Lyapunov-Based Anomaly Detection in Highly-Clustered Networks

Diego Ruiz; Jorge Finke

doi:10.1007/s10955-018-2089-7

Journal of Statistical Physics

August 2018, Volume 172, Issue 4, pp 1127–1146 | Cite as

Lyapunov-Based Anomaly Detection in Highly-Clustered Networks

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Diego Ruiz
Jorge Finke

Article

First Online: 21 June 2018

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Abstract

Network formation models explain the dynamics of the structure of connections using mechanisms that operate under different principles for establishing and removing edges. The Jackson–Rogers model is a generic framework that applies the principle of triadic closure to networks that grow by the addition of new nodes and new edges over time. Past work describes the limit distribution of the in-degree of the nodes based on a continuous-time approximation. Here, we introduce a discrete-time approach of the dynamics of the in- and out-degree distributions of a variation of the model. Furthermore, we characterize the limit distributions and the expected value of the average degree as equilibria, and prove that the equilibria are asymptotically stable. Finally, we use the stability properties of the model to propose a detection criterion for anomalies in the edge formation process.

Keywords

Network formation models Stability Anomalous event detection

Mathematics Subject Classification

05C82 37B25 94C12

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Notes

Acknowledgements

This research was supported in part by the Center of Excellence and Appropriation in Big Data and Data Analytics (CAOBf at the Pontificia Universidad Javeriana, the Ministry of Information Technologies and Telecommunications of Colombia (MinTIC), and the Colombian Administrative Department of Science, Technology and Innovation (COLCIENCIAS), under Grant No. FP44842-546-2015.

Appendix A: Proof of Lemma 1

Consider the following lemma.

Lemma 1

Let $\{s_t\}$ and $\{u_t\}$ be two equivalent sequences of positive real numbers, denoted by $s_t\sim u_t$. That is, $\lim _{t\rightarrow \infty }s_t/u_t=1$. If $\lim _{t\rightarrow \infty } u_t=L<\infty $, then $\lim _{t\rightarrow \infty } s_t=L$. Moreover, if $v_t\sim w_t$, then $s_t+v_t\sim u_t+w_t$.

Proof

First, if $s_t\sim u_t$, then for all $\varepsilon >0$ there exists $T_1\in {\mathbb {N}}$ such that for all $t>T_1$, $\left| \frac{s_t}{u_t}-1\right| <\varepsilon $. If $\lim _{t\rightarrow \infty } u_t=L$, then for all $\varepsilon >0$ there exists $T_2\in {\mathbb {N}}$ such for all $t>T_2$, $\left| u_t-L\right| <\varepsilon $. So

$$\begin{aligned} |s_t-L|&\le |s_t-u_t|+|u_t-L|\\&< |s_t-u_t|+\varepsilon \\&= |u_t|\left| \frac{s_t}{u_t}-1\right| +\varepsilon \\&< (\varepsilon +L)\varepsilon +\varepsilon =\varepsilon ^\prime \end{aligned}$$

Therefore, for all $\varepsilon ^\prime >0$, there exists $T=\max \{T_1,T_2\}$ such that for all $t>T$, $|s_t-L|<\varepsilon ^\prime $.

Second, because $\{s_t\},\{u_t\},\{v_t\}$ and $\{w_t\}$ are sequences of positive real numbers, if $\frac{s_t}{u_t}<\frac{v_t}{w_t}$, then

$$\begin{aligned} \frac{s_t}{u_t}<\frac{s_t+v_t}{u_t+w_t}<\frac{v_t}{w_t} \end{aligned}$$

Since $s_t\sim u_t$ and $v_t\sim w_t$, applying the Squeeze Theorem, we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{s_t+v_t}{u_t+w_t}=1 \end{aligned}$$

That is, $s_t+v_t\sim u_t+w_t$. The same reasoning applies for $\frac{v_t}{w_t}\le \frac{s_t}{u_t}$. $\square $

Appendix B: Proof of Lemma 2

Consider the following lemma.

Lemma 2

Consider the function $\rho :{\mathcal {X}}\times {\mathcal {X}}\rightarrow {\mathbb {R}}^+_0$ defined by

$$\begin{aligned} \rho (x,y)=\left| \sum _{k=0}^\infty (x(k)-y(k))\right| \end{aligned}$$

Moreover, define the following equivalence relation on ${\mathcal {X}}$. The state x is related to y if $\rho (x,y)=0$. Let $\llbracket x \rrbracket $ denote the equivalence class of x and ${\mathcal {X}}^*=\{\llbracket x \rrbracket :x\in {\mathcal {X}}\}$ the set of all equivalence classes. If $\rho ^*:{\mathcal {X}}^*\times {\mathcal {X}}^*\rightarrow {\mathbb {R}}^+_0$ is defined as $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=\rho (x,y)$, then $(\rho ^*,{\mathcal {X}}^*)$ is a metric space.

Proof

Let $w,x,y,z\in {\mathcal {X}}$. First, we show that $\rho $ is a pseudometric. In particular, $\rho (x,y)\ge 0$ and $\rho (x,y)=\rho (y,x)$. To verify that $\rho $ satisfies the triangle inequality, note that

$$\begin{aligned} \rho (x,y)&=\left| \sum _{k=0}^\infty (x(k)-y(k))\right| \\&=\left| \sum _{k=0}^\infty (x(k)-z(k))+\sum _{k=0}^\infty (z(k)-y(k))\right| \\&\le \rho (x,z)+\rho (z,y) \end{aligned}$$

In general, $x\ne y$ does not imply that $\rho (x,y)\ne 0$ (i.e., $\rho $ is a pseudometric). Second, we show that for the equivalence relation over ${\mathcal {X}}$, $\rho ^*$ is well-defined, that is, if $(\llbracket x \rrbracket ,\llbracket y \rrbracket )=(\llbracket z \rrbracket ,\llbracket w \rrbracket )$ then $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=\rho ^*(\llbracket z \rrbracket ,\llbracket w \rrbracket )$. In particular, if $(\llbracket x \rrbracket ,\llbracket y \rrbracket )=(\llbracket z \rrbracket ,\llbracket w \rrbracket )$, then $\llbracket x \rrbracket =\llbracket z \rrbracket $ and $\llbracket y \rrbracket =\llbracket w \rrbracket $. Because $\rho $ satisfies the triangle inequality, we have $\rho (x,y)\le \rho (z,w)$ and $\rho (z,w)\le \rho (x,y)$. That is, $\rho (x,y)=\rho (z,w)$, which implies that $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=\rho ^*(\llbracket z \rrbracket ,\llbracket w \rrbracket )$.

Finally, we verify sufficient conditions for $(\rho ^*,{\mathcal {X}}^*)$ to be a metric space. Let $\llbracket x \rrbracket ,\llbracket y \rrbracket ,\llbracket z \rrbracket \in {\mathcal {X}}^*$. In particular, since $\rho $ is a pseudometric, $\rho ^*$ satisfies:

(1)

For $\llbracket x \rrbracket \ne \llbracket y \rrbracket $, we know that $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=\rho (x,y)=\left| \sum _{k=0}^\infty (x(k)-y(k))\right| >0$.
(2)

For $x\in \llbracket x \rrbracket $ and $y\in \llbracket y \rrbracket $, $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=0$ if and only if $\rho (x,y)=0$, that is
$$\begin{aligned} \left| \sum _{k=0}^\infty (x(k)-y(k))\right| =0 \end{aligned}$$
This implies that $y\in \llbracket x \rrbracket $ and $x\in \llbracket y \rrbracket $. Therefore, $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=0$ if and only if $\llbracket x \rrbracket =\llbracket y \rrbracket $.
(3)

For $x,y\in {\mathcal {X}}$, we know that $\rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )=\rho ^*(\llbracket y \rrbracket ,\llbracket x \rrbracket )$ because $\rho (x,y)=\rho (y,x)$.
(4)

For $x,y,z\in {\mathcal {X}}$
$$\begin{aligned} \rho ^*(\llbracket x \rrbracket ,\llbracket y \rrbracket )&=\rho (x,y)\\&\le \rho (x,z)+\rho (z,y)\\&=\rho ^*(\llbracket x \rrbracket ,\llbracket z \rrbracket )+\rho ^*(\llbracket z \rrbracket ,\llbracket y \rrbracket ) \end{aligned}$$

$\square $

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Authors and Affiliations

Diego Ruiz
- 1
- 2
Email authorView author's OrcID profile
Jorge Finke
- 3

1.Universidad del CaucaPopayánColombia
2.Universidad del ValleCaliColombia
3.Pontificia Universidad JaverianaCaliColombia

Lyapunov-Based Anomaly Detection in Highly-Clustered Networks

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