A New Methodology for Generalizing Unweighted Network Measures

Several important complex network measures that helped discovering common patterns across real-world networks ignore edge weights, an important information in real-world networks. We propose a new methodology for generalizing measures of unweighted networks through a generalization of the cardinality concept of a set of weights. The key observation here is that many measures of unweighted networks use the cardinality (the size) of some subset of edges in their computation. For example, the node degree is the number of edges incident to a node. We define the effective cardinality, a new metric that quantifies how many edges are effectively being used, assuming that an edge's weight reflects the amount of interaction across that edge. We prove that a generalized measure, using our method, reduces to the original unweighted measure if there is no disparity between weights, which ensures that the laws that govern the original unweighted measure will also govern the generalized measure when the weights are equal. We also prove that our generalization ensures a partial ordering (among sets of weighted edges) that is consistent with the original unweighted measure, unlike previously developed generalizations. We illustrate the applicability of our method by generalizing four unweighted network measures. As a case study, we analyze four real-world weighted networks using our generalized degree and clustering coefficient. The analysis shows that the generalized degree distribution is consistent with the power-law hypothesis but with steeper decline and that there is a common pattern governing the ratio between the generalized degree and the traditional degree. The analysis also shows that nodes with more uniform weights tend to cluster with nodes that also have more uniform weights among themselves.Comment: 23 pages, 10 figure

Assessing Code Authorship: The Case of the Linux Kernel

Code authorship is a key information in large-scale open source systems. Among others, it allows maintainers to assess division of work and identify key collaborators. Interestingly, open-source communities lack guidelines on how to manage authorship. This could be mitigated by setting to build an empirical body of knowledge on how authorship-related measures evolve in successful open-source communities. Towards that direction, we perform a case study on the Linux kernel. Our results show that: (a) only a small portion of developers (26 %) makes significant contributions to the code base; (b) the distribution of the number of files per author is highly skewed --- a small group of top authors (3 %) is responsible for hundreds of files, while most authors (75 %) are responsible for at most 11 files; (c) most authors (62 %) have a specialist profile; (d) authors with a high number of co-authorship connections tend to collaborate with others with less connections.Comment: Accepted at 13th International Conference on Open Source Systems (OSS). 12 page

The Routing of Complex Contagion in Kleinberg's Small-World Networks

In Kleinberg's small-world network model, strong ties are modeled as deterministic edges in the underlying base grid and weak ties are modeled as random edges connecting remote nodes. The probability of connecting a node $u$ with node $v$ through a weak tie is proportional to $1/|uv|^\alpha$, where $|uv|$ is the grid distance between $u$ and $v$ and $\alpha\ge 0$ is the parameter of the model. Complex contagion refers to the propagation mechanism in a network where each node is activated only after $k \ge 2$ neighbors of the node are activated. In this paper, we propose the concept of routing of complex contagion (or complex routing), where we can activate one node at one time step with the goal of activating the targeted node in the end. We consider decentralized routing scheme where only the weak ties from the activated nodes are revealed. We study the routing time of complex contagion and compare the result with simple routing and complex diffusion (the diffusion of complex contagion, where all nodes that could be activated are activated immediately in the same step with the goal of activating all nodes in the end). We show that for decentralized complex routing, the routing time is lower bounded by a polynomial in $n$ (the number of nodes in the network) for all range of $\alpha$ both in expectation and with high probability (in particular, $\Omega(n^{\frac{1}{\alpha+2}})$ for $\alpha \le 2$ and $\Omega(n^{\frac{\alpha}{2(\alpha+2)}})$ for $\alpha > 2$ in expectation), while the routing time of simple contagion has polylogarithmic upper bound when $\alpha = 2$. Our results indicate that complex routing is harder than complex diffusion and the routing time of complex contagion differs exponentially compared to simple contagion at sweetspot.Comment: Conference version will appear in COCOON 201

Graph Metrics for Temporal Networks

Temporal networks, i.e., networks in which the interactions among a set of elementary units change over time, can be modelled in terms of time-varying graphs, which are time-ordered sequences of graphs over a set of nodes. In such graphs, the concepts of node adjacency and reachability crucially depend on the exact temporal ordering of the links. Consequently, all the concepts and metrics proposed and used for the characterisation of static complex networks have to be redefined or appropriately extended to time-varying graphs, in order to take into account the effects of time ordering on causality. In this chapter we discuss how to represent temporal networks and we review the definitions of walks, paths, connectedness and connected components valid for graphs in which the links fluctuate over time. We then focus on temporal node-node distance, and we discuss how to characterise link persistence and the temporal small-world behaviour in this class of networks. Finally, we discuss the extension of classic centrality measures, including closeness, betweenness and spectral centrality, to the case of time-varying graphs, and we review the work on temporal motifs analysis and the definition of modularity for temporal graphs.Comment: 26 pages, 5 figures, Chapter in Temporal Networks (Petter Holme and Jari Saram\"aki editors). Springer. Berlin, Heidelberg 201

Detecting rich-club ordering in complex networks

Uncovering the hidden regularities and organizational principles of networks arising in physical systems ranging from the molecular level to the scale of large communication infrastructures is the key issue for the understanding of their fabric and dynamical properties [1-5]. The ``rich-club'' phenomenon refers to the tendency of nodes with high centrality, the dominant elements of the system, to form tightly interconnected communities and it is one of the crucial properties accounting for the formation of dominant communities in both computer and social sciences [4-8]. Here we provide the analytical expression and the correct null models which allow for a quantitative discussion of the rich-club phenomenon. The presented analysis enables the measurement of the rich-club ordering and its relation with the function and dynamics of networks in examples drawn from the biological, social and technological domains.Comment: 1 table, 3 figure

Outlier Edge Detection Using Random Graph Generation Models and Applications

Outliers are samples that are generated by different mechanisms from other normal data samples. Graphs, in particular social network graphs, may contain nodes and edges that are made by scammers, malicious programs or mistakenly by normal users. Detecting outlier nodes and edges is important for data mining and graph analytics. However, previous research in the field has merely focused on detecting outlier nodes. In this article, we study the properties of edges and propose outlier edge detection algorithms using two random graph generation models. We found that the edge-ego-network, which can be defined as the induced graph that contains two end nodes of an edge, their neighboring nodes and the edges that link these nodes, contains critical information to detect outlier edges. We evaluated the proposed algorithms by injecting outlier edges into some real-world graph data. Experiment results show that the proposed algorithms can effectively detect outlier edges. In particular, the algorithm based on the Preferential Attachment Random Graph Generation model consistently gives good performance regardless of the test graph data. Further more, the proposed algorithms are not limited in the area of outlier edge detection. We demonstrate three different applications that benefit from the proposed algorithms: 1) a preprocessing tool that improves the performance of graph clustering algorithms; 2) an outlier node detection algorithm; and 3) a novel noisy data clustering algorithm. These applications show the great potential of the proposed outlier edge detection techniques.Comment: 14 pages, 5 figures, journal pape

Network 'small-world-ness': a quantitative method for determining canonical network equivalence

Background: Many technological, biological, social, and information networks fall into the broad class of 'small-world' networks: they have tightly interconnected clusters of nodes, and a shortest mean path length that is similar to a matched random graph (same number of nodes and edges). This semi-quantitative definition leads to a categorical distinction ('small/not-small') rather than a quantitative, continuous grading of networks, and can lead to uncertainty about a network's small-world status. Moreover, systems described by small-world networks are often studied using an equivalent canonical network model-the Watts-Strogatz (WS) model. However, the process of establishing an equivalent WS model is imprecise and there is a pressing need to discover ways in which this equivalence may be quantified. Methodology/Principal Findings: We defined a precise measure of 'small-world-ness' S based on the trade off between high local clustering and short path length. A network is now deemed a 'small-world' if S. 1-an assertion which may be tested statistically. We then examined the behavior of S on a large data-set of real-world systems. We found that all these systems were linked by a linear relationship between their S values and the network size n. Moreover, we show a method for assigning a unique Watts-Strogatz (WS) model to any real-world network, and show analytically that the WS models associated with our sample of networks also show linearity between S and n. Linearity between S and n is not, however, inevitable, and neither is S maximal for an arbitrary network of given size. Linearity may, however, be explained by a common limiting growth process. Conclusions/Significance: We have shown how the notion of a small-world network may be quantified. Several key properties of the metric are described and the use of WS canonical models is placed on a more secure footing

Community Aliveness: Discovering Interaction Decay Patterns in Online Social Communities

Online Social Communities (OSCs) provide a medium for connecting people, sharing news, eliciting information, and finding jobs, among others. The dynamics of the interaction among the members of OSCs is not always growth dynamics. Instead, a $\textit{decay}$ or $\textit{inactivity}$ dynamics often happens, which makes an OSC obsolete. Understanding the behavior and the characteristics of the members of an inactive community help to sustain the growth dynamics of these communities and, possibly, prevents them from being out of service. In this work, we provide two prediction models for predicting the interaction decay of community members, namely: a Simple Threshold Model (STM) and a supervised machine learning classification framework. We conducted evaluation experiments for our prediction models supported by a $\textit{ground truth}$ of decayed communities extracted from the StackExchange platform. The results of the experiments revealed that it is possible, with satisfactory prediction performance in terms of the F1-score and the accuracy, to predict the decay of the activity of the members of these communities using network-based attributes and network-exogenous attributes of the members. The upper bound of the prediction performance of the methods we used is $0.91$ and $0.83$ for the F1-score and the accuracy, respectively. These results indicate that network-based attributes are correlated with the activity of the members and that we can find decay patterns in terms of these attributes. The results also showed that the structure of the decayed communities can be used to support the alive communities by discovering inactive members.Comment: pre-print for the 4th European Network Intelligence Conference - 11-12 September 2017 Duisburg, German

Worldwide food recall patterns over an eleven month period: A country perspective.

<p>Abstract</p> <p>Background</p> <p>Following the World Health Organization Forum in November 2007, the Beijing Declaration recognized the importance of food safety along with the rights of all individuals to a safe and adequate diet. The aim of this study is to retrospectively analyze the patterns in food alert and recall by countries to identify the principal hazard generators and gatekeepers of food safety in the eleven months leading up to the Declaration.</p> <p>Methods</p> <p>The food recall data set was collected by the Laboratory of the Government Chemist (LGC, UK) over the period from January to November 2007. Statistics were computed with the focus reporting patterns by the 117 countries. The complexity of the recorded interrelations was depicted as a network constructed from structural properties contained in the data. The analysed network properties included degrees, weighted degrees, modularity and <it>k</it>-core decomposition. Network analyses of the reports, based on 'country making report' (<it>detector</it>) and 'country reported on' (<it>transgressor</it>), revealed that the network is organized around a dominant core.</p> <p>Results</p> <p>Ten countries were reported for sixty per cent of all faulty products marketed, with the top 5 countries having received between 100 to 281 reports. Further analysis of the dominant core revealed that out of the top five transgressors three made no reports (in the order China > Turkey > Iran). The top ten detectors account for three quarters of reports with three > 300 (Italy: 406, Germany: 340, United Kingdom: 322).</p> <p>Conclusion</p> <p>Of the 117 countries studied, the vast majority of food reports are made by 10 countries, with EU countries predominating. The majority of the faulty foodstuffs originate in ten countries with four major producers making no reports. This pattern is very distant from that proposed by the Beijing Declaration which urges all countries to take responsibility for the provision of safe and adequate diets for their nationals.</p