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SCIENTIA SINICA Informationis, Volume 49 , Issue 12 : 1572-1585(2019) https://doi.org/10.1360/SSI-2019-0117

Maximum average entropy-rate based correlation clustering for big data

Liwen ZHANG 1, Tao WANG 1, Jian LUO 1, Shusen YANG 2, Zongben XU 2,*
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  • ReceivedJun 4, 2019
  • AcceptedJul 10, 2019
  • PublishedDec 16, 2019
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Abstract


Funded by

国家自然科学基金(61772410,61802298,11690011,U1811461)

国家重点研发计划(2017YFB1010004)


References

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  • Click to view Download high-quality image

    Figure 1

    (Color online) Illustration of correlation clustering

  • Click to view Download high-quality image

    Figure 2

    (Color online) Comparison of neighbors by original $\sigma_{0}$ and optimized $\hat{\sigma}$. Taking points in Aggregation for instance, neighbors by original $\sigma_{0}=10$ are shown in (a) and neighbors by optimized $\hat{\sigma}=0.4$ are shown in (b).

  • Click to view Download high-quality image

    Figure 3

    (Color online) Clustering results comparison on standard 2D datasets

  • Click to view Download high-quality image

    Figure 4

    (Color online) Comparison on running time of ECC and $k$-means: running time is the average of 3 times running; values of $k$ for $k$-means are $1003,~2119,~3141,~4207,~5181,~6209,~7232,~8252$ and 9306; parameters for ECC are $\sigma_{0}=0.5,{\rm~iter}_{\rm~max}=10,l=1$; parameters for $k$-NN+Louvain are $l=0,k=10$.

  • Table 1   Clustering metrics comparison on standard 2D datasets, including f1-measure and purity
    Datasets Metrics ECC $k$-means[4] AP[25] Mean-shift[5] Spectral[26] HAC[27] Ncut[7]
    2*Flame $F1$ 1.0000 0.8406 0.8243 0.8528 0.9832 0.7732 0.9917
    $P$ 1.0000 0.8647 0.8488 0.8586 0.8787 0.8595 0.9917
    2*Spiral $F1$ 1.0000 0.3502 0.3225 0.3562 1.0000 0.3917 0.5823
    $P$ 1.0000 0.3447 0.3566 0.3475 1.0000 0.3337 0.5992
    2*Aggregation $F1$ 1.0000 0.8504 0.8618 0.9595 0.7491 0.9097 0.9925
    $P$ 1.0000 0.9453 0.9415 0.9685 0.8242 0.9646 0.9930
    2*Face $F1$ 0.9669 0.8637 0.8867 0.6883 0.3478 0.9605 0.1644
    $P$ 0.9727 0.7134 0.7583 0.5445 0.1738 0.8335 0.1952
  •   

    Algorithm 1 Neighboring by greedy algorithm-${{\rm~nbr}_{\bar{\mathcal{H}}}}(\cdot,\cdot)$

    Require:Node $~i~$, initial edge set of $i$ $\boldsymbol{E}_{i}~=~\{e_{ij}\in\boldsymbol{E}\}$;

    Output:Edge set based on maximum average entropy rate $\mathcal{N}_{i}$;

    $\mathcal{N}_{i}~\Leftarrow~\emptyset$;

    $\boldsymbol{E}'_{i}~\Leftarrow~$Order edges in $\boldsymbol{E}_{i}$ by weight in descending order;

    $\bar{\mathcal{H}}_{\rm~max}~\Leftarrow~0$; % Variable to store the value of maximum average entropy rate.

    for Edge $e_{ij}$ in $\boldsymbol{E}'_{i}$

    $\bar{\mathcal{H}}_{\rm~temp}~\Leftarrow\bar{\mathcal{H}}(\mathcal{N}_{i})$; % Average entropy rate of current edge set.

    if $\bar{\mathcal{H}}_{\rm~temp}\ge\bar{\mathcal{H}}_{\rm~max}$ then

    Add $e_{ij}$ to $\mathcal{N}_{i}$;

    $\bar{\mathcal{H}}_{\rm~max}~\Leftarrow~\bar{\mathcal{H}}_{\rm~temp}$;

    else

    Return $\mathcal{N}_{i}$;

    end if

    end for
  •   

    Algorithm 2 Correlation clustering on single cluster-${{\rm~Cc_{s}}}(\cdot,~\cdot)$

    Require:Graph $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E})$: $v\in~\boldsymbol{V},~v.{\text~{cluster~=~`unset'}},~e\in\boldsymbol{E},~e.{\rm~label}=-$, maximum iteration ${\rm~iter}_{\rm~max}$;

    Output:Node set of cluster $\boldsymbol{c}$;

    $\boldsymbol{V}_{\rm~temp}\Leftarrow$ nodes from $\boldsymbol{V}$ with .cluster = `unset'; % Unclustered node set.

    $i~\Leftarrow~$ node from $\boldsymbol{V}_{\rm~temp}$ with maximum degree;

    Set edge in $\boldsymbol{E}_{i}$ .label = +;

    ${\rm~iter}\Leftarrow0$;

    while ${\rm~iter<~iter}_{\rm~max}$ do

    for each $v$ in $\boldsymbol{V}$

    if ${\boldsymbol~f}^{+}(v)\ge{\boldsymbol~f}^{-}(v)$ then

    Set $v.{\rm~cluster}=i$, set edge label in $\boldsymbol{E}_{v}$ to $+$; % $v$ is contained by cluster ${\boldsymbol~c}$.

    else

    Set $v.{\rm~cluster=`unset}$', set edge label in $\boldsymbol{E}_{v}$ to $-$; % $v$ is not contained by cluster ${\boldsymbol~c}$.

    end if

    end for

    ${\rm~iter}\Leftarrow~{\rm~iter}+1$;

    end while

    $\boldsymbol{c}\Leftarrow$ nodes from $\boldsymbol{V}$ with .cluster = $i$;

    Return $\boldsymbol{c}$.
  •   

    Algorithm 3 Correlation clustering on multiple clusters-${\rm~Cc_{m}}(\cdot,~\cdot,~\cdot)$

    Require:Dataset $\boldsymbol{X}$, initial neighbor parameter $\sigma_{0}$, maximum iteration ${\rm~iter}_{\rm~max}$;

    Output:Clustering result $\boldsymbol{C}$;

    $\boldsymbol{E}\Leftarrow~\emptyset$;

    for each $x$ in $\boldsymbol{X}$

    $\boldsymbol{E}_{x}\Leftarrow\mathcal{N}_{\sigma_{0}}(x)$; % $\mathcal{N}_{\sigma_{0}}(\cdot)$ return initial neighbor with parameter $\sigma_{0}$.

    $\mathcal{N}_{x}\Leftarrow{{\rm~nbr}_{\bar{\mathcal{H}}}}(x,\boldsymbol{E}_{x})$;

    Add $\mathcal{N}_{x}$ to ${\boldsymbol~E}$;

    end for

    Generate graph $\boldsymbol{G}(\boldsymbol{V}=\boldsymbol{X},\boldsymbol{E})$;

    Initialize $\boldsymbol{G}$: $v\in~\boldsymbol{V},~v.{\text{cluster~=~`unset&apos;}},~e\in\boldsymbol{E},~e.{\rm~label}=-$;

    $\boldsymbol{C}\Leftarrow\{\}$;

    while $\{v\in~\boldsymbol{V},~v.{\text{cluster=`unset&apos;}}\}\ne~\emptyset$ do

    $\boldsymbol{c}\Leftarrow{\rm~Cc_{s}}(\boldsymbol{G},~{\rm~iter}_{\rm~max})$;

    Add $\boldsymbol{c}$ to $\boldsymbol{C}$;

    Reset $\boldsymbol{G}$: $e\in\boldsymbol{E},~e.{\rm~label}=-$;

    end while

    Return $\boldsymbol{C}$.
  •   

    Algorithm 4 Start points initialization-${\rm~Initspts}(\cdot)$

    Require:Graph $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E})$: $v\in~\boldsymbol{V},~{\text{v.cluster~=~`unset&apos;}},~e\in\boldsymbol{E},~e.{\text{label~=~`unset&apos;}}$;

    Output:Initialized graph $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E}&apos;)$;

    $\boldsymbol{E}&apos;\Leftarrow\{\}$;

    for each $e_{ij}$ in $\boldsymbol{E}$

    if $e_{ij}.{\text{label~==~`unset&apos;}}$ then

    $e_{ij}.{\text{label}}\Leftarrow$ node with maximum degree between node $i,j$;

    else

    $e_{ij}.{\rm~label}\Leftarrow$ node with maximum degree among node $i,j$ and $e_{ij}.{\rm~label}$;

    end if

    Add $e_{ij}$ to $\boldsymbol{E}&apos;$;

    end for

    Return $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E}&apos;)$.
  •   

    Algorithm 5 Clusters generation-${\rm~Gencluster}(\cdot,\cdot,\cdot)$

    Require:Edge initialized graph $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E})$, overlapping parameter $l$, maximum iteration ${\rm~iter}_{\rm~max}$;

    Output:Node set of clusters $\boldsymbol{C}$;

    Initialize $\boldsymbol{V}$ of $\boldsymbol{G}(\boldsymbol{V},\boldsymbol{E})$: $v\in\boldsymbol{V},~v.{\rm~cluster}\leftarrow~{\text{empty~list~with}}~~l~~{\rm~legnth}$;

    while ${\rm~iter}<{\rm~iter}_{\rm~max}$ do

    for each $v$ in $\boldsymbol{V}$

    $\boldsymbol{K}\Leftarrow$ collection of .label value of edge in $\boldsymbol{E}_{v}$;

    ${\rm~templist}\Leftarrow$ empty list;

    for each $k$ in $\boldsymbol{K}$

    if ${\boldsymbol~f}_{k}(v)\ge\sum\nolimits_{i\ne~k}{\boldsymbol~f}_{i}(v)$ then

    Add $k$ to templist;

    end if

    end for

    $v.{\rm~cluster}\Leftarrow$ Keep the top $l$ items in templist ordered by its force ${\boldsymbol~f}_{k}(v)$;

    end for

    ${\rm~iter}\Leftarrow~{\rm~iter}+1$;

    end while

    $\boldsymbol{C}\Leftarrow$ aggregate nodes by their label;

    Return $\boldsymbol{C}$.
  •   

    Algorithm 6 Correlation clustering on big data-${\rm~Cc_{b}}(\cdot,\cdot,\cdot,\cdot)$

    Require:Dataset $\boldsymbol{X}$, initial neighbor parameter $\sigma_{0}$, maximum iteration ${\rm~iter}_{\rm~max}$, overlapping parameter $l$;

    Output:Node set of clusters $\boldsymbol{C}$;

    $\boldsymbol{E}\Leftarrow\emptyset$;

    for each $x$ in $\boldsymbol{X}$

    $\boldsymbol{E}_{x}\Leftarrow\mathcal{N}_{\sigma_{0}}(x)$;

    $\mathcal{N}_{x}\Leftarrow{{\rm~nbr}_{\bar{\mathcal{H}}}}(x,\boldsymbol{E}_{x})$;

    Add $\mathcal{N}_{x}$ to ${\boldsymbol~E}$;

    end for

    Generate graph $\boldsymbol{G}(\boldsymbol{V}=\boldsymbol{X},\boldsymbol{E})$;

    ${\rm~Initespts}(\boldsymbol{G})$;

    $\boldsymbol{C}\Leftarrow{\rm~Gencluster}(\boldsymbol{G},l,{\rm~iter}_{\rm~max})$;

    Return $\boldsymbol{C}$.
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