The multi-walker chain and its application in local community detection

Abstract

Local community detection (or local clustering) is of fundamental importance in large network analysis. Random walk-based methods have been routinely used in this task. Most existing random walk methods are based on the single-walker model. However, without any guidance, a single walker may not be adequate to effectively capture the local cluster. In this paper, we study a multi-walker chain (MWC) model, which allows multiple walkers to explore the network. Each walker is influenced (or pulled back) by all other walkers when deciding the next steps. This helps the walkers to stay as a group and within the cluster. We introduce two measures based on the mean and standard deviation of the visiting probabilities of the walkers. These measures not only can accurately identify the local cluster, but also help detect the cluster center and boundary, which cannot be achieved by the existing single-walker methods. We provide rigorous theoretical foundation for MWC and devise efficient algorithms to compute it. Extensive experimental results on a variety of real-world and synthetic networks demonstrate that MWC outperforms the state-of-the-art local community detection methods by a large margin.

Appendix A

1.1 A.1 The Proof of Theorem 1

Before proving Theorem 1, we first prove the following lemma. Without loss of generality, we use $W_k$ as an example and omit the subscript k in our discussion.

Lemma 1

For any walker in MWC, if ${\mathbf {P}}$ is stochastic, irreducible and aperiodic (SIA), then for any $\tau \ge 1$, ${\mathbb {P}}^{(1,\tau )}={\mathbb {P}}^{(1)}{\mathbb {P}}^{(2)}\ldots {\mathbb {P}}^{(\tau )}$ is also SIA.

Proof

Based on Eq. (5), for any $\tau \ge 1$, ${\mathbb {P}}^{(\tau )}$ is stochastic, so is ${\mathbb {P}}^{(1,\tau )}$ since the product of stochastic matrices is also stochastic.

Furthermore, we have

$$\begin{aligned} {\mathbb {P}}^{(1,\tau )}= & {} {\mathbb {P}}^{(1,\tau -1)}{\mathbb {P}}^{(\tau )}\\= & {} {\mathbb {P}}^{(1,\tau -1)}[\alpha {\mathbf {P}}+(1-\alpha )\mathbf {1(v}^{(\tau -1)})^\intercal ]\\= & {} \alpha {\mathbb {P}}^{(1,\tau -1)}{\mathbf {P}}+(1-\alpha )\mathbf {1(v}^{(\tau -1)})^\intercal \\= & {} \alpha ^\tau {\mathbf {P}}^\tau +\sum _{m=1}^{\tau -1}\alpha ^m(1-\alpha )\mathbf {1(v}^{(\tau -1-m)})^\intercal {\mathbf {P}}^m+(1-\alpha ) \mathbf {1(v}^{(\tau -1)})^\intercal \end{aligned}$$

Due to the last term, $(1-\alpha ) \mathbf {1(v}^{(\tau -1)})^\intercal$, we know that there is at least one column of ${\mathbb {P}}^{(1,\tau )}$ whose entries are all positive. Thus, ${\mathbb {P}}^{(1,\tau )}$ is irreducible (Please refer to Corollary 4 in [31]).

The self-loop represented by the diagonal of $(1-\alpha ) \mathbf {1(v}^{(\tau -1)})^\intercal$ can guarantee that ${\mathbb {P}}^{(1,\tau )}$ is aperiodic. $\square$

If T exists, ${\mathbb {P}}^{(\tau +1)}$=${\mathbb {P}}^{(\tau +T+1)}$ ($\tau \ge \tau _p$), since ${\mathbf {v}}^{(\tau )}$=${\mathbf {v}}^{(\tau +T)}$ in Eq. (3). Thus, ${\mathbb {P}}^{(\tau +1,\tau +T)} ={\mathbb {P}}^{(\tau +T+1,\tau +2T)}$. Now we only discuss the sequence $\{{\mathbb {P}}^{(\tau )}\}_{\tau =\tau _p+1}^\infty$ after $\tau _p$, and define

$$\begin{aligned} \begin{aligned} {\mathcal {P}}={{\mathbb {P}}}^{(\tau _p+1,\tau _p+T)}={{\mathbb {P}}}^{(\tau _p+T+1,\tau _p+2T)} \end{aligned} \end{aligned}$$

Then we know that the chain $\{{\mathcal {P}}\}$ is homogeneous. Suppose the graph has finite nodes set, and it’s undirected, connected, and its corresponding ${\mathbf {P}}$ is SIA. Then based on Lemma 1, in MWC, the modified transition matrices ${\mathbb {P}}^{(\tau )}$ and ${\mathcal {P}}$ are also SIA. Then we have $\lim \limits _{n\rightarrow \infty }|| {\mathcal {P}}^{n+1}-{\mathcal {P}}^{n}||_\infty =0$ [10, 14].

Proof of Theorem 1

For $1\le \mu \le T$, we have

$$\begin{aligned} \begin{aligned}&\parallel {\mathbf {x}}^{(\tau _p+(n+1)T+\mu )}-{\mathbf {x}}^{(\tau _p+nT+\mu )}\parallel _1\\&\quad =\parallel ({\mathcal {P}}^{n+1})^\intercal {\mathbf {x}}^{(\tau _p+\mu )}-({\mathcal {P}}^{n})^\intercal {\mathbf {x}}^{(\tau _p+\mu )}\parallel _1\\&\quad \le \parallel {\mathcal {P}}^{n+1}-{\mathcal {P}}^{n}\parallel _\infty \parallel {\mathbf {x}}^{(\tau _p+\mu )}\parallel _1\\&\quad \le \parallel {\mathcal {P}}^{n+1}-{\mathcal {P}}^{n}\parallel _\infty \end{aligned} \end{aligned}$$

For a sufficient large n such that ${\mathcal {P}}^{n+1}={\mathcal {P}}^{n}={\mathcal {Q}}$, let $\tau '_c=\tau _p+nT$.

Then we have ${\mathbf {x}}^{(\tau '_c+T+\mu )}={\mathbf {x}}^{(\tau '_c+\mu )}$. That is, for any $\tau \ge \tau '_c, {\mathbf {x}}^{(\tau +T)}={\mathbf {x}}^{(\tau )}$.

Next, we estimate $\tau _c'$. To reach a computational tolerance $\epsilon >0$, i.e., $|| {\mathcal {P}}^{n+1}-{\mathcal {P}}^{n}||_\infty <\epsilon$, a rough estimate of n is $\log \epsilon /\log (\alpha ^T)$ [19], because

$$\begin{aligned} \begin{aligned} {\mathcal {P}}=&\,{\mathbb {P}}^{(\tau _p+1,\tau _p+T)}\\ =&\,\alpha ^T{\mathbf {P}}^T+\sum _{m=0}^{T-1}\alpha ^m(1-\alpha )\mathbf {1(v}^{(\tau _p+T-1-m)})^\intercal {\mathbf {P}}^m\\ \end{aligned} \end{aligned}$$

Then $\tau '_c=\tau _p+nT\le \tau _p+\lfloor \log \epsilon /\log \alpha \rfloor$.

For the entire group of K walkers, we set $\tau _c$ as the largest $\tau '_c$ of all walkers and we have $\tau _p\le \tau _c\le \tau _p+\lfloor \log \epsilon /\log \alpha \rfloor$. $\square$

1.2 A.2 The Proof of Theorem 2

Theorem 2 directly follows from the following lemma.

Lemma 2

[30] Let $\Omega = \{{\mathbb {A}}^{(1)},\ldots ,{\mathbb {A}}^{(\tau )}\}$ be a set of square stochastic matrices of the same order such that for any $m\ge 1$, the product ${\mathbb {B}}={\mathbb {A}}^{(i_1)}\ldots {\mathbb {A}}^{(i_m)}$ ($1\le i_j\le \tau$ for $1\le j\le m$) is SIA. Then for any $\epsilon >0$, there exists an integer $\nu (\epsilon )$ such that any ${\mathbb {B}}$ of length $n\ge \nu (\epsilon )$ satisfies $\delta ({\mathbb {B}})<\epsilon$.

Proof of Theorem 2

Let $\Omega$ be the set that contains all possible modified transition matrices in MWC. From Lemma 1, we know that the product of matrices from $\Omega$ with any length and order is SIA. Based on Lemma 2, we know that for any $\epsilon >0$, there exists an integer $\nu (\epsilon )$ such that when $\tau _s\ge \nu (\epsilon )$, $\delta ({\mathbb {P}}^{(1,\tau _s)})<\epsilon$.

For $\nu (\epsilon )$, based on Lemma 2 in [30], we know that $\delta ({\mathbb {P}}^{(1,\tau _s)})\le \prod _{\tau =1}^{\tau _s}\lambda ({\mathbb {P}}^{(\tau )})$. Because the rows in ${\mathbf {1}}({\mathbf {v}}^{(\tau -1)})^\intercal$ are identical, then

$$\begin{aligned} \begin{aligned}&\lambda ({\mathbb {P}}^{(\tau )})=1-\min _{v,v'}\sum \limits _{w}\min \{{\mathbb {P}}^{(\tau )}(v,w),{\mathbb {P}}^{(\tau )}(v',w)\}\\&=1-\min _{v,v'}\sum \limits _{w}\min \{[\alpha {\mathbf {P}}+(1-\alpha ){\mathbf {1}}({\mathbf {v}}^{(\tau -1)})^\intercal ](v,w),\\&\qquad [\alpha {\mathbf {P}}+(1-\alpha ){\mathbf {1}}({\mathbf {v}}^{(\tau -1)})^\intercal ](v',w)\}\\&=1-\alpha \min _{v,v'}\sum \limits _{w}\min \{{\mathbf {P}}(v,w),{\mathbf {P}}(v',w)\}-(1-\alpha )\\&=\alpha (1-\min _{v,v'}\sum \limits _{w}\min \{{\mathbf {P}}(v,w),{\mathbf {P}}(v',w)\})\\&=\alpha \lambda ({\mathbf {P}}) \end{aligned} \end{aligned}$$

So $\delta ({\mathbb {P}}^{(1,\tau _s)})\le (\alpha \lambda ({\mathbf {P}}))^{\tau _s}$. To satisfy the tolerance $\epsilon$, we can set $\nu (\epsilon )=\log \epsilon /\log (\alpha \lambda ({\mathbf {P}}))$. $\square$

For a stochastic matrix ${\mathbf {P}}$, the measure $\lambda ({\mathbf {P}})$ shows the difference of its row vectors and we have $0\le \lambda ({\mathbf {P}})\le 1$. In practice, the transition matrix ${\mathbf {P}}$ is sparse and has block structure. We can select two rows from ${\mathbb {P}}$ that have no overlapping nonzero entries to make $\lambda ({\mathbf {P}})$ reach the maximum value 1. So in the worst case, we can set $\tau _s=\lfloor \log \epsilon /\log \alpha \rfloor$ to stop the algorithm.

1.3 A.3 The Proof of Theorem 3

In this section, we analyze the error bound on the difference between the exact and estimated probability vectors for $W_k$ in the $(\tau +1)$th group iteration. We first define the probability flow, and provide a tight error bound in Theorem 4. The proof of Theorem 3 then follows. For simplicity, we omit the subscript “k” and use ${\mathbf {x}}^{(\tau +1)}$ and $\hat{{\mathbf {x}}}^{(\tau +1)}$ to represent the exact and estimated probability vectors, respectively.

Let $U^{(\tau )}$ represent the set of nodes whose probability will be updated in the $(\tau +1)$th iteration, i.e., $U^{(\tau )} = C^{(\tau )}\cup \varGamma (C^{(\tau )})$, where $\varGamma (C)$ represents the direct neighbors of the nodes in C, and let $S^{(\tau )}$ represent the remaining nodes. We only update the scores for nodes in $U^{(\tau )}$ in the $(\tau +1)$th iteration, i.e.,

$$\begin{aligned} \hat{{\mathbf {x}}}^{(\tau +1)}(i)= {\left\{ \begin{array}{ll} \hbox {update based on Eq. (2)} &{}\quad \text {if}\ i\in U^{(\tau )};\\ {\mathbf {x}}^{(\tau )}(i) &{} \quad \text {otherwise}.\\ \end{array}\right. } \end{aligned}$$

(11)

The estimation error is thus $\Delta =||\frac{\hat{{\mathbf {x}}}^{(\tau +1)}}{||\hat{{\mathbf {x}}}^{(\tau +1)}||_1}-{\mathbf {x}}^{(\tau +1)}||_1$

Definition 5

The probability flow from a set of nodes B to another set A in the $(\tau +1)$th group iteration is defined as

$$\begin{aligned} F_{B\rightarrow A}=\sum \limits _{g\in A}\sum \limits _{i\in N_g \cap B}{\mathbf {P}}(i,g){\mathbf {x}}^{(\tau )}(i),\end{aligned}$$

where $N_g$ represents the direct neighbors of g.

The probability flows during the $(\tau +1)$th iteration are (we omit the superscript “${(\tau )}$” for $U^{(\tau )}$ and $S^{(\tau )}$ below):

$$\begin{aligned} {\left\{ \begin{array}{ll} a:=F_{U\rightarrow S}=\alpha \sum \nolimits _{g\in S}\sum \nolimits _{i\in N_g \cap U}{\mathbf {P}}(i,g){\mathbf {x}}^{(\tau )}(i);\\ b:=F_{S\rightarrow S}=\alpha \sum \nolimits _{g\in S}\sum \nolimits _{j\in N_g \cap S}{\mathbf {P}}(j,g){\mathbf {x}}^{(\tau )}(j);\\ c:=F_{S\rightarrow U}=\alpha \sum \nolimits _{g\in U}\sum \nolimits _{j\in N_g \cap S}{\mathbf {P}}(j,g){\mathbf {x}}^{(\tau )}(j). \end{array}\right. } \end{aligned}$$

(12)

Theorem 4

$$\begin{aligned} \begin{aligned}\Delta \le \frac{1}{1-\gamma }[(\gamma +|\gamma |)(1-\gamma -\beta )+2\beta ],\end{aligned} \end{aligned}$$

where $\gamma =1-\parallel \hat{{\mathbf {x}}}^{(\tau +1)} \parallel _1=a+b-\beta$ and $\beta =(c+b)/\alpha$.

Proof

According to Eqs. (11) and (12), we have

$$\begin{aligned} \begin{aligned} \gamma =&1-\parallel \hat{{\mathbf {x}}}^{(\tau +1)} \parallel _1=1-\parallel [\begin{array}{c}{\mathbf {x}}_{U}^{(\tau +1)}\\ {\mathbf {x}}_{S}^{(\tau )}\end{array}]\parallel _1\\ =&\,\parallel {\mathbf {x}}_{S}^{(\tau +1)}\parallel _1-\parallel {\mathbf {x}}_{S}^{(\tau )}\parallel _1\\ =&\,F_{U\rightarrow S}+F_{S\rightarrow S}-\sum \nolimits _{g\in S} {\mathbf {x}}^{(\tau )}(g)\\ =&\,a+b-(c+b)/\alpha \\ =&\,a+b-\beta \\ \end{aligned} \end{aligned}$$

where ${\mathbf {x}}_{U}$ and ${\mathbf {x}}_{S}$ are the sub-vector projections of ${\mathbf {x}}$ on U and S, respectively, and $\beta =(c+b)/\alpha$.

Thus, we have

$$\begin{aligned} \begin{aligned} \Delta&=\,\parallel \frac{\hat{{\mathbf {x}}}^{(\tau +1)}}{||\hat{{\mathbf {x}}}^{(\tau +1)}||_1}-{\mathbf {x}}^{(\tau +1)} \parallel _1\\&=\,\parallel [\begin{array}{c}{\mathbf {x}}_{U}^{(\tau +1)}\\ {\mathbf {x}}_{S}^{(\tau )}\end{array}]/(1-\gamma ) -[\begin{array}{c}{\mathbf {x}}_{U}^{(\tau +1)}\\ {\mathbf {x}}_{S}^{(\tau +1)}\end{array}]\parallel _1\\&=\frac{|\gamma |}{1-\gamma }\parallel {\mathbf {x}}_{U}^{(\tau +1)}\parallel _1+\parallel \frac{{\mathbf {x}}_{S}^{(\tau )}}{1-\gamma }- {\mathbf {x}}_{S}^{(\tau +1)}\parallel _1\\&\le \frac{|\gamma |}{1-\gamma }(1- \parallel {\mathbf {x}}_{S}^{(\tau +1)}\parallel _1)+ \frac{\parallel {\mathbf {x}}_{S}^{(\tau )}\parallel _1}{1-\gamma }+\parallel {\mathbf {x}}_{S}^{(\tau +1)}\parallel _1\\&=\frac{|\gamma |}{1-\gamma } (1-\gamma -\beta )+\frac{\beta }{1-\gamma }+(\gamma +\beta )\\&=\frac{1}{1-\gamma }[(\gamma +|\gamma |)(1-\gamma -\beta )+2\beta ] \end{aligned} \end{aligned}$$

$\square$

The proof of Theorem 3 is as follows.

Proof

(1) If $\gamma \le 0$, we have

$$\begin{aligned} \begin{aligned} \Delta \le&\frac{2\beta }{1-\gamma }\le 2\beta =2\sum \nolimits _{g\in S} {\mathbf {x}}^{(\tau )}(g)\\&\le \, 2 \sum \nolimits _{g\in V\backslash C^{(\tau )}} {\mathbf {x}}^{(\tau )}(g)=2(1-Pr(C^{(\tau )})) \le 2(1-\theta ) \end{aligned} \end{aligned}$$

(2) If $\gamma >0$, we have $\Delta \le 2(\beta +\gamma )=2(a+b)$.

Let $\delta U=\{g\in U|\exists i\in S, s.t., (i,g)\in E\}$ be the set of boundary nodes of U. We have $a+b\le F_{\delta U\cup S\rightarrow S}\le F_{V\backslash C^{(\tau )}\rightarrow V}\le 1-\theta$.

Thus, $\Delta \le 2(1-\theta )$. $\square$

Algorithm 4 shows the overall process of the partial node set updating strategy for walker $W_k$, which can be used to replace the UPDATE in Algorithm 2. The algorithm first finds the core node set $C_{k}^{(\tau )}$ by a breadth first search starting from the influential nodes of walker $W_k$ (Line 1). In Line 2, the updating node set $U_k^{(\tau )}$ can be obtained as the union of the nodes in $C_{k}^{(\tau )}$ and their direct neighbors. Lines 3 and 4 update the node visiting probabilities according to Eq. (11). Line 5 returns the normalized probability vector.

The breath first search costs $O(d|U_k^{(\tau )}|)$, where d is the average degree of nodes in $U_k^{(\tau )}$. In each iteration, the updating time is reduced from $O(|V|+|E|)$ to $O(d|U_k^{(\tau )}|)$ for $W_k$.