analysis provides another evidence for the concepts of strong/weak ties and structur...