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Memory-based random walk for multi-query local community detection

A Correction to this article was published on 11 October 2019

This article has been updated


Local community detection, which aims to find a target community containing a set of query nodes, has recently drawn intense research interest. The existing local community detection methods usually assume all query nodes are from the same community and only find a single target community. This is a strict requirement and does not allow much flexibility. In many real-world applications, however, we may not have any prior knowledge about the community memberships of the query nodes, and different query nodes may be from different communities. To address this limitation of the existing methods, we propose a novel memory-based random walk method, MRW, that can simultaneously identify multiple target local communities to which the query nodes belong. In MRW, each query node is associated with a random walker. Different from commonly used memoryless random walk models, MRW records the entire visiting history of each walker. The visiting histories of walkers can help unravel whether they are from the same community or not. Intuitively, walkers with similar visiting histories are more likely to be in the same community. Moreover, MRW allows walkers with similar visiting histories to reinforce each other so that they can better capture the community structure instead of being biased to the query nodes. We provide rigorous theoretical foundations for the proposed method and develop efficient algorithms to identify multiple target local communities simultaneously. Comprehensive experimental evaluations on a variety of real-world datasets demonstrate the effectiveness and efficiency of the proposed method.

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Change history

  • 11 October 2019

    In the published article, Figure 9(a) and Figure 9(b) are the same figure.


  1. The Lambert-W function is a transcendental function defined by solutions of the equation \(W(x)e^{W(x)}=x\). For real values of the argument, x, it has two branches, \(W_0\) and \(W_{-1}\), the principal and the negative branches [11].

  2. The running time doesn’t include the preprocessing time of computing RWR score for the query node. RWR score can be obtained fast by applying the “push” strategy in [6] (for online computation) and the approximation strategy in [38] (for offline computation).


  4. The functionalities of the Brodmann areas can be obtained from


  1. Ahn Y-Y, Ahnert SE, Bagrow JP, Barabási A-L (2011) Flavor network and the principles of food pairing. Sci Rep 1:196

    Article  Google Scholar 

  2. Akakin HC, Gurcan MN (2012) Content-based microscopic image retrieval system for multi-image queries. IEEE Trans Inf Technol Biomed 16(4):758–769

    Article  Google Scholar 

  3. Akbas E, Zhao P (2017) Truss-based community search: a truss-equivalence based indexing approach. Proc VLDB Endow 10(11):1298–1309

    Article  Google Scholar 

  4. Akoglu L, Chau DH, Vreeken J, Tatti N, Tong H, Faloutsos C (2013) Mining connection pathways for marked nodes in large graphs. In: SDM

  5. Alamgir M, Von Luxburg U (2010) Multi-agent random walks for local clustering on graphs. In: ICDM

  6. Andersen R, Chung F, Lang K (2006) Local graph partitioning using pagerank vectors. In: FOCS

  7. Andersen R, Lang KJ (2006) Communities from seed sets. In: WWW

  8. Bagrow JP, Bollt EM (2005) Local method for detecting communities. Phys Rev E 72(4):046108

    Article  Google Scholar 

  9. Baranzini SE, Galwey NW, Wang J et al (2009) Pathway and network-based analysis of genome-wide association studies in multiple sclerosis. Human Mol Genet 18(11):2078–2090

    Article  Google Scholar 

  10. Barbieri N, Bonchi F, Galimberti E, Gullo F (2015) Efficient and effective community search. Data Min Knowl Discov 29(5):1406–1433

    Article  MathSciNet  Google Scholar 

  11. Barry D, Parlange J-Y, Li L, Prommer H, Cunningham C, Stagnitti F (2000) Analytical approximations for real values of the lambert w-function. Math Comput Simul 53(1):95–103

    Article  MathSciNet  Google Scholar 

  12. Benson AR, Gleich DF, Leskovec J (2015) Tensor spectral clustering for partitioning higher-order network structures. In: SDM

  13. Bian Y, Ni J, Cheng W, Zhang X (2017) Many heads are better than one: local community detection by the multi-walker chain. In: ICDM

  14. Bian Y, Ni J, Cheng W, Zhang X (2019) The multi-walker chain and its application in local community detection. Knowl Inf Syst 60(3):1663–1691

    Article  Google Scholar 

  15. Cho D-Y, Kim Y-A, Przytycka TM (2012) Network biology approach to complex diseases. PLoS Comput Biol 8(12):e1002820

    Article  Google Scholar 

  16. Clauset A (2005) Finding local community structure in networks. Phys Rev E 72(2):026132

    Article  Google Scholar 

  17. Crossley NA, Mechelli A, Vértes PE, Winton-Brown TT, Patel AX, Ginestet CE, McGuire P, Bullmore ET (2013) Cognitive relevance of the community structure of the human brain functional coactivation network. Proc Natl Acad Sci 110:11583–11588

    Article  Google Scholar 

  18. Cui W, Xiao Y, Wang H, Lu Y, Wang W (2013) Online search of overlapping communities. In: SIGMOD, pp 277–288

  19. Fang Y, Cheng R, Li X, Luo S, Hu J (2017) Effective community search over large spatial graphs. Proc VLDB Endow 10(6):709–720

    Article  Google Scholar 

  20. Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174

    Article  MathSciNet  Google Scholar 

  21. He K, Shi P, Hopcroft JE, Bindel D (2016) Local spectral diffusion for robust community detection. In: 12th workshop on mining and learning with graphs

  22. He K, Sun Y, Bindel D, Hopcroft J, Li Y (2015) Detecting overlapping communities from local spectral subspaces. In: ICDM

  23. Huang X, Cheng H, Qin L, Tian W, Yu JX (2014) Querying k-truss community in large and dynamic graphs. In: SIGMOD

  24. Huang X, Lakshmanan LV, Xu J (2017) Community search over big graphs: Models, algorithms, and opportunities. In: ICDE

  25. Huang X, Lakshmanan LV, Yu JX, Cheng H (2015) Approximate closest community search in networks. Proc VLDB Endow 9(4):276–287

    Article  Google Scholar 

  26. Kloster K, Gleich DF (2014) Heat kernel based community detection. In: KDD

  27. Kloumann IM, Kleinberg JM (2014) Community membership identification from small seed sets. In: KDD

  28. Kong X, Yu PS (2014) Brain network analysis: a data mining perspective. ACM SIGKDD Explor Newsl 15(2):30–38

    Article  Google Scholar 

  29. Lee C, Jang W-D, Sim J-Y, Kim C-S (2015) Multiple random walkers and their application to image cosegmentation. In: CVPR

  30. Lee S-H, Jang W-D, Park BK, Kim C-S (2016) Rgb-d image segmentation based on multiple random walkers. In: ICIP

  31. Li Y, He K, Bindel D, Hopcroft JE (2015) Uncovering the small community structure in large networks: a local spectral approach. In: WWW

  32. Newman ME (2001) The structure of scientific collaboration networks. Proc Natl Acad Sci 98(2):404–409

    Article  MathSciNet  Google Scholar 

  33. Ruchansky N, Bonchi F, García-Soriano D, Gullo F, Kourtellis N (2015) The minimum wiener connector problem. In: SIGMOD

  34. Ruchansky N, Bonchi F, Garcia-Soriano D, Gullo F, Kourtellis N (2017) To be connected, or not to be connected: that is the minimum inefficiency subgraph problem. In: CIKM

  35. Shan J, Shen D, Nie T, Kou Y, Yu G (2016) Searching overlapping communities for group query. World Wide Web 19(6):1179–1202

    Article  Google Scholar 

  36. Spielman DA, Teng S-H (2013) A local clustering algorithm for massive graphs and its application to nearly-linear time graph partitioning. SIAM J Comput 42(1):1–26

    Article  MathSciNet  Google Scholar 

  37. Tong H, Faloutsos C (2006) Center-piece subgraphs: problem definition and fast solutions. In: KDD

  38. Tong H, Faloutsos C, Pan J-Y (2006) Fast random walk with restart and its applications. In: ICDM

  39. Wang K, Li M, Hakonarson H (2010) Analysing biological pathways in genome-wide association studies. Nat Rev Genet 11(12):843

    Article  Google Scholar 

  40. Whang JJ, Gleich DF, Dhillon IS (2013) Overlapping community detection using seed set expansion. In: CIKM

  41. Wu Y, Bian Y, Zhang X (2016) Remember where you came from: on the second-order random walk based proximity measures. Proc VLDB Endow 10(1):13–24

    Article  Google Scholar 

  42. Wu Y, Jin R, Li J, Zhang X (2015) Robust local community detection: on free rider effect and its elimination. In: VLDB

  43. Wu Y, Zhang X, Bian Y, Cai Z, Lian X, Liao X, Zhao F (2018) Second-order random walk-based proximity measures in graph analysis: formulations and algorithms. VLDB J 27(1):127–152

    Article  Google Scholar 

  44. Xia M, Wang J, He Y (2013) Brainnet viewer: a network visualization tool for human brain connectomics. PLoS ONE 8(7):e68910

    Article  Google Scholar 

  45. Yan Y, Bian Y, Luo D, Lee D, Zhang X (2019) Constrained local graph clustering by colored random walk. In: The world wide web conference, pp 2137–2146

  46. Yang J, Leskovec J (2012) Defining and evaluating network communities based on ground-truth. In: ICDM

  47. Yin H, Benson AR, Leskovec J, Gleich DF (2017) Local higher-order graph clustering. In: KDD

  48. Yuan L, Qin L, Zhang W, Chang L, Yang J (2018) Index-based densest clique percolation community search in networks. TKDE 30(5):922–935

    Google Scholar 

  49. Zhou D, Zhang S, Yildirim MY, Alcorn S, Tong H, Davulcu H, He J (2017) A local algorithm for structure-preserving graph cut. In: KDD

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This work was partially supported by the National Science Foundation grants IIS-1664629 and CAREER.

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Correspondence to Yuchen Bian.

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1.1 More theoretical results of the visiting history vector

In this section, we first show that the visiting history vector \(\mathbf {v}^{(t)}\) (we omit the subscript “i” for simplicity) is a weighted sum of \(\mathbf {e}^{(\tau )}(1\le \tau \le t)\). Then we provide a tighter error bound on \(\Vert \mathbf {v}^{(t+1)}-\mathbf {v}^{(t)}\Vert _1\). These results are used in the speeding-up strategies introduced in Sect. 5.

1.1.1 Expansion of the visiting history vector

We expand the visiting history vector \(\mathbf {v}^{(t)}\) into a linear combination of vectors \(\mathbf {e}^{(\tau )}(0\le \tau \le t)\) and show that \(\mathbf {v}^{(t)}\) is stochastic.

Theorem 4

Based on Eq. (3), for \(K\ge 1\) and \(t\ge 1\), we have

$$\begin{aligned} \mathbf {v}^{(t)} =\sum \limits _{\tau =0}^{t}\psi ^{(t)}_\tau (\beta )\mathbf {e}^{(\tau )} \end{aligned}$$


$$\begin{aligned} \psi ^{(t)}_{\tau }(\beta ) = {\left\{ \begin{array}{ll} 1-\frac{1}{K}\sum \limits _{j=1}^{K}\theta ^{(t)}_{t+1-j}(\beta )&{} \text {if } \tau =0\\ \frac{1}{K}[\theta ^{(t)}_{t+1-\tau }(\beta )-\theta ^{(t)}_{t+1-K-\tau }(\beta )] &{} \text {if } 1\le \tau \le t\\ \end{array}\right. } \end{aligned}$$


$$\begin{aligned} \theta ^{(t)}_{\tau }(\beta )= {\left\{ \begin{array}{ll} 1-\prod \limits _{j=1}^\tau (1-\beta ^{t-j}) &{} \text {if } 1\le \tau \le t\\ 0 &{} \text {otherwise}\\ \end{array}\right. } \end{aligned}$$

Please refer to Appendix 8.4 for the proof.

Next we show \(\mathbf {v}^{(t)}\) is stochastic. Note that in the following analysis, we use \(\theta ^{(t)}_{\tau }\) to represent \(\theta ^{(t)}_{\tau }(\beta ) \) for simplicity.

Lemma 2

Let \(\varvec{\psi }^{(t)}\) be the column vector \([\psi ^{(t)}_{0},\dots ,\psi ^{(t)}_{t}]^\intercal \). For \(t\ge 1\), \(\varvec{\psi }^{(t)}\) is positive and \(\parallel \varvec{\psi }^{(t)}\parallel _1=1\).


From Theorem 4, we have \(0<\theta ^{(t)}_{\tau }\le 1\) for \(1\le \tau \le t\) and \(\theta ^{(t)}_{\tau }>\theta ^{(t)}_{\tau '}\) for \(1\le \tau '<\tau \le t\). Thus, \(\varvec{\psi }^{(t)}\) is positive. Furthermore, we have

$$\begin{aligned} \parallel \varvec{\psi }^{(t)}\parallel _1=1-\frac{1}{K}\sum \limits _{j=1}^{K}\theta ^{(t)}_{t+1-j}+\frac{1}{K}\sum \limits _{\tau =1}^{t}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau }) \end{aligned}$$

If \(1\le t\le K\),

$$\begin{aligned} \parallel \varvec{\psi }^{(t)}\parallel _1=1-\frac{1}{K}\sum \limits _{j=1}^{t}\theta ^{(t)}_{t+1-j}+\frac{1}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t)}_{t+1-\tau }=1 \end{aligned}$$

If \(t\ge K+1\),

$$\begin{aligned}&\parallel \varvec{\psi }^{(t)}\parallel _1\\&\quad = 1-\frac{1}{K}\sum \limits _{j=1}^{K}\theta ^{(t)}_{t+1-j}+\frac{1}{K}\sum \limits _{\tau =t-K+1}^{t}\theta ^{(t)}_{t+1-\tau }+\frac{1}{K}\sum \limits _{\tau =1}^{t-K}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })\\&\quad = 1+\frac{1}{K}\left( -\sum \limits _{\tau =t-K+1}^{t}\theta ^{(t)}_{\tau }+\sum \limits _{\tau =1}^{K}\theta ^{(t)}_{\tau }+\sum \limits _{\tau =K+1}^{t}\theta ^{(t)}_{\tau }-\sum \limits _{\tau =1}^{t-K}\theta ^{(t)}_{\tau }\right) \\&\quad = 1 \end{aligned}$$

Therefore, \(\parallel \varvec{\psi }^{(t)}\parallel _1=1\). \(\square \)

The stochastic property of \(\mathbf {v}^{(t)}\) can be directly derived from Lemma 2 and Eq. (7) which is summarized in the following theorem.

Theorem 5

For \(t\ge 0\), \(\mathbf {v}^{(t)}\) is nonnegative and \(\parallel \mathbf {v}^{(t)}\parallel _1=1\).

1.1.2 A tighter error bound

Next we provide a tight error bound on \(\parallel \mathbf {v}^{(t+1)}-\mathbf {v}^{(t)}\parallel _1\) to show its fast convergence. Based on Eqs. (8) and (9), when t is large, \(\theta ^{(t)}_{\tau }(\beta )\) and \(\psi ^{(t)}_{\tau }(\beta )\) will be small. For simplicity, in the following analysis, we use \(\theta ^{(t)}_{\tau }\) and \(\psi ^{(t)}_{\tau }\)to represent \(\theta ^{(t)}_{\tau }(\beta ) \) and \(\psi ^{(t)}_{\tau }(\beta )\), respectively. The following theorem first provides some properties of \(\psi ^{(t)}_{\tau }\) regarding t and then gives the error bound on \(\parallel \mathbf {v}^{(t+1)}-\mathbf {v}^{(t)}\parallel _1\).

Theorem 6

  1. (i)

    When \(t>t_\beta =\log \epsilon /\log \beta \) for a \(\epsilon >0\), we have \(|\psi ^{(t+1)}_{0}-\psi ^{(t)}_{0}|<\epsilon \), \(\psi ^{(t+1)}_{t+1}<\frac{\epsilon }{K}\) and \(|\psi ^{(t+1)}_{\tau }-\psi ^{(t)}_{\tau }|<\frac{\epsilon }{K}\) for \(1\le \tau \le t\).

  2. (ii)

    \(\parallel \mathbf {v}^{(t+1)}-\mathbf {v}^{(t)}\parallel _1 \le 2\beta ^t(1-\frac{1}{K}\sum \nolimits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau })\)

To prove Theorem 6, we first show some properties of \(\theta ^{(t)}_{\tau }\).

Lemma 3

For \(t\ge \tau \ge 1\), \(t>\tau _0\ge 1\), we have

  1. (i)

    \((1-\theta ^{(t)}_{\tau -1})(1-\beta ^{t-\tau }) = 1 - \theta ^{(t)}_{\tau }\)

  2. (ii)

    \((1-\theta ^{(t)}_{\tau _0})(1-\theta ^{(t-\tau _0)}_{\tau }) = 1 - \theta ^{(t)}_{\tau _0+\tau }\)

Lemma 3 can be directly derived from the definition of \(\theta ^{(t)}_{\tau }\) in Eq. (9). From Lemma 3 we have the following additional properties.

Corollary 1

For \(t\ge \tau \ge 1\), \(t>\tau _0\ge 1\),

  1. (i)

    \(\theta ^{(t)}_{1}=\beta ^{t-1}\); \(\theta ^{(t)}_{t}=1\)

  2. (ii)

    \((1-\theta ^{(t)}_{\tau -1})\beta ^{t-\tau } = \theta ^{(t)}_{\tau } - \theta ^{(t)}_{\tau -1}\)

  3. (iii)

    \((1-\theta ^{(t)}_{\tau _0})\theta ^{(t-\tau _0)}_{\tau } = \theta ^{(t)}_{\tau _0+\tau } - \theta ^{(t)}_{\tau _0}\);

       \((1-\theta ^{(t-\tau _0)}_{\tau })\theta ^{(t)}_{\tau _0} = \theta ^{(t)}_{\tau _0+\tau } - \theta ^{(t-\tau _0)}_{\tau }\)

Now we prove Theorem 6.


(of Theorem 6) Based on the properties of Corollary 1, we have \(\theta ^{(t+1)}_{\tau +1}-\theta ^{(t)}_{\tau }=(1-\theta ^{(t)}_{\tau })\theta ^{(t+1)}_{1}\) for \(t\ge \tau \ge 1\).

Now, we divide the proof into several cases based on the value of \(\tau \).

  1. (a)

    When \(\tau =0\), \(\psi ^{(t)}_{0}=1-\frac{1}{K}\sum \limits _{j=1}^{K}\theta ^{(t)}_{t+1-j}\),

    $$\begin{aligned} \psi ^{(t+1)}_{0}-\psi ^{(t)}_{0}=\frac{-1}{K}\sum \limits _{j=1}^{K}(\theta ^{(t+1)}_{t+2-j}-\theta ^{(t)}_{t+1-j})=\frac{-1}{K}\sum \limits _{j=1}^{K}(1-\theta ^{(t)}_{t+1-j})\theta ^{(t+1)}_{1}<0 \end{aligned}$$
  2. (b)

    When \(1\le \tau \le t-K\), \(\psi ^{(t)}_{\tau }=\frac{1}{K}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })\),

    $$\begin{aligned} \psi ^{(t+1)}_{\tau }-\psi ^{(t)}_{\tau }&=\frac{1}{K}[(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t)}_{t+1-\tau })-(\theta ^{(t+1)}_{t+2-K-\tau }-\theta ^{(t)}_{t+1-K-\tau })]\\&=\frac{1}{K}[(1-\theta ^{(t)}_{t+1-\tau })-(1-\theta ^{(t)}_{t+1-K-\tau })]\theta ^{(t+1)}_{1}\\&=\frac{1}{K}(\theta ^{(t)}_{t+1-K-\tau }-\theta ^{(t)}_{t+1-\tau })\theta ^{(t+1)}_{1}\\&<0 \end{aligned}$$
  3. (c)

    When \(\tau =t-K+1\),

    $$\begin{aligned} \psi ^{(t+1)}_{t-K+1}-\psi ^{(t)}_{t-K+1}&=\frac{1}{K}[(\theta ^{(t+1)}_{K+1}-\theta ^{(t+1)}_{1})-\theta ^{(t)}_{K}]\\&=\frac{1}{K}[(1-\theta ^{(t)}_{K})\theta ^{(t+1)}_{1}-\theta ^{(t+1)}_{1}]\\&=-\frac{1}{K}\theta ^{(t)}_{K}\theta ^{(t+1)}_{1}\\&<0 \end{aligned}$$
  4. (d)

    When \(t-K+2\le \tau \le t\), \(\psi ^{(t)}_{\tau }=\frac{1}{K}\theta ^{(t)}_{t+1-\tau }\),

    $$\begin{aligned} \psi ^{(t+1)}_{\tau }-\psi ^{(t)}_{\tau }&=\frac{1}{K}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t)}_{t+1-\tau })\\&=\frac{1}{K}(1-\theta ^{(t)}_{t+1-\tau })\theta ^{(t+1)}_{1}\\&>0 \end{aligned}$$
  5. (e)

    When \(\tau =t+1\), \(\psi ^{(t+1)}_{t+1}=\frac{1}{K}\theta ^{(t+1)}_{1}\).

For statement (i) in Theorem 6, when \(t>t_\beta =\log \epsilon /\log \beta \), \(\theta ^{(t+1)}_{1}=\beta ^t<\epsilon \), so \(|\psi ^{(t+1)}_{0}-\psi ^{(t)}_{0}|<\epsilon \), \(\psi ^{(t+1)}_{t+1}<\frac{\epsilon }{K}\) and \(|\psi ^{(t+1)}_{\tau }-\psi ^{(t)}_{\tau }|<\frac{\epsilon }{K}\) for \(1\le \tau \le t\).

For statement (ii), since \(\mathbf {v}^{(t)} =\sum \nolimits _{\tau =0}^{t}\psi ^{(t)}_{\tau }\mathbf {e}^{(\tau )}\), we have

$$\begin{aligned}&\parallel \mathbf {v}^{(t+1)}-\mathbf {v}^{(t)}\parallel _1\\&\quad \le \psi ^{(t+1)}_{t+1}+\sum \limits _{\tau =0}^{t}|\psi ^{(t+1)}_{\tau }-\psi ^{(t)}_{\tau }|\\&\quad \le \frac{\theta ^{(t+1)}_{1}}{K}\left[ \sum \limits _{j=1}^{K}(1-\theta ^{(t)}_{t+1-j})\right. \\&\qquad \left. +\sum \limits _{\tau =1}^{t-K}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })+\theta ^{(t)}_{K}+\sum \limits _{\tau =t-K+2}^{t}(1-\theta ^{(t)}_{t+1-\tau })+1\right] \\&\quad =\frac{\theta ^{(t+1)}_{1}}{K}\left[ K-\sum \limits _{\tau =t-K+1}^{t}\theta ^{(t)}_{\tau }+\sum \limits _{\tau =K+1}^{t}\theta ^{(t)}_{\tau }-\sum \limits _{\tau =1}^{t-K}\theta ^{(t)}_{\tau }+\theta ^{(t)}_{K}-\sum \limits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau }+K-1+1\right] \\&\quad =\frac{\theta ^{(t+1)}_{1}}{K}\left[ 2K-\sum \limits _{\tau =1}^{t}\theta ^{(t)}_{\tau }+\sum \limits _{\tau =K}^{t}\theta ^{(t)}_{\tau }+\sum \limits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau }-2\sum \limits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau }\right] \\&\quad =\frac{\theta ^{(t+1)}_{1}}{K}\left[ 2K-2\sum \limits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau }\right] =2\beta ^t\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K-1}\theta ^{(t)}_{\tau }\right) \end{aligned}$$

\(\square \)

1.2 The Proof of Theorem 2

In the following proof, we omit the subscript “i” for simplicity.


$$\begin{aligned}&\Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t_\beta )}\Vert _1\\&\quad \le \Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t-1)}\Vert _1+\cdots +\Vert \mathbf {x}^{(t_\beta +1)}-\mathbf {x}^{(t_\beta )}\Vert _1\\&\quad =\sum _{\tau =t_{\beta }+1}^{t}{\Delta }^{(\tau )} \end{aligned}$$

where \({\Delta }^{(\tau +1)}=\Vert \mathbf {x}^{(\tau +1)}-\mathbf {x}^{(\tau )}\Vert _1\).

From Lemma 1, we know that

$$\begin{aligned} {\Delta }^{(t+1)}\le {\left\{ \begin{array}{ll} \frac{2}{|\beta -\alpha |}\max \{\beta ^t,\alpha ^t\} &{} \text {if } \alpha \ne \beta \\ 2t\beta ^{t-1} &{} \text {if } \alpha =\beta \\ \end{array}\right. } \end{aligned}$$

If \(\alpha \ne \beta \), let \(c=\max \{\alpha ,\beta \}\), then

$$\begin{aligned}&\Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t_\beta )}\Vert _1\\&\quad \le \sum _{\tau =t_{\beta }+1}^{t}{\Delta }^{(\tau )}=\frac{2}{|\beta -\alpha |}\sum _{\tau =t_{\beta }+1}^{t}c^{\tau -1}\\&\quad =\frac{2(c^{t_\beta }-c^t)}{|\beta -\alpha |(1-c)}\le \frac{2c^{t_\beta }}{|\beta -\alpha |(1-c)} \end{aligned}$$

Since \(t_\beta =\log _\beta \epsilon \), we have \(\alpha ^{t_\beta }=\epsilon ^{\log _\beta \alpha }\) and \(\beta ^{t_\beta }=\epsilon \), then \(c^{t_\beta }=O(\epsilon ^{\min \{1,\log _\beta \alpha \}})\)

As a result, if \(\alpha \ne \beta \), \(\Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t_\beta )}\Vert _1=O(\epsilon ^{\min \{1,\log _\beta \alpha \}})\).

If \(\alpha =\beta \), we have

$$\begin{aligned} \Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t_\beta )}\Vert _1&\le \sum _{\tau =t_{\beta }+1}^{t}{\Delta }^{(\tau )}=2\sum _{\tau =t_{\beta }}^{t-1}\tau \beta ^{\tau -1}\\&=\frac{2}{(1-\beta )^2}[\beta ^{t_\beta -1}(t_\beta (1-\beta )+\beta )-\beta ^{t-1}(t(1-\beta )+\beta )]\\&\le \frac{2}{(1-\beta )^2}(t_\beta \beta ^{t_\beta -1}+\beta ^{t_\beta })\\&=\frac{2}{(1-\beta )^2}\left( \frac{\epsilon \log \epsilon }{\beta \log \beta }+\epsilon \right) \end{aligned}$$

Therefore, \(\Vert \mathbf {x}^{(t)}-\mathbf {x}^{(t_\beta )}\Vert _1=O(\epsilon |\log \epsilon |)\) if \(\alpha =\beta \). \(\square \)

1.3 Convergence property of the reinforcement process

In this section, we show the convergence property of \(\mathbf {x}_i^{(t)}\).

Theorem 7

\(\lim \nolimits _{t\rightarrow \infty } \Vert \mathbf {x}_i^{(t+1)}-\mathbf {x}_i^{(t)}\Vert _1 = 0\) for \(q_i\in Q\).


From \(\mathbf {x}_i^{(t-1)}\) to \(\mathbf {x}_i^{(t)}\), there are two steps: (1) form \(\mathbf {x}_i^{(t-1)}\) to \(\mathbb {x}_i^{(t-1)}\) based on Eq. (6), and (2) from \(\mathbb {x}_i^{(t-1)}\) to \(\mathbf {x}_i^{(t)}\) based on Eq. (2).

In the first step, we can rewrite Eq. (6) as \(\mathbb {x}_i^{(t-1)}={\mathbb {Q}_i^{(t-1)}}^\intercal \mathbf {x}_i^{(t-1)}\), where

$$\begin{aligned} \mathbb {Q}_{i}^{(t)}= {\left\{ \begin{array}{ll} (1-\gamma )\mathbf {I}+\gamma \mathbf {1}{\mathbf {y}_{i}^{(t)}}^\intercal &{} \text {if } \sum \nolimits _{j=1}^S\mathbf {R}^{(t)}(j,i)=1\\ \mathbf {I} &{} \text {if } \sum \nolimits _{j=1}^S\mathbf {R}^{(t)}(j,i)=0\\ \end{array}\right. } \end{aligned}$$

and \(\mathbf {y}_{i}^{(t)}=\sum \nolimits _{j=1}^S\mathbf {R}^{(t)}(j,i)\mathbf {x}_{j}^{(t)}\). We know that \({\mathbf {y}_{i}^{(t)}}^\intercal \mathbf {1}=1\).

In the second step, we can rewrite Eq. (2) as \(\mathbf {x}_i^{(t)}={\mathbb {P}_i^{(t)}}^\intercal \mathbb {x}_i^{(t-1)}\), where \(\mathbb {P}_i^{(t)}=\alpha \mathbf {P}+(1-\alpha )\mathbf {1}{\mathbf {v}_i^{(t-1)}}^\intercal \). According to Eq. (3), after a enough long time \(\tau _1\), for \(t>\tau _1\), \(\mathbf {v}_i^{(t)}\) will converge, so will \(\mathbb {P}_i^{(t)}\). Let \(\mathbb {P}_i\) denote the converged \(\mathbb {P}_i^{(t)}\).

As a result, \(\mathbf {x}_i^{(t)}=\mathbb {P}_i^\intercal \mathbb {x}_i^{(t-1)}=[\mathbb {Q}_i^{(t-1)}\mathbb {P}_i]^\intercal \mathbf {x}_i^{(t-1)}\) and \(\mathbb {Q}_i^{(t-1)}\) and \(\mathbb {P}_i\) are stochastic, i.e., \(\mathbb {Q}_i^{(t-1)}\mathbf {1}=\mathbb {P}_i\mathbf {1}=\mathbf {1}\).

Since the reinforcement process is a dynamic process between time points 0 and t, we can construct a graph to record the intermediate steps as follows. If at certain time point \(\tau \) (\(0\le \tau \le t\)), the score vectors of \(q_i\) and \(q_j\) have positive similarity (bigger than a very small threshold, e.g., 0.01), then there is an edge between them. In this graph, if a node \(q_i\) is isolated, that means the score vector of \(q_i\) has no positive similarity (bigger than a very small threshold, e.g., 0.01) with the score vector of any other query node at all time. Its convergence is guaranteed according to Theorem 1.

Let \(Q'\) represent the node set of any other connected component containing at least 2 nodes. For a query node \(q_i\in Q'\), let \(T=\{\tau |\sum \nolimits _{q_j\in Q'}\mathbf {R}^{(\tau )}(j,i)=0\}\). Then for \(\tau \in T\), \(\mathbb {Q}_i^{(\tau )}=\mathbf {I}\). Note that \(T\ne \emptyset \), since we have \(\mathbb {Q}_i^{(0)}=\mathbf {I}\). Next we extend \(\mathbf {x}_i^{(t)}\).

$$\begin{aligned} \mathbf {x}_i^{(t)}&=[\mathbb {Q}_i^{(t-1)}\mathbb {P}_i]^\intercal \mathbf {x}_i^{(t-1)}=[\mathbb {Q}_i^{(t-2)}\mathbb {P}_i\mathbb {Q}_i^{(t-1)}\mathbb {P}_i]^\intercal \mathbf {x}_i^{(t-2)}\\&=[\mathbb {Q}_i^{(t-2)}[(1-\gamma )\mathbb {P}_i+\gamma \mathbf {1}{\mathbf {y}_i^{(t-1)}}^\intercal ]\mathbb {P}_i]^\intercal \mathbf {x}_i^{(t-2)}\\&=[(1-\gamma )\mathbb {Q}_i^{(t-2)}\mathbb {P}_i^2+\gamma \mathbf {1}{\mathbf {y}_i^{(t-1)}}^\intercal \mathbb {P}_i]^\intercal \mathbf {x}_i^{(t-2)}\\&=[(1-\gamma )^{t-1-|T|}\mathbb {Q}_i^{(0)}\mathbb {P}_i^t+\gamma \sum \limits _{k=1,k\notin T}^{t-1}(1-\gamma )^{k-1}\mathbf {1}{\mathbf {y}_i^{(t-k)}}^\intercal \mathbb {P}_i^k]^\intercal \mathbf {x}_i^{(0)} \end{aligned}$$

Based on the Perron–Frobenius theorem, after a enough long time \(\tau _2\), for \(t>\tau _2\), \(\mathbb {P}_i^t=\mathbf {1x}_i^\intercal \), i.e., the row vectors of \(\mathbb {P}_i^t\) will be an identical vector \(\mathbf {x}_i^\intercal \) [13]. We set \(\tau _0 = \max \{\tau _1,\tau _2\}\). Then when \(t>\tau _0\), we have

$$\begin{aligned} \mathbf {x}_i^{(t)} =&\,(1-\gamma )^{t-1-|T|}\mathbf {x}_i+\gamma \sum \limits _{k=\tau _0+1,k\notin T}^{t-1}(1-\gamma )^{k-1}\mathbf {x}_i\\&+\gamma \sum \limits _{k=1,k\notin T}^{\tau _0}(1-\gamma )^{k-1}[\mathbb {P}_i^k]^\intercal {\mathbf {y}_i^{(t-k)}}\\ \end{aligned}$$

Next we check the difference between \(\mathbf {x}_i^{(t+1)}\) and \(\mathbf {x}_i^{(t)}\).

$$\begin{aligned} \Delta _i^{(t+1)}&=\Vert \mathbf {x}_i^{(t+1)}-\mathbf {x}_i^{(t)}\Vert _1\\&\le -\gamma (1-\gamma )^{t-1-|T|}+\gamma (1-\gamma )^{t-1}+\gamma \sum \limits _{k=1,k\notin T}^{\tau _0}(1-\gamma )^{k-1}\Vert \mathbf {y}_i^{(t+1-k)}-\mathbf {y}_i^{(t-k)}\Vert _1\\&\le \gamma \sum \limits _{k=1,k\notin T}^{\tau _0}(1-\gamma )^{k-1}\Vert \mathbf {y}_i^{(t+1-k)}-\mathbf {y}_i^{(t-k)}\Vert _1 \end{aligned}$$

Since after \(t_c\) steps, \(\mathbf {R}^{(t)}\) does not change, we have

$$\begin{aligned}&\Vert \mathbf {y}_i^{(t+1)}-\mathbf {y}_i^{(t)}\Vert _1\\&\quad =\sum \limits _{q_j\in Q'}\Vert \mathbf {R}^{(t+1)}(j,i)\mathbf {x}_j^{(t+1)}-\mathbf {R}^{(t)}(j,i)\mathbf {x}_{j}^{(t)}\Vert _1\\&\quad =\sum \limits _{q_j\in Q'}\mathbf {R}^{(t_c)}(j,i)\Vert \mathbf {x}_j^{(t+1)}-\mathbf {x}_{j}^{(t)}\Vert _1\\&\quad \le \max _{q_j\in {Q'}}\Delta _j^{(t+1)} \end{aligned}$$

Let \(\Delta ^{(t+1)}=\max _{q_j\in {Q'}}\Delta _j^{(t+1)}\), we have

$$\begin{aligned} \Delta ^{(t+1)}\le \gamma \sum \limits _{k=1,k\notin T}^{\tau _0}(1-\gamma )^{k-1}\Delta ^{(t+1-k)}<\gamma \sum \limits _{k=1}^{\tau _0}(1-\gamma )^{k-1}\Delta ^{(t+1-k)} \end{aligned}$$

Let \(\Delta '^{(t+1)}=\gamma \sum \nolimits _{k=1}^{\tau _0}(1-\gamma )^{k-1}\Delta ^{(t+1-k)}\), we have \(\Delta ^{(t+1)}<\Delta '^{(t+1)}\). Moreover,

$$\begin{aligned}&\Delta '^{(t+1)}\\&\quad =\gamma \sum \nolimits _{k=1}^{\tau _0}(1-\gamma )^{k-1}\Delta ^{(t+1-k)}\\&\quad =\gamma \Delta ^{(t)}+(1-\gamma )\left[ \gamma \sum \limits _{k=2}^{\tau _0+1}(1-\gamma )^{k-2}\Delta ^{(t+1-k)}\right] -\gamma (1-\gamma )^{\tau _0}\Delta ^{(t-\tau _0)}\\&\quad =\gamma \Delta ^{(t)}+(1-\gamma )\Delta '^{(t)}-\gamma (1-\gamma )^{\tau _0}\Delta ^{(t-\tau _0)}\\&\quad<\gamma \Delta '^{(t)}+(1-\gamma )\Delta '^{(t)}\\&\quad <\Delta '^{(t)} \end{aligned}$$

Thus, for any \(q_i\in Q'\), \(\lim \nolimits _{t\rightarrow \infty }\Delta _i^{(t)}=\lim \nolimits _{t_\rightarrow \infty }\Delta ^{(t)}=\lim \nolimits _{t_\rightarrow \infty }\Delta '^{(t)}=0\). \(\square \)

1.4 The Proof of Theorem 4


We use induction to prove

$$\begin{aligned} \mathbf {v}^{(t)} = \left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K}\theta ^{(t)}_{t+1-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })\mathbf {e}^{(\tau )} \end{aligned}$$

When \(t=1\), based on Eq. (3) and \(\mathbf {v}^{(0)}=\mathbf {e}^{(0)}\), we have

$$\begin{aligned} \mathbf {v}^{(1)}= & {} (1-\beta ^{1-1})\mathbf {v}^{(0)}+\beta ^{1-1}\frac{1}{K}(\mathbf {e}^{(1)}+\mathbf {e}^{(0)}+\cdots +\mathbf {e}^{(0)})\\= & {} \frac{K-1}{K}\mathbf {e}^{(0)}+\frac{1}{K}\mathbf {e}^{(1)} \end{aligned}$$

On the other hand, based on Eq. (11), we have

$$\begin{aligned} \mathbf {v}^{(1)} = \left( 1-\frac{\theta ^{(1)}_{2-1}}{K}\right) \mathbf {e}^{(0)}+\frac{\theta ^{(1)}_{2-1}}{K}\mathbf {e}^{(1)}=\frac{K-1}{K}\mathbf {e}^{(0)}+\frac{1}{K}\mathbf {e}^{(1)} \end{aligned}$$

Suppose that for \(t\ge 1\), Eq. (11) holds. Now we check \(\mathbf {v}^{(t+1)}\). When \(1\le t\le K-2\),

$$\begin{aligned} \mathbf {v}^{(t+1)}=(1-\beta ^{t})\mathbf {v}^{(t)}+\frac{\beta ^{t}}{K}\left( \sum \limits _{\tau =1}^{t+1}\mathbf {e}^{(\tau )}+(K-t-1)\mathbf {e}^{(0)}\right) \end{aligned}$$

Since \( t\le K-2\), then for \(1\le \tau \le t\), \(\theta ^{(t)}_{t+1-K-\tau }=0 \). Next, based on Eq. (11),

$$\begin{aligned} \mathbf {v}^{(t)}= \left( 1-\frac{1}{K}\sum \limits _{i=1}^{t}\theta ^{(t)}_{t+1-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t)}_{t+1-\tau }\mathbf {e}^{(\tau )} \end{aligned}$$

According to Corollaries 1 (i) and (iii), we have \(\beta ^t=\theta ^{(t+1)}_{1}\) and \((1-\theta ^{(t+1)}_{1})\theta ^{(t+1-1)}_{t+1-\tau }=\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1}\) for \(1\le \tau \le t\). Then we substitute \(\mathbf {v}^{(t)}\) into \(\mathbf {v}^{(t+1)}\).

$$\begin{aligned} \mathbf {v}^{(t+1)}&=(1-\theta ^{(t+1)}_{1})\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t)}_{t+1-\tau }\right) \mathbf {e}^{(0)}+\frac{1-\theta ^{(t+1)}_{1}}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t)}_{t+1-\tau }\mathbf {e}^{(\tau )}\\&\quad +\theta ^{(t+1)}_{1}\frac{K-t-1}{K}\mathbf {e}^{(0)}+\frac{1}{K}\theta ^{(t+1)}_{1}\sum \limits _{\tau =1}^{t+1}\mathbf {e}^{(\tau )}\\&=\left[ 1-\theta ^{(t+1)}_{1}\left( 1-\frac{K-t-1}{K}\right) -\frac{1}{K}\sum \limits _{\tau =1}^{t}(1-\theta ^{(t+1)}_{1})\theta ^{(t)}_{t+1-\tau }\right] \mathbf {e}^{(0)}\\&\quad +\frac{1}{K}\sum \limits _{\tau =1}^{t}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1})\mathbf {e}^{(\tau )}+\frac{\theta ^{(t+1)}_{1}}{K}\sum \limits _{\tau =1}^{t}\mathbf {e}^{(\tau )}+\frac{\theta ^{(t+1)}_{1}}{K}\mathbf {e}^{(t+1)}\\&=\left[ 1-\theta ^{(t+1)}_{1}\left( 1-\frac{K-t-1}{K}-\frac{t}{K}\right) -\frac{1}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t+1)}_{t+2-\tau }\right] \mathbf {e}^{(0)}\\&\quad +\frac{1}{K}\sum \limits _{\tau =1}^{t}\theta ^{(t+1)}_{t+2-\tau }\mathbf {e}^{(\tau )}+\frac{1}{K}\theta ^{(t+1)}_{1}\mathbf {e}^{(t+1)}\\&=\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{t+1}\theta ^{(t+1)}_{t+2-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t+1}\theta ^{(t+1)}_{t+2-\tau }\mathbf {e}^{(\tau )}\\ \end{aligned}$$

This means when \(1\le t\le K-1\), Eq. (11) holds. In the following inductive steps, we check \(\mathbf {v}^{(t+1)}\) for \(t\ge K-1\).

Based on Eq. (3), we have

$$\begin{aligned} \mathbf {v}^{(t+1)} =&(1-\beta ^{t})\mathbf {v}^{(t)}+\beta ^{t}\frac{1}{K}(\mathbf {e}^{(t+1)}+\mathbf {e}^{(t)}+\cdots + \mathbf {e}^{(t+2-K)})\\ =&(1-\theta ^{(t+1)}_{1})\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K}\theta ^{(t)}_{t+1-\tau }\right) \mathbf {e}^{(0)}\\&+\frac{(1-\theta ^{(t+1)}_{1})}{K}\sum \limits _{\tau =1}^{t}(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })\mathbf {e}^{(\tau )}\\&+\frac{1}{K}\theta ^{(t+1)}_{1}\mathbf {e}^{(t+1)}+\frac{1}{K}\theta ^{(t+1)}_{1}\sum \limits _{\tau =t-K+2}^{t}\mathbf {e}^{(\tau )} \end{aligned}$$

If \(t-K+1\le \tau \le t\), then \(\theta ^{(t)}_{t+1-K-\tau }=0\). Now we divide the second term into two parts: \(1\sim (t-K)\) and \((t-K+1)\sim t\).

$$\begin{aligned} \mathbf {v}^{(t+1)}&=\left[ 1-\theta ^{(t+1)}_{1}-\frac{1}{K}\sum \limits _{\tau =1}^{K}(1-\theta ^{(t+1)}_{1})\theta ^{(t)}_{t+1-\tau }\right] \mathbf {e}^{(0)}\\&\quad +\frac{1}{K}\sum \limits _{\tau =1}^{t-K}(1-\theta ^{(t+1)}_{1})(\theta ^{(t)}_{t+1-\tau }-\theta ^{(t)}_{t+1-K-\tau })\mathbf {e}^{(\tau )}\\&\quad +\frac{1}{K}\sum \limits _{\tau =t-K+1}^{t}(1-\theta ^{(t+1)}_{1})\theta ^{(t)}_{t+1-\tau }\mathbf {e}^{(\tau )}+\frac{1}{K}\theta ^{(t+1)}_{1}\mathbf {e}^{(t+1)}+\frac{1}{K}\theta ^{(t+1)}_{1}\sum \limits _{\tau =t-K+2}^{t}\mathbf {e}^{(\tau )}\\&=\left[ 1-\theta ^{(t+1)}_{1}-\frac{1}{K}\sum \limits _{\tau =1}^{K}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1})\right] \mathbf {e}^{(0)}\\&\quad +\frac{1}{K}\sum \limits _{\tau =1}^{t-K}[(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1})-(\theta ^{(t+1)}_{t+2-K-\tau }-\theta ^{(t+1)}_{1})]\mathbf {e}^{(\tau )}\\&\quad +\frac{1}{K}\theta ^{(t+1)}_{1}\mathbf {e}^{(t+1)}+\frac{1}{K}\sum \limits _{\tau =t-K+1}^{t}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1})\mathbf {e}^{(\tau )}+\frac{1}{K}\theta ^{(t+1)}_{1}\sum \limits _{\tau =t-K+2}^{t}\mathbf {e}^{(\tau )}\\&=\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K}\theta ^{(t+1)}_{t+2-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t-K}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{t+2-K-\tau })\mathbf {e}^{(\tau )}+\frac{1}{K}\theta ^{(t+1)}_{1}\mathbf {e}^{(t+1)}\\&\quad +\frac{1}{K}\sum \limits _{\tau =t-K+2}^{t}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{1}+\theta ^{(t+1)}_{1})\mathbf {e}^{(\tau )}+\frac{1}{K}(\theta ^{(t+1)}_{K+1}-\theta ^{(t+1)}_{1})\mathbf {e}^{(t-K+1)}\\&=\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K}\theta ^{(t+1)}_{t+2-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t-K+1}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{t+2-K-\tau })\mathbf {e}^{(\tau )}\\&\quad +\frac{1}{K}\sum \limits _{\tau =t-K+2}^{t+1}\theta ^{(t+1)}_{t+2-\tau }\mathbf {e}^{(\tau )} \end{aligned}$$

Since when \(t-K+2\le \tau \le t+1\), \(\theta ^{(t+1)}_{t+2-K-\tau }=0\), we can combine the last two items to get

$$\begin{aligned} \mathbf {v}^{(t+1)} =\left( 1-\frac{1}{K}\sum \limits _{\tau =1}^{K}\theta ^{(t+1)}_{t+2-\tau }\right) \mathbf {e}^{(0)}+\frac{1}{K}\sum \limits _{\tau =1}^{t+1}(\theta ^{(t+1)}_{t+2-\tau }-\theta ^{(t+1)}_{t+2-K-\tau })\mathbf {e}^{(\tau )} \end{aligned}$$

Therefore, when \(t\ge 1\), Eq. (11) is always valid. \(\square \)

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Bian, Y., Luo, D., Yan, Y. et al. Memory-based random walk for multi-query local community detection. Knowl Inf Syst 62, 2067–2101 (2020).

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  • Local community detection
  • Local clustering
  • Memory-based random walk
  • Visiting history