\begin{algorithmic}
    \State \textbf{Input:} Two graphs $\mathcal{G}=(V, E, \boldsymbol{X})$ and $\mathcal{H}=(P, F, \boldsymbol{Y})$
    \State $c_v^{(0)} \leftarrow \textsc{Hash}(\boldsymbol{x}_v), \forall v \in V$
    \State $d_p^{(0)} \leftarrow \textsc{Hash}(\boldsymbol{y}_p), \forall p \in P$
    \Repeat \ ($\ell=1,2,\cdots$)
    \If{$\{\!\{c_v^{(\ell-1)}|v \in V\}\!\} \neq \{\!\{d_p^{(\ell-1)}|p \in P\}\!\}$}
    \State \Return $\mathcal{G} \not\simeq \mathcal{H}$
    \EndIf
    \For{$v\in V$}
    \State $c_v^{(\ell)} \leftarrow \textsc{Hash}\Bigl(c_v^{(\ell-1)}, \{\!\{c_u^{(\ell-1)}|u\in\mathcal{N}_v\}\!\},
    \underline{\{\!\{\{\!\{c_{u_1}^{(\ell-1)}, c_{u_2}^{(\ell-1)}\}\!\}|u_1,u_2\in\mathcal{N}_v, (u_1,u_2)\in E\}\!\}} \Bigr)$
    \EndFor
    \For{$p\in P$}
    \State $d_p^{(\ell)} \leftarrow \textsc{Hash}\Bigl(d_p^{(\ell-1)}, \{\!\{d_q^{(\ell-1)}|q\in\mathcal{N}_p\}\!\}, \underline{\{\!\{\{\!\{d_{q_1}^{(\ell-1)}, d_{q_2}^{(\ell-1)}\}\!\}|q_1,q_2\in\mathcal{N}_p, (q_1,q_2)\in F\}\!\}}\Bigr)$
    \EndFor
    \Until convergence
    \State \Return $\mathcal{G} \simeq \mathcal{H}$
\end{algorithmic}

\begin{algorithmic}
\State \textbf{Input:} Two graphs $\mathcal{G}=(V, E, \boldsymbol{X})$ and $\mathcal{H}=(P, F, \boldsymbol{Y})$
\State $c_v^{(0)} \leftarrow \textsc{Hash}(\boldsymbol{x}_v), \forall v \in V$
\State $d_p^{(0)} \leftarrow \textsc{Hash}(\boldsymbol{y}_p), \forall p \in P$
\Repeat \ ($\ell=1,2,\cdots$)
\If{$\{\!\{c_v^{(\ell-1)}|v \in V\}\!\} \neq \{\!\{d_p^{(\ell-1)}|p \in P\}\!\}$}
\State \Return $\mathcal{G} \not\simeq \mathcal{H}$
\EndIf
\For{$v\in V$}
\State $c_v^{(\ell)} \leftarrow \textsc{Hash}\Bigl(c_v^{(\ell-1)}, \{\!\{c_u^{(\ell-1)}|u\in\mathcal{N}_v\}\!\},
\underline{\{\!\{\{\!\{c_{u_1}^{(\ell-1)}, c_{u_2}^{(\ell-1)}\}\!\}|u_1,u_2\in\mathcal{N}_v, (u_1,u_2)\in E\}\!\}} \Bigr)$
\EndFor
\For{$p\in P$}
\State $d_p^{(\ell)} \leftarrow \textsc{Hash}\Bigl(d_p^{(\ell-1)}, \{\!\{d_q^{(\ell-1)}|q\in\mathcal{N}_p\}\!\}, \underline{\{\!\{\{\!\{d_{q_1}^{(\ell-1)}, d_{q_2}^{(\ell-1)}\}\!\}|q_1,q_2\in\mathcal{N}_p, (q_1,q_2)\in F\}\!\}}\Bigr)$
\EndFor
\Until convergence
\State \Return $\mathcal{G} \simeq \mathcal{H}$
\end{algorithmic}
{
    "title": "empowering gnns via edge-aware weisfeiler-leman algorithm",
    "id": "2206.02059",
    "abstract": "message passing graph neural networks (gnns) are known to have their expressiveness upper-bounded by 1-dimensional weisfeiler-leman (1-wl) algorithm. to achieve more powerful gnns, existing attempts either require ad hoc features, or involve operations that incur high time and space complexities. in this work, we propose a general and provably powerful gnn framework that preserves the scalability of the message passing scheme. in particular, we first propose to empower 1-wl for graph isomorphism test by considering edges among neighbors, giving rise to nc-1-wl. the expressiveness of nc-1-wl is shown to be strictly above 1-wl and below 3-wl theoretically. further, we propose the nc-gnn framework as a differentiable neural version of nc-1-wl. our simple implementation of nc-gnn is provably as powerful as nc-1-wl. experiments demonstrate that our nc-gnn performs effectively and efficiently on various benchmarks.",
    "categories": "cs.lg cs.ai",
    "doi": "",
    "created": "2022-06-04",
    "updated": "2024-01-23",
    "authors": [
        "meng liu",
        "haiyang yu",
        "shuiwang ji"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2206.02059"
}
{
    "ToxicityFilter": {
        "text_to_toxicity_attributes": {
            "math_commands.tex": {
                "toxicity_score": 0.01005285,
                "severe_toxicity_score": 0.00084400177,
                "identity_attack_score": 0.0023678627,
                "insult_score": 0.0074368757,
                "profanity_score": 0.011526455,
                "threat_score": 0.006602164,
                "sexually_explicit_score": 0.0059272056,
                "flirtation_score": 0.37736848
            },
            "tmlr.bst": {
                "toxicity_score": 0.02048268,
                "severe_toxicity_score": 0.0017642975,
                "identity_attack_score": 0.003921773,
                "insult_score": 0.00983089,
                "profanity_score": 0.023379711,
                "threat_score": 0.008129764,
                "sexually_explicit_score": 0.021585643,
                "flirtation_score": 0.53546125
            },
            "fancyhdr.sty": {
                "toxicity_score": 0.06065326,
                "severe_toxicity_score": 0.0034332275,
                "identity_attack_score": 0.0056236745,
                "insult_score": 0.022255141,
                "profanity_score": 0.05268429,
                "threat_score": 0.007922632,
                "sexually_explicit_score": 0.036801755,
                "flirtation_score": 0.3852209
            },
            "main.bib": {
                "toxicity_score": 0.0131943645,
                "severe_toxicity_score": 0.0012588501,
                "identity_attack_score": 0.0032373124,
                "insult_score": 0.007797878,
                "profanity_score": 0.016752819,
                "threat_score": 0.008039144,
                "sexually_explicit_score": 0.0136827035,
                "flirtation_score": 0.3190224
            },
            "NCGNN.bbl": {
                "toxicity_score": 0.013320025,
                "severe_toxicity_score": 0.0012159348,
                "identity_attack_score": 0.00414376,
                "insult_score": 0.007797878,
                "profanity_score": 0.01535229,
                "threat_score": 0.0076507707,
                "sexually_explicit_score": 0.012739069,
                "flirtation_score": 0.195667
            },
            "tmlr.sty": {
                "toxicity_score": 0.006754258,
                "severe_toxicity_score": 0.00047922134,
                "identity_attack_score": 0.0013781702,
                "insult_score": 0.006999873,
                "profanity_score": 0.011458136,
                "threat_score": 0.0057606893,
                "sexually_explicit_score": 0.005543854,
                "flirtation_score": 0.48204565
            },
            "LICENSE": {
                "toxicity_score": 0.00823077,
                "severe_toxicity_score": 0.0007581711,
                "identity_attack_score": 0.001951637,
                "insult_score": 0.0069713728,
                "profanity_score": 0.012175481,
                "threat_score": 0.0068028234,
                "sexually_explicit_score": 0.0065759546,
                "flirtation_score": 0.17881343
            },
            "NCGNN.tex": {
                "toxicity_score": 0.017969469,
                "severe_toxicity_score": 0.0016021729,
                "identity_attack_score": 0.0061416444,
                "insult_score": 0.009127886,
                "profanity_score": 0.018392462,
                "threat_score": 0.008764107,
                "sexually_explicit_score": 0.01875474,
                "flirtation_score": 0.36146027
            },
            "README.md": {
                "toxicity_score": 0.013822667,
                "severe_toxicity_score": 0.0014019012,
                "identity_attack_score": 0.0036997858,
                "insult_score": 0.00810188,
                "profanity_score": 0.016342908,
                "threat_score": 0.00780612,
                "sexually_explicit_score": 0.015334064,
                "flirtation_score": 0.31661955
            }
        },
        "reason": null
    }
}
{
    "num_done": {
        "figure": 0,
        "algorithm": 3
    }
}
{
    "NonTrivialRenderingFilter": {
        "white_pixels_ratio": 91.82406955040274,
        "hash": "0f0f1f1f871f853f",
        "most_frequent_color_ratio": 91.82406955040274
    }
}