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.

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 walks⁴ 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).