Estimating robustness in large social graphs

Abstract

Given a large social graph, what can we say about its robustness? Broadly speaking, the property of robustness is crucial in real graphs, since it is related to the structural behavior of graphs to retain their connectivity properties after losing a portion of their edges/nodes. Can we estimate a robustness index for a graph quickly? Additionally, if the graph evolves over time, how this property changes? In this work, we are trying to answer the above questions studying the expansion properties of large social graphs. First, we present a measure that characterizes the robustness properties of a graph and also serves as global measure of the community structure (or lack thereof). We show how to compute this measure efficiently by exploiting the special spectral properties of real-world networks. We apply our method on several diverse real networks with millions of nodes, and we observe interesting properties for both static and time-evolving social graphs. As an application example, we show how to spot outliers and anomalies in graphs over time. Finally, we examine how graph generating models that mimic several properties of real-world graphs and behave in terms of robustness dynamics.

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Notes

  1. 1.

    This property simply means that the \(\sinh (\cdot )\) function retains the signs of the eigenvalues.

  2. 2.

    www.imdb.com.

  3. 3.

    Personal communication with Michael Ley and Florian Reitz from DBLP.

  4. 4.

    The bipartite graphs do not have odd length closed walks, and thus, the \(\mathrm{SC}\) is computed based on the even length closed walks. This happens replacing the \(\sinh (\cdot )\) function with the \(\cosh (\cdot )\) [15]. But then the \(\mathrm{SC}\) for the bipartite graphs cannot be efficiently approximated using similar ideas with the proposed \(\mathrm{NSC}_k\) (Sect. 4), because of the fact that the \(\cosh (\cdot )\) is an even function. However, our approach for bipartite graphs (Sect. 4, Proposition 4.1) overcomes this bottleneck and can be efficiently computed for large-scale graphs.

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Acknowledgments

Fragkiskos D. Malliaros is a recipient of the Google Europe Fellowship in Graph Mining, and this research is supported in part by this Google Fellowship. Vasileios Megalooikonomou is partially supported by the ARMOR Project (FP7-ICT-2011-5.1-287720) that is co-funded by the European Commission under the Seventh Framework Programme and by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the NSRF—Research Funding Program: Thales. Investing in knowledge society through the European Social Fund. Christos Faloutsos is supported by the National Science Foundation under Grants No. IIS-1217559 CNS-1314632, by the Army Research Laboratory under Cooperative Agreement Number W911NF-09-2-0053 and under Contract Number W911NF-11-C-0088, by an IBM Faculty Award and a Google Focused Research Award. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation, or other funding parties. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.

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Appendix

Appendix

In this Appendix, we provide a more detailed description of how the property of large spectral gap along with the subgraph centrality measure leads to the measure \(\xi (G)\) [17] as presented in Sect. 3. First of all, the subgraph centrality measure is defined as [15]

$$\begin{aligned} \mathrm{SC}(i) = \sum _{\ell = 0}^{\infty } \dfrac{A^\ell _{ii}}{\ell !},\quad \forall i \in V, \end{aligned}$$
(5)

where the diagonal entry \(A_{ii}\) of the matrix \(\mathbf {A}^\ell \) contains the number of closed walks of length \(\ell \) that begin and end at the same node \(i\). Focusing on unipartite graphs and keeping only the odd length closed walksFootnote 4 In order to avoid cycles in acyclic graphs, the \(\mathrm{SC}\) can be expressed as

$$\begin{aligned} \mathrm{SC}(i) = u_{i1}^2 \sinh (\lambda _1) + \sum _{j=2}^{|V|} u_{ij}^2 \sinh (\lambda _j). \end{aligned}$$
(6)

If the graph has good expansion properties (and thus high robustness), it means that \(\lambda _1 \gg \lambda _2\), and then \( u_{i1}^2 \sinh (\lambda _1) \gg \sum _{j=2}^{|V|} u_{ij}^2 \sinh (\lambda _j)\). Thus, Eq. (6) could be written as

$$\begin{aligned} \mathrm{SC}(i) \approx u_{i1}^2 \sinh (\lambda _1), ~ \forall i \in V. \end{aligned}$$
(7)

This means that for graphs with high robustness, the principal eigenvector \(u_{i1}\) will be related to \(\mathrm{SC}(i)\) as

$$\begin{aligned} u_{i1} \propto \sinh ^{-1/2}(\lambda _1) ~ \mathrm{SC}(i)^{1/2}. \end{aligned}$$
(8)

This relation suggests that if the graph shows high robustness, \(u_{i1}\) will be proportional to \(\mathrm{SC}(i)\) and a log–log plot of \(u_{i1}\) versus \(\mathrm{SC}(i), ~ \forall i \in V\) will show a linear fit with slope \(1/2\) (the discrepancy plot).

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Malliaros, F.D., Megalooikonomou, V. & Faloutsos, C. Estimating robustness in large social graphs. Knowl Inf Syst 45, 645–678 (2015). https://doi.org/10.1007/s10115-014-0810-7

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Keywords

  • Network robustness
  • Expansion properties
  • Temporal evolution
  • Graph generating models
  • Social network analysis
  • Graph mining