Bound and exact methods for assessing link vulnerability in complex networks

Abstract

Assessing network systems for failures is critical to mitigate the risk and develop proactive responses. In this paper, we investigate devastating consequences of link failures in networks. We propose an exact algorithm and a spectral lower-bound on the minimum number of removed links to incur a significant level of disruption. Our exact solution can identify optimal solutions in both uniform and weighted networks through solving a well-constructed mixed integer program. Also, our spectral lower-bound derives from the Laplacian eigenvalues an estimation on the vulnerability of large networks that are intractable for exact methods. Through experiments on both synthetic and real-world networks, we demonstrate the efficiency of the proposed methods.

Appendix

1.1 Proof of Lemma 2 from Bissias (2010)

Let $\varvec{H}$ be an $n \times k$ indicator matrix where

$$\begin{aligned} H_{ij} = \left\{ \begin{matrix} 1, &{} \text{ vertex } i \text{ in } j\mathrm{th}\; \text{ subset } \text{(or } \text{ component) }\\ 0, &{} \text{ otherwise } \end{matrix} \right. \end{aligned}$$

and $\varvec{h}_j$ denote the $j$th column of $\varvec{H}$, i.e., the membership vector for the $j$th subset.

We have

$$\begin{aligned} \varvec{h}_j^T \times \varvec{h}_j = s_j, \end{aligned}$$

and the total weights of edges going out of the $j$th subset is given by

$$\begin{aligned} \varvec{h}_j^T \varvec{L} h_j. \end{aligned}$$

Therefore

$$\begin{aligned} E_{cut} = \frac{1}{2}\sum _{j=1}^k \varvec{h}_j^T \varvec{L} h_j. \end{aligned}$$

Since $\varvec{L}$ is symmetric and positive semidefinite, it has $n$ real non-negative eigenvalues $\lambda _1 \le \lambda _2 \le \cdots \le \lambda _n$, and the corresponding eigenvectors $u_1, u_2, \ldots , u_n$ form a complete orthonormal basis, i.e., $\varvec{L} = \sum _{i=1}^n \lambda _i \varvec{u}_i \varvec{u}_i^T$.

Hence

$$\begin{aligned} E_{cut}&= \frac{1}{2}\sum _{j=1}^k \varvec{h}_j^T \left( \sum _{i=1}^n \lambda _i \varvec{u}_i \varvec{u}_i^T\right) \varvec{h}_j\end{aligned}$$

(34)

$$\begin{aligned}&= \frac{1}{2}\sum _{j=1}^k \sum _{i=1}^n \lambda _i \left( \varvec{u}_i^T \varvec{h}_j \right) ^2 \end{aligned}$$

(35)

Let $x_{ij} = \left( \varvec{u}_i^T \varvec{h}_j \right) ^2/m_j$. Substituting $x_{ij}$, we have

$$\begin{aligned} E_{cut} = \frac{1}{2}\sum _{j=1}^k m_j \sum _{i=1}^n \lambda _i x_{ij} \end{aligned}$$

(36)

We can verify that

$$\begin{aligned} \sum _{i=1}^n x_{ij} = 1 \text{ and } \sum _{j=1}^k x_{ij} \le 1 \end{aligned}$$

Since $\lambda _1$ is the smallest and $m_1$ is the largest, the Eq. (36) is minimized when $x_{ii}=1$ and $x_{ij} = 0\ \forall i \ne j$, i.e.,

$$\begin{aligned} E_{cut}&= \frac{1}{2}\sum _{j=1}^k \sum _{i=1}^n \lambda _i \left( \varvec{u}_i^T \varvec{h}_j \right) ^2 \ge \frac{1}{2}\sum _{j=1}^k \lambda _j m_j. \end{aligned}$$

1.2 Dynamic programming algorithm (ILB)

We present a dynamic programming algorithm to compute the exact solution of the QP (14–17).

For $k \le l \le n$ and $p \le \beta {n \atopwithdelims ()2}$, define $\mathcal {L}_k(l, p)$ to be the minimum spectral bound obtained by first $k$ subsets that the total sizes is $l$ and the total pairwise connectivity is at most $p$. That is

$$\begin{aligned} \mathcal {L}_k(l, p)= \displaystyle \min _{\varvec{s}^{(k)} \in \mathbb N^k} \left\{ \varvec{s}^{(k) T} \varvec{\lambda }^{(k)}\ : \ \Vert \varvec{s}^{(k)}\Vert _1 = l,\ \sum _{i=1}^{k} {s_i \atopwithdelims ()2}\le p \right\} , \end{aligned}$$

Then the optimal objective value QP (14–17) shall be given by $Q_\beta = \mathcal {L}_{n}(n, \beta {n \atopwithdelims ()2})$.

By Lemma 3, we pay attention only to partitions satisfying $s_1 \ge s_2 \ge \ldots \ge s_n$. We now derive the recursive formula for $\mathcal {L}_p(l, k)$ based on the sub-optimal structure of the QP problem. Consider two possible cases of $s_k$

• $s_k = 0$ There are at most $k-1$ partitions whose sizes sum up to $l$. Hence, for this case $\mathcal {L}(l, k)=\mathcal {L}_{k-1}(l, p)$.

• $s_k > 0$ Since $s_1 \ge s_2 \ge \ldots \ge s_k > 0$. Let $\tilde{s}_i = s_i -1 \ge 0$, the vector $\tilde{s}=\{ \tilde{s}_1, \tilde{s}_2,\ldots ,\tilde{s}_k\}$ satisfies simultaneously the following

  $$\begin{aligned} \sum _{i=1}^k \lambda _i \tilde{s}_i&= \sum _{i=1}^k \lambda _i s_i - \sum _{i=1}^{k} \lambda _i \\ \sum _{i=1}^k \tilde{s}_i&= \sum _{i=1}^k s_i - k = l - k\\ \sum _{i=1}^k { \tilde{s}_i \atopwithdelims ()2}&= \sum _{i=1}^k \left[ { s_i \atopwithdelims ()2} - s_i+1 \right] \\&= \sum _{i=1}^k { s_i \atopwithdelims ()2} - l + k\le p - l + k \end{aligned}$$

  Therefore, in this case $\mathcal {L}_k(l, p)= \mathcal {L}_k(l - k, p - l + k) + \sum _{i=1}^k \lambda _i$ In summary, we have

  $$\begin{aligned} \mathcal {L}_k(l, p)= \min \left\{ \begin{array}{l} \mathcal {L}_{k-1}(l, p),\\ \mathcal {L}_k(l - k, p - l + k) + \sum _{i=1}^k \lambda _i \end{array} \right\} \end{aligned}$$

We compute value of $\mathcal {L}_p(l, k)$ in increasing order of $p$ and $l$ but in decreasing order of $k$. The base cases for $\mathcal {L}_p(l, k)$ are as follow.

$$\begin{aligned} \mathcal {L}_k(l, p)= \left\{ \begin{array}{l@{\quad }l} +\infty ,&{} \text{ if } p < p_{\min }(l, k)\\ \lambda _1 l = 0,&{} \text{ if } p \ge p_{\max }(l, k) \end{array} \right. \end{aligned}$$

(37)

where $p_{\min }(l, k) = { \lceil l/k\rceil \atopwithdelims ()2 } (l \,\hbox {mod}\, k) + { \lfloor l/k\rfloor \atopwithdelims ()2 } (k - l \,\hbox {mod}\, k)$ and $p_{\max }(l, k) = {l \atopwithdelims ()2}$ that are the minimum and maximum pairwise connectivity of a graph with $l$ vertices and $k$ connected components, respectively.